id	sid	tid	token	lemma	pos
ejpam-6690	1	1	european	european	PROPN
ejpam-6690	1	2	journal	journal	PROPN
ejpam-6690	1	3	of	of	ADP
ejpam-6690	1	4	pure	pure	ADJ
ejpam-6690	1	5	and	and	CCONJ
ejpam-6690	1	6	applied	applied	ADJ
ejpam-6690	1	7	mathematics	mathematic	NOUN
ejpam-6690	1	8	2025	2025	NUM
ejpam-6690	1	9	,	,	PUNCT
ejpam-6690	1	10	vol	vol	NOUN
ejpam-6690	1	11	.	.	PROPN
ejpam-6690	1	12	18	18	NUM
ejpam-6690	1	13	,	,	PUNCT
ejpam-6690	1	14	issue	issue	NOUN
ejpam-6690	1	15	4	4	NUM
ejpam-6690	1	16	,	,	PUNCT
ejpam-6690	1	17	article	article	NOUN
ejpam-6690	1	18	number	number	NOUN
ejpam-6690	1	19	6690	6690	NUM
ejpam-6690	1	20	issn	issn	PROPN
ejpam-6690	1	21	1307	1307	NUM
ejpam-6690	1	22	-	-	SYM
ejpam-6690	1	23	5543	5543	NUM
ejpam-6690	1	24	–	–	PUNCT
ejpam-6690	1	25	ejpam.com	ejpam.com	X
ejpam-6690	1	26	published	publish	VERB
ejpam-6690	1	27	by	by	ADP
ejpam-6690	1	28	new	new	PROPN
ejpam-6690	1	29	york	york	PROPN
ejpam-6690	1	30	business	business	PROPN
ejpam-6690	1	31	global	global	ADJ
ejpam-6690	1	32	zariski	zariski	PROPN
ejpam-6690	1	33	topology	topology	NOUN
ejpam-6690	1	34	of	of	ADP
ejpam-6690	1	35	(	(	PUNCT
ejpam-6690	1	36	krasner	krasner	NOUN
ejpam-6690	1	37	)	)	PUNCT
ejpam-6690	1	38	hyperrings	hyperring	NOUN
ejpam-6690	1	39	behnam	behnam	PROPN
ejpam-6690	1	40	afshar1	afshar1	PROPN
ejpam-6690	1	41	,	,	PUNCT
ejpam-6690	1	42	reza	reza	PROPN
ejpam-6690	1	43	ameri1,∗	ameri1,∗	NOUN
ejpam-6690	1	44	,	,	PUNCT
ejpam-6690	1	45	madeleine	madeleine	PROPN
ejpam-6690	1	46	al	al	PROPN
ejpam-6690	1	47	-	-	PUNCT
ejpam-6690	1	48	tahan2	tahan2	PROPN
ejpam-6690	1	49	1	1	NUM
ejpam-6690	1	50	department	department	NOUN
ejpam-6690	1	51	of	of	ADP
ejpam-6690	1	52	mathematics	mathematic	NOUN
ejpam-6690	1	53	,	,	PUNCT
ejpam-6690	1	54	statistics	statistic	NOUN
ejpam-6690	1	55	and	and	CCONJ
ejpam-6690	1	56	computer	computer	NOUN
ejpam-6690	1	57	science	science	NOUN
ejpam-6690	1	58	,	,	PUNCT
ejpam-6690	1	59	college	college	NOUN
ejpam-6690	1	60	of	of	ADP
ejpam-6690	1	61	science	science	NOUN
ejpam-6690	1	62	,	,	PUNCT
ejpam-6690	1	63	university	university	NOUN
ejpam-6690	1	64	of	of	ADP
ejpam-6690	1	65	tehran	tehran	PROPN
ejpam-6690	1	66	,	,	PUNCT
ejpam-6690	1	67	tehran	tehran	PROPN
ejpam-6690	1	68	,	,	PUNCT
ejpam-6690	1	69	iran	iran	PROPN
ejpam-6690	1	70	2	2	NUM
ejpam-6690	1	71	department	department	NOUN
ejpam-6690	1	72	of	of	ADP
ejpam-6690	1	73	mathematics	mathematic	NOUN
ejpam-6690	1	74	and	and	CCONJ
ejpam-6690	1	75	statistics	statistic	NOUN
ejpam-6690	1	76	,	,	PUNCT
ejpam-6690	1	77	college	college	NOUN
ejpam-6690	1	78	of	of	ADP
ejpam-6690	1	79	engineering	engineering	NOUN
ejpam-6690	1	80	,	,	PUNCT
ejpam-6690	1	81	abu	abu	PROPN
ejpam-6690	1	82	dhabi	dhabi	PROPN
ejpam-6690	1	83	university	university	PROPN
ejpam-6690	1	84	,	,	PUNCT
ejpam-6690	1	85	abu	abu	PROPN
ejpam-6690	1	86	dhabi	dhabi	PROPN
ejpam-6690	1	87	,	,	PUNCT
ejpam-6690	1	88	united	united	PROPN
ejpam-6690	1	89	arab	arab	PROPN
ejpam-6690	1	90	emirates	emirates	PROPN
ejpam-6690	1	91	abstract	abstract	PROPN
ejpam-6690	1	92	.	.	PUNCT
ejpam-6690	2	1	in	in	ADP
ejpam-6690	2	2	this	this	DET
ejpam-6690	2	3	paper	paper	NOUN
ejpam-6690	2	4	,	,	PUNCT
ejpam-6690	2	5	we	we	PRON
ejpam-6690	2	6	investigate	investigate	VERB
ejpam-6690	2	7	the	the	DET
ejpam-6690	2	8	zariski	zariski	ADJ
ejpam-6690	2	9	topology	topology	NOUN
ejpam-6690	2	10	on	on	ADP
ejpam-6690	2	11	the	the	DET
ejpam-6690	2	12	prime	prime	ADJ
ejpam-6690	2	13	spectrum	spectrum	NOUN
ejpam-6690	2	14	of	of	ADP
ejpam-6690	2	15	commutative	commutative	ADJ
ejpam-6690	2	16	krasner	krasner	NOUN
ejpam-6690	2	17	hyperrings	hyperring	NOUN
ejpam-6690	2	18	and	and	CCONJ
ejpam-6690	2	19	explore	explore	VERB
ejpam-6690	2	20	its	its	PRON
ejpam-6690	2	21	interplay	interplay	NOUN
ejpam-6690	2	22	with	with	ADP
ejpam-6690	2	23	the	the	DET
ejpam-6690	2	24	underlying	underlie	VERB
ejpam-6690	2	25	algebraic	algebraic	ADJ
ejpam-6690	2	26	structure	structure	NOUN
ejpam-6690	2	27	.	.	PUNCT
ejpam-6690	3	1	we	we	PRON
ejpam-6690	3	2	characterize	characterize	VERB
ejpam-6690	3	3	the	the	DET
ejpam-6690	3	4	topological	topological	ADJ
ejpam-6690	3	5	properties	property	NOUN
ejpam-6690	3	6	of	of	ADP
ejpam-6690	3	7	the	the	DET
ejpam-6690	3	8	spectrum	spectrum	NOUN
ejpam-6690	3	9	,	,	PUNCT
ejpam-6690	3	10	such	such	ADJ
ejpam-6690	3	11	as	as	ADP
ejpam-6690	3	12	connectedness	connectedness	NOUN
ejpam-6690	3	13	,	,	PUNCT
ejpam-6690	3	14	irreducibility	irreducibility	NOUN
ejpam-6690	3	15	,	,	PUNCT
ejpam-6690	3	16	compactness	compactness	NOUN
ejpam-6690	3	17	,	,	PUNCT
ejpam-6690	3	18	and	and	CCONJ
ejpam-6690	3	19	separation	separation	NOUN
ejpam-6690	3	20	axioms	axiom	VERB
ejpam-6690	3	21	,	,	PUNCT
ejpam-6690	3	22	and	and	CCONJ
ejpam-6690	3	23	provide	provide	VERB
ejpam-6690	3	24	necessary	necessary	ADJ
ejpam-6690	3	25	and	and	CCONJ
ejpam-6690	3	26	sufficient	sufficient	ADJ
ejpam-6690	3	27	conditions	condition	NOUN
ejpam-6690	3	28	for	for	ADP
ejpam-6690	3	29	each	each	PRON
ejpam-6690	3	30	.	.	PUNCT
ejpam-6690	4	1	notably	notably	ADV
ejpam-6690	4	2	,	,	PUNCT
ejpam-6690	4	3	we	we	PRON
ejpam-6690	4	4	show	show	VERB
ejpam-6690	4	5	that	that	SCONJ
ejpam-6690	4	6	the	the	DET
ejpam-6690	4	7	spectrum	spectrum	NOUN
ejpam-6690	4	8	is	be	AUX
ejpam-6690	4	9	irreducible	irreducible	ADJ
ejpam-6690	4	10	if	if	SCONJ
ejpam-6690	4	11	and	and	CCONJ
ejpam-6690	4	12	only	only	ADV
ejpam-6690	4	13	if	if	SCONJ
ejpam-6690	4	14	the	the	DET
ejpam-6690	4	15	nilradical	nilradical	ADJ
ejpam-6690	4	16	is	be	AUX
ejpam-6690	4	17	a	a	DET
ejpam-6690	4	18	prime	prime	ADJ
ejpam-6690	4	19	hyperideal	hyperideal	NOUN
ejpam-6690	4	20	,	,	PUNCT
ejpam-6690	4	21	and	and	CCONJ
ejpam-6690	4	22	it	it	PRON
ejpam-6690	4	23	is	be	AUX
ejpam-6690	4	24	connected	connect	VERB
ejpam-6690	4	25	precisely	precisely	ADV
ejpam-6690	4	26	when	when	SCONJ
ejpam-6690	4	27	the	the	DET
ejpam-6690	4	28	hyperring	hyperring	NOUN
ejpam-6690	4	29	is	be	AUX
ejpam-6690	4	30	not	not	PART
ejpam-6690	4	31	a	a	DET
ejpam-6690	4	32	nontrivial	nontrivial	ADJ
ejpam-6690	4	33	product	product	NOUN
ejpam-6690	4	34	.	.	PUNCT
ejpam-6690	5	1	we	we	PRON
ejpam-6690	5	2	also	also	ADV
ejpam-6690	5	3	study	study	VERB
ejpam-6690	5	4	functorial	functorial	NOUN
ejpam-6690	5	5	behavior	behavior	NOUN
ejpam-6690	5	6	of	of	ADP
ejpam-6690	5	7	the	the	DET
ejpam-6690	5	8	zariski	zariski	ADJ
ejpam-6690	5	9	topology	topology	NOUN
ejpam-6690	5	10	in	in	ADP
ejpam-6690	5	11	the	the	DET
ejpam-6690	5	12	category	category	NOUN
ejpam-6690	5	13	of	of	ADP
ejpam-6690	5	14	hyperrings	hyperring	NOUN
ejpam-6690	5	15	and	and	CCONJ
ejpam-6690	5	16	analyze	analyze	VERB
ejpam-6690	5	17	its	its	PRON
ejpam-6690	5	18	correspondence	correspondence	NOUN
ejpam-6690	5	19	with	with	ADP
ejpam-6690	5	20	classical	classical	ADJ
ejpam-6690	5	21	ring	ring	NOUN
ejpam-6690	5	22	theory	theory	NOUN
ejpam-6690	5	23	via	via	ADP
ejpam-6690	5	24	the	the	DET
ejpam-6690	5	25	fundamental	fundamental	ADJ
ejpam-6690	5	26	relation	relation	NOUN
ejpam-6690	5	27	γ∗.	γ∗.	ADV
ejpam-6690	5	28	furthermore	furthermore	ADV
ejpam-6690	5	29	,	,	PUNCT
ejpam-6690	5	30	we	we	PRON
ejpam-6690	5	31	define	define	VERB
ejpam-6690	5	32	a	a	DET
ejpam-6690	5	33	topology	topology	NOUN
ejpam-6690	5	34	on	on	ADP
ejpam-6690	5	35	the	the	DET
ejpam-6690	5	36	space	space	NOUN
ejpam-6690	5	37	of	of	ADP
ejpam-6690	5	38	prime	prime	ADJ
ejpam-6690	5	39	strongly	strongly	ADV
ejpam-6690	5	40	regular	regular	ADJ
ejpam-6690	5	41	relations	relation	NOUN
ejpam-6690	5	42	and	and	CCONJ
ejpam-6690	5	43	establish	establish	VERB
ejpam-6690	5	44	a	a	DET
ejpam-6690	5	45	homeomorphism	homeomorphism	NOUN
ejpam-6690	5	46	with	with	ADP
ejpam-6690	5	47	a	a	DET
ejpam-6690	5	48	subspace	subspace	NOUN
ejpam-6690	5	49	of	of	ADP
ejpam-6690	5	50	the	the	DET
ejpam-6690	5	51	classical	classical	ADJ
ejpam-6690	5	52	spectrum	spectrum	NOUN
ejpam-6690	5	53	.	.	PUNCT
ejpam-6690	6	1	these	these	DET
ejpam-6690	6	2	results	result	NOUN
ejpam-6690	6	3	contribute	contribute	VERB
ejpam-6690	6	4	to	to	ADP
ejpam-6690	6	5	the	the	DET
ejpam-6690	6	6	categorical	categorical	ADJ
ejpam-6690	6	7	and	and	CCONJ
ejpam-6690	6	8	topological	topological	ADJ
ejpam-6690	6	9	foundations	foundation	NOUN
ejpam-6690	6	10	necessary	necessary	ADJ
ejpam-6690	6	11	for	for	ADP
ejpam-6690	6	12	developing	develop	VERB
ejpam-6690	6	13	a	a	DET
ejpam-6690	6	14	sheaf	sheaf	NOUN
ejpam-6690	6	15	-	-	PUNCT
ejpam-6690	6	16	theoretic	theoretic	NOUN
ejpam-6690	6	17	framework	framework	NOUN
ejpam-6690	6	18	in	in	ADP
ejpam-6690	6	19	the	the	DET
ejpam-6690	6	20	context	context	NOUN
ejpam-6690	6	21	of	of	ADP
ejpam-6690	6	22	hyperrings	hyperring	NOUN
ejpam-6690	6	23	.	.	PUNCT
ejpam-6690	7	1	2020	2020	NUM
ejpam-6690	7	2	mathematics	mathematic	NOUN
ejpam-6690	7	3	subject	subject	NOUN
ejpam-6690	7	4	classifications	classification	NOUN
ejpam-6690	7	5	:	:	PUNCT
ejpam-6690	7	6	20n20	20n20	NUM
ejpam-6690	7	7	,	,	PUNCT
ejpam-6690	7	8	16y99	16y99	NUM
ejpam-6690	7	9	key	key	ADJ
ejpam-6690	7	10	words	word	NOUN
ejpam-6690	7	11	and	and	CCONJ
ejpam-6690	7	12	phrases	phrase	NOUN
ejpam-6690	7	13	:	:	PUNCT
ejpam-6690	7	14	zariski	zariski	NOUN
ejpam-6690	7	15	topology	topology	NOUN
ejpam-6690	7	16	,	,	PUNCT
ejpam-6690	7	17	strongly	strongly	ADV
ejpam-6690	7	18	regular	regular	ADJ
ejpam-6690	7	19	relation	relation	NOUN
ejpam-6690	7	20	,	,	PUNCT
ejpam-6690	7	21	spectrum	spectrum	NOUN
ejpam-6690	7	22	,	,	PUNCT
ejpam-6690	7	23	krasner	krasner	NOUN
ejpam-6690	7	24	hyperring	hyperre	VERB
ejpam-6690	7	25	1	1	NUM
ejpam-6690	7	26	.	.	PUNCT
ejpam-6690	7	27	introduction	introduction	NOUN
ejpam-6690	7	28	the	the	DET
ejpam-6690	7	29	theory	theory	NOUN
ejpam-6690	7	30	of	of	ADP
ejpam-6690	7	31	algebraic	algebraic	ADJ
ejpam-6690	7	32	hyperstructures	hyperstructure	NOUN
ejpam-6690	7	33	,	,	PUNCT
ejpam-6690	7	34	initiated	initiate	VERB
ejpam-6690	7	35	by	by	ADP
ejpam-6690	7	36	f.	f.	PROPN
ejpam-6690	7	37	marty	marty	PROPN
ejpam-6690	7	38	in	in	ADP
ejpam-6690	7	39	1934	1934	NUM
ejpam-6690	8	1	[	[	X
ejpam-6690	8	2	1	1	X
ejpam-6690	8	3	]	]	PUNCT
ejpam-6690	8	4	through	through	ADP
ejpam-6690	8	5	the	the	DET
ejpam-6690	8	6	concept	concept	NOUN
ejpam-6690	8	7	of	of	ADP
ejpam-6690	8	8	hypergroups	hypergroup	NOUN
ejpam-6690	8	9	,	,	PUNCT
ejpam-6690	8	10	extends	extend	VERB
ejpam-6690	8	11	classical	classical	ADJ
ejpam-6690	8	12	algebraic	algebraic	ADJ
ejpam-6690	8	13	systems	system	NOUN
ejpam-6690	8	14	by	by	ADP
ejpam-6690	8	15	allowing	allow	VERB
ejpam-6690	8	16	multi	multi	ADJ
ejpam-6690	8	17	-	-	ADJ
ejpam-6690	8	18	valued	value	VERB
ejpam-6690	8	19	operations	operation	NOUN
ejpam-6690	8	20	.	.	PUNCT
ejpam-6690	9	1	this	this	DET
ejpam-6690	9	2	framework	framework	NOUN
ejpam-6690	9	3	has	have	AUX
ejpam-6690	9	4	found	find	VERB
ejpam-6690	9	5	applications	application	NOUN
ejpam-6690	9	6	in	in	ADP
ejpam-6690	9	7	group	group	NOUN
ejpam-6690	9	8	theory	theory	NOUN
ejpam-6690	9	9	,	,	PUNCT
ejpam-6690	9	10	rational	rational	ADJ
ejpam-6690	9	11	and	and	CCONJ
ejpam-6690	9	12	algebraic	algebraic	ADJ
ejpam-6690	9	13	functions	function	NOUN
ejpam-6690	9	14	,	,	PUNCT
ejpam-6690	9	15	and	and	CCONJ
ejpam-6690	9	16	has	have	AUX
ejpam-6690	9	17	since	since	SCONJ
ejpam-6690	9	18	evolved	evolve	VERB
ejpam-6690	9	19	into	into	ADP
ejpam-6690	9	20	a	a	DET
ejpam-6690	9	21	rich	rich	ADJ
ejpam-6690	9	22	area	area	NOUN
ejpam-6690	9	23	of	of	ADP
ejpam-6690	9	24	research	research	NOUN
ejpam-6690	9	25	with	with	ADP
ejpam-6690	9	26	implications	implication	NOUN
ejpam-6690	9	27	in	in	ADP
ejpam-6690	9	28	algebraic	algebraic	ADJ
ejpam-6690	9	29	geometry	geometry	NOUN
ejpam-6690	9	30	,	,	PUNCT
ejpam-6690	9	31	notably	notably	ADV
ejpam-6690	9	32	through	through	ADP
ejpam-6690	9	33	the	the	DET
ejpam-6690	9	34	works	work	NOUN
ejpam-6690	9	35	of	of	ADP
ejpam-6690	9	36	a.	a.	NOUN
ejpam-6690	9	37	connes	conne	NOUN
ejpam-6690	9	38	and	and	CCONJ
ejpam-6690	9	39	c.	c.	PROPN
ejpam-6690	9	40	consani	consani	PROPN
ejpam-6690	10	1	[	[	X
ejpam-6690	10	2	2–4	2–4	NUM
ejpam-6690	10	3	]	]	X
ejpam-6690	10	4	,	,	PUNCT
ejpam-6690	10	5	who	who	PRON
ejpam-6690	10	6	revealed	reveal	VERB
ejpam-6690	10	7	deep	deep	ADJ
ejpam-6690	10	8	connections	connection	NOUN
ejpam-6690	10	9	between	between	ADP
ejpam-6690	10	10	hyperstructures	hyperstructure	NOUN
ejpam-6690	10	11	and	and	CCONJ
ejpam-6690	10	12	arithmetic	arithmetic	ADJ
ejpam-6690	10	13	geometry	geometry	NOUN
ejpam-6690	10	14	.	.	PUNCT
ejpam-6690	11	1	the	the	DET
ejpam-6690	11	2	results	result	NOUN
ejpam-6690	11	3	established	establish	VERB
ejpam-6690	11	4	in	in	ADP
ejpam-6690	11	5	this	this	DET
ejpam-6690	11	6	paper	paper	NOUN
ejpam-6690	11	7	enable	enable	VERB
ejpam-6690	11	8	us	we	PRON
ejpam-6690	11	9	,	,	PUNCT
ejpam-6690	11	10	by	by	ADP
ejpam-6690	11	11	defining	define	VERB
ejpam-6690	11	12	sheaves	sheaf	NOUN
ejpam-6690	11	13	of	of	ADP
ejpam-6690	11	14	hyperrings	hyperring	NOUN
ejpam-6690	11	15	,	,	PUNCT
ejpam-6690	11	16	hyperringed	hyperringe	VERB
ejpam-6690	11	17	spaces	space	NOUN
ejpam-6690	11	18	,	,	PUNCT
ejpam-6690	11	19	and	and	CCONJ
ejpam-6690	11	20	hyperschemes	hyperscheme	NOUN
ejpam-6690	11	21	based	base	VERB
ejpam-6690	11	22	on	on	ADP
ejpam-6690	11	23	the	the	DET
ejpam-6690	11	24	topology	topology	NOUN
ejpam-6690	11	25	induced	induce	VERB
ejpam-6690	11	26	by	by	ADP
ejpam-6690	11	27	regular	regular	ADJ
ejpam-6690	11	28	relations	relation	NOUN
ejpam-6690	11	29	,	,	PUNCT
ejpam-6690	11	30	to	to	PART
ejpam-6690	11	31	prove	prove	VERB
ejpam-6690	11	32	results	result	NOUN
ejpam-6690	11	33	analogous	analogous	ADJ
ejpam-6690	11	34	to	to	ADP
ejpam-6690	11	35	those	those	PRON
ejpam-6690	11	36	in	in	ADP
ejpam-6690	11	37	[	[	X
ejpam-6690	11	38	2–4	2–4	NUM
ejpam-6690	11	39	]	]	X
ejpam-6690	11	40	from	from	ADP
ejpam-6690	11	41	the	the	DET
ejpam-6690	11	42	perspective	perspective	NOUN
ejpam-6690	11	43	of	of	ADP
ejpam-6690	11	44	regular	regular	ADJ
ejpam-6690	11	45	relations	relation	NOUN
ejpam-6690	11	46	.	.	PUNCT
ejpam-6690	12	1	one	one	NUM
ejpam-6690	12	2	of	of	ADP
ejpam-6690	12	3	the	the	DET
ejpam-6690	12	4	advantages	advantage	NOUN
ejpam-6690	12	5	of	of	ADP
ejpam-6690	12	6	∗corresponding	∗corresponde	VERB
ejpam-6690	12	7	author	author	NOUN
ejpam-6690	12	8	.	.	PUNCT
ejpam-6690	13	1	doi	doi	PROPN
ejpam-6690	13	2	:	:	PUNCT
ejpam-6690	13	3	https://doi.org/10.29020/nybg.ejpam.v18i4.6690	https://doi.org/10.29020/nybg.ejpam.v18i4.6690	PROPN
ejpam-6690	13	4	email	email	NOUN
ejpam-6690	13	5	addresses	address	NOUN
ejpam-6690	13	6	:	:	PUNCT
ejpam-6690	13	7	behnamafshar@ut.ac.ir	behnamafshar@ut.ac.ir	PROPN
ejpam-6690	13	8	(	(	PUNCT
ejpam-6690	13	9	b.	b.	PROPN
ejpam-6690	13	10	afshar	afshar	ADJ
ejpam-6690	13	11	)	)	PUNCT
ejpam-6690	13	12	,	,	PUNCT
ejpam-6690	13	13	rameri@ut.ac.ir	rameri@ut.ac.ir	PROPN
ejpam-6690	13	14	(	(	PUNCT
ejpam-6690	13	15	r.	r.	PROPN
ejpam-6690	13	16	ameri	ameri	PROPN
ejpam-6690	13	17	)	)	PUNCT
ejpam-6690	13	18	,	,	PUNCT
ejpam-6690	13	19	altahan.madeleine@gmail.com	altahan.madeleine@gmail.com	X
ejpam-6690	13	20	(	(	PUNCT
ejpam-6690	13	21	m.	m.	PROPN
ejpam-6690	13	22	al	al	PROPN
ejpam-6690	13	23	-	-	PUNCT
ejpam-6690	13	24	tahan	tahan	PROPN
ejpam-6690	13	25	)	)	PUNCT
ejpam-6690	13	26	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6690	14	1	1	1	NUM
ejpam-6690	14	2	copyright	copyright	NOUN
ejpam-6690	14	3	:	:	PUNCT
ejpam-6690	14	4	©	©	PROPN
ejpam-6690	14	5	2025	2025	NUM
ejpam-6690	14	6	the	the	DET
ejpam-6690	14	7	author(s	author(s	NOUN
ejpam-6690	14	8	)	)	PUNCT
ejpam-6690	14	9	.	.	PUNCT
ejpam-6690	15	1	(	(	PUNCT
ejpam-6690	15	2	cc	cc	NOUN
ejpam-6690	15	3	by	by	ADP
ejpam-6690	15	4	-	-	PUNCT
ejpam-6690	15	5	nc	nc	PROPN
ejpam-6690	15	6	4.0	4.0	NUM
ejpam-6690	15	7	)	)	PUNCT
ejpam-6690	15	8	b.	b.	NOUN
ejpam-6690	15	9	afshar	afshar	PROPN
ejpam-6690	15	10	,	,	PUNCT
ejpam-6690	15	11	r.	r.	PROPN
ejpam-6690	15	12	ameri	ameri	PROPN
ejpam-6690	15	13	,	,	PUNCT
ejpam-6690	15	14	m.	m.	PROPN
ejpam-6690	15	15	al	al	PROPN
ejpam-6690	15	16	-	-	PUNCT
ejpam-6690	15	17	tahan	tahan	PROPN
ejpam-6690	15	18	/	/	SYM
ejpam-6690	15	19	eur	eur	PROPN
ejpam-6690	15	20	.	.	PUNCT
ejpam-6690	16	1	j.	j.	PROPN
ejpam-6690	16	2	pure	pure	PROPN
ejpam-6690	16	3	appl	appl	PROPN
ejpam-6690	16	4	.	.	PROPN
ejpam-6690	16	5	math	math	PROPN
ejpam-6690	16	6	,	,	PUNCT
ejpam-6690	16	7	18	18	NUM
ejpam-6690	16	8	(	(	PUNCT
ejpam-6690	16	9	4	4	NUM
ejpam-6690	16	10	)	)	PUNCT
ejpam-6690	16	11	(	(	PUNCT
ejpam-6690	16	12	2025	2025	NUM
ejpam-6690	16	13	)	)	PUNCT
ejpam-6690	16	14	,	,	PUNCT
ejpam-6690	16	15	6690	6690	NUM
ejpam-6690	16	16	2	2	NUM
ejpam-6690	16	17	of	of	ADP
ejpam-6690	16	18	19	19	NUM
ejpam-6690	16	19	this	this	DET
ejpam-6690	16	20	approach	approach	NOUN
ejpam-6690	16	21	is	be	AUX
ejpam-6690	16	22	that	that	SCONJ
ejpam-6690	16	23	it	it	PRON
ejpam-6690	16	24	provides	provide	VERB
ejpam-6690	16	25	a	a	DET
ejpam-6690	16	26	more	more	ADV
ejpam-6690	16	27	tangible	tangible	ADJ
ejpam-6690	16	28	understanding	understanding	NOUN
ejpam-6690	16	29	of	of	ADP
ejpam-6690	16	30	the	the	DET
ejpam-6690	16	31	connection	connection	NOUN
ejpam-6690	16	32	between	between	ADP
ejpam-6690	16	33	the	the	DET
ejpam-6690	16	34	obtained	obtain	VERB
ejpam-6690	16	35	results	result	NOUN
ejpam-6690	16	36	and	and	CCONJ
ejpam-6690	16	37	the	the	DET
ejpam-6690	16	38	classical	classical	ADJ
ejpam-6690	16	39	case	case	NOUN
ejpam-6690	16	40	.	.	PUNCT
ejpam-6690	17	1	in	in	ADP
ejpam-6690	17	2	1956	1956	NUM
ejpam-6690	17	3	m.	m.	NOUN
ejpam-6690	17	4	krasner	krasner	NOUN
ejpam-6690	17	5	introduced	introduce	VERB
ejpam-6690	17	6	the	the	DET
ejpam-6690	17	7	concept	concept	NOUN
ejpam-6690	17	8	of	of	ADP
ejpam-6690	17	9	hyperrings	hyperring	NOUN
ejpam-6690	17	10	to	to	PART
ejpam-6690	17	11	use	use	VERB
ejpam-6690	17	12	them	they	PRON
ejpam-6690	17	13	as	as	ADP
ejpam-6690	17	14	a	a	DET
ejpam-6690	17	15	technical	technical	ADJ
ejpam-6690	17	16	tool	tool	NOUN
ejpam-6690	17	17	on	on	ADP
ejpam-6690	17	18	the	the	DET
ejpam-6690	17	19	approximation	approximation	NOUN
ejpam-6690	17	20	of	of	ADP
ejpam-6690	17	21	valued	value	VERB
ejpam-6690	17	22	fields	field	NOUN
ejpam-6690	17	23	[	[	X
ejpam-6690	17	24	5	5	NUM
ejpam-6690	17	25	]	]	PUNCT
ejpam-6690	17	26	.	.	PUNCT
ejpam-6690	18	1	a	a	DET
ejpam-6690	18	2	general	general	ADJ
ejpam-6690	18	3	hyperring	hyperring	NOUN
ejpam-6690	18	4	is	be	AUX
ejpam-6690	18	5	a	a	DET
ejpam-6690	18	6	hyperstructure	hyperstructure	NOUN
ejpam-6690	18	7	(	(	PUNCT
ejpam-6690	18	8	r,+	r,+	NUM
ejpam-6690	18	9	,	,	PUNCT
ejpam-6690	18	10	·	·	PUNCT
ejpam-6690	18	11	)	)	PUNCT
ejpam-6690	18	12	where	where	SCONJ
ejpam-6690	18	13	(	(	PUNCT
ejpam-6690	18	14	r,+	r,+	NUM
ejpam-6690	18	15	)	)	PUNCT
ejpam-6690	18	16	is	be	AUX
ejpam-6690	18	17	a	a	DET
ejpam-6690	18	18	hypergroup	hypergroup	NOUN
ejpam-6690	18	19	and	and	CCONJ
ejpam-6690	18	20	(	(	PUNCT
ejpam-6690	18	21	r	r	NOUN
ejpam-6690	18	22	,	,	PUNCT
ejpam-6690	18	23	·	·	PUNCT
ejpam-6690	18	24	)	)	PUNCT
ejpam-6690	18	25	is	be	AUX
ejpam-6690	18	26	semihypergroup	semihypergroup	VERB
ejpam-6690	18	27	such	such	ADJ
ejpam-6690	18	28	that	that	SCONJ
ejpam-6690	18	29	”	"	PUNCT
ejpam-6690	18	30	·	·	PUNCT
ejpam-6690	18	31	”	"	PUNCT
ejpam-6690	18	32	is	be	AUX
ejpam-6690	18	33	distributive	distributive	ADJ
ejpam-6690	18	34	with	with	ADP
ejpam-6690	18	35	respect	respect	NOUN
ejpam-6690	18	36	to	to	ADP
ejpam-6690	18	37	”	"	PUNCT
ejpam-6690	18	38	+	+	NOUN
ejpam-6690	18	39	”	"	PUNCT
ejpam-6690	18	40	.	.	PUNCT
ejpam-6690	19	1	if	if	SCONJ
ejpam-6690	19	2	(	(	PUNCT
ejpam-6690	19	3	r,+	r,+	NUM
ejpam-6690	19	4	)	)	PUNCT
ejpam-6690	19	5	is	be	AUX
ejpam-6690	19	6	a	a	DET
ejpam-6690	19	7	canonical	canonical	ADJ
ejpam-6690	19	8	hypergroup	hypergroup	NOUN
ejpam-6690	19	9	and	and	CCONJ
ejpam-6690	19	10	(	(	PUNCT
ejpam-6690	19	11	r	r	NOUN
ejpam-6690	19	12	,	,	PUNCT
ejpam-6690	19	13	·	·	PUNCT
ejpam-6690	19	14	)	)	PUNCT
ejpam-6690	19	15	is	be	AUX
ejpam-6690	19	16	a	a	DET
ejpam-6690	19	17	semigroup	semigroup	NOUN
ejpam-6690	19	18	such	such	ADJ
ejpam-6690	19	19	that	that	SCONJ
ejpam-6690	19	20	zero	zero	NUM
ejpam-6690	19	21	element	element	NOUN
ejpam-6690	19	22	is	be	AUX
ejpam-6690	19	23	absorbing	absorbing	ADJ
ejpam-6690	19	24	,	,	PUNCT
ejpam-6690	19	25	then	then	ADV
ejpam-6690	19	26	(	(	PUNCT
ejpam-6690	19	27	r,+	r,+	NUM
ejpam-6690	19	28	,	,	PUNCT
ejpam-6690	19	29	·	·	PUNCT
ejpam-6690	19	30	)	)	PUNCT
ejpam-6690	20	1	is	be	AUX
ejpam-6690	20	2	the	the	DET
ejpam-6690	20	3	krasner	krasner	NOUN
ejpam-6690	20	4	hyperring	hyperre	VERB
ejpam-6690	20	5	[	[	X
ejpam-6690	20	6	6	6	NUM
ejpam-6690	20	7	]	]	PUNCT
ejpam-6690	20	8	.	.	PUNCT
ejpam-6690	21	1	the	the	DET
ejpam-6690	21	2	fundamental	fundamental	ADJ
ejpam-6690	21	3	relations	relation	NOUN
ejpam-6690	21	4	are	be	AUX
ejpam-6690	21	5	one	one	NUM
ejpam-6690	21	6	of	of	ADP
ejpam-6690	21	7	the	the	DET
ejpam-6690	21	8	most	most	ADV
ejpam-6690	21	9	important	important	ADJ
ejpam-6690	21	10	and	and	CCONJ
ejpam-6690	21	11	interesting	interesting	ADJ
ejpam-6690	21	12	concepts	concept	NOUN
ejpam-6690	21	13	in	in	ADP
ejpam-6690	21	14	algebraic	algebraic	PROPN
ejpam-6690	21	15	hyperstructures	hyperstructure	NOUN
ejpam-6690	21	16	that	that	SCONJ
ejpam-6690	21	17	ordinary	ordinary	ADJ
ejpam-6690	21	18	algebraic	algebraic	ADJ
ejpam-6690	21	19	structures	structure	NOUN
ejpam-6690	21	20	can	can	AUX
ejpam-6690	21	21	be	be	AUX
ejpam-6690	21	22	derived	derive	VERB
ejpam-6690	21	23	from	from	ADP
ejpam-6690	21	24	algebraic	algebraic	PROPN
ejpam-6690	21	25	hyperstructures	hyperstructure	NOUN
ejpam-6690	21	26	through	through	ADP
ejpam-6690	21	27	them	they	PRON
ejpam-6690	21	28	.	.	PUNCT
ejpam-6690	22	1	the	the	DET
ejpam-6690	22	2	fundamental	fundamental	ADJ
ejpam-6690	22	3	relation	relation	NOUN
ejpam-6690	22	4	β∗	β∗	NOUN
ejpam-6690	22	5	on	on	ADP
ejpam-6690	22	6	hypergroups	hypergroup	NOUN
ejpam-6690	22	7	was	be	AUX
ejpam-6690	22	8	defined	define	VERB
ejpam-6690	22	9	by	by	ADP
ejpam-6690	22	10	koskas	koskas	NOUN
ejpam-6690	23	1	[	[	X
ejpam-6690	23	2	7	7	NUM
ejpam-6690	23	3	]	]	PUNCT
ejpam-6690	23	4	,	,	PUNCT
ejpam-6690	23	5	corsini	corsini	X
ejpam-6690	24	1	[	[	X
ejpam-6690	24	2	6	6	NUM
ejpam-6690	24	3	]	]	PUNCT
ejpam-6690	24	4	,	,	PUNCT
ejpam-6690	24	5	ferni	ferni	PROPN
ejpam-6690	25	1	[	[	X
ejpam-6690	25	2	8	8	NUM
ejpam-6690	25	3	]	]	PUNCT
ejpam-6690	25	4	,	,	PUNCT
ejpam-6690	25	5	and	and	CCONJ
ejpam-6690	25	6	vogiouklis	vogioukli	NOUN
ejpam-6690	26	1	[	[	X
ejpam-6690	26	2	9	9	NUM
ejpam-6690	26	3	]	]	PUNCT
ejpam-6690	26	4	.	.	PUNCT
ejpam-6690	27	1	then	then	ADV
ejpam-6690	27	2	d.	d.	PROPN
ejpam-6690	27	3	ferni	ferni	PROPN
ejpam-6690	27	4	introduced	introduce	VERB
ejpam-6690	27	5	the	the	DET
ejpam-6690	27	6	fundamental	fundamental	ADJ
ejpam-6690	27	7	relation	relation	NOUN
ejpam-6690	27	8	γ∗	γ∗	NOUN
ejpam-6690	27	9	which	which	PRON
ejpam-6690	27	10	is	be	AUX
ejpam-6690	27	11	the	the	DET
ejpam-6690	27	12	transitive	transitive	ADJ
ejpam-6690	27	13	closure	closure	NOUN
ejpam-6690	27	14	of	of	ADP
ejpam-6690	27	15	γ	γ	PROPN
ejpam-6690	27	16	and	and	CCONJ
ejpam-6690	27	17	is	be	AUX
ejpam-6690	27	18	the	the	DET
ejpam-6690	27	19	smallest	small	ADJ
ejpam-6690	27	20	relation	relation	NOUN
ejpam-6690	27	21	such	such	ADJ
ejpam-6690	27	22	that	that	DET
ejpam-6690	27	23	h	h	NOUN
ejpam-6690	27	24	/	/	SYM
ejpam-6690	27	25	γ∗	γ∗	PROPN
ejpam-6690	27	26	is	be	AUX
ejpam-6690	27	27	an	an	DET
ejpam-6690	27	28	abelian	abelian	ADJ
ejpam-6690	27	29	group	group	NOUN
ejpam-6690	27	30	.	.	PUNCT
ejpam-6690	28	1	t.	t.	PROPN
ejpam-6690	28	2	vogiouklis	vogiouklis	PROPN
ejpam-6690	28	3	generalized	generalize	VERB
ejpam-6690	28	4	the	the	DET
ejpam-6690	28	5	fundamental	fundamental	ADJ
ejpam-6690	28	6	relations	relation	NOUN
ejpam-6690	28	7	in	in	ADP
ejpam-6690	28	8	[	[	X
ejpam-6690	28	9	9	9	NUM
ejpam-6690	28	10	]	]	PUNCT
ejpam-6690	28	11	to	to	PART
ejpam-6690	28	12	use	use	VERB
ejpam-6690	28	13	on	on	ADP
ejpam-6690	28	14	hyperrings	hyperring	NOUN
ejpam-6690	28	15	and	and	CCONJ
ejpam-6690	28	16	in	in	ADP
ejpam-6690	28	17	[	[	X
ejpam-6690	28	18	10	10	NUM
ejpam-6690	28	19	]	]	PUNCT
ejpam-6690	28	20	,	,	PUNCT
ejpam-6690	28	21	it	it	PRON
ejpam-6690	28	22	has	have	AUX
ejpam-6690	28	23	been	be	AUX
ejpam-6690	28	24	demonstrated	demonstrate	VERB
ejpam-6690	28	25	that	that	SCONJ
ejpam-6690	28	26	relations	relation	NOUN
ejpam-6690	28	27	β∗	β∗	NOUN
ejpam-6690	28	28	and	and	CCONJ
ejpam-6690	28	29	γ∗	γ∗	NOUN
ejpam-6690	28	30	are	be	AUX
ejpam-6690	28	31	related	relate	VERB
ejpam-6690	28	32	together	together	ADV
ejpam-6690	28	33	in	in	ADP
ejpam-6690	28	34	the	the	DET
ejpam-6690	28	35	form	form	NOUN
ejpam-6690	28	36	of	of	ADP
ejpam-6690	28	37	γ∗	γ∗	NOUN
ejpam-6690	28	38	=	=	PROPN
ejpam-6690	28	39	δ	δ	PROPN
ejpam-6690	28	40	∗	∗	NOUN
ejpam-6690	28	41	β∗	β∗	PROPN
ejpam-6690	28	42	,	,	PUNCT
ejpam-6690	28	43	where	where	SCONJ
ejpam-6690	28	44	δ	δ	PROPN
ejpam-6690	28	45	is	be	AUX
ejpam-6690	28	46	the	the	DET
ejpam-6690	28	47	congruence	congruence	PROPN
ejpam-6690	28	48	relation	relation	NOUN
ejpam-6690	28	49	with	with	ADP
ejpam-6690	28	50	respect	respect	NOUN
ejpam-6690	28	51	to	to	ADP
ejpam-6690	28	52	the	the	DET
ejpam-6690	28	53	commutator	commutator	NOUN
ejpam-6690	28	54	subgroup	subgroup	NOUN
ejpam-6690	28	55	.	.	PUNCT
ejpam-6690	29	1	in	in	ADP
ejpam-6690	29	2	[	[	X
ejpam-6690	29	3	11	11	NUM
ejpam-6690	29	4	]	]	PUNCT
ejpam-6690	29	5	,	,	PUNCT
ejpam-6690	29	6	a	a	DET
ejpam-6690	29	7	one	one	NUM
ejpam-6690	29	8	-	-	PUNCT
ejpam-6690	29	9	to	to	ADP
ejpam-6690	29	10	-	-	PUNCT
ejpam-6690	29	11	one	one	NUM
ejpam-6690	29	12	correspondence	correspondence	NOUN
ejpam-6690	29	13	was	be	AUX
ejpam-6690	29	14	established	establish	VERB
ejpam-6690	29	15	between	between	ADP
ejpam-6690	29	16	the	the	DET
ejpam-6690	29	17	lattice	lattice	NOUN
ejpam-6690	29	18	of	of	ADP
ejpam-6690	29	19	strongly	strongly	ADV
ejpam-6690	29	20	regular	regular	ADJ
ejpam-6690	29	21	relations	relation	NOUN
ejpam-6690	29	22	on	on	ADP
ejpam-6690	29	23	a	a	DET
ejpam-6690	29	24	regular	regular	ADJ
ejpam-6690	29	25	hypergroup	hypergroup	NOUN
ejpam-6690	29	26	h	h	NOUN
ejpam-6690	29	27	and	and	CCONJ
ejpam-6690	29	28	the	the	DET
ejpam-6690	29	29	lattice	lattice	NOUN
ejpam-6690	29	30	of	of	ADP
ejpam-6690	29	31	normal	normal	ADJ
ejpam-6690	29	32	subhypergroups	subhypergroup	NOUN
ejpam-6690	29	33	of	of	ADP
ejpam-6690	29	34	h	h	NOUN
ejpam-6690	29	35	containing	contain	VERB
ejpam-6690	29	36	ωh	ωh	X
ejpam-6690	29	37	.	.	PUNCT
ejpam-6690	30	1	references	reference	NOUN
ejpam-6690	30	2	[	[	X
ejpam-6690	30	3	12	12	NUM
ejpam-6690	30	4	]	]	PUNCT
ejpam-6690	30	5	and	and	CCONJ
ejpam-6690	30	6	[	[	X
ejpam-6690	30	7	13	13	NUM
ejpam-6690	30	8	]	]	PUNCT
ejpam-6690	30	9	explore	explore	VERB
ejpam-6690	30	10	the	the	DET
ejpam-6690	30	11	fundamental	fundamental	ADJ
ejpam-6690	30	12	relations	relation	NOUN
ejpam-6690	30	13	on	on	ADP
ejpam-6690	30	14	hv	hv	NOUN
ejpam-6690	30	15	-	-	PUNCT
ejpam-6690	30	16	modules	module	NOUN
ejpam-6690	30	17	and	and	CCONJ
ejpam-6690	30	18	highlight	highlight	VERB
ejpam-6690	30	19	significant	significant	ADJ
ejpam-6690	30	20	applications	application	NOUN
ejpam-6690	30	21	of	of	ADP
ejpam-6690	30	22	hyperstructure	hyperstructure	NOUN
ejpam-6690	30	23	theory	theory	NOUN
ejpam-6690	30	24	in	in	ADP
ejpam-6690	30	25	chemistry	chemistry	NOUN
ejpam-6690	30	26	,	,	PUNCT
ejpam-6690	30	27	respectively	respectively	ADV
ejpam-6690	30	28	.	.	PUNCT
ejpam-6690	31	1	more	more	ADV
ejpam-6690	31	2	recently	recently	ADV
ejpam-6690	31	3	,	,	PUNCT
ejpam-6690	31	4	considerable	considerable	ADJ
ejpam-6690	31	5	attention	attention	NOUN
ejpam-6690	31	6	has	have	AUX
ejpam-6690	31	7	been	be	AUX
ejpam-6690	31	8	devoted	devote	VERB
ejpam-6690	31	9	to	to	ADP
ejpam-6690	31	10	studying	study	VERB
ejpam-6690	31	11	the	the	DET
ejpam-6690	31	12	zariski	zariski	ADJ
ejpam-6690	31	13	topology	topology	NOUN
ejpam-6690	31	14	of	of	ADP
ejpam-6690	31	15	algebraic	algebraic	PROPN
ejpam-6690	31	16	hyperstructures	hyperstructure	NOUN
ejpam-6690	31	17	.	.	PUNCT
ejpam-6690	32	1	for	for	ADP
ejpam-6690	32	2	a	a	DET
ejpam-6690	32	3	detailed	detailed	ADJ
ejpam-6690	32	4	examination	examination	NOUN
ejpam-6690	32	5	of	of	ADP
ejpam-6690	32	6	the	the	DET
ejpam-6690	32	7	zariski	zariski	ADJ
ejpam-6690	32	8	topology	topology	NOUN
ejpam-6690	32	9	in	in	ADP
ejpam-6690	32	10	the	the	DET
ejpam-6690	32	11	context	context	NOUN
ejpam-6690	32	12	of	of	ADP
ejpam-6690	32	13	multiplicative	multiplicative	ADJ
ejpam-6690	32	14	hyperrings	hyperring	NOUN
ejpam-6690	32	15	,	,	PUNCT
ejpam-6690	32	16	see	see	VERB
ejpam-6690	32	17	[	[	X
ejpam-6690	32	18	14	14	NUM
ejpam-6690	32	19	]	]	PUNCT
ejpam-6690	32	20	.	.	PUNCT
ejpam-6690	33	1	our	our	PRON
ejpam-6690	33	2	paper	paper	NOUN
ejpam-6690	33	3	is	be	AUX
ejpam-6690	33	4	organized	organize	VERB
ejpam-6690	33	5	as	as	SCONJ
ejpam-6690	33	6	follows	follow	VERB
ejpam-6690	33	7	.	.	PUNCT
ejpam-6690	34	1	after	after	ADP
ejpam-6690	34	2	this	this	DET
ejpam-6690	34	3	introduction	introduction	NOUN
ejpam-6690	34	4	,	,	PUNCT
ejpam-6690	34	5	section	section	NOUN
ejpam-6690	34	6	2	2	NUM
ejpam-6690	34	7	presents	present	VERB
ejpam-6690	34	8	the	the	DET
ejpam-6690	34	9	necessary	necessary	ADJ
ejpam-6690	34	10	background	background	NOUN
ejpam-6690	34	11	on	on	ADP
ejpam-6690	34	12	krasner	krasner	PROPN
ejpam-6690	34	13	hyperrings	hyperring	NOUN
ejpam-6690	34	14	,	,	PUNCT
ejpam-6690	34	15	hyperideals	hyperideal	NOUN
ejpam-6690	34	16	,	,	PUNCT
ejpam-6690	34	17	and	and	CCONJ
ejpam-6690	34	18	topological	topological	ADJ
ejpam-6690	34	19	preliminaries	preliminary	NOUN
ejpam-6690	34	20	.	.	PUNCT
ejpam-6690	35	1	section	section	NOUN
ejpam-6690	35	2	3	3	NUM
ejpam-6690	35	3	develops	develop	VERB
ejpam-6690	35	4	the	the	DET
ejpam-6690	35	5	zariski	zariski	ADJ
ejpam-6690	35	6	topology	topology	NOUN
ejpam-6690	35	7	on	on	ADP
ejpam-6690	35	8	the	the	DET
ejpam-6690	35	9	spectrum	spectrum	NOUN
ejpam-6690	35	10	of	of	ADP
ejpam-6690	35	11	prime	prime	ADJ
ejpam-6690	35	12	hyperideals	hyperideal	NOUN
ejpam-6690	35	13	,	,	PUNCT
ejpam-6690	35	14	focusing	focus	VERB
ejpam-6690	35	15	on	on	ADP
ejpam-6690	35	16	its	its	PRON
ejpam-6690	35	17	fundamental	fundamental	ADJ
ejpam-6690	35	18	topological	topological	ADJ
ejpam-6690	35	19	properties	property	NOUN
ejpam-6690	35	20	.	.	PUNCT
ejpam-6690	36	1	in	in	ADP
ejpam-6690	36	2	section	section	NOUN
ejpam-6690	36	3	4	4	NUM
ejpam-6690	36	4	,	,	PUNCT
ejpam-6690	36	5	we	we	PRON
ejpam-6690	36	6	examine	examine	VERB
ejpam-6690	36	7	the	the	DET
ejpam-6690	36	8	categorical	categorical	ADJ
ejpam-6690	36	9	and	and	CCONJ
ejpam-6690	36	10	functorial	functorial	NOUN
ejpam-6690	36	11	aspects	aspect	NOUN
ejpam-6690	36	12	of	of	ADP
ejpam-6690	36	13	zariski	zariski	ADJ
ejpam-6690	36	14	topology	topology	NOUN
ejpam-6690	36	15	,	,	PUNCT
ejpam-6690	36	16	establish	establish	VERB
ejpam-6690	36	17	its	its	PRON
ejpam-6690	36	18	connection	connection	NOUN
ejpam-6690	36	19	with	with	ADP
ejpam-6690	36	20	classical	classical	ADJ
ejpam-6690	36	21	ring	ring	NOUN
ejpam-6690	36	22	theory	theory	NOUN
ejpam-6690	36	23	,	,	PUNCT
ejpam-6690	36	24	and	and	CCONJ
ejpam-6690	36	25	define	define	VERB
ejpam-6690	36	26	a	a	DET
ejpam-6690	36	27	topology	topology	NOUN
ejpam-6690	36	28	on	on	ADP
ejpam-6690	36	29	prime	prime	ADJ
ejpam-6690	36	30	strongly	strongly	ADV
ejpam-6690	36	31	regular	regular	ADJ
ejpam-6690	36	32	relations	relation	NOUN
ejpam-6690	36	33	.	.	PUNCT
ejpam-6690	37	1	section	section	NOUN
ejpam-6690	37	2	5	5	NUM
ejpam-6690	37	3	offers	offer	VERB
ejpam-6690	37	4	further	further	ADJ
ejpam-6690	37	5	discussion	discussion	NOUN
ejpam-6690	37	6	and	and	CCONJ
ejpam-6690	37	7	proposes	propose	VERB
ejpam-6690	37	8	directions	direction	NOUN
ejpam-6690	37	9	for	for	ADP
ejpam-6690	37	10	future	future	ADJ
ejpam-6690	37	11	work	work	NOUN
ejpam-6690	37	12	,	,	PUNCT
ejpam-6690	37	13	while	while	SCONJ
ejpam-6690	37	14	section	section	NOUN
ejpam-6690	37	15	6	6	NUM
ejpam-6690	37	16	concludes	conclude	VERB
ejpam-6690	37	17	the	the	DET
ejpam-6690	37	18	study	study	NOUN
ejpam-6690	37	19	with	with	ADP
ejpam-6690	37	20	a	a	DET
ejpam-6690	37	21	summary	summary	NOUN
ejpam-6690	37	22	of	of	ADP
ejpam-6690	37	23	our	our	PRON
ejpam-6690	37	24	main	main	ADJ
ejpam-6690	37	25	findings	finding	NOUN
ejpam-6690	37	26	.	.	PUNCT
ejpam-6690	38	1	this	this	DET
ejpam-6690	38	2	work	work	NOUN
ejpam-6690	38	3	lays	lay	VERB
ejpam-6690	38	4	the	the	DET
ejpam-6690	38	5	groundwork	groundwork	NOUN
ejpam-6690	38	6	for	for	ADP
ejpam-6690	38	7	future	future	ADJ
ejpam-6690	38	8	research	research	NOUN
ejpam-6690	38	9	on	on	ADP
ejpam-6690	38	10	sheaf	sheaf	NOUN
ejpam-6690	38	11	-	-	PUNCT
ejpam-6690	38	12	theoretic	theoretic	NOUN
ejpam-6690	38	13	constructions	construction	NOUN
ejpam-6690	38	14	in	in	ADP
ejpam-6690	38	15	hyperalgebraic	hyperalgebraic	PROPN
ejpam-6690	38	16	geometry	geometry	NOUN
ejpam-6690	38	17	and	and	CCONJ
ejpam-6690	38	18	provides	provide	VERB
ejpam-6690	38	19	a	a	DET
ejpam-6690	38	20	categorical	categorical	ADJ
ejpam-6690	38	21	bridge	bridge	NOUN
ejpam-6690	38	22	between	between	ADP
ejpam-6690	38	23	hyperstructures	hyperstructure	NOUN
ejpam-6690	38	24	and	and	CCONJ
ejpam-6690	38	25	classical	classical	ADJ
ejpam-6690	38	26	ring	ring	NOUN
ejpam-6690	38	27	-	-	PUNCT
ejpam-6690	38	28	theoretic	theoretic	NOUN
ejpam-6690	38	29	frameworks	framework	NOUN
ejpam-6690	38	30	,	,	PUNCT
ejpam-6690	38	31	with	with	ADP
ejpam-6690	38	32	promising	promising	ADJ
ejpam-6690	38	33	applications	application	NOUN
ejpam-6690	38	34	in	in	ADP
ejpam-6690	38	35	the	the	DET
ejpam-6690	38	36	study	study	NOUN
ejpam-6690	38	37	of	of	ADP
ejpam-6690	38	38	hypermodules	hypermodule	NOUN
ejpam-6690	38	39	and	and	CCONJ
ejpam-6690	38	40	beyond	beyond	ADP
ejpam-6690	38	41	.	.	PUNCT
ejpam-6690	39	1	also	also	ADV
ejpam-6690	39	2	with	with	ADP
ejpam-6690	39	3	the	the	DET
ejpam-6690	39	4	help	help	NOUN
ejpam-6690	39	5	of	of	ADP
ejpam-6690	39	6	equivalence	equivalence	NOUN
ejpam-6690	39	7	relation	relation	NOUN
ejpam-6690	39	8	γ∗	γ∗	NOUN
ejpam-6690	39	9	we	we	PRON
ejpam-6690	39	10	will	will	AUX
ejpam-6690	39	11	examine	examine	VERB
ejpam-6690	39	12	the	the	DET
ejpam-6690	39	13	relationship	relationship	NOUN
ejpam-6690	39	14	between	between	ADP
ejpam-6690	39	15	this	this	DET
ejpam-6690	39	16	functor	functor	NOUN
ejpam-6690	39	17	and	and	CCONJ
ejpam-6690	39	18	its	its	PRON
ejpam-6690	39	19	classical	classical	ADJ
ejpam-6690	39	20	zariski	zariski	NOUN
ejpam-6690	39	21	topology	topology	NOUN
ejpam-6690	39	22	functor	functor	NOUN
ejpam-6690	39	23	.	.	PUNCT
ejpam-6690	40	1	furthermore	furthermore	ADV
ejpam-6690	40	2	,	,	PUNCT
ejpam-6690	40	3	a	a	DET
ejpam-6690	40	4	one	one	NUM
ejpam-6690	40	5	-	-	PUNCT
ejpam-6690	40	6	to	to	ADP
ejpam-6690	40	7	-	-	PUNCT
ejpam-6690	40	8	one	one	NUM
ejpam-6690	40	9	correspondence	correspondence	NOUN
ejpam-6690	40	10	between	between	ADP
ejpam-6690	40	11	strongly	strongly	ADV
ejpam-6690	40	12	regular	regular	ADJ
ejpam-6690	40	13	relations	relation	NOUN
ejpam-6690	40	14	and	and	CCONJ
ejpam-6690	40	15	hyperideals	hyperideal	NOUN
ejpam-6690	40	16	containing	contain	VERB
ejpam-6690	40	17	γ∗(0	γ∗(0	NOUN
ejpam-6690	40	18	)	)	PUNCT
ejpam-6690	40	19	will	will	AUX
ejpam-6690	40	20	be	be	AUX
ejpam-6690	40	21	introduced	introduce	VERB
ejpam-6690	40	22	.	.	PUNCT
ejpam-6690	41	1	also	also	ADV
ejpam-6690	41	2	,	,	PUNCT
ejpam-6690	41	3	by	by	ADP
ejpam-6690	41	4	introducing	introduce	VERB
ejpam-6690	41	5	prime	prime	ADJ
ejpam-6690	41	6	and	and	CCONJ
ejpam-6690	41	7	primary	primary	ADJ
ejpam-6690	41	8	strongly	strongly	ADV
ejpam-6690	41	9	regular	regular	ADJ
ejpam-6690	41	10	relations	relation	NOUN
ejpam-6690	41	11	,	,	PUNCT
ejpam-6690	41	12	a	a	DET
ejpam-6690	41	13	topology	topology	NOUN
ejpam-6690	41	14	is	be	AUX
ejpam-6690	41	15	defined	define	VERB
ejpam-6690	41	16	on	on	ADP
ejpam-6690	41	17	the	the	DET
ejpam-6690	41	18	set	set	NOUN
ejpam-6690	41	19	of	of	ADP
ejpam-6690	41	20	all	all	DET
ejpam-6690	41	21	strongly	strongly	ADV
ejpam-6690	41	22	regular	regular	ADJ
ejpam-6690	41	23	relations	relation	NOUN
ejpam-6690	41	24	,	,	PUNCT
ejpam-6690	41	25	which	which	PRON
ejpam-6690	41	26	in	in	ADP
ejpam-6690	41	27	the	the	DET
ejpam-6690	41	28	next	next	ADJ
ejpam-6690	41	29	researches	research	NOUN
ejpam-6690	41	30	we	we	PRON
ejpam-6690	41	31	intend	intend	VERB
ejpam-6690	41	32	to	to	PART
ejpam-6690	41	33	examine	examine	VERB
ejpam-6690	41	34	its	its	PRON
ejpam-6690	41	35	connection	connection	NOUN
ejpam-6690	41	36	with	with	ADP
ejpam-6690	41	37	zariski	zariski	PROPN
ejpam-6690	41	38	topology	topology	NOUN
ejpam-6690	41	39	.	.	PUNCT
ejpam-6690	42	1	b.	b.	PROPN
ejpam-6690	42	2	afshar	afshar	PROPN
ejpam-6690	42	3	,	,	PUNCT
ejpam-6690	42	4	r.	r.	PROPN
ejpam-6690	42	5	ameri	ameri	PROPN
ejpam-6690	42	6	,	,	PUNCT
ejpam-6690	42	7	m.	m.	PROPN
ejpam-6690	42	8	al	al	PROPN
ejpam-6690	42	9	-	-	PUNCT
ejpam-6690	42	10	tahan	tahan	PROPN
ejpam-6690	42	11	/	/	SYM
ejpam-6690	42	12	eur	eur	PROPN
ejpam-6690	42	13	.	.	PUNCT
ejpam-6690	43	1	j.	j.	PROPN
ejpam-6690	43	2	pure	pure	PROPN
ejpam-6690	43	3	appl	appl	PROPN
ejpam-6690	43	4	.	.	PROPN
ejpam-6690	43	5	math	math	PROPN
ejpam-6690	43	6	,	,	PUNCT
ejpam-6690	43	7	18	18	NUM
ejpam-6690	43	8	(	(	PUNCT
ejpam-6690	43	9	4	4	NUM
ejpam-6690	43	10	)	)	PUNCT
ejpam-6690	43	11	(	(	PUNCT
ejpam-6690	43	12	2025	2025	NUM
ejpam-6690	43	13	)	)	PUNCT
ejpam-6690	43	14	,	,	PUNCT
ejpam-6690	43	15	6690	6690	NUM
ejpam-6690	43	16	3	3	NUM
ejpam-6690	43	17	of	of	ADP
ejpam-6690	43	18	19	19	NUM
ejpam-6690	43	19	2	2	NUM
ejpam-6690	43	20	.	.	PUNCT
ejpam-6690	43	21	preliminaries	preliminary	NOUN
ejpam-6690	43	22	this	this	DET
ejpam-6690	43	23	section	section	NOUN
ejpam-6690	43	24	reviews	review	VERB
ejpam-6690	43	25	the	the	DET
ejpam-6690	43	26	fundamental	fundamental	ADJ
ejpam-6690	43	27	concepts	concept	NOUN
ejpam-6690	43	28	and	and	CCONJ
ejpam-6690	43	29	definitions	definition	NOUN
ejpam-6690	43	30	needed	need	VERB
ejpam-6690	43	31	for	for	ADP
ejpam-6690	43	32	the	the	DET
ejpam-6690	43	33	remainder	remainder	NOUN
ejpam-6690	43	34	of	of	ADP
ejpam-6690	43	35	the	the	DET
ejpam-6690	43	36	paper	paper	NOUN
ejpam-6690	43	37	.	.	PUNCT
ejpam-6690	44	1	we	we	PRON
ejpam-6690	44	2	recall	recall	VERB
ejpam-6690	44	3	basic	basic	ADJ
ejpam-6690	44	4	properties	property	NOUN
ejpam-6690	44	5	of	of	ADP
ejpam-6690	44	6	krasner	krasner	NOUN
ejpam-6690	44	7	hyperrings	hyperring	NOUN
ejpam-6690	44	8	,	,	PUNCT
ejpam-6690	44	9	hyperideals	hyperideal	NOUN
ejpam-6690	44	10	,	,	PUNCT
ejpam-6690	44	11	and	and	CCONJ
ejpam-6690	44	12	regular	regular	ADJ
ejpam-6690	44	13	relations	relation	NOUN
ejpam-6690	44	14	[	[	X
ejpam-6690	44	15	15	15	NUM
ejpam-6690	44	16	]	]	PUNCT
ejpam-6690	44	17	,	,	PUNCT
ejpam-6690	44	18	as	as	ADV
ejpam-6690	44	19	well	well	ADV
ejpam-6690	44	20	as	as	ADP
ejpam-6690	44	21	essential	essential	ADJ
ejpam-6690	44	22	topological	topological	ADJ
ejpam-6690	44	23	notions	notion	NOUN
ejpam-6690	44	24	relevant	relevant	ADJ
ejpam-6690	44	25	to	to	ADP
ejpam-6690	44	26	the	the	DET
ejpam-6690	44	27	zariski	zariski	NOUN
ejpam-6690	44	28	framework	framework	NOUN
ejpam-6690	44	29	[	[	X
ejpam-6690	44	30	16	16	NUM
ejpam-6690	44	31	]	]	PUNCT
ejpam-6690	44	32	.	.	PUNCT
ejpam-6690	45	1	a	a	DET
ejpam-6690	45	2	krasner	krasner	NOUN
ejpam-6690	45	3	hyperring	hyperring	NOUN
ejpam-6690	45	4	is	be	AUX
ejpam-6690	45	5	an	an	DET
ejpam-6690	45	6	algebraic	algebraic	ADJ
ejpam-6690	45	7	structure	structure	NOUN
ejpam-6690	45	8	(	(	PUNCT
ejpam-6690	45	9	r,+	r,+	NUM
ejpam-6690	45	10	,	,	PUNCT
ejpam-6690	45	11	·	·	PUNCT
ejpam-6690	45	12	)	)	PUNCT
ejpam-6690	45	13	which	which	PRON
ejpam-6690	45	14	satisfies	satisfy	VERB
ejpam-6690	45	15	the	the	DET
ejpam-6690	45	16	following	follow	VERB
ejpam-6690	45	17	axioms	axiom	NOUN
ejpam-6690	45	18	:	:	PUNCT
ejpam-6690	45	19	•	•	NOUN
ejpam-6690	45	20	for	for	ADP
ejpam-6690	45	21	every	every	DET
ejpam-6690	45	22	x	x	NOUN
ejpam-6690	45	23	,	,	PUNCT
ejpam-6690	45	24	y	y	PROPN
ejpam-6690	45	25	,	,	PUNCT
ejpam-6690	45	26	z	z	NOUN
ejpam-6690	45	27	∈	∈	PROPN
ejpam-6690	45	28	r	r	NOUN
ejpam-6690	45	29	,	,	PUNCT
ejpam-6690	45	30	x+	x+	NUM
ejpam-6690	45	31	(	(	PUNCT
ejpam-6690	45	32	y	y	PROPN
ejpam-6690	45	33	+	+	PROPN
ejpam-6690	45	34	z	z	NOUN
ejpam-6690	45	35	)	)	PUNCT
ejpam-6690	45	36	=	=	SYM
ejpam-6690	46	1	(	(	PUNCT
ejpam-6690	46	2	x+	x+	X
ejpam-6690	46	3	y	y	NOUN
ejpam-6690	46	4	)	)	PUNCT
ejpam-6690	47	1	+	+	CCONJ
ejpam-6690	48	1	z	z	NOUN
ejpam-6690	48	2	;	;	PUNCT
ejpam-6690	48	3	•	•	ADP
ejpam-6690	48	4	for	for	ADP
ejpam-6690	48	5	every	every	DET
ejpam-6690	48	6	x	x	NOUN
ejpam-6690	48	7	,	,	PUNCT
ejpam-6690	48	8	y	y	PROPN
ejpam-6690	48	9	∈	∈	PROPN
ejpam-6690	48	10	r	r	PROPN
ejpam-6690	48	11	,	,	PUNCT
ejpam-6690	48	12	x+	x+	ADJ
ejpam-6690	48	13	y	y	NOUN
ejpam-6690	48	14	=	=	SYM
ejpam-6690	48	15	y	y	PROPN
ejpam-6690	49	1	+	+	NUM
ejpam-6690	49	2	x	x	SYM
ejpam-6690	49	3	;	;	PUNCT
ejpam-6690	49	4	•	•	X
ejpam-6690	49	5	there	there	PRON
ejpam-6690	49	6	exists	exist	VERB
ejpam-6690	49	7	0	0	NUM
ejpam-6690	49	8	∈	∈	NOUN
ejpam-6690	49	9	r	r	NOUN
ejpam-6690	50	1	such	such	ADJ
ejpam-6690	50	2	that	that	PRON
ejpam-6690	50	3	0	0	NUM
ejpam-6690	51	1	+	+	NUM
ejpam-6690	51	2	x	x	X
ejpam-6690	51	3	=	=	SYM
ejpam-6690	51	4	{	{	PUNCT
ejpam-6690	51	5	x	x	NOUN
ejpam-6690	51	6	}	}	PUNCT
ejpam-6690	51	7	,	,	PUNCT
ejpam-6690	51	8	for	for	ADP
ejpam-6690	51	9	every	every	DET
ejpam-6690	51	10	x	x	SYM
ejpam-6690	51	11	∈	∈	PROPN
ejpam-6690	51	12	r	r	NOUN
ejpam-6690	51	13	;	;	PUNCT
ejpam-6690	51	14	•	•	NOUN
ejpam-6690	51	15	for	for	ADP
ejpam-6690	51	16	every	every	DET
ejpam-6690	51	17	x	x	SYM
ejpam-6690	51	18	∈	∈	NOUN
ejpam-6690	51	19	r	r	NOUN
ejpam-6690	51	20	there	there	PRON
ejpam-6690	51	21	is	be	VERB
ejpam-6690	51	22	a	a	DET
ejpam-6690	51	23	unique	unique	ADJ
ejpam-6690	51	24	x′	x′	PROPN
ejpam-6690	51	25	∈	∈	NOUN
ejpam-6690	51	26	r	r	NOUN
ejpam-6690	51	27	that	that	SCONJ
ejpam-6690	51	28	0	0	NUM
ejpam-6690	51	29	∈	∈	NOUN
ejpam-6690	51	30	x+	x+	X
ejpam-6690	51	31	x′	x′	PROPN
ejpam-6690	52	1	(	(	PUNCT
ejpam-6690	52	2	we	we	PRON
ejpam-6690	52	3	use	use	VERB
ejpam-6690	52	4	−x	−x	NOUN
ejpam-6690	52	5	for	for	ADP
ejpam-6690	52	6	x′	x′	NUM
ejpam-6690	52	7	)	)	PUNCT
ejpam-6690	52	8	;	;	PUNCT
ejpam-6690	52	9	•	•	X
ejpam-6690	52	10	if	if	SCONJ
ejpam-6690	52	11	z	z	NOUN
ejpam-6690	52	12	∈	∈	PROPN
ejpam-6690	52	13	x+	x+	PUNCT
ejpam-6690	53	1	y	y	NOUN
ejpam-6690	53	2	then	then	ADV
ejpam-6690	53	3	y	y	PROPN
ejpam-6690	53	4	∈	∈	PROPN
ejpam-6690	53	5	−x+	−x+	NOUN
ejpam-6690	54	1	z	z	PROPN
ejpam-6690	55	1	and	and	CCONJ
ejpam-6690	55	2	x	x	SYM
ejpam-6690	55	3	∈	∈	PROPN
ejpam-6690	55	4	z	z	NOUN
ejpam-6690	56	1	−	−	PROPN
ejpam-6690	56	2	y	y	NOUN
ejpam-6690	56	3	;	;	PUNCT
ejpam-6690	56	4	•	•	PRON
ejpam-6690	56	5	(	(	PUNCT
ejpam-6690	56	6	r	r	NOUN
ejpam-6690	56	7	,	,	PUNCT
ejpam-6690	56	8	·	·	PUNCT
ejpam-6690	56	9	)	)	PUNCT
ejpam-6690	56	10	is	be	AUX
ejpam-6690	56	11	a	a	DET
ejpam-6690	56	12	semigroup	semigroup	NOUN
ejpam-6690	56	13	,	,	PUNCT
ejpam-6690	56	14	and	and	CCONJ
ejpam-6690	56	15	for	for	ADP
ejpam-6690	56	16	every	every	DET
ejpam-6690	56	17	x	x	SYM
ejpam-6690	56	18	∈	∈	PROPN
ejpam-6690	56	19	r	r	NOUN
ejpam-6690	56	20	,	,	PUNCT
ejpam-6690	56	21	x	x	X
ejpam-6690	56	22	·	·	PUNCT
ejpam-6690	56	23	0	0	PUNCT
ejpam-6690	57	1	=	=	SYM
ejpam-6690	57	2	0	0	PUNCT
ejpam-6690	57	3	·	·	PUNCT
ejpam-6690	57	4	x	x	PUNCT
ejpam-6690	57	5	=	=	PUNCT
ejpam-6690	57	6	0	0	NUM
ejpam-6690	57	7	;	;	PUNCT
ejpam-6690	57	8	•	•	NUM
ejpam-6690	57	9	the	the	DET
ejpam-6690	57	10	operation	operation	NOUN
ejpam-6690	57	11	”	"	PUNCT
ejpam-6690	57	12	·	·	PUNCT
ejpam-6690	57	13	”	"	PUNCT
ejpam-6690	57	14	is	be	AUX
ejpam-6690	57	15	bilaterally	bilaterally	ADV
ejpam-6690	57	16	distributive	distributive	ADJ
ejpam-6690	57	17	with	with	ADP
ejpam-6690	57	18	respect	respect	NOUN
ejpam-6690	57	19	to	to	ADP
ejpam-6690	57	20	the	the	DET
ejpam-6690	57	21	hyperoperation	hyperoperation	NOUN
ejpam-6690	57	22	”	"	PUNCT
ejpam-6690	57	23	+	+	PROPN
ejpam-6690	57	24	”	"	PUNCT
ejpam-6690	57	25	.	.	PUNCT
ejpam-6690	58	1	a	a	DET
ejpam-6690	58	2	krasner	krasner	NOUN
ejpam-6690	58	3	hyperring	hyperring	NOUN
ejpam-6690	58	4	(	(	PUNCT
ejpam-6690	58	5	r,+	r,+	NUM
ejpam-6690	58	6	,	,	PUNCT
ejpam-6690	58	7	·	·	PUNCT
ejpam-6690	58	8	)	)	PUNCT
ejpam-6690	58	9	is	be	AUX
ejpam-6690	58	10	called	call	VERB
ejpam-6690	58	11	commutative	commutative	ADJ
ejpam-6690	58	12	(	(	PUNCT
ejpam-6690	58	13	with	with	ADP
ejpam-6690	58	14	a	a	DET
ejpam-6690	58	15	unit	unit	NOUN
ejpam-6690	58	16	element	element	NOUN
ejpam-6690	58	17	)	)	PUNCT
ejpam-6690	58	18	if	if	SCONJ
ejpam-6690	58	19	(	(	PUNCT
ejpam-6690	58	20	r	r	NOUN
ejpam-6690	58	21	,	,	PUNCT
ejpam-6690	58	22	·	·	PUNCT
ejpam-6690	58	23	)	)	PUNCT
ejpam-6690	58	24	is	be	AUX
ejpam-6690	58	25	commutative	commutative	ADJ
ejpam-6690	58	26	(	(	PUNCT
ejpam-6690	58	27	with	with	ADP
ejpam-6690	58	28	a	a	DET
ejpam-6690	58	29	unit	unit	NOUN
ejpam-6690	58	30	element	element	NOUN
ejpam-6690	58	31	)	)	PUNCT
ejpam-6690	58	32	semigroup	semigroup	PROPN
ejpam-6690	58	33	.	.	PUNCT
ejpam-6690	59	1	if	if	SCONJ
ejpam-6690	59	2	(	(	PUNCT
ejpam-6690	59	3	r	r	NOUN
ejpam-6690	59	4	−	−	PROPN
ejpam-6690	59	5	{	{	PUNCT
ejpam-6690	59	6	0	0	NUM
ejpam-6690	59	7	}	}	PUNCT
ejpam-6690	59	8	,	,	PUNCT
ejpam-6690	59	9	·	·	PUNCT
ejpam-6690	59	10	)	)	PUNCT
ejpam-6690	59	11	is	be	AUX
ejpam-6690	59	12	a	a	DET
ejpam-6690	59	13	group	group	NOUN
ejpam-6690	59	14	then	then	ADV
ejpam-6690	59	15	(	(	PUNCT
ejpam-6690	59	16	r,+	r,+	NUM
ejpam-6690	59	17	,	,	PUNCT
ejpam-6690	59	18	·	·	PUNCT
ejpam-6690	59	19	)	)	PUNCT
ejpam-6690	59	20	is	be	AUX
ejpam-6690	59	21	called	call	VERB
ejpam-6690	59	22	a	a	DET
ejpam-6690	59	23	krasner	krasner	NOUN
ejpam-6690	59	24	hyperfield	hyperfield	NOUN
ejpam-6690	59	25	and	and	CCONJ
ejpam-6690	59	26	if	if	SCONJ
ejpam-6690	59	27	(	(	PUNCT
ejpam-6690	59	28	r,+	r,+	NUM
ejpam-6690	59	29	,	,	PUNCT
ejpam-6690	59	30	·	·	PUNCT
ejpam-6690	59	31	)	)	PUNCT
ejpam-6690	59	32	is	be	AUX
ejpam-6690	59	33	commutative	commutative	ADJ
ejpam-6690	59	34	krasner	krasner	NOUN
ejpam-6690	59	35	hyperring	hyperre	VERB
ejpam-6690	59	36	with	with	ADP
ejpam-6690	59	37	a	a	DET
ejpam-6690	59	38	unit	unit	NOUN
ejpam-6690	59	39	element	element	NOUN
ejpam-6690	59	40	and	and	CCONJ
ejpam-6690	59	41	ab	ab	NOUN
ejpam-6690	59	42	=	=	NOUN
ejpam-6690	59	43	0	0	NUM
ejpam-6690	59	44	implies	imply	VERB
ejpam-6690	59	45	that	that	SCONJ
ejpam-6690	59	46	a	a	DET
ejpam-6690	59	47	=	=	SYM
ejpam-6690	59	48	0	0	NUM
ejpam-6690	59	49	or	or	CCONJ
ejpam-6690	59	50	b	b	NOUN
ejpam-6690	59	51	=	=	NOUN
ejpam-6690	59	52	0	0	NUM
ejpam-6690	59	53	for	for	ADP
ejpam-6690	59	54	all	all	DET
ejpam-6690	59	55	a	a	PRON
ejpam-6690	59	56	,	,	PUNCT
ejpam-6690	59	57	b	b	X
ejpam-6690	59	58	∈	∈	PROPN
ejpam-6690	59	59	r	r	NOUN
ejpam-6690	59	60	,	,	PUNCT
ejpam-6690	59	61	then	then	ADV
ejpam-6690	59	62	r	r	NOUN
ejpam-6690	59	63	is	be	AUX
ejpam-6690	59	64	called	call	VERB
ejpam-6690	59	65	hyperdomain	hyperdomain	NOUN
ejpam-6690	59	66	.	.	PUNCT
ejpam-6690	60	1	throughout	throughout	ADP
ejpam-6690	60	2	this	this	DET
ejpam-6690	60	3	paper	paper	NOUN
ejpam-6690	60	4	,	,	PUNCT
ejpam-6690	60	5	hyperring	hyperre	VERB
ejpam-6690	60	6	refers	refer	VERB
ejpam-6690	60	7	to	to	ADP
ejpam-6690	60	8	commutative	commutative	ADJ
ejpam-6690	60	9	krasner	krasner	NOUN
ejpam-6690	60	10	hyperring	hyperre	VERB
ejpam-6690	60	11	with	with	ADP
ejpam-6690	60	12	a	a	DET
ejpam-6690	60	13	unit	unit	NOUN
ejpam-6690	60	14	element	element	NOUN
ejpam-6690	60	15	.	.	PUNCT
ejpam-6690	61	1	definition	definition	NOUN
ejpam-6690	61	2	1	1	NUM
ejpam-6690	61	3	.	.	PUNCT
ejpam-6690	62	1	[	[	X
ejpam-6690	62	2	15	15	NUM
ejpam-6690	62	3	]	]	PUNCT
ejpam-6690	62	4	let	let	VERB
ejpam-6690	62	5	r	r	PRON
ejpam-6690	62	6	be	be	AUX
ejpam-6690	62	7	a	a	DET
ejpam-6690	62	8	hyperring	hyperring	NOUN
ejpam-6690	62	9	and	and	CCONJ
ejpam-6690	62	10	let	let	VERB
ejpam-6690	62	11	i	i	PRON
ejpam-6690	62	12	⊆	⊆	NUM
ejpam-6690	62	13	r	r	NOUN
ejpam-6690	62	14	be	be	VERB
ejpam-6690	62	15	a	a	DET
ejpam-6690	62	16	subhyperring	subhyperring	NOUN
ejpam-6690	62	17	.	.	PUNCT
ejpam-6690	63	1	•	•	INTJ
ejpam-6690	63	2	i	i	PRON
ejpam-6690	63	3	is	be	AUX
ejpam-6690	63	4	called	call	VERB
ejpam-6690	63	5	a	a	DET
ejpam-6690	63	6	left	left	ADJ
ejpam-6690	63	7	hyperideal	hyperideal	NOUN
ejpam-6690	63	8	of	of	ADP
ejpam-6690	63	9	r	r	NOUN
ejpam-6690	63	10	if	if	SCONJ
ejpam-6690	63	11	ra	ra	PROPN
ejpam-6690	63	12	∈	∈	PROPN
ejpam-6690	63	13	i	i	PRON
ejpam-6690	63	14	for	for	ADP
ejpam-6690	63	15	all	all	DET
ejpam-6690	63	16	r	r	NOUN
ejpam-6690	63	17	∈	∈	NOUN
ejpam-6690	63	18	r	r	NOUN
ejpam-6690	63	19	,	,	PUNCT
ejpam-6690	63	20	a	a	DET
ejpam-6690	63	21	∈	∈	PROPN
ejpam-6690	63	22	i.	i.	NOUN
ejpam-6690	64	1	•	•	NOUN
ejpam-6690	65	1	i	i	PRON
ejpam-6690	65	2	is	be	AUX
ejpam-6690	65	3	called	call	VERB
ejpam-6690	65	4	a	a	DET
ejpam-6690	65	5	right	right	ADJ
ejpam-6690	65	6	hyperideal	hyperideal	NOUN
ejpam-6690	65	7	of	of	ADP
ejpam-6690	65	8	r	r	NOUN
ejpam-6690	65	9	if	if	SCONJ
ejpam-6690	65	10	ar	ar	PROPN
ejpam-6690	65	11	∈	∈	PROPN
ejpam-6690	65	12	i	i	PRON
ejpam-6690	65	13	for	for	ADP
ejpam-6690	65	14	all	all	DET
ejpam-6690	65	15	r	r	NOUN
ejpam-6690	65	16	∈	∈	NOUN
ejpam-6690	65	17	r	r	NOUN
ejpam-6690	65	18	,	,	PUNCT
ejpam-6690	65	19	a	a	DET
ejpam-6690	65	20	∈	∈	PROPN
ejpam-6690	65	21	i.	i.	NOUN
ejpam-6690	65	22	•	•	INTJ
ejpam-6690	65	23	if	if	SCONJ
ejpam-6690	65	24	i	i	PRON
ejpam-6690	65	25	is	be	AUX
ejpam-6690	65	26	both	both	CCONJ
ejpam-6690	65	27	a	a	DET
ejpam-6690	65	28	left	left	NOUN
ejpam-6690	65	29	and	and	CCONJ
ejpam-6690	65	30	a	a	DET
ejpam-6690	65	31	right	right	ADJ
ejpam-6690	65	32	hyperideal	hyperideal	NOUN
ejpam-6690	65	33	,	,	PUNCT
ejpam-6690	65	34	it	it	PRON
ejpam-6690	65	35	is	be	AUX
ejpam-6690	65	36	called	call	VERB
ejpam-6690	65	37	a	a	DET
ejpam-6690	65	38	(	(	PUNCT
ejpam-6690	65	39	two	two	NUM
ejpam-6690	65	40	-	-	PUNCT
ejpam-6690	65	41	sided	sided	ADJ
ejpam-6690	65	42	)	)	PUNCT
ejpam-6690	65	43	hyperideal	hyperideal	NOUN
ejpam-6690	65	44	of	of	ADP
ejpam-6690	65	45	r.	r.	PROPN
ejpam-6690	65	46	a	a	DET
ejpam-6690	65	47	proper	proper	ADJ
ejpam-6690	65	48	hyperideal	hyperideal	NOUN
ejpam-6690	65	49	m	m	VERB
ejpam-6690	65	50	⊊	⊊	AUX
ejpam-6690	65	51	r	r	NOUN
ejpam-6690	65	52	is	be	AUX
ejpam-6690	65	53	called	call	VERB
ejpam-6690	65	54	a	a	DET
ejpam-6690	65	55	maximal	maximal	ADJ
ejpam-6690	65	56	hyperideal	hyperideal	NOUN
ejpam-6690	65	57	if	if	SCONJ
ejpam-6690	65	58	the	the	DET
ejpam-6690	65	59	only	only	ADJ
ejpam-6690	65	60	hyperideals	hyperideal	NOUN
ejpam-6690	65	61	of	of	ADP
ejpam-6690	65	62	r	r	NOUN
ejpam-6690	65	63	that	that	PRON
ejpam-6690	65	64	contain	contain	VERB
ejpam-6690	65	65	m	m	VERB
ejpam-6690	65	66	are	be	AUX
ejpam-6690	65	67	m	m	ADJ
ejpam-6690	65	68	and	and	CCONJ
ejpam-6690	65	69	r	r	VERB
ejpam-6690	65	70	itself	itself	PRON
ejpam-6690	65	71	.	.	PUNCT
ejpam-6690	66	1	a	a	DET
ejpam-6690	66	2	proper	proper	ADJ
ejpam-6690	66	3	hyperideal	hyperideal	NOUN
ejpam-6690	66	4	p	p	NOUN
ejpam-6690	66	5	⊊	⊊	NOUN
ejpam-6690	66	6	r	r	NOUN
ejpam-6690	66	7	is	be	AUX
ejpam-6690	66	8	called	call	VERB
ejpam-6690	66	9	a	a	DET
ejpam-6690	66	10	prime	prime	ADJ
ejpam-6690	66	11	hyperideal	hyperideal	NOUN
ejpam-6690	66	12	if	if	SCONJ
ejpam-6690	66	13	for	for	ADP
ejpam-6690	66	14	all	all	DET
ejpam-6690	66	15	hyperideals	hyperideal	NOUN
ejpam-6690	66	16	a	a	DET
ejpam-6690	66	17	,	,	PUNCT
ejpam-6690	66	18	b	b	NOUN
ejpam-6690	66	19	⊆	⊆	NUM
ejpam-6690	66	20	r	r	NOUN
ejpam-6690	66	21	,	,	PUNCT
ejpam-6690	66	22	the	the	DET
ejpam-6690	66	23	condition	condition	NOUN
ejpam-6690	66	24	ab	ab	PROPN
ejpam-6690	66	25	⊆	⊆	NUM
ejpam-6690	66	26	p	p	PROPN
ejpam-6690	66	27	implies	imply	VERB
ejpam-6690	66	28	a	a	DET
ejpam-6690	66	29	⊆	⊆	NUM
ejpam-6690	66	30	p	p	NOUN
ejpam-6690	66	31	or	or	CCONJ
ejpam-6690	66	32	b	b	NOUN
ejpam-6690	66	33	⊆	⊆	NUM
ejpam-6690	66	34	p	p	NOUN
ejpam-6690	66	35	.	.	PUNCT
ejpam-6690	67	1	lemma	lemma	PROPN
ejpam-6690	67	2	1	1	NUM
ejpam-6690	67	3	.	.	PUNCT
ejpam-6690	68	1	[	[	X
ejpam-6690	68	2	15	15	NUM
ejpam-6690	68	3	]	]	X
ejpam-6690	68	4	a	a	DET
ejpam-6690	68	5	nonempty	nonempty	NOUN
ejpam-6690	68	6	subset	subset	VERB
ejpam-6690	68	7	i	i	PRON
ejpam-6690	68	8	of	of	ADP
ejpam-6690	68	9	a	a	DET
ejpam-6690	68	10	hyperring	hyperring	NOUN
ejpam-6690	68	11	r	r	NOUN
ejpam-6690	68	12	is	be	AUX
ejpam-6690	68	13	a	a	DET
ejpam-6690	68	14	left(right	left(right	PROPN
ejpam-6690	68	15	)	)	PUNCT
ejpam-6690	68	16	hyperideal	hyperideal	NOUN
ejpam-6690	68	17	if	if	SCONJ
ejpam-6690	68	18	and	and	CCONJ
ejpam-6690	68	19	only	only	ADV
ejpam-6690	68	20	if	if	SCONJ
ejpam-6690	68	21	a−	a−	PROPN
ejpam-6690	68	22	b	b	NOUN
ejpam-6690	68	23	⊆	⊆	NUM
ejpam-6690	68	24	i	i	PRON
ejpam-6690	68	25	and	and	CCONJ
ejpam-6690	68	26	ra	ra	PROPN
ejpam-6690	68	27	∈	∈	PROPN
ejpam-6690	68	28	i(ra	i(ra	PROPN
ejpam-6690	68	29	∈	∈	PROPN
ejpam-6690	68	30	i	i	PROPN
ejpam-6690	68	31	)	)	PUNCT
ejpam-6690	68	32	,	,	PUNCT
ejpam-6690	68	33	for	for	ADP
ejpam-6690	68	34	all	all	DET
ejpam-6690	68	35	a	a	PRON
ejpam-6690	68	36	,	,	PUNCT
ejpam-6690	68	37	b	b	X
ejpam-6690	68	38	∈	∈	PROPN
ejpam-6690	68	39	i	i	PRON
ejpam-6690	68	40	,	,	PUNCT
ejpam-6690	68	41	r	r	PROPN
ejpam-6690	68	42	∈	∈	PROPN
ejpam-6690	68	43	r.	r.	PROPN
ejpam-6690	68	44	b.	b.	PROPN
ejpam-6690	68	45	afshar	afshar	PROPN
ejpam-6690	68	46	,	,	PUNCT
ejpam-6690	68	47	r.	r.	PROPN
ejpam-6690	68	48	ameri	ameri	PROPN
ejpam-6690	68	49	,	,	PUNCT
ejpam-6690	68	50	m.	m.	PROPN
ejpam-6690	68	51	al	al	PROPN
ejpam-6690	68	52	-	-	PUNCT
ejpam-6690	68	53	tahan	tahan	PROPN
ejpam-6690	68	54	/	/	SYM
ejpam-6690	68	55	eur	eur	PROPN
ejpam-6690	68	56	.	.	PUNCT
ejpam-6690	69	1	j.	j.	PROPN
ejpam-6690	69	2	pure	pure	PROPN
ejpam-6690	69	3	appl	appl	PROPN
ejpam-6690	69	4	.	.	PROPN
ejpam-6690	69	5	math	math	PROPN
ejpam-6690	69	6	,	,	PUNCT
ejpam-6690	69	7	18	18	NUM
ejpam-6690	69	8	(	(	PUNCT
ejpam-6690	69	9	4	4	NUM
ejpam-6690	69	10	)	)	PUNCT
ejpam-6690	69	11	(	(	PUNCT
ejpam-6690	69	12	2025	2025	NUM
ejpam-6690	69	13	)	)	PUNCT
ejpam-6690	69	14	,	,	PUNCT
ejpam-6690	69	15	6690	6690	NUM
ejpam-6690	69	16	4	4	NUM
ejpam-6690	69	17	of	of	ADP
ejpam-6690	69	18	19	19	NUM
ejpam-6690	69	19	an	an	DET
ejpam-6690	69	20	equivalence	equivalence	NOUN
ejpam-6690	69	21	relation	relation	NOUN
ejpam-6690	69	22	θ	θ	PROPN
ejpam-6690	69	23	on	on	ADP
ejpam-6690	69	24	a	a	DET
ejpam-6690	69	25	krasner	krasner	NOUN
ejpam-6690	69	26	hyperring	hyperre	VERB
ejpam-6690	69	27	r	r	NOUN
ejpam-6690	69	28	is	be	AUX
ejpam-6690	69	29	called	call	VERB
ejpam-6690	69	30	regular	regular	ADJ
ejpam-6690	69	31	if	if	SCONJ
ejpam-6690	69	32	the	the	DET
ejpam-6690	69	33	following	follow	VERB
ejpam-6690	69	34	implication	implication	NOUN
ejpam-6690	69	35	holds	hold	VERB
ejpam-6690	69	36	:	:	PUNCT
ejpam-6690	69	37	a	a	DET
ejpam-6690	69	38	θ	θ	PROPN
ejpam-6690	69	39	b	b	PROPN
ejpam-6690	69	40	,	,	PUNCT
ejpam-6690	69	41	c	c	NOUN
ejpam-6690	69	42	θ	θ	PROPN
ejpam-6690	69	43	d	d	NOUN
ejpam-6690	69	44	⇒	⇒	PROPN
ejpam-6690	69	45	(	(	PUNCT
ejpam-6690	69	46	a+	a+	PUNCT
ejpam-6690	69	47	c	c	X
ejpam-6690	69	48	)	)	PUNCT
ejpam-6690	69	49	θ̄	θ̄	NOUN
ejpam-6690	69	50	(	(	PUNCT
ejpam-6690	69	51	b+	b+	X
ejpam-6690	69	52	d	d	X
ejpam-6690	69	53	)	)	PUNCT
ejpam-6690	69	54	and	and	CCONJ
ejpam-6690	69	55	ac	ac	PROPN
ejpam-6690	69	56	θ	θ	PROPN
ejpam-6690	69	57	bd	bd	PROPN
ejpam-6690	69	58	,	,	PUNCT
ejpam-6690	69	59	(	(	PUNCT
ejpam-6690	69	60	1	1	NUM
ejpam-6690	69	61	)	)	PUNCT
ejpam-6690	69	62	and	and	CCONJ
ejpam-6690	69	63	is	be	AUX
ejpam-6690	69	64	called	call	VERB
ejpam-6690	69	65	strongly	strongly	ADV
ejpam-6690	69	66	regular	regular	ADJ
ejpam-6690	69	67	if	if	SCONJ
ejpam-6690	69	68	a	a	DET
ejpam-6690	69	69	θ	θ	PROPN
ejpam-6690	69	70	b	b	PROPN
ejpam-6690	69	71	,	,	PUNCT
ejpam-6690	70	1	c	c	NOUN
ejpam-6690	70	2	θ	θ	PROPN
ejpam-6690	70	3	d	d	NOUN
ejpam-6690	70	4	⇒	⇒	PROPN
ejpam-6690	70	5	(	(	PUNCT
ejpam-6690	70	6	a+	a+	PUNCT
ejpam-6690	70	7	c	c	X
ejpam-6690	70	8	)	)	PUNCT
ejpam-6690	70	9	¯̄θ	¯̄θ	PROPN
ejpam-6690	70	10	(	(	PUNCT
ejpam-6690	70	11	b+	b+	X
ejpam-6690	70	12	d	d	X
ejpam-6690	70	13	)	)	PUNCT
ejpam-6690	70	14	and	and	CCONJ
ejpam-6690	70	15	ac	ac	PROPN
ejpam-6690	70	16	θ	θ	PROPN
ejpam-6690	70	17	bd	bd	PROPN
ejpam-6690	70	18	,	,	PUNCT
ejpam-6690	70	19	(	(	PUNCT
ejpam-6690	70	20	2	2	NUM
ejpam-6690	70	21	)	)	PUNCT
ejpam-6690	70	22	for	for	ADP
ejpam-6690	70	23	every	every	DET
ejpam-6690	70	24	a	a	DET
ejpam-6690	70	25	,	,	PUNCT
ejpam-6690	70	26	b	b	NOUN
ejpam-6690	70	27	,	,	PUNCT
ejpam-6690	70	28	c	c	NOUN
ejpam-6690	70	29	,	,	PUNCT
ejpam-6690	70	30	d	d	PROPN
ejpam-6690	70	31	∈	∈	PROPN
ejpam-6690	70	32	r.	r.	NOUN
ejpam-6690	70	33	definition	definition	NOUN
ejpam-6690	70	34	2	2	NUM
ejpam-6690	70	35	.	.	PUNCT
ejpam-6690	71	1	[	[	X
ejpam-6690	71	2	15	15	NUM
ejpam-6690	71	3	]	]	PUNCT
ejpam-6690	71	4	let	let	VERB
ejpam-6690	71	5	r	r	NOUN
ejpam-6690	71	6	and	and	CCONJ
ejpam-6690	71	7	s	s	VERB
ejpam-6690	71	8	be	be	AUX
ejpam-6690	71	9	hyperrings	hyperring	NOUN
ejpam-6690	71	10	.	.	PUNCT
ejpam-6690	72	1	a	a	DET
ejpam-6690	72	2	mapping	mapping	NOUN
ejpam-6690	72	3	f	f	NOUN
ejpam-6690	72	4	from	from	ADP
ejpam-6690	72	5	r	r	NOUN
ejpam-6690	72	6	to	to	ADP
ejpam-6690	72	7	s	s	NOUN
ejpam-6690	72	8	is	be	AUX
ejpam-6690	72	9	said	say	VERB
ejpam-6690	72	10	to	to	PART
ejpam-6690	72	11	be	be	AUX
ejpam-6690	72	12	a	a	DET
ejpam-6690	72	13	good	good	ADJ
ejpam-6690	72	14	homomorphism	homomorphism	NOUN
ejpam-6690	72	15	if	if	SCONJ
ejpam-6690	72	16	for	for	ADP
ejpam-6690	72	17	every	every	DET
ejpam-6690	72	18	a	a	PROPN
ejpam-6690	72	19	,	,	PUNCT
ejpam-6690	72	20	b	b	X
ejpam-6690	72	21	∈	∈	PROPN
ejpam-6690	72	22	r	r	NOUN
ejpam-6690	72	23	:	:	PUNCT
ejpam-6690	72	24	f(a	f(a	NOUN
ejpam-6690	72	25	+	+	NUM
ejpam-6690	72	26	b	b	X
ejpam-6690	72	27	)	)	PUNCT
ejpam-6690	72	28	=	=	SYM
ejpam-6690	72	29	f(a	f(a	NOUN
ejpam-6690	72	30	)	)	PUNCT
ejpam-6690	73	1	+	+	CCONJ
ejpam-6690	73	2	f(b	f(b	PROPN
ejpam-6690	73	3	)	)	PUNCT
ejpam-6690	73	4	,	,	PUNCT
ejpam-6690	73	5	f(ab	f(ab	NOUN
ejpam-6690	73	6	)	)	PUNCT
ejpam-6690	73	7	=	=	SYM
ejpam-6690	73	8	f(a)f(b	f(a)f(b	NOUN
ejpam-6690	73	9	)	)	PUNCT
ejpam-6690	73	10	and	and	CCONJ
ejpam-6690	73	11	f(0	f(0	NOUN
ejpam-6690	73	12	)	)	PUNCT
ejpam-6690	73	13	=	=	SYM
ejpam-6690	74	1	0	0	X
ejpam-6690	74	2	.	.	PUNCT
ejpam-6690	75	1	let	let	VERB
ejpam-6690	75	2	r	r	PRON
ejpam-6690	75	3	be	be	AUX
ejpam-6690	75	4	a	a	DET
ejpam-6690	75	5	commutative	commutative	ADJ
ejpam-6690	75	6	hyperring	hyperring	NOUN
ejpam-6690	75	7	with	with	ADP
ejpam-6690	75	8	a	a	DET
ejpam-6690	75	9	unit	unit	NOUN
ejpam-6690	75	10	element	element	NOUN
ejpam-6690	75	11	and	and	CCONJ
ejpam-6690	75	12	i	i	PRON
ejpam-6690	75	13	be	be	VERB
ejpam-6690	75	14	a	a	DET
ejpam-6690	75	15	proper	proper	ADJ
ejpam-6690	75	16	hyperideal	hyperideal	NOUN
ejpam-6690	75	17	of	of	ADP
ejpam-6690	75	18	r.	r.	PROPN
ejpam-6690	75	19	then	then	ADV
ejpam-6690	75	20	there	there	PRON
ejpam-6690	75	21	exists	exist	VERB
ejpam-6690	75	22	a	a	DET
ejpam-6690	75	23	maximal	maximal	ADJ
ejpam-6690	75	24	hyperideal	hyperideal	NOUN
ejpam-6690	75	25	of	of	ADP
ejpam-6690	75	26	r	r	NOUN
ejpam-6690	75	27	containing	contain	VERB
ejpam-6690	75	28	i	i	PRON
ejpam-6690	75	29	and	and	CCONJ
ejpam-6690	75	30	each	each	DET
ejpam-6690	75	31	maximal	maximal	ADJ
ejpam-6690	75	32	hyperideal	hyperideal	NOUN
ejpam-6690	75	33	is	be	AUX
ejpam-6690	75	34	a	a	DET
ejpam-6690	75	35	prime	prime	ADJ
ejpam-6690	75	36	hyperideal	hyperideal	NOUN
ejpam-6690	75	37	.	.	PUNCT
ejpam-6690	76	1	additionally	additionally	ADV
ejpam-6690	76	2	,	,	PUNCT
ejpam-6690	76	3	i	i	PRON
ejpam-6690	76	4	is	be	AUX
ejpam-6690	76	5	prime	prime	ADJ
ejpam-6690	76	6	if	if	SCONJ
ejpam-6690	76	7	ab	ab	PROPN
ejpam-6690	76	8	∈	∈	PROPN
ejpam-6690	76	9	p	p	PROPN
ejpam-6690	76	10	implies	imply	VERB
ejpam-6690	76	11	that	that	SCONJ
ejpam-6690	76	12	a	a	DET
ejpam-6690	76	13	∈	∈	PROPN
ejpam-6690	76	14	p	p	NOUN
ejpam-6690	76	15	or	or	CCONJ
ejpam-6690	76	16	b	b	NOUN
ejpam-6690	76	17	∈	∈	PROPN
ejpam-6690	76	18	p	p	NOUN
ejpam-6690	76	19	,	,	PUNCT
ejpam-6690	76	20	for	for	ADP
ejpam-6690	76	21	every	every	DET
ejpam-6690	76	22	a	a	PROPN
ejpam-6690	76	23	,	,	PUNCT
ejpam-6690	76	24	b	b	PROPN
ejpam-6690	76	25	∈	∈	PROPN
ejpam-6690	76	26	r.	r.	NOUN
ejpam-6690	76	27	proposition	proposition	NOUN
ejpam-6690	76	28	1	1	NUM
ejpam-6690	76	29	.	.	PUNCT
ejpam-6690	77	1	[	[	X
ejpam-6690	77	2	15	15	NUM
ejpam-6690	77	3	]	]	PUNCT
ejpam-6690	77	4	let	let	VERB
ejpam-6690	77	5	r	r	PRON
ejpam-6690	77	6	be	be	AUX
ejpam-6690	77	7	a	a	DET
ejpam-6690	77	8	commutative	commutative	ADJ
ejpam-6690	77	9	hyperring	hyperring	NOUN
ejpam-6690	77	10	with	with	ADP
ejpam-6690	77	11	a	a	DET
ejpam-6690	77	12	unit	unit	NOUN
ejpam-6690	77	13	element	element	NOUN
ejpam-6690	77	14	and	and	CCONJ
ejpam-6690	77	15	i	i	PRON
ejpam-6690	77	16	be	be	VERB
ejpam-6690	77	17	a	a	DET
ejpam-6690	77	18	proper	proper	ADJ
ejpam-6690	77	19	hyperideal	hyperideal	NOUN
ejpam-6690	77	20	of	of	ADP
ejpam-6690	77	21	r.	r.	PROPN
ejpam-6690	77	22	then	then	ADV
ejpam-6690	77	23	:	:	PUNCT
ejpam-6690	77	24	(	(	PUNCT
ejpam-6690	77	25	i	i	NOUN
ejpam-6690	77	26	)	)	PUNCT
ejpam-6690	78	1	i	i	PRON
ejpam-6690	78	2	is	be	AUX
ejpam-6690	78	3	prime	prime	ADJ
ejpam-6690	78	4	hyperideal	hyperideal	NOUN
ejpam-6690	78	5	if	if	SCONJ
ejpam-6690	79	1	and	and	CCONJ
ejpam-6690	79	2	only	only	ADV
ejpam-6690	79	3	if	if	SCONJ
ejpam-6690	79	4	r	r	X
ejpam-6690	79	5	/	/	SYM
ejpam-6690	79	6	i	i	PRON
ejpam-6690	79	7	is	be	AUX
ejpam-6690	79	8	a	a	DET
ejpam-6690	79	9	hyperdomain	hyperdomain	NOUN
ejpam-6690	79	10	.	.	PUNCT
ejpam-6690	80	1	(	(	PUNCT
ejpam-6690	80	2	ii	ii	X
ejpam-6690	80	3	)	)	PUNCT
ejpam-6690	81	1	i	i	PRON
ejpam-6690	81	2	is	be	AUX
ejpam-6690	81	3	maximal	maximal	ADJ
ejpam-6690	81	4	hyperideal	hyperideal	ADJ
ejpam-6690	81	5	if	if	SCONJ
ejpam-6690	82	1	and	and	CCONJ
ejpam-6690	82	2	only	only	ADV
ejpam-6690	82	3	if	if	SCONJ
ejpam-6690	82	4	r	r	X
ejpam-6690	82	5	/	/	SYM
ejpam-6690	82	6	i	i	PRON
ejpam-6690	82	7	is	be	AUX
ejpam-6690	82	8	a	a	DET
ejpam-6690	82	9	hyperfield	hyperfield	NOUN
ejpam-6690	82	10	.	.	PUNCT
ejpam-6690	83	1	for	for	ADP
ejpam-6690	83	2	any	any	DET
ejpam-6690	83	3	regular	regular	ADJ
ejpam-6690	83	4	hypergroup	hypergroup	NOUN
ejpam-6690	83	5	h	h	NOUN
ejpam-6690	83	6	,	,	PUNCT
ejpam-6690	83	7	if	if	SCONJ
ejpam-6690	83	8	sr(h	sr(h	PUNCT
ejpam-6690	83	9	)	)	PUNCT
ejpam-6690	83	10	is	be	AUX
ejpam-6690	83	11	the	the	DET
ejpam-6690	83	12	set	set	NOUN
ejpam-6690	83	13	of	of	ADP
ejpam-6690	83	14	all	all	DET
ejpam-6690	83	15	strongly	strongly	ADV
ejpam-6690	83	16	regular	regular	ADJ
ejpam-6690	83	17	relations	relation	NOUN
ejpam-6690	83	18	on	on	ADP
ejpam-6690	83	19	h	h	NOUN
ejpam-6690	83	20	and	and	CCONJ
ejpam-6690	83	21	n(sβ	n(sβ	NOUN
ejpam-6690	83	22	)	)	PUNCT
ejpam-6690	83	23	is	be	AUX
ejpam-6690	83	24	the	the	DET
ejpam-6690	83	25	set	set	NOUN
ejpam-6690	83	26	of	of	ADP
ejpam-6690	83	27	all	all	DET
ejpam-6690	83	28	normal	normal	ADJ
ejpam-6690	83	29	subhypergroups	subhypergroup	NOUN
ejpam-6690	83	30	of	of	ADP
ejpam-6690	83	31	h	h	NOUN
ejpam-6690	83	32	,	,	PUNCT
ejpam-6690	83	33	containing	contain	VERB
ejpam-6690	83	34	sβ	sβ	PRON
ejpam-6690	83	35	(=	(=	ADJ
ejpam-6690	83	36	ωh	ωh	PROPN
ejpam-6690	83	37	)	)	PUNCT
ejpam-6690	83	38	,	,	PUNCT
ejpam-6690	83	39	then	then	ADV
ejpam-6690	83	40	the	the	DET
ejpam-6690	83	41	map	map	NOUN
ejpam-6690	83	42	φ	φ	X
ejpam-6690	83	43	:	:	PUNCT
ejpam-6690	83	44	sr(h	sr(h	NUM
ejpam-6690	83	45	)	)	PUNCT
ejpam-6690	83	46	→	→	SYM
ejpam-6690	83	47	n(sβ	n(sβ	NOUN
ejpam-6690	83	48	)	)	PUNCT
ejpam-6690	83	49	ρ	ρ	PROPN
ejpam-6690	83	50	7→	7→	NUM
ejpam-6690	83	51	sρ	sρ	PRON
ejpam-6690	83	52	(	(	PUNCT
ejpam-6690	83	53	3	3	NUM
ejpam-6690	83	54	)	)	PUNCT
ejpam-6690	83	55	where	where	SCONJ
ejpam-6690	83	56	sρ	sρ	X
ejpam-6690	83	57	=	=	PRON
ejpam-6690	83	58	{	{	PUNCT
ejpam-6690	83	59	x	x	PUNCT
ejpam-6690	83	60	∈	∈	PROPN
ejpam-6690	83	61	h	h	NOUN
ejpam-6690	83	62	;	;	PUNCT
ejpam-6690	83	63	ρ(x	ρ(x	NUM
ejpam-6690	83	64	)	)	PUNCT
ejpam-6690	83	65	=	=	PUNCT
ejpam-6690	83	66	eh	eh	INTJ
ejpam-6690	83	67	/	/	SYM
ejpam-6690	83	68	ρ	ρ	NOUN
ejpam-6690	83	69	}	}	PUNCT
ejpam-6690	83	70	,	,	PUNCT
ejpam-6690	83	71	is	be	AUX
ejpam-6690	83	72	an	an	DET
ejpam-6690	83	73	isomorphism	isomorphism	NOUN
ejpam-6690	83	74	of	of	ADP
ejpam-6690	83	75	complete	complete	ADJ
ejpam-6690	83	76	lattices	lattice	NOUN
ejpam-6690	83	77	[	[	X
ejpam-6690	83	78	11	11	NUM
ejpam-6690	83	79	]	]	PUNCT
ejpam-6690	83	80	.	.	PUNCT
ejpam-6690	84	1	definition	definition	NOUN
ejpam-6690	84	2	3	3	NUM
ejpam-6690	84	3	.	.	PUNCT
ejpam-6690	85	1	[	[	X
ejpam-6690	85	2	16	16	NUM
ejpam-6690	85	3	]	]	PUNCT
ejpam-6690	85	4	let	let	VERB
ejpam-6690	85	5	t	t	NOUN
ejpam-6690	85	6	be	be	AUX
ejpam-6690	85	7	a	a	DET
ejpam-6690	85	8	topological	topological	ADJ
ejpam-6690	85	9	space	space	NOUN
ejpam-6690	85	10	.	.	PUNCT
ejpam-6690	86	1	•	•	NUM
ejpam-6690	86	2	t	t	PROPN
ejpam-6690	86	3	is	be	AUX
ejpam-6690	86	4	called	call	VERB
ejpam-6690	86	5	disconnected	disconnected	ADJ
ejpam-6690	86	6	if	if	SCONJ
ejpam-6690	86	7	it	it	PRON
ejpam-6690	86	8	can	can	AUX
ejpam-6690	86	9	be	be	AUX
ejpam-6690	86	10	written	write	VERB
ejpam-6690	86	11	as	as	ADP
ejpam-6690	86	12	the	the	DET
ejpam-6690	86	13	disjoint	disjoint	PROPN
ejpam-6690	86	14	union	union	NOUN
ejpam-6690	86	15	of	of	ADP
ejpam-6690	86	16	two	two	NUM
ejpam-6690	86	17	nonempty	nonempty	ADJ
ejpam-6690	86	18	closed	close	VERB
ejpam-6690	86	19	subsets	subset	NOUN
ejpam-6690	86	20	.	.	PUNCT
ejpam-6690	87	1	•	•	NUM
ejpam-6690	87	2	t	t	PROPN
ejpam-6690	87	3	is	be	AUX
ejpam-6690	87	4	called	call	VERB
ejpam-6690	87	5	irreducible	irreducible	ADJ
ejpam-6690	87	6	if	if	SCONJ
ejpam-6690	87	7	every	every	DET
ejpam-6690	87	8	pair	pair	NOUN
ejpam-6690	87	9	of	of	ADP
ejpam-6690	87	10	nonempty	nonempty	X
ejpam-6690	87	11	open	open	ADJ
ejpam-6690	87	12	subsets	subset	NOUN
ejpam-6690	87	13	of	of	ADP
ejpam-6690	87	14	t	t	PROPN
ejpam-6690	87	15	has	have	VERB
ejpam-6690	87	16	nonempty	nonempty	ADJ
ejpam-6690	87	17	intersection	intersection	NOUN
ejpam-6690	87	18	.	.	PUNCT
ejpam-6690	88	1	•	•	NUM
ejpam-6690	88	2	t	t	PROPN
ejpam-6690	88	3	is	be	AUX
ejpam-6690	88	4	a	a	DET
ejpam-6690	88	5	t0	t0	NOUN
ejpam-6690	88	6	-	-	NOUN
ejpam-6690	88	7	space	space	NOUN
ejpam-6690	88	8	if	if	SCONJ
ejpam-6690	88	9	for	for	ADP
ejpam-6690	88	10	every	every	DET
ejpam-6690	88	11	pair	pair	NOUN
ejpam-6690	88	12	of	of	ADP
ejpam-6690	88	13	distinct	distinct	ADJ
ejpam-6690	88	14	points	point	NOUN
ejpam-6690	88	15	a	a	DET
ejpam-6690	88	16	,	,	PUNCT
ejpam-6690	88	17	b	b	PROPN
ejpam-6690	88	18	∈	∈	PROPN
ejpam-6690	88	19	t	t	NOUN
ejpam-6690	88	20	,	,	PUNCT
ejpam-6690	88	21	there	there	PRON
ejpam-6690	88	22	exists	exist	VERB
ejpam-6690	88	23	an	an	DET
ejpam-6690	88	24	open	open	ADJ
ejpam-6690	88	25	set	set	NOUN
ejpam-6690	88	26	that	that	PRON
ejpam-6690	88	27	contains	contain	VERB
ejpam-6690	88	28	one	one	NUM
ejpam-6690	88	29	of	of	ADP
ejpam-6690	88	30	them	they	PRON
ejpam-6690	88	31	but	but	CCONJ
ejpam-6690	88	32	not	not	PART
ejpam-6690	88	33	the	the	DET
ejpam-6690	88	34	other	other	ADJ
ejpam-6690	88	35	.	.	PUNCT
ejpam-6690	89	1	•	•	NUM
ejpam-6690	89	2	t	t	PROPN
ejpam-6690	89	3	is	be	AUX
ejpam-6690	89	4	a	a	DET
ejpam-6690	89	5	t1	t1	NOUN
ejpam-6690	89	6	-	-	PUNCT
ejpam-6690	89	7	space	space	NOUN
ejpam-6690	89	8	if	if	SCONJ
ejpam-6690	89	9	for	for	ADP
ejpam-6690	89	10	every	every	DET
ejpam-6690	89	11	pair	pair	NOUN
ejpam-6690	89	12	of	of	ADP
ejpam-6690	89	13	distinct	distinct	ADJ
ejpam-6690	89	14	points	point	NOUN
ejpam-6690	89	15	a	a	DET
ejpam-6690	89	16	,	,	PUNCT
ejpam-6690	89	17	b	b	PROPN
ejpam-6690	89	18	∈	∈	PROPN
ejpam-6690	89	19	t	t	NOUN
ejpam-6690	89	20	,	,	PUNCT
ejpam-6690	89	21	there	there	PRON
ejpam-6690	89	22	exist	exist	VERB
ejpam-6690	89	23	open	open	ADJ
ejpam-6690	89	24	sets	set	NOUN
ejpam-6690	89	25	u	u	NOUN
ejpam-6690	89	26	,	,	PUNCT
ejpam-6690	89	27	v	v	ADP
ejpam-6690	89	28	⊆	⊆	NUM
ejpam-6690	89	29	t	t	NOUN
ejpam-6690	89	30	such	such	ADJ
ejpam-6690	89	31	that	that	SCONJ
ejpam-6690	89	32	a	a	DET
ejpam-6690	89	33	∈	∈	PROPN
ejpam-6690	89	34	u	u	NOUN
ejpam-6690	89	35	,	,	PUNCT
ejpam-6690	89	36	b	b	PROPN
ejpam-6690	89	37	/∈	/∈	PUNCT
ejpam-6690	89	38	u	u	NOUN
ejpam-6690	89	39	,	,	PUNCT
ejpam-6690	89	40	and	and	CCONJ
ejpam-6690	89	41	b	b	X
ejpam-6690	89	42	∈	∈	PROPN
ejpam-6690	89	43	v	v	NOUN
ejpam-6690	89	44	,	,	PUNCT
ejpam-6690	89	45	a	a	DET
ejpam-6690	89	46	/∈	/∈	NOUN
ejpam-6690	89	47	v	v	NOUN
ejpam-6690	89	48	.	.	PUNCT
ejpam-6690	89	49	•	•	NUM
ejpam-6690	89	50	t	t	PROPN
ejpam-6690	89	51	is	be	AUX
ejpam-6690	89	52	a	a	DET
ejpam-6690	89	53	t2	t2	NOUN
ejpam-6690	89	54	-	-	PUNCT
ejpam-6690	89	55	space	space	NOUN
ejpam-6690	89	56	(	(	PUNCT
ejpam-6690	89	57	or	or	CCONJ
ejpam-6690	89	58	hausdorff	hausdorff	NOUN
ejpam-6690	89	59	)	)	PUNCT
ejpam-6690	89	60	if	if	SCONJ
ejpam-6690	89	61	for	for	ADP
ejpam-6690	89	62	every	every	DET
ejpam-6690	89	63	pair	pair	NOUN
ejpam-6690	89	64	of	of	ADP
ejpam-6690	89	65	distinct	distinct	ADJ
ejpam-6690	89	66	points	point	NOUN
ejpam-6690	89	67	a	a	PRON
ejpam-6690	89	68	,	,	PUNCT
ejpam-6690	89	69	b	b	PROPN
ejpam-6690	89	70	∈	∈	PROPN
ejpam-6690	89	71	t	t	NOUN
ejpam-6690	89	72	,	,	PUNCT
ejpam-6690	89	73	there	there	PRON
ejpam-6690	89	74	exist	exist	VERB
ejpam-6690	89	75	open	open	ADJ
ejpam-6690	89	76	sets	set	NOUN
ejpam-6690	89	77	u	u	NOUN
ejpam-6690	89	78	,	,	PUNCT
ejpam-6690	89	79	v	v	ADP
ejpam-6690	89	80	⊆	⊆	NUM
ejpam-6690	89	81	t	t	NOUN
ejpam-6690	89	82	such	such	ADJ
ejpam-6690	89	83	that	that	SCONJ
ejpam-6690	89	84	a	a	DET
ejpam-6690	89	85	∈	∈	PROPN
ejpam-6690	89	86	u	u	NOUN
ejpam-6690	89	87	,	,	PUNCT
ejpam-6690	89	88	b	b	PROPN
ejpam-6690	89	89	∈	∈	PROPN
ejpam-6690	89	90	v	v	NOUN
ejpam-6690	89	91	,	,	PUNCT
ejpam-6690	89	92	and	and	CCONJ
ejpam-6690	89	93	u	u	NOUN
ejpam-6690	89	94	∩	∩	NOUN
ejpam-6690	89	95	v	v	NOUN
ejpam-6690	89	96	=	=	PUNCT
ejpam-6690	89	97	∅.	∅.	PROPN
ejpam-6690	89	98	b.	b.	PROPN
ejpam-6690	89	99	afshar	afshar	PROPN
ejpam-6690	89	100	,	,	PUNCT
ejpam-6690	89	101	r.	r.	PROPN
ejpam-6690	89	102	ameri	ameri	PROPN
ejpam-6690	89	103	,	,	PUNCT
ejpam-6690	89	104	m.	m.	PROPN
ejpam-6690	89	105	al	al	PROPN
ejpam-6690	89	106	-	-	PUNCT
ejpam-6690	89	107	tahan	tahan	PROPN
ejpam-6690	89	108	/	/	SYM
ejpam-6690	89	109	eur	eur	PROPN
ejpam-6690	89	110	.	.	PUNCT
ejpam-6690	90	1	j.	j.	PROPN
ejpam-6690	90	2	pure	pure	PROPN
ejpam-6690	90	3	appl	appl	PROPN
ejpam-6690	90	4	.	.	PROPN
ejpam-6690	90	5	math	math	PROPN
ejpam-6690	90	6	,	,	PUNCT
ejpam-6690	90	7	18	18	NUM
ejpam-6690	90	8	(	(	PUNCT
ejpam-6690	90	9	4	4	NUM
ejpam-6690	90	10	)	)	PUNCT
ejpam-6690	90	11	(	(	PUNCT
ejpam-6690	90	12	2025	2025	NUM
ejpam-6690	90	13	)	)	PUNCT
ejpam-6690	90	14	,	,	PUNCT
ejpam-6690	90	15	6690	6690	NUM
ejpam-6690	90	16	5	5	NUM
ejpam-6690	90	17	of	of	ADP
ejpam-6690	90	18	19	19	NUM
ejpam-6690	90	19	theorem	theorem	NOUN
ejpam-6690	90	20	1	1	NUM
ejpam-6690	90	21	.	.	PUNCT
ejpam-6690	91	1	[	[	X
ejpam-6690	91	2	16	16	NUM
ejpam-6690	91	3	]	]	PUNCT
ejpam-6690	91	4	let	let	VERB
ejpam-6690	91	5	t	t	NOUN
ejpam-6690	91	6	be	be	AUX
ejpam-6690	91	7	a	a	DET
ejpam-6690	91	8	topological	topological	ADJ
ejpam-6690	91	9	space	space	NOUN
ejpam-6690	91	10	.	.	PUNCT
ejpam-6690	92	1	then	then	ADV
ejpam-6690	92	2	:	:	PUNCT
ejpam-6690	92	3	(	(	PUNCT
ejpam-6690	92	4	i	i	NOUN
ejpam-6690	92	5	)	)	PUNCT
ejpam-6690	92	6	if	if	SCONJ
ejpam-6690	92	7	s	s	NOUN
ejpam-6690	92	8	is	be	AUX
ejpam-6690	92	9	an	an	DET
ejpam-6690	92	10	irreducible	irreducible	ADJ
ejpam-6690	92	11	subspace	subspace	NOUN
ejpam-6690	92	12	of	of	ADP
ejpam-6690	92	13	t	t	PROPN
ejpam-6690	92	14	then	then	ADV
ejpam-6690	92	15	s̄	s̄	NOUN
ejpam-6690	92	16	is	be	AUX
ejpam-6690	92	17	irreducible	irreducible	ADJ
ejpam-6690	92	18	.	.	PUNCT
ejpam-6690	93	1	(	(	PUNCT
ejpam-6690	93	2	ii	ii	NOUN
ejpam-6690	93	3	)	)	PUNCT
ejpam-6690	93	4	every	every	DET
ejpam-6690	93	5	irreducible	irreducible	ADJ
ejpam-6690	93	6	subspace	subspace	NOUN
ejpam-6690	93	7	is	be	AUX
ejpam-6690	93	8	contained	contain	VERB
ejpam-6690	93	9	in	in	ADP
ejpam-6690	93	10	a	a	DET
ejpam-6690	93	11	maximal	maximal	ADJ
ejpam-6690	93	12	irreducible	irreducible	ADJ
ejpam-6690	93	13	subspace	subspace	NOUN
ejpam-6690	93	14	.	.	PUNCT
ejpam-6690	94	1	(	(	PUNCT
ejpam-6690	94	2	iii	iii	X
ejpam-6690	94	3	)	)	PUNCT
ejpam-6690	94	4	the	the	DET
ejpam-6690	94	5	irreducible	irreducible	ADJ
ejpam-6690	94	6	components	component	NOUN
ejpam-6690	94	7	of	of	ADP
ejpam-6690	94	8	t	t	PROPN
ejpam-6690	94	9	are	be	AUX
ejpam-6690	94	10	closed	close	VERB
ejpam-6690	94	11	and	and	CCONJ
ejpam-6690	94	12	cover	cover	VERB
ejpam-6690	94	13	t	t	NOUN
ejpam-6690	94	14	.	.	PUNCT
ejpam-6690	95	1	3	3	X
ejpam-6690	95	2	.	.	X
ejpam-6690	95	3	zariski	zariski	NOUN
ejpam-6690	95	4	topology	topology	NOUN
ejpam-6690	95	5	of	of	ADP
ejpam-6690	95	6	krasner	krasner	NOUN
ejpam-6690	95	7	hyperrings	hyperring	NOUN
ejpam-6690	95	8	in	in	ADP
ejpam-6690	95	9	this	this	DET
ejpam-6690	95	10	section	section	NOUN
ejpam-6690	95	11	,	,	PUNCT
ejpam-6690	95	12	we	we	PRON
ejpam-6690	95	13	construct	construct	VERB
ejpam-6690	95	14	and	and	CCONJ
ejpam-6690	95	15	analyze	analyze	VERB
ejpam-6690	95	16	the	the	DET
ejpam-6690	95	17	zariski	zariski	NOUN
ejpam-6690	95	18	topology	topology	NOUN
ejpam-6690	95	19	on	on	ADP
ejpam-6690	95	20	the	the	DET
ejpam-6690	95	21	spectrum	spectrum	NOUN
ejpam-6690	95	22	of	of	ADP
ejpam-6690	95	23	commutative	commutative	ADJ
ejpam-6690	95	24	krasner	krasner	NOUN
ejpam-6690	95	25	hyperring	hyperring	NOUN
ejpam-6690	95	26	.	.	PUNCT
ejpam-6690	96	1	we	we	PRON
ejpam-6690	96	2	investigate	investigate	VERB
ejpam-6690	96	3	key	key	ADJ
ejpam-6690	96	4	topological	topological	ADJ
ejpam-6690	96	5	properties	property	NOUN
ejpam-6690	96	6	such	such	ADJ
ejpam-6690	96	7	as	as	ADP
ejpam-6690	96	8	irreducibility	irreducibility	NOUN
ejpam-6690	96	9	,	,	PUNCT
ejpam-6690	96	10	connectedness	connectedness	NOUN
ejpam-6690	96	11	,	,	PUNCT
ejpam-6690	96	12	compactness	compactness	NOUN
ejpam-6690	96	13	,	,	PUNCT
ejpam-6690	96	14	and	and	CCONJ
ejpam-6690	96	15	separation	separation	NOUN
ejpam-6690	96	16	,	,	PUNCT
ejpam-6690	96	17	and	and	CCONJ
ejpam-6690	96	18	relate	relate	VERB
ejpam-6690	96	19	them	they	PRON
ejpam-6690	96	20	to	to	ADP
ejpam-6690	96	21	the	the	DET
ejpam-6690	96	22	algebraic	algebraic	ADJ
ejpam-6690	96	23	structure	structure	NOUN
ejpam-6690	96	24	of	of	ADP
ejpam-6690	96	25	the	the	DET
ejpam-6690	96	26	hyperring	hyperring	NOUN
ejpam-6690	96	27	.	.	PUNCT
ejpam-6690	97	1	everywhere	everywhere	ADV
ejpam-6690	97	2	in	in	ADP
ejpam-6690	97	3	this	this	DET
ejpam-6690	97	4	section	section	NOUN
ejpam-6690	97	5	r	r	NOUN
ejpam-6690	97	6	is	be	AUX
ejpam-6690	97	7	a	a	DET
ejpam-6690	97	8	commutative	commutative	ADJ
ejpam-6690	97	9	krasner	krasner	NOUN
ejpam-6690	97	10	hyperring	hyperre	VERB
ejpam-6690	97	11	with	with	ADP
ejpam-6690	97	12	a	a	DET
ejpam-6690	97	13	unit	unit	NOUN
ejpam-6690	97	14	element	element	NOUN
ejpam-6690	97	15	.	.	PUNCT
ejpam-6690	98	1	let	let	VERB
ejpam-6690	98	2	r	r	PRON
ejpam-6690	98	3	be	be	AUX
ejpam-6690	98	4	a	a	DET
ejpam-6690	98	5	hyperring	hyperring	NOUN
ejpam-6690	98	6	and	and	CCONJ
ejpam-6690	98	7	spec(r	spec(r	ADJ
ejpam-6690	98	8	)	)	PUNCT
ejpam-6690	98	9	be	be	VERB
ejpam-6690	98	10	the	the	DET
ejpam-6690	98	11	set	set	NOUN
ejpam-6690	98	12	of	of	ADP
ejpam-6690	98	13	all	all	DET
ejpam-6690	98	14	prime	prime	ADJ
ejpam-6690	98	15	hyperideals	hyperideal	NOUN
ejpam-6690	98	16	of	of	ADP
ejpam-6690	98	17	r	r	NOUN
ejpam-6690	98	18	and	and	CCONJ
ejpam-6690	98	19	mspec(r	mspec(r	PROPN
ejpam-6690	98	20	)	)	PUNCT
ejpam-6690	98	21	be	be	VERB
ejpam-6690	98	22	the	the	DET
ejpam-6690	98	23	set	set	NOUN
ejpam-6690	98	24	of	of	ADP
ejpam-6690	98	25	all	all	DET
ejpam-6690	98	26	maximal	maximal	ADJ
ejpam-6690	98	27	hyperideals	hyperideal	NOUN
ejpam-6690	98	28	of	of	ADP
ejpam-6690	98	29	r.	r.	NOUN
ejpam-6690	98	30	for	for	ADP
ejpam-6690	98	31	all	all	PRON
ejpam-6690	98	32	x	x	X
ejpam-6690	99	1	=	=	PUNCT
ejpam-6690	99	2	p	p	X
ejpam-6690	99	3	∈	∈	PROPN
ejpam-6690	99	4	spec(r	spec(r	PROPN
ejpam-6690	99	5	)	)	PUNCT
ejpam-6690	99	6	let	let	VERB
ejpam-6690	99	7	k(x	k(x	NOUN
ejpam-6690	99	8	)	)	PUNCT
ejpam-6690	99	9	be	be	AUX
ejpam-6690	99	10	the	the	DET
ejpam-6690	99	11	quotient	quotient	NOUN
ejpam-6690	99	12	hyperfield	hyperfield	NOUN
ejpam-6690	99	13	of	of	ADP
ejpam-6690	99	14	the	the	DET
ejpam-6690	99	15	hyperdomain	hyperdomain	NOUN
ejpam-6690	99	16	r	r	NOUN
ejpam-6690	99	17	/	/	SYM
ejpam-6690	99	18	p	p	NOUN
ejpam-6690	99	19	.	.	PUNCT
ejpam-6690	100	1	for	for	ADP
ejpam-6690	100	2	all	all	DET
ejpam-6690	100	3	f	f	PROPN
ejpam-6690	100	4	∈	∈	NOUN
ejpam-6690	100	5	r	r	NOUN
ejpam-6690	100	6	we	we	PRON
ejpam-6690	100	7	have	have	VERB
ejpam-6690	100	8	r	r	NOUN
ejpam-6690	100	9	→	→	SYM
ejpam-6690	100	10	r	r	NOUN
ejpam-6690	100	11	/	/	SYM
ejpam-6690	100	12	p	p	X
ejpam-6690	100	13	→	→	SYM
ejpam-6690	100	14	k(x	k(x	PROPN
ejpam-6690	100	15	)	)	PUNCT
ejpam-6690	101	1	where	where	SCONJ
ejpam-6690	101	2	f	f	PROPN
ejpam-6690	101	3	→	→	SYM
ejpam-6690	101	4	f	f	PROPN
ejpam-6690	101	5	+	+	CCONJ
ejpam-6690	101	6	p	p	X
ejpam-6690	101	7	→	→	PUNCT
ejpam-6690	101	8	(	(	PUNCT
ejpam-6690	101	9	f+p	f+p	PROPN
ejpam-6690	101	10	)	)	PUNCT
ejpam-6690	101	11	(	(	PUNCT
ejpam-6690	101	12	1+p	1+p	NUM
ejpam-6690	101	13	)	)	PUNCT
ejpam-6690	101	14	1+p	1+p	NUM
ejpam-6690	101	15	.	.	PUNCT
ejpam-6690	101	16	remark	remark	PROPN
ejpam-6690	101	17	1	1	NUM
ejpam-6690	101	18	.	.	PUNCT
ejpam-6690	102	1	since	since	SCONJ
ejpam-6690	102	2	every	every	DET
ejpam-6690	102	3	maximal	maximal	ADJ
ejpam-6690	102	4	hyperideal	hyperideal	NOUN
ejpam-6690	102	5	is	be	AUX
ejpam-6690	102	6	a	a	DET
ejpam-6690	102	7	prime	prime	ADJ
ejpam-6690	102	8	hyperideal	hyperideal	NOUN
ejpam-6690	102	9	,	,	PUNCT
ejpam-6690	102	10	it	it	PRON
ejpam-6690	102	11	is	be	AUX
ejpam-6690	102	12	always	always	ADV
ejpam-6690	102	13	true	true	ADJ
ejpam-6690	102	14	that	that	SCONJ
ejpam-6690	102	15	mspec(r	mspec(r	PROPN
ejpam-6690	102	16	)	)	PUNCT
ejpam-6690	102	17	⊆	⊆	NUM
ejpam-6690	102	18	spec(r	spec(r	PROPN
ejpam-6690	102	19	)	)	PUNCT
ejpam-6690	102	20	.	.	PUNCT
ejpam-6690	103	1	also	also	ADV
ejpam-6690	103	2	,	,	PUNCT
ejpam-6690	103	3	r	r	NOUN
ejpam-6690	103	4	is	be	AUX
ejpam-6690	103	5	a	a	DET
ejpam-6690	103	6	hyperfield	hyperfield	NOUN
ejpam-6690	103	7	if	if	SCONJ
ejpam-6690	103	8	and	and	CCONJ
ejpam-6690	103	9	only	only	ADV
ejpam-6690	103	10	if	if	SCONJ
ejpam-6690	103	11	0	0	NUM
ejpam-6690	103	12	∈	∈	PROPN
ejpam-6690	103	13	mspec(r	mspec(r	PROPN
ejpam-6690	103	14	)	)	PUNCT
ejpam-6690	103	15	,	,	PUNCT
ejpam-6690	103	16	i.e.	i.e.	X
ejpam-6690	103	17	r	r	NOUN
ejpam-6690	103	18	has	have	VERB
ejpam-6690	103	19	only	only	ADV
ejpam-6690	103	20	trivial	trivial	ADJ
ejpam-6690	103	21	hyperideals	hyperideal	NOUN
ejpam-6690	103	22	.	.	PUNCT
ejpam-6690	104	1	definition	definition	NOUN
ejpam-6690	104	2	4	4	NUM
ejpam-6690	104	3	.	.	PUNCT
ejpam-6690	105	1	for	for	ADP
ejpam-6690	105	2	every	every	DET
ejpam-6690	105	3	nonempty	nonempty	NOUN
ejpam-6690	105	4	subset	subset	VERB
ejpam-6690	105	5	s	s	VERB
ejpam-6690	105	6	⊆	⊆	NUM
ejpam-6690	105	7	r	r	NOUN
ejpam-6690	105	8	,	,	PUNCT
ejpam-6690	105	9	let	let	VERB
ejpam-6690	105	10	v	v	NOUN
ejpam-6690	105	11	(	(	PUNCT
ejpam-6690	105	12	s	s	X
ejpam-6690	105	13	)	)	PUNCT
ejpam-6690	105	14	=	=	SYM
ejpam-6690	106	1	{	{	PUNCT
ejpam-6690	106	2	x	x	PUNCT
ejpam-6690	106	3	∈	∈	PROPN
ejpam-6690	106	4	spec(r	spec(r	PROPN
ejpam-6690	106	5	)	)	PUNCT
ejpam-6690	106	6	;	;	PUNCT
ejpam-6690	107	1	f(x	f(x	PROPN
ejpam-6690	107	2	)	)	PUNCT
ejpam-6690	107	3	=	=	SYM
ejpam-6690	107	4	0	0	NUM
ejpam-6690	107	5	,	,	PUNCT
ejpam-6690	107	6	∀f	∀f	PROPN
ejpam-6690	107	7	∈	∈	PROPN
ejpam-6690	107	8	s	s	PART
ejpam-6690	107	9	}	}	PUNCT
ejpam-6690	107	10	=	=	PUNCT
ejpam-6690	107	11	{	{	PUNCT
ejpam-6690	107	12	p	p	NOUN
ejpam-6690	107	13	∈	∈	PROPN
ejpam-6690	107	14	spec(r	spec(r	PROPN
ejpam-6690	107	15	)	)	PUNCT
ejpam-6690	107	16	;	;	PUNCT
ejpam-6690	107	17	s	s	VERB
ejpam-6690	107	18	⊆	⊆	NUM
ejpam-6690	107	19	p	p	NOUN
ejpam-6690	107	20	}	}	PUNCT
ejpam-6690	107	21	,	,	PUNCT
ejpam-6690	107	22	vm(s	vm(s	PROPN
ejpam-6690	107	23	)	)	PUNCT
ejpam-6690	107	24	=	=	PRON
ejpam-6690	108	1	{	{	PUNCT
ejpam-6690	108	2	x	x	PUNCT
ejpam-6690	108	3	∈	∈	PROPN
ejpam-6690	108	4	mspec(r	mspec(r	PROPN
ejpam-6690	108	5	)	)	PUNCT
ejpam-6690	108	6	;	;	PUNCT
ejpam-6690	108	7	f(x	f(x	PROPN
ejpam-6690	108	8	)	)	PUNCT
ejpam-6690	108	9	=	=	SYM
ejpam-6690	109	1	0	0	NUM
ejpam-6690	109	2	,	,	PUNCT
ejpam-6690	109	3	∀f	∀f	PROPN
ejpam-6690	109	4	∈	∈	PROPN
ejpam-6690	109	5	s	s	PART
ejpam-6690	109	6	}	}	PUNCT
ejpam-6690	109	7	=	=	SYM
ejpam-6690	109	8	{	{	PUNCT
ejpam-6690	109	9	m	m	PROPN
ejpam-6690	109	10	∈	∈	PROPN
ejpam-6690	109	11	mspec(r	mspec(r	PROPN
ejpam-6690	109	12	)	)	PUNCT
ejpam-6690	109	13	;	;	PUNCT
ejpam-6690	109	14	s	s	VERB
ejpam-6690	109	15	⊆	⊆	NUM
ejpam-6690	109	16	m	m	NUM
ejpam-6690	109	17	}	}	PUNCT
ejpam-6690	109	18	.	.	PUNCT
ejpam-6690	110	1	lemma	lemma	PROPN
ejpam-6690	110	2	2	2	NUM
ejpam-6690	110	3	.	.	PUNCT
ejpam-6690	111	1	the	the	DET
ejpam-6690	111	2	following	follow	VERB
ejpam-6690	111	3	properties	property	NOUN
ejpam-6690	111	4	hold	hold	VERB
ejpam-6690	111	5	:	:	PUNCT
ejpam-6690	111	6	(	(	PUNCT
ejpam-6690	111	7	i	i	NOUN
ejpam-6690	111	8	)	)	PUNCT
ejpam-6690	111	9	if	if	SCONJ
ejpam-6690	111	10	i	i	PRON
ejpam-6690	111	11	=	=	PUNCT
ejpam-6690	111	12	i(s	i(s	NOUN
ejpam-6690	111	13	)	)	PUNCT
ejpam-6690	111	14	is	be	AUX
ejpam-6690	111	15	the	the	DET
ejpam-6690	111	16	hyperideal	hyperideal	NOUN
ejpam-6690	111	17	generated	generate	VERB
ejpam-6690	111	18	by	by	ADP
ejpam-6690	111	19	s	s	PROPN
ejpam-6690	111	20	,	,	PUNCT
ejpam-6690	111	21	then	then	ADV
ejpam-6690	111	22	v	v	X
ejpam-6690	111	23	(	(	PUNCT
ejpam-6690	111	24	s	s	NOUN
ejpam-6690	111	25	)	)	PUNCT
ejpam-6690	111	26	=	=	SYM
ejpam-6690	111	27	v	v	X
ejpam-6690	111	28	(	(	PUNCT
ejpam-6690	111	29	i	i	NOUN
ejpam-6690	111	30	)	)	PUNCT
ejpam-6690	111	31	.	.	PUNCT
ejpam-6690	112	1	(	(	PUNCT
ejpam-6690	112	2	ii	ii	NOUN
ejpam-6690	112	3	)	)	PUNCT
ejpam-6690	112	4	if	if	SCONJ
ejpam-6690	112	5	s1	s1	PROPN
ejpam-6690	112	6	⊆	⊆	NUM
ejpam-6690	112	7	s2	s2	PROPN
ejpam-6690	112	8	,	,	PUNCT
ejpam-6690	112	9	then	then	ADV
ejpam-6690	112	10	v	v	X
ejpam-6690	112	11	(	(	PUNCT
ejpam-6690	112	12	s2	s2	PROPN
ejpam-6690	112	13	)	)	PUNCT
ejpam-6690	112	14	⊆	⊆	NUM
ejpam-6690	112	15	v	v	NOUN
ejpam-6690	112	16	(	(	PUNCT
ejpam-6690	112	17	s1	s1	NOUN
ejpam-6690	112	18	)	)	PUNCT
ejpam-6690	112	19	.	.	PUNCT
ejpam-6690	113	1	(	(	PUNCT
ejpam-6690	113	2	iii	iii	X
ejpam-6690	113	3	)	)	PUNCT
ejpam-6690	113	4	v	v	NOUN
ejpam-6690	113	5	(	(	PUNCT
ejpam-6690	113	6	s	s	NOUN
ejpam-6690	113	7	)	)	PUNCT
ejpam-6690	113	8	=	=	NOUN
ejpam-6690	113	9	∅	∅	NOUN
ejpam-6690	113	10	if	if	SCONJ
ejpam-6690	113	11	and	and	CCONJ
ejpam-6690	113	12	only	only	ADV
ejpam-6690	113	13	if	if	SCONJ
ejpam-6690	113	14	1	1	NUM
ejpam-6690	113	15	∈	∈	PROPN
ejpam-6690	113	16	i(s	i(s	NOUN
ejpam-6690	113	17	)	)	PUNCT
ejpam-6690	113	18	.	.	PUNCT
ejpam-6690	114	1	(	(	PUNCT
ejpam-6690	114	2	iv	iv	X
ejpam-6690	114	3	)	)	PUNCT
ejpam-6690	114	4	v	v	NOUN
ejpam-6690	114	5	(	(	PUNCT
ejpam-6690	114	6	m	m	NOUN
ejpam-6690	114	7	)	)	PUNCT
ejpam-6690	115	1	=	=	PUNCT
ejpam-6690	115	2	m	m	NOUN
ejpam-6690	115	3	if	if	SCONJ
ejpam-6690	115	4	and	and	CCONJ
ejpam-6690	115	5	only	only	ADV
ejpam-6690	115	6	if	if	SCONJ
ejpam-6690	115	7	m	m	PROPN
ejpam-6690	115	8	∈	∈	PROPN
ejpam-6690	115	9	mspec(r	mspec(r	PROPN
ejpam-6690	115	10	)	)	PUNCT
ejpam-6690	115	11	.	.	PUNCT
ejpam-6690	116	1	proof	proof	NOUN
ejpam-6690	116	2	.	.	PUNCT
ejpam-6690	117	1	the	the	DET
ejpam-6690	117	2	proof	proof	NOUN
ejpam-6690	117	3	is	be	AUX
ejpam-6690	117	4	straightforward	straightforward	ADJ
ejpam-6690	117	5	and	and	CCONJ
ejpam-6690	117	6	similar	similar	ADJ
ejpam-6690	117	7	to	to	ADP
ejpam-6690	117	8	the	the	DET
ejpam-6690	117	9	classical	classical	ADJ
ejpam-6690	117	10	case	case	NOUN
ejpam-6690	117	11	.	.	PUNCT
ejpam-6690	118	1	theorem	theorem	NOUN
ejpam-6690	118	2	2	2	NUM
ejpam-6690	118	3	.	.	PUNCT
ejpam-6690	119	1	the	the	DET
ejpam-6690	119	2	family	family	NOUN
ejpam-6690	119	3	of	of	ADP
ejpam-6690	119	4	sets	set	NOUN
ejpam-6690	119	5	{	{	PUNCT
ejpam-6690	119	6	v	v	NOUN
ejpam-6690	119	7	(	(	PUNCT
ejpam-6690	119	8	i)}i◁r	i)}i◁r	ADV
ejpam-6690	119	9	,	,	PUNCT
ejpam-6690	119	10	satisfy	satisfy	VERB
ejpam-6690	119	11	the	the	DET
ejpam-6690	119	12	axioms	axiom	NOUN
ejpam-6690	119	13	for	for	ADP
ejpam-6690	119	14	closed	closed	ADJ
ejpam-6690	119	15	sets	set	NOUN
ejpam-6690	119	16	in	in	ADP
ejpam-6690	119	17	a	a	DET
ejpam-6690	119	18	topological	topological	ADJ
ejpam-6690	119	19	space	space	NOUN
ejpam-6690	119	20	.	.	PUNCT
ejpam-6690	120	1	proof	proof	NOUN
ejpam-6690	120	2	.	.	PUNCT
ejpam-6690	121	1	we	we	PRON
ejpam-6690	121	2	have	have	VERB
ejpam-6690	121	3	v	v	NUM
ejpam-6690	121	4	(	(	PUNCT
ejpam-6690	121	5	r	r	NOUN
ejpam-6690	121	6	)	)	PUNCT
ejpam-6690	121	7	=	=	NOUN
ejpam-6690	121	8	∅	∅	NOUN
ejpam-6690	121	9	and	and	CCONJ
ejpam-6690	121	10	v	v	NOUN
ejpam-6690	121	11	(	(	PUNCT
ejpam-6690	121	12	{	{	PUNCT
ejpam-6690	121	13	0	0	NUM
ejpam-6690	121	14	}	}	PUNCT
ejpam-6690	121	15	)	)	PUNCT
ejpam-6690	122	1	=	=	SYM
ejpam-6690	122	2	spec(r	spec(r	PROPN
ejpam-6690	122	3	)	)	PUNCT
ejpam-6690	122	4	.	.	PUNCT
ejpam-6690	123	1	let	let	VERB
ejpam-6690	123	2	{	{	PUNCT
ejpam-6690	123	3	v	v	NOUN
ejpam-6690	123	4	(	(	PUNCT
ejpam-6690	123	5	ij)}j∈j	ij)}j∈j	NOUN
ejpam-6690	123	6	be	be	VERB
ejpam-6690	123	7	a	a	DET
ejpam-6690	123	8	family	family	NOUN
ejpam-6690	123	9	of	of	ADP
ejpam-6690	123	10	closed	closed	ADJ
ejpam-6690	123	11	sets	set	NOUN
ejpam-6690	123	12	and	and	CCONJ
ejpam-6690	123	13	p	p	NOUN
ejpam-6690	123	14	∈	∈	PROPN
ejpam-6690	123	15	v	v	ADP
ejpam-6690	123	16	(	(	PUNCT
ejpam-6690	123	17	∑	∑	INTJ
ejpam-6690	123	18	j∈j	j∈j	NOUN
ejpam-6690	123	19	ij	ij	NOUN
ejpam-6690	123	20	)	)	PUNCT
ejpam-6690	123	21	.	.	PUNCT
ejpam-6690	124	1	then	then	ADV
ejpam-6690	124	2	ii	ii	PROPN
ejpam-6690	124	3	⊆	⊆	NUM
ejpam-6690	124	4	∑	∑	ADP
ejpam-6690	124	5	j∈j	j∈j	NOUN
ejpam-6690	124	6	ij	ij	NOUN
ejpam-6690	124	7	⊆	⊆	NUM
ejpam-6690	124	8	p	p	NOUN
ejpam-6690	124	9	,	,	PUNCT
ejpam-6690	124	10	for	for	ADP
ejpam-6690	124	11	every	every	DET
ejpam-6690	124	12	i	i	PROPN
ejpam-6690	124	13	∈	∈	PROPN
ejpam-6690	124	14	j	j	PROPN
ejpam-6690	124	15	.	.	PUNCT
ejpam-6690	125	1	hence	hence	ADV
ejpam-6690	125	2	v	v	X
ejpam-6690	125	3	(	(	PUNCT
ejpam-6690	125	4	∑	∑	INTJ
ejpam-6690	125	5	j∈j	j∈j	NOUN
ejpam-6690	125	6	ij	ij	NOUN
ejpam-6690	125	7	)	)	PUNCT
ejpam-6690	125	8	⊆⋂	⊆⋂	PROPN
ejpam-6690	125	9	j∈j	j∈j	NOUN
ejpam-6690	125	10	v	v	PROPN
ejpam-6690	125	11	(	(	PUNCT
ejpam-6690	125	12	ij	ij	NOUN
ejpam-6690	125	13	)	)	PUNCT
ejpam-6690	125	14	.	.	PUNCT
ejpam-6690	126	1	if	if	SCONJ
ejpam-6690	126	2	p	p	PROPN
ejpam-6690	126	3	∈	∈	PROPN
ejpam-6690	126	4	⋂	⋂	PROPN
ejpam-6690	126	5	j∈j	j∈j	NOUN
ejpam-6690	126	6	v	v	NOUN
ejpam-6690	126	7	(	(	PUNCT
ejpam-6690	126	8	ij	ij	NOUN
ejpam-6690	126	9	)	)	PUNCT
ejpam-6690	126	10	,	,	PUNCT
ejpam-6690	126	11	then	then	ADV
ejpam-6690	126	12	ii	ii	VERB
ejpam-6690	126	13	⊆	⊆	NUM
ejpam-6690	126	14	p	p	NOUN
ejpam-6690	126	15	,	,	PUNCT
ejpam-6690	126	16	for	for	ADP
ejpam-6690	126	17	every	every	DET
ejpam-6690	126	18	i	i	PROPN
ejpam-6690	126	19	∈	∈	PROPN
ejpam-6690	126	20	j	j	PROPN
ejpam-6690	126	21	.	.	PUNCT
ejpam-6690	127	1	so	so	ADV
ejpam-6690	127	2	ii	ii	PROPN
ejpam-6690	127	3	⊆	⊆	NUM
ejpam-6690	127	4	∑	∑	ADP
ejpam-6690	127	5	j∈j	j∈j	NOUN
ejpam-6690	127	6	ij	ij	NOUN
ejpam-6690	127	7	⊆	⊆	NUM
ejpam-6690	127	8	p	p	NOUN
ejpam-6690	127	9	,	,	PUNCT
ejpam-6690	127	10	and	and	CCONJ
ejpam-6690	127	11	p	p	NOUN
ejpam-6690	127	12	∈	∈	PROPN
ejpam-6690	127	13	v	v	ADP
ejpam-6690	127	14	(	(	PUNCT
ejpam-6690	127	15	∑	∑	INTJ
ejpam-6690	127	16	j∈j	j∈j	NOUN
ejpam-6690	127	17	ij	ij	NOUN
ejpam-6690	127	18	)	)	PUNCT
ejpam-6690	127	19	.	.	PUNCT
ejpam-6690	128	1	therefore	therefore	ADV
ejpam-6690	128	2	,	,	PUNCT
ejpam-6690	128	3	v	v	INTJ
ejpam-6690	128	4	(	(	PUNCT
ejpam-6690	128	5	∑	∑	INTJ
ejpam-6690	128	6	j∈j	j∈j	NOUN
ejpam-6690	128	7	ij	ij	NOUN
ejpam-6690	128	8	)	)	PUNCT
ejpam-6690	128	9	⊆	⊆	NUM
ejpam-6690	128	10	⋂	⋂	PROPN
ejpam-6690	128	11	j∈j	j∈j	NOUN
ejpam-6690	128	12	v	v	NOUN
ejpam-6690	128	13	(	(	PUNCT
ejpam-6690	128	14	ij	ij	NOUN
ejpam-6690	128	15	)	)	PUNCT
ejpam-6690	128	16	.	.	PUNCT
ejpam-6690	129	1	now	now	ADV
ejpam-6690	129	2	let	let	VERB
ejpam-6690	129	3	i	i	PRON
ejpam-6690	129	4	and	and	CCONJ
ejpam-6690	129	5	j	j	PROPN
ejpam-6690	129	6	be	be	VERB
ejpam-6690	129	7	hyperideals	hyperideal	NOUN
ejpam-6690	129	8	of	of	ADP
ejpam-6690	129	9	r	r	NOUN
ejpam-6690	129	10	and	and	CCONJ
ejpam-6690	129	11	p	p	NOUN
ejpam-6690	129	12	∈	∈	PROPN
ejpam-6690	129	13	v	v	ADP
ejpam-6690	129	14	(	(	PUNCT
ejpam-6690	129	15	ij	ij	NOUN
ejpam-6690	129	16	)	)	PUNCT
ejpam-6690	129	17	.	.	PUNCT
ejpam-6690	130	1	since	since	SCONJ
ejpam-6690	130	2	p	p	NOUN
ejpam-6690	130	3	is	be	AUX
ejpam-6690	130	4	prime	prime	ADJ
ejpam-6690	130	5	and	and	CCONJ
ejpam-6690	130	6	ij	ij	NOUN
ejpam-6690	130	7	⊆	⊆	NUM
ejpam-6690	130	8	p	p	NOUN
ejpam-6690	130	9	,	,	PUNCT
ejpam-6690	130	10	then	then	ADV
ejpam-6690	130	11	i	i	PRON
ejpam-6690	130	12	⊆	⊆	NUM
ejpam-6690	130	13	p	p	NOUN
ejpam-6690	130	14	or	or	CCONJ
ejpam-6690	130	15	j	j	PROPN
ejpam-6690	130	16	⊆	⊆	NUM
ejpam-6690	130	17	p	p	NOUN
ejpam-6690	130	18	.	.	PUNCT
ejpam-6690	131	1	so	so	ADV
ejpam-6690	131	2	p	p	ADP
ejpam-6690	131	3	∈	∈	PROPN
ejpam-6690	131	4	v	v	ADP
ejpam-6690	131	5	(	(	PUNCT
ejpam-6690	131	6	i	i	NOUN
ejpam-6690	131	7	)	)	PUNCT
ejpam-6690	131	8	∪	∪	ADP
ejpam-6690	131	9	v	v	PROPN
ejpam-6690	131	10	(	(	PUNCT
ejpam-6690	131	11	j	j	NOUN
ejpam-6690	131	12	)	)	PUNCT
ejpam-6690	131	13	.	.	PUNCT
ejpam-6690	132	1	let	let	VERB
ejpam-6690	132	2	p	p	PRON
ejpam-6690	132	3	∈	∈	PROPN
ejpam-6690	132	4	v	v	ADP
ejpam-6690	132	5	(	(	PUNCT
ejpam-6690	132	6	i	i	NOUN
ejpam-6690	132	7	)	)	PUNCT
ejpam-6690	132	8	∪	∪	ADP
ejpam-6690	132	9	v	v	PROPN
ejpam-6690	132	10	(	(	PUNCT
ejpam-6690	132	11	j	j	NOUN
ejpam-6690	132	12	)	)	PUNCT
ejpam-6690	132	13	,	,	PUNCT
ejpam-6690	132	14	then	then	ADV
ejpam-6690	132	15	i	i	PRON
ejpam-6690	132	16	⊆	⊆	NUM
ejpam-6690	132	17	p	p	NOUN
ejpam-6690	132	18	or	or	CCONJ
ejpam-6690	132	19	j	j	PROPN
ejpam-6690	132	20	⊆	⊆	NUM
ejpam-6690	132	21	p	p	NOUN
ejpam-6690	132	22	,	,	PUNCT
ejpam-6690	132	23	so	so	ADV
ejpam-6690	132	24	ij	ij	ADP
ejpam-6690	132	25	⊆	⊆	NUM
ejpam-6690	132	26	p	p	NOUN
ejpam-6690	133	1	and	and	CCONJ
ejpam-6690	133	2	p	p	NOUN
ejpam-6690	133	3	∈	∈	PROPN
ejpam-6690	133	4	v	v	ADP
ejpam-6690	133	5	(	(	PUNCT
ejpam-6690	133	6	ij	ij	NOUN
ejpam-6690	133	7	)	)	PUNCT
ejpam-6690	133	8	.	.	PUNCT
ejpam-6690	134	1	therefore	therefore	ADV
ejpam-6690	134	2	,	,	PUNCT
ejpam-6690	134	3	v	v	INTJ
ejpam-6690	134	4	(	(	PUNCT
ejpam-6690	134	5	ij	ij	NOUN
ejpam-6690	134	6	)	)	PUNCT
ejpam-6690	134	7	=	=	NOUN
ejpam-6690	134	8	v	v	X
ejpam-6690	134	9	(	(	PUNCT
ejpam-6690	134	10	i	i	NOUN
ejpam-6690	134	11	)	)	PUNCT
ejpam-6690	134	12	∪	∪	ADP
ejpam-6690	134	13	v	v	PROPN
ejpam-6690	134	14	(	(	PUNCT
ejpam-6690	134	15	j	j	NOUN
ejpam-6690	134	16	)	)	PUNCT
ejpam-6690	134	17	.	.	PUNCT
ejpam-6690	135	1	b.	b.	PROPN
ejpam-6690	135	2	afshar	afshar	PROPN
ejpam-6690	135	3	,	,	PUNCT
ejpam-6690	135	4	r.	r.	PROPN
ejpam-6690	135	5	ameri	ameri	PROPN
ejpam-6690	135	6	,	,	PUNCT
ejpam-6690	135	7	m.	m.	PROPN
ejpam-6690	135	8	al	al	PROPN
ejpam-6690	135	9	-	-	PUNCT
ejpam-6690	135	10	tahan	tahan	PROPN
ejpam-6690	135	11	/	/	SYM
ejpam-6690	135	12	eur	eur	PROPN
ejpam-6690	135	13	.	.	PUNCT
ejpam-6690	136	1	j.	j.	PROPN
ejpam-6690	136	2	pure	pure	PROPN
ejpam-6690	136	3	appl	appl	PROPN
ejpam-6690	136	4	.	.	PROPN
ejpam-6690	136	5	math	math	PROPN
ejpam-6690	136	6	,	,	PUNCT
ejpam-6690	136	7	18	18	NUM
ejpam-6690	136	8	(	(	PUNCT
ejpam-6690	136	9	4	4	NUM
ejpam-6690	136	10	)	)	PUNCT
ejpam-6690	136	11	(	(	PUNCT
ejpam-6690	136	12	2025	2025	NUM
ejpam-6690	136	13	)	)	PUNCT
ejpam-6690	136	14	,	,	PUNCT
ejpam-6690	136	15	6690	6690	NUM
ejpam-6690	136	16	6	6	NUM
ejpam-6690	136	17	of	of	ADP
ejpam-6690	136	18	19	19	NUM
ejpam-6690	136	19	the	the	DET
ejpam-6690	136	20	resulting	result	VERB
ejpam-6690	136	21	topology	topology	NOUN
ejpam-6690	136	22	on	on	ADP
ejpam-6690	136	23	spec(r	spec(r	PROPN
ejpam-6690	136	24	)	)	PUNCT
ejpam-6690	136	25	is	be	AUX
ejpam-6690	136	26	called	call	VERB
ejpam-6690	136	27	the	the	DET
ejpam-6690	136	28	zariski	zariski	ADJ
ejpam-6690	136	29	topology	topology	NOUN
ejpam-6690	136	30	of	of	ADP
ejpam-6690	136	31	r	r	NOUN
ejpam-6690	136	32	,	,	PUNCT
ejpam-6690	136	33	and	and	CCONJ
ejpam-6690	136	34	the	the	DET
ejpam-6690	136	35	family	family	NOUN
ejpam-6690	136	36	{	{	PUNCT
ejpam-6690	136	37	vm(i)}i◁r	vm(i)}i◁r	VERB
ejpam-6690	136	38	forms	form	VERB
ejpam-6690	136	39	a	a	DET
ejpam-6690	136	40	subspace	subspace	NOUN
ejpam-6690	136	41	for	for	ADP
ejpam-6690	136	42	it	it	PRON
ejpam-6690	136	43	.	.	PUNCT
ejpam-6690	137	1	remark	remark	PROPN
ejpam-6690	137	2	2	2	NUM
ejpam-6690	137	3	.	.	PUNCT
ejpam-6690	138	1	let	let	VERB
ejpam-6690	138	2	{	{	PUNCT
ejpam-6690	138	3	sj}j∈j	sj}j∈j	PART
ejpam-6690	138	4	be	be	AUX
ejpam-6690	138	5	a	a	DET
ejpam-6690	138	6	family	family	NOUN
ejpam-6690	138	7	of	of	ADP
ejpam-6690	138	8	subsets	subset	NOUN
ejpam-6690	138	9	of	of	ADP
ejpam-6690	138	10	hyperring	hyperre	VERB
ejpam-6690	138	11	r.	r.	PROPN
ejpam-6690	138	12	since	since	SCONJ
ejpam-6690	138	13	i	i	PRON
ejpam-6690	138	14	(	(	PUNCT
ejpam-6690	138	15	⋃	⋃	PROPN
ejpam-6690	138	16	j∈j	j∈j	NOUN
ejpam-6690	138	17	sj	sj	NOUN
ejpam-6690	138	18	)	)	PUNCT
ejpam-6690	138	19	=	=	SYM
ejpam-6690	138	20	∑	∑	PUNCT
ejpam-6690	138	21	j∈j	j∈j	PROPN
ejpam-6690	138	22	i(sj	i(sj	PROPN
ejpam-6690	138	23	)	)	PUNCT
ejpam-6690	138	24	,	,	PUNCT
ejpam-6690	138	25	(	(	PUNCT
ejpam-6690	138	26	4	4	X
ejpam-6690	138	27	)	)	PUNCT
ejpam-6690	138	28	then	then	ADV
ejpam-6690	138	29	by	by	ADP
ejpam-6690	138	30	lemma	lemma	PROPN
ejpam-6690	138	31	2	2	NUM
ejpam-6690	138	32	,	,	PUNCT
ejpam-6690	138	33	for	for	ADP
ejpam-6690	138	34	every	every	DET
ejpam-6690	138	35	i	i	PROPN
ejpam-6690	138	36	,	,	PUNCT
ejpam-6690	138	37	j	j	PROPN
ejpam-6690	138	38	∈	∈	PROPN
ejpam-6690	138	39	j	j	NOUN
ejpam-6690	138	40	we	we	PRON
ejpam-6690	138	41	have	have	VERB
ejpam-6690	138	42	v	v	NUM
ejpam-6690	138	43	(	(	PUNCT
ejpam-6690	138	44	⋃	⋃	NOUN
ejpam-6690	138	45	j∈j	j∈j	NOUN
ejpam-6690	138	46	sj	sj	NOUN
ejpam-6690	138	47	)	)	PUNCT
ejpam-6690	138	48	=	=	SYM
ejpam-6690	139	1	⋂	⋂	PROPN
ejpam-6690	139	2	j∈j	j∈j	NOUN
ejpam-6690	139	3	v	v	NOUN
ejpam-6690	139	4	(	(	PUNCT
ejpam-6690	139	5	sj	sj	INTJ
ejpam-6690	139	6	)	)	PUNCT
ejpam-6690	139	7	=	=	SYM
ejpam-6690	139	8	⋂	⋂	PROPN
ejpam-6690	139	9	j∈j	j∈j	NOUN
ejpam-6690	139	10	v	v	PROPN
ejpam-6690	139	11	(	(	PUNCT
ejpam-6690	139	12	i(sj	i(sj	PROPN
ejpam-6690	139	13	)	)	PUNCT
ejpam-6690	139	14	)	)	PUNCT
ejpam-6690	140	1	=	=	SYM
ejpam-6690	140	2	v	v	X
ejpam-6690	140	3	(	(	PUNCT
ejpam-6690	140	4	∑	∑	INTJ
ejpam-6690	140	5	j∈j	j∈j	NOUN
ejpam-6690	140	6	i(sj	i(sj	PROPN
ejpam-6690	140	7	)	)	PUNCT
ejpam-6690	140	8	)	)	PUNCT
ejpam-6690	140	9	,	,	PUNCT
ejpam-6690	140	10	v	v	X
ejpam-6690	140	11	(	(	PUNCT
ejpam-6690	140	12	si	si	X
ejpam-6690	140	13	∩	∩	X
ejpam-6690	140	14	sj	sj	NOUN
ejpam-6690	140	15	)	)	PUNCT
ejpam-6690	140	16	=	=	SYM
ejpam-6690	140	17	v	v	NOUN
ejpam-6690	140	18	(	(	PUNCT
ejpam-6690	140	19	si	si	NOUN
ejpam-6690	140	20	)	)	PUNCT
ejpam-6690	140	21	∪	∪	NOUN
ejpam-6690	140	22	v	v	NOUN
ejpam-6690	140	23	(	(	PUNCT
ejpam-6690	140	24	sj	sj	NOUN
ejpam-6690	140	25	)	)	PUNCT
ejpam-6690	140	26	=	=	SYM
ejpam-6690	140	27	v	v	NOUN
ejpam-6690	140	28	(	(	PUNCT
ejpam-6690	140	29	i(si	i(si	PROPN
ejpam-6690	140	30	)	)	PUNCT
ejpam-6690	140	31	)	)	PUNCT
ejpam-6690	140	32	∪	∪	ADP
ejpam-6690	140	33	v	v	NOUN
ejpam-6690	140	34	(	(	PUNCT
ejpam-6690	140	35	i(sj	i(sj	NOUN
ejpam-6690	140	36	)	)	PUNCT
ejpam-6690	140	37	)	)	PUNCT
ejpam-6690	141	1	=	=	SYM
ejpam-6690	141	2	v	v	X
ejpam-6690	141	3	(	(	PUNCT
ejpam-6690	141	4	i(si	i(si	PROPN
ejpam-6690	141	5	)	)	PUNCT
ejpam-6690	141	6	∩	∩	NOUN
ejpam-6690	141	7	i(sj	i(sj	NOUN
ejpam-6690	141	8	)	)	PUNCT
ejpam-6690	141	9	)	)	PUNCT
ejpam-6690	141	10	.	.	PUNCT
ejpam-6690	142	1	also	also	ADV
ejpam-6690	142	2	,	,	PUNCT
ejpam-6690	142	3	if	if	SCONJ
ejpam-6690	142	4	i	i	PRON
ejpam-6690	142	5	and	and	CCONJ
ejpam-6690	142	6	j	j	PROPN
ejpam-6690	142	7	are	be	AUX
ejpam-6690	142	8	hyperideals	hyperideal	NOUN
ejpam-6690	142	9	of	of	ADP
ejpam-6690	142	10	r	r	NOUN
ejpam-6690	142	11	then	then	ADV
ejpam-6690	142	12	by	by	ADP
ejpam-6690	142	13	lemma	lemma	PROPN
ejpam-6690	142	14	2	2	NUM
ejpam-6690	142	15	,	,	PUNCT
ejpam-6690	142	16	v	v	PROPN
ejpam-6690	142	17	(	(	PUNCT
ejpam-6690	142	18	i	i	NOUN
ejpam-6690	142	19	)	)	PUNCT
ejpam-6690	142	20	∪	∪	ADP
ejpam-6690	142	21	v	v	PROPN
ejpam-6690	142	22	(	(	PUNCT
ejpam-6690	142	23	j	j	NOUN
ejpam-6690	142	24	)	)	PUNCT
ejpam-6690	142	25	⊆	⊆	NUM
ejpam-6690	142	26	v	v	NOUN
ejpam-6690	142	27	(	(	PUNCT
ejpam-6690	142	28	i	i	PROPN
ejpam-6690	142	29	∩	∩	PROPN
ejpam-6690	142	30	j	j	PROPN
ejpam-6690	142	31	)	)	PUNCT
ejpam-6690	142	32	.	.	PUNCT
ejpam-6690	143	1	let	let	VERB
ejpam-6690	143	2	p	p	PRON
ejpam-6690	143	3	∈	∈	PROPN
ejpam-6690	143	4	v	v	NOUN
ejpam-6690	143	5	(	(	PUNCT
ejpam-6690	143	6	i	i	PROPN
ejpam-6690	143	7	∩	∩	PROPN
ejpam-6690	143	8	j	j	PROPN
ejpam-6690	143	9	)	)	PUNCT
ejpam-6690	144	1	but	but	CCONJ
ejpam-6690	144	2	p	p	NOUN
ejpam-6690	144	3	/∈	/∈	NOUN
ejpam-6690	145	1	v	v	INTJ
ejpam-6690	145	2	(	(	PUNCT
ejpam-6690	145	3	i	i	NOUN
ejpam-6690	145	4	)	)	PUNCT
ejpam-6690	145	5	∪	∪	ADP
ejpam-6690	145	6	v	v	PROPN
ejpam-6690	145	7	(	(	PUNCT
ejpam-6690	145	8	j	j	NOUN
ejpam-6690	145	9	)	)	PUNCT
ejpam-6690	145	10	,	,	PUNCT
ejpam-6690	145	11	then	then	ADV
ejpam-6690	145	12	there	there	PRON
ejpam-6690	145	13	are	be	VERB
ejpam-6690	145	14	x	x	X
ejpam-6690	145	15	∈	∈	PROPN
ejpam-6690	146	1	i	i	PRON
ejpam-6690	146	2	−	−	PROPN
ejpam-6690	147	1	p	p	PROPN
ejpam-6690	147	2	and	and	CCONJ
ejpam-6690	147	3	y	y	PROPN
ejpam-6690	147	4	∈	∈	PROPN
ejpam-6690	148	1	j	j	PROPN
ejpam-6690	149	1	−	−	PROPN
ejpam-6690	149	2	p	p	PRON
ejpam-6690	149	3	such	such	ADJ
ejpam-6690	149	4	that	that	SCONJ
ejpam-6690	149	5	xy	xy	PROPN
ejpam-6690	149	6	∈	∈	PROPN
ejpam-6690	150	1	i	i	PRON
ejpam-6690	150	2	∩	∩	PROPN
ejpam-6690	150	3	j	j	PROPN
ejpam-6690	150	4	.	.	PUNCT
ejpam-6690	151	1	so	so	ADV
ejpam-6690	151	2	xy	xy	PROPN
ejpam-6690	151	3	∈	∈	PROPN
ejpam-6690	151	4	p	p	X
ejpam-6690	151	5	,	,	PUNCT
ejpam-6690	151	6	and	and	CCONJ
ejpam-6690	151	7	it	it	PRON
ejpam-6690	151	8	is	be	AUX
ejpam-6690	151	9	a	a	DET
ejpam-6690	151	10	contradiction	contradiction	NOUN
ejpam-6690	151	11	.	.	PUNCT
ejpam-6690	152	1	therefore	therefore	ADV
ejpam-6690	152	2	,	,	PUNCT
ejpam-6690	152	3	v	v	X
ejpam-6690	152	4	(	(	PUNCT
ejpam-6690	152	5	ij	ij	NOUN
ejpam-6690	152	6	)	)	PUNCT
ejpam-6690	152	7	=	=	NOUN
ejpam-6690	152	8	v	v	X
ejpam-6690	152	9	(	(	PUNCT
ejpam-6690	152	10	i	i	PROPN
ejpam-6690	152	11	∩	∩	PROPN
ejpam-6690	152	12	j	j	PROPN
ejpam-6690	152	13	)	)	PUNCT
ejpam-6690	152	14	.	.	PUNCT
ejpam-6690	153	1	proposition	proposition	NOUN
ejpam-6690	153	2	2	2	NUM
ejpam-6690	153	3	.	.	PUNCT
ejpam-6690	154	1	if	if	SCONJ
ejpam-6690	154	2	i	i	PRON
ejpam-6690	154	3	and	and	CCONJ
ejpam-6690	154	4	j	j	PROPN
ejpam-6690	154	5	are	be	AUX
ejpam-6690	154	6	hyperideals	hyperideal	NOUN
ejpam-6690	154	7	of	of	ADP
ejpam-6690	154	8	r	r	NOUN
ejpam-6690	154	9	,	,	PUNCT
ejpam-6690	154	10	then	then	ADV
ejpam-6690	154	11	:	:	PUNCT
ejpam-6690	154	12	(	(	PUNCT
ejpam-6690	154	13	i	i	NOUN
ejpam-6690	154	14	)	)	PUNCT
ejpam-6690	154	15	v	v	PROPN
ejpam-6690	154	16	(	(	PUNCT
ejpam-6690	154	17	i	i	NOUN
ejpam-6690	154	18	)	)	PUNCT
ejpam-6690	155	1	=	=	SYM
ejpam-6690	155	2	v	v	X
ejpam-6690	155	3	(	(	PUNCT
ejpam-6690	155	4	√	√	PROPN
ejpam-6690	155	5	i	i	PRON
ejpam-6690	155	6	)	)	PUNCT
ejpam-6690	155	7	.	.	PUNCT
ejpam-6690	156	1	(	(	PUNCT
ejpam-6690	156	2	ii	ii	NOUN
ejpam-6690	156	3	)	)	PUNCT
ejpam-6690	156	4	v	v	NOUN
ejpam-6690	156	5	(	(	PUNCT
ejpam-6690	156	6	i	i	NOUN
ejpam-6690	156	7	)	)	PUNCT
ejpam-6690	156	8	⊆	⊆	NUM
ejpam-6690	156	9	v	v	X
ejpam-6690	156	10	(	(	PUNCT
ejpam-6690	156	11	j	j	PROPN
ejpam-6690	156	12	)	)	PUNCT
ejpam-6690	156	13	⇔	⇔	NOUN
ejpam-6690	156	14	√	√	NUM
ejpam-6690	156	15	j	j	PROPN
ejpam-6690	156	16	⊆	⊆	NUM
ejpam-6690	156	17	√	√	PROPN
ejpam-6690	156	18	i.	i.	NOUN
ejpam-6690	156	19	proof	proof	NOUN
ejpam-6690	156	20	.	.	PUNCT
ejpam-6690	157	1	√	√	INTJ
ejpam-6690	158	1	i	i	PRON
ejpam-6690	158	2	=	=	SYM
ejpam-6690	158	3	⋂	⋂	PROPN
ejpam-6690	158	4	p∈v	p∈v	NOUN
ejpam-6690	158	5	(	(	PUNCT
ejpam-6690	158	6	i	i	NOUN
ejpam-6690	158	7	)	)	PUNCT
ejpam-6690	158	8	p	p	NOUN
ejpam-6690	159	1	=	=	PUNCT
ejpam-6690	159	2	{	{	PUNCT
ejpam-6690	159	3	x	x	SYM
ejpam-6690	159	4	∈	∈	PROPN
ejpam-6690	159	5	r	r	NOUN
ejpam-6690	159	6	;	;	PUNCT
ejpam-6690	159	7	xn	xn	PROPN
ejpam-6690	159	8	∈	∈	PROPN
ejpam-6690	160	1	i	i	PRON
ejpam-6690	160	2	,	,	PUNCT
ejpam-6690	160	3	for	for	ADP
ejpam-6690	160	4	some	some	DET
ejpam-6690	160	5	n	n	PRON
ejpam-6690	160	6	∈	∈	PROPN
ejpam-6690	160	7	n	n	CCONJ
ejpam-6690	160	8	}	}	PUNCT
ejpam-6690	160	9	,	,	PUNCT
ejpam-6690	160	10	and	and	CCONJ
ejpam-6690	160	11	the	the	DET
ejpam-6690	160	12	proof	proof	NOUN
ejpam-6690	160	13	is	be	AUX
ejpam-6690	160	14	similar	similar	ADJ
ejpam-6690	160	15	to	to	ADP
ejpam-6690	160	16	that	that	PRON
ejpam-6690	160	17	of	of	ADP
ejpam-6690	160	18	classic	classic	ADJ
ejpam-6690	160	19	rings	ring	NOUN
ejpam-6690	160	20	.	.	PUNCT
ejpam-6690	161	1	theorem	theorem	NOUN
ejpam-6690	161	2	3	3	NUM
ejpam-6690	161	3	.	.	PUNCT
ejpam-6690	162	1	if	if	SCONJ
ejpam-6690	162	2	f	f	PROPN
ejpam-6690	162	3	:	:	PUNCT
ejpam-6690	162	4	r	r	X
ejpam-6690	162	5	→	→	SYM
ejpam-6690	162	6	s	s	PART
ejpam-6690	162	7	is	be	AUX
ejpam-6690	162	8	a	a	DET
ejpam-6690	162	9	good	good	ADJ
ejpam-6690	162	10	homomorphism	homomorphism	NOUN
ejpam-6690	162	11	of	of	ADP
ejpam-6690	162	12	hyperrings	hyperring	NOUN
ejpam-6690	162	13	,	,	PUNCT
ejpam-6690	162	14	then	then	ADV
ejpam-6690	162	15	f̄	f̄	NOUN
ejpam-6690	162	16	:	:	PUNCT
ejpam-6690	162	17	spec(s	spec(s	PROPN
ejpam-6690	162	18	)	)	PUNCT
ejpam-6690	162	19	→	→	SYM
ejpam-6690	162	20	spec(r	spec(r	PROPN
ejpam-6690	162	21	)	)	PUNCT
ejpam-6690	162	22	,	,	PUNCT
ejpam-6690	162	23	defined	define	VERB
ejpam-6690	162	24	by	by	ADP
ejpam-6690	162	25	f̄(p	f̄(p	NOUN
ejpam-6690	162	26	)	)	PUNCT
ejpam-6690	163	1	=	=	SYM
ejpam-6690	163	2	f−1(p	f−1(p	PROPN
ejpam-6690	163	3	)	)	PUNCT
ejpam-6690	163	4	,	,	PUNCT
ejpam-6690	163	5	is	be	AUX
ejpam-6690	163	6	continuous	continuous	ADJ
ejpam-6690	163	7	.	.	PUNCT
ejpam-6690	164	1	proof	proof	NOUN
ejpam-6690	164	2	.	.	PUNCT
ejpam-6690	165	1	let	let	VERB
ejpam-6690	165	2	v	v	X
ejpam-6690	165	3	(	(	PUNCT
ejpam-6690	165	4	i	i	NOUN
ejpam-6690	165	5	)	)	PUNCT
ejpam-6690	165	6	be	be	VERB
ejpam-6690	165	7	a	a	DET
ejpam-6690	165	8	closed	closed	ADJ
ejpam-6690	165	9	set	set	NOUN
ejpam-6690	165	10	of	of	ADP
ejpam-6690	165	11	spec(r	spec(r	PROPN
ejpam-6690	165	12	)	)	PUNCT
ejpam-6690	165	13	.	.	PUNCT
ejpam-6690	166	1	then	then	ADV
ejpam-6690	166	2	:	:	PUNCT
ejpam-6690	166	3	f̄	f̄	PROPN
ejpam-6690	166	4	−1	−1	NOUN
ejpam-6690	166	5	(	(	PUNCT
ejpam-6690	166	6	v	v	NOUN
ejpam-6690	166	7	(	(	PUNCT
ejpam-6690	166	8	i	i	NOUN
ejpam-6690	166	9	)	)	PUNCT
ejpam-6690	166	10	)	)	PUNCT
ejpam-6690	167	1	=	=	PRON
ejpam-6690	167	2	{	{	PUNCT
ejpam-6690	167	3	p	p	X
ejpam-6690	167	4	∈	∈	PROPN
ejpam-6690	167	5	spec(s	spec(s	NOUN
ejpam-6690	167	6	)	)	PUNCT
ejpam-6690	167	7	;	;	PUNCT
ejpam-6690	167	8	f̄(p	f̄(p	NOUN
ejpam-6690	167	9	)	)	PUNCT
ejpam-6690	167	10	∈	∈	PROPN
ejpam-6690	167	11	v	v	NOUN
ejpam-6690	167	12	(	(	PUNCT
ejpam-6690	167	13	i	i	NOUN
ejpam-6690	167	14	)	)	PUNCT
ejpam-6690	167	15	}	}	PUNCT
ejpam-6690	167	16	=	=	PUNCT
ejpam-6690	167	17	{	{	PUNCT
ejpam-6690	167	18	p	p	NOUN
ejpam-6690	167	19	∈	∈	PROPN
ejpam-6690	167	20	spec(s	spec(s	PROPN
ejpam-6690	167	21	)	)	PUNCT
ejpam-6690	167	22	;	;	PUNCT
ejpam-6690	167	23	i	i	PRON
ejpam-6690	167	24	⊆	⊆	NUM
ejpam-6690	167	25	f−1(p	f−1(p	NOUN
ejpam-6690	167	26	)	)	PUNCT
ejpam-6690	167	27	}	}	PUNCT
ejpam-6690	167	28	=	=	PUNCT
ejpam-6690	167	29	{	{	PUNCT
ejpam-6690	167	30	p	p	NOUN
ejpam-6690	167	31	∈	∈	PROPN
ejpam-6690	167	32	spec(s	spec(s	PROPN
ejpam-6690	167	33	)	)	PUNCT
ejpam-6690	167	34	;	;	PUNCT
ejpam-6690	167	35	f(i	f(i	X
ejpam-6690	167	36	)	)	PUNCT
ejpam-6690	167	37	⊆	⊆	NUM
ejpam-6690	167	38	p	p	X
ejpam-6690	167	39	}	}	PUNCT
ejpam-6690	167	40	=	=	SYM
ejpam-6690	167	41	v	v	NOUN
ejpam-6690	167	42	(	(	PUNCT
ejpam-6690	167	43	f(i	f(i	PROPN
ejpam-6690	167	44	)	)	PUNCT
ejpam-6690	167	45	)	)	PUNCT
ejpam-6690	167	46	.	.	PUNCT
ejpam-6690	168	1	if	if	SCONJ
ejpam-6690	168	2	r	r	NOUN
ejpam-6690	168	3	is	be	AUX
ejpam-6690	168	4	a	a	DET
ejpam-6690	168	5	hyperring	hyperring	NOUN
ejpam-6690	168	6	and	and	CCONJ
ejpam-6690	168	7	i	i	PRON
ejpam-6690	168	8	is	be	AUX
ejpam-6690	168	9	a	a	DET
ejpam-6690	168	10	hyperideal	hyperideal	NOUN
ejpam-6690	168	11	of	of	ADP
ejpam-6690	168	12	r	r	NOUN
ejpam-6690	168	13	,	,	PUNCT
ejpam-6690	168	14	then	then	ADV
ejpam-6690	168	15	π	π	X
ejpam-6690	168	16	:	:	PUNCT
ejpam-6690	168	17	r	r	NOUN
ejpam-6690	168	18	→	→	SYM
ejpam-6690	168	19	r	r	NOUN
ejpam-6690	168	20	/	/	SYM
ejpam-6690	168	21	i	i	PRON
ejpam-6690	168	22	is	be	AUX
ejpam-6690	168	23	projection	projection	NOUN
ejpam-6690	168	24	map	map	NOUN
ejpam-6690	168	25	and	and	CCONJ
ejpam-6690	168	26	π̄	π̄	VERB
ejpam-6690	168	27	is	be	AUX
ejpam-6690	168	28	a	a	DET
ejpam-6690	168	29	continuous	continuous	ADJ
ejpam-6690	168	30	map	map	NOUN
ejpam-6690	168	31	from	from	ADP
ejpam-6690	168	32	spec(r	spec(r	PROPN
ejpam-6690	168	33	/	/	SYM
ejpam-6690	168	34	i	i	NOUN
ejpam-6690	168	35	)	)	PUNCT
ejpam-6690	168	36	to	to	ADP
ejpam-6690	168	37	spec(r	spec(r	PROPN
ejpam-6690	168	38	)	)	PUNCT
ejpam-6690	168	39	.	.	PUNCT
ejpam-6690	169	1	theorem	theorem	ADJ
ejpam-6690	169	2	4	4	NUM
ejpam-6690	169	3	.	.	PUNCT
ejpam-6690	170	1	let	let	VERB
ejpam-6690	170	2	r	r	PRON
ejpam-6690	170	3	be	be	AUX
ejpam-6690	170	4	a	a	DET
ejpam-6690	170	5	hyperring	hyperring	NOUN
ejpam-6690	171	1	and	and	CCONJ
ejpam-6690	171	2	i	i	PRON
ejpam-6690	171	3	be	be	VERB
ejpam-6690	171	4	a	a	DET
ejpam-6690	171	5	hyperideal	hyperideal	NOUN
ejpam-6690	171	6	of	of	ADP
ejpam-6690	171	7	r.	r.	PROPN
ejpam-6690	171	8	then	then	ADV
ejpam-6690	171	9	:	:	PUNCT
ejpam-6690	171	10	(	(	PUNCT
ejpam-6690	171	11	i	i	NOUN
ejpam-6690	171	12	)	)	PUNCT
ejpam-6690	171	13	π̄(spec(r	π̄(spec(r	PROPN
ejpam-6690	171	14	/	/	SYM
ejpam-6690	171	15	i	i	NOUN
ejpam-6690	171	16	)	)	PUNCT
ejpam-6690	171	17	)	)	PUNCT
ejpam-6690	172	1	=	=	SYM
ejpam-6690	172	2	v	v	X
ejpam-6690	172	3	(	(	PUNCT
ejpam-6690	172	4	i	i	NOUN
ejpam-6690	172	5	)	)	PUNCT
ejpam-6690	172	6	.	.	PUNCT
ejpam-6690	173	1	(	(	PUNCT
ejpam-6690	173	2	ii	ii	NOUN
ejpam-6690	173	3	)	)	PUNCT
ejpam-6690	173	4	π̄	π̄	VERB
ejpam-6690	173	5	is	be	AUX
ejpam-6690	173	6	injective	injective	ADJ
ejpam-6690	173	7	.	.	PUNCT
ejpam-6690	174	1	(	(	PUNCT
ejpam-6690	174	2	iii	iii	X
ejpam-6690	174	3	)	)	PUNCT
ejpam-6690	174	4	spec(r	spec(r	PROPN
ejpam-6690	174	5	/	/	SYM
ejpam-6690	174	6	i	i	NOUN
ejpam-6690	174	7	)	)	PUNCT
ejpam-6690	174	8	is	be	AUX
ejpam-6690	174	9	homeomorphic	homeomorphic	ADJ
ejpam-6690	174	10	to	to	ADP
ejpam-6690	174	11	v	v	PROPN
ejpam-6690	174	12	(	(	PUNCT
ejpam-6690	174	13	i	i	NOUN
ejpam-6690	174	14	)	)	PUNCT
ejpam-6690	174	15	by	by	ADP
ejpam-6690	174	16	subspace	subspace	NOUN
ejpam-6690	174	17	topology	topology	NOUN
ejpam-6690	174	18	.	.	PUNCT
ejpam-6690	175	1	proof	proof	NOUN
ejpam-6690	175	2	.	.	PUNCT
ejpam-6690	176	1	we	we	PRON
ejpam-6690	176	2	have	have	VERB
ejpam-6690	176	3	π̄(spec(r	π̄(spec(r	NOUN
ejpam-6690	176	4	/	/	SYM
ejpam-6690	176	5	i	i	NOUN
ejpam-6690	176	6	)	)	PUNCT
ejpam-6690	176	7	)	)	PUNCT
ejpam-6690	177	1	=	=	PRON
ejpam-6690	177	2	{	{	PUNCT
ejpam-6690	177	3	p	p	NOUN
ejpam-6690	177	4	∈	∈	PROPN
ejpam-6690	177	5	spec(r	spec(r	PROPN
ejpam-6690	177	6	)	)	PUNCT
ejpam-6690	177	7	;	;	PUNCT
ejpam-6690	178	1	p	p	X
ejpam-6690	178	2	/	/	SYM
ejpam-6690	178	3	i	i	NOUN
ejpam-6690	178	4	∈	∈	PROPN
ejpam-6690	179	1	spec(r	spec(r	PROPN
ejpam-6690	179	2	/	/	SYM
ejpam-6690	179	3	i	i	NOUN
ejpam-6690	179	4	)	)	PUNCT
ejpam-6690	179	5	}	}	PUNCT
ejpam-6690	180	1	=	=	PUNCT
ejpam-6690	180	2	{	{	PUNCT
ejpam-6690	180	3	p	p	NOUN
ejpam-6690	180	4	∈	∈	PROPN
ejpam-6690	180	5	spec(r	spec(r	PROPN
ejpam-6690	180	6	)	)	PUNCT
ejpam-6690	180	7	;	;	PUNCT
ejpam-6690	180	8	i	i	PRON
ejpam-6690	180	9	⊆	⊆	NUM
ejpam-6690	180	10	p	p	X
ejpam-6690	180	11	}	}	PUNCT
ejpam-6690	180	12	=	=	SYM
ejpam-6690	180	13	v	v	NOUN
ejpam-6690	180	14	(	(	PUNCT
ejpam-6690	180	15	i	i	NOUN
ejpam-6690	180	16	)	)	PUNCT
ejpam-6690	180	17	.	.	PUNCT
ejpam-6690	181	1	b.	b.	PROPN
ejpam-6690	181	2	afshar	afshar	PROPN
ejpam-6690	181	3	,	,	PUNCT
ejpam-6690	181	4	r.	r.	PROPN
ejpam-6690	181	5	ameri	ameri	PROPN
ejpam-6690	181	6	,	,	PUNCT
ejpam-6690	181	7	m.	m.	PROPN
ejpam-6690	181	8	al	al	PROPN
ejpam-6690	181	9	-	-	PUNCT
ejpam-6690	181	10	tahan	tahan	PROPN
ejpam-6690	181	11	/	/	SYM
ejpam-6690	181	12	eur	eur	PROPN
ejpam-6690	181	13	.	.	PUNCT
ejpam-6690	182	1	j.	j.	PROPN
ejpam-6690	182	2	pure	pure	PROPN
ejpam-6690	182	3	appl	appl	PROPN
ejpam-6690	182	4	.	.	PROPN
ejpam-6690	182	5	math	math	PROPN
ejpam-6690	182	6	,	,	PUNCT
ejpam-6690	182	7	18	18	NUM
ejpam-6690	182	8	(	(	PUNCT
ejpam-6690	182	9	4	4	NUM
ejpam-6690	182	10	)	)	PUNCT
ejpam-6690	182	11	(	(	PUNCT
ejpam-6690	182	12	2025	2025	NUM
ejpam-6690	182	13	)	)	PUNCT
ejpam-6690	182	14	,	,	PUNCT
ejpam-6690	182	15	6690	6690	NUM
ejpam-6690	182	16	7	7	NUM
ejpam-6690	182	17	of	of	ADP
ejpam-6690	182	18	19	19	NUM
ejpam-6690	182	19	since	since	SCONJ
ejpam-6690	182	20	there	there	PRON
ejpam-6690	182	21	is	be	VERB
ejpam-6690	182	22	a	a	DET
ejpam-6690	182	23	bijection	bijection	NOUN
ejpam-6690	182	24	between	between	ADP
ejpam-6690	182	25	the	the	DET
ejpam-6690	182	26	hyperideals	hyperideal	NOUN
ejpam-6690	182	27	of	of	ADP
ejpam-6690	182	28	r	r	NOUN
ejpam-6690	182	29	/	/	SYM
ejpam-6690	182	30	i	i	PRON
ejpam-6690	182	31	and	and	CCONJ
ejpam-6690	182	32	the	the	DET
ejpam-6690	182	33	hyperideals	hyperideal	NOUN
ejpam-6690	182	34	of	of	ADP
ejpam-6690	182	35	r	r	NOUN
ejpam-6690	182	36	containing	contain	VERB
ejpam-6690	182	37	i	i	PRON
ejpam-6690	182	38	,	,	PUNCT
ejpam-6690	182	39	then	then	ADV
ejpam-6690	182	40	π̄	π̄	VERB
ejpam-6690	182	41	is	be	AUX
ejpam-6690	182	42	a	a	DET
ejpam-6690	182	43	bijection	bijection	NOUN
ejpam-6690	182	44	between	between	ADP
ejpam-6690	182	45	spec(r	spec(r	PROPN
ejpam-6690	182	46	/	/	SYM
ejpam-6690	182	47	i	i	PROPN
ejpam-6690	182	48	)	)	PUNCT
ejpam-6690	182	49	and	and	CCONJ
ejpam-6690	182	50	v	v	X
ejpam-6690	182	51	(	(	PUNCT
ejpam-6690	182	52	i	i	NOUN
ejpam-6690	182	53	)	)	PUNCT
ejpam-6690	182	54	.	.	PUNCT
ejpam-6690	183	1	by	by	ADP
ejpam-6690	183	2	theorem	theorem	NOUN
ejpam-6690	183	3	3	3	NUM
ejpam-6690	183	4	π̄	π̄	NOUN
ejpam-6690	183	5	is	be	AUX
ejpam-6690	183	6	continuous	continuous	ADJ
ejpam-6690	183	7	and	and	CCONJ
ejpam-6690	183	8	also	also	ADV
ejpam-6690	183	9	π̄	π̄	VERB
ejpam-6690	183	10	−1	−1	NOUN
ejpam-6690	183	11	:	:	PUNCT
ejpam-6690	183	12	v	v	X
ejpam-6690	183	13	(	(	PUNCT
ejpam-6690	183	14	i	i	NOUN
ejpam-6690	183	15	)	)	PUNCT
ejpam-6690	183	16	→	→	PUNCT
ejpam-6690	183	17	spec(r	spec(r	PROPN
ejpam-6690	183	18	/	/	SYM
ejpam-6690	183	19	i	i	NOUN
ejpam-6690	183	20	)	)	PUNCT
ejpam-6690	183	21	defined	define	VERB
ejpam-6690	183	22	by	by	ADP
ejpam-6690	183	23	π̄	π̄	VERB
ejpam-6690	183	24	−1	−1	NOUN
ejpam-6690	183	25	(	(	PUNCT
ejpam-6690	183	26	p	p	NOUN
ejpam-6690	183	27	)	)	PUNCT
ejpam-6690	184	1	=	=	SYM
ejpam-6690	185	1	p	p	X
ejpam-6690	185	2	/	/	SYM
ejpam-6690	185	3	i	i	PRON
ejpam-6690	185	4	is	be	AUX
ejpam-6690	185	5	continuous	continuous	ADJ
ejpam-6690	185	6	,	,	PUNCT
ejpam-6690	185	7	because	because	SCONJ
ejpam-6690	185	8	every	every	DET
ejpam-6690	185	9	prime	prime	ADJ
ejpam-6690	185	10	hyperideal	hyperideal	NOUN
ejpam-6690	185	11	of	of	ADP
ejpam-6690	185	12	r	r	NOUN
ejpam-6690	185	13	/	/	SYM
ejpam-6690	185	14	i	i	PRON
ejpam-6690	185	15	is	be	AUX
ejpam-6690	185	16	of	of	ADP
ejpam-6690	185	17	the	the	DET
ejpam-6690	185	18	form	form	NOUN
ejpam-6690	185	19	p	p	X
ejpam-6690	185	20	/	/	SYM
ejpam-6690	185	21	i	i	NOUN
ejpam-6690	185	22	,	,	PUNCT
ejpam-6690	185	23	where	where	SCONJ
ejpam-6690	185	24	p	p	PROPN
ejpam-6690	185	25	∈	∈	PROPN
ejpam-6690	185	26	v	v	ADP
ejpam-6690	185	27	(	(	PUNCT
ejpam-6690	185	28	i	i	NOUN
ejpam-6690	185	29	)	)	PUNCT
ejpam-6690	185	30	.	.	PUNCT
ejpam-6690	186	1	corollary	corollary	ADJ
ejpam-6690	186	2	1	1	NUM
ejpam-6690	186	3	.	.	PUNCT
ejpam-6690	187	1	if	if	SCONJ
ejpam-6690	187	2	r	r	NOUN
ejpam-6690	187	3	is	be	AUX
ejpam-6690	187	4	a	a	DET
ejpam-6690	187	5	hyperring	hyperring	NOUN
ejpam-6690	187	6	,	,	PUNCT
ejpam-6690	187	7	then	then	ADV
ejpam-6690	187	8	spec(r	spec(r	PROPN
ejpam-6690	187	9	)	)	PUNCT
ejpam-6690	187	10	∼=	∼=	PROPN
ejpam-6690	187	11	spec(r	spec(r	PROPN
ejpam-6690	187	12	/	/	SYM
ejpam-6690	187	13	nil(r	nil(r	NOUN
ejpam-6690	187	14	)	)	PUNCT
ejpam-6690	187	15	)	)	PUNCT
ejpam-6690	187	16	.	.	PUNCT
ejpam-6690	188	1	proof	proof	NOUN
ejpam-6690	188	2	.	.	PUNCT
ejpam-6690	189	1	by	by	ADP
ejpam-6690	189	2	theorem	theorem	NOUN
ejpam-6690	189	3	4	4	NUM
ejpam-6690	189	4	,	,	PUNCT
ejpam-6690	189	5	spec(r	spec(r	PROPN
ejpam-6690	189	6	/	/	SYM
ejpam-6690	189	7	nil(r	nil(r	NOUN
ejpam-6690	189	8	)	)	PUNCT
ejpam-6690	189	9	)	)	PUNCT
ejpam-6690	190	1	=	=	SYM
ejpam-6690	190	2	v	v	X
ejpam-6690	190	3	(	(	PUNCT
ejpam-6690	190	4	nil(r	nil(r	NOUN
ejpam-6690	190	5	)	)	PUNCT
ejpam-6690	190	6	)	)	PUNCT
ejpam-6690	190	7	and	and	CCONJ
ejpam-6690	190	8	since	since	SCONJ
ejpam-6690	190	9	nil(r	nil(r	PROPN
ejpam-6690	190	10	)	)	PUNCT
ejpam-6690	190	11	⊆	⊆	NUM
ejpam-6690	190	12	p	p	NOUN
ejpam-6690	190	13	,	,	PUNCT
ejpam-6690	190	14	for	for	ADP
ejpam-6690	190	15	every	every	DET
ejpam-6690	190	16	p	p	PROPN
ejpam-6690	190	17	∈	∈	PROPN
ejpam-6690	190	18	spec(r	spec(r	PROPN
ejpam-6690	190	19	)	)	PUNCT
ejpam-6690	190	20	we	we	PRON
ejpam-6690	190	21	have	have	VERB
ejpam-6690	190	22	v	v	NUM
ejpam-6690	190	23	(	(	PUNCT
ejpam-6690	190	24	nil(r	nil(r	NOUN
ejpam-6690	190	25	)	)	PUNCT
ejpam-6690	190	26	)	)	PUNCT
ejpam-6690	191	1	=	=	SYM
ejpam-6690	191	2	spec(r	spec(r	PROPN
ejpam-6690	191	3	)	)	PUNCT
ejpam-6690	191	4	.	.	PUNCT
ejpam-6690	192	1	theorem	theorem	NOUN
ejpam-6690	192	2	5	5	NUM
ejpam-6690	192	3	.	.	PUNCT
ejpam-6690	193	1	let	let	VERB
ejpam-6690	193	2	r1	r1	PROPN
ejpam-6690	193	3	and	and	CCONJ
ejpam-6690	193	4	r2	r2	PROPN
ejpam-6690	193	5	be	be	VERB
ejpam-6690	193	6	hyperrings	hyperring	NOUN
ejpam-6690	193	7	,	,	PUNCT
ejpam-6690	193	8	then	then	ADV
ejpam-6690	193	9	:	:	PUNCT
ejpam-6690	193	10	spec(r1	spec(r1	NOUN
ejpam-6690	193	11	×r2	×r2	PROPN
ejpam-6690	193	12	)	)	PUNCT
ejpam-6690	193	13	=	=	SYM
ejpam-6690	193	14	spec(r1	spec(r1	NOUN
ejpam-6690	193	15	)	)	PUNCT
ejpam-6690	193	16	∪̇	∪̇	X
ejpam-6690	193	17	spec(r2	spec(r2	NOUN
ejpam-6690	193	18	)	)	PUNCT
ejpam-6690	193	19	.	.	PUNCT
ejpam-6690	194	1	(	(	PUNCT
ejpam-6690	194	2	5	5	NUM
ejpam-6690	194	3	)	)	PUNCT
ejpam-6690	194	4	where	where	SCONJ
ejpam-6690	194	5	∪̇	∪̇	PROPN
ejpam-6690	194	6	means	mean	VERB
ejpam-6690	194	7	disjoint	disjoint	PROPN
ejpam-6690	194	8	union	union	NOUN
ejpam-6690	194	9	.	.	PUNCT
ejpam-6690	195	1	proof	proof	NOUN
ejpam-6690	195	2	.	.	PUNCT
ejpam-6690	196	1	let	let	VERB
ejpam-6690	196	2	p1	p1	PROPN
ejpam-6690	196	3	×	×	NOUN
ejpam-6690	196	4	p2	p2	PROPN
ejpam-6690	196	5	be	be	VERB
ejpam-6690	196	6	a	a	DET
ejpam-6690	196	7	prime	prime	ADJ
ejpam-6690	196	8	hyperideal	hyperideal	NOUN
ejpam-6690	196	9	of	of	ADP
ejpam-6690	196	10	r1	r1	PROPN
ejpam-6690	196	11	×	×	PROPN
ejpam-6690	196	12	r2	r2	PROPN
ejpam-6690	196	13	.	.	PUNCT
ejpam-6690	197	1	then	then	ADV
ejpam-6690	197	2	r1	r1	PROPN
ejpam-6690	197	3	/	/	SYM
ejpam-6690	197	4	p1	p1	PROPN
ejpam-6690	197	5	×	×	NOUN
ejpam-6690	197	6	r2	r2	NOUN
ejpam-6690	197	7	/	/	SYM
ejpam-6690	197	8	p2	p2	NOUN
ejpam-6690	197	9	,	,	PUNCT
ejpam-6690	197	10	is	be	AUX
ejpam-6690	197	11	hyperdomain	hyperdomain	ADJ
ejpam-6690	197	12	.	.	PUNCT
ejpam-6690	198	1	so	so	ADV
ejpam-6690	198	2	p1	p1	PROPN
ejpam-6690	198	3	=	=	SYM
ejpam-6690	198	4	r1	r1	PROPN
ejpam-6690	198	5	and	and	CCONJ
ejpam-6690	198	6	p2	p2	PROPN
ejpam-6690	198	7	∈	∈	PROPN
ejpam-6690	198	8	spec(r2	spec(r2	NOUN
ejpam-6690	198	9	)	)	PUNCT
ejpam-6690	198	10	or	or	CCONJ
ejpam-6690	198	11	p2	p2	PROPN
ejpam-6690	198	12	=	=	SYM
ejpam-6690	198	13	r2	r2	PROPN
ejpam-6690	198	14	and	and	CCONJ
ejpam-6690	198	15	p1	p1	PROPN
ejpam-6690	198	16	∈	∈	PROPN
ejpam-6690	198	17	spec(r1	spec(r1	NOUN
ejpam-6690	198	18	)	)	PUNCT
ejpam-6690	198	19	.	.	PUNCT
ejpam-6690	199	1	definition	definition	NOUN
ejpam-6690	199	2	5	5	NUM
ejpam-6690	199	3	.	.	PUNCT
ejpam-6690	200	1	[	[	X
ejpam-6690	200	2	17	17	NUM
ejpam-6690	200	3	]	]	PUNCT
ejpam-6690	200	4	the	the	DET
ejpam-6690	200	5	hyperideals	hyperideal	NOUN
ejpam-6690	201	1	i	i	PRON
ejpam-6690	201	2	and	and	CCONJ
ejpam-6690	201	3	j	j	PROPN
ejpam-6690	201	4	of	of	ADP
ejpam-6690	201	5	a	a	DET
ejpam-6690	201	6	hyperring	hyperring	NOUN
ejpam-6690	201	7	r	r	NOUN
ejpam-6690	201	8	are	be	AUX
ejpam-6690	201	9	said	say	VERB
ejpam-6690	201	10	to	to	PART
ejpam-6690	201	11	be	be	AUX
ejpam-6690	201	12	comaximal	comaximal	ADJ
ejpam-6690	201	13	if	if	SCONJ
ejpam-6690	201	14	i	i	PRON
ejpam-6690	201	15	+	+	NUM
ejpam-6690	201	16	j	j	PROPN
ejpam-6690	201	17	=	=	SYM
ejpam-6690	201	18	r.	r.	PROPN
ejpam-6690	201	19	theorem	theorem	VERB
ejpam-6690	201	20	6	6	NUM
ejpam-6690	201	21	(	(	PUNCT
ejpam-6690	201	22	chinese	chinese	ADJ
ejpam-6690	201	23	remainder	remainder	NOUN
ejpam-6690	201	24	theorem	theorem	NOUN
ejpam-6690	201	25	)	)	PUNCT
ejpam-6690	201	26	.	.	PUNCT
ejpam-6690	202	1	if	if	SCONJ
ejpam-6690	202	2	i1	i1	PROPN
ejpam-6690	202	3	,	,	PUNCT
ejpam-6690	202	4	i2	i2	PROPN
ejpam-6690	202	5	,	,	PUNCT
ejpam-6690	202	6	...	...	PUNCT
ejpam-6690	202	7	,	,	PUNCT
ejpam-6690	202	8	ik	ik	PROPN
ejpam-6690	202	9	are	be	AUX
ejpam-6690	202	10	hyperrings	hyperring	NOUN
ejpam-6690	202	11	of	of	ADP
ejpam-6690	202	12	r	r	NOUN
ejpam-6690	202	13	,	,	PUNCT
ejpam-6690	202	14	then	then	ADV
ejpam-6690	202	15	the	the	DET
ejpam-6690	202	16	map	map	NOUN
ejpam-6690	202	17	r	r	NOUN
ejpam-6690	202	18	→	→	SYM
ejpam-6690	202	19	r	r	NOUN
ejpam-6690	202	20	/	/	SYM
ejpam-6690	202	21	i1	i1	NOUN
ejpam-6690	202	22	×	×	PROPN
ejpam-6690	202	23	r	r	PROPN
ejpam-6690	202	24	/	/	SYM
ejpam-6690	202	25	i2	i2	PROPN
ejpam-6690	202	26	×	×	NOUN
ejpam-6690	202	27	...	...	PUNCT
ejpam-6690	202	28	×	×	NOUN
ejpam-6690	202	29	r	r	NOUN
ejpam-6690	202	30	/	/	SYM
ejpam-6690	202	31	ik	ik	PROPN
ejpam-6690	202	32	defined	define	VERB
ejpam-6690	202	33	by	by	ADP
ejpam-6690	202	34	r	r	NOUN
ejpam-6690	202	35	7−→	7−→	PROPN
ejpam-6690	202	36	(	(	PUNCT
ejpam-6690	202	37	r	r	NOUN
ejpam-6690	202	38	+	+	NUM
ejpam-6690	202	39	i1	i1	PROPN
ejpam-6690	202	40	,	,	PUNCT
ejpam-6690	202	41	r	r	NOUN
ejpam-6690	202	42	+	+	NUM
ejpam-6690	202	43	i2	i2	PROPN
ejpam-6690	202	44	,	,	PUNCT
ejpam-6690	202	45	...	...	PUNCT
ejpam-6690	202	46	,	,	PUNCT
ejpam-6690	202	47	r	r	NOUN
ejpam-6690	202	48	+	+	X
ejpam-6690	202	49	ik	ik	NOUN
ejpam-6690	202	50	)	)	PUNCT
ejpam-6690	202	51	is	be	AUX
ejpam-6690	202	52	a	a	DET
ejpam-6690	202	53	good	good	ADJ
ejpam-6690	202	54	homomorphism	homomorphism	NOUN
ejpam-6690	202	55	with	with	ADP
ejpam-6690	202	56	kernel	kernel	PROPN
ejpam-6690	202	57	i1	i1	PROPN
ejpam-6690	202	58	∩	∩	PROPN
ejpam-6690	202	59	i2	i2	PROPN
ejpam-6690	202	60	∩	∩	NOUN
ejpam-6690	202	61	...	...	PUNCT
ejpam-6690	202	62	∩	∩	PROPN
ejpam-6690	202	63	ik	ik	PROPN
ejpam-6690	202	64	.	.	PROPN
ejpam-6690	203	1	if	if	SCONJ
ejpam-6690	203	2	for	for	ADP
ejpam-6690	203	3	each	each	DET
ejpam-6690	203	4	i	i	PRON
ejpam-6690	203	5	,	,	PUNCT
ejpam-6690	203	6	j	j	PROPN
ejpam-6690	203	7	∈	∈	PROPN
ejpam-6690	203	8	{	{	PUNCT
ejpam-6690	203	9	1	1	NUM
ejpam-6690	203	10	,	,	PUNCT
ejpam-6690	203	11	2	2	NUM
ejpam-6690	203	12	,	,	PUNCT
ejpam-6690	203	13	...	...	PUNCT
ejpam-6690	203	14	,	,	PUNCT
ejpam-6690	203	15	k	k	NOUN
ejpam-6690	203	16	}	}	PUNCT
ejpam-6690	203	17	with	with	ADP
ejpam-6690	203	18	i	i	PRON
ejpam-6690	203	19	̸=	̸=	PROPN
ejpam-6690	203	20	j	j	PROPN
ejpam-6690	203	21	the	the	DET
ejpam-6690	203	22	hyperideals	hyperideal	NOUN
ejpam-6690	203	23	ii	ii	PROPN
ejpam-6690	203	24	and	and	CCONJ
ejpam-6690	203	25	ij	ij	NOUN
ejpam-6690	203	26	are	be	AUX
ejpam-6690	203	27	comaximal	comaximal	ADJ
ejpam-6690	203	28	,	,	PUNCT
ejpam-6690	203	29	then	then	ADV
ejpam-6690	203	30	this	this	DET
ejpam-6690	203	31	map	map	NOUN
ejpam-6690	203	32	is	be	AUX
ejpam-6690	203	33	surjective	surjective	ADJ
ejpam-6690	203	34	and	and	CCONJ
ejpam-6690	203	35	i1	i1	PROPN
ejpam-6690	203	36	∩	∩	PROPN
ejpam-6690	203	37	i2	i2	PROPN
ejpam-6690	203	38	∩	∩	NOUN
ejpam-6690	203	39	...	...	PUNCT
ejpam-6690	203	40	∩	∩	ADJ
ejpam-6690	203	41	ik	ik	PROPN
ejpam-6690	203	42	=	=	SYM
ejpam-6690	203	43	i1i2	i1i2	PROPN
ejpam-6690	203	44	...	...	PUNCT
ejpam-6690	203	45	ik	ik	X
ejpam-6690	203	46	,	,	PUNCT
ejpam-6690	203	47	so	so	ADV
ejpam-6690	203	48	:	:	PUNCT
ejpam-6690	203	49	r/(i1i2	r/(i1i2	NOUN
ejpam-6690	203	50	...	...	SYM
ejpam-6690	203	51	ik	ik	PROPN
ejpam-6690	203	52	)	)	PUNCT
ejpam-6690	203	53	=	=	SYM
ejpam-6690	203	54	r/(i1	r/(i1	NOUN
ejpam-6690	203	55	∩	∩	ADJ
ejpam-6690	203	56	i2	i2	PROPN
ejpam-6690	203	57	∩	∩	NOUN
ejpam-6690	203	58	...	...	PUNCT
ejpam-6690	203	59	∩	∩	NOUN
ejpam-6690	203	60	ik	ik	ADJ
ejpam-6690	203	61	)	)	PUNCT
ejpam-6690	203	62	=	=	SYM
ejpam-6690	203	63	r	r	X
ejpam-6690	203	64	/	/	SYM
ejpam-6690	203	65	i1	i1	PROPN
ejpam-6690	203	66	×r	×r	PROPN
ejpam-6690	203	67	/	/	SYM
ejpam-6690	203	68	i2	i2	PROPN
ejpam-6690	203	69	×	×	NOUN
ejpam-6690	203	70	...	...	PUNCT
ejpam-6690	203	71	×r	×r	PROPN
ejpam-6690	203	72	/	/	SYM
ejpam-6690	203	73	ik	ik	PROPN
ejpam-6690	203	74	.	.	PROPN
ejpam-6690	204	1	(	(	PUNCT
ejpam-6690	204	2	6	6	NUM
ejpam-6690	204	3	)	)	PUNCT
ejpam-6690	204	4	proof	proof	NOUN
ejpam-6690	204	5	.	.	PUNCT
ejpam-6690	205	1	let	let	VERB
ejpam-6690	205	2	i	i	PRON
ejpam-6690	205	3	=	=	PROPN
ejpam-6690	205	4	i1	i1	PROPN
ejpam-6690	205	5	and	and	CCONJ
ejpam-6690	205	6	j	j	PROPN
ejpam-6690	205	7	=	=	PROPN
ejpam-6690	205	8	i2	i2	PROPN
ejpam-6690	205	9	.	.	PUNCT
ejpam-6690	205	10	consider	consider	VERB
ejpam-6690	205	11	the	the	DET
ejpam-6690	205	12	map	map	NOUN
ejpam-6690	205	13	φ	φ	X
ejpam-6690	205	14	:	:	PUNCT
ejpam-6690	205	15	r	r	NOUN
ejpam-6690	205	16	→	→	SYM
ejpam-6690	205	17	r	r	NOUN
ejpam-6690	205	18	/	/	SYM
ejpam-6690	205	19	i	i	PRON
ejpam-6690	205	20	×	×	NOUN
ejpam-6690	205	21	r	r	NOUN
ejpam-6690	205	22	/	/	SYM
ejpam-6690	205	23	j	j	PROPN
ejpam-6690	205	24	,	,	PUNCT
ejpam-6690	205	25	defined	define	VERB
ejpam-6690	205	26	by	by	ADP
ejpam-6690	205	27	φ(r	φ(r	ADJ
ejpam-6690	205	28	)	)	PUNCT
ejpam-6690	205	29	=	=	SYM
ejpam-6690	206	1	(	(	PUNCT
ejpam-6690	206	2	r	r	NOUN
ejpam-6690	206	3	+	+	CCONJ
ejpam-6690	206	4	i	i	PROPN
ejpam-6690	206	5	,	,	PUNCT
ejpam-6690	206	6	r	r	NOUN
ejpam-6690	206	7	+	+	PROPN
ejpam-6690	206	8	j	j	NOUN
ejpam-6690	206	9	)	)	PUNCT
ejpam-6690	206	10	.	.	PUNCT
ejpam-6690	207	1	then	then	ADV
ejpam-6690	207	2	φ	φ	PROPN
ejpam-6690	207	3	is	be	AUX
ejpam-6690	207	4	good	good	ADJ
ejpam-6690	207	5	homomorphism	homomorphism	NOUN
ejpam-6690	207	6	of	of	ADP
ejpam-6690	207	7	hyperrings	hyperring	NOUN
ejpam-6690	207	8	and	and	CCONJ
ejpam-6690	207	9	kerφ	kerφ	PROPN
ejpam-6690	207	10	=	=	PROPN
ejpam-6690	208	1	i	i	PROPN
ejpam-6690	208	2	∩	∩	PROPN
ejpam-6690	208	3	j	j	PROPN
ejpam-6690	208	4	.	.	PUNCT
ejpam-6690	209	1	since	since	SCONJ
ejpam-6690	209	2	r	r	NOUN
ejpam-6690	209	3	is	be	AUX
ejpam-6690	209	4	a	a	DET
ejpam-6690	209	5	krasner	krasner	NOUN
ejpam-6690	209	6	hyperring	hyperre	VERB
ejpam-6690	209	7	with	with	ADP
ejpam-6690	209	8	a	a	DET
ejpam-6690	209	9	unit	unit	NOUN
ejpam-6690	209	10	element	element	NOUN
ejpam-6690	209	11	and	and	CCONJ
ejpam-6690	209	12	i+j	i+j	NUM
ejpam-6690	209	13	=	=	SYM
ejpam-6690	209	14	r	r	NOUN
ejpam-6690	209	15	,	,	PUNCT
ejpam-6690	209	16	then	then	ADV
ejpam-6690	209	17	there	there	PRON
ejpam-6690	209	18	are	be	VERB
ejpam-6690	209	19	elements	element	NOUN
ejpam-6690	209	20	x	x	SYM
ejpam-6690	209	21	∈	∈	PROPN
ejpam-6690	210	1	i	i	PRON
ejpam-6690	210	2	and	and	CCONJ
ejpam-6690	210	3	y	y	PROPN
ejpam-6690	210	4	∈	∈	PROPN
ejpam-6690	210	5	j	j	PROPN
ejpam-6690	210	6	such	such	ADJ
ejpam-6690	210	7	that	that	SCONJ
ejpam-6690	210	8	1	1	NUM
ejpam-6690	210	9	∈	∈	NOUN
ejpam-6690	210	10	x	x	PUNCT
ejpam-6690	210	11	+	+	CCONJ
ejpam-6690	210	12	y.	y.	NOUN
ejpam-6690	210	13	hence	hence	ADV
ejpam-6690	210	14	x	x	SYM
ejpam-6690	210	15	∈	∈	NOUN
ejpam-6690	210	16	1	1	NUM
ejpam-6690	210	17	−	−	NOUN
ejpam-6690	210	18	y	y	PROPN
ejpam-6690	210	19	⊆	⊆	NUM
ejpam-6690	210	20	1	1	NUM
ejpam-6690	210	21	+	+	CCONJ
ejpam-6690	210	22	j	j	PROPN
ejpam-6690	210	23	and	and	CCONJ
ejpam-6690	210	24	y	y	PROPN
ejpam-6690	210	25	∈	∈	PROPN
ejpam-6690	210	26	1	1	NUM
ejpam-6690	210	27	−	−	NOUN
ejpam-6690	210	28	x	x	SYM
ejpam-6690	210	29	⊆	⊆	SYM
ejpam-6690	210	30	1	1	NUM
ejpam-6690	210	31	+	+	NUM
ejpam-6690	210	32	i.	i.	NOUN
ejpam-6690	210	33	also	also	ADV
ejpam-6690	210	34	1	1	NUM
ejpam-6690	210	35	−	−	PROPN
ejpam-6690	210	36	y	y	PROPN
ejpam-6690	210	37	+	+	CCONJ
ejpam-6690	210	38	j	j	NOUN
ejpam-6690	210	39	=	=	SYM
ejpam-6690	210	40	1	1	NUM
ejpam-6690	210	41	+	+	CCONJ
ejpam-6690	210	42	j	j	PROPN
ejpam-6690	210	43	and	and	CCONJ
ejpam-6690	210	44	1	1	NUM
ejpam-6690	210	45	−	−	NOUN
ejpam-6690	210	46	x	x	PUNCT
ejpam-6690	211	1	+	+	CCONJ
ejpam-6690	211	2	i	i	NOUN
ejpam-6690	211	3	=	=	NOUN
ejpam-6690	211	4	1	1	NUM
ejpam-6690	211	5	+	+	NUM
ejpam-6690	211	6	i.	i.	NOUN
ejpam-6690	211	7	so	so	SCONJ
ejpam-6690	211	8	φ(x	φ(x	NOUN
ejpam-6690	211	9	)	)	PUNCT
ejpam-6690	211	10	=	=	SYM
ejpam-6690	212	1	(	(	PUNCT
ejpam-6690	212	2	i	i	INTJ
ejpam-6690	212	3	,	,	PUNCT
ejpam-6690	212	4	1	1	NUM
ejpam-6690	212	5	+	+	CCONJ
ejpam-6690	212	6	j	j	NOUN
ejpam-6690	212	7	)	)	PUNCT
ejpam-6690	212	8	=	=	PRON
ejpam-6690	212	9	(	(	PUNCT
ejpam-6690	212	10	0r	0r	NUM
ejpam-6690	212	11	/	/	SYM
ejpam-6690	212	12	i	i	PROPN
ejpam-6690	212	13	,	,	PUNCT
ejpam-6690	212	14	1r	1r	NUM
ejpam-6690	212	15	/	/	SYM
ejpam-6690	212	16	j	j	NOUN
ejpam-6690	212	17	)	)	PUNCT
ejpam-6690	212	18	and	and	CCONJ
ejpam-6690	212	19	φ(y	φ(y	NOUN
ejpam-6690	212	20	)	)	PUNCT
ejpam-6690	212	21	=	=	PUNCT
ejpam-6690	212	22	(	(	PUNCT
ejpam-6690	212	23	1	1	NUM
ejpam-6690	212	24	+	+	CCONJ
ejpam-6690	212	25	i	i	PROPN
ejpam-6690	212	26	,	,	PUNCT
ejpam-6690	212	27	j	j	PROPN
ejpam-6690	212	28	)	)	PUNCT
ejpam-6690	212	29	=	=	PUNCT
ejpam-6690	213	1	(	(	PUNCT
ejpam-6690	213	2	1r	1r	NUM
ejpam-6690	213	3	/	/	SYM
ejpam-6690	213	4	i	i	PRON
ejpam-6690	213	5	,	,	PUNCT
ejpam-6690	213	6	0r	0r	PROPN
ejpam-6690	213	7	/	/	SYM
ejpam-6690	213	8	j	j	PROPN
ejpam-6690	213	9	)	)	PUNCT
ejpam-6690	213	10	.	.	PUNCT
ejpam-6690	214	1	let	let	VERB
ejpam-6690	214	2	(	(	PUNCT
ejpam-6690	214	3	r1	r1	NOUN
ejpam-6690	214	4	+	+	CCONJ
ejpam-6690	215	1	i	i	PROPN
ejpam-6690	215	2	,	,	PUNCT
ejpam-6690	215	3	r2	r2	PROPN
ejpam-6690	215	4	+	+	CCONJ
ejpam-6690	215	5	j	j	PROPN
ejpam-6690	215	6	)	)	PUNCT
ejpam-6690	215	7	∈	∈	PROPN
ejpam-6690	215	8	r	r	X
ejpam-6690	215	9	/	/	SYM
ejpam-6690	215	10	i	i	PROPN
ejpam-6690	215	11	×r	×r	PROPN
ejpam-6690	215	12	/	/	SYM
ejpam-6690	215	13	j	j	PROPN
ejpam-6690	215	14	,	,	PUNCT
ejpam-6690	215	15	then	then	ADV
ejpam-6690	215	16	(	(	PUNCT
ejpam-6690	215	17	r1	r1	PROPN
ejpam-6690	215	18	+	+	CCONJ
ejpam-6690	215	19	i	i	PROPN
ejpam-6690	215	20	,	,	PUNCT
ejpam-6690	215	21	r2	r2	PROPN
ejpam-6690	215	22	+	+	CCONJ
ejpam-6690	215	23	j	j	PROPN
ejpam-6690	215	24	)	)	PUNCT
ejpam-6690	215	25	=	=	PUNCT
ejpam-6690	216	1	(	(	PUNCT
ejpam-6690	216	2	r1	r1	PROPN
ejpam-6690	216	3	+	+	CCONJ
ejpam-6690	216	4	i	i	PROPN
ejpam-6690	216	5	,	,	PUNCT
ejpam-6690	216	6	0	0	NUM
ejpam-6690	216	7	)	)	PUNCT
ejpam-6690	217	1	+	+	CCONJ
ejpam-6690	217	2	(	(	PUNCT
ejpam-6690	217	3	0	0	NUM
ejpam-6690	217	4	,	,	PUNCT
ejpam-6690	217	5	r2	r2	PROPN
ejpam-6690	217	6	+	+	CCONJ
ejpam-6690	217	7	j	j	PROPN
ejpam-6690	217	8	)	)	PUNCT
ejpam-6690	217	9	=	=	PUNCT
ejpam-6690	217	10	(	(	PUNCT
ejpam-6690	217	11	r1	r1	PROPN
ejpam-6690	217	12	+	+	CCONJ
ejpam-6690	217	13	i	i	PROPN
ejpam-6690	217	14	,	,	PUNCT
ejpam-6690	217	15	r1	r1	PROPN
ejpam-6690	217	16	+	+	CCONJ
ejpam-6690	217	17	j)(1	j)(1	PROPN
ejpam-6690	217	18	,	,	PUNCT
ejpam-6690	217	19	0	0	NUM
ejpam-6690	217	20	)	)	PUNCT
ejpam-6690	217	21	+	+	CCONJ
ejpam-6690	217	22	(	(	PUNCT
ejpam-6690	217	23	r2	r2	PROPN
ejpam-6690	217	24	+	+	CCONJ
ejpam-6690	217	25	i	i	PROPN
ejpam-6690	217	26	,	,	PUNCT
ejpam-6690	217	27	r2	r2	PROPN
ejpam-6690	217	28	+	+	CCONJ
ejpam-6690	217	29	j)(0	j)(0	ADJ
ejpam-6690	217	30	,	,	PUNCT
ejpam-6690	217	31	1	1	NUM
ejpam-6690	217	32	)	)	PUNCT
ejpam-6690	217	33	=	=	SYM
ejpam-6690	217	34	φ(r1)φ(y	φ(r1)φ(y	NOUN
ejpam-6690	217	35	)	)	PUNCT
ejpam-6690	217	36	+	+	CCONJ
ejpam-6690	217	37	φ(r2)φ(x	φ(r2)φ(x	NOUN
ejpam-6690	217	38	)	)	PUNCT
ejpam-6690	217	39	=	=	PUNCT
ejpam-6690	217	40	φ(r1y	φ(r1y	NOUN
ejpam-6690	217	41	+	+	CCONJ
ejpam-6690	217	42	r2x	r2x	NOUN
ejpam-6690	217	43	)	)	PUNCT
ejpam-6690	217	44	.	.	PUNCT
ejpam-6690	218	1	we	we	PRON
ejpam-6690	218	2	know	know	VERB
ejpam-6690	218	3	that	that	PRON
ejpam-6690	218	4	ij	ij	NOUN
ejpam-6690	218	5	⊆	⊆	NUM
ejpam-6690	218	6	i	i	PROPN
ejpam-6690	218	7	∩	∩	PROPN
ejpam-6690	218	8	j	j	PROPN
ejpam-6690	218	9	.	.	PUNCT
ejpam-6690	219	1	let	let	VERB
ejpam-6690	219	2	z	z	NOUN
ejpam-6690	219	3	∈	∈	PROPN
ejpam-6690	220	1	i	i	PRON
ejpam-6690	220	2	∩	∩	PROPN
ejpam-6690	220	3	j	j	PROPN
ejpam-6690	220	4	,	,	PUNCT
ejpam-6690	220	5	then	then	ADV
ejpam-6690	220	6	z	z	NOUN
ejpam-6690	220	7	=	=	PUNCT
ejpam-6690	220	8	z.1	z.1	PROPN
ejpam-6690	220	9	∈	∈	PROPN
ejpam-6690	220	10	z(x+	z(x+	X
ejpam-6690	220	11	y	y	NOUN
ejpam-6690	220	12	)	)	PUNCT
ejpam-6690	221	1	=	=	SYM
ejpam-6690	221	2	zx+	zx+	PROPN
ejpam-6690	221	3	zy	zy	PROPN
ejpam-6690	221	4	∈	∈	PROPN
ejpam-6690	221	5	ij	ij	INTJ
ejpam-6690	221	6	.	.	PUNCT
ejpam-6690	222	1	hence	hence	ADV
ejpam-6690	222	2	i	i	PRON
ejpam-6690	222	3	∩	∩	PROPN
ejpam-6690	222	4	j	j	PROPN
ejpam-6690	222	5	⊆	⊆	NUM
ejpam-6690	222	6	ij	ij	NOUN
ejpam-6690	222	7	.	.	PUNCT
ejpam-6690	223	1	the	the	DET
ejpam-6690	223	2	general	general	ADJ
ejpam-6690	223	3	case	case	NOUN
ejpam-6690	223	4	follows	follow	VERB
ejpam-6690	223	5	by	by	ADP
ejpam-6690	223	6	induction	induction	NOUN
ejpam-6690	223	7	from	from	ADP
ejpam-6690	223	8	the	the	DET
ejpam-6690	223	9	case	case	NOUN
ejpam-6690	223	10	of	of	ADP
ejpam-6690	223	11	two	two	NUM
ejpam-6690	223	12	hyperideals	hyperideal	NOUN
ejpam-6690	223	13	using	use	VERB
ejpam-6690	223	14	i	i	PROPN
ejpam-6690	223	15	=	=	PROPN
ejpam-6690	223	16	i1	i1	PROPN
ejpam-6690	223	17	and	and	CCONJ
ejpam-6690	223	18	j	j	PROPN
ejpam-6690	223	19	=	=	PUNCT
ejpam-6690	223	20	i2i3	i2i3	PROPN
ejpam-6690	223	21	...	...	NOUN
ejpam-6690	223	22	ik	ik	PROPN
ejpam-6690	223	23	.	.	PROPN
ejpam-6690	223	24	b.	b.	PROPN
ejpam-6690	223	25	afshar	afshar	PROPN
ejpam-6690	223	26	,	,	PUNCT
ejpam-6690	223	27	r.	r.	PROPN
ejpam-6690	223	28	ameri	ameri	PROPN
ejpam-6690	223	29	,	,	PUNCT
ejpam-6690	223	30	m.	m.	PROPN
ejpam-6690	223	31	al	al	PROPN
ejpam-6690	223	32	-	-	PUNCT
ejpam-6690	223	33	tahan	tahan	PROPN
ejpam-6690	223	34	/	/	SYM
ejpam-6690	223	35	eur	eur	PROPN
ejpam-6690	223	36	.	.	PUNCT
ejpam-6690	224	1	j.	j.	PROPN
ejpam-6690	224	2	pure	pure	PROPN
ejpam-6690	224	3	appl	appl	PROPN
ejpam-6690	224	4	.	.	PROPN
ejpam-6690	224	5	math	math	PROPN
ejpam-6690	224	6	,	,	PUNCT
ejpam-6690	224	7	18	18	NUM
ejpam-6690	224	8	(	(	PUNCT
ejpam-6690	224	9	4	4	NUM
ejpam-6690	224	10	)	)	PUNCT
ejpam-6690	224	11	(	(	PUNCT
ejpam-6690	224	12	2025	2025	NUM
ejpam-6690	224	13	)	)	PUNCT
ejpam-6690	224	14	,	,	PUNCT
ejpam-6690	224	15	6690	6690	NUM
ejpam-6690	224	16	8	8	NUM
ejpam-6690	224	17	of	of	ADP
ejpam-6690	224	18	19	19	NUM
ejpam-6690	224	19	corollary	corollary	ADJ
ejpam-6690	224	20	2	2	NUM
ejpam-6690	224	21	.	.	PUNCT
ejpam-6690	225	1	the	the	DET
ejpam-6690	225	2	hyperring	hyperring	NOUN
ejpam-6690	225	3	r	r	NOUN
ejpam-6690	225	4	is	be	AUX
ejpam-6690	225	5	a	a	DET
ejpam-6690	225	6	product	product	NOUN
ejpam-6690	225	7	of	of	ADP
ejpam-6690	225	8	hyperrings	hyperring	NOUN
ejpam-6690	225	9	if	if	SCONJ
ejpam-6690	225	10	and	and	CCONJ
ejpam-6690	225	11	only	only	ADV
ejpam-6690	225	12	if	if	SCONJ
ejpam-6690	225	13	r	r	NOUN
ejpam-6690	225	14	has	have	VERB
ejpam-6690	225	15	nontrivial	nontrivial	ADJ
ejpam-6690	225	16	idempotents	idempotent	NOUN
ejpam-6690	225	17	.	.	PUNCT
ejpam-6690	226	1	proof	proof	NOUN
ejpam-6690	226	2	.	.	PUNCT
ejpam-6690	227	1	let	let	VERB
ejpam-6690	227	2	x	x	PRON
ejpam-6690	227	3	be	be	AUX
ejpam-6690	227	4	a	a	DET
ejpam-6690	227	5	nontrivial	nontrivial	ADJ
ejpam-6690	227	6	idempotent	idempotent	NOUN
ejpam-6690	227	7	of	of	ADP
ejpam-6690	227	8	r.	r.	PROPN
ejpam-6690	227	9	hence	hence	ADV
ejpam-6690	227	10	(	(	PUNCT
ejpam-6690	227	11	x	x	X
ejpam-6690	227	12	)	)	PUNCT
ejpam-6690	227	13	and	and	CCONJ
ejpam-6690	227	14	(	(	PUNCT
ejpam-6690	227	15	1	1	NUM
ejpam-6690	227	16	−	−	NOUN
ejpam-6690	227	17	x	x	X
ejpam-6690	227	18	)	)	PUNCT
ejpam-6690	227	19	are	be	AUX
ejpam-6690	227	20	comaximal	comaximal	ADJ
ejpam-6690	227	21	hyperideals	hyperideal	NOUN
ejpam-6690	227	22	of	of	ADP
ejpam-6690	227	23	r	r	NOUN
ejpam-6690	227	24	and	and	CCONJ
ejpam-6690	227	25	(	(	PUNCT
ejpam-6690	227	26	x)(1	x)(1	NUM
ejpam-6690	228	1	−	−	NUM
ejpam-6690	229	1	x	x	X
ejpam-6690	229	2	)	)	PUNCT
ejpam-6690	229	3	=	=	SYM
ejpam-6690	229	4	(	(	PUNCT
ejpam-6690	229	5	0	0	NUM
ejpam-6690	229	6	)	)	PUNCT
ejpam-6690	229	7	.	.	PUNCT
ejpam-6690	230	1	now	now	ADV
ejpam-6690	230	2	by	by	ADP
ejpam-6690	230	3	the	the	DET
ejpam-6690	230	4	chinese	chinese	ADJ
ejpam-6690	230	5	remainder	remainder	NOUN
ejpam-6690	230	6	theorem	theorem	VERB
ejpam-6690	230	7	:	:	PUNCT
ejpam-6690	230	8	r	r	NOUN
ejpam-6690	230	9	∼=	∼=	NOUN
ejpam-6690	230	10	r/(0	r/(0	NOUN
ejpam-6690	230	11	)	)	PUNCT
ejpam-6690	230	12	∼=	∼=	PROPN
ejpam-6690	230	13	r/(x)×r/(1−	r/(x)×r/(1−	NOUN
ejpam-6690	230	14	x	x	NOUN
ejpam-6690	230	15	)	)	PUNCT
ejpam-6690	230	16	.	.	PUNCT
ejpam-6690	231	1	conversely	conversely	ADV
ejpam-6690	231	2	suppose	suppose	VERB
ejpam-6690	231	3	that	that	SCONJ
ejpam-6690	231	4	r	r	NOUN
ejpam-6690	231	5	=	=	SYM
ejpam-6690	231	6	r1	r1	PROPN
ejpam-6690	231	7	×r2	×r2	PROPN
ejpam-6690	231	8	,	,	PUNCT
ejpam-6690	231	9	then	then	ADV
ejpam-6690	231	10	(	(	PUNCT
ejpam-6690	231	11	1	1	NUM
ejpam-6690	231	12	,	,	PUNCT
ejpam-6690	231	13	0)2	0)2	NOUN
ejpam-6690	231	14	=	=	SYM
ejpam-6690	231	15	(	(	PUNCT
ejpam-6690	231	16	1	1	NUM
ejpam-6690	231	17	,	,	PUNCT
ejpam-6690	231	18	0	0	NUM
ejpam-6690	231	19	)	)	PUNCT
ejpam-6690	231	20	.	.	PUNCT
ejpam-6690	232	1	lemma	lemma	PROPN
ejpam-6690	233	1	3	3	X
ejpam-6690	233	2	.	.	PUNCT
ejpam-6690	234	1	if	if	SCONJ
ejpam-6690	234	2	r	r	NOUN
ejpam-6690	234	3	/	/	SYM
ejpam-6690	234	4	nil(r	nil(r	NOUN
ejpam-6690	234	5	)	)	PUNCT
ejpam-6690	234	6	has	have	VERB
ejpam-6690	234	7	nontrivial	nontrivial	ADJ
ejpam-6690	234	8	idempotent	idempotent	NOUN
ejpam-6690	234	9	then	then	ADV
ejpam-6690	234	10	r	r	NOUN
ejpam-6690	234	11	has	have	VERB
ejpam-6690	234	12	nontrivial	nontrivial	ADJ
ejpam-6690	234	13	idempotent	idempotent	NOUN
ejpam-6690	234	14	.	.	PUNCT
ejpam-6690	235	1	proof	proof	NOUN
ejpam-6690	235	2	.	.	PUNCT
ejpam-6690	236	1	let	let	VERB
ejpam-6690	236	2	x+nil(r	x+nil(r	NUM
ejpam-6690	236	3	)	)	PUNCT
ejpam-6690	236	4	∈	∈	PROPN
ejpam-6690	236	5	r	r	NOUN
ejpam-6690	236	6	/	/	SYM
ejpam-6690	236	7	nil(r	nil(r	ADJ
ejpam-6690	236	8	)	)	PUNCT
ejpam-6690	236	9	be	be	AUX
ejpam-6690	236	10	nontrivial	nontrivial	ADJ
ejpam-6690	236	11	idempotent	idempotent	NOUN
ejpam-6690	236	12	.	.	PUNCT
ejpam-6690	237	1	so	so	ADV
ejpam-6690	237	2	x2+nil(r	x2+nil(r	PROPN
ejpam-6690	237	3	)	)	PUNCT
ejpam-6690	238	1	=	=	PUNCT
ejpam-6690	239	1	x+nil(r	x+nil(r	PROPN
ejpam-6690	239	2	)	)	PUNCT
ejpam-6690	239	3	and	and	CCONJ
ejpam-6690	239	4	x2	x2	NUM
ejpam-6690	240	1	−	−	NOUN
ejpam-6690	240	2	x	x	SYM
ejpam-6690	241	1	⊆	⊆	NUM
ejpam-6690	241	2	nil(r	nil(r	NUM
ejpam-6690	241	3	)	)	PUNCT
ejpam-6690	241	4	.	.	PUNCT
ejpam-6690	242	1	hence	hence	ADV
ejpam-6690	242	2	there	there	PRON
ejpam-6690	242	3	is	be	VERB
ejpam-6690	242	4	n	n	DET
ejpam-6690	242	5	∈	∈	PROPN
ejpam-6690	242	6	n	n	PRON
ejpam-6690	242	7	such	such	ADJ
ejpam-6690	242	8	that	that	SCONJ
ejpam-6690	242	9	(	(	PUNCT
ejpam-6690	242	10	x2	x2	INTJ
ejpam-6690	242	11	−	−	PROPN
ejpam-6690	242	12	x)n	x)n	PUNCT
ejpam-6690	243	1	=	=	PRON
ejpam-6690	243	2	xn(x	xn(x	PUNCT
ejpam-6690	244	1	−	−	PROPN
ejpam-6690	244	2	1)n	1)n	X
ejpam-6690	244	3	=	=	SYM
ejpam-6690	244	4	0	0	X
ejpam-6690	244	5	.	.	PUNCT
ejpam-6690	245	1	also	also	ADV
ejpam-6690	245	2	(	(	PUNCT
ejpam-6690	245	3	xn	xn	X
ejpam-6690	245	4	)	)	PUNCT
ejpam-6690	245	5	+	+	CCONJ
ejpam-6690	245	6	(	(	PUNCT
ejpam-6690	245	7	(	(	PUNCT
ejpam-6690	245	8	x	x	SYM
ejpam-6690	245	9	−	−	PROPN
ejpam-6690	245	10	1)n	1)n	NUM
ejpam-6690	245	11	)	)	PUNCT
ejpam-6690	245	12	=	=	SYM
ejpam-6690	245	13	r	r	NOUN
ejpam-6690	245	14	,	,	PUNCT
ejpam-6690	245	15	and	and	CCONJ
ejpam-6690	245	16	(	(	PUNCT
ejpam-6690	245	17	xn).((x	xn).((x	NOUN
ejpam-6690	245	18	−	−	PROPN
ejpam-6690	245	19	1)n	1)n	NUM
ejpam-6690	245	20	)	)	PUNCT
ejpam-6690	245	21	=	=	SYM
ejpam-6690	245	22	(	(	PUNCT
ejpam-6690	245	23	0	0	NUM
ejpam-6690	245	24	)	)	PUNCT
ejpam-6690	245	25	.	.	PUNCT
ejpam-6690	246	1	now	now	ADV
ejpam-6690	246	2	by	by	ADP
ejpam-6690	246	3	chinese	chinese	ADJ
ejpam-6690	246	4	remainder	remainder	NOUN
ejpam-6690	246	5	theorem	theorem	VERB
ejpam-6690	246	6	φ	φ	NOUN
ejpam-6690	246	7	:	:	PUNCT
ejpam-6690	246	8	r	r	NOUN
ejpam-6690	246	9	→	→	SYM
ejpam-6690	246	10	r/(xn	r/(xn	NOUN
ejpam-6690	246	11	)	)	PUNCT
ejpam-6690	246	12	×	×	NOUN
ejpam-6690	246	13	r/((x	r/((x	NOUN
ejpam-6690	246	14	−	−	PROPN
ejpam-6690	246	15	1)n	1)n	NUM
ejpam-6690	246	16	)	)	PUNCT
ejpam-6690	246	17	,	,	PUNCT
ejpam-6690	246	18	defined	define	VERB
ejpam-6690	246	19	by	by	ADP
ejpam-6690	246	20	φ(x	φ(x	NOUN
ejpam-6690	246	21	)	)	PUNCT
ejpam-6690	247	1	=	=	NOUN
ejpam-6690	247	2	(	(	PUNCT
ejpam-6690	247	3	r	r	NOUN
ejpam-6690	247	4	+	+	CCONJ
ejpam-6690	247	5	(	(	PUNCT
ejpam-6690	247	6	xn	xn	PROPN
ejpam-6690	247	7	)	)	PUNCT
ejpam-6690	247	8	,	,	PUNCT
ejpam-6690	248	1	r	r	NOUN
ejpam-6690	248	2	+	+	CCONJ
ejpam-6690	248	3	(	(	PUNCT
ejpam-6690	248	4	(	(	PUNCT
ejpam-6690	248	5	x	x	SYM
ejpam-6690	248	6	−	−	PROPN
ejpam-6690	248	7	1)n	1)n	NUM
ejpam-6690	248	8	)	)	PUNCT
ejpam-6690	248	9	)	)	PUNCT
ejpam-6690	248	10	is	be	AUX
ejpam-6690	248	11	an	an	DET
ejpam-6690	248	12	isomorphism	isomorphism	NOUN
ejpam-6690	248	13	and	and	CCONJ
ejpam-6690	248	14	(	(	PUNCT
ejpam-6690	248	15	0	0	NUM
ejpam-6690	248	16	,	,	PUNCT
ejpam-6690	248	17	1	1	X
ejpam-6690	248	18	)	)	PUNCT
ejpam-6690	248	19	∈	∈	PROPN
ejpam-6690	248	20	r/(xn)×r/((x−1)n	r/(xn)×r/((x−1)n	NOUN
ejpam-6690	248	21	)	)	PUNCT
ejpam-6690	248	22	is	be	AUX
ejpam-6690	248	23	nontrivial	nontrivial	ADJ
ejpam-6690	248	24	idempotent	idempotent	NOUN
ejpam-6690	248	25	.	.	PUNCT
ejpam-6690	249	1	so	so	ADV
ejpam-6690	249	2	φ−1((0	φ−1((0	NOUN
ejpam-6690	249	3	,	,	PUNCT
ejpam-6690	249	4	1	1	NUM
ejpam-6690	249	5	)	)	PUNCT
ejpam-6690	249	6	)	)	PUNCT
ejpam-6690	250	1	∈	∈	NOUN
ejpam-6690	250	2	r	r	NOUN
ejpam-6690	250	3	is	be	AUX
ejpam-6690	250	4	nontrivial	nontrivial	ADJ
ejpam-6690	250	5	idempotent	idempotent	NOUN
ejpam-6690	250	6	.	.	PUNCT
ejpam-6690	251	1	theorem	theorem	VERB
ejpam-6690	251	2	7	7	NUM
ejpam-6690	251	3	.	.	PUNCT
ejpam-6690	252	1	if	if	SCONJ
ejpam-6690	252	2	spec(r	spec(r	PROPN
ejpam-6690	252	3	)	)	PUNCT
ejpam-6690	252	4	is	be	AUX
ejpam-6690	252	5	a	a	DET
ejpam-6690	252	6	disconnected	disconnected	ADJ
ejpam-6690	252	7	space	space	NOUN
ejpam-6690	252	8	and	and	CCONJ
ejpam-6690	252	9	spec(r	spec(r	ADJ
ejpam-6690	252	10	)	)	PUNCT
ejpam-6690	252	11	=	=	SYM
ejpam-6690	252	12	c	c	NOUN
ejpam-6690	252	13	∪̇	∪̇	X
ejpam-6690	253	1	d	d	NOUN
ejpam-6690	253	2	,	,	PUNCT
ejpam-6690	253	3	then	then	ADV
ejpam-6690	253	4	r	r	NOUN
ejpam-6690	253	5	∼=	∼=	PROPN
ejpam-6690	253	6	r1×r2	r1×r2	PROPN
ejpam-6690	253	7	.	.	PUNCT
ejpam-6690	254	1	here	here	ADV
ejpam-6690	254	2	,	,	PUNCT
ejpam-6690	254	3	c	c	NOUN
ejpam-6690	254	4	=	=	SYM
ejpam-6690	254	5	spec(r1	spec(r1	NOUN
ejpam-6690	254	6	)	)	PUNCT
ejpam-6690	254	7	and	and	CCONJ
ejpam-6690	254	8	d	d	NOUN
ejpam-6690	254	9	=	=	SYM
ejpam-6690	254	10	spec(r2	spec(r2	NOUN
ejpam-6690	254	11	)	)	PUNCT
ejpam-6690	254	12	.	.	PUNCT
ejpam-6690	255	1	proof	proof	NOUN
ejpam-6690	255	2	.	.	PUNCT
ejpam-6690	256	1	let	let	VERB
ejpam-6690	256	2	i	i	PRON
ejpam-6690	256	3	and	and	CCONJ
ejpam-6690	256	4	j	j	PROPN
ejpam-6690	256	5	be	be	VERB
ejpam-6690	256	6	hyperideals	hyperideal	NOUN
ejpam-6690	256	7	of	of	ADP
ejpam-6690	256	8	r	r	NOUN
ejpam-6690	256	9	such	such	ADJ
ejpam-6690	256	10	that	that	DET
ejpam-6690	256	11	c	c	PROPN
ejpam-6690	256	12	=	=	SYM
ejpam-6690	256	13	v	v	PROPN
ejpam-6690	256	14	(	(	PUNCT
ejpam-6690	256	15	i	i	NOUN
ejpam-6690	256	16	)	)	PUNCT
ejpam-6690	256	17	and	and	CCONJ
ejpam-6690	256	18	d	d	X
ejpam-6690	256	19	=	=	SYM
ejpam-6690	256	20	v	v	PROPN
ejpam-6690	256	21	(	(	PUNCT
ejpam-6690	256	22	j	j	NOUN
ejpam-6690	256	23	)	)	PUNCT
ejpam-6690	256	24	.	.	PUNCT
ejpam-6690	257	1	since	since	SCONJ
ejpam-6690	257	2	v	v	NOUN
ejpam-6690	257	3	(	(	PUNCT
ejpam-6690	257	4	r	r	NOUN
ejpam-6690	257	5	)	)	PUNCT
ejpam-6690	257	6	=	=	NOUN
ejpam-6690	257	7	∅	∅	NOUN
ejpam-6690	257	8	and	and	CCONJ
ejpam-6690	257	9	v	v	NOUN
ejpam-6690	257	10	(	(	PUNCT
ejpam-6690	257	11	0	0	NUM
ejpam-6690	257	12	)	)	PUNCT
ejpam-6690	257	13	=	=	SYM
ejpam-6690	257	14	spec(r	spec(r	PROPN
ejpam-6690	257	15	)	)	PUNCT
ejpam-6690	257	16	,	,	PUNCT
ejpam-6690	257	17	then	then	ADV
ejpam-6690	257	18	∅	∅	NOUN
ejpam-6690	257	19	=	=	PUNCT
ejpam-6690	257	20	c	c	NOUN
ejpam-6690	257	21	∩d	∩d	NOUN
ejpam-6690	257	22	=	=	NOUN
ejpam-6690	258	1	v	v	X
ejpam-6690	258	2	(	(	PUNCT
ejpam-6690	258	3	i	i	NOUN
ejpam-6690	258	4	)	)	PUNCT
ejpam-6690	258	5	∩	∩	PROPN
ejpam-6690	258	6	v	v	X
ejpam-6690	258	7	(	(	PUNCT
ejpam-6690	258	8	j	j	NOUN
ejpam-6690	258	9	)	)	PUNCT
ejpam-6690	258	10	=	=	SYM
ejpam-6690	258	11	v	v	X
ejpam-6690	258	12	(	(	PUNCT
ejpam-6690	258	13	i	i	PROPN
ejpam-6690	258	14	+	+	NUM
ejpam-6690	258	15	j	j	NOUN
ejpam-6690	258	16	)	)	PUNCT
ejpam-6690	258	17	=	=	NOUN
ejpam-6690	258	18	v	v	X
ejpam-6690	258	19	(	(	PUNCT
ejpam-6690	258	20	r	r	NOUN
ejpam-6690	258	21	)	)	PUNCT
ejpam-6690	258	22	;	;	PUNCT
ejpam-6690	258	23	spec(r	spec(r	X
ejpam-6690	258	24	)	)	PUNCT
ejpam-6690	258	25	=	=	SYM
ejpam-6690	258	26	c	c	NOUN
ejpam-6690	258	27	∪d	∪d	X
ejpam-6690	258	28	=	=	SYM
ejpam-6690	258	29	v	v	NOUN
ejpam-6690	258	30	(	(	PUNCT
ejpam-6690	258	31	i	i	NOUN
ejpam-6690	258	32	)	)	PUNCT
ejpam-6690	258	33	∪	∪	ADP
ejpam-6690	258	34	v	v	PROPN
ejpam-6690	258	35	(	(	PUNCT
ejpam-6690	258	36	j	j	NOUN
ejpam-6690	258	37	)	)	PUNCT
ejpam-6690	258	38	=	=	NOUN
ejpam-6690	258	39	v	v	NOUN
ejpam-6690	258	40	(	(	PUNCT
ejpam-6690	258	41	ij	ij	NOUN
ejpam-6690	258	42	)	)	PUNCT
ejpam-6690	258	43	=	=	NOUN
ejpam-6690	258	44	v	v	X
ejpam-6690	258	45	(	(	PUNCT
ejpam-6690	258	46	0	0	NUM
ejpam-6690	258	47	)	)	PUNCT
ejpam-6690	258	48	.	.	PUNCT
ejpam-6690	259	1	by	by	ADP
ejpam-6690	259	2	lemma	lemma	PROPN
ejpam-6690	259	3	2	2	NUM
ejpam-6690	259	4	we	we	PRON
ejpam-6690	259	5	have	have	VERB
ejpam-6690	259	6	i+j	i+j	NUM
ejpam-6690	259	7	=	=	SYM
ejpam-6690	259	8	r	r	NOUN
ejpam-6690	259	9	and	and	CCONJ
ejpam-6690	259	10	by	by	ADP
ejpam-6690	259	11	chinese	chinese	ADJ
ejpam-6690	259	12	remainder	remainder	NOUN
ejpam-6690	259	13	theorem	theorem	NOUN
ejpam-6690	259	14	r	r	NOUN
ejpam-6690	259	15	/	/	SYM
ejpam-6690	259	16	ij	ij	NOUN
ejpam-6690	259	17	∼=	∼=	ADV
ejpam-6690	259	18	r	r	NOUN
ejpam-6690	259	19	/	/	SYM
ejpam-6690	259	20	i×r	i×r	PROPN
ejpam-6690	259	21	/	/	SYM
ejpam-6690	259	22	j	j	PROPN
ejpam-6690	259	23	.	.	PUNCT
ejpam-6690	260	1	if	if	SCONJ
ejpam-6690	260	2	r	r	NOUN
ejpam-6690	260	3	has	have	VERB
ejpam-6690	260	4	no	no	DET
ejpam-6690	260	5	nilpotent	nilpotent	ADJ
ejpam-6690	260	6	elements	element	NOUN
ejpam-6690	260	7	,	,	PUNCT
ejpam-6690	260	8	then	then	ADV
ejpam-6690	260	9	nil(r	nil(r	NUM
ejpam-6690	260	10	)	)	PUNCT
ejpam-6690	260	11	=	=	SYM
ejpam-6690	260	12	(	(	PUNCT
ejpam-6690	260	13	0	0	NUM
ejpam-6690	260	14	)	)	PUNCT
ejpam-6690	260	15	,	,	PUNCT
ejpam-6690	260	16	and	and	CCONJ
ejpam-6690	260	17	by	by	ADP
ejpam-6690	260	18	proposition	proposition	NOUN
ejpam-6690	260	19	2	2	NUM
ejpam-6690	260	20	ij	ij	NOUN
ejpam-6690	260	21	=	=	NOUN
ejpam-6690	260	22	0	0	PROPN
ejpam-6690	260	23	.	.	PUNCT
ejpam-6690	261	1	so	so	ADV
ejpam-6690	261	2	r/(0	r/(0	NOUN
ejpam-6690	261	3	)	)	PUNCT
ejpam-6690	261	4	∼=	∼=	NOUN
ejpam-6690	261	5	r	r	NOUN
ejpam-6690	261	6	∼=	∼=	NOUN
ejpam-6690	261	7	r	r	NOUN
ejpam-6690	261	8	/	/	SYM
ejpam-6690	261	9	i	i	NOUN
ejpam-6690	261	10	×	×	NOUN
ejpam-6690	261	11	r	r	NOUN
ejpam-6690	261	12	/	/	SYM
ejpam-6690	261	13	j	j	NOUN
ejpam-6690	261	14	,	,	PUNCT
ejpam-6690	261	15	and	and	CCONJ
ejpam-6690	261	16	by	by	ADP
ejpam-6690	261	17	theorem	theorem	NOUN
ejpam-6690	261	18	4	4	NUM
ejpam-6690	261	19	we	we	PRON
ejpam-6690	261	20	have	have	VERB
ejpam-6690	261	21	v	v	NOUN
ejpam-6690	261	22	(	(	PUNCT
ejpam-6690	261	23	i	i	NOUN
ejpam-6690	261	24	)	)	PUNCT
ejpam-6690	262	1	=	=	PUNCT
ejpam-6690	262	2	spec(r	spec(r	PROPN
ejpam-6690	262	3	/	/	SYM
ejpam-6690	262	4	i	i	PROPN
ejpam-6690	262	5	)	)	PUNCT
ejpam-6690	262	6	and	and	CCONJ
ejpam-6690	262	7	v	v	X
ejpam-6690	262	8	(	(	PUNCT
ejpam-6690	262	9	j	j	NOUN
ejpam-6690	262	10	)	)	PUNCT
ejpam-6690	263	1	=	=	PUNCT
ejpam-6690	263	2	spec(r	spec(r	PROPN
ejpam-6690	263	3	/	/	SYM
ejpam-6690	263	4	j	j	NOUN
ejpam-6690	263	5	)	)	PUNCT
ejpam-6690	263	6	.	.	PUNCT
ejpam-6690	264	1	now	now	ADV
ejpam-6690	264	2	since	since	SCONJ
ejpam-6690	264	3	spec(r	spec(r	PROPN
ejpam-6690	264	4	)	)	PUNCT
ejpam-6690	264	5	=	=	SYM
ejpam-6690	264	6	spec(r	spec(r	PROPN
ejpam-6690	264	7	/	/	SYM
ejpam-6690	264	8	nil(r	nil(r	NOUN
ejpam-6690	264	9	)	)	PUNCT
ejpam-6690	264	10	)	)	PUNCT
ejpam-6690	265	1	so	so	ADV
ejpam-6690	265	2	spec(r	spec(r	PROPN
ejpam-6690	265	3	/	/	SYM
ejpam-6690	265	4	nil(r	nil(r	NOUN
ejpam-6690	265	5	)	)	PUNCT
ejpam-6690	265	6	)	)	PUNCT
ejpam-6690	265	7	is	be	AUX
ejpam-6690	265	8	disconnected	disconnect	VERB
ejpam-6690	265	9	and	and	CCONJ
ejpam-6690	265	10	r	r	NOUN
ejpam-6690	265	11	/	/	SYM
ejpam-6690	265	12	nil(r	nil(r	NOUN
ejpam-6690	265	13	)	)	PUNCT
ejpam-6690	265	14	has	have	VERB
ejpam-6690	265	15	no	no	DET
ejpam-6690	265	16	nilpotent	nilpotent	ADJ
ejpam-6690	265	17	elements	element	NOUN
ejpam-6690	265	18	.	.	PUNCT
ejpam-6690	266	1	hence	hence	ADV
ejpam-6690	266	2	r	r	NOUN
ejpam-6690	266	3	/	/	SYM
ejpam-6690	266	4	nil(r	nil(r	NOUN
ejpam-6690	266	5	)	)	PUNCT
ejpam-6690	266	6	∼=	∼=	PROPN
ejpam-6690	266	7	s	s	PART
ejpam-6690	266	8	×	×	NOUN
ejpam-6690	266	9	t	t	NOUN
ejpam-6690	266	10	,	,	PUNCT
ejpam-6690	266	11	for	for	ADP
ejpam-6690	266	12	some	some	DET
ejpam-6690	266	13	hyperrings	hyperring	NOUN
ejpam-6690	266	14	s	s	NOUN
ejpam-6690	266	15	and	and	CCONJ
ejpam-6690	266	16	t	t	PROPN
ejpam-6690	266	17	,	,	PUNCT
ejpam-6690	266	18	and	and	CCONJ
ejpam-6690	266	19	by	by	ADP
ejpam-6690	266	20	corollary	corollary	ADJ
ejpam-6690	266	21	2	2	NUM
ejpam-6690	266	22	,	,	PUNCT
ejpam-6690	266	23	r	r	NOUN
ejpam-6690	266	24	/	/	SYM
ejpam-6690	266	25	nil(r	nil(r	NOUN
ejpam-6690	266	26	)	)	PUNCT
ejpam-6690	266	27	has	have	VERB
ejpam-6690	266	28	nontrivial	nontrivial	ADJ
ejpam-6690	266	29	idempotents	idempotent	NOUN
ejpam-6690	266	30	.	.	PUNCT
ejpam-6690	267	1	therefore	therefore	ADV
ejpam-6690	267	2	,	,	PUNCT
ejpam-6690	267	3	by	by	ADP
ejpam-6690	267	4	lemma	lemma	PROPN
ejpam-6690	267	5	3	3	NUM
ejpam-6690	267	6	,	,	PUNCT
ejpam-6690	267	7	r	r	NOUN
ejpam-6690	267	8	has	have	VERB
ejpam-6690	267	9	nontrivial	nontrivial	ADJ
ejpam-6690	267	10	idempotent	idempotent	NOUN
ejpam-6690	267	11	and	and	CCONJ
ejpam-6690	267	12	by	by	ADP
ejpam-6690	267	13	corollary	corollary	ADJ
ejpam-6690	267	14	2	2	NUM
ejpam-6690	267	15	,	,	PUNCT
ejpam-6690	267	16	r	r	NOUN
ejpam-6690	267	17	is	be	AUX
ejpam-6690	267	18	product	product	NOUN
ejpam-6690	267	19	of	of	ADP
ejpam-6690	267	20	hyperrings	hyperring	NOUN
ejpam-6690	267	21	.	.	PUNCT
ejpam-6690	268	1	remark	remark	PROPN
ejpam-6690	268	2	3	3	NUM
ejpam-6690	268	3	.	.	PUNCT
ejpam-6690	269	1	consider	consider	VERB
ejpam-6690	269	2	w	w	NOUN
ejpam-6690	269	3	(	(	PUNCT
ejpam-6690	269	4	f	f	X
ejpam-6690	269	5	)	)	PUNCT
ejpam-6690	269	6	=	=	PRON
ejpam-6690	269	7	{	{	PUNCT
ejpam-6690	269	8	p	p	X
ejpam-6690	269	9	∈	∈	PROPN
ejpam-6690	269	10	spec(r	spec(r	PROPN
ejpam-6690	269	11	)	)	PUNCT
ejpam-6690	269	12	;	;	PUNCT
ejpam-6690	270	1	f	f	X
ejpam-6690	270	2	/∈	/∈	PUNCT
ejpam-6690	271	1	p	p	X
ejpam-6690	271	2	}	}	PUNCT
ejpam-6690	271	3	,	,	PUNCT
ejpam-6690	271	4	for	for	ADP
ejpam-6690	271	5	every	every	DET
ejpam-6690	271	6	f	f	PROPN
ejpam-6690	271	7	∈	∈	PROPN
ejpam-6690	271	8	r.	r.	PROPN
ejpam-6690	271	9	then	then	ADV
ejpam-6690	271	10	w	w	PROPN
ejpam-6690	271	11	(	(	PUNCT
ejpam-6690	271	12	0	0	NUM
ejpam-6690	271	13	)	)	PUNCT
ejpam-6690	271	14	=	=	NOUN
ejpam-6690	271	15	∅	∅	NOUN
ejpam-6690	271	16	,	,	PUNCT
ejpam-6690	271	17	w	w	PROPN
ejpam-6690	271	18	(	(	PUNCT
ejpam-6690	271	19	1	1	NUM
ejpam-6690	271	20	)	)	PUNCT
ejpam-6690	271	21	=	=	NOUN
ejpam-6690	271	22	x	x	X
ejpam-6690	271	23	=	=	SYM
ejpam-6690	271	24	spec(r	spec(r	PROPN
ejpam-6690	271	25	)	)	PUNCT
ejpam-6690	271	26	and	and	CCONJ
ejpam-6690	271	27	x	x	SYM
ejpam-6690	271	28	−	−	NOUN
ejpam-6690	271	29	v	v	X
ejpam-6690	271	30	(	(	PUNCT
ejpam-6690	271	31	e	e	NOUN
ejpam-6690	271	32	)	)	PUNCT
ejpam-6690	271	33	=	=	SYM
ejpam-6690	271	34	⋃	⋃	NOUN
ejpam-6690	271	35	f∈e	f∈e	NOUN
ejpam-6690	271	36	w	w	PROPN
ejpam-6690	271	37	(	(	PUNCT
ejpam-6690	271	38	f	f	NOUN
ejpam-6690	271	39	)	)	PUNCT
ejpam-6690	271	40	,	,	PUNCT
ejpam-6690	271	41	for	for	ADP
ejpam-6690	271	42	e	e	PROPN
ejpam-6690	271	43	⊆	⊆	PROPN
ejpam-6690	271	44	r.	r.	PROPN
ejpam-6690	271	45	therefore	therefore	ADV
ejpam-6690	271	46	,	,	PUNCT
ejpam-6690	271	47	{	{	PUNCT
ejpam-6690	271	48	w	w	NOUN
ejpam-6690	271	49	(	(	PUNCT
ejpam-6690	271	50	f	f	PROPN
ejpam-6690	271	51	)	)	PUNCT
ejpam-6690	271	52	;	;	PUNCT
ejpam-6690	271	53	f	f	PROPN
ejpam-6690	271	54	∈	∈	PROPN
ejpam-6690	271	55	r	r	X
ejpam-6690	271	56	}	}	PUNCT
ejpam-6690	271	57	is	be	AUX
ejpam-6690	271	58	a	a	DET
ejpam-6690	271	59	basis	basis	NOUN
ejpam-6690	271	60	for	for	ADP
ejpam-6690	271	61	zariski	zariski	ADJ
ejpam-6690	271	62	topology	topology	NOUN
ejpam-6690	271	63	on	on	ADP
ejpam-6690	271	64	spec(r	spec(r	PROPN
ejpam-6690	271	65	)	)	PUNCT
ejpam-6690	271	66	,	,	PUNCT
ejpam-6690	271	67	and	and	CCONJ
ejpam-6690	271	68	it	it	PRON
ejpam-6690	271	69	is	be	AUX
ejpam-6690	271	70	clear	clear	ADJ
ejpam-6690	271	71	that	that	SCONJ
ejpam-6690	271	72	w	w	PROPN
ejpam-6690	271	73	(	(	PUNCT
ejpam-6690	271	74	f)∩w	f)∩w	ADJ
ejpam-6690	271	75	(	(	PUNCT
ejpam-6690	271	76	g	g	NOUN
ejpam-6690	271	77	)	)	PUNCT
ejpam-6690	271	78	=	=	SYM
ejpam-6690	271	79	w	w	PROPN
ejpam-6690	271	80	(	(	PUNCT
ejpam-6690	271	81	fg	fg	PROPN
ejpam-6690	271	82	)	)	PUNCT
ejpam-6690	271	83	.	.	PUNCT
ejpam-6690	272	1	also	also	ADV
ejpam-6690	272	2	w	w	PROPN
ejpam-6690	272	3	(	(	PUNCT
ejpam-6690	272	4	f	f	X
ejpam-6690	272	5	)	)	PUNCT
ejpam-6690	272	6	=	=	SYM
ejpam-6690	272	7	w	w	PROPN
ejpam-6690	272	8	(	(	PUNCT
ejpam-6690	272	9	g	g	NOUN
ejpam-6690	272	10	)	)	PUNCT
ejpam-6690	272	11	if	if	SCONJ
ejpam-6690	273	1	and	and	CCONJ
ejpam-6690	273	2	only	only	ADV
ejpam-6690	273	3	if	if	SCONJ
ejpam-6690	273	4	√	√	PROPN
ejpam-6690	273	5	(	(	PUNCT
ejpam-6690	273	6	f	f	X
ejpam-6690	273	7	)	)	PUNCT
ejpam-6690	273	8	=	=	SYM
ejpam-6690	273	9	√	√	NUM
ejpam-6690	273	10	(	(	PUNCT
ejpam-6690	273	11	g	g	NOUN
ejpam-6690	273	12	)	)	PUNCT
ejpam-6690	273	13	,	,	PUNCT
ejpam-6690	273	14	and	and	CCONJ
ejpam-6690	273	15	f	f	X
ejpam-6690	273	16	∈	∈	PROPN
ejpam-6690	273	17	r	r	NOUN
ejpam-6690	273	18	is	be	AUX
ejpam-6690	273	19	nilpotent	nilpotent	ADJ
ejpam-6690	273	20	if	if	SCONJ
ejpam-6690	273	21	and	and	CCONJ
ejpam-6690	273	22	only	only	ADV
ejpam-6690	273	23	if	if	SCONJ
ejpam-6690	273	24	w	w	PROPN
ejpam-6690	273	25	(	(	PUNCT
ejpam-6690	273	26	f	f	X
ejpam-6690	273	27	)	)	PUNCT
ejpam-6690	273	28	=	=	NOUN
ejpam-6690	273	29	∅	∅	NOUN
ejpam-6690	273	30	,	,	PUNCT
ejpam-6690	273	31	and	and	CCONJ
ejpam-6690	273	32	f	f	PROPN
ejpam-6690	273	33	∈	∈	PROPN
ejpam-6690	273	34	r	r	NOUN
ejpam-6690	273	35	is	be	AUX
ejpam-6690	273	36	unit	unit	NOUN
ejpam-6690	273	37	if	if	SCONJ
ejpam-6690	273	38	and	and	CCONJ
ejpam-6690	273	39	only	only	ADV
ejpam-6690	273	40	if	if	SCONJ
ejpam-6690	273	41	w	w	PROPN
ejpam-6690	273	42	(	(	PUNCT
ejpam-6690	273	43	f	f	X
ejpam-6690	273	44	)	)	PUNCT
ejpam-6690	274	1	=	=	PUNCT
ejpam-6690	274	2	x.	x.	NOUN
ejpam-6690	274	3	here	here	ADV
ejpam-6690	274	4	too	too	ADV
ejpam-6690	274	5	like	like	ADP
ejpam-6690	274	6	the	the	DET
ejpam-6690	274	7	theory	theory	NOUN
ejpam-6690	274	8	of	of	ADP
ejpam-6690	274	9	classical	classical	ADJ
ejpam-6690	274	10	rings	ring	NOUN
ejpam-6690	274	11	,	,	PUNCT
ejpam-6690	274	12	it	it	PRON
ejpam-6690	274	13	can	can	AUX
ejpam-6690	274	14	be	be	AUX
ejpam-6690	274	15	proved	prove	VERB
ejpam-6690	274	16	that	that	SCONJ
ejpam-6690	274	17	w	w	PROPN
ejpam-6690	274	18	(	(	PUNCT
ejpam-6690	274	19	f	f	X
ejpam-6690	274	20	)	)	PUNCT
ejpam-6690	274	21	is	be	AUX
ejpam-6690	274	22	quasi	quasi	ADJ
ejpam-6690	274	23	-	-	ADJ
ejpam-6690	274	24	compact	compact	ADJ
ejpam-6690	274	25	for	for	ADP
ejpam-6690	274	26	every	every	DET
ejpam-6690	274	27	f	f	PROPN
ejpam-6690	274	28	∈	∈	PROPN
ejpam-6690	274	29	r	r	NOUN
ejpam-6690	274	30	,	,	PUNCT
ejpam-6690	274	31	and	and	CCONJ
ejpam-6690	274	32	an	an	DET
ejpam-6690	274	33	open	open	ADJ
ejpam-6690	274	34	subset	subset	NOUN
ejpam-6690	274	35	of	of	ADP
ejpam-6690	274	36	x	x	PUNCT
ejpam-6690	274	37	is	be	AUX
ejpam-6690	274	38	quasi	quasi	ADJ
ejpam-6690	274	39	-	-	ADJ
ejpam-6690	274	40	compact	compact	ADJ
ejpam-6690	274	41	if	if	SCONJ
ejpam-6690	274	42	and	and	CCONJ
ejpam-6690	274	43	only	only	ADV
ejpam-6690	274	44	if	if	SCONJ
ejpam-6690	274	45	it	it	PRON
ejpam-6690	274	46	is	be	AUX
ejpam-6690	274	47	a	a	DET
ejpam-6690	274	48	finite	finite	ADJ
ejpam-6690	274	49	union	union	NOUN
ejpam-6690	274	50	of	of	ADP
ejpam-6690	274	51	sets	set	NOUN
ejpam-6690	274	52	w	w	PROPN
ejpam-6690	274	53	(	(	PUNCT
ejpam-6690	274	54	f	f	X
ejpam-6690	274	55	)	)	PUNCT
ejpam-6690	275	1	[	[	X
ejpam-6690	275	2	17	17	NUM
ejpam-6690	275	3	]	]	PUNCT
ejpam-6690	275	4	.	.	PUNCT
ejpam-6690	276	1	on	on	ADP
ejpam-6690	276	2	the	the	DET
ejpam-6690	276	3	other	other	ADJ
ejpam-6690	276	4	hand	hand	NOUN
ejpam-6690	276	5	if	if	SCONJ
ejpam-6690	276	6	we	we	PRON
ejpam-6690	276	7	define	define	VERB
ejpam-6690	276	8	zariski	zariski	NOUN
ejpam-6690	276	9	topology	topology	NOUN
ejpam-6690	276	10	based	base	VERB
ejpam-6690	276	11	on	on	ADP
ejpam-6690	276	12	closed	closed	ADJ
ejpam-6690	276	13	subsets	subset	NOUN
ejpam-6690	276	14	,	,	PUNCT
ejpam-6690	276	15	then	then	ADV
ejpam-6690	276	16	b	b	X
ejpam-6690	276	17	=	=	PRON
ejpam-6690	276	18	{	{	PUNCT
ejpam-6690	276	19	b(x);x	b(x);x	PROPN
ejpam-6690	276	20	∈	∈	PROPN
ejpam-6690	276	21	r	r	NOUN
ejpam-6690	276	22	}	}	PUNCT
ejpam-6690	276	23	is	be	AUX
ejpam-6690	276	24	a	a	DET
ejpam-6690	276	25	basis	basis	NOUN
ejpam-6690	276	26	for	for	ADP
ejpam-6690	276	27	zariski	zariski	ADJ
ejpam-6690	276	28	topology	topology	NOUN
ejpam-6690	276	29	on	on	ADP
ejpam-6690	276	30	spec(r	spec(r	PROPN
ejpam-6690	276	31	)	)	PUNCT
ejpam-6690	276	32	,	,	PUNCT
ejpam-6690	276	33	where	where	SCONJ
ejpam-6690	276	34	b(x	b(x	NOUN
ejpam-6690	276	35	)	)	PUNCT
ejpam-6690	276	36	=	=	SYM
ejpam-6690	276	37	v	v	NOUN
ejpam-6690	276	38	(	(	PUNCT
ejpam-6690	276	39	(	(	PUNCT
ejpam-6690	276	40	x	x	NOUN
ejpam-6690	276	41	)	)	PUNCT
ejpam-6690	276	42	)	)	PUNCT
ejpam-6690	277	1	=	=	PRON
ejpam-6690	277	2	{	{	PUNCT
ejpam-6690	277	3	p	p	X
ejpam-6690	277	4	∈	∈	PROPN
ejpam-6690	277	5	spec(r);x	spec(r);x	ADP
ejpam-6690	277	6	∈	∈	PROPN
ejpam-6690	277	7	p	p	X
ejpam-6690	277	8	}	}	PUNCT
ejpam-6690	277	9	.	.	PUNCT
ejpam-6690	278	1	theorem	theorem	ADJ
ejpam-6690	278	2	8	8	NUM
ejpam-6690	278	3	.	.	PUNCT
ejpam-6690	279	1	the	the	DET
ejpam-6690	279	2	hyperideal	hyperideal	PROPN
ejpam-6690	279	3	nil(r	nil(r	NOUN
ejpam-6690	279	4	)	)	PUNCT
ejpam-6690	279	5	of	of	ADP
ejpam-6690	279	6	r	r	NOUN
ejpam-6690	279	7	is	be	AUX
ejpam-6690	279	8	prime	prime	ADJ
ejpam-6690	279	9	if	if	SCONJ
ejpam-6690	279	10	and	and	CCONJ
ejpam-6690	279	11	only	only	ADV
ejpam-6690	280	1	if	if	SCONJ
ejpam-6690	280	2	spec(r	spec(r	PROPN
ejpam-6690	280	3	)	)	PUNCT
ejpam-6690	280	4	is	be	AUX
ejpam-6690	280	5	irreducible	irreducible	ADJ
ejpam-6690	280	6	,	,	PUNCT
ejpam-6690	280	7	(	(	PUNCT
ejpam-6690	280	8	especially	especially	ADV
ejpam-6690	280	9	that	that	SCONJ
ejpam-6690	280	10	if	if	SCONJ
ejpam-6690	280	11	r	r	NOUN
ejpam-6690	280	12	is	be	AUX
ejpam-6690	280	13	hyperdomain	hyperdomain	ADJ
ejpam-6690	280	14	,	,	PUNCT
ejpam-6690	280	15	then	then	ADV
ejpam-6690	280	16	spec(r	spec(r	PROPN
ejpam-6690	280	17	)	)	PUNCT
ejpam-6690	280	18	is	be	AUX
ejpam-6690	280	19	irreducible	irreducible	ADJ
ejpam-6690	280	20	)	)	PUNCT
ejpam-6690	280	21	.	.	PUNCT
ejpam-6690	281	1	b.	b.	PROPN
ejpam-6690	281	2	afshar	afshar	PROPN
ejpam-6690	281	3	,	,	PUNCT
ejpam-6690	281	4	r.	r.	PROPN
ejpam-6690	281	5	ameri	ameri	PROPN
ejpam-6690	281	6	,	,	PUNCT
ejpam-6690	281	7	m.	m.	PROPN
ejpam-6690	281	8	al	al	PROPN
ejpam-6690	281	9	-	-	PUNCT
ejpam-6690	281	10	tahan	tahan	PROPN
ejpam-6690	281	11	/	/	SYM
ejpam-6690	281	12	eur	eur	PROPN
ejpam-6690	281	13	.	.	PUNCT
ejpam-6690	282	1	j.	j.	PROPN
ejpam-6690	282	2	pure	pure	PROPN
ejpam-6690	282	3	appl	appl	PROPN
ejpam-6690	282	4	.	.	PROPN
ejpam-6690	282	5	math	math	PROPN
ejpam-6690	282	6	,	,	PUNCT
ejpam-6690	282	7	18	18	NUM
ejpam-6690	282	8	(	(	PUNCT
ejpam-6690	282	9	4	4	NUM
ejpam-6690	282	10	)	)	PUNCT
ejpam-6690	282	11	(	(	PUNCT
ejpam-6690	282	12	2025	2025	NUM
ejpam-6690	282	13	)	)	PUNCT
ejpam-6690	282	14	,	,	PUNCT
ejpam-6690	282	15	6690	6690	NUM
ejpam-6690	282	16	9	9	NUM
ejpam-6690	282	17	of	of	ADP
ejpam-6690	282	18	19	19	NUM
ejpam-6690	282	19	proof	proof	NOUN
ejpam-6690	282	20	.	.	PUNCT
ejpam-6690	283	1	we	we	PRON
ejpam-6690	283	2	know	know	VERB
ejpam-6690	283	3	that	that	SCONJ
ejpam-6690	283	4	{	{	PUNCT
ejpam-6690	283	5	w	w	NOUN
ejpam-6690	283	6	(	(	PUNCT
ejpam-6690	283	7	f)}f∈r	f)}f∈r	NOUN
ejpam-6690	283	8	,	,	PUNCT
ejpam-6690	283	9	is	be	AUX
ejpam-6690	283	10	a	a	DET
ejpam-6690	283	11	basis	basis	NOUN
ejpam-6690	283	12	for	for	ADP
ejpam-6690	283	13	zariski	zariski	ADJ
ejpam-6690	283	14	topology	topology	NOUN
ejpam-6690	283	15	on	on	ADP
ejpam-6690	283	16	spec(r	spec(r	PROPN
ejpam-6690	283	17	)	)	PUNCT
ejpam-6690	283	18	and	and	CCONJ
ejpam-6690	283	19	any	any	DET
ejpam-6690	283	20	two	two	NUM
ejpam-6690	283	21	non	non	ADJ
ejpam-6690	283	22	-	-	ADJ
ejpam-6690	283	23	empty	empty	ADJ
ejpam-6690	283	24	sets	set	NOUN
ejpam-6690	283	25	will	will	AUX
ejpam-6690	283	26	intersect	intersect	VERB
ejpam-6690	283	27	if	if	SCONJ
ejpam-6690	283	28	and	and	CCONJ
ejpam-6690	283	29	only	only	ADV
ejpam-6690	283	30	if	if	SCONJ
ejpam-6690	283	31	any	any	DET
ejpam-6690	283	32	tow	tow	NOUN
ejpam-6690	283	33	non	non	ADJ
ejpam-6690	283	34	-	-	ADJ
ejpam-6690	283	35	empty	empty	ADJ
ejpam-6690	283	36	basis	basis	NOUN
ejpam-6690	283	37	elements	element	NOUN
ejpam-6690	283	38	intersect	intersect	ADJ
ejpam-6690	283	39	.	.	PUNCT
ejpam-6690	284	1	so	so	ADV
ejpam-6690	284	2	spec(r	spec(r	PROPN
ejpam-6690	284	3	)	)	PUNCT
ejpam-6690	284	4	is	be	AUX
ejpam-6690	284	5	irreducible	irreducible	ADJ
ejpam-6690	284	6	if	if	SCONJ
ejpam-6690	284	7	and	and	CCONJ
ejpam-6690	284	8	only	only	ADV
ejpam-6690	284	9	if	if	SCONJ
ejpam-6690	284	10	any	any	DET
ejpam-6690	284	11	tow	tow	NOUN
ejpam-6690	284	12	non	non	ADJ
ejpam-6690	284	13	-	-	ADJ
ejpam-6690	284	14	empty	empty	ADJ
ejpam-6690	284	15	basis	basis	NOUN
ejpam-6690	284	16	elements	element	NOUN
ejpam-6690	284	17	intersect	intersect	ADJ
ejpam-6690	284	18	,	,	PUNCT
ejpam-6690	284	19	that	that	PRON
ejpam-6690	284	20	is	be	AUX
ejpam-6690	284	21	for	for	ADP
ejpam-6690	284	22	every	every	DET
ejpam-6690	284	23	w	w	NOUN
ejpam-6690	284	24	(	(	PUNCT
ejpam-6690	284	25	f	f	X
ejpam-6690	284	26	)	)	PUNCT
ejpam-6690	284	27	̸=	̸=	PROPN
ejpam-6690	284	28	∅	∅	NOUN
ejpam-6690	284	29	and	and	CCONJ
ejpam-6690	284	30	w	w	PROPN
ejpam-6690	284	31	(	(	PUNCT
ejpam-6690	284	32	g	g	NOUN
ejpam-6690	284	33	)	)	PUNCT
ejpam-6690	284	34	̸=	̸=	PROPN
ejpam-6690	284	35	∅	∅	NOUN
ejpam-6690	284	36	we	we	PRON
ejpam-6690	284	37	have	have	AUX
ejpam-6690	284	38	w	w	PROPN
ejpam-6690	284	39	(	(	PUNCT
ejpam-6690	284	40	fg	fg	NOUN
ejpam-6690	284	41	)	)	PUNCT
ejpam-6690	284	42	=	=	SYM
ejpam-6690	284	43	w	w	PROPN
ejpam-6690	284	44	(	(	PUNCT
ejpam-6690	284	45	f	f	X
ejpam-6690	284	46	)	)	PUNCT
ejpam-6690	284	47	∩	∩	PROPN
ejpam-6690	284	48	w	w	PROPN
ejpam-6690	284	49	(	(	PUNCT
ejpam-6690	284	50	g	g	NOUN
ejpam-6690	284	51	)	)	PUNCT
ejpam-6690	284	52	̸=	̸=	PROPN
ejpam-6690	284	53	∅.	∅.	VERB
ejpam-6690	284	54	therefore	therefore	ADV
ejpam-6690	284	55	,	,	PUNCT
ejpam-6690	284	56	spec(r	spec(r	PROPN
ejpam-6690	284	57	)	)	PUNCT
ejpam-6690	284	58	is	be	AUX
ejpam-6690	284	59	irreducible	irreducible	ADJ
ejpam-6690	284	60	if	if	SCONJ
ejpam-6690	284	61	and	and	CCONJ
ejpam-6690	284	62	only	only	ADV
ejpam-6690	284	63	if	if	SCONJ
ejpam-6690	284	64	for	for	ADP
ejpam-6690	284	65	every	every	DET
ejpam-6690	284	66	f	f	NOUN
ejpam-6690	284	67	,	,	PUNCT
ejpam-6690	285	1	g	g	PROPN
ejpam-6690	285	2	∈	∈	PROPN
ejpam-6690	285	3	r	r	NOUN
ejpam-6690	285	4	,	,	PUNCT
ejpam-6690	285	5	if	if	SCONJ
ejpam-6690	285	6	f	f	X
ejpam-6690	285	7	,	,	PUNCT
ejpam-6690	285	8	g	g	PROPN
ejpam-6690	285	9	/∈	/∈	PUNCT
ejpam-6690	286	1	nil(r	nil(r	PROPN
ejpam-6690	286	2	)	)	PUNCT
ejpam-6690	287	1	then	then	ADV
ejpam-6690	287	2	fg	fg	PROPN
ejpam-6690	287	3	/∈	/∈	PUNCT
ejpam-6690	287	4	nil(r	nil(r	PROPN
ejpam-6690	287	5	)	)	PUNCT
ejpam-6690	287	6	,	,	PUNCT
ejpam-6690	287	7	which	which	PRON
ejpam-6690	287	8	means	mean	VERB
ejpam-6690	287	9	that	that	SCONJ
ejpam-6690	287	10	nil(r	nil(r	PROPN
ejpam-6690	287	11	)	)	PUNCT
ejpam-6690	287	12	is	be	AUX
ejpam-6690	287	13	the	the	DET
ejpam-6690	287	14	prime	prime	ADJ
ejpam-6690	287	15	hyperideal	hyperideal	NOUN
ejpam-6690	287	16	of	of	ADP
ejpam-6690	287	17	r.	r.	PROPN
ejpam-6690	287	18	the	the	DET
ejpam-6690	287	19	following	following	ADJ
ejpam-6690	287	20	result	result	NOUN
ejpam-6690	287	21	is	be	AUX
ejpam-6690	287	22	obtained	obtain	VERB
ejpam-6690	287	23	immediately	immediately	ADV
ejpam-6690	287	24	from	from	ADP
ejpam-6690	287	25	theorem	theorem	ADJ
ejpam-6690	287	26	8	8	NUM
ejpam-6690	287	27	regarding	regard	VERB
ejpam-6690	287	28	quotient	quotient	NOUN
ejpam-6690	287	29	hyperrings	hyperring	NOUN
ejpam-6690	287	30	.	.	PUNCT
ejpam-6690	288	1	corollary	corollary	ADJ
ejpam-6690	288	2	3	3	X
ejpam-6690	288	3	.	.	PUNCT
ejpam-6690	289	1	let	let	VERB
ejpam-6690	289	2	r	r	PRON
ejpam-6690	289	3	be	be	AUX
ejpam-6690	289	4	a	a	DET
ejpam-6690	289	5	hyperring	hyperring	NOUN
ejpam-6690	290	1	and	and	CCONJ
ejpam-6690	290	2	i	i	PRON
ejpam-6690	290	3	be	be	VERB
ejpam-6690	290	4	a	a	DET
ejpam-6690	290	5	hyperideal	hyperideal	NOUN
ejpam-6690	290	6	of	of	ADP
ejpam-6690	290	7	r.	r.	PROPN
ejpam-6690	290	8	then	then	ADV
ejpam-6690	290	9	v	v	PROPN
ejpam-6690	290	10	(	(	PUNCT
ejpam-6690	290	11	i	i	NOUN
ejpam-6690	290	12	)	)	PUNCT
ejpam-6690	290	13	is	be	AUX
ejpam-6690	290	14	an	an	DET
ejpam-6690	290	15	irreducible	irreducible	ADJ
ejpam-6690	290	16	subset	subset	NOUN
ejpam-6690	290	17	of	of	ADP
ejpam-6690	290	18	spec(r	spec(r	PROPN
ejpam-6690	290	19	/	/	SYM
ejpam-6690	290	20	i	i	NOUN
ejpam-6690	290	21	)	)	PUNCT
ejpam-6690	290	22	if	if	SCONJ
ejpam-6690	290	23	and	and	CCONJ
ejpam-6690	290	24	only	only	ADV
ejpam-6690	290	25	if	if	SCONJ
ejpam-6690	290	26	√	√	VERB
ejpam-6690	290	27	i	i	PRON
ejpam-6690	290	28	is	be	AUX
ejpam-6690	290	29	prime	prime	ADJ
ejpam-6690	290	30	.	.	PUNCT
ejpam-6690	291	1	proof	proof	NOUN
ejpam-6690	291	2	.	.	PUNCT
ejpam-6690	292	1	by	by	ADP
ejpam-6690	292	2	theorem	theorem	NOUN
ejpam-6690	292	3	8	8	NUM
ejpam-6690	292	4	,	,	PUNCT
ejpam-6690	292	5	spec(r	spec(r	PROPN
ejpam-6690	292	6	/	/	SYM
ejpam-6690	292	7	i	i	NOUN
ejpam-6690	292	8	)	)	PUNCT
ejpam-6690	292	9	is	be	AUX
ejpam-6690	292	10	irreducible	irreducible	ADJ
ejpam-6690	292	11	if	if	SCONJ
ejpam-6690	292	12	and	and	CCONJ
ejpam-6690	292	13	only	only	ADV
ejpam-6690	292	14	if	if	SCONJ
ejpam-6690	292	15	√	√	VERB
ejpam-6690	292	16	i	i	PRON
ejpam-6690	292	17	is	be	AUX
ejpam-6690	292	18	prime	prime	ADJ
ejpam-6690	292	19	.	.	PUNCT
ejpam-6690	293	1	also	also	ADV
ejpam-6690	293	2	spec(r	spec(r	PROPN
ejpam-6690	293	3	/	/	SYM
ejpam-6690	293	4	i	i	NOUN
ejpam-6690	293	5	)	)	PUNCT
ejpam-6690	293	6	∼=	∼=	PROPN
ejpam-6690	293	7	v	v	NOUN
ejpam-6690	293	8	(	(	PUNCT
ejpam-6690	293	9	i	i	NOUN
ejpam-6690	293	10	)	)	PUNCT
ejpam-6690	293	11	.	.	PUNCT
ejpam-6690	294	1	remark	remark	PROPN
ejpam-6690	294	2	4	4	NUM
ejpam-6690	294	3	.	.	PUNCT
ejpam-6690	295	1	the	the	DET
ejpam-6690	295	2	subset	subset	NOUN
ejpam-6690	295	3	{	{	PUNCT
ejpam-6690	295	4	x	x	NOUN
ejpam-6690	295	5	}	}	PUNCT
ejpam-6690	295	6	=	=	SYM
ejpam-6690	295	7	{	{	PUNCT
ejpam-6690	295	8	p	p	NOUN
ejpam-6690	295	9	}	}	PUNCT
ejpam-6690	295	10	of	of	ADP
ejpam-6690	295	11	spec(r	spec(r	PROPN
ejpam-6690	295	12	)	)	PUNCT
ejpam-6690	295	13	is	be	AUX
ejpam-6690	295	14	closed	close	VERB
ejpam-6690	295	15	(	(	PUNCT
ejpam-6690	295	16	closed	closed	ADJ
ejpam-6690	295	17	point	point	NOUN
ejpam-6690	295	18	)	)	PUNCT
ejpam-6690	295	19	,	,	PUNCT
ejpam-6690	295	20	if	if	SCONJ
ejpam-6690	295	21	and	and	CCONJ
ejpam-6690	295	22	only	only	ADV
ejpam-6690	295	23	if	if	SCONJ
ejpam-6690	295	24	p	p	PROPN
ejpam-6690	295	25	∈	∈	PROPN
ejpam-6690	295	26	mspec(r	mspec(r	PROPN
ejpam-6690	295	27	)	)	PUNCT
ejpam-6690	295	28	.	.	PUNCT
ejpam-6690	296	1	so	so	ADV
ejpam-6690	296	2	{	{	PUNCT
ejpam-6690	296	3	x	x	NOUN
ejpam-6690	296	4	}	}	PUNCT
ejpam-6690	296	5	=	=	SYM
ejpam-6690	296	6	v	v	NOUN
ejpam-6690	296	7	(	(	PUNCT
ejpam-6690	296	8	p	p	NOUN
ejpam-6690	296	9	)	)	PUNCT
ejpam-6690	296	10	and	and	CCONJ
ejpam-6690	296	11	y	y	NOUN
ejpam-6690	296	12	=	=	PUNCT
ejpam-6690	296	13	q	q	PROPN
ejpam-6690	296	14	∈	∈	PROPN
ejpam-6690	296	15	{	{	PUNCT
ejpam-6690	296	16	x	x	NOUN
ejpam-6690	296	17	}	}	PUNCT
ejpam-6690	296	18	if	if	SCONJ
ejpam-6690	296	19	and	and	CCONJ
ejpam-6690	296	20	only	only	ADV
ejpam-6690	296	21	if	if	SCONJ
ejpam-6690	296	22	p	p	PROPN
ejpam-6690	296	23	⊆	⊆	NUM
ejpam-6690	296	24	q.	q.	NOUN
ejpam-6690	296	25	also	also	ADV
ejpam-6690	296	26	spec(r	spec(r	PROPN
ejpam-6690	296	27	)	)	PUNCT
ejpam-6690	296	28	is	be	AUX
ejpam-6690	296	29	a	a	DET
ejpam-6690	296	30	t0	t0	NOUN
ejpam-6690	296	31	-	-	NOUN
ejpam-6690	296	32	space	space	NOUN
ejpam-6690	296	33	,	,	PUNCT
ejpam-6690	296	34	because	because	SCONJ
ejpam-6690	296	35	if	if	SCONJ
ejpam-6690	296	36	x	x	X
ejpam-6690	296	37	=	=	PUNCT
ejpam-6690	296	38	p	p	PROPN
ejpam-6690	296	39	and	and	CCONJ
ejpam-6690	296	40	y	y	PROPN
ejpam-6690	296	41	=	=	PUNCT
ejpam-6690	296	42	q	q	NOUN
ejpam-6690	296	43	are	be	AUX
ejpam-6690	296	44	distinct	distinct	ADJ
ejpam-6690	296	45	points	point	NOUN
ejpam-6690	296	46	of	of	ADP
ejpam-6690	296	47	spec(r	spec(r	PROPN
ejpam-6690	296	48	)	)	PUNCT
ejpam-6690	296	49	,	,	PUNCT
ejpam-6690	296	50	then	then	ADV
ejpam-6690	296	51	p	p	X
ejpam-6690	296	52	⊈	⊈	PROPN
ejpam-6690	296	53	q	q	NOUN
ejpam-6690	296	54	or	or	CCONJ
ejpam-6690	296	55	q	q	ADJ
ejpam-6690	296	56	⊈	⊈	PROPN
ejpam-6690	296	57	p	p	NOUN
ejpam-6690	296	58	.	.	PUNCT
ejpam-6690	297	1	without	without	ADP
ejpam-6690	297	2	loss	loss	NOUN
ejpam-6690	297	3	of	of	ADP
ejpam-6690	297	4	generality	generality	NOUN
ejpam-6690	297	5	assume	assume	VERB
ejpam-6690	297	6	that	that	SCONJ
ejpam-6690	297	7	q	q	X
ejpam-6690	297	8	⊈	⊈	PROPN
ejpam-6690	297	9	p	p	NOUN
ejpam-6690	297	10	.	.	PUNCT
ejpam-6690	298	1	then	then	ADV
ejpam-6690	298	2	x	x	X
ejpam-6690	298	3	/∈	/∈	PUNCT
ejpam-6690	298	4	{	{	PUNCT
ejpam-6690	298	5	y	y	NOUN
ejpam-6690	298	6	}	}	PUNCT
ejpam-6690	298	7	and	and	CCONJ
ejpam-6690	298	8	so	so	ADV
ejpam-6690	298	9	spec(r)−{y	spec(r)−{y	PROPN
ejpam-6690	298	10	}	}	PUNCT
ejpam-6690	298	11	is	be	AUX
ejpam-6690	298	12	an	an	DET
ejpam-6690	298	13	open	open	ADJ
ejpam-6690	298	14	set	set	NOUN
ejpam-6690	298	15	that	that	PRON
ejpam-6690	298	16	contains	contain	VERB
ejpam-6690	298	17	x	x	PRON
ejpam-6690	298	18	,	,	PUNCT
ejpam-6690	298	19	and	and	CCONJ
ejpam-6690	298	20	y	y	PROPN
ejpam-6690	298	21	/∈	/∈	PUNCT
ejpam-6690	298	22	spec(r)−	spec(r)−	NOUN
ejpam-6690	298	23	{	{	PUNCT
ejpam-6690	298	24	y	y	NOUN
ejpam-6690	298	25	}	}	PUNCT
ejpam-6690	298	26	.	.	PUNCT
ejpam-6690	299	1	if	if	SCONJ
ejpam-6690	299	2	p	p	PROPN
ejpam-6690	299	3	∈	∈	PROPN
ejpam-6690	299	4	spec(r	spec(r	PROPN
ejpam-6690	299	5	)	)	PUNCT
ejpam-6690	299	6	,	,	PUNCT
ejpam-6690	299	7	the	the	DET
ejpam-6690	299	8	height	height	NOUN
ejpam-6690	299	9	of	of	ADP
ejpam-6690	299	10	p	p	PRON
ejpam-6690	299	11	denoted	denote	VERB
ejpam-6690	299	12	by	by	ADP
ejpam-6690	299	13	h(p	h(p	PROPN
ejpam-6690	299	14	)	)	PUNCT
ejpam-6690	299	15	and	and	CCONJ
ejpam-6690	299	16	defined	define	VERB
ejpam-6690	299	17	to	to	PART
ejpam-6690	299	18	be	be	AUX
ejpam-6690	299	19	the	the	DET
ejpam-6690	299	20	supremum	supremum	NOUN
ejpam-6690	299	21	of	of	ADP
ejpam-6690	299	22	lengths	length	NOUN
ejpam-6690	299	23	of	of	ADP
ejpam-6690	299	24	chains	chain	NOUN
ejpam-6690	299	25	p0	p0	PROPN
ejpam-6690	299	26	⊊	⊊	VERB
ejpam-6690	299	27	p1	p1	PROPN
ejpam-6690	299	28	⊊	⊊	NOUN
ejpam-6690	299	29	...	...	PUNCT
ejpam-6690	299	30	⊊	⊊	VERB
ejpam-6690	299	31	pn	pn	NOUN
ejpam-6690	299	32	=	=	SYM
ejpam-6690	299	33	p	p	X
ejpam-6690	299	34	,	,	PUNCT
ejpam-6690	299	35	if	if	SCONJ
ejpam-6690	299	36	this	this	DET
ejpam-6690	299	37	supremum	supremum	ADJ
ejpam-6690	299	38	exists	exist	VERB
ejpam-6690	299	39	,	,	PUNCT
ejpam-6690	299	40	and	and	CCONJ
ejpam-6690	299	41	∞	∞	NUM
ejpam-6690	299	42	otherwise	otherwise	ADV
ejpam-6690	299	43	.	.	PUNCT
ejpam-6690	300	1	the	the	DET
ejpam-6690	300	2	dimension	dimension	NOUN
ejpam-6690	300	3	of	of	ADP
ejpam-6690	300	4	r	r	NOUN
ejpam-6690	300	5	,	,	PUNCT
ejpam-6690	300	6	denoted	denote	VERB
ejpam-6690	300	7	by	by	ADP
ejpam-6690	300	8	dim(r	dim(r	PROPN
ejpam-6690	300	9	)	)	PUNCT
ejpam-6690	300	10	and	and	CCONJ
ejpam-6690	300	11	dim(r	dim(r	PROPN
ejpam-6690	300	12	)	)	PUNCT
ejpam-6690	301	1	=	=	SYM
ejpam-6690	301	2	sup{h(p	sup{h(p	PROPN
ejpam-6690	301	3	)	)	PUNCT
ejpam-6690	301	4	;	;	PUNCT
ejpam-6690	301	5	p	p	PROPN
ejpam-6690	301	6	∈	∈	PROPN
ejpam-6690	301	7	spec(r	spec(r	PROPN
ejpam-6690	301	8	)	)	PUNCT
ejpam-6690	301	9	}	}	PUNCT
ejpam-6690	301	10	.	.	PUNCT
ejpam-6690	302	1	it	it	PRON
ejpam-6690	302	2	is	be	AUX
ejpam-6690	302	3	clear	clear	ADJ
ejpam-6690	302	4	that	that	SCONJ
ejpam-6690	302	5	if	if	SCONJ
ejpam-6690	302	6	r	r	NOUN
ejpam-6690	302	7	is	be	AUX
ejpam-6690	302	8	a	a	DET
ejpam-6690	302	9	hyperfield	hyperfield	NOUN
ejpam-6690	302	10	then	then	ADV
ejpam-6690	302	11	dim(r	dim(r	PROPN
ejpam-6690	302	12	)	)	PUNCT
ejpam-6690	303	1	=	=	SYM
ejpam-6690	303	2	0	0	X
ejpam-6690	303	3	.	.	PUNCT
ejpam-6690	304	1	clearly	clearly	ADV
ejpam-6690	304	2	,	,	PUNCT
ejpam-6690	304	3	spec(r	spec(r	PROPN
ejpam-6690	304	4	)	)	PUNCT
ejpam-6690	304	5	is	be	AUX
ejpam-6690	304	6	a	a	DET
ejpam-6690	304	7	t1	t1	NOUN
ejpam-6690	304	8	-	-	PUNCT
ejpam-6690	304	9	space	space	NOUN
ejpam-6690	304	10	if	if	SCONJ
ejpam-6690	304	11	and	and	CCONJ
ejpam-6690	304	12	only	only	ADV
ejpam-6690	304	13	if	if	SCONJ
ejpam-6690	304	14	dim(r	dim(r	PROPN
ejpam-6690	304	15	)	)	PUNCT
ejpam-6690	304	16	=	=	PUNCT
ejpam-6690	305	1	0	0	X
ejpam-6690	305	2	.	.	PUNCT
ejpam-6690	306	1	because	because	SCONJ
ejpam-6690	306	2	,	,	PUNCT
ejpam-6690	306	3	if	if	SCONJ
ejpam-6690	306	4	dim(r	dim(r	PROPN
ejpam-6690	306	5	)	)	PUNCT
ejpam-6690	306	6	̸=	̸=	PROPN
ejpam-6690	306	7	0	0	NUM
ejpam-6690	306	8	,	,	PUNCT
ejpam-6690	306	9	then	then	ADV
ejpam-6690	306	10	there	there	PRON
ejpam-6690	306	11	are	be	VERB
ejpam-6690	306	12	p1	p1	NOUN
ejpam-6690	306	13	,	,	PUNCT
ejpam-6690	306	14	p2	p2	PROPN
ejpam-6690	306	15	∈	∈	PROPN
ejpam-6690	306	16	spec(r	spec(r	PROPN
ejpam-6690	306	17	)	)	PUNCT
ejpam-6690	306	18	such	such	ADJ
ejpam-6690	306	19	that	that	SCONJ
ejpam-6690	306	20	p1	p1	PROPN
ejpam-6690	306	21	⊊	⊊	VERB
ejpam-6690	306	22	p2	p2	NOUN
ejpam-6690	306	23	.	.	PUNCT
ejpam-6690	307	1	so	so	ADV
ejpam-6690	307	2	every	every	DET
ejpam-6690	307	3	neighborhood	neighborhood	NOUN
ejpam-6690	307	4	containing	contain	VERB
ejpam-6690	307	5	p2	p2	NOUN
ejpam-6690	307	6	is	be	AUX
ejpam-6690	307	7	contain	contain	NOUN
ejpam-6690	307	8	p1	p1	NOUN
ejpam-6690	307	9	,	,	PUNCT
ejpam-6690	307	10	and	and	CCONJ
ejpam-6690	307	11	it	it	PRON
ejpam-6690	307	12	is	be	AUX
ejpam-6690	307	13	contradiction	contradiction	NOUN
ejpam-6690	307	14	.	.	PUNCT
ejpam-6690	308	1	now	now	ADV
ejpam-6690	308	2	suppose	suppose	VERB
ejpam-6690	308	3	that	that	SCONJ
ejpam-6690	308	4	dim(r	dim(r	NOUN
ejpam-6690	308	5	)	)	PUNCT
ejpam-6690	308	6	=	=	PUNCT
ejpam-6690	309	1	0	0	X
ejpam-6690	309	2	.	.	PUNCT
ejpam-6690	310	1	so	so	ADV
ejpam-6690	310	2	there	there	PRON
ejpam-6690	310	3	are	be	VERB
ejpam-6690	310	4	p1	p1	NOUN
ejpam-6690	310	5	,	,	PUNCT
ejpam-6690	310	6	p2	p2	PROPN
ejpam-6690	310	7	∈	∈	PROPN
ejpam-6690	310	8	spec(r	spec(r	PROPN
ejpam-6690	310	9	)	)	PUNCT
ejpam-6690	310	10	such	such	ADJ
ejpam-6690	310	11	that	that	SCONJ
ejpam-6690	310	12	x	x	SYM
ejpam-6690	310	13	∈	∈	PROPN
ejpam-6690	310	14	p1	p1	NOUN
ejpam-6690	310	15	−	−	PROPN
ejpam-6690	310	16	p2	p2	PROPN
ejpam-6690	310	17	and	and	CCONJ
ejpam-6690	310	18	y	y	PROPN
ejpam-6690	310	19	∈	∈	PROPN
ejpam-6690	310	20	p2	p2	PROPN
ejpam-6690	310	21	−	−	PROPN
ejpam-6690	310	22	p1	p1	PROPN
ejpam-6690	310	23	.	.	PUNCT
ejpam-6690	311	1	therefore	therefore	ADV
ejpam-6690	311	2	,	,	PUNCT
ejpam-6690	311	3	(	(	PUNCT
ejpam-6690	311	4	x	x	X
ejpam-6690	311	5	)	)	PUNCT
ejpam-6690	311	6	⊆	⊆	NUM
ejpam-6690	311	7	p1	p1	NOUN
ejpam-6690	311	8	,	,	PUNCT
ejpam-6690	311	9	(	(	PUNCT
ejpam-6690	311	10	x	x	X
ejpam-6690	311	11	)	)	PUNCT
ejpam-6690	311	12	⊈	⊈	PROPN
ejpam-6690	311	13	p2	p2	PROPN
ejpam-6690	311	14	and	and	CCONJ
ejpam-6690	311	15	(	(	PUNCT
ejpam-6690	311	16	y	y	NOUN
ejpam-6690	311	17	)	)	PUNCT
ejpam-6690	311	18	⊆	⊆	NUM
ejpam-6690	311	19	p2	p2	NOUN
ejpam-6690	311	20	,	,	PUNCT
ejpam-6690	311	21	(	(	PUNCT
ejpam-6690	311	22	y	y	NOUN
ejpam-6690	311	23	)	)	PUNCT
ejpam-6690	311	24	⊈	⊈	PROPN
ejpam-6690	311	25	p1	p1	NOUN
ejpam-6690	311	26	.	.	PUNCT
ejpam-6690	312	1	so	so	ADV
ejpam-6690	312	2	p1	p1	PROPN
ejpam-6690	312	3	∈	∈	PROPN
ejpam-6690	312	4	spec(r	spec(r	PROPN
ejpam-6690	312	5	)	)	PUNCT
ejpam-6690	312	6	−	−	PROPN
ejpam-6690	312	7	v	v	NOUN
ejpam-6690	312	8	(	(	PUNCT
ejpam-6690	312	9	(	(	PUNCT
ejpam-6690	312	10	y	y	NOUN
ejpam-6690	312	11	)	)	PUNCT
ejpam-6690	312	12	)	)	PUNCT
ejpam-6690	312	13	,	,	PUNCT
ejpam-6690	312	14	p1	p1	PROPN
ejpam-6690	312	15	/∈	/∈	PUNCT
ejpam-6690	313	1	spec(r	spec(r	ADP
ejpam-6690	313	2	)	)	PUNCT
ejpam-6690	313	3	−	−	PROPN
ejpam-6690	313	4	v	v	NOUN
ejpam-6690	313	5	(	(	PUNCT
ejpam-6690	313	6	(	(	PUNCT
ejpam-6690	313	7	x	x	NOUN
ejpam-6690	313	8	)	)	PUNCT
ejpam-6690	313	9	)	)	PUNCT
ejpam-6690	313	10	and	and	CCONJ
ejpam-6690	313	11	p2	p2	PROPN
ejpam-6690	313	12	∈	∈	PROPN
ejpam-6690	313	13	spec(r	spec(r	PROPN
ejpam-6690	313	14	)	)	PUNCT
ejpam-6690	313	15	−	−	PROPN
ejpam-6690	313	16	v	v	NOUN
ejpam-6690	313	17	(	(	PUNCT
ejpam-6690	313	18	(	(	PUNCT
ejpam-6690	313	19	x	x	NOUN
ejpam-6690	313	20	)	)	PUNCT
ejpam-6690	313	21	)	)	PUNCT
ejpam-6690	313	22	,	,	PUNCT
ejpam-6690	313	23	p2	p2	PROPN
ejpam-6690	313	24	/∈	/∈	PUNCT
ejpam-6690	314	1	spec(r	spec(r	ADP
ejpam-6690	314	2	)	)	PUNCT
ejpam-6690	314	3	−	−	PROPN
ejpam-6690	314	4	v	v	NOUN
ejpam-6690	314	5	(	(	PUNCT
ejpam-6690	314	6	(	(	PUNCT
ejpam-6690	314	7	y	y	NOUN
ejpam-6690	314	8	)	)	PUNCT
ejpam-6690	314	9	)	)	PUNCT
ejpam-6690	314	10	.	.	PUNCT
ejpam-6690	315	1	in	in	ADP
ejpam-6690	315	2	addition	addition	NOUN
ejpam-6690	315	3	,	,	PUNCT
ejpam-6690	315	4	we	we	PRON
ejpam-6690	315	5	can	can	AUX
ejpam-6690	315	6	also	also	ADV
ejpam-6690	315	7	state	state	VERB
ejpam-6690	315	8	the	the	DET
ejpam-6690	315	9	following	follow	VERB
ejpam-6690	315	10	theorem	theorem	PROPN
ejpam-6690	315	11	.	.	PUNCT
ejpam-6690	315	12	theorem	theorem	NOUN
ejpam-6690	315	13	9	9	NUM
ejpam-6690	315	14	.	.	PUNCT
ejpam-6690	316	1	spec(r	spec(r	NOUN
ejpam-6690	316	2	)	)	PUNCT
ejpam-6690	316	3	is	be	AUX
ejpam-6690	316	4	a	a	DET
ejpam-6690	316	5	hausdorff	hausdorff	NOUN
ejpam-6690	316	6	space	space	NOUN
ejpam-6690	316	7	if	if	SCONJ
ejpam-6690	316	8	and	and	CCONJ
ejpam-6690	316	9	only	only	ADV
ejpam-6690	316	10	if	if	SCONJ
ejpam-6690	316	11	dim(r	dim(r	PROPN
ejpam-6690	316	12	)	)	PUNCT
ejpam-6690	317	1	=	=	SYM
ejpam-6690	317	2	0	0	X
ejpam-6690	317	3	.	.	PUNCT
ejpam-6690	318	1	proof	proof	NOUN
ejpam-6690	318	2	.	.	PUNCT
ejpam-6690	319	1	every	every	DET
ejpam-6690	319	2	hausdorff	hausdorff	NOUN
ejpam-6690	319	3	space	space	NOUN
ejpam-6690	319	4	(	(	PUNCT
ejpam-6690	319	5	t2	t2	NOUN
ejpam-6690	319	6	-	-	PUNCT
ejpam-6690	319	7	space	space	NOUN
ejpam-6690	319	8	)	)	PUNCT
ejpam-6690	319	9	,	,	PUNCT
ejpam-6690	319	10	is	be	AUX
ejpam-6690	319	11	a	a	DET
ejpam-6690	319	12	t1	t1	NOUN
ejpam-6690	319	13	-	-	PUNCT
ejpam-6690	319	14	space	space	NOUN
ejpam-6690	319	15	.	.	PUNCT
ejpam-6690	320	1	so	so	ADV
ejpam-6690	320	2	assume	assume	VERB
ejpam-6690	320	3	that	that	SCONJ
ejpam-6690	320	4	dimr	dimr	NOUN
ejpam-6690	320	5	=	=	SYM
ejpam-6690	320	6	0	0	NUM
ejpam-6690	320	7	,	,	PUNCT
ejpam-6690	320	8	and	and	CCONJ
ejpam-6690	320	9	p	p	X
ejpam-6690	320	10	,	,	PUNCT
ejpam-6690	320	11	q	q	PROPN
ejpam-6690	320	12	∈	∈	PROPN
ejpam-6690	320	13	spec(r	spec(r	PROPN
ejpam-6690	320	14	)	)	PUNCT
ejpam-6690	320	15	and	and	CCONJ
ejpam-6690	320	16	x	x	PUNCT
ejpam-6690	320	17	∈	∈	PROPN
ejpam-6690	321	1	p	p	NOUN
ejpam-6690	321	2	−	−	PROPN
ejpam-6690	321	3	q.	q.	NOUN
ejpam-6690	321	4	we	we	PRON
ejpam-6690	321	5	know	know	VERB
ejpam-6690	321	6	that	that	SCONJ
ejpam-6690	322	1	pp	pp	ADV
ejpam-6690	322	2	is	be	AUX
ejpam-6690	322	3	maximal	maximal	ADJ
ejpam-6690	322	4	hyperideal	hyperideal	NOUN
ejpam-6690	322	5	of	of	ADP
ejpam-6690	322	6	local	local	ADJ
ejpam-6690	322	7	hyperring	hyperring	NOUN
ejpam-6690	322	8	rp	rp	NOUN
ejpam-6690	322	9	,	,	PUNCT
ejpam-6690	322	10	and	and	CCONJ
ejpam-6690	322	11	pp	pp	ADV
ejpam-6690	322	12	=	=	PUNCT
ejpam-6690	322	13	nil(rp	nil(rp	NUM
ejpam-6690	322	14	)	)	PUNCT
ejpam-6690	322	15	.	.	PUNCT
ejpam-6690	323	1	let	let	VERB
ejpam-6690	323	2	x	x	SYM
ejpam-6690	323	3	1	1	NUM
ejpam-6690	323	4	∈	∈	NOUN
ejpam-6690	323	5	pp	pp	ADP
ejpam-6690	323	6	=	=	PUNCT
ejpam-6690	323	7	nil(rp	nil(rp	PROPN
ejpam-6690	323	8	)	)	PUNCT
ejpam-6690	323	9	.	.	PUNCT
ejpam-6690	324	1	then	then	ADV
ejpam-6690	324	2	there	there	PRON
ejpam-6690	324	3	is	be	VERB
ejpam-6690	324	4	n	n	DET
ejpam-6690	324	5	∈	∈	PROPN
ejpam-6690	324	6	n	n	PRON
ejpam-6690	324	7	such	such	ADJ
ejpam-6690	324	8	that	that	SCONJ
ejpam-6690	324	9	xn	xn	PROPN
ejpam-6690	324	10	1	1	NUM
ejpam-6690	324	11	=	=	NOUN
ejpam-6690	324	12	0rp	0rp	NOUN
ejpam-6690	324	13	.	.	PUNCT
ejpam-6690	325	1	hence	hence	ADV
ejpam-6690	325	2	for	for	ADP
ejpam-6690	325	3	some	some	DET
ejpam-6690	325	4	s	s	X
ejpam-6690	325	5	∈	∈	PROPN
ejpam-6690	325	6	r−	r−	PROPN
ejpam-6690	325	7	p	p	NOUN
ejpam-6690	325	8	we	we	PRON
ejpam-6690	325	9	have	have	VERB
ejpam-6690	325	10	sxn	sxn	NOUN
ejpam-6690	325	11	=	=	NOUN
ejpam-6690	325	12	0	0	X
ejpam-6690	325	13	.	.	PUNCT
ejpam-6690	326	1	since	since	SCONJ
ejpam-6690	326	2	x	x	PROPN
ejpam-6690	326	3	/∈	/∈	NOUN
ejpam-6690	326	4	q	q	NOUN
ejpam-6690	326	5	and	and	CCONJ
ejpam-6690	326	6	s	s	VERB
ejpam-6690	326	7	/∈	/∈	PUNCT
ejpam-6690	327	1	p	p	X
ejpam-6690	327	2	then	then	ADV
ejpam-6690	327	3	q	q	PROPN
ejpam-6690	327	4	∈	∈	PROPN
ejpam-6690	327	5	spec(r)−v	spec(r)−v	PROPN
ejpam-6690	327	6	(	(	PUNCT
ejpam-6690	327	7	(	(	PUNCT
ejpam-6690	327	8	x	x	NOUN
ejpam-6690	327	9	)	)	PUNCT
ejpam-6690	327	10	)	)	PUNCT
ejpam-6690	328	1	=	=	SYM
ejpam-6690	328	2	w	w	X
ejpam-6690	328	3	(	(	PUNCT
ejpam-6690	328	4	(	(	PUNCT
ejpam-6690	328	5	x	x	NOUN
ejpam-6690	328	6	)	)	PUNCT
ejpam-6690	328	7	)	)	PUNCT
ejpam-6690	328	8	and	and	CCONJ
ejpam-6690	328	9	p	p	PROPN
ejpam-6690	328	10	∈	∈	PROPN
ejpam-6690	328	11	spec(r)−v	spec(r)−v	PROPN
ejpam-6690	328	12	(	(	PUNCT
ejpam-6690	328	13	(	(	PUNCT
ejpam-6690	328	14	s	s	NOUN
ejpam-6690	328	15	)	)	PUNCT
ejpam-6690	328	16	)	)	PUNCT
ejpam-6690	329	1	=	=	SYM
ejpam-6690	329	2	w	w	X
ejpam-6690	329	3	(	(	PUNCT
ejpam-6690	329	4	(	(	PUNCT
ejpam-6690	329	5	s	s	NOUN
ejpam-6690	329	6	)	)	PUNCT
ejpam-6690	329	7	)	)	PUNCT
ejpam-6690	329	8	.	.	PUNCT
ejpam-6690	330	1	also	also	ADV
ejpam-6690	330	2	w	w	PROPN
ejpam-6690	330	3	(	(	PUNCT
ejpam-6690	330	4	(	(	PUNCT
ejpam-6690	330	5	x	x	NOUN
ejpam-6690	330	6	)	)	PUNCT
ejpam-6690	330	7	)	)	PUNCT
ejpam-6690	331	1	∩w	∩w	NOUN
ejpam-6690	331	2	(	(	PUNCT
ejpam-6690	331	3	(	(	PUNCT
ejpam-6690	331	4	s	s	NOUN
ejpam-6690	331	5	)	)	PUNCT
ejpam-6690	331	6	)	)	PUNCT
ejpam-6690	332	1	=	=	PRON
ejpam-6690	332	2	{	{	PUNCT
ejpam-6690	332	3	i	i	PROPN
ejpam-6690	332	4	∈	∈	PROPN
ejpam-6690	332	5	spec(r	spec(r	PROPN
ejpam-6690	332	6	)	)	PUNCT
ejpam-6690	332	7	;	;	PUNCT
ejpam-6690	332	8	s	s	X
ejpam-6690	332	9	/∈	/∈	PUNCT
ejpam-6690	333	1	i	i	PRON
ejpam-6690	333	2	,	,	PUNCT
ejpam-6690	333	3	x	x	PROPN
ejpam-6690	333	4	/∈	/∈	PUNCT
ejpam-6690	334	1	i	i	PRON
ejpam-6690	334	2	}	}	PUNCT
ejpam-6690	334	3	=	=	PUNCT
ejpam-6690	334	4	{	{	PUNCT
ejpam-6690	334	5	i	i	PROPN
ejpam-6690	334	6	∈	∈	PROPN
ejpam-6690	334	7	spec(r	spec(r	PROPN
ejpam-6690	334	8	)	)	PUNCT
ejpam-6690	334	9	;	;	PUNCT
ejpam-6690	334	10	s	s	X
ejpam-6690	334	11	/∈	/∈	PUNCT
ejpam-6690	335	1	i	i	PRON
ejpam-6690	335	2	,	,	PUNCT
ejpam-6690	335	3	xn	xn	PROPN
ejpam-6690	335	4	/∈	/∈	PUNCT
ejpam-6690	336	1	i	i	PRON
ejpam-6690	336	2	}	}	PUNCT
ejpam-6690	336	3	=	=	PUNCT
ejpam-6690	336	4	{	{	PUNCT
ejpam-6690	336	5	i	i	PROPN
ejpam-6690	336	6	∈	∈	PROPN
ejpam-6690	336	7	spec(r	spec(r	PROPN
ejpam-6690	336	8	)	)	PUNCT
ejpam-6690	336	9	;	;	PUNCT
ejpam-6690	336	10	sxn	sxn	X
ejpam-6690	336	11	/∈	/∈	PUNCT
ejpam-6690	337	1	i	i	PRON
ejpam-6690	337	2	}	}	PUNCT
ejpam-6690	337	3	=	=	SYM
ejpam-6690	337	4	spec(r	spec(r	PROPN
ejpam-6690	337	5	)	)	PUNCT
ejpam-6690	337	6	−	−	PROPN
ejpam-6690	337	7	v	v	NOUN
ejpam-6690	337	8	(	(	PUNCT
ejpam-6690	337	9	(	(	PUNCT
ejpam-6690	337	10	sxn	sxn	NOUN
ejpam-6690	337	11	)	)	PUNCT
ejpam-6690	337	12	)	)	PUNCT
ejpam-6690	338	1	=	=	NOUN
ejpam-6690	338	2	∅	∅	NOUN
ejpam-6690	338	3	,	,	PUNCT
ejpam-6690	338	4	since	since	SCONJ
ejpam-6690	338	5	v	v	NOUN
ejpam-6690	338	6	(	(	PUNCT
ejpam-6690	338	7	(	(	PUNCT
ejpam-6690	338	8	sxn	sxn	NOUN
ejpam-6690	338	9	)	)	PUNCT
ejpam-6690	338	10	)	)	PUNCT
ejpam-6690	339	1	=	=	SYM
ejpam-6690	339	2	v	v	X
ejpam-6690	339	3	(	(	PUNCT
ejpam-6690	339	4	(	(	PUNCT
ejpam-6690	339	5	0	0	NUM
ejpam-6690	339	6	)	)	PUNCT
ejpam-6690	339	7	)	)	PUNCT
ejpam-6690	340	1	=	=	SYM
ejpam-6690	340	2	spec(r	spec(r	PROPN
ejpam-6690	340	3	)	)	PUNCT
ejpam-6690	340	4	.	.	PUNCT
ejpam-6690	341	1	b.	b.	PROPN
ejpam-6690	341	2	afshar	afshar	PROPN
ejpam-6690	341	3	,	,	PUNCT
ejpam-6690	341	4	r.	r.	PROPN
ejpam-6690	341	5	ameri	ameri	PROPN
ejpam-6690	341	6	,	,	PUNCT
ejpam-6690	341	7	m.	m.	PROPN
ejpam-6690	341	8	al	al	PROPN
ejpam-6690	341	9	-	-	PUNCT
ejpam-6690	341	10	tahan	tahan	PROPN
ejpam-6690	341	11	/	/	SYM
ejpam-6690	341	12	eur	eur	PROPN
ejpam-6690	341	13	.	.	PUNCT
ejpam-6690	342	1	j.	j.	PROPN
ejpam-6690	342	2	pure	pure	PROPN
ejpam-6690	342	3	appl	appl	PROPN
ejpam-6690	342	4	.	.	PROPN
ejpam-6690	342	5	math	math	PROPN
ejpam-6690	342	6	,	,	PUNCT
ejpam-6690	342	7	18	18	NUM
ejpam-6690	342	8	(	(	PUNCT
ejpam-6690	342	9	4	4	NUM
ejpam-6690	342	10	)	)	PUNCT
ejpam-6690	342	11	(	(	PUNCT
ejpam-6690	342	12	2025	2025	NUM
ejpam-6690	342	13	)	)	PUNCT
ejpam-6690	342	14	,	,	PUNCT
ejpam-6690	342	15	6690	6690	NUM
ejpam-6690	342	16	10	10	NUM
ejpam-6690	342	17	of	of	ADP
ejpam-6690	342	18	19	19	NUM
ejpam-6690	342	19	corollary	corollary	ADJ
ejpam-6690	342	20	4	4	NUM
ejpam-6690	342	21	.	.	PUNCT
ejpam-6690	342	22	spec(p	spec(p	NOUN
ejpam-6690	342	23	)	)	PUNCT
ejpam-6690	342	24	is	be	AUX
ejpam-6690	342	25	t1	t1	NOUN
ejpam-6690	342	26	-	-	PUNCT
ejpam-6690	342	27	space	space	NOUN
ejpam-6690	342	28	if	if	SCONJ
ejpam-6690	342	29	and	and	CCONJ
ejpam-6690	342	30	only	only	ADV
ejpam-6690	342	31	if	if	SCONJ
ejpam-6690	342	32	mspec(r	mspec(r	PROPN
ejpam-6690	342	33	)	)	PUNCT
ejpam-6690	342	34	=	=	SYM
ejpam-6690	342	35	spec(r	spec(r	PROPN
ejpam-6690	342	36	)	)	PUNCT
ejpam-6690	342	37	.	.	PUNCT
ejpam-6690	343	1	proposition	proposition	NOUN
ejpam-6690	343	2	3	3	NUM
ejpam-6690	343	3	.	.	X
ejpam-6690	343	4	for	for	ADP
ejpam-6690	343	5	any	any	DET
ejpam-6690	343	6	hyperring	hyperring	NOUN
ejpam-6690	343	7	r	r	NOUN
ejpam-6690	343	8	,	,	PUNCT
ejpam-6690	343	9	spec(r	spec(r	PROPN
ejpam-6690	343	10	)	)	PUNCT
ejpam-6690	343	11	is	be	AUX
ejpam-6690	343	12	a	a	DET
ejpam-6690	343	13	compact	compact	ADJ
ejpam-6690	343	14	space	space	NOUN
ejpam-6690	343	15	under	under	ADP
ejpam-6690	343	16	the	the	DET
ejpam-6690	343	17	zariski	zariski	NOUN
ejpam-6690	343	18	topology	topology	NOUN
ejpam-6690	343	19	.	.	PUNCT
ejpam-6690	344	1	proof	proof	NOUN
ejpam-6690	344	2	.	.	PUNCT
ejpam-6690	345	1	let	let	VERB
ejpam-6690	345	2	{	{	PUNCT
ejpam-6690	345	3	ij}j∈j	ij}j∈j	PART
ejpam-6690	345	4	be	be	AUX
ejpam-6690	345	5	a	a	DET
ejpam-6690	345	6	family	family	NOUN
ejpam-6690	345	7	of	of	ADP
ejpam-6690	345	8	hyperideals	hyperideal	NOUN
ejpam-6690	345	9	of	of	ADP
ejpam-6690	345	10	r	r	NOUN
ejpam-6690	345	11	and	and	CCONJ
ejpam-6690	345	12	{	{	PUNCT
ejpam-6690	345	13	w	w	NOUN
ejpam-6690	345	14	(	(	PUNCT
ejpam-6690	345	15	ij)}j∈j	ij)}j∈j	NOUN
ejpam-6690	345	16	be	be	VERB
ejpam-6690	345	17	a	a	DET
ejpam-6690	345	18	family	family	NOUN
ejpam-6690	345	19	of	of	ADP
ejpam-6690	345	20	open	open	ADJ
ejpam-6690	345	21	sets	set	NOUN
ejpam-6690	345	22	that	that	SCONJ
ejpam-6690	345	23	spec(r	spec(r	ADP
ejpam-6690	345	24	)	)	PUNCT
ejpam-6690	345	25	=	=	SYM
ejpam-6690	345	26	⋃	⋃	NOUN
ejpam-6690	345	27	j∈j	j∈j	NOUN
ejpam-6690	345	28	w	w	PROPN
ejpam-6690	345	29	(	(	PUNCT
ejpam-6690	345	30	ij	ij	NOUN
ejpam-6690	345	31	)	)	PUNCT
ejpam-6690	345	32	and	and	CCONJ
ejpam-6690	345	33	w	w	PROPN
ejpam-6690	345	34	(	(	PUNCT
ejpam-6690	345	35	ij	ij	NOUN
ejpam-6690	345	36	)	)	PUNCT
ejpam-6690	345	37	=	=	SYM
ejpam-6690	345	38	spec(r	spec(r	PROPN
ejpam-6690	345	39	)	)	PUNCT
ejpam-6690	345	40	−	−	PROPN
ejpam-6690	345	41	v	v	INTJ
ejpam-6690	345	42	(	(	PUNCT
ejpam-6690	345	43	ij	ij	NOUN
ejpam-6690	345	44	)	)	PUNCT
ejpam-6690	345	45	,	,	PUNCT
ejpam-6690	345	46	for	for	ADP
ejpam-6690	345	47	any	any	DET
ejpam-6690	345	48	j	j	PROPN
ejpam-6690	345	49	∈	∈	PROPN
ejpam-6690	345	50	j	j	PROPN
ejpam-6690	345	51	.	.	PUNCT
ejpam-6690	346	1	so	so	ADV
ejpam-6690	346	2	spec(r	spec(r	ADJ
ejpam-6690	346	3	)	)	PUNCT
ejpam-6690	346	4	=	=	PUNCT
ejpam-6690	346	5	⋃	⋃	NOUN
ejpam-6690	346	6	j∈j(spec(r	j∈j(spec(r	NOUN
ejpam-6690	346	7	)	)	PUNCT
ejpam-6690	346	8	−	−	PROPN
ejpam-6690	346	9	v	v	X
ejpam-6690	346	10	(	(	PUNCT
ejpam-6690	346	11	ij	ij	NOUN
ejpam-6690	346	12	)	)	PUNCT
ejpam-6690	346	13	)	)	PUNCT
ejpam-6690	347	1	=	=	SYM
ejpam-6690	347	2	spec(r	spec(r	PROPN
ejpam-6690	347	3	)	)	PUNCT
ejpam-6690	347	4	−	−	PROPN
ejpam-6690	348	1	⋂	⋂	PROPN
ejpam-6690	348	2	j∈j	j∈j	NOUN
ejpam-6690	348	3	v	v	NOUN
ejpam-6690	348	4	(	(	PUNCT
ejpam-6690	348	5	ij	ij	NOUN
ejpam-6690	348	6	)	)	PUNCT
ejpam-6690	348	7	=	=	SYM
ejpam-6690	348	8	spec(r	spec(r	PROPN
ejpam-6690	348	9	)	)	PUNCT
ejpam-6690	348	10	−	−	PROPN
ejpam-6690	348	11	v	v	NOUN
ejpam-6690	348	12	(	(	PUNCT
ejpam-6690	348	13	∑	∑	INTJ
ejpam-6690	348	14	j∈j	j∈j	NOUN
ejpam-6690	348	15	ij	ij	NOUN
ejpam-6690	348	16	)	)	PUNCT
ejpam-6690	348	17	.	.	PUNCT
ejpam-6690	349	1	hence	hence	ADV
ejpam-6690	349	2	v	v	NOUN
ejpam-6690	349	3	(	(	PUNCT
ejpam-6690	349	4	∑	∑	INTJ
ejpam-6690	349	5	j∈j	j∈j	NOUN
ejpam-6690	349	6	ij	ij	NOUN
ejpam-6690	349	7	)	)	PUNCT
ejpam-6690	349	8	=	=	NOUN
ejpam-6690	349	9	∅	∅	NOUN
ejpam-6690	349	10	and	and	CCONJ
ejpam-6690	349	11	∑	∑	ADP
ejpam-6690	349	12	j∈j	j∈j	NOUN
ejpam-6690	349	13	ij	ij	NOUN
ejpam-6690	349	14	=	=	PUNCT
ejpam-6690	349	15	r.	r.	PROPN
ejpam-6690	350	1	so	so	ADV
ejpam-6690	350	2	1	1	NUM
ejpam-6690	350	3	∈	∈	PROPN
ejpam-6690	350	4	∑	∑	PUNCT
ejpam-6690	350	5	k∈k	k∈k	PROPN
ejpam-6690	350	6	akxk	akxk	PROPN
ejpam-6690	350	7	where	where	SCONJ
ejpam-6690	350	8	ak	ak	PROPN
ejpam-6690	350	9	∈	∈	PROPN
ejpam-6690	350	10	r	r	PROPN
ejpam-6690	350	11	,	,	PUNCT
ejpam-6690	350	12	xk	xk	PROPN
ejpam-6690	350	13	∈	∈	PROPN
ejpam-6690	350	14	ik	ik	PROPN
ejpam-6690	350	15	and	and	CCONJ
ejpam-6690	350	16	k	k	PROPN
ejpam-6690	350	17	is	be	AUX
ejpam-6690	350	18	a	a	DET
ejpam-6690	350	19	finite	finite	NOUN
ejpam-6690	350	20	subset	subset	NOUN
ejpam-6690	350	21	of	of	ADP
ejpam-6690	350	22	j	j	PROPN
ejpam-6690	350	23	.	.	PUNCT
ejpam-6690	351	1	therefore	therefore	ADV
ejpam-6690	351	2	,	,	PUNCT
ejpam-6690	351	3	∑	∑	ADP
ejpam-6690	351	4	k∈k	k∈k	NOUN
ejpam-6690	351	5	ik	ik	PROPN
ejpam-6690	351	6	=	=	SYM
ejpam-6690	351	7	r	r	NOUN
ejpam-6690	351	8	and	and	CCONJ
ejpam-6690	351	9	v	v	NOUN
ejpam-6690	351	10	(	(	PUNCT
ejpam-6690	351	11	∑	∑	INTJ
ejpam-6690	351	12	k∈k	k∈k	NOUN
ejpam-6690	351	13	ik	ik	NOUN
ejpam-6690	351	14	)	)	PUNCT
ejpam-6690	351	15	=	=	PUNCT
ejpam-6690	351	16	∅.	∅.	VERB
ejpam-6690	351	17	so	so	ADV
ejpam-6690	351	18	spec(r	spec(r	ADJ
ejpam-6690	351	19	)	)	PUNCT
ejpam-6690	351	20	=	=	SYM
ejpam-6690	351	21	⋃	⋃	NOUN
ejpam-6690	351	22	k∈k	k∈k	NOUN
ejpam-6690	351	23	w	w	PROPN
ejpam-6690	351	24	(	(	PUNCT
ejpam-6690	351	25	ik	ik	PROPN
ejpam-6690	351	26	)	)	PUNCT
ejpam-6690	351	27	.	.	PUNCT
ejpam-6690	352	1	proposition	proposition	NOUN
ejpam-6690	352	2	4	4	NUM
ejpam-6690	352	3	.	.	PUNCT
ejpam-6690	353	1	let	let	VERB
ejpam-6690	353	2	r	r	PRON
ejpam-6690	353	3	be	be	AUX
ejpam-6690	353	4	a	a	DET
ejpam-6690	353	5	hyperring	hyperring	NOUN
ejpam-6690	353	6	and	and	CCONJ
ejpam-6690	353	7	x	x	SYM
ejpam-6690	353	8	∈	∈	PROPN
ejpam-6690	353	9	r.	r.	PROPN
ejpam-6690	353	10	then	then	ADV
ejpam-6690	353	11	w	w	PROPN
ejpam-6690	353	12	(	(	PUNCT
ejpam-6690	353	13	x	x	X
ejpam-6690	353	14	)	)	PUNCT
ejpam-6690	353	15	is	be	AUX
ejpam-6690	353	16	a	a	DET
ejpam-6690	353	17	compact	compact	ADJ
ejpam-6690	353	18	subset	subset	NOUN
ejpam-6690	353	19	of	of	ADP
ejpam-6690	353	20	spec(r	spec(r	PROPN
ejpam-6690	353	21	)	)	PUNCT
ejpam-6690	353	22	.	.	PUNCT
ejpam-6690	354	1	also	also	ADV
ejpam-6690	354	2	an	an	DET
ejpam-6690	354	3	open	open	ADJ
ejpam-6690	354	4	subset	subset	NOUN
ejpam-6690	354	5	of	of	ADP
ejpam-6690	354	6	spec(r	spec(r	PROPN
ejpam-6690	354	7	)	)	PUNCT
ejpam-6690	354	8	is	be	AUX
ejpam-6690	354	9	compact	compact	ADJ
ejpam-6690	354	10	if	if	SCONJ
ejpam-6690	355	1	and	and	CCONJ
ejpam-6690	355	2	only	only	ADV
ejpam-6690	355	3	if	if	SCONJ
ejpam-6690	355	4	it	it	PRON
ejpam-6690	355	5	is	be	AUX
ejpam-6690	355	6	a	a	DET
ejpam-6690	355	7	finite	finite	ADJ
ejpam-6690	355	8	union	union	NOUN
ejpam-6690	355	9	of	of	ADP
ejpam-6690	355	10	sets	set	NOUN
ejpam-6690	355	11	w	w	PROPN
ejpam-6690	355	12	(	(	PUNCT
ejpam-6690	355	13	xi	xi	PROPN
ejpam-6690	355	14	)	)	PUNCT
ejpam-6690	355	15	,	,	PUNCT
ejpam-6690	355	16	where	where	SCONJ
ejpam-6690	355	17	xi	xi	PROPN
ejpam-6690	355	18	∈	∈	PROPN
ejpam-6690	355	19	r.	r.	PROPN
ejpam-6690	355	20	proof	proof	NOUN
ejpam-6690	355	21	.	.	PUNCT
ejpam-6690	356	1	consider	consider	VERB
ejpam-6690	356	2	w	w	NOUN
ejpam-6690	356	3	(	(	PUNCT
ejpam-6690	356	4	x	x	NOUN
ejpam-6690	356	5	)	)	PUNCT
ejpam-6690	356	6	=	=	PUNCT
ejpam-6690	356	7	⋃	⋃	ADP
ejpam-6690	356	8	i∈i	i∈i	ADJ
ejpam-6690	356	9	w	w	PROPN
ejpam-6690	356	10	(	(	PUNCT
ejpam-6690	356	11	xi	xi	PROPN
ejpam-6690	356	12	)	)	PUNCT
ejpam-6690	356	13	.	.	PUNCT
ejpam-6690	357	1	for	for	ADP
ejpam-6690	357	2	every	every	DET
ejpam-6690	357	3	p	p	PROPN
ejpam-6690	357	4	∈	∈	PROPN
ejpam-6690	357	5	spec(r	spec(r	PROPN
ejpam-6690	357	6	)	)	PUNCT
ejpam-6690	357	7	,	,	PUNCT
ejpam-6690	357	8	if	if	SCONJ
ejpam-6690	357	9	x	x	PROPN
ejpam-6690	357	10	/∈	/∈	PUNCT
ejpam-6690	358	1	p	p	X
ejpam-6690	358	2	then	then	ADV
ejpam-6690	358	3	there	there	PRON
ejpam-6690	358	4	is	be	VERB
ejpam-6690	358	5	i	i	PRON
ejpam-6690	358	6	∈	∈	PROPN
ejpam-6690	358	7	i	i	PRON
ejpam-6690	358	8	such	such	VERB
ejpam-6690	358	9	that	that	PRON
ejpam-6690	358	10	xi	xi	PROPN
ejpam-6690	358	11	/∈	/∈	PUNCT
ejpam-6690	359	1	p	p	X
ejpam-6690	359	2	.	.	PUNCT
ejpam-6690	360	1	let	let	VERB
ejpam-6690	360	2	k	k	NOUN
ejpam-6690	360	3	=	=	PUNCT
ejpam-6690	360	4	∑	∑	PUNCT
ejpam-6690	360	5	i∈i(xi	i∈i(xi	PROPN
ejpam-6690	360	6	)	)	PUNCT
ejpam-6690	360	7	.	.	PUNCT
ejpam-6690	361	1	since	since	SCONJ
ejpam-6690	361	2	spec(r	spec(r	PROPN
ejpam-6690	361	3	)	)	PUNCT
ejpam-6690	361	4	−	−	PROPN
ejpam-6690	361	5	v	v	X
ejpam-6690	361	6	(	(	PUNCT
ejpam-6690	361	7	x	x	NOUN
ejpam-6690	361	8	)	)	PUNCT
ejpam-6690	361	9	=	=	SYM
ejpam-6690	361	10	spec(r	spec(r	PROPN
ejpam-6690	361	11	)	)	PUNCT
ejpam-6690	361	12	−	−	PROPN
ejpam-6690	361	13	v	v	X
ejpam-6690	361	14	(	(	PUNCT
ejpam-6690	361	15	k	k	NOUN
ejpam-6690	361	16	)	)	PUNCT
ejpam-6690	361	17	,	,	PUNCT
ejpam-6690	361	18	then	then	ADV
ejpam-6690	361	19	for	for	ADP
ejpam-6690	361	20	every	every	DET
ejpam-6690	361	21	prime	prime	ADJ
ejpam-6690	361	22	hyperideal	hyperideal	NOUN
ejpam-6690	361	23	p	p	NOUN
ejpam-6690	361	24	,	,	PUNCT
ejpam-6690	361	25	that	that	SCONJ
ejpam-6690	361	26	k	k	PROPN
ejpam-6690	361	27	⊆	⊆	NUM
ejpam-6690	361	28	p	p	NOUN
ejpam-6690	361	29	,	,	PUNCT
ejpam-6690	361	30	we	we	PRON
ejpam-6690	361	31	have	have	VERB
ejpam-6690	361	32	x	x	X
ejpam-6690	361	33	∈	∈	PROPN
ejpam-6690	361	34	p	p	NOUN
ejpam-6690	361	35	.	.	PUNCT
ejpam-6690	362	1	so	so	ADV
ejpam-6690	362	2	x	x	SYM
ejpam-6690	362	3	∈	∈	PROPN
ejpam-6690	362	4	√	√	NUM
ejpam-6690	363	1	k	k	NOUN
ejpam-6690	363	2	and	and	CCONJ
ejpam-6690	363	3	xm	xm	PROPN
ejpam-6690	363	4	∈	∈	PROPN
ejpam-6690	364	1	k	k	PROPN
ejpam-6690	364	2	,	,	PUNCT
ejpam-6690	364	3	for	for	ADP
ejpam-6690	364	4	some	some	DET
ejpam-6690	364	5	m	m	NOUN
ejpam-6690	364	6	∈	∈	NOUN
ejpam-6690	364	7	n.	n.	NOUN
ejpam-6690	364	8	hence	hence	ADV
ejpam-6690	364	9	for	for	ADP
ejpam-6690	364	10	some	some	DET
ejpam-6690	364	11	x1	x1	PROPN
ejpam-6690	364	12	,	,	PUNCT
ejpam-6690	364	13	x2	x2	PROPN
ejpam-6690	364	14	,	,	PUNCT
ejpam-6690	364	15	...	...	PUNCT
ejpam-6690	364	16	,	,	PUNCT
ejpam-6690	364	17	xn	xn	PROPN
ejpam-6690	364	18	and	and	CCONJ
ejpam-6690	364	19	ri	ri	PROPN
ejpam-6690	364	20	∈	∈	PROPN
ejpam-6690	364	21	r(1	r(1	PROPN
ejpam-6690	364	22	≤	≤	NOUN
ejpam-6690	365	1	i	i	PRON
ejpam-6690	365	2	≤	≤	PROPN
ejpam-6690	365	3	n	n	CCONJ
ejpam-6690	365	4	)	)	PUNCT
ejpam-6690	365	5	,	,	PUNCT
ejpam-6690	365	6	we	we	PRON
ejpam-6690	365	7	have	have	VERB
ejpam-6690	365	8	xm	xm	NOUN
ejpam-6690	365	9	=	=	SYM
ejpam-6690	366	1	∑n	∑n	PROPN
ejpam-6690	366	2	i=1	i=1	PROPN
ejpam-6690	366	3	rixi	rixi	NOUN
ejpam-6690	366	4	.	.	PUNCT
ejpam-6690	367	1	if	if	SCONJ
ejpam-6690	367	2	p	p	PROPN
ejpam-6690	367	3	∈	∈	PROPN
ejpam-6690	367	4	spec(r	spec(r	PROPN
ejpam-6690	367	5	)	)	PUNCT
ejpam-6690	367	6	and	and	CCONJ
ejpam-6690	367	7	{	{	PUNCT
ejpam-6690	367	8	x1	x1	PROPN
ejpam-6690	367	9	,	,	PUNCT
ejpam-6690	367	10	x2	x2	PROPN
ejpam-6690	367	11	,	,	PUNCT
ejpam-6690	367	12	...	...	PUNCT
ejpam-6690	367	13	,	,	PUNCT
ejpam-6690	367	14	xn	xn	X
ejpam-6690	367	15	}	}	PUNCT
ejpam-6690	367	16	⊆	⊆	NUM
ejpam-6690	367	17	p	p	NOUN
ejpam-6690	367	18	,	,	PUNCT
ejpam-6690	367	19	then	then	ADV
ejpam-6690	367	20	xm	xm	PROPN
ejpam-6690	367	21	∈	∈	PROPN
ejpam-6690	368	1	p	p	PROPN
ejpam-6690	369	1	and	and	CCONJ
ejpam-6690	369	2	so	so	ADV
ejpam-6690	369	3	x	x	SYM
ejpam-6690	369	4	∈	∈	PROPN
ejpam-6690	369	5	p	p	NOUN
ejpam-6690	369	6	.	.	PUNCT
ejpam-6690	370	1	also	also	ADV
ejpam-6690	370	2	if	if	SCONJ
ejpam-6690	370	3	x	x	X
ejpam-6690	370	4	/∈	/∈	PUNCT
ejpam-6690	371	1	p	p	X
ejpam-6690	372	1	then	then	ADV
ejpam-6690	372	2	xi	xi	INTJ
ejpam-6690	372	3	/∈	/∈	PUNCT
ejpam-6690	373	1	p	p	X
ejpam-6690	373	2	,	,	PUNCT
ejpam-6690	373	3	for	for	ADP
ejpam-6690	373	4	some	some	DET
ejpam-6690	373	5	i	i	PRON
ejpam-6690	373	6	∈	∈	PROPN
ejpam-6690	373	7	{	{	PUNCT
ejpam-6690	373	8	1	1	NUM
ejpam-6690	373	9	,	,	PUNCT
ejpam-6690	373	10	2	2	NUM
ejpam-6690	373	11	,	,	PUNCT
ejpam-6690	373	12	...	...	PUNCT
ejpam-6690	373	13	,	,	PUNCT
ejpam-6690	373	14	n	n	CCONJ
ejpam-6690	373	15	}	}	PUNCT
ejpam-6690	373	16	.	.	PUNCT
ejpam-6690	374	1	therefore	therefore	ADV
ejpam-6690	374	2	,	,	PUNCT
ejpam-6690	374	3	w	w	PROPN
ejpam-6690	374	4	(	(	PUNCT
ejpam-6690	374	5	x	x	NOUN
ejpam-6690	374	6	)	)	PUNCT
ejpam-6690	374	7	=	=	SYM
ejpam-6690	374	8	⋃n	⋃n	NOUN
ejpam-6690	374	9	i=1w	i=1w	NOUN
ejpam-6690	374	10	(	(	PUNCT
ejpam-6690	374	11	xi	xi	NOUN
ejpam-6690	374	12	)	)	PUNCT
ejpam-6690	374	13	.	.	PUNCT
ejpam-6690	375	1	for	for	ADP
ejpam-6690	375	2	the	the	DET
ejpam-6690	375	3	last	last	ADJ
ejpam-6690	375	4	part	part	NOUN
ejpam-6690	375	5	,	,	PUNCT
ejpam-6690	375	6	let	let	VERB
ejpam-6690	375	7	x	x	PRON
ejpam-6690	375	8	be	be	AUX
ejpam-6690	375	9	a	a	DET
ejpam-6690	375	10	compact	compact	ADJ
ejpam-6690	375	11	and	and	CCONJ
ejpam-6690	375	12	open	open	ADJ
ejpam-6690	375	13	subset	subset	NOUN
ejpam-6690	375	14	of	of	ADP
ejpam-6690	375	15	spec(r	spec(r	PROPN
ejpam-6690	375	16	)	)	PUNCT
ejpam-6690	375	17	and	and	CCONJ
ejpam-6690	375	18	consider	consider	VERB
ejpam-6690	375	19	x	x	NOUN
ejpam-6690	375	20	=	=	SYM
ejpam-6690	375	21	⋃	⋃	VERB
ejpam-6690	375	22	i∈i	i∈i	ADJ
ejpam-6690	375	23	w	w	PROPN
ejpam-6690	375	24	(	(	PUNCT
ejpam-6690	375	25	xi	xi	PROPN
ejpam-6690	375	26	)	)	PUNCT
ejpam-6690	375	27	.	.	PUNCT
ejpam-6690	376	1	then	then	ADV
ejpam-6690	376	2	there	there	PRON
ejpam-6690	376	3	is	be	VERB
ejpam-6690	376	4	a	a	DET
ejpam-6690	376	5	finite	finite	ADJ
ejpam-6690	376	6	subcover	subcover	NOUN
ejpam-6690	376	7	x	x	PUNCT
ejpam-6690	376	8	=	=	SYM
ejpam-6690	376	9	⋃n	⋃n	PROPN
ejpam-6690	376	10	i=1w	i=1w	NOUN
ejpam-6690	376	11	(	(	PUNCT
ejpam-6690	376	12	xi	xi	NOUN
ejpam-6690	376	13	)	)	PUNCT
ejpam-6690	376	14	for	for	ADP
ejpam-6690	376	15	some	some	DET
ejpam-6690	376	16	x1	x1	PROPN
ejpam-6690	376	17	,	,	PUNCT
ejpam-6690	376	18	x2	x2	PROPN
ejpam-6690	376	19	,	,	PUNCT
ejpam-6690	376	20	...	...	PUNCT
ejpam-6690	376	21	,	,	PUNCT
ejpam-6690	377	1	xn	xn	PROPN
ejpam-6690	377	2	.	.	PUNCT
ejpam-6690	378	1	also	also	ADV
ejpam-6690	378	2	x	x	X
ejpam-6690	378	3	=	=	SYM
ejpam-6690	378	4	⋃n	⋃n	NOUN
ejpam-6690	378	5	i=1w	i=1w	NOUN
ejpam-6690	378	6	(	(	PUNCT
ejpam-6690	378	7	xi	xi	NOUN
ejpam-6690	378	8	)	)	PUNCT
ejpam-6690	378	9	is	be	AUX
ejpam-6690	378	10	compact	compact	ADJ
ejpam-6690	378	11	,	,	PUNCT
ejpam-6690	378	12	since	since	SCONJ
ejpam-6690	378	13	the	the	DET
ejpam-6690	378	14	finite	finite	PROPN
ejpam-6690	378	15	union	union	NOUN
ejpam-6690	378	16	of	of	ADP
ejpam-6690	378	17	compact	compact	ADJ
ejpam-6690	378	18	sets	set	NOUN
ejpam-6690	378	19	is	be	AUX
ejpam-6690	378	20	compact	compact	ADJ
ejpam-6690	378	21	.	.	PUNCT
ejpam-6690	379	1	remark	remark	NOUN
ejpam-6690	379	2	5	5	NUM
ejpam-6690	379	3	.	.	PUNCT
ejpam-6690	380	1	[	[	X
ejpam-6690	380	2	17	17	NUM
ejpam-6690	380	3	]	]	PUNCT
ejpam-6690	380	4	for	for	ADP
ejpam-6690	380	5	a	a	DET
ejpam-6690	380	6	hyperring	hyperring	NOUN
ejpam-6690	380	7	r	r	NOUN
ejpam-6690	380	8	,	,	PUNCT
ejpam-6690	380	9	similar	similar	ADJ
ejpam-6690	380	10	to	to	ADP
ejpam-6690	380	11	a	a	DET
ejpam-6690	380	12	ring	ring	NOUN
ejpam-6690	380	13	it	it	PRON
ejpam-6690	380	14	can	can	AUX
ejpam-6690	380	15	be	be	AUX
ejpam-6690	380	16	shown	show	VERB
ejpam-6690	380	17	that	that	SCONJ
ejpam-6690	380	18	if	if	SCONJ
ejpam-6690	380	19	r	r	NOUN
ejpam-6690	380	20	is	be	AUX
ejpam-6690	380	21	artinian	artinian	ADJ
ejpam-6690	380	22	,	,	PUNCT
ejpam-6690	380	23	then	then	ADV
ejpam-6690	380	24	every	every	DET
ejpam-6690	380	25	prime	prime	ADJ
ejpam-6690	380	26	hyperideal	hyperideal	NOUN
ejpam-6690	380	27	is	be	AUX
ejpam-6690	380	28	maximal	maximal	ADJ
ejpam-6690	380	29	,	,	PUNCT
ejpam-6690	380	30	and	and	CCONJ
ejpam-6690	380	31	the	the	DET
ejpam-6690	380	32	number	number	NOUN
ejpam-6690	380	33	of	of	ADP
ejpam-6690	380	34	maximal	maximal	ADJ
ejpam-6690	380	35	hyperideals	hyperideal	NOUN
ejpam-6690	380	36	is	be	AUX
ejpam-6690	380	37	finite	finite	ADJ
ejpam-6690	380	38	.	.	PUNCT
ejpam-6690	381	1	also	also	ADV
ejpam-6690	381	2	,	,	PUNCT
ejpam-6690	381	3	r	r	NOUN
ejpam-6690	381	4	is	be	AUX
ejpam-6690	381	5	artinian	artinian	ADJ
ejpam-6690	381	6	if	if	SCONJ
ejpam-6690	381	7	and	and	CCONJ
ejpam-6690	381	8	only	only	ADV
ejpam-6690	381	9	if	if	SCONJ
ejpam-6690	381	10	r	r	NOUN
ejpam-6690	381	11	is	be	AUX
ejpam-6690	381	12	noetherian	noetherian	ADJ
ejpam-6690	381	13	and	and	CCONJ
ejpam-6690	381	14	dim(r	dim(r	NOUN
ejpam-6690	381	15	)	)	PUNCT
ejpam-6690	382	1	=	=	SYM
ejpam-6690	382	2	0	0	X
ejpam-6690	382	3	.	.	PUNCT
ejpam-6690	383	1	by	by	ADP
ejpam-6690	383	2	proposition	proposition	NOUN
ejpam-6690	383	3	2	2	NUM
ejpam-6690	383	4	,	,	PUNCT
ejpam-6690	383	5	it	it	PRON
ejpam-6690	383	6	conclude	conclude	VERB
ejpam-6690	383	7	that	that	SCONJ
ejpam-6690	383	8	if	if	SCONJ
ejpam-6690	383	9	r	r	NOUN
ejpam-6690	383	10	is	be	AUX
ejpam-6690	383	11	a	a	DET
ejpam-6690	383	12	noetherian	noetherian	ADJ
ejpam-6690	383	13	hyperring	hyperring	NOUN
ejpam-6690	383	14	,	,	PUNCT
ejpam-6690	383	15	then	then	ADV
ejpam-6690	383	16	spec(r	spec(r	PROPN
ejpam-6690	383	17	)	)	PUNCT
ejpam-6690	383	18	is	be	AUX
ejpam-6690	383	19	noetherian	noetherian	ADJ
ejpam-6690	383	20	space	space	NOUN
ejpam-6690	383	21	.	.	PUNCT
ejpam-6690	384	1	proposition	proposition	NOUN
ejpam-6690	384	2	5	5	NUM
ejpam-6690	384	3	.	.	PUNCT
ejpam-6690	385	1	let	let	VERB
ejpam-6690	385	2	r	r	PRON
ejpam-6690	385	3	be	be	AUX
ejpam-6690	385	4	a	a	DET
ejpam-6690	385	5	noetherian	noetherian	ADJ
ejpam-6690	385	6	hyperring	hyperring	NOUN
ejpam-6690	385	7	.	.	PUNCT
ejpam-6690	386	1	the	the	DET
ejpam-6690	386	2	following	follow	VERB
ejpam-6690	386	3	assertions	assertion	NOUN
ejpam-6690	386	4	are	be	AUX
ejpam-6690	386	5	equivalent	equivalent	ADJ
ejpam-6690	386	6	:	:	PUNCT
ejpam-6690	386	7	(	(	PUNCT
ejpam-6690	386	8	i	i	NOUN
ejpam-6690	386	9	)	)	PUNCT
ejpam-6690	386	10	r	r	NOUN
ejpam-6690	386	11	is	be	AUX
ejpam-6690	386	12	artinian	artinian	ADJ
ejpam-6690	386	13	.	.	PUNCT
ejpam-6690	387	1	(	(	PUNCT
ejpam-6690	387	2	ii	ii	NOUN
ejpam-6690	387	3	)	)	PUNCT
ejpam-6690	387	4	spec(r	spec(r	PROPN
ejpam-6690	387	5	)	)	PUNCT
ejpam-6690	387	6	is	be	AUX
ejpam-6690	387	7	a	a	DET
ejpam-6690	387	8	discrete	discrete	ADJ
ejpam-6690	387	9	and	and	CCONJ
ejpam-6690	387	10	finite	finite	ADJ
ejpam-6690	387	11	space	space	NOUN
ejpam-6690	387	12	.	.	PUNCT
ejpam-6690	388	1	(	(	PUNCT
ejpam-6690	388	2	iii	iii	X
ejpam-6690	388	3	)	)	PUNCT
ejpam-6690	388	4	spec(r	spec(r	PROPN
ejpam-6690	388	5	)	)	PUNCT
ejpam-6690	388	6	is	be	AUX
ejpam-6690	388	7	a	a	DET
ejpam-6690	388	8	discrete	discrete	ADJ
ejpam-6690	388	9	space	space	NOUN
ejpam-6690	388	10	.	.	PUNCT
ejpam-6690	389	1	proof	proof	NOUN
ejpam-6690	389	2	.	.	PUNCT
ejpam-6690	390	1	1	1	NUM
ejpam-6690	390	2	⇒	⇒	NOUN
ejpam-6690	390	3	2	2	NUM
ejpam-6690	390	4	:	:	PUNCT
ejpam-6690	390	5	if	if	SCONJ
ejpam-6690	390	6	r	r	NOUN
ejpam-6690	390	7	is	be	AUX
ejpam-6690	390	8	artinian	artinian	ADJ
ejpam-6690	390	9	hyperring	hyperre	VERB
ejpam-6690	390	10	by	by	ADP
ejpam-6690	390	11	remark	remark	NOUN
ejpam-6690	390	12	5	5	NUM
ejpam-6690	390	13	,	,	PUNCT
ejpam-6690	390	14	mspec(r	mspec(r	PROPN
ejpam-6690	390	15	)	)	PUNCT
ejpam-6690	390	16	=	=	SYM
ejpam-6690	390	17	spec(r	spec(r	PROPN
ejpam-6690	390	18	)	)	PUNCT
ejpam-6690	390	19	and	and	CCONJ
ejpam-6690	390	20	each	each	DET
ejpam-6690	390	21	point	point	NOUN
ejpam-6690	390	22	of	of	ADP
ejpam-6690	390	23	spec(r	spec(r	PROPN
ejpam-6690	390	24	)	)	PUNCT
ejpam-6690	390	25	is	be	AUX
ejpam-6690	390	26	closed	close	VERB
ejpam-6690	390	27	and	and	CCONJ
ejpam-6690	390	28	the	the	DET
ejpam-6690	390	29	number	number	NOUN
ejpam-6690	390	30	of	of	ADP
ejpam-6690	390	31	maximal	maximal	ADJ
ejpam-6690	390	32	hyperideals	hyperideal	NOUN
ejpam-6690	390	33	of	of	ADP
ejpam-6690	390	34	r	r	NOUN
ejpam-6690	390	35	is	be	AUX
ejpam-6690	390	36	finite	finite	ADJ
ejpam-6690	390	37	.	.	PUNCT
ejpam-6690	391	1	hence	hence	ADV
ejpam-6690	391	2	spec(r	spec(r	PROPN
ejpam-6690	391	3	)	)	PUNCT
ejpam-6690	391	4	is	be	AUX
ejpam-6690	391	5	a	a	DET
ejpam-6690	391	6	discrete	discrete	ADJ
ejpam-6690	391	7	and	and	CCONJ
ejpam-6690	391	8	finite	finite	ADJ
ejpam-6690	391	9	space	space	NOUN
ejpam-6690	391	10	.	.	PUNCT
ejpam-6690	392	1	3	3	NUM
ejpam-6690	392	2	⇒	⇒	NOUN
ejpam-6690	392	3	1	1	NUM
ejpam-6690	392	4	:	:	PUNCT
ejpam-6690	392	5	since	since	SCONJ
ejpam-6690	392	6	spec(r	spec(r	PROPN
ejpam-6690	392	7	)	)	PUNCT
ejpam-6690	392	8	is	be	AUX
ejpam-6690	392	9	a	a	DET
ejpam-6690	392	10	discrete	discrete	ADJ
ejpam-6690	392	11	space	space	NOUN
ejpam-6690	392	12	,	,	PUNCT
ejpam-6690	392	13	then	then	ADV
ejpam-6690	392	14	each	each	DET
ejpam-6690	392	15	point	point	NOUN
ejpam-6690	392	16	is	be	AUX
ejpam-6690	392	17	closed	close	VERB
ejpam-6690	392	18	.	.	PUNCT
ejpam-6690	393	1	by	by	ADP
ejpam-6690	393	2	lemma	lemma	PROPN
ejpam-6690	393	3	2	2	NUM
ejpam-6690	393	4	,	,	PUNCT
ejpam-6690	393	5	any	any	DET
ejpam-6690	393	6	closed	closed	ADJ
ejpam-6690	393	7	point	point	NOUN
ejpam-6690	393	8	is	be	AUX
ejpam-6690	393	9	maximal	maximal	ADJ
ejpam-6690	393	10	hyperideal	hyperideal	NOUN
ejpam-6690	393	11	.	.	PUNCT
ejpam-6690	394	1	so	so	ADV
ejpam-6690	394	2	mspec(r	mspec(r	PROPN
ejpam-6690	394	3	)	)	PUNCT
ejpam-6690	394	4	=	=	SYM
ejpam-6690	394	5	spec(r	spec(r	PROPN
ejpam-6690	394	6	)	)	PUNCT
ejpam-6690	394	7	and	and	CCONJ
ejpam-6690	394	8	dim(r	dim(r	PROPN
ejpam-6690	394	9	)	)	PUNCT
ejpam-6690	395	1	=	=	SYM
ejpam-6690	395	2	0	0	X
ejpam-6690	395	3	.	.	PUNCT
ejpam-6690	395	4	now	now	ADV
ejpam-6690	395	5	by	by	ADP
ejpam-6690	395	6	remark	remark	NOUN
ejpam-6690	395	7	5	5	NUM
ejpam-6690	395	8	,	,	PUNCT
ejpam-6690	395	9	r	r	NOUN
ejpam-6690	395	10	is	be	AUX
ejpam-6690	395	11	artinian	artinian	ADJ
ejpam-6690	395	12	.	.	PUNCT
ejpam-6690	396	1	b.	b.	PROPN
ejpam-6690	396	2	afshar	afshar	PROPN
ejpam-6690	396	3	,	,	PUNCT
ejpam-6690	396	4	r.	r.	PROPN
ejpam-6690	396	5	ameri	ameri	PROPN
ejpam-6690	396	6	,	,	PUNCT
ejpam-6690	396	7	m.	m.	PROPN
ejpam-6690	396	8	al	al	PROPN
ejpam-6690	396	9	-	-	PUNCT
ejpam-6690	396	10	tahan	tahan	PROPN
ejpam-6690	396	11	/	/	SYM
ejpam-6690	396	12	eur	eur	PROPN
ejpam-6690	396	13	.	.	PUNCT
ejpam-6690	397	1	j.	j.	PROPN
ejpam-6690	397	2	pure	pure	PROPN
ejpam-6690	397	3	appl	appl	PROPN
ejpam-6690	397	4	.	.	PROPN
ejpam-6690	397	5	math	math	PROPN
ejpam-6690	397	6	,	,	PUNCT
ejpam-6690	397	7	18	18	NUM
ejpam-6690	397	8	(	(	PUNCT
ejpam-6690	397	9	4	4	NUM
ejpam-6690	397	10	)	)	PUNCT
ejpam-6690	397	11	(	(	PUNCT
ejpam-6690	397	12	2025	2025	NUM
ejpam-6690	397	13	)	)	PUNCT
ejpam-6690	397	14	,	,	PUNCT
ejpam-6690	397	15	6690	6690	NUM
ejpam-6690	397	16	11	11	NUM
ejpam-6690	397	17	of	of	ADP
ejpam-6690	397	18	19	19	NUM
ejpam-6690	397	19	proposition	proposition	NOUN
ejpam-6690	397	20	6	6	NUM
ejpam-6690	397	21	.	.	PUNCT
ejpam-6690	398	1	let	let	VERB
ejpam-6690	398	2	r	r	PRON
ejpam-6690	398	3	be	be	AUX
ejpam-6690	398	4	a	a	DET
ejpam-6690	398	5	hyprring	hyprring	NOUN
ejpam-6690	398	6	.	.	PUNCT
ejpam-6690	399	1	then	then	ADV
ejpam-6690	399	2	the	the	DET
ejpam-6690	399	3	irreducible	irreducible	ADJ
ejpam-6690	399	4	components	component	NOUN
ejpam-6690	399	5	of	of	ADP
ejpam-6690	399	6	spec(r	spec(r	PROPN
ejpam-6690	399	7	)	)	PUNCT
ejpam-6690	399	8	are	be	AUX
ejpam-6690	399	9	closed	close	VERB
ejpam-6690	399	10	sets	set	NOUN
ejpam-6690	399	11	v	v	X
ejpam-6690	399	12	(	(	PUNCT
ejpam-6690	399	13	i	i	NOUN
ejpam-6690	399	14	)	)	PUNCT
ejpam-6690	399	15	,	,	PUNCT
ejpam-6690	399	16	where	where	SCONJ
ejpam-6690	399	17	i	i	PRON
ejpam-6690	399	18	is	be	AUX
ejpam-6690	399	19	a	a	DET
ejpam-6690	399	20	minimal	minimal	ADJ
ejpam-6690	399	21	prime	prime	ADJ
ejpam-6690	399	22	hyperideal	hyperideal	NOUN
ejpam-6690	399	23	of	of	ADP
ejpam-6690	399	24	r.	r.	PROPN
ejpam-6690	399	25	proof	proof	NOUN
ejpam-6690	399	26	.	.	PUNCT
ejpam-6690	400	1	if	if	SCONJ
ejpam-6690	400	2	x	x	PRON
ejpam-6690	400	3	is	be	AUX
ejpam-6690	400	4	a	a	DET
ejpam-6690	400	5	maximal	maximal	ADJ
ejpam-6690	400	6	irreducible	irreducible	ADJ
ejpam-6690	400	7	subset	subset	NOUN
ejpam-6690	400	8	of	of	ADP
ejpam-6690	400	9	spec(r	spec(r	PROPN
ejpam-6690	400	10	)	)	PUNCT
ejpam-6690	400	11	,	,	PUNCT
ejpam-6690	400	12	then	then	ADV
ejpam-6690	400	13	by	by	ADP
ejpam-6690	400	14	theorem	theorem	NOUN
ejpam-6690	400	15	1	1	NUM
ejpam-6690	400	16	,	,	PUNCT
ejpam-6690	400	17	x	x	PRON
ejpam-6690	400	18	is	be	AUX
ejpam-6690	400	19	closed	closed	ADJ
ejpam-6690	400	20	and	and	CCONJ
ejpam-6690	400	21	so	so	ADV
ejpam-6690	400	22	x	x	X
ejpam-6690	400	23	=	=	SYM
ejpam-6690	400	24	v	v	X
ejpam-6690	400	25	(	(	PUNCT
ejpam-6690	400	26	i	i	NOUN
ejpam-6690	400	27	)	)	PUNCT
ejpam-6690	400	28	,	,	PUNCT
ejpam-6690	400	29	for	for	ADP
ejpam-6690	400	30	some	some	DET
ejpam-6690	400	31	hyperideal	hyperideal	NOUN
ejpam-6690	400	32	i	i	PRON
ejpam-6690	400	33	of	of	ADP
ejpam-6690	400	34	r.	r.	PROPN
ejpam-6690	400	35	by	by	ADP
ejpam-6690	400	36	corollary	corollary	ADJ
ejpam-6690	400	37	3	3	NUM
ejpam-6690	400	38	,	,	PUNCT
ejpam-6690	400	39	√	√	PUNCT
ejpam-6690	401	1	i	i	PRON
ejpam-6690	401	2	is	be	AUX
ejpam-6690	401	3	prime	prime	ADJ
ejpam-6690	401	4	hyperideal	hyperideal	NOUN
ejpam-6690	401	5	and	and	CCONJ
ejpam-6690	401	6	if	if	SCONJ
ejpam-6690	401	7	p	p	PROPN
ejpam-6690	401	8	∈	∈	PROPN
ejpam-6690	401	9	spec(r	spec(r	PROPN
ejpam-6690	401	10	)	)	PUNCT
ejpam-6690	401	11	such	such	ADJ
ejpam-6690	401	12	that	that	PRON
ejpam-6690	401	13	v	v	NOUN
ejpam-6690	401	14	(	(	PUNCT
ejpam-6690	401	15	p	p	NOUN
ejpam-6690	401	16	)	)	PUNCT
ejpam-6690	401	17	is	be	AUX
ejpam-6690	401	18	irreducible	irreducible	ADJ
ejpam-6690	401	19	and	and	CCONJ
ejpam-6690	401	20	p	p	NOUN
ejpam-6690	401	21	⊆	⊆	NUM
ejpam-6690	401	22	i	i	PRON
ejpam-6690	401	23	,	,	PUNCT
ejpam-6690	401	24	then	then	ADV
ejpam-6690	401	25	x	x	X
ejpam-6690	401	26	=	=	SYM
ejpam-6690	401	27	v	v	X
ejpam-6690	401	28	(	(	PUNCT
ejpam-6690	401	29	i	i	NOUN
ejpam-6690	401	30	)	)	PUNCT
ejpam-6690	401	31	⊆	⊆	NUM
ejpam-6690	401	32	v	v	X
ejpam-6690	401	33	(	(	PUNCT
ejpam-6690	401	34	p	p	NOUN
ejpam-6690	401	35	)	)	PUNCT
ejpam-6690	401	36	.	.	PUNCT
ejpam-6690	402	1	so	so	ADV
ejpam-6690	402	2	v	v	INTJ
ejpam-6690	402	3	(	(	PUNCT
ejpam-6690	402	4	i	i	NOUN
ejpam-6690	402	5	)	)	PUNCT
ejpam-6690	402	6	=	=	SYM
ejpam-6690	402	7	v	v	X
ejpam-6690	402	8	(	(	PUNCT
ejpam-6690	402	9	p	p	NOUN
ejpam-6690	402	10	)	)	PUNCT
ejpam-6690	402	11	and	and	CCONJ
ejpam-6690	402	12	therefore	therefore	ADV
ejpam-6690	402	13	,	,	PUNCT
ejpam-6690	402	14	√	√	PROPN
ejpam-6690	403	1	p	p	NOUN
ejpam-6690	403	2	=	=	NOUN
ejpam-6690	403	3	√	√	PROPN
ejpam-6690	403	4	i.	i.	NOUN
ejpam-6690	403	5	theorem	theorem	VERB
ejpam-6690	403	6	10	10	NUM
ejpam-6690	403	7	.	.	PUNCT
ejpam-6690	404	1	if	if	SCONJ
ejpam-6690	404	2	r	r	NOUN
ejpam-6690	404	3	is	be	AUX
ejpam-6690	404	4	a	a	DET
ejpam-6690	404	5	local	local	ADJ
ejpam-6690	404	6	hyprring	hyprring	NOUN
ejpam-6690	404	7	,	,	PUNCT
ejpam-6690	404	8	then	then	ADV
ejpam-6690	404	9	spec(r	spec(r	PROPN
ejpam-6690	404	10	)	)	PUNCT
ejpam-6690	404	11	is	be	AUX
ejpam-6690	404	12	connected	connect	VERB
ejpam-6690	404	13	.	.	PUNCT
ejpam-6690	405	1	proof	proof	NOUN
ejpam-6690	405	2	.	.	PUNCT
ejpam-6690	406	1	let	let	VERB
ejpam-6690	406	2	x	x	PUNCT
ejpam-6690	406	3	∈	∈	NOUN
ejpam-6690	406	4	r	r	NOUN
ejpam-6690	406	5	be	be	AUX
ejpam-6690	406	6	a	a	DET
ejpam-6690	406	7	nontrivial	nontrivial	ADJ
ejpam-6690	406	8	idempotent	idempotent	NOUN
ejpam-6690	406	9	and	and	CCONJ
ejpam-6690	406	10	m	m	AUX
ejpam-6690	406	11	be	be	AUX
ejpam-6690	406	12	a	a	DET
ejpam-6690	406	13	maximal	maximal	ADJ
ejpam-6690	406	14	hyperideal	hyperideal	NOUN
ejpam-6690	406	15	of	of	ADP
ejpam-6690	406	16	r.	r.	PROPN
ejpam-6690	406	17	then	then	ADV
ejpam-6690	406	18	x	x	X
ejpam-6690	406	19	/∈	/∈	PUNCT
ejpam-6690	406	20	m	m	VERB
ejpam-6690	406	21	=	=	ADJ
ejpam-6690	406	22	nil(r	nil(r	PROPN
ejpam-6690	406	23	)	)	PUNCT
ejpam-6690	406	24	.	.	PUNCT
ejpam-6690	407	1	since	since	SCONJ
ejpam-6690	407	2	x	x	PRON
ejpam-6690	407	3	is	be	AUX
ejpam-6690	407	4	not	not	PART
ejpam-6690	407	5	unit	unit	NOUN
ejpam-6690	407	6	,	,	PUNCT
ejpam-6690	407	7	then	then	ADV
ejpam-6690	407	8	x	x	PUNCT
ejpam-6690	407	9	is	be	AUX
ejpam-6690	407	10	contained	contain	VERB
ejpam-6690	407	11	in	in	ADP
ejpam-6690	407	12	some	some	DET
ejpam-6690	407	13	maximal	maximal	ADJ
ejpam-6690	407	14	hyperideal	hyperideal	NOUN
ejpam-6690	407	15	.	.	PUNCT
ejpam-6690	408	1	so	so	ADV
ejpam-6690	408	2	x	x	SYM
ejpam-6690	408	3	∈	∈	PROPN
ejpam-6690	408	4	m	m	NOUN
ejpam-6690	408	5	and	and	CCONJ
ejpam-6690	408	6	it	it	PRON
ejpam-6690	408	7	is	be	AUX
ejpam-6690	408	8	contradiction	contradiction	NOUN
ejpam-6690	408	9	.	.	PUNCT
ejpam-6690	409	1	remark	remark	NOUN
ejpam-6690	409	2	6	6	NUM
ejpam-6690	409	3	.	.	PUNCT
ejpam-6690	410	1	[	[	X
ejpam-6690	410	2	6	6	NUM
ejpam-6690	410	3	]	]	PUNCT
ejpam-6690	410	4	let	let	VERB
ejpam-6690	410	5	s	s	PRON
ejpam-6690	410	6	be	be	AUX
ejpam-6690	410	7	a	a	DET
ejpam-6690	410	8	multiplicative	multiplicative	ADJ
ejpam-6690	410	9	subset	subset	NOUN
ejpam-6690	410	10	of	of	ADP
ejpam-6690	410	11	hyperring	hyperre	VERB
ejpam-6690	410	12	r	r	NOUN
ejpam-6690	410	13	,	,	PUNCT
ejpam-6690	410	14	and	and	CCONJ
ejpam-6690	410	15	i	i	PRON
ejpam-6690	410	16	be	be	VERB
ejpam-6690	410	17	a	a	DET
ejpam-6690	410	18	hyperideal	hyperideal	NOUN
ejpam-6690	410	19	of	of	ADP
ejpam-6690	410	20	r.	r.	PROPN
ejpam-6690	410	21	then	then	ADV
ejpam-6690	410	22	s∩	s∩	VERB
ejpam-6690	410	23	i	i	PRON
ejpam-6690	410	24	̸=	̸=	PROPN
ejpam-6690	410	25	∅	∅	NOUN
ejpam-6690	410	26	if	if	SCONJ
ejpam-6690	410	27	and	and	CCONJ
ejpam-6690	410	28	only	only	ADV
ejpam-6690	410	29	if	if	SCONJ
ejpam-6690	410	30	s−1i	s−1i	NOUN
ejpam-6690	410	31	=	=	SYM
ejpam-6690	410	32	s−1r	s−1r	PROPN
ejpam-6690	410	33	.	.	PUNCT
ejpam-6690	411	1	also	also	ADV
ejpam-6690	411	2	there	there	PRON
ejpam-6690	411	3	is	be	VERB
ejpam-6690	411	4	a	a	DET
ejpam-6690	411	5	one	one	NUM
ejpam-6690	411	6	to	to	ADP
ejpam-6690	411	7	one	one	NUM
ejpam-6690	411	8	correspondence	correspondence	NOUN
ejpam-6690	411	9	between	between	ADP
ejpam-6690	411	10	the	the	DET
ejpam-6690	411	11	sets	set	NOUN
ejpam-6690	411	12	{	{	PUNCT
ejpam-6690	411	13	p	p	NOUN
ejpam-6690	411	14	;	;	PUNCT
ejpam-6690	411	15	p	p	PROPN
ejpam-6690	411	16	∈	∈	PROPN
ejpam-6690	411	17	spec(r	spec(r	PROPN
ejpam-6690	411	18	)	)	PUNCT
ejpam-6690	411	19	,	,	PUNCT
ejpam-6690	411	20	p	p	PROPN
ejpam-6690	411	21	∩	∩	NOUN
ejpam-6690	411	22	s	s	PART
ejpam-6690	411	23	=	=	NOUN
ejpam-6690	411	24	∅	∅	NOUN
ejpam-6690	411	25	}	}	PUNCT
ejpam-6690	411	26	and	and	CCONJ
ejpam-6690	411	27	spec(s−1r	spec(s−1r	NOUN
ejpam-6690	411	28	)	)	PUNCT
ejpam-6690	411	29	,	,	PUNCT
ejpam-6690	411	30	under	under	ADP
ejpam-6690	411	31	the	the	DET
ejpam-6690	411	32	mapping	mapping	NOUN
ejpam-6690	411	33	p	p	PROPN
ejpam-6690	411	34	7→	7→	NUM
ejpam-6690	411	35	s−1p	s−1p	NOUN
ejpam-6690	411	36	.	.	PUNCT
ejpam-6690	412	1	proposition	proposition	NOUN
ejpam-6690	412	2	7	7	NUM
ejpam-6690	412	3	.	.	PUNCT
ejpam-6690	413	1	let	let	VERB
ejpam-6690	413	2	s	s	PRON
ejpam-6690	413	3	be	be	AUX
ejpam-6690	413	4	a	a	DET
ejpam-6690	413	5	multiplicative	multiplicative	ADJ
ejpam-6690	413	6	subset	subset	NOUN
ejpam-6690	413	7	of	of	ADP
ejpam-6690	413	8	hyperring	hyperre	VERB
ejpam-6690	413	9	r.	r.	PROPN
ejpam-6690	413	10	then	then	ADV
ejpam-6690	413	11	spec(r	spec(r	PROPN
ejpam-6690	413	12	)	)	PUNCT
ejpam-6690	413	13	is	be	AUX
ejpam-6690	413	14	irreducible	irreducible	ADJ
ejpam-6690	413	15	and	and	CCONJ
ejpam-6690	413	16	nil(r	nil(r	ADJ
ejpam-6690	413	17	)	)	PUNCT
ejpam-6690	413	18	∩	∩	NOUN
ejpam-6690	413	19	s	s	PART
ejpam-6690	413	20	=	=	PUNCT
ejpam-6690	413	21	∅	∅	NOUN
ejpam-6690	413	22	if	if	SCONJ
ejpam-6690	413	23	and	and	CCONJ
ejpam-6690	413	24	only	only	ADV
ejpam-6690	413	25	if	if	SCONJ
ejpam-6690	413	26	spec(s−1r	spec(s−1r	NOUN
ejpam-6690	413	27	)	)	PUNCT
ejpam-6690	413	28	is	be	AUX
ejpam-6690	413	29	irreducible	irreducible	ADJ
ejpam-6690	413	30	.	.	PUNCT
ejpam-6690	414	1	proof	proof	NOUN
ejpam-6690	414	2	.	.	PUNCT
ejpam-6690	415	1	if	if	SCONJ
ejpam-6690	415	2	spec(r	spec(r	PROPN
ejpam-6690	415	3	)	)	PUNCT
ejpam-6690	415	4	is	be	AUX
ejpam-6690	415	5	irreducible	irreducible	ADJ
ejpam-6690	415	6	,	,	PUNCT
ejpam-6690	415	7	then	then	ADV
ejpam-6690	415	8	by	by	ADP
ejpam-6690	415	9	theorem	theorem	NOUN
ejpam-6690	415	10	8	8	NUM
ejpam-6690	415	11	,	,	PUNCT
ejpam-6690	415	12	nil(r	nil(r	NOUN
ejpam-6690	415	13	)	)	PUNCT
ejpam-6690	415	14	∈	∈	PROPN
ejpam-6690	415	15	spec(r	spec(r	PROPN
ejpam-6690	415	16	)	)	PUNCT
ejpam-6690	415	17	and	and	CCONJ
ejpam-6690	415	18	since	since	SCONJ
ejpam-6690	415	19	nil(r	nil(r	NOUN
ejpam-6690	415	20	)	)	PUNCT
ejpam-6690	415	21	∩	∩	NOUN
ejpam-6690	415	22	s	s	PART
ejpam-6690	415	23	=	=	SYM
ejpam-6690	415	24	∅	∅	NOUN
ejpam-6690	415	25	and	and	CCONJ
ejpam-6690	415	26	nil(s−1r	nil(s−1r	NUM
ejpam-6690	415	27	)	)	PUNCT
ejpam-6690	415	28	=	=	SYM
ejpam-6690	415	29	s−1nil(r	s−1nil(r	NOUN
ejpam-6690	415	30	)	)	PUNCT
ejpam-6690	415	31	,	,	PUNCT
ejpam-6690	415	32	then	then	ADV
ejpam-6690	415	33	nil(s−1r	nil(s−1r	NUM
ejpam-6690	415	34	)	)	PUNCT
ejpam-6690	415	35	∈	∈	PROPN
ejpam-6690	415	36	spec(s−1r	spec(s−1r	NOUN
ejpam-6690	415	37	)	)	PUNCT
ejpam-6690	415	38	and	and	CCONJ
ejpam-6690	415	39	so	so	ADV
ejpam-6690	415	40	spec(s−1r	spec(s−1r	ADV
ejpam-6690	415	41	)	)	PUNCT
ejpam-6690	415	42	is	be	AUX
ejpam-6690	415	43	irreducible	irreducible	ADJ
ejpam-6690	415	44	.	.	PUNCT
ejpam-6690	416	1	if	if	SCONJ
ejpam-6690	416	2	spec(s−1r	spec(s−1r	NOUN
ejpam-6690	416	3	)	)	PUNCT
ejpam-6690	416	4	is	be	AUX
ejpam-6690	416	5	irreducible	irreducible	ADJ
ejpam-6690	416	6	,	,	PUNCT
ejpam-6690	416	7	then	then	ADV
ejpam-6690	416	8	nil(s−1r	nil(s−1r	NUM
ejpam-6690	416	9	)	)	PUNCT
ejpam-6690	416	10	=	=	SYM
ejpam-6690	416	11	s−1nil(r	s−1nil(r	NOUN
ejpam-6690	416	12	)	)	PUNCT
ejpam-6690	416	13	∈	∈	PROPN
ejpam-6690	416	14	spec(s−1r	spec(s−1r	NOUN
ejpam-6690	416	15	)	)	PUNCT
ejpam-6690	416	16	.	.	PUNCT
ejpam-6690	417	1	hence	hence	ADV
ejpam-6690	417	2	nil(r)∩	nil(r)∩	NOUN
ejpam-6690	417	3	s	s	PART
ejpam-6690	417	4	=	=	NOUN
ejpam-6690	417	5	∅	∅	NOUN
ejpam-6690	417	6	and	and	CCONJ
ejpam-6690	417	7	nil(r	nil(r	NOUN
ejpam-6690	417	8	)	)	PUNCT
ejpam-6690	417	9	∈	∈	PROPN
ejpam-6690	417	10	spec(r	spec(r	PROPN
ejpam-6690	417	11	)	)	PUNCT
ejpam-6690	417	12	.	.	PUNCT
ejpam-6690	418	1	proposition	proposition	NOUN
ejpam-6690	418	2	8	8	NUM
ejpam-6690	418	3	.	.	PUNCT
ejpam-6690	418	4	spec(s−1r	spec(s−1r	X
ejpam-6690	418	5	)	)	PUNCT
ejpam-6690	418	6	is	be	AUX
ejpam-6690	418	7	disconnected	disconnect	VERB
ejpam-6690	418	8	if	if	SCONJ
ejpam-6690	418	9	and	and	CCONJ
ejpam-6690	418	10	only	only	ADV
ejpam-6690	418	11	if	if	SCONJ
ejpam-6690	418	12	spec(r	spec(r	PROPN
ejpam-6690	418	13	)	)	PUNCT
ejpam-6690	418	14	is	be	AUX
ejpam-6690	418	15	disconnected	disconnect	VERB
ejpam-6690	418	16	.	.	PUNCT
ejpam-6690	419	1	proof	proof	NOUN
ejpam-6690	419	2	.	.	PUNCT
ejpam-6690	420	1	since	since	SCONJ
ejpam-6690	420	2	s−1(r1	s−1(r1	PROPN
ejpam-6690	420	3	×	×	PROPN
ejpam-6690	420	4	r2	r2	NOUN
ejpam-6690	420	5	)	)	PUNCT
ejpam-6690	420	6	∼=	∼=	PART
ejpam-6690	420	7	s−1r1	s−1r1	ADJ
ejpam-6690	420	8	×	×	NOUN
ejpam-6690	420	9	s−1r2	s−1r2	VERB
ejpam-6690	420	10	,	,	PUNCT
ejpam-6690	420	11	then	then	ADV
ejpam-6690	420	12	by	by	ADP
ejpam-6690	420	13	theorem	theorem	NOUN
ejpam-6690	420	14	5	5	NUM
ejpam-6690	420	15	,	,	PUNCT
ejpam-6690	420	16	and	and	CCONJ
ejpam-6690	420	17	theorem	theorem	VERB
ejpam-6690	420	18	7	7	NUM
ejpam-6690	420	19	,	,	PUNCT
ejpam-6690	420	20	the	the	DET
ejpam-6690	420	21	result	result	NOUN
ejpam-6690	420	22	immediately	immediately	ADV
ejpam-6690	420	23	holds	hold	VERB
ejpam-6690	420	24	.	.	PUNCT
ejpam-6690	420	25	example	example	NOUN
ejpam-6690	421	1	1	1	NUM
ejpam-6690	421	2	.	.	PUNCT
ejpam-6690	422	1	let	let	VERB
ejpam-6690	422	2	(	(	PUNCT
ejpam-6690	422	3	r,+	r,+	NUM
ejpam-6690	422	4	,	,	PUNCT
ejpam-6690	422	5	.	.	PUNCT
ejpam-6690	422	6	)	)	PUNCT
ejpam-6690	423	1	be	be	AUX
ejpam-6690	423	2	a	a	DET
ejpam-6690	423	3	commutative	commutative	ADJ
ejpam-6690	423	4	ring	ring	NOUN
ejpam-6690	423	5	with	with	ADP
ejpam-6690	423	6	a	a	DET
ejpam-6690	423	7	unit	unit	NOUN
ejpam-6690	423	8	element	element	NOUN
ejpam-6690	423	9	and	and	CCONJ
ejpam-6690	423	10	g	g	NOUN
ejpam-6690	423	11	be	be	AUX
ejpam-6690	423	12	a	a	DET
ejpam-6690	423	13	subgroup	subgroup	NOUN
ejpam-6690	423	14	of	of	ADP
ejpam-6690	423	15	monoid	monoid	PROPN
ejpam-6690	423	16	(	(	PUNCT
ejpam-6690	423	17	r−	r−	PROPN
ejpam-6690	423	18	{	{	PUNCT
ejpam-6690	423	19	0	0	NUM
ejpam-6690	423	20	}	}	PUNCT
ejpam-6690	423	21	,	,	PUNCT
ejpam-6690	423	22	.	.	PUNCT
ejpam-6690	423	23	)	)	PUNCT
ejpam-6690	424	1	and	and	CCONJ
ejpam-6690	424	2	i	i	PRON
ejpam-6690	424	3	be	be	VERB
ejpam-6690	424	4	an	an	DET
ejpam-6690	424	5	ideal	ideal	NOUN
ejpam-6690	424	6	of	of	ADP
ejpam-6690	424	7	r.	r.	PROPN
ejpam-6690	424	8	(	(	PUNCT
ejpam-6690	424	9	1	1	X
ejpam-6690	424	10	)	)	PUNCT
ejpam-6690	424	11	consider	consider	VERB
ejpam-6690	424	12	r̄	r̄	NOUN
ejpam-6690	424	13	=	=	NOUN
ejpam-6690	424	14	r	r	NOUN
ejpam-6690	424	15	/	/	SYM
ejpam-6690	424	16	g	g	NOUN
ejpam-6690	424	17	=	=	SYM
ejpam-6690	424	18	{	{	PUNCT
ejpam-6690	424	19	rg	rg	NOUN
ejpam-6690	424	20	;	;	PUNCT
ejpam-6690	424	21	r	r	NOUN
ejpam-6690	424	22	∈	∈	PROPN
ejpam-6690	424	23	r	r	NOUN
ejpam-6690	424	24	}	}	PUNCT
ejpam-6690	424	25	and	and	CCONJ
ejpam-6690	424	26	rg⊕sg	rg⊕sg	NOUN
ejpam-6690	425	1	=	=	PRON
ejpam-6690	425	2	{	{	PUNCT
ejpam-6690	425	3	tg	tg	PROPN
ejpam-6690	425	4	;	;	PUNCT
ejpam-6690	425	5	t	t	PROPN
ejpam-6690	425	6	∈	∈	PROPN
ejpam-6690	425	7	rg+sg	rg+sg	PROPN
ejpam-6690	425	8	}	}	PUNCT
ejpam-6690	425	9	,	,	PUNCT
ejpam-6690	425	10	rg⊙sg	rg⊙sg	PROPN
ejpam-6690	425	11	=	=	SYM
ejpam-6690	425	12	rsg	rsg	PROPN
ejpam-6690	425	13	.	.	PUNCT
ejpam-6690	426	1	then	then	ADV
ejpam-6690	426	2	(	(	PUNCT
ejpam-6690	426	3	r̄,⊕,⊙	r̄,⊕,⊙	X
ejpam-6690	426	4	)	)	PUNCT
ejpam-6690	426	5	is	be	AUX
ejpam-6690	426	6	hyperring	hyperre	VERB
ejpam-6690	426	7	and	and	CCONJ
ejpam-6690	426	8	0r̄	0r̄	NUM
ejpam-6690	427	1	=	=	PRON
ejpam-6690	427	2	{	{	PUNCT
ejpam-6690	427	3	0	0	NUM
ejpam-6690	427	4	}	}	PUNCT
ejpam-6690	427	5	,	,	PUNCT
ejpam-6690	427	6	1r̄	1r̄	NUM
ejpam-6690	427	7	=	=	PUNCT
ejpam-6690	428	1	g.	g.	PROPN
ejpam-6690	428	2	moreover	moreover	ADV
ejpam-6690	428	3	if	if	SCONJ
ejpam-6690	428	4	r	r	NOUN
ejpam-6690	428	5	is	be	AUX
ejpam-6690	428	6	a	a	DET
ejpam-6690	428	7	field	field	NOUN
ejpam-6690	428	8	,	,	PUNCT
ejpam-6690	428	9	then	then	ADV
ejpam-6690	428	10	(	(	PUNCT
ejpam-6690	428	11	r̄,⊕,⊙	r̄,⊕,⊙	X
ejpam-6690	428	12	)	)	PUNCT
ejpam-6690	428	13	is	be	AUX
ejpam-6690	428	14	a	a	DET
ejpam-6690	428	15	hyperfield,(for	hyperfield,(for	PROPN
ejpam-6690	428	16	more	more	ADJ
ejpam-6690	428	17	details	detail	NOUN
ejpam-6690	428	18	see	see	VERB
ejpam-6690	428	19	[	[	X
ejpam-6690	428	20	7	7	NUM
ejpam-6690	428	21	]	]	NUM
ejpam-6690	428	22	)	)	PUNCT
ejpam-6690	428	23	.	.	PUNCT
ejpam-6690	429	1	then	then	ADV
ejpam-6690	429	2	it	it	PRON
ejpam-6690	429	3	is	be	AUX
ejpam-6690	429	4	easy	easy	ADJ
ejpam-6690	429	5	to	to	PART
ejpam-6690	429	6	verify	verify	VERB
ejpam-6690	429	7	r	r	NOUN
ejpam-6690	429	8	are	be	AUX
ejpam-6690	429	9	of	of	ADP
ejpam-6690	429	10	the	the	DET
ejpam-6690	429	11	form	form	NOUN
ejpam-6690	429	12	i	i	X
ejpam-6690	429	13	/	/	SYM
ejpam-6690	429	14	g	g	PROPN
ejpam-6690	429	15	such	such	ADJ
ejpam-6690	429	16	that	that	SCONJ
ejpam-6690	429	17	i	i	PRON
ejpam-6690	429	18	is	be	AUX
ejpam-6690	429	19	ideal	ideal	ADJ
ejpam-6690	429	20	of	of	ADP
ejpam-6690	429	21	r.	r.	PROPN
ejpam-6690	429	22	let	let	VERB
ejpam-6690	429	23	p	p	PROPN
ejpam-6690	429	24	∈	∈	PROPN
ejpam-6690	429	25	spec(r	spec(r	PROPN
ejpam-6690	429	26	)	)	PUNCT
ejpam-6690	429	27	and	and	CCONJ
ejpam-6690	429	28	rg	rg	PROPN
ejpam-6690	429	29	⊙	⊙	PROPN
ejpam-6690	429	30	sg	sg	PROPN
ejpam-6690	430	1	=	=	PUNCT
ejpam-6690	430	2	rsg	rsg	PROPN
ejpam-6690	430	3	∈	∈	PROPN
ejpam-6690	430	4	p	p	X
ejpam-6690	430	5	/	/	SYM
ejpam-6690	430	6	g	g	NOUN
ejpam-6690	430	7	,	,	PUNCT
ejpam-6690	430	8	then	then	ADV
ejpam-6690	430	9	rs	rs	PROPN
ejpam-6690	430	10	∈	∈	PROPN
ejpam-6690	430	11	p	p	PROPN
ejpam-6690	430	12	.	.	PUNCT
ejpam-6690	431	1	so	so	ADV
ejpam-6690	431	2	r	r	NOUN
ejpam-6690	431	3	∈	∈	PROPN
ejpam-6690	431	4	p	p	NOUN
ejpam-6690	431	5	or	or	CCONJ
ejpam-6690	431	6	s	s	NOUN
ejpam-6690	431	7	∈	∈	PROPN
ejpam-6690	431	8	p	p	NOUN
ejpam-6690	431	9	and	and	CCONJ
ejpam-6690	431	10	therefore	therefore	ADV
ejpam-6690	431	11	,	,	PUNCT
ejpam-6690	431	12	rg	rg	PROPN
ejpam-6690	431	13	∈	∈	PROPN
ejpam-6690	431	14	p	p	X
ejpam-6690	431	15	/	/	SYM
ejpam-6690	431	16	g	g	NOUN
ejpam-6690	431	17	or	or	CCONJ
ejpam-6690	431	18	sg	sg	ADP
ejpam-6690	431	19	∈	∈	PROPN
ejpam-6690	431	20	p	p	PROPN
ejpam-6690	431	21	/	/	SYM
ejpam-6690	431	22	g.	g.	PROPN
ejpam-6690	432	1	and	and	CCONJ
ejpam-6690	432	2	p	p	X
ejpam-6690	432	3	/	/	SYM
ejpam-6690	432	4	g	g	PROPN
ejpam-6690	432	5	∈	∈	PROPN
ejpam-6690	432	6	spec(r̄	spec(r̄	PROPN
ejpam-6690	432	7	)	)	PUNCT
ejpam-6690	432	8	.	.	PUNCT
ejpam-6690	433	1	also	also	ADV
ejpam-6690	433	2	for	for	ADP
ejpam-6690	433	3	any	any	DET
ejpam-6690	433	4	a	a	DET
ejpam-6690	433	5	∈	∈	PROPN
ejpam-6690	433	6	spec(r̄	spec(r̄	NOUN
ejpam-6690	433	7	)	)	PUNCT
ejpam-6690	433	8	,	,	PUNCT
ejpam-6690	433	9	there	there	PRON
ejpam-6690	433	10	is	be	VERB
ejpam-6690	433	11	a	a	DET
ejpam-6690	433	12	prime	prime	ADJ
ejpam-6690	433	13	ideal	ideal	NOUN
ejpam-6690	433	14	p	p	PROPN
ejpam-6690	433	15	∈	∈	PROPN
ejpam-6690	433	16	spec(r	spec(r	PROPN
ejpam-6690	433	17	)	)	PUNCT
ejpam-6690	433	18	such	such	ADJ
ejpam-6690	433	19	that	that	SCONJ
ejpam-6690	433	20	a	a	DET
ejpam-6690	433	21	=	=	X
ejpam-6690	433	22	p	p	X
ejpam-6690	433	23	/	/	SYM
ejpam-6690	433	24	g.	g.	NOUN
ejpam-6690	433	25	so	so	ADV
ejpam-6690	433	26	spec(r̄	spec(r̄	PROPN
ejpam-6690	433	27	)	)	PUNCT
ejpam-6690	434	1	=	=	PRON
ejpam-6690	434	2	{	{	PUNCT
ejpam-6690	434	3	p	p	X
ejpam-6690	434	4	/	/	SYM
ejpam-6690	434	5	g;p	g;p	PROPN
ejpam-6690	434	6	∈	∈	PROPN
ejpam-6690	434	7	spec(r	spec(r	PROPN
ejpam-6690	434	8	)	)	PUNCT
ejpam-6690	434	9	}	}	PUNCT
ejpam-6690	434	10	.	.	PUNCT
ejpam-6690	435	1	(	(	PUNCT
ejpam-6690	435	2	2	2	X
ejpam-6690	435	3	)	)	PUNCT
ejpam-6690	435	4	consider	consider	VERB
ejpam-6690	435	5	r̄	r̄	NOUN
ejpam-6690	435	6	=	=	NOUN
ejpam-6690	435	7	r	r	NOUN
ejpam-6690	435	8	/	/	SYM
ejpam-6690	435	9	i	i	NOUN
ejpam-6690	435	10	=	=	PUNCT
ejpam-6690	435	11	{	{	PUNCT
ejpam-6690	435	12	r	r	NOUN
ejpam-6690	436	1	+	+	CCONJ
ejpam-6690	436	2	i	i	NOUN
ejpam-6690	436	3	;	;	PUNCT
ejpam-6690	436	4	r	r	NOUN
ejpam-6690	436	5	∈	∈	PROPN
ejpam-6690	436	6	r	r	NOUN
ejpam-6690	436	7	}	}	PUNCT
ejpam-6690	436	8	and	and	CCONJ
ejpam-6690	436	9	define	define	VERB
ejpam-6690	436	10	r	r	NOUN
ejpam-6690	437	1	+	+	CCONJ
ejpam-6690	437	2	i	i	PROPN
ejpam-6690	437	3	⊕	⊕	NOUN
ejpam-6690	437	4	s	s	PART
ejpam-6690	438	1	+	+	NUM
ejpam-6690	438	2	i	i	NOUN
ejpam-6690	438	3	=	=	PUNCT
ejpam-6690	438	4	(	(	PUNCT
ejpam-6690	438	5	r	r	NOUN
ejpam-6690	438	6	+	+	SYM
ejpam-6690	438	7	s	s	X
ejpam-6690	438	8	)	)	PUNCT
ejpam-6690	439	1	+	+	CCONJ
ejpam-6690	439	2	i	i	PRON
ejpam-6690	439	3	and	and	CCONJ
ejpam-6690	439	4	r+	r+	NOUN
ejpam-6690	439	5	i	i	PRON
ejpam-6690	439	6	⊙	⊙	VERB
ejpam-6690	439	7	s+	s+	PUNCT
ejpam-6690	440	1	i	i	PRON
ejpam-6690	440	2	=	=	PUNCT
ejpam-6690	440	3	{	{	PUNCT
ejpam-6690	440	4	t+	t+	NOUN
ejpam-6690	440	5	i	i	PRON
ejpam-6690	440	6	;	;	PUNCT
ejpam-6690	440	7	t	t	PROPN
ejpam-6690	440	8	∈	∈	PROPN
ejpam-6690	440	9	rs+	rs+	NOUN
ejpam-6690	440	10	i	i	PRON
ejpam-6690	440	11	}	}	PUNCT
ejpam-6690	440	12	.	.	PUNCT
ejpam-6690	441	1	then	then	ADV
ejpam-6690	441	2	(	(	PUNCT
ejpam-6690	441	3	r̄,⊕,⊙	r̄,⊕,⊙	X
ejpam-6690	441	4	)	)	PUNCT
ejpam-6690	441	5	is	be	AUX
ejpam-6690	441	6	multiplicative	multiplicative	ADJ
ejpam-6690	441	7	hyperring	hyperring	NOUN
ejpam-6690	441	8	with	with	ADP
ejpam-6690	441	9	0r̄	0r̄	PROPN
ejpam-6690	442	1	=	=	SYM
ejpam-6690	443	1	i	i	PROPN
ejpam-6690	443	2	,	,	PUNCT
ejpam-6690	443	3	1r̄	1r̄	NUM
ejpam-6690	443	4	=	=	SYM
ejpam-6690	443	5	1	1	NUM
ejpam-6690	443	6	+	+	NUM
ejpam-6690	443	7	i.	i.	NOUN
ejpam-6690	443	8	it	it	PRON
ejpam-6690	443	9	is	be	AUX
ejpam-6690	443	10	clear	clear	ADJ
ejpam-6690	443	11	that	that	SCONJ
ejpam-6690	443	12	spec(r̄	spec(r̄	NOUN
ejpam-6690	443	13	)	)	PUNCT
ejpam-6690	443	14	=	=	PRON
ejpam-6690	444	1	{	{	PUNCT
ejpam-6690	444	2	p	p	X
ejpam-6690	444	3	/	/	SYM
ejpam-6690	444	4	i;p	i;p	NOUN
ejpam-6690	444	5	∈	∈	PROPN
ejpam-6690	444	6	spec(r	spec(r	PROPN
ejpam-6690	444	7	)	)	PUNCT
ejpam-6690	444	8	}	}	PUNCT
ejpam-6690	444	9	∼=	∼=	NOUN
ejpam-6690	444	10	v	v	NOUN
ejpam-6690	444	11	(	(	PUNCT
ejpam-6690	444	12	i	i	NOUN
ejpam-6690	444	13	)	)	PUNCT
ejpam-6690	444	14	.	.	PUNCT
ejpam-6690	445	1	(	(	PUNCT
ejpam-6690	445	2	3	3	X
ejpam-6690	445	3	)	)	PUNCT
ejpam-6690	445	4	if	if	SCONJ
ejpam-6690	445	5	r̄	r̄	NOUN
ejpam-6690	445	6	=	=	SYM
ejpam-6690	445	7	{	{	PUNCT
ejpam-6690	445	8	r+i	r+i	PROPN
ejpam-6690	445	9	;	;	PUNCT
ejpam-6690	445	10	r	r	NOUN
ejpam-6690	445	11	∈	∈	PROPN
ejpam-6690	445	12	r	r	NOUN
ejpam-6690	445	13	}	}	PUNCT
ejpam-6690	445	14	,	,	PUNCT
ejpam-6690	445	15	r+i⊕s+i	r+i⊕s+i	NOUN
ejpam-6690	445	16	=	=	SYM
ejpam-6690	445	17	{	{	PUNCT
ejpam-6690	445	18	t+i	t+i	PROPN
ejpam-6690	445	19	;	;	PUNCT
ejpam-6690	445	20	t	t	PROPN
ejpam-6690	445	21	∈	∈	PROPN
ejpam-6690	445	22	r+s+i	r+s+i	PROPN
ejpam-6690	445	23	}	}	PUNCT
ejpam-6690	445	24	and	and	CCONJ
ejpam-6690	445	25	r+i⊙s+i	r+i⊙s+i	NOUN
ejpam-6690	445	26	=	=	SYM
ejpam-6690	445	27	{	{	PUNCT
ejpam-6690	445	28	t+i	t+i	NUM
ejpam-6690	445	29	;	;	PUNCT
ejpam-6690	445	30	t	t	PROPN
ejpam-6690	445	31	∈	∈	PROPN
ejpam-6690	445	32	rs+	rs+	NOUN
ejpam-6690	445	33	i	i	NOUN
ejpam-6690	445	34	}	}	PUNCT
ejpam-6690	445	35	,	,	PUNCT
ejpam-6690	445	36	then	then	ADV
ejpam-6690	445	37	(	(	PUNCT
ejpam-6690	445	38	r̄,⊕,⊙	r̄,⊕,⊙	X
ejpam-6690	445	39	)	)	PUNCT
ejpam-6690	445	40	is	be	AUX
ejpam-6690	445	41	general	general	ADJ
ejpam-6690	445	42	hyperring	hyperring	NOUN
ejpam-6690	445	43	and	and	CCONJ
ejpam-6690	445	44	spec(r̄	spec(r̄	PROPN
ejpam-6690	445	45	)	)	PUNCT
ejpam-6690	446	1	=	=	PRON
ejpam-6690	446	2	{	{	PUNCT
ejpam-6690	446	3	p	p	X
ejpam-6690	446	4	/	/	SYM
ejpam-6690	446	5	i;p	i;p	NOUN
ejpam-6690	446	6	∈	∈	PROPN
ejpam-6690	446	7	spec(r	spec(r	PROPN
ejpam-6690	446	8	)	)	PUNCT
ejpam-6690	446	9	}	}	PUNCT
ejpam-6690	446	10	∼=	∼=	NOUN
ejpam-6690	446	11	v	v	NOUN
ejpam-6690	446	12	(	(	PUNCT
ejpam-6690	446	13	i	i	NOUN
ejpam-6690	446	14	)	)	PUNCT
ejpam-6690	446	15	.	.	PUNCT
ejpam-6690	447	1	b.	b.	PROPN
ejpam-6690	447	2	afshar	afshar	PROPN
ejpam-6690	447	3	,	,	PUNCT
ejpam-6690	447	4	r.	r.	PROPN
ejpam-6690	447	5	ameri	ameri	PROPN
ejpam-6690	447	6	,	,	PUNCT
ejpam-6690	447	7	m.	m.	PROPN
ejpam-6690	447	8	al	al	PROPN
ejpam-6690	447	9	-	-	PUNCT
ejpam-6690	447	10	tahan	tahan	PROPN
ejpam-6690	447	11	/	/	SYM
ejpam-6690	447	12	eur	eur	PROPN
ejpam-6690	447	13	.	.	PUNCT
ejpam-6690	448	1	j.	j.	PROPN
ejpam-6690	448	2	pure	pure	PROPN
ejpam-6690	448	3	appl	appl	PROPN
ejpam-6690	448	4	.	.	PROPN
ejpam-6690	448	5	math	math	PROPN
ejpam-6690	448	6	,	,	PUNCT
ejpam-6690	448	7	18	18	NUM
ejpam-6690	448	8	(	(	PUNCT
ejpam-6690	448	9	4	4	NUM
ejpam-6690	448	10	)	)	PUNCT
ejpam-6690	448	11	(	(	PUNCT
ejpam-6690	448	12	2025	2025	NUM
ejpam-6690	448	13	)	)	PUNCT
ejpam-6690	448	14	,	,	PUNCT
ejpam-6690	448	15	6690	6690	NUM
ejpam-6690	448	16	12	12	NUM
ejpam-6690	448	17	of	of	ADP
ejpam-6690	448	18	19	19	NUM
ejpam-6690	448	19	corollary	corollary	ADJ
ejpam-6690	448	20	5	5	NUM
ejpam-6690	448	21	.	.	PUNCT
ejpam-6690	448	22	spec(r̄	spec(r̄	NOUN
ejpam-6690	448	23	)	)	PUNCT
ejpam-6690	448	24	is	be	AUX
ejpam-6690	448	25	irreducible	irreducible	ADJ
ejpam-6690	448	26	(	(	PUNCT
ejpam-6690	448	27	disconnected	disconnected	ADJ
ejpam-6690	448	28	)	)	PUNCT
ejpam-6690	448	29	if	if	SCONJ
ejpam-6690	449	1	and	and	CCONJ
ejpam-6690	449	2	only	only	ADV
ejpam-6690	449	3	if	if	SCONJ
ejpam-6690	449	4	spec(r	spec(r	PROPN
ejpam-6690	449	5	)	)	PUNCT
ejpam-6690	449	6	is	be	AUX
ejpam-6690	449	7	irreducible	irreducible	ADJ
ejpam-6690	449	8	(	(	PUNCT
ejpam-6690	449	9	disconnected	disconnected	ADJ
ejpam-6690	449	10	)	)	PUNCT
ejpam-6690	449	11	.	.	PUNCT
ejpam-6690	450	1	proof	proof	NOUN
ejpam-6690	450	2	.	.	PUNCT
ejpam-6690	451	1	clearly	clearly	ADV
ejpam-6690	451	2	,	,	PUNCT
ejpam-6690	451	3	p	p	PROPN
ejpam-6690	451	4	∈	∈	PROPN
ejpam-6690	451	5	spec(r	spec(r	PROPN
ejpam-6690	451	6	)	)	PUNCT
ejpam-6690	451	7	if	if	SCONJ
ejpam-6690	451	8	and	and	CCONJ
ejpam-6690	451	9	only	only	ADV
ejpam-6690	451	10	if	if	SCONJ
ejpam-6690	451	11	p	p	X
ejpam-6690	451	12	/	/	SYM
ejpam-6690	451	13	g	g	PROPN
ejpam-6690	451	14	∈	∈	PROPN
ejpam-6690	451	15	spec(r̄	spec(r̄	PROPN
ejpam-6690	451	16	)	)	PUNCT
ejpam-6690	451	17	.	.	PUNCT
ejpam-6690	452	1	since	since	SCONJ
ejpam-6690	452	2	1r	1r	NUM
ejpam-6690	452	3	∈	∈	PROPN
ejpam-6690	452	4	g	g	NOUN
ejpam-6690	452	5	,	,	PUNCT
ejpam-6690	452	6	then	then	ADV
ejpam-6690	452	7	nil(r̄	nil(r̄	PROPN
ejpam-6690	452	8	)	)	PUNCT
ejpam-6690	452	9	=	=	PRON
ejpam-6690	452	10	{	{	PUNCT
ejpam-6690	452	11	rg	rg	NOUN
ejpam-6690	452	12	;	;	PUNCT
ejpam-6690	452	13	rmg	rmg	PROPN
ejpam-6690	452	14	=	=	PROPN
ejpam-6690	452	15	0	0	PROPN
ejpam-6690	452	16	,	,	PUNCT
ejpam-6690	452	17	for	for	ADP
ejpam-6690	452	18	some	some	DET
ejpam-6690	452	19	m	m	NOUN
ejpam-6690	452	20	∈	∈	NOUN
ejpam-6690	452	21	n	n	CCONJ
ejpam-6690	452	22	}	}	PUNCT
ejpam-6690	452	23	=	=	SYM
ejpam-6690	452	24	{	{	PUNCT
ejpam-6690	452	25	rg	rg	NOUN
ejpam-6690	452	26	;	;	PUNCT
ejpam-6690	452	27	rm	rm	PROPN
ejpam-6690	452	28	=	=	SYM
ejpam-6690	452	29	0	0	PROPN
ejpam-6690	452	30	,	,	PUNCT
ejpam-6690	452	31	for	for	ADP
ejpam-6690	452	32	some	some	DET
ejpam-6690	452	33	m	m	NOUN
ejpam-6690	452	34	∈	∈	NOUN
ejpam-6690	452	35	n	n	CCONJ
ejpam-6690	452	36	}	}	PUNCT
ejpam-6690	452	37	=	=	SYM
ejpam-6690	452	38	{	{	PUNCT
ejpam-6690	452	39	rg	rg	NOUN
ejpam-6690	452	40	;	;	PUNCT
ejpam-6690	452	41	r	r	NOUN
ejpam-6690	452	42	∈	∈	PROPN
ejpam-6690	452	43	nil(r	nil(r	NOUN
ejpam-6690	452	44	)	)	PUNCT
ejpam-6690	452	45	}	}	PUNCT
ejpam-6690	453	1	=	=	SYM
ejpam-6690	453	2	nil(r)/g	nil(r)/g	PROPN
ejpam-6690	453	3	.	.	PUNCT
ejpam-6690	454	1	so	so	ADV
ejpam-6690	454	2	nil(r	nil(r	ADJ
ejpam-6690	454	3	)	)	PUNCT
ejpam-6690	454	4	∈	∈	PROPN
ejpam-6690	454	5	spec(r	spec(r	PROPN
ejpam-6690	454	6	)	)	PUNCT
ejpam-6690	454	7	if	if	SCONJ
ejpam-6690	454	8	and	and	CCONJ
ejpam-6690	454	9	only	only	ADV
ejpam-6690	454	10	if	if	SCONJ
ejpam-6690	454	11	nil(r̄	nil(r̄	NOUN
ejpam-6690	454	12	)	)	PUNCT
ejpam-6690	454	13	=	=	PUNCT
ejpam-6690	455	1	nil(r)/g	nil(r)/g	PROPN
ejpam-6690	455	2	∈	∈	PROPN
ejpam-6690	455	3	spec(r̄	spec(r̄	PROPN
ejpam-6690	455	4	)	)	PUNCT
ejpam-6690	455	5	.	.	PUNCT
ejpam-6690	456	1	by	by	ADP
ejpam-6690	456	2	theorem	theorem	NOUN
ejpam-6690	456	3	8	8	NUM
ejpam-6690	456	4	,	,	PUNCT
ejpam-6690	456	5	spec(r	spec(r	PROPN
ejpam-6690	456	6	)	)	PUNCT
ejpam-6690	456	7	is	be	AUX
ejpam-6690	456	8	irreducible	irreducible	ADJ
ejpam-6690	456	9	if	if	SCONJ
ejpam-6690	456	10	and	and	CCONJ
ejpam-6690	456	11	only	only	ADV
ejpam-6690	456	12	if	if	SCONJ
ejpam-6690	456	13	spec(r̄	spec(r̄	PROPN
ejpam-6690	456	14	)	)	PUNCT
ejpam-6690	456	15	is	be	AUX
ejpam-6690	456	16	irreducible	irreducible	ADJ
ejpam-6690	456	17	.	.	PUNCT
ejpam-6690	457	1	since	since	SCONJ
ejpam-6690	457	2	(	(	PUNCT
ejpam-6690	457	3	r1	r1	PROPN
ejpam-6690	457	4	×	×	PROPN
ejpam-6690	457	5	r2)/g	r2)/g	ADV
ejpam-6690	457	6	∼=	∼=	PROPN
ejpam-6690	457	7	r1	r1	PROPN
ejpam-6690	457	8	/	/	SYM
ejpam-6690	457	9	g	g	NOUN
ejpam-6690	457	10	×	×	NOUN
ejpam-6690	457	11	r2	r2	PROPN
ejpam-6690	457	12	/	/	SYM
ejpam-6690	457	13	g	g	PROPN
ejpam-6690	457	14	,	,	PUNCT
ejpam-6690	457	15	it	it	PRON
ejpam-6690	457	16	is	be	AUX
ejpam-6690	457	17	clear	clear	ADJ
ejpam-6690	457	18	that	that	SCONJ
ejpam-6690	457	19	spec(r	spec(r	PROPN
ejpam-6690	457	20	)	)	PUNCT
ejpam-6690	457	21	is	be	AUX
ejpam-6690	457	22	disconnected	disconnect	VERB
ejpam-6690	457	23	if	if	SCONJ
ejpam-6690	457	24	and	and	CCONJ
ejpam-6690	457	25	only	only	ADV
ejpam-6690	457	26	if	if	SCONJ
ejpam-6690	457	27	spec(r̄	spec(r̄	PROPN
ejpam-6690	457	28	)	)	PUNCT
ejpam-6690	457	29	is	be	AUX
ejpam-6690	457	30	disconnected	disconnect	VERB
ejpam-6690	457	31	.	.	PUNCT
ejpam-6690	457	32	example	example	NOUN
ejpam-6690	458	1	2	2	NUM
ejpam-6690	458	2	.	.	PUNCT
ejpam-6690	458	3	let	let	VERB
ejpam-6690	458	4	r	r	NOUN
ejpam-6690	458	5	=	=	PUNCT
ejpam-6690	458	6	z	z	PROPN
ejpam-6690	458	7	and	and	CCONJ
ejpam-6690	458	8	g	g	NOUN
ejpam-6690	458	9	=	=	PUNCT
ejpam-6690	458	10	{	{	PUNCT
ejpam-6690	458	11	±1	±1	NOUN
ejpam-6690	458	12	}	}	PUNCT
ejpam-6690	458	13	.	.	PUNCT
ejpam-6690	459	1	then	then	ADV
ejpam-6690	459	2	r̄	r̄	NOUN
ejpam-6690	459	3	=	=	SYM
ejpam-6690	459	4	{	{	PUNCT
ejpam-6690	459	5	{	{	PUNCT
ejpam-6690	459	6	±a	±a	PROPN
ejpam-6690	459	7	}	}	PUNCT
ejpam-6690	459	8	;	;	PUNCT
ejpam-6690	459	9	a	a	DET
ejpam-6690	459	10	∈	∈	PROPN
ejpam-6690	459	11	z	z	X
ejpam-6690	459	12	}	}	PUNCT
ejpam-6690	459	13	and	and	CCONJ
ejpam-6690	459	14	{	{	PUNCT
ejpam-6690	459	15	±a	±a	PROPN
ejpam-6690	459	16	}	}	PUNCT
ejpam-6690	459	17	⊕	⊕	PROPN
ejpam-6690	459	18	{	{	PUNCT
ejpam-6690	459	19	±b	±b	PROPN
ejpam-6690	459	20	}	}	PUNCT
ejpam-6690	459	21	=	=	SYM
ejpam-6690	459	22	{	{	PUNCT
ejpam-6690	459	23	{	{	PUNCT
ejpam-6690	459	24	±(a	±(a	NOUN
ejpam-6690	459	25	+	+	X
ejpam-6690	459	26	b	b	X
ejpam-6690	459	27	)	)	PUNCT
ejpam-6690	459	28	}	}	PUNCT
ejpam-6690	459	29	,	,	PUNCT
ejpam-6690	459	30	{	{	PUNCT
ejpam-6690	459	31	±(a	±(a	NOUN
ejpam-6690	459	32	−	−	PROPN
ejpam-6690	459	33	b	b	NOUN
ejpam-6690	459	34	)	)	PUNCT
ejpam-6690	459	35	}	}	PUNCT
ejpam-6690	459	36	}	}	PUNCT
ejpam-6690	459	37	,	,	PUNCT
ejpam-6690	459	38	{	{	PUNCT
ejpam-6690	459	39	±a	±a	PROPN
ejpam-6690	459	40	}	}	PUNCT
ejpam-6690	459	41	⊙	⊙	X
ejpam-6690	459	42	{	{	PUNCT
ejpam-6690	459	43	±b	±b	PROPN
ejpam-6690	459	44	}	}	PUNCT
ejpam-6690	459	45	=	=	SYM
ejpam-6690	459	46	{	{	PUNCT
ejpam-6690	459	47	±ab	±ab	NOUN
ejpam-6690	459	48	}	}	PUNCT
ejpam-6690	459	49	.	.	PUNCT
ejpam-6690	460	1	for	for	ADP
ejpam-6690	460	2	every	every	DET
ejpam-6690	460	3	prime	prime	NOUN
ejpam-6690	460	4	p	p	PROPN
ejpam-6690	460	5	∈	∈	PROPN
ejpam-6690	460	6	z	z	NOUN
ejpam-6690	460	7	;	;	PUNCT
ejpam-6690	460	8	(	(	PUNCT
ejpam-6690	460	9	p)/g	p)/g	PROPN
ejpam-6690	460	10	=	=	SYM
ejpam-6690	460	11	{	{	PUNCT
ejpam-6690	460	12	{	{	PUNCT
ejpam-6690	460	13	±kp	±kp	PROPN
ejpam-6690	460	14	}	}	PUNCT
ejpam-6690	460	15	;	;	PUNCT
ejpam-6690	460	16	k	k	PROPN
ejpam-6690	460	17	∈	∈	PROPN
ejpam-6690	460	18	n	n	PART
ejpam-6690	460	19	∪	∪	X
ejpam-6690	460	20	{	{	PUNCT
ejpam-6690	460	21	0	0	NUM
ejpam-6690	460	22	}	}	PUNCT
ejpam-6690	460	23	}	}	PUNCT
ejpam-6690	460	24	.	.	PUNCT
ejpam-6690	461	1	also	also	ADV
ejpam-6690	461	2	z	z	PROPN
ejpam-6690	461	3	is	be	AUX
ejpam-6690	461	4	integral	integral	ADJ
ejpam-6690	461	5	domain	domain	NOUN
ejpam-6690	461	6	therefore	therefore	ADV
ejpam-6690	461	7	,	,	PUNCT
ejpam-6690	461	8	r̄	r̄	NOUN
ejpam-6690	461	9	is	be	AUX
ejpam-6690	461	10	hyperdomain	hyperdomain	ADJ
ejpam-6690	461	11	and	and	CCONJ
ejpam-6690	461	12	nil(r̄	nil(r̄	NOUN
ejpam-6690	461	13	)	)	PUNCT
ejpam-6690	461	14	is	be	AUX
ejpam-6690	461	15	a	a	DET
ejpam-6690	461	16	prime	prime	ADJ
ejpam-6690	461	17	hyperideal	hyperideal	NOUN
ejpam-6690	461	18	of	of	ADP
ejpam-6690	461	19	r̄.	r̄.	PROPN
ejpam-6690	461	20	so	so	ADV
ejpam-6690	461	21	spec(r̄	spec(r̄	PROPN
ejpam-6690	461	22	)	)	PUNCT
ejpam-6690	461	23	is	be	AUX
ejpam-6690	461	24	irreducible	irreducible	ADJ
ejpam-6690	461	25	and	and	CCONJ
ejpam-6690	461	26	connected	connect	VERB
ejpam-6690	461	27	.	.	PUNCT
ejpam-6690	462	1	example	example	NOUN
ejpam-6690	463	1	3	3	NUM
ejpam-6690	463	2	.	.	PUNCT
ejpam-6690	464	1	in	in	ADP
ejpam-6690	464	2	example	example	NOUN
ejpam-6690	464	3	2	2	NUM
ejpam-6690	464	4	,	,	PUNCT
ejpam-6690	464	5	if	if	SCONJ
ejpam-6690	464	6	f	f	PROPN
ejpam-6690	464	7	is	be	AUX
ejpam-6690	464	8	a	a	DET
ejpam-6690	464	9	field	field	NOUN
ejpam-6690	464	10	and	and	CCONJ
ejpam-6690	464	11	r	r	NOUN
ejpam-6690	464	12	=	=	SYM
ejpam-6690	464	13	f	f	X
ejpam-6690	465	1	[	[	X
ejpam-6690	465	2	x	x	X
ejpam-6690	465	3	]	]	X
ejpam-6690	465	4	and	and	CCONJ
ejpam-6690	465	5	g	g	NOUN
ejpam-6690	465	6	=	=	PUNCT
ejpam-6690	465	7	f−{0	f−{0	ADV
ejpam-6690	465	8	}	}	PUNCT
ejpam-6690	465	9	,	,	PUNCT
ejpam-6690	465	10	then	then	ADV
ejpam-6690	465	11	spec(r̄	spec(r̄	PROPN
ejpam-6690	465	12	)	)	PUNCT
ejpam-6690	466	1	=	=	NOUN
ejpam-6690	466	2	{	{	PUNCT
ejpam-6690	466	3	(	(	PUNCT
ejpam-6690	466	4	f(x))/g	f(x))/g	PROPN
ejpam-6690	466	5	;	;	PUNCT
ejpam-6690	466	6	f(x	f(x	PROPN
ejpam-6690	466	7	)	)	PUNCT
ejpam-6690	466	8	∈	∈	PROPN
ejpam-6690	467	1	f	f	X
ejpam-6690	468	1	[	[	X
ejpam-6690	468	2	x	x	X
ejpam-6690	468	3	]	]	X
ejpam-6690	468	4	is	be	AUX
ejpam-6690	468	5	irreducible	irreducible	ADJ
ejpam-6690	468	6	}	}	PUNCT
ejpam-6690	468	7	.	.	PUNCT
ejpam-6690	469	1	4	4	X
ejpam-6690	469	2	.	.	X
ejpam-6690	469	3	from	from	ADP
ejpam-6690	469	4	zariski	zariski	NOUN
ejpam-6690	469	5	topology	topology	NOUN
ejpam-6690	469	6	of	of	ADP
ejpam-6690	469	7	hyperrings	hyperring	NOUN
ejpam-6690	469	8	to	to	PART
ejpam-6690	469	9	zariski	zariski	VERB
ejpam-6690	469	10	topology	topology	NOUN
ejpam-6690	469	11	of	of	ADP
ejpam-6690	469	12	rings	ring	NOUN
ejpam-6690	469	13	this	this	DET
ejpam-6690	469	14	section	section	NOUN
ejpam-6690	469	15	develops	develop	VERB
ejpam-6690	469	16	a	a	DET
ejpam-6690	469	17	categorical	categorical	ADJ
ejpam-6690	469	18	and	and	CCONJ
ejpam-6690	469	19	functorial	functorial	NOUN
ejpam-6690	469	20	approach	approach	NOUN
ejpam-6690	469	21	to	to	ADP
ejpam-6690	469	22	the	the	DET
ejpam-6690	469	23	zariski	zariski	ADJ
ejpam-6690	469	24	topology	topology	NOUN
ejpam-6690	469	25	of	of	ADP
ejpam-6690	469	26	hyperrings	hyperring	NOUN
ejpam-6690	469	27	.	.	PUNCT
ejpam-6690	470	1	we	we	PRON
ejpam-6690	470	2	explore	explore	VERB
ejpam-6690	470	3	the	the	DET
ejpam-6690	470	4	role	role	NOUN
ejpam-6690	470	5	of	of	ADP
ejpam-6690	470	6	the	the	DET
ejpam-6690	470	7	fundamental	fundamental	ADJ
ejpam-6690	470	8	relation	relation	NOUN
ejpam-6690	470	9	γ∗	γ∗	NOUN
ejpam-6690	470	10	in	in	ADP
ejpam-6690	470	11	connecting	connect	VERB
ejpam-6690	470	12	hyperring	hyperre	VERB
ejpam-6690	470	13	spectra	spectra	NOUN
ejpam-6690	470	14	to	to	ADP
ejpam-6690	470	15	their	their	PRON
ejpam-6690	470	16	classical	classical	ADJ
ejpam-6690	470	17	counterparts	counterpart	NOUN
ejpam-6690	470	18	and	and	CCONJ
ejpam-6690	470	19	introduce	introduce	VERB
ejpam-6690	470	20	a	a	DET
ejpam-6690	470	21	topology	topology	NOUN
ejpam-6690	470	22	on	on	ADP
ejpam-6690	470	23	prime	prime	ADJ
ejpam-6690	470	24	strongly	strongly	ADV
ejpam-6690	470	25	regular	regular	ADJ
ejpam-6690	470	26	relations	relation	NOUN
ejpam-6690	470	27	.	.	PUNCT
ejpam-6690	471	1	definition	definition	NOUN
ejpam-6690	471	2	6	6	NUM
ejpam-6690	471	3	.	.	PUNCT
ejpam-6690	472	1	[	[	X
ejpam-6690	472	2	18	18	NUM
ejpam-6690	472	3	]	]	PUNCT
ejpam-6690	472	4	let	let	VERB
ejpam-6690	472	5	u	u	PRON
ejpam-6690	472	6	be	be	AUX
ejpam-6690	472	7	the	the	DET
ejpam-6690	472	8	set	set	NOUN
ejpam-6690	472	9	of	of	ADP
ejpam-6690	472	10	all	all	DET
ejpam-6690	472	11	finite	finite	ADJ
ejpam-6690	472	12	sums	sum	NOUN
ejpam-6690	472	13	of	of	ADP
ejpam-6690	472	14	finite	finite	ADJ
ejpam-6690	472	15	products	product	NOUN
ejpam-6690	472	16	of	of	ADP
ejpam-6690	472	17	elements	element	NOUN
ejpam-6690	472	18	of	of	ADP
ejpam-6690	472	19	a	a	DET
ejpam-6690	472	20	general	general	ADJ
ejpam-6690	472	21	hyperring	hyperre	VERB
ejpam-6690	472	22	r.	r.	PROPN
ejpam-6690	472	23	then	then	ADV
ejpam-6690	472	24	(	(	PUNCT
ejpam-6690	472	25	a	a	PRON
ejpam-6690	472	26	,	,	PUNCT
ejpam-6690	472	27	b	b	NOUN
ejpam-6690	472	28	)	)	PUNCT
ejpam-6690	472	29	∈	∈	NOUN
ejpam-6690	472	30	γ∗	γ∗	NOUN
ejpam-6690	472	31	if	if	SCONJ
ejpam-6690	472	32	and	and	CCONJ
ejpam-6690	472	33	only	only	ADV
ejpam-6690	472	34	if	if	SCONJ
ejpam-6690	472	35	there	there	PRON
ejpam-6690	472	36	exists	exist	VERB
ejpam-6690	472	37	(	(	PUNCT
ejpam-6690	472	38	a	a	DET
ejpam-6690	472	39	=	=	SYM
ejpam-6690	472	40	z1	z1	VERB
ejpam-6690	472	41	,	,	PUNCT
ejpam-6690	472	42	z2	z2	PROPN
ejpam-6690	472	43	,	,	PUNCT
ejpam-6690	472	44	...	...	PUNCT
ejpam-6690	472	45	,	,	PUNCT
ejpam-6690	472	46	b	b	X
ejpam-6690	472	47	=	=	SYM
ejpam-6690	472	48	zn+1	zn+1	X
ejpam-6690	472	49	)	)	PUNCT
ejpam-6690	472	50	∈	∈	NOUN
ejpam-6690	472	51	rn+1	rn+1	NUM
ejpam-6690	472	52	and	and	CCONJ
ejpam-6690	472	53	u1	u1	NOUN
ejpam-6690	472	54	,	,	PUNCT
ejpam-6690	472	55	u2	u2	PROPN
ejpam-6690	472	56	,	,	PUNCT
ejpam-6690	472	57	...	...	PUNCT
ejpam-6690	472	58	,	,	PUNCT
ejpam-6690	472	59	un	un	PROPN
ejpam-6690	472	60	∈	∈	PROPN
ejpam-6690	472	61	u	u	PROPN
ejpam-6690	472	62	such	such	ADJ
ejpam-6690	472	63	that	that	SCONJ
ejpam-6690	472	64	{	{	PUNCT
ejpam-6690	472	65	zi	zi	NOUN
ejpam-6690	472	66	,	,	PUNCT
ejpam-6690	472	67	zi+1	zi+1	CCONJ
ejpam-6690	472	68	}	}	PUNCT
ejpam-6690	472	69	⊆	⊆	NUM
ejpam-6690	472	70	ui	ui	NOUN
ejpam-6690	472	71	,	,	PUNCT
ejpam-6690	472	72	for	for	ADP
ejpam-6690	472	73	any	any	DET
ejpam-6690	472	74	i	i	PROPN
ejpam-6690	472	75	∈	∈	PROPN
ejpam-6690	472	76	{	{	PUNCT
ejpam-6690	472	77	1	1	NUM
ejpam-6690	472	78	,	,	PUNCT
ejpam-6690	472	79	2	2	NUM
ejpam-6690	472	80	,	,	PUNCT
ejpam-6690	472	81	...	...	PUNCT
ejpam-6690	472	82	,	,	PUNCT
ejpam-6690	472	83	n	n	CCONJ
ejpam-6690	472	84	}	}	PUNCT
ejpam-6690	472	85	.	.	PUNCT
ejpam-6690	473	1	the	the	DET
ejpam-6690	473	2	relation	relation	NOUN
ejpam-6690	473	3	γ∗	γ∗	NOUN
ejpam-6690	473	4	is	be	AUX
ejpam-6690	473	5	the	the	DET
ejpam-6690	473	6	smallest	small	ADJ
ejpam-6690	473	7	equivalence	equivalence	NOUN
ejpam-6690	473	8	relation	relation	NOUN
ejpam-6690	473	9	on	on	ADP
ejpam-6690	473	10	the	the	DET
ejpam-6690	473	11	general	general	ADJ
ejpam-6690	473	12	hyperring	hyperring	NOUN
ejpam-6690	473	13	r	r	NOUN
ejpam-6690	473	14	such	such	ADJ
ejpam-6690	473	15	that	that	SCONJ
ejpam-6690	473	16	the	the	DET
ejpam-6690	473	17	quotient	quotient	NOUN
ejpam-6690	473	18	r	r	NOUN
ejpam-6690	473	19	/	/	SYM
ejpam-6690	473	20	γ∗	γ∗	NOUN
ejpam-6690	473	21	is	be	AUX
ejpam-6690	473	22	a	a	DET
ejpam-6690	473	23	ring	ring	NOUN
ejpam-6690	473	24	.	.	PUNCT
ejpam-6690	474	1	r	r	X
ejpam-6690	474	2	/	/	SYM
ejpam-6690	474	3	γ∗	γ∗	NOUN
ejpam-6690	474	4	is	be	AUX
ejpam-6690	474	5	called	call	VERB
ejpam-6690	474	6	the	the	DET
ejpam-6690	474	7	fundamental	fundamental	ADJ
ejpam-6690	474	8	ring	ring	NOUN
ejpam-6690	474	9	[	[	X
ejpam-6690	474	10	15	15	NUM
ejpam-6690	474	11	]	]	PUNCT
ejpam-6690	474	12	.	.	PUNCT
ejpam-6690	475	1	proposition	proposition	NOUN
ejpam-6690	475	2	9	9	NUM
ejpam-6690	475	3	.	.	PUNCT
ejpam-6690	476	1	if	if	SCONJ
ejpam-6690	476	2	ρ	ρ	PROPN
ejpam-6690	476	3	is	be	AUX
ejpam-6690	476	4	a	a	DET
ejpam-6690	476	5	strongly	strongly	ADV
ejpam-6690	476	6	regular	regular	ADJ
ejpam-6690	476	7	relation	relation	NOUN
ejpam-6690	476	8	on	on	ADP
ejpam-6690	476	9	krasner	krasner	NOUN
ejpam-6690	476	10	hyperring	hyperre	VERB
ejpam-6690	476	11	r	r	NOUN
ejpam-6690	476	12	,	,	PUNCT
ejpam-6690	476	13	then	then	ADV
ejpam-6690	476	14	ρ(0	ρ(0	PROPN
ejpam-6690	476	15	)	)	PUNCT
ejpam-6690	476	16	=	=	PRON
ejpam-6690	476	17	{	{	PUNCT
ejpam-6690	476	18	x	x	PUNCT
ejpam-6690	476	19	∈	∈	PROPN
ejpam-6690	476	20	r	r	NOUN
ejpam-6690	476	21	;	;	PUNCT
ejpam-6690	476	22	ρ(x	ρ(x	NUM
ejpam-6690	476	23	)	)	PUNCT
ejpam-6690	476	24	=	=	SYM
ejpam-6690	476	25	0r	0r	X
ejpam-6690	476	26	/	/	SYM
ejpam-6690	476	27	ρ	ρ	PROPN
ejpam-6690	476	28	}	}	PUNCT
ejpam-6690	476	29	is	be	AUX
ejpam-6690	476	30	a	a	DET
ejpam-6690	476	31	normal	normal	ADJ
ejpam-6690	476	32	hyperideal	hyperideal	NOUN
ejpam-6690	476	33	of	of	ADP
ejpam-6690	476	34	r.	r.	PROPN
ejpam-6690	476	35	proof	proof	NOUN
ejpam-6690	476	36	.	.	PUNCT
ejpam-6690	477	1	for	for	ADP
ejpam-6690	477	2	every	every	DET
ejpam-6690	477	3	x	x	NOUN
ejpam-6690	477	4	,	,	PUNCT
ejpam-6690	477	5	y	y	PROPN
ejpam-6690	477	6	∈	∈	PROPN
ejpam-6690	477	7	ρ(0	ρ(0	PROPN
ejpam-6690	477	8	)	)	PUNCT
ejpam-6690	477	9	;	;	PUNCT
ejpam-6690	477	10	ρ(x	ρ(x	PROPN
ejpam-6690	477	11	−	−	PROPN
ejpam-6690	477	12	y	y	NOUN
ejpam-6690	477	13	)	)	PUNCT
ejpam-6690	477	14	=	=	SYM
ejpam-6690	477	15	ρ(x	ρ(x	NOUN
ejpam-6690	477	16	)	)	PUNCT
ejpam-6690	477	17	−	−	ADP
ejpam-6690	477	18	ρ(y	ρ(y	NOUN
ejpam-6690	477	19	)	)	PUNCT
ejpam-6690	477	20	=	=	SYM
ejpam-6690	477	21	0r	0r	X
ejpam-6690	477	22	/	/	SYM
ejpam-6690	477	23	ρ	ρ	PROPN
ejpam-6690	477	24	.	.	PUNCT
ejpam-6690	478	1	so	so	ADV
ejpam-6690	478	2	x	x	PUNCT
ejpam-6690	478	3	−	−	NOUN
ejpam-6690	478	4	y	y	PROPN
ejpam-6690	478	5	⊆	⊆	NUM
ejpam-6690	478	6	ρ(0	ρ(0	PROPN
ejpam-6690	478	7	)	)	PUNCT
ejpam-6690	478	8	.	.	PUNCT
ejpam-6690	479	1	let	let	VERB
ejpam-6690	479	2	r	r	NOUN
ejpam-6690	479	3	∈	∈	NOUN
ejpam-6690	479	4	r	r	NOUN
ejpam-6690	479	5	,	,	PUNCT
ejpam-6690	479	6	then	then	ADV
ejpam-6690	479	7	ρ(rx	ρ(rx	NUM
ejpam-6690	479	8	)	)	PUNCT
ejpam-6690	479	9	=	=	SYM
ejpam-6690	479	10	ρ(r)ρ(x	ρ(r)ρ(x	NOUN
ejpam-6690	479	11	)	)	PUNCT
ejpam-6690	479	12	=	=	PUNCT
ejpam-6690	479	13	ρ(r)0r	ρ(r)0r	PROPN
ejpam-6690	479	14	/	/	SYM
ejpam-6690	479	15	ρ	ρ	PROPN
ejpam-6690	479	16	=	=	SYM
ejpam-6690	479	17	0r	0r	PROPN
ejpam-6690	479	18	/	/	SYM
ejpam-6690	479	19	ρ	ρ	PROPN
ejpam-6690	479	20	.	.	PUNCT
ejpam-6690	480	1	so	so	ADV
ejpam-6690	480	2	rx	rx	VERB
ejpam-6690	480	3	∈	∈	PROPN
ejpam-6690	480	4	ρ(0	ρ(0	PROPN
ejpam-6690	480	5	)	)	PUNCT
ejpam-6690	480	6	,	,	PUNCT
ejpam-6690	480	7	and	and	CCONJ
ejpam-6690	480	8	similarly	similarly	ADV
ejpam-6690	480	9	xr	xr	PROPN
ejpam-6690	480	10	∈	∈	PROPN
ejpam-6690	480	11	ρ(0	ρ(0	PROPN
ejpam-6690	480	12	)	)	PUNCT
ejpam-6690	480	13	.	.	PUNCT
ejpam-6690	481	1	also	also	ADV
ejpam-6690	481	2	,	,	PUNCT
ejpam-6690	481	3	since	since	SCONJ
ejpam-6690	481	4	γ∗	γ∗	PROPN
ejpam-6690	481	5	⊆	⊆	NUM
ejpam-6690	481	6	ρ	ρ	NOUN
ejpam-6690	481	7	and	and	CCONJ
ejpam-6690	481	8	r	r	NOUN
ejpam-6690	481	9	−	−	NOUN
ejpam-6690	481	10	r	r	NOUN
ejpam-6690	481	11	⊆	⊆	NUM
ejpam-6690	481	12	γ∗(0	γ∗(0	NOUN
ejpam-6690	481	13	)	)	PUNCT
ejpam-6690	481	14	,	,	PUNCT
ejpam-6690	481	15	then	then	ADV
ejpam-6690	481	16	r	r	NOUN
ejpam-6690	481	17	−	−	NOUN
ejpam-6690	481	18	r	r	NOUN
ejpam-6690	481	19	⊆	⊆	NUM
ejpam-6690	481	20	ρ(0	ρ(0	NOUN
ejpam-6690	481	21	)	)	PUNCT
ejpam-6690	481	22	and	and	CCONJ
ejpam-6690	481	23	r	r	NOUN
ejpam-6690	481	24	+	+	CCONJ
ejpam-6690	481	25	ρ(0	ρ(0	PROPN
ejpam-6690	481	26	)	)	PUNCT
ejpam-6690	481	27	−	−	NOUN
ejpam-6690	481	28	r	r	NOUN
ejpam-6690	481	29	⊆	⊆	NUM
ejpam-6690	481	30	ρ(0	ρ(0	NOUN
ejpam-6690	481	31	)	)	PUNCT
ejpam-6690	481	32	.	.	PUNCT
ejpam-6690	482	1	thus	thus	ADV
ejpam-6690	482	2	ρ(0	ρ(0	NOUN
ejpam-6690	482	3	)	)	PUNCT
ejpam-6690	482	4	is	be	AUX
ejpam-6690	482	5	normal	normal	ADJ
ejpam-6690	482	6	.	.	PUNCT
ejpam-6690	483	1	clearly	clearly	ADV
ejpam-6690	483	2	every	every	DET
ejpam-6690	483	3	hyperideal	hyperideal	NOUN
ejpam-6690	483	4	i	i	PRON
ejpam-6690	483	5	of	of	ADP
ejpam-6690	483	6	r	r	NOUN
ejpam-6690	483	7	containing	contain	VERB
ejpam-6690	483	8	γ∗(0	γ∗(0	NOUN
ejpam-6690	483	9	)	)	PUNCT
ejpam-6690	483	10	is	be	AUX
ejpam-6690	483	11	normal	normal	ADJ
ejpam-6690	483	12	,	,	PUNCT
ejpam-6690	483	13	and	and	CCONJ
ejpam-6690	483	14	r	r	X
ejpam-6690	483	15	/	/	SYM
ejpam-6690	483	16	i	i	PRON
ejpam-6690	483	17	is	be	AUX
ejpam-6690	483	18	krasner	krasner	NOUN
ejpam-6690	483	19	hyperring	hyperring	PROPN
ejpam-6690	483	20	.	.	PUNCT
ejpam-6690	484	1	denote	denote	VERB
ejpam-6690	484	2	the	the	DET
ejpam-6690	484	3	set	set	NOUN
ejpam-6690	484	4	of	of	ADP
ejpam-6690	484	5	all	all	DET
ejpam-6690	484	6	strongly	strongly	ADV
ejpam-6690	484	7	regular	regular	ADJ
ejpam-6690	484	8	relations	relation	NOUN
ejpam-6690	484	9	on	on	ADP
ejpam-6690	484	10	r	r	NOUN
ejpam-6690	484	11	,	,	PUNCT
ejpam-6690	484	12	by	by	ADP
ejpam-6690	484	13	sr(r	sr(r	NOUN
ejpam-6690	484	14	)	)	PUNCT
ejpam-6690	484	15	and	and	CCONJ
ejpam-6690	484	16	the	the	DET
ejpam-6690	484	17	set	set	NOUN
ejpam-6690	484	18	of	of	ADP
ejpam-6690	484	19	all	all	DET
ejpam-6690	484	20	hyperideals	hyperideal	NOUN
ejpam-6690	484	21	containing	contain	VERB
ejpam-6690	484	22	γ∗(0	γ∗(0	NOUN
ejpam-6690	484	23	)	)	PUNCT
ejpam-6690	484	24	,	,	PUNCT
ejpam-6690	484	25	by	by	ADP
ejpam-6690	484	26	i(γ∗(0	i(γ∗(0	NOUN
ejpam-6690	484	27	)	)	PUNCT
ejpam-6690	484	28	)	)	PUNCT
ejpam-6690	484	29	.	.	PUNCT
ejpam-6690	485	1	lemma	lemma	PROPN
ejpam-6690	485	2	4	4	X
ejpam-6690	485	3	.	.	PUNCT
ejpam-6690	486	1	if	if	SCONJ
ejpam-6690	486	2	ρ	ρ	PROPN
ejpam-6690	486	3	∈	∈	PROPN
ejpam-6690	486	4	sr(r	sr(r	NOUN
ejpam-6690	486	5	)	)	PUNCT
ejpam-6690	486	6	,	,	PUNCT
ejpam-6690	486	7	then	then	ADV
ejpam-6690	486	8	ρ(x	ρ(x	NUM
ejpam-6690	486	9	)	)	PUNCT
ejpam-6690	486	10	=	=	SYM
ejpam-6690	487	1	x+	x+	PUNCT
ejpam-6690	487	2	ρ(0	ρ(0	PROPN
ejpam-6690	487	3	)	)	PUNCT
ejpam-6690	487	4	for	for	ADP
ejpam-6690	487	5	every	every	DET
ejpam-6690	487	6	x	x	PROPN
ejpam-6690	487	7	∈	∈	PROPN
ejpam-6690	487	8	r.	r.	NOUN
ejpam-6690	487	9	proof	proof	NOUN
ejpam-6690	487	10	.	.	PUNCT
ejpam-6690	488	1	if	if	SCONJ
ejpam-6690	488	2	z	z	PROPN
ejpam-6690	488	3	∈	∈	PROPN
ejpam-6690	488	4	ρ(x	ρ(x	PROPN
ejpam-6690	488	5	)	)	PUNCT
ejpam-6690	488	6	,	,	PUNCT
ejpam-6690	488	7	then	then	ADV
ejpam-6690	488	8	−x+z	−x+z	PROPN
ejpam-6690	488	9	⊆	⊆	NUM
ejpam-6690	488	10	ρ(0	ρ(0	NOUN
ejpam-6690	488	11	)	)	PUNCT
ejpam-6690	488	12	and	and	CCONJ
ejpam-6690	488	13	z	z	NOUN
ejpam-6690	488	14	∈	∈	PROPN
ejpam-6690	488	15	x+(−x+z	x+(−x+z	VERB
ejpam-6690	488	16	)	)	PUNCT
ejpam-6690	488	17	⊆	⊆	NUM
ejpam-6690	488	18	x+ρ(0	x+ρ(0	NUM
ejpam-6690	488	19	)	)	PUNCT
ejpam-6690	488	20	.	.	PUNCT
ejpam-6690	489	1	so	so	ADV
ejpam-6690	489	2	z	z	PROPN
ejpam-6690	489	3	∈	∈	PROPN
ejpam-6690	489	4	x+ρ(0	x+ρ(0	PROPN
ejpam-6690	489	5	)	)	PUNCT
ejpam-6690	489	6	and	and	CCONJ
ejpam-6690	489	7	ρ(x	ρ(x	NUM
ejpam-6690	489	8	)	)	PUNCT
ejpam-6690	490	1	⊆	⊆	NUM
ejpam-6690	490	2	x	x	SYM
ejpam-6690	490	3	+	+	PUNCT
ejpam-6690	490	4	ρ(0	ρ(0	PROPN
ejpam-6690	490	5	)	)	PUNCT
ejpam-6690	490	6	.	.	PUNCT
ejpam-6690	491	1	conversely	conversely	ADV
ejpam-6690	491	2	,	,	PUNCT
ejpam-6690	491	3	if	if	SCONJ
ejpam-6690	491	4	z	z	NOUN
ejpam-6690	491	5	∈	∈	VERB
ejpam-6690	491	6	x	x	PUNCT
ejpam-6690	491	7	+	+	PUNCT
ejpam-6690	491	8	ρ(0	ρ(0	PROPN
ejpam-6690	491	9	)	)	PUNCT
ejpam-6690	491	10	,	,	PUNCT
ejpam-6690	491	11	then	then	ADV
ejpam-6690	491	12	z	z	NOUN
ejpam-6690	491	13	∈	∈	PROPN
ejpam-6690	491	14	x	x	PUNCT
ejpam-6690	492	1	+	+	NUM
ejpam-6690	492	2	y	y	NOUN
ejpam-6690	492	3	,	,	PUNCT
ejpam-6690	492	4	for	for	ADP
ejpam-6690	492	5	some	some	DET
ejpam-6690	492	6	y	y	PROPN
ejpam-6690	492	7	∈	∈	PROPN
ejpam-6690	492	8	ρ(0	ρ(0	PROPN
ejpam-6690	492	9	)	)	PUNCT
ejpam-6690	492	10	.	.	PUNCT
ejpam-6690	493	1	so	so	ADV
ejpam-6690	493	2	ρ(z	ρ(z	NUM
ejpam-6690	493	3	)	)	PUNCT
ejpam-6690	493	4	=	=	SYM
ejpam-6690	493	5	ρ(x	ρ(x	NOUN
ejpam-6690	493	6	)	)	PUNCT
ejpam-6690	493	7	and	and	CCONJ
ejpam-6690	493	8	z	z	NOUN
ejpam-6690	493	9	∈	∈	PROPN
ejpam-6690	493	10	ρ(x	ρ(x	PROPN
ejpam-6690	493	11	)	)	PUNCT
ejpam-6690	493	12	.	.	PUNCT
ejpam-6690	494	1	therefore	therefore	ADV
ejpam-6690	494	2	,	,	PUNCT
ejpam-6690	494	3	x+	x+	PROPN
ejpam-6690	494	4	ρ(0	ρ(0	PROPN
ejpam-6690	494	5	)	)	PUNCT
ejpam-6690	494	6	⊆	⊆	NUM
ejpam-6690	494	7	ρ(x	ρ(x	NUM
ejpam-6690	494	8	)	)	PUNCT
ejpam-6690	494	9	.	.	PUNCT
ejpam-6690	495	1	b.	b.	PROPN
ejpam-6690	495	2	afshar	afshar	PROPN
ejpam-6690	495	3	,	,	PUNCT
ejpam-6690	495	4	r.	r.	PROPN
ejpam-6690	495	5	ameri	ameri	PROPN
ejpam-6690	495	6	,	,	PUNCT
ejpam-6690	495	7	m.	m.	PROPN
ejpam-6690	495	8	al	al	PROPN
ejpam-6690	495	9	-	-	PUNCT
ejpam-6690	495	10	tahan	tahan	PROPN
ejpam-6690	495	11	/	/	SYM
ejpam-6690	495	12	eur	eur	PROPN
ejpam-6690	495	13	.	.	PUNCT
ejpam-6690	496	1	j.	j.	PROPN
ejpam-6690	496	2	pure	pure	PROPN
ejpam-6690	496	3	appl	appl	PROPN
ejpam-6690	496	4	.	.	PROPN
ejpam-6690	496	5	math	math	PROPN
ejpam-6690	496	6	,	,	PUNCT
ejpam-6690	496	7	18	18	NUM
ejpam-6690	496	8	(	(	PUNCT
ejpam-6690	496	9	4	4	NUM
ejpam-6690	496	10	)	)	PUNCT
ejpam-6690	496	11	(	(	PUNCT
ejpam-6690	496	12	2025	2025	NUM
ejpam-6690	496	13	)	)	PUNCT
ejpam-6690	496	14	,	,	PUNCT
ejpam-6690	496	15	6690	6690	NUM
ejpam-6690	496	16	13	13	NUM
ejpam-6690	496	17	of	of	ADP
ejpam-6690	496	18	19	19	NUM
ejpam-6690	496	19	lemma	lemma	PROPN
ejpam-6690	496	20	5	5	NUM
ejpam-6690	496	21	.	.	PUNCT
ejpam-6690	497	1	if	if	SCONJ
ejpam-6690	497	2	i	i	PRON
ejpam-6690	497	3	∈	∈	PROPN
ejpam-6690	497	4	i(γ∗(0	i(γ∗(0	NOUN
ejpam-6690	497	5	)	)	PUNCT
ejpam-6690	497	6	)	)	PUNCT
ejpam-6690	497	7	,	,	PUNCT
ejpam-6690	497	8	then	then	ADV
ejpam-6690	497	9	the	the	DET
ejpam-6690	497	10	congruence	congruence	PROPN
ejpam-6690	497	11	relation	relation	NOUN
ejpam-6690	497	12	modulo	modulo	VERB
ejpam-6690	497	13	i	i	PRON
ejpam-6690	497	14	is	be	AUX
ejpam-6690	497	15	strongly	strongly	ADV
ejpam-6690	497	16	regular	regular	ADJ
ejpam-6690	497	17	.	.	PUNCT
ejpam-6690	498	1	proof	proof	NOUN
ejpam-6690	498	2	.	.	PUNCT
ejpam-6690	499	1	let	let	VERB
ejpam-6690	499	2	x	x	PRON
ejpam-6690	499	3	,	,	PUNCT
ejpam-6690	499	4	y	y	PROPN
ejpam-6690	499	5	,	,	PUNCT
ejpam-6690	499	6	z	z	NOUN
ejpam-6690	499	7	∈	∈	PROPN
ejpam-6690	499	8	r	r	NOUN
ejpam-6690	499	9	and	and	CCONJ
ejpam-6690	499	10	x	x	PUNCT
ejpam-6690	500	1	+	+	CCONJ
ejpam-6690	500	2	i	i	NOUN
ejpam-6690	500	3	=	=	SYM
ejpam-6690	500	4	y	y	PROPN
ejpam-6690	500	5	+	+	NUM
ejpam-6690	500	6	i.	i.	NOUN
ejpam-6690	500	7	then	then	ADV
ejpam-6690	500	8	(	(	PUNCT
ejpam-6690	500	9	z	z	NOUN
ejpam-6690	500	10	+	+	NOUN
ejpam-6690	500	11	x	x	X
ejpam-6690	500	12	)	)	PUNCT
ejpam-6690	501	1	+	+	CCONJ
ejpam-6690	501	2	i	i	PRON
ejpam-6690	501	3	=	=	PUNCT
ejpam-6690	501	4	(	(	PUNCT
ejpam-6690	501	5	z	z	X
ejpam-6690	501	6	+	+	NOUN
ejpam-6690	501	7	y	y	NOUN
ejpam-6690	501	8	)	)	PUNCT
ejpam-6690	502	1	+	+	CCONJ
ejpam-6690	502	2	i	i	PRON
ejpam-6690	502	3	and	and	CCONJ
ejpam-6690	502	4	since	since	SCONJ
ejpam-6690	502	5	sβ	sβ	NUM
ejpam-6690	502	6	⊆	⊆	NUM
ejpam-6690	502	7	i	i	PROPN
ejpam-6690	502	8	,	,	PUNCT
ejpam-6690	502	9	then	then	ADV
ejpam-6690	502	10	for	for	ADP
ejpam-6690	502	11	every	every	DET
ejpam-6690	502	12	r	r	NOUN
ejpam-6690	502	13	∈	∈	NOUN
ejpam-6690	502	14	z+x	z+x	NUM
ejpam-6690	502	15	and	and	CCONJ
ejpam-6690	502	16	s	s	NOUN
ejpam-6690	502	17	∈	∈	PROPN
ejpam-6690	502	18	z+	z+	NUM
ejpam-6690	502	19	y	y	NOUN
ejpam-6690	502	20	;	;	PUNCT
ejpam-6690	502	21	(	(	PUNCT
ejpam-6690	502	22	z+x)+	z+x)+	X
ejpam-6690	502	23	i	i	NOUN
ejpam-6690	502	24	=	=	PUNCT
ejpam-6690	502	25	r+	r+	PUNCT
ejpam-6690	502	26	i	i	PRON
ejpam-6690	502	27	and	and	CCONJ
ejpam-6690	502	28	(	(	PUNCT
ejpam-6690	502	29	z+	z+	NUM
ejpam-6690	502	30	y)+	y)+	NOUN
ejpam-6690	503	1	i	i	PRON
ejpam-6690	503	2	=	=	PUNCT
ejpam-6690	503	3	s+	s+	PUNCT
ejpam-6690	503	4	i.	i.	PROPN
ejpam-6690	503	5	hence	hence	ADV
ejpam-6690	503	6	the	the	DET
ejpam-6690	503	7	congruence	congruence	PROPN
ejpam-6690	503	8	relation	relation	NOUN
ejpam-6690	503	9	modulo	modulo	VERB
ejpam-6690	503	10	i	i	PRON
ejpam-6690	503	11	is	be	AUX
ejpam-6690	503	12	strongly	strongly	ADV
ejpam-6690	503	13	regular	regular	ADJ
ejpam-6690	503	14	.	.	PUNCT
ejpam-6690	504	1	let	let	AUX
ejpam-6690	504	2	sr(r	sr(r	NOUN
ejpam-6690	504	3	)	)	PUNCT
ejpam-6690	504	4	be	be	AUX
ejpam-6690	504	5	the	the	DET
ejpam-6690	504	6	set	set	NOUN
ejpam-6690	504	7	of	of	ADP
ejpam-6690	504	8	all	all	DET
ejpam-6690	504	9	strongly	strongly	ADV
ejpam-6690	504	10	regular	regular	ADJ
ejpam-6690	504	11	relations	relation	NOUN
ejpam-6690	504	12	on	on	ADP
ejpam-6690	504	13	canonical	canonical	ADJ
ejpam-6690	504	14	hypergroup	hypergroup	NOUN
ejpam-6690	504	15	(	(	PUNCT
ejpam-6690	504	16	r,+	r,+	NUM
ejpam-6690	504	17	)	)	PUNCT
ejpam-6690	504	18	.	.	PUNCT
ejpam-6690	505	1	if	if	SCONJ
ejpam-6690	505	2	ρ	ρ	PROPN
ejpam-6690	505	3	,	,	PUNCT
ejpam-6690	505	4	σ	σ	PROPN
ejpam-6690	505	5	∈	∈	PROPN
ejpam-6690	505	6	sr(r	sr(r	NOUN
ejpam-6690	505	7	)	)	PUNCT
ejpam-6690	505	8	,	,	PUNCT
ejpam-6690	505	9	then	then	ADV
ejpam-6690	505	10	ρ	ρ	PROPN
ejpam-6690	505	11	∨	∨	PROPN
ejpam-6690	505	12	σ	σ	PROPN
ejpam-6690	505	13	∈	∈	PROPN
ejpam-6690	505	14	sr(r	sr(r	NOUN
ejpam-6690	505	15	)	)	PUNCT
ejpam-6690	505	16	,	,	PUNCT
ejpam-6690	505	17	and	and	CCONJ
ejpam-6690	505	18	the	the	DET
ejpam-6690	505	19	lattices	lattice	NOUN
ejpam-6690	505	20	n(sβ	n(sβ	PROPN
ejpam-6690	505	21	)	)	PUNCT
ejpam-6690	505	22	=	=	PRON
ejpam-6690	505	23	{	{	PUNCT
ejpam-6690	505	24	h	h	NOUN
ejpam-6690	505	25	;	;	PUNCT
ejpam-6690	505	26	sβ	sβ	NUM
ejpam-6690	505	27	⊆	⊆	NUM
ejpam-6690	505	28	h	h	NOUN
ejpam-6690	505	29	◁	◁	NOUN
ejpam-6690	505	30	r	r	NOUN
ejpam-6690	505	31	}	}	PUNCT
ejpam-6690	505	32	and	and	CCONJ
ejpam-6690	505	33	sr(r	sr(r	NOUN
ejpam-6690	505	34	)	)	PUNCT
ejpam-6690	505	35	are	be	AUX
ejpam-6690	505	36	isomorph	isomorph	NOUN
ejpam-6690	505	37	,	,	PUNCT
ejpam-6690	505	38	[	[	X
ejpam-6690	505	39	11	11	NUM
ejpam-6690	505	40	]	]	PUNCT
ejpam-6690	505	41	.	.	PUNCT
ejpam-6690	506	1	theorem	theorem	VERB
ejpam-6690	506	2	11	11	NUM
ejpam-6690	506	3	.	.	PUNCT
ejpam-6690	507	1	for	for	ADP
ejpam-6690	507	2	every	every	DET
ejpam-6690	507	3	hyperring	hyperring	NOUN
ejpam-6690	507	4	r	r	NOUN
ejpam-6690	507	5	,	,	PUNCT
ejpam-6690	507	6	sr(r	sr(r	NOUN
ejpam-6690	507	7	)	)	PUNCT
ejpam-6690	507	8	∼=	∼=	PROPN
ejpam-6690	507	9	i(γ∗(0	i(γ∗(0	NOUN
ejpam-6690	507	10	)	)	PUNCT
ejpam-6690	507	11	)	)	PUNCT
ejpam-6690	507	12	is	be	AUX
ejpam-6690	507	13	an	an	DET
ejpam-6690	507	14	isomorphism	isomorphism	NOUN
ejpam-6690	507	15	of	of	ADP
ejpam-6690	507	16	complete	complete	ADJ
ejpam-6690	507	17	lattices	lattice	NOUN
ejpam-6690	507	18	.	.	PUNCT
ejpam-6690	508	1	proof	proof	NOUN
ejpam-6690	508	2	.	.	PUNCT
ejpam-6690	509	1	since	since	SCONJ
ejpam-6690	509	2	r	r	NOUN
ejpam-6690	509	3	is	be	AUX
ejpam-6690	509	4	a	a	DET
ejpam-6690	509	5	krasner	krasner	NOUN
ejpam-6690	509	6	hyperring	hyperring	NOUN
ejpam-6690	509	7	,	,	PUNCT
ejpam-6690	509	8	then	then	ADV
ejpam-6690	509	9	for	for	ADP
ejpam-6690	509	10	every	every	DET
ejpam-6690	509	11	ρ	ρ	PROPN
ejpam-6690	509	12	,	,	PUNCT
ejpam-6690	509	13	σ	σ	PROPN
ejpam-6690	509	14	∈	∈	PROPN
ejpam-6690	509	15	sr(r	sr(r	NOUN
ejpam-6690	509	16	)	)	PUNCT
ejpam-6690	509	17	,	,	PUNCT
ejpam-6690	509	18	a	a	DET
ejpam-6690	509	19	⊆	⊆	NUM
ejpam-6690	509	20	sr(r	sr(r	NOUN
ejpam-6690	509	21	)	)	PUNCT
ejpam-6690	509	22	,	,	PUNCT
ejpam-6690	509	23	i	i	PRON
ejpam-6690	509	24	,	,	PUNCT
ejpam-6690	509	25	j	j	PROPN
ejpam-6690	509	26	∈	∈	PROPN
ejpam-6690	509	27	i(γ∗(0	i(γ∗(0	NOUN
ejpam-6690	509	28	)	)	PUNCT
ejpam-6690	509	29	)	)	PUNCT
ejpam-6690	509	30	and	and	CCONJ
ejpam-6690	509	31	b	b	X
ejpam-6690	509	32	⊆	⊆	NUM
ejpam-6690	509	33	i(γ∗(0	i(γ∗(0	NOUN
ejpam-6690	509	34	)	)	PUNCT
ejpam-6690	509	35	)	)	PUNCT
ejpam-6690	509	36	,	,	PUNCT
ejpam-6690	509	37	we	we	PRON
ejpam-6690	509	38	have	have	VERB
ejpam-6690	509	39	ρ∨σ	ρ∨σ	NOUN
ejpam-6690	509	40	,	,	PUNCT
ejpam-6690	509	41	ρ∩σ	ρ∩σ	NOUN
ejpam-6690	509	42	∈	∈	NOUN
ejpam-6690	509	43	sr(r	sr(r	NOUN
ejpam-6690	509	44	)	)	PUNCT
ejpam-6690	509	45	and	and	CCONJ
ejpam-6690	509	46	i	i	PRON
ejpam-6690	509	47	∨j	∨j	VERB
ejpam-6690	509	48	=	=	SYM
ejpam-6690	509	49	i+j	i+j	NUM
ejpam-6690	509	50	,	,	PUNCT
ejpam-6690	509	51	i	i	PRON
ejpam-6690	509	52	∩j	∩j	VERB
ejpam-6690	509	53	∈	∈	PROPN
ejpam-6690	509	54	i(γ∗(0	i(γ∗(0	NOUN
ejpam-6690	509	55	)	)	PUNCT
ejpam-6690	509	56	)	)	PUNCT
ejpam-6690	509	57	.	.	PUNCT
ejpam-6690	510	1	also∨	also∨	PROPN
ejpam-6690	510	2	ρ∈a	ρ∈a	VERB
ejpam-6690	510	3	ρ	ρ	PRON
ejpam-6690	510	4	∈	∈	PROPN
ejpam-6690	510	5	sr(r	sr(r	NOUN
ejpam-6690	510	6	)	)	PUNCT
ejpam-6690	510	7	,	,	PUNCT
ejpam-6690	510	8	⋂	⋂	PROPN
ejpam-6690	510	9	ρ∈sr(r	ρ∈sr(r	PROPN
ejpam-6690	510	10	)	)	PUNCT
ejpam-6690	510	11	ρ	ρ	PROPN
ejpam-6690	510	12	=	=	SYM
ejpam-6690	510	13	γ∗	γ∗	PROPN
ejpam-6690	510	14	,	,	PUNCT
ejpam-6690	510	15	∨	∨	NUM
ejpam-6690	510	16	i∈b	i∈b	VERB
ejpam-6690	510	17	i	i	PRON
ejpam-6690	510	18	=	=	PUNCT
ejpam-6690	510	19	∑	∑	PUNCT
ejpam-6690	510	20	i∈b	i∈b	VERB
ejpam-6690	510	21	i	i	PRON
ejpam-6690	510	22	∈	∈	PROPN
ejpam-6690	510	23	i(γ∗(0	i(γ∗(0	NOUN
ejpam-6690	510	24	)	)	PUNCT
ejpam-6690	510	25	)	)	PUNCT
ejpam-6690	510	26	,	,	PUNCT
ejpam-6690	510	27	⋂	⋂	PROPN
ejpam-6690	510	28	i∈i(γ∗(0	i∈i(γ∗(0	NOUN
ejpam-6690	510	29	)	)	PUNCT
ejpam-6690	510	30	)	)	PUNCT
ejpam-6690	511	1	i	i	PRON
ejpam-6690	511	2	=	=	SYM
ejpam-6690	511	3	γ∗(0	γ∗(0	PROPN
ejpam-6690	511	4	)	)	PUNCT
ejpam-6690	511	5	.	.	PUNCT
ejpam-6690	512	1	let	let	VERB
ejpam-6690	512	2	f	f	NOUN
ejpam-6690	512	3	:	:	PUNCT
ejpam-6690	512	4	sr(r	sr(r	NOUN
ejpam-6690	512	5	)	)	PUNCT
ejpam-6690	512	6	→	→	SYM
ejpam-6690	512	7	i(γ∗(0	i(γ∗(0	NOUN
ejpam-6690	512	8	)	)	PUNCT
ejpam-6690	512	9	)	)	PUNCT
ejpam-6690	512	10	by	by	ADP
ejpam-6690	512	11	ρ	ρ	PROPN
ejpam-6690	512	12	7→	7→	PROPN
ejpam-6690	512	13	ρ(0	ρ(0	PROPN
ejpam-6690	512	14	)	)	PUNCT
ejpam-6690	512	15	and	and	CCONJ
ejpam-6690	512	16	g	g	NOUN
ejpam-6690	512	17	:	:	PUNCT
ejpam-6690	512	18	i(γ∗(0	i(γ∗(0	NOUN
ejpam-6690	512	19	)	)	PUNCT
ejpam-6690	512	20	)	)	PUNCT
ejpam-6690	512	21	→	→	SYM
ejpam-6690	512	22	sr(r	sr(r	NOUN
ejpam-6690	512	23	)	)	PUNCT
ejpam-6690	512	24	by	by	ADP
ejpam-6690	512	25	i	i	PROPN
ejpam-6690	512	26	7→	7→	NUM
ejpam-6690	512	27	ρi	ρi	NOUN
ejpam-6690	512	28	:	:	PUNCT
ejpam-6690	512	29	=	=	SYM
ejpam-6690	512	30	{	{	PUNCT
ejpam-6690	512	31	(	(	PUNCT
ejpam-6690	512	32	x	x	NOUN
ejpam-6690	512	33	,	,	PUNCT
ejpam-6690	512	34	y	y	NOUN
ejpam-6690	512	35	)	)	PUNCT
ejpam-6690	512	36	∈	∈	NOUN
ejpam-6690	512	37	r2	r2	NOUN
ejpam-6690	512	38	;	;	PUNCT
ejpam-6690	512	39	x+	x+	PUNCT
ejpam-6690	512	40	i	i	NOUN
ejpam-6690	512	41	=	=	SYM
ejpam-6690	512	42	y	y	PROPN
ejpam-6690	512	43	+	+	NUM
ejpam-6690	512	44	i	i	PROPN
ejpam-6690	512	45	}	}	PUNCT
ejpam-6690	512	46	.	.	PUNCT
ejpam-6690	513	1	by	by	ADP
ejpam-6690	513	2	lemmas	lemmas	PROPN
ejpam-6690	513	3	4	4	NUM
ejpam-6690	513	4	and	and	CCONJ
ejpam-6690	513	5	5	5	NUM
ejpam-6690	513	6	,	,	PUNCT
ejpam-6690	513	7	f	f	PROPN
ejpam-6690	513	8	and	and	CCONJ
ejpam-6690	513	9	g	g	PROPN
ejpam-6690	513	10	are	be	AUX
ejpam-6690	513	11	well	well	ADV
ejpam-6690	513	12	defined	define	VERB
ejpam-6690	513	13	,	,	PUNCT
ejpam-6690	513	14	and	and	CCONJ
ejpam-6690	513	15	f	f	PROPN
ejpam-6690	513	16	◦	◦	NOUN
ejpam-6690	513	17	g(i	g(i	PROPN
ejpam-6690	513	18	)	)	PUNCT
ejpam-6690	513	19	=	=	PUNCT
ejpam-6690	514	1	f(ρi	f(ρi	X
ejpam-6690	514	2	)	)	PUNCT
ejpam-6690	514	3	=	=	PRON
ejpam-6690	514	4	{	{	PUNCT
ejpam-6690	514	5	r	r	NOUN
ejpam-6690	514	6	∈	∈	PROPN
ejpam-6690	514	7	r	r	NOUN
ejpam-6690	514	8	;	;	PUNCT
ejpam-6690	514	9	(	(	PUNCT
ejpam-6690	514	10	x+	x+	X
ejpam-6690	514	11	i	i	NOUN
ejpam-6690	514	12	)	)	PUNCT
ejpam-6690	515	1	+	+	CCONJ
ejpam-6690	515	2	(	(	PUNCT
ejpam-6690	515	3	r	r	NOUN
ejpam-6690	515	4	+	+	PROPN
ejpam-6690	515	5	i	i	NOUN
ejpam-6690	515	6	)	)	PUNCT
ejpam-6690	516	1	=	=	PUNCT
ejpam-6690	516	2	x+	x+	PROPN
ejpam-6690	517	1	i	i	PRON
ejpam-6690	517	2	,	,	PUNCT
ejpam-6690	517	3	∀x	∀x	VERB
ejpam-6690	517	4	∈	∈	PROPN
ejpam-6690	517	5	r	r	NOUN
ejpam-6690	517	6	}	}	PUNCT
ejpam-6690	517	7	=	=	NOUN
ejpam-6690	517	8	{	{	PUNCT
ejpam-6690	517	9	r	r	NOUN
ejpam-6690	517	10	∈	∈	PROPN
ejpam-6690	517	11	r	r	NOUN
ejpam-6690	517	12	;	;	PUNCT
ejpam-6690	517	13	(	(	PUNCT
ejpam-6690	517	14	x+	x+	X
ejpam-6690	517	15	r	r	NOUN
ejpam-6690	517	16	)	)	PUNCT
ejpam-6690	518	1	+	+	CCONJ
ejpam-6690	518	2	i	i	PRON
ejpam-6690	518	3	=	=	SYM
ejpam-6690	519	1	x+	x+	PROPN
ejpam-6690	519	2	i	i	PRON
ejpam-6690	519	3	,	,	PUNCT
ejpam-6690	519	4	∀x	∀x	VERB
ejpam-6690	519	5	∈	∈	PROPN
ejpam-6690	519	6	r	r	NOUN
ejpam-6690	519	7	}	}	PUNCT
ejpam-6690	519	8	=	=	NOUN
ejpam-6690	519	9	i	i	PROPN
ejpam-6690	519	10	,	,	PUNCT
ejpam-6690	519	11	because	because	SCONJ
ejpam-6690	519	12	,	,	PUNCT
ejpam-6690	519	13	for	for	ADP
ejpam-6690	519	14	every	every	DET
ejpam-6690	519	15	x	x	SYM
ejpam-6690	519	16	∈	∈	PROPN
ejpam-6690	519	17	r	r	NOUN
ejpam-6690	519	18	,	,	PUNCT
ejpam-6690	519	19	0	0	NUM
ejpam-6690	519	20	∈	∈	NOUN
ejpam-6690	519	21	−x+	−x+	NOUN
ejpam-6690	519	22	x	x	SYM
ejpam-6690	519	23	⊆	⊆	NUM
ejpam-6690	519	24	i.	i.	NOUN
ejpam-6690	519	25	also	also	ADV
ejpam-6690	519	26	g	g	ADP
ejpam-6690	519	27	◦	◦	NOUN
ejpam-6690	519	28	f(ρ	f(ρ	NOUN
ejpam-6690	519	29	)	)	PUNCT
ejpam-6690	519	30	=	=	SYM
ejpam-6690	519	31	g(ρ(0	g(ρ(0	NOUN
ejpam-6690	519	32	)	)	PUNCT
ejpam-6690	519	33	)	)	PUNCT
ejpam-6690	520	1	=	=	PRON
ejpam-6690	520	2	{	{	PUNCT
ejpam-6690	520	3	(	(	PUNCT
ejpam-6690	520	4	x	x	NOUN
ejpam-6690	520	5	,	,	PUNCT
ejpam-6690	520	6	y	y	NOUN
ejpam-6690	520	7	)	)	PUNCT
ejpam-6690	520	8	∈	∈	NOUN
ejpam-6690	520	9	r2	r2	NOUN
ejpam-6690	520	10	;	;	PUNCT
ejpam-6690	520	11	x+	x+	X
ejpam-6690	520	12	ρ(0	ρ(0	PROPN
ejpam-6690	520	13	)	)	PUNCT
ejpam-6690	520	14	=	=	SYM
ejpam-6690	520	15	y	y	PROPN
ejpam-6690	520	16	+	+	CCONJ
ejpam-6690	520	17	ρ(0	ρ(0	PROPN
ejpam-6690	520	18	)	)	PUNCT
ejpam-6690	520	19	}	}	PUNCT
ejpam-6690	520	20	=	=	SYM
ejpam-6690	520	21	{	{	PUNCT
ejpam-6690	520	22	(	(	PUNCT
ejpam-6690	520	23	x	x	NOUN
ejpam-6690	520	24	,	,	PUNCT
ejpam-6690	520	25	y	y	NOUN
ejpam-6690	520	26	)	)	PUNCT
ejpam-6690	520	27	∈	∈	NOUN
ejpam-6690	520	28	r2	r2	NOUN
ejpam-6690	520	29	;	;	PUNCT
ejpam-6690	520	30	ρ(x	ρ(x	NUM
ejpam-6690	520	31	)	)	PUNCT
ejpam-6690	520	32	=	=	SYM
ejpam-6690	520	33	ρ(y	ρ(y	NOUN
ejpam-6690	520	34	)	)	PUNCT
ejpam-6690	520	35	}	}	PUNCT
ejpam-6690	520	36	=	=	SYM
ejpam-6690	520	37	ρ	ρ	PROPN
ejpam-6690	520	38	.	.	PUNCT
ejpam-6690	520	39	now	now	ADV
ejpam-6690	520	40	let	let	VERB
ejpam-6690	520	41	ρ	ρ	NOUN
ejpam-6690	520	42	,	,	PUNCT
ejpam-6690	520	43	σ	σ	PROPN
ejpam-6690	520	44	∈	∈	PROPN
ejpam-6690	520	45	sr(r	sr(r	NOUN
ejpam-6690	520	46	)	)	PUNCT
ejpam-6690	520	47	and	and	CCONJ
ejpam-6690	520	48	i	i	PRON
ejpam-6690	520	49	,	,	PUNCT
ejpam-6690	520	50	j	j	PROPN
ejpam-6690	520	51	∈	∈	PROPN
ejpam-6690	520	52	i(γ∗(0	i(γ∗(0	NOUN
ejpam-6690	520	53	)	)	PUNCT
ejpam-6690	520	54	)	)	PUNCT
ejpam-6690	520	55	such	such	ADJ
ejpam-6690	520	56	that	that	SCONJ
ejpam-6690	520	57	ρ	ρ	PROPN
ejpam-6690	520	58	⊆	⊆	NUM
ejpam-6690	520	59	σ	σ	NOUN
ejpam-6690	520	60	and	and	CCONJ
ejpam-6690	520	61	i	i	PRON
ejpam-6690	520	62	⊆	⊆	NUM
ejpam-6690	520	63	j	j	PROPN
ejpam-6690	520	64	.	.	PUNCT
ejpam-6690	521	1	then	then	ADV
ejpam-6690	521	2	f(ρ	f(ρ	NOUN
ejpam-6690	521	3	)	)	PUNCT
ejpam-6690	521	4	=	=	SYM
ejpam-6690	521	5	ρ(0	ρ(0	PROPN
ejpam-6690	521	6	)	)	PUNCT
ejpam-6690	521	7	⊆	⊆	NUM
ejpam-6690	521	8	σ(0	σ(0	PROPN
ejpam-6690	521	9	)	)	PUNCT
ejpam-6690	521	10	=	=	SYM
ejpam-6690	521	11	f(σ	f(σ	NOUN
ejpam-6690	521	12	)	)	PUNCT
ejpam-6690	521	13	and	and	CCONJ
ejpam-6690	521	14	if	if	SCONJ
ejpam-6690	521	15	(	(	PUNCT
ejpam-6690	521	16	x	x	NOUN
ejpam-6690	521	17	,	,	PUNCT
ejpam-6690	521	18	y	y	NOUN
ejpam-6690	521	19	)	)	PUNCT
ejpam-6690	521	20	∈	∈	PROPN
ejpam-6690	521	21	r2	r2	NOUN
ejpam-6690	521	22	such	such	ADJ
ejpam-6690	521	23	that	that	SCONJ
ejpam-6690	521	24	x+	x+	PROPN
ejpam-6690	522	1	i	i	NOUN
ejpam-6690	522	2	=	=	PUNCT
ejpam-6690	522	3	y+	y+	PROPN
ejpam-6690	522	4	i	i	PRON
ejpam-6690	522	5	,	,	PUNCT
ejpam-6690	522	6	then	then	ADV
ejpam-6690	522	7	x+	x+	PUNCT
ejpam-6690	522	8	i	i	PRON
ejpam-6690	522	9	+	+	NUM
ejpam-6690	522	10	j	j	NOUN
ejpam-6690	522	11	=	=	VERB
ejpam-6690	523	1	y+	y+	PROPN
ejpam-6690	523	2	i	i	PROPN
ejpam-6690	523	3	+	+	PROPN
ejpam-6690	523	4	j	j	PROPN
ejpam-6690	523	5	.	.	PUNCT
ejpam-6690	524	1	therefore	therefore	ADV
ejpam-6690	524	2	,	,	PUNCT
ejpam-6690	524	3	x+	x+	PROPN
ejpam-6690	524	4	j	j	PROPN
ejpam-6690	524	5	=	=	SYM
ejpam-6690	524	6	y	y	PROPN
ejpam-6690	524	7	+	+	CCONJ
ejpam-6690	524	8	j	j	PROPN
ejpam-6690	524	9	and	and	CCONJ
ejpam-6690	524	10	f−1(i	f−1(i	PROPN
ejpam-6690	524	11	)	)	PUNCT
ejpam-6690	524	12	⊆	⊆	NUM
ejpam-6690	524	13	f−1(j	f−1(j	NOUN
ejpam-6690	524	14	)	)	PUNCT
ejpam-6690	524	15	.	.	PUNCT
ejpam-6690	525	1	corollary	corollary	ADJ
ejpam-6690	525	2	6	6	NUM
ejpam-6690	525	3	.	.	PUNCT
ejpam-6690	526	1	if	if	SCONJ
ejpam-6690	526	2	ρ	ρ	PROPN
ejpam-6690	526	3	∈	∈	PROPN
ejpam-6690	526	4	sr(r	sr(r	NOUN
ejpam-6690	526	5	)	)	PUNCT
ejpam-6690	526	6	,	,	PUNCT
ejpam-6690	526	7	then	then	ADV
ejpam-6690	526	8	r	r	X
ejpam-6690	526	9	/	/	SYM
ejpam-6690	526	10	ρ	ρ	NOUN
ejpam-6690	526	11	∼=	∼=	NOUN
ejpam-6690	526	12	r	r	NOUN
ejpam-6690	526	13	/	/	SYM
ejpam-6690	526	14	ρ(0	ρ(0	PROPN
ejpam-6690	526	15	)	)	PUNCT
ejpam-6690	526	16	is	be	AUX
ejpam-6690	526	17	a	a	DET
ejpam-6690	526	18	ring	ring	NOUN
ejpam-6690	526	19	isomorphism	isomorphism	NOUN
ejpam-6690	526	20	.	.	PUNCT
ejpam-6690	527	1	proof	proof	NOUN
ejpam-6690	527	2	.	.	PUNCT
ejpam-6690	528	1	since	since	SCONJ
ejpam-6690	528	2	for	for	ADP
ejpam-6690	528	3	every	every	DET
ejpam-6690	528	4	x	x	NOUN
ejpam-6690	528	5	,	,	PUNCT
ejpam-6690	528	6	y	y	PROPN
ejpam-6690	528	7	∈	∈	PROPN
ejpam-6690	528	8	r	r	NOUN
ejpam-6690	528	9	and	and	CCONJ
ejpam-6690	528	10	z	z	NOUN
ejpam-6690	528	11	∈	∈	PROPN
ejpam-6690	528	12	x+y	x+y	NUM
ejpam-6690	528	13	;	;	PUNCT
ejpam-6690	528	14	(	(	PUNCT
ejpam-6690	528	15	x+ρ(0))+(y+ρ(0	x+ρ(0))+(y+ρ(0	NOUN
ejpam-6690	528	16	)	)	PUNCT
ejpam-6690	528	17	)	)	PUNCT
ejpam-6690	528	18	=	=	SYM
ejpam-6690	528	19	(	(	PUNCT
ejpam-6690	528	20	x+y)+ρ(0	x+y)+ρ(0	PROPN
ejpam-6690	528	21	)	)	PUNCT
ejpam-6690	529	1	=	=	SYM
ejpam-6690	529	2	z	z	X
ejpam-6690	529	3	+	+	NUM
ejpam-6690	529	4	ρ(0	ρ(0	PROPN
ejpam-6690	529	5	)	)	PUNCT
ejpam-6690	529	6	,	,	PUNCT
ejpam-6690	529	7	then	then	ADV
ejpam-6690	529	8	r	r	PROPN
ejpam-6690	529	9	/	/	SYM
ejpam-6690	529	10	ρ(0	ρ(0	PROPN
ejpam-6690	529	11	)	)	PUNCT
ejpam-6690	529	12	is	be	AUX
ejpam-6690	529	13	a	a	DET
ejpam-6690	529	14	ring	ring	NOUN
ejpam-6690	529	15	.	.	PUNCT
ejpam-6690	530	1	consider	consider	VERB
ejpam-6690	530	2	the	the	DET
ejpam-6690	530	3	bijection	bijection	NOUN
ejpam-6690	530	4	map	map	NOUN
ejpam-6690	530	5	ϕ	ϕ	NOUN
ejpam-6690	530	6	:	:	PUNCT
ejpam-6690	530	7	r	r	X
ejpam-6690	530	8	/	/	SYM
ejpam-6690	530	9	ρ	ρ	PROPN
ejpam-6690	530	10	→	→	SYM
ejpam-6690	530	11	r	r	NOUN
ejpam-6690	530	12	/	/	SYM
ejpam-6690	530	13	ρ(0	ρ(0	PROPN
ejpam-6690	530	14	)	)	PUNCT
ejpam-6690	530	15	by	by	ADP
ejpam-6690	530	16	ρ(x	ρ(x	NOUN
ejpam-6690	530	17	)	)	PUNCT
ejpam-6690	530	18	7→	7→	NUM
ejpam-6690	530	19	x+	x+	PUNCT
ejpam-6690	530	20	ρ(0	ρ(0	PROPN
ejpam-6690	530	21	)	)	PUNCT
ejpam-6690	530	22	,	,	PUNCT
ejpam-6690	530	23	then	then	ADV
ejpam-6690	530	24	:	:	PUNCT
ejpam-6690	530	25	ϕ(ρ(x	ϕ(ρ(x	X
ejpam-6690	530	26	)	)	PUNCT
ejpam-6690	531	1	+	+	CCONJ
ejpam-6690	531	2	ρ(y	ρ(y	NOUN
ejpam-6690	531	3	)	)	PUNCT
ejpam-6690	531	4	)	)	PUNCT
ejpam-6690	532	1	=	=	PUNCT
ejpam-6690	533	1	ϕ(ρ(x+	ϕ(ρ(x+	NUM
ejpam-6690	533	2	y	y	NOUN
ejpam-6690	533	3	)	)	PUNCT
ejpam-6690	533	4	)	)	PUNCT
ejpam-6690	534	1	=	=	PRON
ejpam-6690	534	2	(	(	PUNCT
ejpam-6690	534	3	x+	x+	X
ejpam-6690	534	4	y	y	NOUN
ejpam-6690	534	5	)	)	PUNCT
ejpam-6690	534	6	+	+	SYM
ejpam-6690	535	1	ρ(0	ρ(0	PROPN
ejpam-6690	535	2	)	)	PUNCT
ejpam-6690	535	3	=	=	SYM
ejpam-6690	535	4	(	(	PUNCT
ejpam-6690	535	5	x+	x+	ADJ
ejpam-6690	535	6	ρ(0	ρ(0	PROPN
ejpam-6690	535	7	)	)	PUNCT
ejpam-6690	535	8	)	)	PUNCT
ejpam-6690	536	1	+	+	CCONJ
ejpam-6690	536	2	(	(	PUNCT
ejpam-6690	536	3	y	y	PROPN
ejpam-6690	536	4	+	+	CCONJ
ejpam-6690	536	5	ρ(0	ρ(0	PROPN
ejpam-6690	536	6	)	)	PUNCT
ejpam-6690	536	7	)	)	PUNCT
ejpam-6690	537	1	=	=	SYM
ejpam-6690	537	2	ϕ(ρ(x	ϕ(ρ(x	PROPN
ejpam-6690	537	3	)	)	PUNCT
ejpam-6690	537	4	)	)	PUNCT
ejpam-6690	538	1	+	+	CCONJ
ejpam-6690	538	2	ϕ(ρ(y	ϕ(ρ(y	X
ejpam-6690	538	3	)	)	PUNCT
ejpam-6690	538	4	)	)	PUNCT
ejpam-6690	538	5	and	and	CCONJ
ejpam-6690	538	6	ϕ(ρ(x)ρ(y	ϕ(ρ(x)ρ(y	NOUN
ejpam-6690	538	7	)	)	PUNCT
ejpam-6690	538	8	)	)	PUNCT
ejpam-6690	539	1	=	=	PUNCT
ejpam-6690	539	2	ϕ(ρ(xy	ϕ(ρ(xy	NOUN
ejpam-6690	539	3	)	)	PUNCT
ejpam-6690	539	4	)	)	PUNCT
ejpam-6690	540	1	=	=	PRON
ejpam-6690	540	2	(	(	PUNCT
ejpam-6690	540	3	xy	xy	NOUN
ejpam-6690	540	4	)	)	PUNCT
ejpam-6690	540	5	+	+	CCONJ
ejpam-6690	541	1	ρ(0	ρ(0	ADJ
ejpam-6690	541	2	)	)	PUNCT
ejpam-6690	541	3	=	=	SYM
ejpam-6690	541	4	(	(	PUNCT
ejpam-6690	541	5	x	x	SYM
ejpam-6690	541	6	+	+	NUM
ejpam-6690	541	7	ρ(0))(y	ρ(0))(y	NUM
ejpam-6690	541	8	+	+	NUM
ejpam-6690	541	9	ρ(0	ρ(0	PROPN
ejpam-6690	541	10	)	)	PUNCT
ejpam-6690	541	11	)	)	PUNCT
ejpam-6690	542	1	=	=	SYM
ejpam-6690	542	2	ϕ(ρ(x))ϕ(ρ(y	ϕ(ρ(x))ϕ(ρ(y	NOUN
ejpam-6690	542	3	)	)	PUNCT
ejpam-6690	542	4	)	)	PUNCT
ejpam-6690	542	5	.	.	PUNCT
ejpam-6690	543	1	therefore	therefore	ADV
ejpam-6690	543	2	,	,	PUNCT
ejpam-6690	543	3	if	if	SCONJ
ejpam-6690	543	4	ρ	ρ	PROPN
ejpam-6690	543	5	,	,	PUNCT
ejpam-6690	543	6	σ	σ	PROPN
ejpam-6690	543	7	∈	∈	PROPN
ejpam-6690	543	8	sr(r	sr(r	NOUN
ejpam-6690	543	9	)	)	PUNCT
ejpam-6690	543	10	and	and	CCONJ
ejpam-6690	543	11	i	i	PRON
ejpam-6690	543	12	,	,	PUNCT
ejpam-6690	543	13	j	j	PROPN
ejpam-6690	543	14	∈	∈	PROPN
ejpam-6690	543	15	i(γ∗(0	i(γ∗(0	NOUN
ejpam-6690	543	16	)	)	PUNCT
ejpam-6690	543	17	)	)	PUNCT
ejpam-6690	543	18	,	,	PUNCT
ejpam-6690	543	19	then	then	ADV
ejpam-6690	543	20	ρ(0	ρ(0	PROPN
ejpam-6690	543	21	)	)	PUNCT
ejpam-6690	543	22	+	+	CCONJ
ejpam-6690	543	23	σ(0	σ(0	PROPN
ejpam-6690	543	24	)	)	PUNCT
ejpam-6690	543	25	=	=	SYM
ejpam-6690	543	26	(	(	PUNCT
ejpam-6690	543	27	ρ	ρ	PROPN
ejpam-6690	543	28	∨	∨	NUM
ejpam-6690	543	29	σ)(0	σ)(0	NUM
ejpam-6690	543	30	)	)	PUNCT
ejpam-6690	543	31	,	,	PUNCT
ejpam-6690	543	32	ρ(0	ρ(0	PROPN
ejpam-6690	543	33	)	)	PUNCT
ejpam-6690	543	34	∩	∩	NOUN
ejpam-6690	543	35	σ(0	σ(0	PROPN
ejpam-6690	543	36	)	)	PUNCT
ejpam-6690	543	37	=	=	SYM
ejpam-6690	543	38	(	(	PUNCT
ejpam-6690	543	39	ρ	ρ	PROPN
ejpam-6690	543	40	∩	∩	X
ejpam-6690	543	41	σ)(0	σ)(0	NUM
ejpam-6690	543	42	)	)	PUNCT
ejpam-6690	543	43	and	and	CCONJ
ejpam-6690	543	44	ρi	ρi	NUM
ejpam-6690	543	45	∨	∨	NOUN
ejpam-6690	543	46	ρj	ρj	X
ejpam-6690	543	47	=	=	SYM
ejpam-6690	543	48	ρi+j	ρi+j	PROPN
ejpam-6690	543	49	,	,	PUNCT
ejpam-6690	543	50	ρi	ρi	NOUN
ejpam-6690	543	51	∩	∩	NOUN
ejpam-6690	543	52	ρj	ρj	NOUN
ejpam-6690	543	53	=	=	SYM
ejpam-6690	543	54	ρi∩j	ρi∩j	PROPN
ejpam-6690	543	55	.	.	PUNCT
ejpam-6690	544	1	the	the	DET
ejpam-6690	544	2	hyperideal	hyperideal	NOUN
ejpam-6690	544	3	i	i	PRON
ejpam-6690	544	4	of	of	ADP
ejpam-6690	544	5	a	a	DET
ejpam-6690	544	6	krasner	krasner	NOUN
ejpam-6690	544	7	hyperring	hyperre	VERB
ejpam-6690	544	8	r	r	NOUN
ejpam-6690	544	9	is	be	AUX
ejpam-6690	544	10	a	a	DET
ejpam-6690	544	11	normal	normal	ADJ
ejpam-6690	544	12	hyperideal	hyperideal	NOUN
ejpam-6690	544	13	if	if	SCONJ
ejpam-6690	545	1	and	and	CCONJ
ejpam-6690	545	2	only	only	ADV
ejpam-6690	545	3	if	if	SCONJ
ejpam-6690	545	4	x+i−x	x+i−x	PROPN
ejpam-6690	545	5	⊆	⊆	NUM
ejpam-6690	545	6	i	i	PROPN
ejpam-6690	545	7	,	,	PUNCT
ejpam-6690	545	8	for	for	ADP
ejpam-6690	545	9	all	all	DET
ejpam-6690	545	10	x	x	SYM
ejpam-6690	545	11	∈	∈	PROPN
ejpam-6690	545	12	r	r	NOUN
ejpam-6690	545	13	,	,	PUNCT
ejpam-6690	545	14	[	[	X
ejpam-6690	545	15	18	18	NUM
ejpam-6690	545	16	]	]	PUNCT
ejpam-6690	545	17	.	.	PUNCT
ejpam-6690	546	1	also	also	ADV
ejpam-6690	546	2	it	it	PRON
ejpam-6690	546	3	is	be	AUX
ejpam-6690	546	4	proved	prove	VERB
ejpam-6690	546	5	that	that	SCONJ
ejpam-6690	546	6	r	r	NOUN
ejpam-6690	546	7	/	/	SYM
ejpam-6690	546	8	i	i	PRON
ejpam-6690	546	9	is	be	AUX
ejpam-6690	546	10	a	a	DET
ejpam-6690	546	11	ring	ring	NOUN
ejpam-6690	546	12	if	if	SCONJ
ejpam-6690	546	13	and	and	CCONJ
ejpam-6690	546	14	only	only	ADV
ejpam-6690	546	15	if	if	SCONJ
ejpam-6690	546	16	i	i	PRON
ejpam-6690	546	17	is	be	AUX
ejpam-6690	546	18	a	a	DET
ejpam-6690	546	19	normal	normal	ADJ
ejpam-6690	546	20	hyperideal	hyperideal	NOUN
ejpam-6690	546	21	of	of	ADP
ejpam-6690	546	22	r	r	NOUN
ejpam-6690	546	23	,	,	PUNCT
ejpam-6690	546	24	[	[	X
ejpam-6690	546	25	18	18	NUM
ejpam-6690	546	26	]	]	PUNCT
ejpam-6690	546	27	.	.	PUNCT
ejpam-6690	547	1	b.	b.	PROPN
ejpam-6690	547	2	afshar	afshar	PROPN
ejpam-6690	547	3	,	,	PUNCT
ejpam-6690	547	4	r.	r.	PROPN
ejpam-6690	547	5	ameri	ameri	PROPN
ejpam-6690	547	6	,	,	PUNCT
ejpam-6690	547	7	m.	m.	PROPN
ejpam-6690	547	8	al	al	PROPN
ejpam-6690	547	9	-	-	PUNCT
ejpam-6690	547	10	tahan	tahan	PROPN
ejpam-6690	547	11	/	/	SYM
ejpam-6690	547	12	eur	eur	PROPN
ejpam-6690	547	13	.	.	PUNCT
ejpam-6690	548	1	j.	j.	PROPN
ejpam-6690	548	2	pure	pure	PROPN
ejpam-6690	548	3	appl	appl	PROPN
ejpam-6690	548	4	.	.	PROPN
ejpam-6690	548	5	math	math	PROPN
ejpam-6690	548	6	,	,	PUNCT
ejpam-6690	548	7	18	18	NUM
ejpam-6690	548	8	(	(	PUNCT
ejpam-6690	548	9	4	4	NUM
ejpam-6690	548	10	)	)	PUNCT
ejpam-6690	548	11	(	(	PUNCT
ejpam-6690	548	12	2025	2025	NUM
ejpam-6690	548	13	)	)	PUNCT
ejpam-6690	548	14	,	,	PUNCT
ejpam-6690	548	15	6690	6690	NUM
ejpam-6690	548	16	14	14	NUM
ejpam-6690	548	17	of	of	ADP
ejpam-6690	548	18	19	19	NUM
ejpam-6690	548	19	corollary	corollary	ADJ
ejpam-6690	548	20	7	7	NUM
ejpam-6690	548	21	.	.	PUNCT
ejpam-6690	549	1	let	let	VERB
ejpam-6690	549	2	i	i	PRON
ejpam-6690	549	3	be	be	AUX
ejpam-6690	549	4	a	a	DET
ejpam-6690	549	5	hyperideal	hyperideal	NOUN
ejpam-6690	549	6	of	of	ADP
ejpam-6690	549	7	krasner	krasner	NOUN
ejpam-6690	549	8	hyperring	hyperre	VERB
ejpam-6690	549	9	r.	r.	PROPN
ejpam-6690	549	10	then	then	ADV
ejpam-6690	549	11	,	,	PUNCT
ejpam-6690	549	12	i	i	PRON
ejpam-6690	549	13	is	be	AUX
ejpam-6690	549	14	normal	normal	ADJ
ejpam-6690	549	15	if	if	SCONJ
ejpam-6690	549	16	and	and	CCONJ
ejpam-6690	549	17	only	only	ADV
ejpam-6690	549	18	if	if	SCONJ
ejpam-6690	549	19	γ∗(0	γ∗(0	NOUN
ejpam-6690	549	20	)	)	PUNCT
ejpam-6690	549	21	⊆	⊆	NUM
ejpam-6690	549	22	i.	i.	NOUN
ejpam-6690	549	23	proof	proof	NOUN
ejpam-6690	549	24	.	.	PUNCT
ejpam-6690	550	1	since	since	SCONJ
ejpam-6690	550	2	0	0	NUM
ejpam-6690	550	3	∈	∈	PROPN
ejpam-6690	550	4	i	i	PRON
ejpam-6690	550	5	,	,	PUNCT
ejpam-6690	550	6	then	then	ADV
ejpam-6690	550	7	the	the	DET
ejpam-6690	550	8	hyperideal	hyperideal	NOUN
ejpam-6690	550	9	i	i	PRON
ejpam-6690	550	10	is	be	AUX
ejpam-6690	550	11	normal	normal	ADJ
ejpam-6690	550	12	if	if	SCONJ
ejpam-6690	550	13	and	and	CCONJ
ejpam-6690	550	14	only	only	ADV
ejpam-6690	550	15	if	if	SCONJ
ejpam-6690	550	16	x	x	PRON
ejpam-6690	550	17	−	−	NOUN
ejpam-6690	550	18	x	x	SYM
ejpam-6690	550	19	⊆	⊆	NUM
ejpam-6690	550	20	i	i	PRON
ejpam-6690	550	21	,	,	PUNCT
ejpam-6690	550	22	for	for	ADP
ejpam-6690	550	23	all	all	DET
ejpam-6690	550	24	x	x	SYM
ejpam-6690	550	25	∈	∈	PROPN
ejpam-6690	550	26	r.	r.	NOUN
ejpam-6690	550	27	if	if	SCONJ
ejpam-6690	550	28	y	y	PROPN
ejpam-6690	550	29	∈	∈	PROPN
ejpam-6690	550	30	γ∗(0	γ∗(0	PROPN
ejpam-6690	550	31	)	)	PUNCT
ejpam-6690	550	32	,	,	PUNCT
ejpam-6690	550	33	then	then	ADV
ejpam-6690	550	34	there	there	PRON
ejpam-6690	550	35	are	be	VERB
ejpam-6690	550	36	n	n	PRON
ejpam-6690	550	37	∈	∈	PROPN
ejpam-6690	550	38	n	n	NOUN
ejpam-6690	550	39	and	and	CCONJ
ejpam-6690	550	40	(	(	PUNCT
ejpam-6690	550	41	x1	x1	PROPN
ejpam-6690	550	42	,	,	PUNCT
ejpam-6690	550	43	...	...	PUNCT
ejpam-6690	550	44	,	,	PUNCT
ejpam-6690	550	45	xn	xn	X
ejpam-6690	550	46	)	)	PUNCT
ejpam-6690	550	47	∈	∈	PROPN
ejpam-6690	551	1	rn	rn	PROPN
ejpam-6690	551	2	such	such	ADJ
ejpam-6690	551	3	that	that	PRON
ejpam-6690	551	4	0	0	NUM
ejpam-6690	551	5	,	,	PUNCT
ejpam-6690	551	6	y	y	PROPN
ejpam-6690	551	7	∈	∈	PROPN
ejpam-6690	552	1	∑n	∑n	PROPN
ejpam-6690	552	2	i=1	i=1	PRON
ejpam-6690	553	1	xi	xi	PROPN
ejpam-6690	553	2	.	.	PUNCT
ejpam-6690	554	1	therefore	therefore	ADV
ejpam-6690	554	2	,	,	PUNCT
ejpam-6690	554	3	y	y	PROPN
ejpam-6690	554	4	∈	∈	PROPN
ejpam-6690	555	1	∑n	∑n	PROPN
ejpam-6690	556	1	i=1	i=1	X
ejpam-6690	556	2	xi	xi	PROPN
ejpam-6690	557	1	−	−	PROPN
ejpam-6690	557	2	∑n	∑n	PROPN
ejpam-6690	557	3	i=1	i=1	PROPN
ejpam-6690	557	4	xi	xi	ADP
ejpam-6690	557	5	⊆	⊆	NUM
ejpam-6690	557	6	i.	i.	NOUN
ejpam-6690	557	7	let	let	VERB
ejpam-6690	557	8	γ∗(0	γ∗(0	NOUN
ejpam-6690	557	9	)	)	PUNCT
ejpam-6690	557	10	⊆	⊆	NUM
ejpam-6690	557	11	i.	i.	NOUN
ejpam-6690	557	12	since	since	SCONJ
ejpam-6690	557	13	x−	x−	PROPN
ejpam-6690	557	14	x	x	PROPN
ejpam-6690	557	15	⊆	⊆	NUM
ejpam-6690	557	16	γ∗(0	γ∗(0	NOUN
ejpam-6690	557	17	)	)	PUNCT
ejpam-6690	557	18	,	,	PUNCT
ejpam-6690	557	19	for	for	ADP
ejpam-6690	557	20	all	all	DET
ejpam-6690	557	21	x	x	SYM
ejpam-6690	557	22	∈	∈	PROPN
ejpam-6690	557	23	r	r	NOUN
ejpam-6690	557	24	,	,	PUNCT
ejpam-6690	557	25	then	then	ADV
ejpam-6690	557	26	i	i	PRON
ejpam-6690	557	27	is	be	AUX
ejpam-6690	557	28	normal	normal	ADJ
ejpam-6690	557	29	.	.	PUNCT
ejpam-6690	558	1	so	so	ADV
ejpam-6690	558	2	we	we	PRON
ejpam-6690	558	3	can	can	AUX
ejpam-6690	558	4	say	say	VERB
ejpam-6690	558	5	that	that	SCONJ
ejpam-6690	558	6	if	if	SCONJ
ejpam-6690	558	7	r	r	NOUN
ejpam-6690	558	8	is	be	AUX
ejpam-6690	558	9	a	a	DET
ejpam-6690	558	10	krasner	krasner	NOUN
ejpam-6690	558	11	hyperring	hyperring	NOUN
ejpam-6690	558	12	,	,	PUNCT
ejpam-6690	558	13	then	then	ADV
ejpam-6690	558	14	r	r	X
ejpam-6690	558	15	/	/	SYM
ejpam-6690	558	16	i	i	PRON
ejpam-6690	558	17	is	be	AUX
ejpam-6690	558	18	ring	re	VERB
ejpam-6690	558	19	if	if	SCONJ
ejpam-6690	558	20	and	and	CCONJ
ejpam-6690	558	21	only	only	ADV
ejpam-6690	558	22	if	if	SCONJ
ejpam-6690	558	23	i	i	PRON
ejpam-6690	558	24	∈	∈	PROPN
ejpam-6690	558	25	i(γ∗(0	i(γ∗(0	NOUN
ejpam-6690	558	26	)	)	PUNCT
ejpam-6690	558	27	)	)	PUNCT
ejpam-6690	558	28	.	.	PUNCT
ejpam-6690	559	1	definition	definition	NOUN
ejpam-6690	559	2	7	7	NUM
ejpam-6690	559	3	.	.	PUNCT
ejpam-6690	559	4	a	a	DET
ejpam-6690	559	5	strongly	strongly	ADV
ejpam-6690	559	6	regular	regular	ADJ
ejpam-6690	559	7	relation	relation	NOUN
ejpam-6690	559	8	ρ	ρ	NOUN
ejpam-6690	559	9	on	on	ADP
ejpam-6690	559	10	the	the	DET
ejpam-6690	559	11	commutative	commutative	ADJ
ejpam-6690	559	12	and	and	CCONJ
ejpam-6690	559	13	of	of	ADP
ejpam-6690	559	14	unity	unity	NOUN
ejpam-6690	559	15	krasner	krasner	NOUN
ejpam-6690	559	16	hyperring	hyperre	VERB
ejpam-6690	559	17	r	r	NOUN
ejpam-6690	559	18	is	be	AUX
ejpam-6690	559	19	called	call	VERB
ejpam-6690	559	20	prime	prime	ADJ
ejpam-6690	559	21	,	,	PUNCT
ejpam-6690	559	22	if	if	SCONJ
ejpam-6690	559	23	for	for	ADP
ejpam-6690	559	24	every	every	DET
ejpam-6690	559	25	x	x	NOUN
ejpam-6690	559	26	and	and	CCONJ
ejpam-6690	559	27	y	y	PROPN
ejpam-6690	559	28	in	in	ADP
ejpam-6690	559	29	r	r	NOUN
ejpam-6690	559	30	,	,	PUNCT
ejpam-6690	559	31	we	we	PRON
ejpam-6690	559	32	have	have	VERB
ejpam-6690	559	33	:	:	PUNCT
ejpam-6690	559	34	(	(	PUNCT
ejpam-6690	559	35	xy	xy	INTJ
ejpam-6690	559	36	,	,	PUNCT
ejpam-6690	559	37	0	0	NUM
ejpam-6690	559	38	)	)	PUNCT
ejpam-6690	559	39	∈	∈	PROPN
ejpam-6690	559	40	ρ	ρ	PROPN
ejpam-6690	559	41	⇒	⇒	PROPN
ejpam-6690	559	42	(	(	PUNCT
ejpam-6690	559	43	x	x	X
ejpam-6690	559	44	,	,	PUNCT
ejpam-6690	559	45	0	0	NUM
ejpam-6690	559	46	)	)	PUNCT
ejpam-6690	559	47	∈	∈	NOUN
ejpam-6690	559	48	ρ	ρ	NOUN
ejpam-6690	559	49	or	or	CCONJ
ejpam-6690	559	50	(	(	PUNCT
ejpam-6690	559	51	y	y	PROPN
ejpam-6690	559	52	,	,	PUNCT
ejpam-6690	559	53	0	0	NUM
ejpam-6690	559	54	)	)	PUNCT
ejpam-6690	559	55	∈	∈	PROPN
ejpam-6690	559	56	ρ	ρ	PROPN
ejpam-6690	559	57	.	.	PUNCT
ejpam-6690	560	1	(	(	PUNCT
ejpam-6690	560	2	7	7	X
ejpam-6690	560	3	)	)	PUNCT
ejpam-6690	560	4	also	also	ADV
ejpam-6690	560	5	ρ	ρ	PROPN
ejpam-6690	560	6	is	be	AUX
ejpam-6690	560	7	primitive	primitive	ADJ
ejpam-6690	560	8	,	,	PUNCT
ejpam-6690	560	9	if	if	SCONJ
ejpam-6690	560	10	for	for	ADP
ejpam-6690	560	11	every	every	DET
ejpam-6690	560	12	x	x	NOUN
ejpam-6690	560	13	and	and	CCONJ
ejpam-6690	560	14	y	y	PROPN
ejpam-6690	560	15	in	in	ADP
ejpam-6690	560	16	r	r	NOUN
ejpam-6690	560	17	,	,	PUNCT
ejpam-6690	560	18	we	we	PRON
ejpam-6690	560	19	have	have	VERB
ejpam-6690	560	20	:	:	PUNCT
ejpam-6690	560	21	(	(	PUNCT
ejpam-6690	560	22	xy	xy	INTJ
ejpam-6690	560	23	,	,	PUNCT
ejpam-6690	560	24	0	0	NUM
ejpam-6690	560	25	)	)	PUNCT
ejpam-6690	560	26	∈	∈	PROPN
ejpam-6690	560	27	ρ	ρ	PROPN
ejpam-6690	560	28	,	,	PUNCT
ejpam-6690	560	29	(	(	PUNCT
ejpam-6690	560	30	x	x	X
ejpam-6690	560	31	,	,	PUNCT
ejpam-6690	560	32	0	0	NUM
ejpam-6690	560	33	)	)	PUNCT
ejpam-6690	560	34	/∈	/∈	PUNCT
ejpam-6690	561	1	ρ	ρ	NOUN
ejpam-6690	561	2	⇒	⇒	NOUN
ejpam-6690	561	3	∃n	∃n	PROPN
ejpam-6690	561	4	∈	∈	PROPN
ejpam-6690	561	5	n	n	PRON
ejpam-6690	561	6	s.t	s.t	PROPN
ejpam-6690	561	7	.	.	PROPN
ejpam-6690	561	8	(	(	PUNCT
ejpam-6690	561	9	yn	yn	PROPN
ejpam-6690	561	10	,	,	PUNCT
ejpam-6690	561	11	0	0	NUM
ejpam-6690	561	12	)	)	PUNCT
ejpam-6690	561	13	∈	∈	PROPN
ejpam-6690	561	14	ρ	ρ	PROPN
ejpam-6690	561	15	.	.	PUNCT
ejpam-6690	562	1	(	(	PUNCT
ejpam-6690	562	2	8)	8)	NUM
ejpam-6690	562	3	since	since	ADV
ejpam-6690	562	4	,	,	PUNCT
ejpam-6690	562	5	(	(	PUNCT
ejpam-6690	562	6	x	x	X
ejpam-6690	562	7	,	,	PUNCT
ejpam-6690	562	8	0	0	NUM
ejpam-6690	562	9	)	)	PUNCT
ejpam-6690	562	10	∈	∈	NOUN
ejpam-6690	562	11	ρ	ρ	NOUN
ejpam-6690	562	12	if	if	SCONJ
ejpam-6690	562	13	and	and	CCONJ
ejpam-6690	562	14	only	only	ADV
ejpam-6690	562	15	if	if	SCONJ
ejpam-6690	562	16	x	x	PROPN
ejpam-6690	562	17	∈	∈	PROPN
ejpam-6690	562	18	ρ(0	ρ(0	PROPN
ejpam-6690	562	19	)	)	PUNCT
ejpam-6690	562	20	,	,	PUNCT
ejpam-6690	562	21	then	then	ADV
ejpam-6690	562	22	ρ	ρ	PROPN
ejpam-6690	562	23	is	be	AUX
ejpam-6690	562	24	prime	prime	ADJ
ejpam-6690	562	25	if	if	SCONJ
ejpam-6690	563	1	and	and	CCONJ
ejpam-6690	563	2	only	only	ADV
ejpam-6690	563	3	if	if	SCONJ
ejpam-6690	563	4	xy	xy	PROPN
ejpam-6690	563	5	∈	∈	PROPN
ejpam-6690	563	6	ρ(0	ρ(0	PROPN
ejpam-6690	563	7	)	)	PUNCT
ejpam-6690	563	8	results	result	NOUN
ejpam-6690	563	9	in	in	ADP
ejpam-6690	563	10	x	x	PUNCT
ejpam-6690	563	11	∈	∈	PROPN
ejpam-6690	563	12	ρ(0	ρ(0	PROPN
ejpam-6690	563	13	)	)	PUNCT
ejpam-6690	563	14	or	or	CCONJ
ejpam-6690	563	15	y	y	PROPN
ejpam-6690	563	16	∈	∈	PROPN
ejpam-6690	563	17	ρ(0	ρ(0	PROPN
ejpam-6690	563	18	)	)	PUNCT
ejpam-6690	563	19	,	,	PUNCT
ejpam-6690	563	20	if	if	SCONJ
ejpam-6690	563	21	and	and	CCONJ
ejpam-6690	563	22	only	only	ADV
ejpam-6690	563	23	if	if	SCONJ
ejpam-6690	563	24	ρ(x)ρ(y	ρ(x)ρ(y	ADJ
ejpam-6690	563	25	)	)	PUNCT
ejpam-6690	563	26	=	=	SYM
ejpam-6690	563	27	ρ(0	ρ(0	PROPN
ejpam-6690	563	28	)	)	PUNCT
ejpam-6690	563	29	results	result	NOUN
ejpam-6690	563	30	in	in	ADP
ejpam-6690	563	31	ρ(x	ρ(x	NOUN
ejpam-6690	563	32	)	)	PUNCT
ejpam-6690	563	33	=	=	SYM
ejpam-6690	564	1	ρ(0	ρ(0	PROPN
ejpam-6690	564	2	)	)	PUNCT
ejpam-6690	564	3	or	or	CCONJ
ejpam-6690	564	4	ρ(y	ρ(y	NOUN
ejpam-6690	564	5	)	)	PUNCT
ejpam-6690	565	1	=	=	SYM
ejpam-6690	565	2	ρ(0	ρ(0	PROPN
ejpam-6690	565	3	)	)	PUNCT
ejpam-6690	565	4	if	if	SCONJ
ejpam-6690	565	5	and	and	CCONJ
ejpam-6690	565	6	only	only	ADV
ejpam-6690	565	7	if	if	SCONJ
ejpam-6690	565	8	r	r	NOUN
ejpam-6690	565	9	/	/	SYM
ejpam-6690	565	10	ρ	ρ	PROPN
ejpam-6690	565	11	is	be	AUX
ejpam-6690	565	12	an	an	DET
ejpam-6690	565	13	integral	integral	ADJ
ejpam-6690	565	14	domain	domain	NOUN
ejpam-6690	565	15	.	.	PUNCT
ejpam-6690	566	1	therefore	therefore	ADV
ejpam-6690	566	2	,	,	PUNCT
ejpam-6690	566	3	ρ	ρ	PROPN
ejpam-6690	566	4	is	be	AUX
ejpam-6690	566	5	prime	prime	ADJ
ejpam-6690	566	6	if	if	SCONJ
ejpam-6690	566	7	and	and	CCONJ
ejpam-6690	566	8	only	only	ADV
ejpam-6690	566	9	if	if	SCONJ
ejpam-6690	566	10	ρ(0	ρ(0	PROPN
ejpam-6690	566	11	)	)	PUNCT
ejpam-6690	566	12	is	be	AUX
ejpam-6690	566	13	a	a	DET
ejpam-6690	566	14	prime	prime	ADJ
ejpam-6690	566	15	hyperideal	hyperideal	NOUN
ejpam-6690	566	16	of	of	ADP
ejpam-6690	566	17	r.	r.	PROPN
ejpam-6690	566	18	additionally	additionally	ADV
ejpam-6690	566	19	,	,	PUNCT
ejpam-6690	566	20	since	since	SCONJ
ejpam-6690	566	21	the	the	DET
ejpam-6690	566	22	lattices	lattice	NOUN
ejpam-6690	566	23	sr(r	sr(r	NOUN
ejpam-6690	566	24	)	)	PUNCT
ejpam-6690	566	25	and	and	CCONJ
ejpam-6690	566	26	i(γ∗(0	i(γ∗(0	NOUN
ejpam-6690	566	27	)	)	PUNCT
ejpam-6690	566	28	)	)	PUNCT
ejpam-6690	566	29	are	be	AUX
ejpam-6690	566	30	isomorph	isomorph	NOUN
ejpam-6690	566	31	,	,	PUNCT
ejpam-6690	566	32	then	then	ADV
ejpam-6690	566	33	ρ	ρ	PROPN
ejpam-6690	566	34	is	be	AUX
ejpam-6690	566	35	maximal	maximal	ADJ
ejpam-6690	566	36	if	if	SCONJ
ejpam-6690	566	37	and	and	CCONJ
ejpam-6690	566	38	only	only	ADV
ejpam-6690	566	39	if	if	SCONJ
ejpam-6690	566	40	ρ(0	ρ(0	PROPN
ejpam-6690	566	41	)	)	PUNCT
ejpam-6690	566	42	is	be	AUX
ejpam-6690	566	43	a	a	DET
ejpam-6690	566	44	maximal	maximal	ADJ
ejpam-6690	566	45	hyperideal	hyperideal	NOUN
ejpam-6690	566	46	of	of	ADP
ejpam-6690	566	47	r.	r.	PROPN
ejpam-6690	566	48	example	example	NOUN
ejpam-6690	567	1	4	4	NUM
ejpam-6690	567	2	.	.	PUNCT
ejpam-6690	567	3	consider	consider	VERB
ejpam-6690	567	4	the	the	DET
ejpam-6690	567	5	krasner	krasner	NOUN
ejpam-6690	567	6	hyperring	hyperre	VERB
ejpam-6690	567	7	r	r	NOUN
ejpam-6690	567	8	as	as	SCONJ
ejpam-6690	567	9	follows	follow	VERB
ejpam-6690	567	10	:	:	PUNCT
ejpam-6690	567	11	+	+	SYM
ejpam-6690	567	12	0	0	NUM
ejpam-6690	567	13	1	1	NUM
ejpam-6690	567	14	a	a	DET
ejpam-6690	567	15	b	b	NOUN
ejpam-6690	567	16	c	c	NOUN
ejpam-6690	567	17	d	d	X
ejpam-6690	567	18	e	e	PROPN
ejpam-6690	567	19	f	f	PROPN
ejpam-6690	567	20	0	0	NUM
ejpam-6690	567	21	0	0	NUM
ejpam-6690	567	22	1	1	NUM
ejpam-6690	567	23	a	a	DET
ejpam-6690	567	24	b	b	NOUN
ejpam-6690	567	25	c	c	NOUN
ejpam-6690	567	26	d	d	X
ejpam-6690	567	27	e	e	PROPN
ejpam-6690	567	28	f	f	PROPN
ejpam-6690	567	29	1	1	NUM
ejpam-6690	567	30	1	1	NUM
ejpam-6690	567	31	a	a	PRON
ejpam-6690	567	32	,	,	PUNCT
ejpam-6690	567	33	d	d	PROPN
ejpam-6690	567	34	b	b	PROPN
ejpam-6690	567	35	,	,	PUNCT
ejpam-6690	567	36	e	e	PROPN
ejpam-6690	567	37	c	c	NOUN
ejpam-6690	567	38	1,f	1,f	PROPN
ejpam-6690	567	39	e	e	X
ejpam-6690	567	40	0,c	0,c	PRON
ejpam-6690	567	41	a	a	DET
ejpam-6690	567	42	a	a	DET
ejpam-6690	567	43	a	a	DET
ejpam-6690	567	44	b	b	NOUN
ejpam-6690	567	45	,	,	PUNCT
ejpam-6690	567	46	e	e	X
ejpam-6690	567	47	0,c	0,c	DET
ejpam-6690	567	48	1	1	NUM
ejpam-6690	567	49	a	a	PRON
ejpam-6690	567	50	,	,	PUNCT
ejpam-6690	568	1	d	d	X
ejpam-6690	568	2	c	c	X
ejpam-6690	569	1	1,f	1,f	PROPN
ejpam-6690	569	2	e	e	X
ejpam-6690	569	3	b	b	PROPN
ejpam-6690	569	4	b	b	PROPN
ejpam-6690	569	5	c	c	PROPN
ejpam-6690	569	6	1	1	NUM
ejpam-6690	569	7	d	d	NOUN
ejpam-6690	569	8	e	e	X
ejpam-6690	569	9	f	f	PROPN
ejpam-6690	569	10	a	a	PRON
ejpam-6690	569	11	0	0	NUM
ejpam-6690	569	12	c	c	NOUN
ejpam-6690	569	13	c	c	NOUN
ejpam-6690	569	14	1,f	1,f	NUM
ejpam-6690	569	15	a	a	DET
ejpam-6690	569	16	,	,	PUNCT
ejpam-6690	569	17	d	d	X
ejpam-6690	569	18	e	e	X
ejpam-6690	569	19	0,c	0,c	DET
ejpam-6690	569	20	a	a	DET
ejpam-6690	569	21	b	b	NOUN
ejpam-6690	569	22	,	,	PUNCT
ejpam-6690	569	23	e	e	PROPN
ejpam-6690	570	1	1	1	NUM
ejpam-6690	570	2	d	d	NOUN
ejpam-6690	570	3	d	d	X
ejpam-6690	570	4	e	e	X
ejpam-6690	570	5	c	c	NOUN
ejpam-6690	570	6	f	f	PROPN
ejpam-6690	570	7	a	a	DET
ejpam-6690	570	8	0	0	NUM
ejpam-6690	570	9	1	1	NUM
ejpam-6690	570	10	b	b	NOUN
ejpam-6690	570	11	e	e	X
ejpam-6690	570	12	e	e	X
ejpam-6690	570	13	0,c	0,c	PRON
ejpam-6690	571	1	1,f	1,f	NUM
ejpam-6690	571	2	a	a	DET
ejpam-6690	571	3	b	b	NOUN
ejpam-6690	571	4	,	,	PUNCT
ejpam-6690	571	5	e	e	PROPN
ejpam-6690	571	6	1	1	NUM
ejpam-6690	571	7	a	a	PRON
ejpam-6690	571	8	,	,	PUNCT
ejpam-6690	572	1	d	d	X
ejpam-6690	572	2	c	c	NOUN
ejpam-6690	572	3	f	f	PROPN
ejpam-6690	572	4	f	f	PROPN
ejpam-6690	572	5	a	a	DET
ejpam-6690	572	6	e	e	NOUN
ejpam-6690	572	7	0	0	NUM
ejpam-6690	572	8	1	1	NUM
ejpam-6690	572	9	b	b	X
ejpam-6690	572	10	c	c	NOUN
ejpam-6690	572	11	d	d	PROPN
ejpam-6690	572	12	.	.	PUNCT
ejpam-6690	572	13	0	0	NUM
ejpam-6690	573	1	1	1	NUM
ejpam-6690	573	2	a	a	DET
ejpam-6690	573	3	b	b	NOUN
ejpam-6690	573	4	c	c	NOUN
ejpam-6690	573	5	d	d	X
ejpam-6690	573	6	e	e	PROPN
ejpam-6690	573	7	f	f	PROPN
ejpam-6690	573	8	0	0	PROPN
ejpam-6690	573	9	0	0	NUM
ejpam-6690	573	10	0	0	NUM
ejpam-6690	573	11	0	0	NUM
ejpam-6690	573	12	0	0	NUM
ejpam-6690	573	13	0	0	NUM
ejpam-6690	573	14	0	0	NUM
ejpam-6690	573	15	0	0	NUM
ejpam-6690	573	16	0	0	NUM
ejpam-6690	573	17	1	1	NUM
ejpam-6690	573	18	0	0	NUM
ejpam-6690	573	19	1	1	NUM
ejpam-6690	573	20	a	a	DET
ejpam-6690	573	21	b	b	NOUN
ejpam-6690	573	22	c	c	NOUN
ejpam-6690	573	23	d	d	PROPN
ejpam-6690	573	24	e	e	X
ejpam-6690	573	25	f	f	PROPN
ejpam-6690	573	26	a	a	DET
ejpam-6690	573	27	0	0	NUM
ejpam-6690	573	28	a	a	DET
ejpam-6690	573	29	c	c	NOUN
ejpam-6690	573	30	d	d	NOUN
ejpam-6690	573	31	c	c	NOUN
ejpam-6690	573	32	0	0	PUNCT
ejpam-6690	573	33	a	a	PRON
ejpam-6690	573	34	d	d	X
ejpam-6690	573	35	b	b	PROPN
ejpam-6690	573	36	0	0	NUM
ejpam-6690	573	37	b	b	PROPN
ejpam-6690	574	1	d	d	NOUN
ejpam-6690	574	2	f	f	PROPN
ejpam-6690	574	3	0	0	PROPN
ejpam-6690	575	1	d	d	NOUN
ejpam-6690	575	2	f	f	PROPN
ejpam-6690	575	3	b	b	PROPN
ejpam-6690	575	4	c	c	NOUN
ejpam-6690	575	5	0	0	PUNCT
ejpam-6690	576	1	c	c	PROPN
ejpam-6690	576	2	c	c	NOUN
ejpam-6690	576	3	0	0	PUNCT
ejpam-6690	577	1	c	c	NOUN
ejpam-6690	577	2	0	0	PUNCT
ejpam-6690	578	1	c	c	NOUN
ejpam-6690	578	2	0	0	PUNCT
ejpam-6690	579	1	d	d	NOUN
ejpam-6690	579	2	0	0	NUM
ejpam-6690	580	1	d	d	NOUN
ejpam-6690	580	2	0	0	PUNCT
ejpam-6690	581	1	d	d	NOUN
ejpam-6690	581	2	0	0	NUM
ejpam-6690	581	3	0	0	NUM
ejpam-6690	582	1	d	d	NOUN
ejpam-6690	582	2	d	d	X
ejpam-6690	582	3	e	e	X
ejpam-6690	582	4	0	0	NUM
ejpam-6690	582	5	e	e	X
ejpam-6690	582	6	a	a	X
ejpam-6690	582	7	f	f	X
ejpam-6690	582	8	c	c	NOUN
ejpam-6690	582	9	d	d	PROPN
ejpam-6690	582	10	1	1	NUM
ejpam-6690	582	11	b	b	X
ejpam-6690	582	12	f	f	NOUN
ejpam-6690	582	13	0	0	PROPN
ejpam-6690	583	1	f	f	PROPN
ejpam-6690	584	1	d	d	PROPN
ejpam-6690	584	2	b	b	PROPN
ejpam-6690	584	3	0	0	PROPN
ejpam-6690	584	4	d	d	PROPN
ejpam-6690	584	5	b	b	PROPN
ejpam-6690	584	6	f	f	PROPN
ejpam-6690	584	7	b.	b.	PROPN
ejpam-6690	584	8	afshar	afshar	PROPN
ejpam-6690	584	9	,	,	PUNCT
ejpam-6690	584	10	r.	r.	PROPN
ejpam-6690	584	11	ameri	ameri	PROPN
ejpam-6690	584	12	,	,	PUNCT
ejpam-6690	584	13	m.	m.	PROPN
ejpam-6690	584	14	al	al	PROPN
ejpam-6690	584	15	-	-	PUNCT
ejpam-6690	584	16	tahan	tahan	PROPN
ejpam-6690	584	17	/	/	SYM
ejpam-6690	584	18	eur	eur	PROPN
ejpam-6690	584	19	.	.	PUNCT
ejpam-6690	585	1	j.	j.	PROPN
ejpam-6690	585	2	pure	pure	PROPN
ejpam-6690	585	3	appl	appl	PROPN
ejpam-6690	585	4	.	.	PROPN
ejpam-6690	585	5	math	math	PROPN
ejpam-6690	585	6	,	,	PUNCT
ejpam-6690	585	7	18	18	NUM
ejpam-6690	585	8	(	(	PUNCT
ejpam-6690	585	9	4	4	NUM
ejpam-6690	585	10	)	)	PUNCT
ejpam-6690	585	11	(	(	PUNCT
ejpam-6690	585	12	2025	2025	NUM
ejpam-6690	585	13	)	)	PUNCT
ejpam-6690	585	14	,	,	PUNCT
ejpam-6690	585	15	6690	6690	NUM
ejpam-6690	585	16	15	15	NUM
ejpam-6690	585	17	of	of	ADP
ejpam-6690	585	18	19	19	NUM
ejpam-6690	585	19	then	then	ADV
ejpam-6690	585	20	γ∗	γ∗	NOUN
ejpam-6690	585	21	=	=	SYM
ejpam-6690	585	22	{	{	PUNCT
ejpam-6690	585	23	(	(	PUNCT
ejpam-6690	585	24	a	a	DET
ejpam-6690	585	25	,	,	PUNCT
ejpam-6690	585	26	d	d	NOUN
ejpam-6690	585	27	)	)	PUNCT
ejpam-6690	585	28	,	,	PUNCT
ejpam-6690	585	29	(	(	PUNCT
ejpam-6690	585	30	d	d	X
ejpam-6690	585	31	,	,	PUNCT
ejpam-6690	585	32	a	a	NOUN
ejpam-6690	585	33	)	)	PUNCT
ejpam-6690	585	34	,	,	PUNCT
ejpam-6690	585	35	(	(	PUNCT
ejpam-6690	585	36	b	b	X
ejpam-6690	585	37	,	,	PUNCT
ejpam-6690	585	38	e	e	NOUN
ejpam-6690	585	39	)	)	PUNCT
ejpam-6690	585	40	,	,	PUNCT
ejpam-6690	585	41	(	(	PUNCT
ejpam-6690	585	42	e	e	NOUN
ejpam-6690	585	43	,	,	PUNCT
ejpam-6690	585	44	b	b	NOUN
ejpam-6690	585	45	)	)	PUNCT
ejpam-6690	585	46	,	,	PUNCT
ejpam-6690	585	47	(	(	PUNCT
ejpam-6690	585	48	1	1	NUM
ejpam-6690	585	49	,	,	PUNCT
ejpam-6690	585	50	f	f	NOUN
ejpam-6690	585	51	)	)	PUNCT
ejpam-6690	585	52	,	,	PUNCT
ejpam-6690	585	53	(	(	PUNCT
ejpam-6690	585	54	f	f	X
ejpam-6690	585	55	,	,	PUNCT
ejpam-6690	585	56	1	1	NUM
ejpam-6690	585	57	)	)	PUNCT
ejpam-6690	585	58	,	,	PUNCT
ejpam-6690	585	59	(	(	PUNCT
ejpam-6690	585	60	0	0	NUM
ejpam-6690	585	61	,	,	PUNCT
ejpam-6690	585	62	c	c	NOUN
ejpam-6690	585	63	)	)	PUNCT
ejpam-6690	585	64	,	,	PUNCT
ejpam-6690	585	65	(	(	PUNCT
ejpam-6690	585	66	c	c	X
ejpam-6690	585	67	,	,	PUNCT
ejpam-6690	585	68	0)}∪∆r	0)}∪∆r	NUM
ejpam-6690	585	69	is	be	AUX
ejpam-6690	585	70	a	a	DET
ejpam-6690	585	71	primitive	primitive	ADJ
ejpam-6690	585	72	strongly	strongly	ADV
ejpam-6690	585	73	regular	regular	ADJ
ejpam-6690	585	74	relation	relation	NOUN
ejpam-6690	585	75	on	on	ADP
ejpam-6690	585	76	r	r	NOUN
ejpam-6690	585	77	that	that	PRON
ejpam-6690	585	78	is	be	AUX
ejpam-6690	585	79	not	not	PART
ejpam-6690	585	80	prime	prime	ADJ
ejpam-6690	585	81	because	because	SCONJ
ejpam-6690	585	82	(	(	PUNCT
ejpam-6690	585	83	ad	ad	NOUN
ejpam-6690	585	84	,	,	PUNCT
ejpam-6690	585	85	0	0	NUM
ejpam-6690	585	86	)	)	PUNCT
ejpam-6690	585	87	∈	∈	NOUN
ejpam-6690	585	88	γ∗	γ∗	NOUN
ejpam-6690	585	89	but	but	CCONJ
ejpam-6690	585	90	(	(	PUNCT
ejpam-6690	585	91	a	a	PRON
ejpam-6690	585	92	,	,	PUNCT
ejpam-6690	585	93	0	0	NUM
ejpam-6690	585	94	)	)	PUNCT
ejpam-6690	585	95	,	,	PUNCT
ejpam-6690	585	96	(	(	PUNCT
ejpam-6690	585	97	d	d	NOUN
ejpam-6690	585	98	,	,	PUNCT
ejpam-6690	585	99	0	0	NUM
ejpam-6690	585	100	)	)	PUNCT
ejpam-6690	585	101	/∈	/∈	PUNCT
ejpam-6690	586	1	γ∗.	γ∗.	ADV
ejpam-6690	586	2	equivalently	equivalently	ADV
ejpam-6690	586	3	,	,	PUNCT
ejpam-6690	586	4	γ∗(0	γ∗(0	NOUN
ejpam-6690	586	5	)	)	PUNCT
ejpam-6690	586	6	=	=	SYM
ejpam-6690	586	7	{	{	PUNCT
ejpam-6690	586	8	0	0	NUM
ejpam-6690	586	9	,	,	PUNCT
ejpam-6690	586	10	c	c	NOUN
ejpam-6690	586	11	}	}	PUNCT
ejpam-6690	586	12	is	be	AUX
ejpam-6690	586	13	a	a	DET
ejpam-6690	586	14	primitive	primitive	ADJ
ejpam-6690	586	15	hyperideal	hyperideal	NOUN
ejpam-6690	586	16	that	that	PRON
ejpam-6690	586	17	is	be	AUX
ejpam-6690	586	18	not	not	PART
ejpam-6690	586	19	prime	prime	ADJ
ejpam-6690	586	20	.	.	PUNCT
ejpam-6690	587	1	also	also	ADV
ejpam-6690	587	2	,	,	PUNCT
ejpam-6690	587	3	r	r	NOUN
ejpam-6690	587	4	/	/	SYM
ejpam-6690	587	5	γ∗	γ∗	NOUN
ejpam-6690	587	6	=	=	SYM
ejpam-6690	587	7	{	{	PUNCT
ejpam-6690	587	8	{	{	PUNCT
ejpam-6690	587	9	0	0	NUM
ejpam-6690	587	10	,	,	PUNCT
ejpam-6690	587	11	c	c	NOUN
ejpam-6690	587	12	}	}	PUNCT
ejpam-6690	587	13	,	,	PUNCT
ejpam-6690	587	14	{	{	PUNCT
ejpam-6690	587	15	1	1	NUM
ejpam-6690	587	16	,	,	PUNCT
ejpam-6690	587	17	f	f	NOUN
ejpam-6690	587	18	}	}	PUNCT
ejpam-6690	587	19	,	,	PUNCT
ejpam-6690	587	20	{	{	PUNCT
ejpam-6690	587	21	a	a	DET
ejpam-6690	587	22	,	,	PUNCT
ejpam-6690	587	23	d	d	NOUN
ejpam-6690	587	24	}	}	PUNCT
ejpam-6690	587	25	,	,	PUNCT
ejpam-6690	587	26	{	{	PUNCT
ejpam-6690	587	27	b	b	NOUN
ejpam-6690	587	28	,	,	PUNCT
ejpam-6690	587	29	e	e	NOUN
ejpam-6690	587	30	}	}	PUNCT
ejpam-6690	587	31	}	}	PUNCT
ejpam-6690	587	32	∼=	∼=	VERB
ejpam-6690	587	33	z4	z4	NOUN
ejpam-6690	587	34	,	,	PUNCT
ejpam-6690	587	35	which	which	PRON
ejpam-6690	587	36	is	be	AUX
ejpam-6690	587	37	not	not	PART
ejpam-6690	587	38	an	an	DET
ejpam-6690	587	39	integral	integral	ADJ
ejpam-6690	587	40	domain	domain	NOUN
ejpam-6690	587	41	,	,	PUNCT
ejpam-6690	587	42	and	and	CCONJ
ejpam-6690	587	43	i(γ∗(0	i(γ∗(0	NOUN
ejpam-6690	587	44	)	)	PUNCT
ejpam-6690	587	45	)	)	PUNCT
ejpam-6690	588	1	=	=	PRON
ejpam-6690	588	2	{	{	PUNCT
ejpam-6690	588	3	{	{	PUNCT
ejpam-6690	588	4	0	0	NUM
ejpam-6690	588	5	,	,	PUNCT
ejpam-6690	588	6	c	c	NOUN
ejpam-6690	588	7	}	}	PUNCT
ejpam-6690	588	8	,	,	PUNCT
ejpam-6690	588	9	{	{	PUNCT
ejpam-6690	588	10	0	0	NUM
ejpam-6690	588	11	,	,	PUNCT
ejpam-6690	588	12	c	c	NOUN
ejpam-6690	588	13	,	,	PUNCT
ejpam-6690	588	14	a	a	DET
ejpam-6690	588	15	,	,	PUNCT
ejpam-6690	588	16	d	d	NOUN
ejpam-6690	588	17	}	}	PUNCT
ejpam-6690	588	18	}	}	PUNCT
ejpam-6690	588	19	,	,	PUNCT
ejpam-6690	588	20	where	where	SCONJ
ejpam-6690	588	21	{	{	PUNCT
ejpam-6690	588	22	0	0	NUM
ejpam-6690	588	23	,	,	PUNCT
ejpam-6690	588	24	c	c	NOUN
ejpam-6690	588	25	,	,	PUNCT
ejpam-6690	588	26	a	a	PRON
ejpam-6690	588	27	,	,	PUNCT
ejpam-6690	588	28	d	d	NOUN
ejpam-6690	588	29	}	}	PUNCT
ejpam-6690	588	30	=	=	SYM
ejpam-6690	588	31	v	v	NOUN
ejpam-6690	588	32	(	(	PUNCT
ejpam-6690	588	33	γ∗(0	γ∗(0	NOUN
ejpam-6690	588	34	)	)	PUNCT
ejpam-6690	588	35	)	)	PUNCT
ejpam-6690	588	36	=	=	SYM
ejpam-6690	588	37	√	√	NUM
ejpam-6690	588	38	γ∗(0	γ∗(0	NOUN
ejpam-6690	588	39	)	)	PUNCT
ejpam-6690	588	40	is	be	AUX
ejpam-6690	588	41	a	a	DET
ejpam-6690	588	42	maximal	maximal	ADJ
ejpam-6690	588	43	hyperideal	hyperideal	NOUN
ejpam-6690	588	44	of	of	ADP
ejpam-6690	588	45	r	r	NOUN
ejpam-6690	588	46	and	and	CCONJ
ejpam-6690	588	47	r/	r/	ADV
ejpam-6690	588	48	√	√	NUM
ejpam-6690	588	49	γ∗(0	γ∗(0	NOUN
ejpam-6690	588	50	)	)	PUNCT
ejpam-6690	588	51	=	=	PRON
ejpam-6690	588	52	{	{	PUNCT
ejpam-6690	588	53	{	{	PUNCT
ejpam-6690	588	54	0	0	NUM
ejpam-6690	588	55	,	,	PUNCT
ejpam-6690	588	56	c	c	NOUN
ejpam-6690	588	57	,	,	PUNCT
ejpam-6690	588	58	a	a	DET
ejpam-6690	588	59	,	,	PUNCT
ejpam-6690	588	60	d	d	NOUN
ejpam-6690	588	61	}	}	PUNCT
ejpam-6690	588	62	,	,	PUNCT
ejpam-6690	588	63	{	{	PUNCT
ejpam-6690	588	64	1	1	NUM
ejpam-6690	588	65	,	,	PUNCT
ejpam-6690	588	66	f	f	PROPN
ejpam-6690	588	67	,	,	PUNCT
ejpam-6690	588	68	b	b	PROPN
ejpam-6690	588	69	,	,	PUNCT
ejpam-6690	588	70	e	e	NOUN
ejpam-6690	588	71	}	}	PUNCT
ejpam-6690	588	72	}	}	PUNCT
ejpam-6690	588	73	∼=	∼=	PROPN
ejpam-6690	588	74	z2	z2	NOUN
ejpam-6690	588	75	is	be	AUX
ejpam-6690	588	76	a	a	DET
ejpam-6690	588	77	field	field	NOUN
ejpam-6690	588	78	.	.	PUNCT
ejpam-6690	589	1	let	let	VERB
ejpam-6690	589	2	i	i	PRON
ejpam-6690	589	3	=	=	PUNCT
ejpam-6690	589	4	{	{	PUNCT
ejpam-6690	589	5	0	0	NUM
ejpam-6690	589	6	,	,	PUNCT
ejpam-6690	589	7	d	d	NOUN
ejpam-6690	589	8	}	}	PUNCT
ejpam-6690	589	9	and	and	CCONJ
ejpam-6690	589	10	m	m	PROPN
ejpam-6690	589	11	=	=	SYM
ejpam-6690	589	12	{	{	PUNCT
ejpam-6690	589	13	0	0	NUM
ejpam-6690	589	14	,	,	PUNCT
ejpam-6690	589	15	b	b	NOUN
ejpam-6690	589	16	,	,	PUNCT
ejpam-6690	589	17	d	d	PROPN
ejpam-6690	589	18	,	,	PUNCT
ejpam-6690	589	19	f	f	NOUN
ejpam-6690	589	20	}	}	PUNCT
ejpam-6690	589	21	.	.	PUNCT
ejpam-6690	590	1	because	because	SCONJ
ejpam-6690	590	2	a	a	PRON
ejpam-6690	590	3	,	,	PUNCT
ejpam-6690	590	4	f	f	PROPN
ejpam-6690	590	5	/∈	/∈	PUNCT
ejpam-6690	591	1	i	i	PRON
ejpam-6690	591	2	while	while	SCONJ
ejpam-6690	591	3	af	af	PROPN
ejpam-6690	591	4	=	=	SYM
ejpam-6690	591	5	d	d	X
ejpam-6690	591	6	∈	∈	PROPN
ejpam-6690	592	1	i	i	PRON
ejpam-6690	592	2	and	and	CCONJ
ejpam-6690	592	3	γ∗(0	γ∗(0	NOUN
ejpam-6690	592	4	)	)	PUNCT
ejpam-6690	592	5	⊈	⊈	PROPN
ejpam-6690	593	1	i	i	PRON
ejpam-6690	593	2	,	,	PUNCT
ejpam-6690	593	3	then	then	ADV
ejpam-6690	593	4	i	i	PRON
ejpam-6690	593	5	is	be	AUX
ejpam-6690	593	6	neither	neither	CCONJ
ejpam-6690	593	7	prime	prime	ADJ
ejpam-6690	593	8	nor	nor	CCONJ
ejpam-6690	593	9	normal	normal	ADJ
ejpam-6690	593	10	.	.	PUNCT
ejpam-6690	594	1	also	also	ADV
ejpam-6690	594	2	,	,	PUNCT
ejpam-6690	594	3	m	m	VERB
ejpam-6690	594	4	is	be	AUX
ejpam-6690	594	5	a	a	DET
ejpam-6690	594	6	maximal	maximal	ADJ
ejpam-6690	594	7	hyperideal	hyperideal	NOUN
ejpam-6690	594	8	which	which	PRON
ejpam-6690	594	9	is	be	AUX
ejpam-6690	594	10	not	not	PART
ejpam-6690	594	11	normal	normal	ADJ
ejpam-6690	594	12	.	.	PUNCT
ejpam-6690	595	1	so	so	ADV
ejpam-6690	595	2	,	,	PUNCT
ejpam-6690	595	3	spec(r	spec(r	PROPN
ejpam-6690	595	4	)	)	PUNCT
ejpam-6690	595	5	=	=	PUNCT
ejpam-6690	595	6	{	{	PUNCT
ejpam-6690	595	7	m	m	PROPN
ejpam-6690	595	8	,	,	PUNCT
ejpam-6690	595	9	√	√	NUM
ejpam-6690	595	10	γ∗(0	γ∗(0	NOUN
ejpam-6690	595	11	)	)	PUNCT
ejpam-6690	595	12	}	}	PUNCT
ejpam-6690	595	13	.	.	PUNCT
ejpam-6690	596	1	theorem	theorem	NOUN
ejpam-6690	596	2	12	12	NUM
ejpam-6690	596	3	.	.	PUNCT
ejpam-6690	597	1	let	let	VERB
ejpam-6690	597	2	r	r	PRON
ejpam-6690	597	3	be	be	AUX
ejpam-6690	597	4	a	a	DET
ejpam-6690	597	5	hyperring	hyperring	NOUN
ejpam-6690	597	6	.	.	PUNCT
ejpam-6690	598	1	then	then	ADV
ejpam-6690	598	2	spec(r	spec(r	PROPN
ejpam-6690	598	3	/	/	SYM
ejpam-6690	598	4	γ∗	γ∗	PROPN
ejpam-6690	598	5	)	)	PUNCT
ejpam-6690	599	1	=	=	PRON
ejpam-6690	599	2	{	{	PUNCT
ejpam-6690	599	3	p	p	X
ejpam-6690	599	4	/	/	SYM
ejpam-6690	599	5	γ∗;p	γ∗;p	PROPN
ejpam-6690	599	6	∈	∈	PROPN
ejpam-6690	599	7	spec(r	spec(r	PROPN
ejpam-6690	599	8	)	)	PUNCT
ejpam-6690	599	9	}	}	PUNCT
ejpam-6690	599	10	.	.	PUNCT
ejpam-6690	600	1	proof	proof	NOUN
ejpam-6690	600	2	.	.	PUNCT
ejpam-6690	601	1	let	let	VERB
ejpam-6690	601	2	i	i	PRON
ejpam-6690	601	3	be	be	AUX
ejpam-6690	601	4	a	a	DET
ejpam-6690	601	5	prime	prime	ADJ
ejpam-6690	601	6	ideal	ideal	NOUN
ejpam-6690	601	7	of	of	ADP
ejpam-6690	601	8	r	r	NOUN
ejpam-6690	601	9	/	/	SYM
ejpam-6690	601	10	γ∗	γ∗	NOUN
ejpam-6690	601	11	and	and	CCONJ
ejpam-6690	601	12	p	p	NOUN
ejpam-6690	601	13	=	=	X
ejpam-6690	601	14	{	{	PUNCT
ejpam-6690	601	15	s	s	NOUN
ejpam-6690	601	16	∈	∈	NOUN
ejpam-6690	601	17	r	r	NOUN
ejpam-6690	601	18	;	;	PUNCT
ejpam-6690	601	19	γ∗(r	γ∗(r	NUM
ejpam-6690	601	20	)	)	PUNCT
ejpam-6690	601	21	∈	∈	PROPN
ejpam-6690	602	1	i	i	PRON
ejpam-6690	602	2	}	}	PUNCT
ejpam-6690	602	3	.	.	PUNCT
ejpam-6690	603	1	so	so	ADV
ejpam-6690	603	2	i	i	PRON
ejpam-6690	603	3	=	=	SYM
ejpam-6690	604	1	p	p	X
ejpam-6690	604	2	/	/	SYM
ejpam-6690	604	3	γ∗	γ∗	NOUN
ejpam-6690	604	4	and	and	CCONJ
ejpam-6690	604	5	if	if	SCONJ
ejpam-6690	604	6	s	s	PROPN
ejpam-6690	604	7	,	,	PUNCT
ejpam-6690	604	8	t	t	PROPN
ejpam-6690	604	9	∈	∈	PROPN
ejpam-6690	604	10	p	p	PROPN
ejpam-6690	604	11	and	and	CCONJ
ejpam-6690	604	12	r	r	NOUN
ejpam-6690	604	13	,	,	PUNCT
ejpam-6690	604	14	k	k	PROPN
ejpam-6690	604	15	∈	∈	PROPN
ejpam-6690	604	16	r	r	NOUN
ejpam-6690	604	17	,	,	PUNCT
ejpam-6690	604	18	then	then	ADV
ejpam-6690	604	19	γ∗(s	γ∗(s	PROPN
ejpam-6690	604	20	)	)	PUNCT
ejpam-6690	605	1	−	−	PROPN
ejpam-6690	605	2	γ∗(t	γ∗(t	NOUN
ejpam-6690	605	3	)	)	PUNCT
ejpam-6690	605	4	=	=	SYM
ejpam-6690	606	1	γ∗(s	γ∗(s	PROPN
ejpam-6690	606	2	−	−	PROPN
ejpam-6690	606	3	t	t	PROPN
ejpam-6690	606	4	)	)	PUNCT
ejpam-6690	606	5	∈	∈	PROPN
ejpam-6690	606	6	i	i	PRON
ejpam-6690	606	7	and	and	CCONJ
ejpam-6690	606	8	γ∗(r)γ∗(s	γ∗(r)γ∗(s	NUM
ejpam-6690	606	9	)	)	PUNCT
ejpam-6690	606	10	=	=	SYM
ejpam-6690	606	11	γ∗(rs	γ∗(rs	PROPN
ejpam-6690	606	12	)	)	PUNCT
ejpam-6690	606	13	∈	∈	PROPN
ejpam-6690	606	14	i.	i.	NOUN
ejpam-6690	606	15	hence	hence	ADV
ejpam-6690	606	16	s−	s−	PROPN
ejpam-6690	606	17	t	t	PROPN
ejpam-6690	606	18	⊆	⊆	NUM
ejpam-6690	606	19	p	p	NOUN
ejpam-6690	606	20	and	and	CCONJ
ejpam-6690	606	21	rs	rs	PROPN
ejpam-6690	606	22	∈	∈	PROPN
ejpam-6690	606	23	p	p	NOUN
ejpam-6690	606	24	.	.	PUNCT
ejpam-6690	607	1	if	if	SCONJ
ejpam-6690	607	2	γ∗(r)γ∗(k	γ∗(r)γ∗(k	NUM
ejpam-6690	607	3	)	)	PUNCT
ejpam-6690	607	4	=	=	SYM
ejpam-6690	607	5	γ∗(rk	γ∗(rk	NOUN
ejpam-6690	607	6	)	)	PUNCT
ejpam-6690	607	7	∈	∈	PROPN
ejpam-6690	608	1	i	i	PRON
ejpam-6690	608	2	,	,	PUNCT
ejpam-6690	608	3	then	then	ADV
ejpam-6690	608	4	γ∗(r	γ∗(r	NUM
ejpam-6690	608	5	)	)	PUNCT
ejpam-6690	608	6	∈	∈	PROPN
ejpam-6690	608	7	i	i	PRON
ejpam-6690	608	8	or	or	CCONJ
ejpam-6690	608	9	γ∗(k	γ∗(k	NOUN
ejpam-6690	608	10	)	)	PUNCT
ejpam-6690	608	11	∈	∈	PROPN
ejpam-6690	608	12	i.	i.	NOUN
ejpam-6690	609	1	so	so	ADV
ejpam-6690	609	2	if	if	SCONJ
ejpam-6690	609	3	rk	rk	VERB
ejpam-6690	609	4	∈	∈	PROPN
ejpam-6690	609	5	p	p	NOUN
ejpam-6690	609	6	,	,	PUNCT
ejpam-6690	609	7	then	then	ADV
ejpam-6690	609	8	r	r	NOUN
ejpam-6690	609	9	∈	∈	PROPN
ejpam-6690	609	10	p	p	NOUN
ejpam-6690	609	11	or	or	CCONJ
ejpam-6690	609	12	k	k	PROPN
ejpam-6690	609	13	∈	∈	PROPN
ejpam-6690	609	14	p	p	PROPN
ejpam-6690	609	15	.	.	PUNCT
ejpam-6690	610	1	lemma	lemma	PROPN
ejpam-6690	610	2	6	6	NUM
ejpam-6690	610	3	.	.	PUNCT
ejpam-6690	611	1	let	let	VERB
ejpam-6690	611	2	s	s	PRON
ejpam-6690	611	3	⊂	⊂	X
ejpam-6690	611	4	spec(r	spec(r	PROPN
ejpam-6690	611	5	)	)	PUNCT
ejpam-6690	611	6	.	.	PUNCT
ejpam-6690	612	1	then	then	ADV
ejpam-6690	612	2	s̄	s̄	NOUN
ejpam-6690	612	3	=	=	SYM
ejpam-6690	612	4	v	v	NOUN
ejpam-6690	612	5	(	(	PUNCT
ejpam-6690	612	6	⋂	⋂	PROPN
ejpam-6690	612	7	p∈s	p∈	VERB
ejpam-6690	612	8	p	p	NOUN
ejpam-6690	612	9	)	)	PUNCT
ejpam-6690	612	10	.	.	PUNCT
ejpam-6690	613	1	proof	proof	NOUN
ejpam-6690	613	2	.	.	PUNCT
ejpam-6690	614	1	let	let	VERB
ejpam-6690	614	2	i	i	PRON
ejpam-6690	614	3	be	be	AUX
ejpam-6690	614	4	the	the	DET
ejpam-6690	614	5	radical	radical	ADJ
ejpam-6690	614	6	hyperideal	hyperideal	NOUN
ejpam-6690	615	1	such	such	DET
ejpam-6690	615	2	that	that	DET
ejpam-6690	615	3	v	v	NOUN
ejpam-6690	615	4	(	(	PUNCT
ejpam-6690	615	5	i	i	NOUN
ejpam-6690	615	6	)	)	PUNCT
ejpam-6690	615	7	=	=	SYM
ejpam-6690	615	8	s̄	s̄	NOUN
ejpam-6690	615	9	,	,	PUNCT
ejpam-6690	615	10	then	then	ADV
ejpam-6690	615	11	i	i	PRON
ejpam-6690	615	12	⊂	⊂	PROPN
ejpam-6690	615	13	⋂	⋂	PROPN
ejpam-6690	615	14	p∈s	p∈	VERB
ejpam-6690	615	15	p	p	PROPN
ejpam-6690	615	16	.	.	PUNCT
ejpam-6690	616	1	since	since	SCONJ
ejpam-6690	616	2	v	v	NOUN
ejpam-6690	616	3	(	(	PUNCT
ejpam-6690	616	4	⋂	⋂	PROPN
ejpam-6690	616	5	p∈s	p∈	NOUN
ejpam-6690	616	6	p	p	NOUN
ejpam-6690	616	7	)	)	PUNCT
ejpam-6690	616	8	is	be	AUX
ejpam-6690	616	9	closed	close	VERB
ejpam-6690	616	10	,	,	PUNCT
ejpam-6690	616	11	then	then	ADV
ejpam-6690	616	12	s	s	VERB
ejpam-6690	616	13	⊂	⊂	PROPN
ejpam-6690	616	14	v	v	X
ejpam-6690	616	15	(	(	PUNCT
ejpam-6690	616	16	⋂	⋂	PROPN
ejpam-6690	616	17	p∈s	p∈	NOUN
ejpam-6690	616	18	p	p	NOUN
ejpam-6690	616	19	)	)	PUNCT
ejpam-6690	616	20	and	and	CCONJ
ejpam-6690	616	21	so	so	ADV
ejpam-6690	616	22	s̄	s̄	PROPN
ejpam-6690	616	23	⊂	⊂	PROPN
ejpam-6690	616	24	v	v	NOUN
ejpam-6690	616	25	(	(	PUNCT
ejpam-6690	616	26	⋂	⋂	PROPN
ejpam-6690	616	27	p∈s	p∈	VERB
ejpam-6690	616	28	p	p	NOUN
ejpam-6690	616	29	)	)	PUNCT
ejpam-6690	616	30	.	.	PUNCT
ejpam-6690	617	1	hence	hence	ADV
ejpam-6690	617	2	v	v	NOUN
ejpam-6690	617	3	(	(	PUNCT
ejpam-6690	617	4	i	i	NOUN
ejpam-6690	617	5	)	)	PUNCT
ejpam-6690	618	1	⊂	⊂	PROPN
ejpam-6690	618	2	v	v	X
ejpam-6690	618	3	(	(	PUNCT
ejpam-6690	618	4	⋂	⋂	PROPN
ejpam-6690	618	5	p∈s	p∈	NOUN
ejpam-6690	618	6	p	p	NOUN
ejpam-6690	618	7	)	)	PUNCT
ejpam-6690	618	8	and	and	CCONJ
ejpam-6690	618	9	by	by	ADP
ejpam-6690	618	10	proposition	proposition	NOUN
ejpam-6690	618	11	2	2	NUM
ejpam-6690	618	12	,	,	PUNCT
ejpam-6690	618	13	⋂	⋂	PROPN
ejpam-6690	618	14	p∈s	p∈	VERB
ejpam-6690	618	15	p	p	PROPN
ejpam-6690	618	16	=	=	NOUN
ejpam-6690	618	17	i.	i.	NOUN
ejpam-6690	618	18	theorem	theorem	VERB
ejpam-6690	618	19	13	13	NUM
ejpam-6690	618	20	.	.	PUNCT
ejpam-6690	619	1	let	let	VERB
ejpam-6690	619	2	f	f	NOUN
ejpam-6690	619	3	:	:	PUNCT
ejpam-6690	619	4	r	r	X
ejpam-6690	619	5	→	→	SYM
ejpam-6690	619	6	s	s	AUX
ejpam-6690	619	7	be	be	AUX
ejpam-6690	619	8	a	a	DET
ejpam-6690	619	9	good	good	ADJ
ejpam-6690	619	10	homomorphism	homomorphism	NOUN
ejpam-6690	619	11	of	of	ADP
ejpam-6690	619	12	hyperrings	hyperring	NOUN
ejpam-6690	619	13	.	.	PUNCT
ejpam-6690	620	1	then	then	ADV
ejpam-6690	620	2	:	:	PUNCT
ejpam-6690	620	3	(	(	PUNCT
ejpam-6690	620	4	i	i	NOUN
ejpam-6690	620	5	)	)	PUNCT
ejpam-6690	620	6	if	if	SCONJ
ejpam-6690	620	7	i	i	PRON
ejpam-6690	620	8	is	be	AUX
ejpam-6690	620	9	a	a	DET
ejpam-6690	620	10	hyperideal	hyperideal	NOUN
ejpam-6690	620	11	of	of	ADP
ejpam-6690	620	12	r	r	NOUN
ejpam-6690	620	13	,	,	PUNCT
ejpam-6690	620	14	then	then	ADV
ejpam-6690	620	15	f̄	f̄	PROPN
ejpam-6690	620	16	−1	−1	NOUN
ejpam-6690	620	17	(	(	PUNCT
ejpam-6690	620	18	v	v	NOUN
ejpam-6690	620	19	(	(	PUNCT
ejpam-6690	620	20	i	i	NOUN
ejpam-6690	620	21	)	)	PUNCT
ejpam-6690	620	22	)	)	PUNCT
ejpam-6690	621	1	=	=	SYM
ejpam-6690	621	2	v	v	X
ejpam-6690	621	3	(	(	PUNCT
ejpam-6690	621	4	ie	ie	X
ejpam-6690	621	5	)	)	PUNCT
ejpam-6690	621	6	.	.	PUNCT
ejpam-6690	622	1	(	(	PUNCT
ejpam-6690	622	2	ii	ii	NOUN
ejpam-6690	622	3	)	)	PUNCT
ejpam-6690	622	4	for	for	ADP
ejpam-6690	622	5	every	every	DET
ejpam-6690	622	6	x	x	SYM
ejpam-6690	622	7	∈	∈	PROPN
ejpam-6690	622	8	r	r	NOUN
ejpam-6690	622	9	,	,	PUNCT
ejpam-6690	622	10	f̄	f̄	NOUN
ejpam-6690	622	11	−1	−1	NOUN
ejpam-6690	622	12	(	(	PUNCT
ejpam-6690	622	13	w	w	PROPN
ejpam-6690	622	14	(	(	PUNCT
ejpam-6690	622	15	x	x	NOUN
ejpam-6690	622	16	)	)	PUNCT
ejpam-6690	622	17	)	)	PUNCT
ejpam-6690	623	1	=	=	SYM
ejpam-6690	623	2	w	w	PROPN
ejpam-6690	623	3	(	(	PUNCT
ejpam-6690	623	4	f(x	f(x	PROPN
ejpam-6690	623	5	)	)	PUNCT
ejpam-6690	623	6	)	)	PUNCT
ejpam-6690	623	7	.	.	PUNCT
ejpam-6690	624	1	(	(	PUNCT
ejpam-6690	624	2	iii	iii	X
ejpam-6690	624	3	)	)	PUNCT
ejpam-6690	624	4	if	if	SCONJ
ejpam-6690	624	5	j	j	PROPN
ejpam-6690	624	6	is	be	AUX
ejpam-6690	624	7	a	a	DET
ejpam-6690	624	8	hyperideal	hyperideal	NOUN
ejpam-6690	624	9	of	of	ADP
ejpam-6690	624	10	s	s	NOUN
ejpam-6690	624	11	,	,	PUNCT
ejpam-6690	624	12	then	then	ADV
ejpam-6690	624	13	f̄(v	f̄(v	PROPN
ejpam-6690	624	14	(	(	PUNCT
ejpam-6690	624	15	j	j	NOUN
ejpam-6690	624	16	)	)	PUNCT
ejpam-6690	624	17	)	)	PUNCT
ejpam-6690	625	1	=	=	SYM
ejpam-6690	625	2	v	v	X
ejpam-6690	625	3	(	(	PUNCT
ejpam-6690	625	4	jc	jc	PROPN
ejpam-6690	625	5	)	)	PUNCT
ejpam-6690	625	6	.	.	PUNCT
ejpam-6690	626	1	(	(	PUNCT
ejpam-6690	626	2	iv	iv	X
ejpam-6690	626	3	)	)	PUNCT
ejpam-6690	626	4	if	if	SCONJ
ejpam-6690	626	5	f	f	PROPN
ejpam-6690	626	6	is	be	AUX
ejpam-6690	626	7	surjective	surjective	ADJ
ejpam-6690	626	8	,	,	PUNCT
ejpam-6690	626	9	then	then	ADV
ejpam-6690	626	10	f̄	f̄	PROPN
ejpam-6690	626	11	is	be	AUX
ejpam-6690	626	12	a	a	DET
ejpam-6690	626	13	homiomorphism	homiomorphism	NOUN
ejpam-6690	626	14	of	of	ADP
ejpam-6690	626	15	spec(s	spec(s	NOUN
ejpam-6690	626	16	)	)	PUNCT
ejpam-6690	626	17	on	on	ADP
ejpam-6690	626	18	to	to	ADP
ejpam-6690	626	19	the	the	DET
ejpam-6690	626	20	closed	closed	ADJ
ejpam-6690	626	21	subset	subset	NOUN
ejpam-6690	626	22	v	v	NOUN
ejpam-6690	626	23	(	(	PUNCT
ejpam-6690	626	24	ker(f	ker(f	PROPN
ejpam-6690	626	25	)	)	PUNCT
ejpam-6690	626	26	)	)	PUNCT
ejpam-6690	626	27	of	of	ADP
ejpam-6690	626	28	spec(r	spec(r	PROPN
ejpam-6690	626	29	)	)	PUNCT
ejpam-6690	626	30	.	.	PUNCT
ejpam-6690	627	1	(	(	PUNCT
ejpam-6690	627	2	v	v	NOUN
ejpam-6690	627	3	)	)	PUNCT
ejpam-6690	627	4	if	if	SCONJ
ejpam-6690	627	5	f	f	PROPN
ejpam-6690	627	6	is	be	AUX
ejpam-6690	627	7	injective	injective	ADJ
ejpam-6690	627	8	,	,	PUNCT
ejpam-6690	627	9	then	then	ADV
ejpam-6690	627	10	f̄(spec(s	f̄(spec(	VERB
ejpam-6690	627	11	)	)	PUNCT
ejpam-6690	627	12	)	)	PUNCT
ejpam-6690	627	13	is	be	AUX
ejpam-6690	627	14	dense	dense	ADJ
ejpam-6690	627	15	in	in	ADP
ejpam-6690	627	16	spec(r	spec(r	PROPN
ejpam-6690	627	17	)	)	PUNCT
ejpam-6690	627	18	.	.	PUNCT
ejpam-6690	628	1	in	in	ADP
ejpam-6690	628	2	fact	fact	NOUN
ejpam-6690	628	3	,	,	PUNCT
ejpam-6690	628	4	the	the	DET
ejpam-6690	628	5	image	image	NOUN
ejpam-6690	628	6	f̄(spec(s	f̄(spec(	VERB
ejpam-6690	628	7	)	)	PUNCT
ejpam-6690	628	8	)	)	PUNCT
ejpam-6690	629	1	is	be	AUX
ejpam-6690	629	2	dense	dense	ADJ
ejpam-6690	629	3	in	in	ADP
ejpam-6690	629	4	spec(r	spec(r	PROPN
ejpam-6690	629	5	)	)	PUNCT
ejpam-6690	629	6	if	if	SCONJ
ejpam-6690	630	1	and	and	CCONJ
ejpam-6690	630	2	only	only	ADV
ejpam-6690	630	3	if	if	SCONJ
ejpam-6690	630	4	ker(f	ker(f	PROPN
ejpam-6690	630	5	)	)	PUNCT
ejpam-6690	630	6	⊂	⊂	PRON
ejpam-6690	631	1	√	√	NOUN
ejpam-6690	631	2	0	0	NUM
ejpam-6690	631	3	.	.	PUNCT
ejpam-6690	632	1	(	(	PUNCT
ejpam-6690	632	2	vi	vi	NOUN
ejpam-6690	632	3	)	)	PUNCT
ejpam-6690	632	4	if	if	SCONJ
ejpam-6690	632	5	g	g	NOUN
ejpam-6690	632	6	:	:	PUNCT
ejpam-6690	632	7	s	s	X
ejpam-6690	632	8	→	→	SYM
ejpam-6690	632	9	t	t	PROPN
ejpam-6690	632	10	is	be	AUX
ejpam-6690	632	11	another	another	DET
ejpam-6690	632	12	homomorphism	homomorphism	NOUN
ejpam-6690	632	13	of	of	ADP
ejpam-6690	632	14	hyperrings	hyperring	NOUN
ejpam-6690	632	15	,	,	PUNCT
ejpam-6690	632	16	then	then	ADV
ejpam-6690	632	17	g	g	PROPN
ejpam-6690	632	18	◦	◦	NOUN
ejpam-6690	632	19	f	f	PROPN
ejpam-6690	633	1	=	=	SYM
ejpam-6690	633	2	f̄	f̄	PROPN
ejpam-6690	633	3	◦	◦	NOUN
ejpam-6690	633	4	ḡ.	ḡ.	X
ejpam-6690	633	5	proof	proof	NOUN
ejpam-6690	633	6	.	.	PUNCT
ejpam-6690	634	1	(	(	PUNCT
ejpam-6690	634	2	i	i	NOUN
ejpam-6690	634	3	)	)	PUNCT
ejpam-6690	634	4	by	by	ADP
ejpam-6690	634	5	theorem	theorem	NOUN
ejpam-6690	634	6	3	3	NUM
ejpam-6690	634	7	,	,	PUNCT
ejpam-6690	634	8	we	we	PRON
ejpam-6690	634	9	have	have	VERB
ejpam-6690	634	10	f̄	f̄	PROPN
ejpam-6690	634	11	−1	−1	NOUN
ejpam-6690	634	12	(	(	PUNCT
ejpam-6690	634	13	v	v	NOUN
ejpam-6690	634	14	(	(	PUNCT
ejpam-6690	634	15	i	i	NOUN
ejpam-6690	634	16	)	)	PUNCT
ejpam-6690	634	17	)	)	PUNCT
ejpam-6690	635	1	=	=	SYM
ejpam-6690	635	2	v	v	X
ejpam-6690	635	3	(	(	PUNCT
ejpam-6690	635	4	f(i	f(i	PROPN
ejpam-6690	635	5	)	)	PUNCT
ejpam-6690	635	6	)	)	PUNCT
ejpam-6690	635	7	and	and	CCONJ
ejpam-6690	635	8	since	since	SCONJ
ejpam-6690	635	9	(	(	PUNCT
ejpam-6690	635	10	f(i	f(i	PROPN
ejpam-6690	635	11	)	)	PUNCT
ejpam-6690	635	12	)	)	PUNCT
ejpam-6690	636	1	=	=	PUNCT
ejpam-6690	636	2	ie	ie	X
ejpam-6690	636	3	,	,	PUNCT
ejpam-6690	636	4	by	by	ADP
ejpam-6690	636	5	lemma	lemma	PROPN
ejpam-6690	636	6	2	2	NUM
ejpam-6690	636	7	,	,	PUNCT
ejpam-6690	636	8	v	v	PROPN
ejpam-6690	636	9	(	(	PUNCT
ejpam-6690	636	10	f(i	f(i	PROPN
ejpam-6690	636	11	)	)	PUNCT
ejpam-6690	636	12	)	)	PUNCT
ejpam-6690	637	1	=	=	SYM
ejpam-6690	637	2	v	v	X
ejpam-6690	637	3	(	(	PUNCT
ejpam-6690	637	4	ie	ie	X
ejpam-6690	637	5	)	)	PUNCT
ejpam-6690	637	6	.	.	PUNCT
ejpam-6690	638	1	(	(	PUNCT
ejpam-6690	638	2	ii	ii	NOUN
ejpam-6690	638	3	)	)	PUNCT
ejpam-6690	638	4	for	for	ADP
ejpam-6690	638	5	every	every	DET
ejpam-6690	638	6	x	x	SYM
ejpam-6690	638	7	∈	∈	NOUN
ejpam-6690	638	8	r	r	NOUN
ejpam-6690	638	9	we	we	PRON
ejpam-6690	638	10	have	have	VERB
ejpam-6690	638	11	:	:	PUNCT
ejpam-6690	638	12	f̄	f̄	PROPN
ejpam-6690	638	13	−1	−1	NOUN
ejpam-6690	638	14	(	(	PUNCT
ejpam-6690	638	15	w	w	PROPN
ejpam-6690	638	16	(	(	PUNCT
ejpam-6690	638	17	x	x	NOUN
ejpam-6690	638	18	)	)	PUNCT
ejpam-6690	638	19	)	)	PUNCT
ejpam-6690	639	1	=	=	PRON
ejpam-6690	639	2	{	{	PUNCT
ejpam-6690	640	1	p	p	X
ejpam-6690	640	2	⊂	⊂	PROPN
ejpam-6690	640	3	s;x	s;x	NOUN
ejpam-6690	640	4	/∈	/∈	PUNCT
ejpam-6690	641	1	f−1(p	f−1(p	PROPN
ejpam-6690	641	2	)	)	PUNCT
ejpam-6690	641	3	}	}	PUNCT
ejpam-6690	642	1	=	=	PUNCT
ejpam-6690	642	2	{	{	PUNCT
ejpam-6690	642	3	p	p	X
ejpam-6690	642	4	⊂	⊂	PROPN
ejpam-6690	642	5	s	s	PART
ejpam-6690	642	6	;	;	PUNCT
ejpam-6690	642	7	f(x	f(x	PROPN
ejpam-6690	642	8	)	)	PUNCT
ejpam-6690	642	9	/∈	/∈	PUNCT
ejpam-6690	643	1	p	p	X
ejpam-6690	643	2	}	}	PUNCT
ejpam-6690	643	3	=	=	SYM
ejpam-6690	643	4	w	w	PROPN
ejpam-6690	643	5	(	(	PUNCT
ejpam-6690	643	6	f(x	f(x	PROPN
ejpam-6690	643	7	)	)	PUNCT
ejpam-6690	643	8	)	)	PUNCT
ejpam-6690	643	9	.	.	PUNCT
ejpam-6690	644	1	(	(	PUNCT
ejpam-6690	644	2	iii	iii	X
ejpam-6690	644	3	)	)	PUNCT
ejpam-6690	644	4	by	by	ADP
ejpam-6690	644	5	lemma	lemma	PROPN
ejpam-6690	644	6	6	6	NUM
ejpam-6690	644	7	,	,	PUNCT
ejpam-6690	644	8	f̄(v	f̄(v	PROPN
ejpam-6690	644	9	(	(	PUNCT
ejpam-6690	644	10	j	j	NOUN
ejpam-6690	644	11	)	)	PUNCT
ejpam-6690	644	12	)	)	PUNCT
ejpam-6690	645	1	=	=	SYM
ejpam-6690	645	2	v	v	X
ejpam-6690	645	3	(	(	PUNCT
ejpam-6690	645	4	⋂	⋂	PROPN
ejpam-6690	645	5	p∈f̄(v	p∈f̄(v	PROPN
ejpam-6690	645	6	(	(	PUNCT
ejpam-6690	645	7	j	j	PROPN
ejpam-6690	645	8	)	)	PUNCT
ejpam-6690	645	9	)	)	PUNCT
ejpam-6690	646	1	p	p	NOUN
ejpam-6690	646	2	)	)	PUNCT
ejpam-6690	646	3	but⋂	but⋂	NOUN
ejpam-6690	646	4	p∈f̄(v	p∈f̄(v	NOUN
ejpam-6690	646	5	(	(	PUNCT
ejpam-6690	646	6	j	j	NOUN
ejpam-6690	646	7	)	)	PUNCT
ejpam-6690	646	8	)	)	PUNCT
ejpam-6690	647	1	p	p	NOUN
ejpam-6690	647	2	=	=	PUNCT
ejpam-6690	647	3	⋂	⋂	PROPN
ejpam-6690	647	4	j⊂q	j⊂q	PROPN
ejpam-6690	647	5	f−1(q	f−1(q	NOUN
ejpam-6690	647	6	)	)	PUNCT
ejpam-6690	648	1	=	=	SYM
ejpam-6690	648	2	f−1	f−1	PROPN
ejpam-6690	648	3	(	(	PUNCT
ejpam-6690	648	4	⋂	⋂	PROPN
ejpam-6690	648	5	j⊂qq	j⊂qq	NOUN
ejpam-6690	648	6	)	)	PUNCT
ejpam-6690	648	7	=	=	SYM
ejpam-6690	649	1	f−1	f−1	PROPN
ejpam-6690	649	2	(	(	PUNCT
ejpam-6690	649	3	√	√	PROPN
ejpam-6690	649	4	j	j	NOUN
ejpam-6690	649	5	)	)	PUNCT
ejpam-6690	649	6	=	=	SYM
ejpam-6690	649	7	√	√	NUM
ejpam-6690	649	8	f−1(j	f−1(j	PROPN
ejpam-6690	649	9	)	)	PUNCT
ejpam-6690	649	10	.	.	PUNCT
ejpam-6690	650	1	b.	b.	PROPN
ejpam-6690	650	2	afshar	afshar	PROPN
ejpam-6690	650	3	,	,	PUNCT
ejpam-6690	650	4	r.	r.	PROPN
ejpam-6690	650	5	ameri	ameri	PROPN
ejpam-6690	650	6	,	,	PUNCT
ejpam-6690	650	7	m.	m.	PROPN
ejpam-6690	650	8	al	al	PROPN
ejpam-6690	650	9	-	-	PUNCT
ejpam-6690	650	10	tahan	tahan	PROPN
ejpam-6690	650	11	/	/	SYM
ejpam-6690	650	12	eur	eur	PROPN
ejpam-6690	650	13	.	.	PUNCT
ejpam-6690	651	1	j.	j.	PROPN
ejpam-6690	651	2	pure	pure	PROPN
ejpam-6690	651	3	appl	appl	PROPN
ejpam-6690	651	4	.	.	PROPN
ejpam-6690	651	5	math	math	PROPN
ejpam-6690	651	6	,	,	PUNCT
ejpam-6690	651	7	18	18	NUM
ejpam-6690	651	8	(	(	PUNCT
ejpam-6690	651	9	4	4	NUM
ejpam-6690	651	10	)	)	PUNCT
ejpam-6690	651	11	(	(	PUNCT
ejpam-6690	651	12	2025	2025	NUM
ejpam-6690	651	13	)	)	PUNCT
ejpam-6690	651	14	,	,	PUNCT
ejpam-6690	651	15	6690	6690	NUM
ejpam-6690	651	16	16	16	NUM
ejpam-6690	651	17	of	of	ADP
ejpam-6690	651	18	19	19	NUM
ejpam-6690	651	19	now	now	ADV
ejpam-6690	651	20	by	by	ADP
ejpam-6690	651	21	proposition	proposition	NOUN
ejpam-6690	651	22	2	2	NUM
ejpam-6690	651	23	,	,	PUNCT
ejpam-6690	651	24	we	we	PRON
ejpam-6690	651	25	have	have	AUX
ejpam-6690	651	26	v	v	NUM
ejpam-6690	651	27	(	(	PUNCT
ejpam-6690	651	28	⋂	⋂	PROPN
ejpam-6690	651	29	p∈f̄(v	p∈f̄(v	PROPN
ejpam-6690	651	30	(	(	PUNCT
ejpam-6690	651	31	j	j	PROPN
ejpam-6690	651	32	)	)	PUNCT
ejpam-6690	651	33	)	)	PUNCT
ejpam-6690	652	1	p	p	NOUN
ejpam-6690	652	2	)	)	PUNCT
ejpam-6690	652	3	=	=	SYM
ejpam-6690	652	4	v	v	X
ejpam-6690	652	5	(	(	PUNCT
ejpam-6690	652	6	f−1(j	f−1(j	NOUN
ejpam-6690	652	7	)	)	PUNCT
ejpam-6690	652	8	)	)	PUNCT
ejpam-6690	652	9	.	.	PUNCT
ejpam-6690	653	1	(	(	PUNCT
ejpam-6690	653	2	iv	iv	X
ejpam-6690	653	3	)	)	PUNCT
ejpam-6690	653	4	since	since	SCONJ
ejpam-6690	653	5	f	f	PROPN
ejpam-6690	653	6	is	be	AUX
ejpam-6690	653	7	surjective	surjective	ADJ
ejpam-6690	653	8	,	,	PUNCT
ejpam-6690	653	9	consider	consider	VERB
ejpam-6690	653	10	s	s	NOUN
ejpam-6690	653	11	=	=	VERB
ejpam-6690	653	12	r	r	X
ejpam-6690	653	13	/	/	SYM
ejpam-6690	653	14	ker(f	ker(f	PROPN
ejpam-6690	653	15	)	)	PUNCT
ejpam-6690	653	16	.	.	PUNCT
ejpam-6690	654	1	we	we	PRON
ejpam-6690	654	2	know	know	VERB
ejpam-6690	654	3	that	that	SCONJ
ejpam-6690	654	4	there	there	PRON
ejpam-6690	654	5	is	be	VERB
ejpam-6690	654	6	an	an	DET
ejpam-6690	654	7	inclusion	inclusion	NOUN
ejpam-6690	654	8	perserving	perserve	VERB
ejpam-6690	654	9	one	one	NUM
ejpam-6690	654	10	to	to	ADP
ejpam-6690	654	11	one	one	NUM
ejpam-6690	654	12	correspondence	correspondence	NOUN
ejpam-6690	654	13	between	between	ADP
ejpam-6690	654	14	prime	prime	ADJ
ejpam-6690	654	15	hyperideals	hyperideal	NOUN
ejpam-6690	654	16	of	of	ADP
ejpam-6690	654	17	r	r	PROPN
ejpam-6690	654	18	/	/	SYM
ejpam-6690	654	19	ker(f	ker(f	PROPN
ejpam-6690	654	20	)	)	PUNCT
ejpam-6690	654	21	and	and	CCONJ
ejpam-6690	654	22	prime	prime	ADJ
ejpam-6690	654	23	hyperideals	hyperideal	NOUN
ejpam-6690	654	24	of	of	ADP
ejpam-6690	654	25	r	r	NOUN
ejpam-6690	654	26	containing	contain	VERB
ejpam-6690	654	27	ker(f	ker(f	PROPN
ejpam-6690	654	28	)	)	PUNCT
ejpam-6690	654	29	.	.	PUNCT
ejpam-6690	655	1	so	so	ADV
ejpam-6690	655	2	f̄	f̄	PROPN
ejpam-6690	655	3	is	be	AUX
ejpam-6690	655	4	a	a	DET
ejpam-6690	655	5	continuous	continuous	ADJ
ejpam-6690	655	6	bijection	bijection	NOUN
ejpam-6690	655	7	onto	onto	ADP
ejpam-6690	655	8	the	the	DET
ejpam-6690	655	9	closed	closed	ADJ
ejpam-6690	655	10	subset	subset	NOUN
ejpam-6690	655	11	v	v	NOUN
ejpam-6690	655	12	(	(	PUNCT
ejpam-6690	655	13	ker(f	ker(f	PROPN
ejpam-6690	655	14	)	)	PUNCT
ejpam-6690	655	15	)	)	PUNCT
ejpam-6690	655	16	,	,	PUNCT
ejpam-6690	655	17	and	and	CCONJ
ejpam-6690	655	18	f̄	f̄	PROPN
ejpam-6690	655	19	−1	−1	NOUN
ejpam-6690	655	20	is	be	AUX
ejpam-6690	655	21	continuous	continuous	ADJ
ejpam-6690	655	22	because	because	SCONJ
ejpam-6690	655	23	:	:	PUNCT
ejpam-6690	655	24	f̄(i	f̄(i	PROPN
ejpam-6690	655	25	/	/	SYM
ejpam-6690	655	26	ker(f	ker(f	PROPN
ejpam-6690	655	27	)	)	PUNCT
ejpam-6690	655	28	)	)	PUNCT
ejpam-6690	656	1	=	=	PRON
ejpam-6690	656	2	{	{	PUNCT
ejpam-6690	656	3	p	p	NOUN
ejpam-6690	656	4	∈	∈	PROPN
ejpam-6690	656	5	spec(r	spec(r	PROPN
ejpam-6690	656	6	)	)	PUNCT
ejpam-6690	656	7	;	;	PUNCT
ejpam-6690	657	1	i	i	PROPN
ejpam-6690	657	2	/	/	SYM
ejpam-6690	657	3	ker(f	ker(f	PROPN
ejpam-6690	657	4	)	)	PUNCT
ejpam-6690	658	1	⊂	⊂	PROPN
ejpam-6690	658	2	p	p	X
ejpam-6690	658	3	/	/	SYM
ejpam-6690	658	4	ker(f	ker(f	PROPN
ejpam-6690	658	5	)	)	PUNCT
ejpam-6690	658	6	∈	∈	PROPN
ejpam-6690	658	7	spec(r	spec(r	PROPN
ejpam-6690	658	8	/	/	SYM
ejpam-6690	658	9	ker(f	ker(f	PROPN
ejpam-6690	658	10	)	)	PUNCT
ejpam-6690	658	11	)	)	PUNCT
ejpam-6690	658	12	}	}	PUNCT
ejpam-6690	659	1	=	=	SYM
ejpam-6690	659	2	v	v	X
ejpam-6690	659	3	(	(	PUNCT
ejpam-6690	659	4	i	i	NOUN
ejpam-6690	659	5	)	)	PUNCT
ejpam-6690	659	6	.	.	PUNCT
ejpam-6690	660	1	(	(	PUNCT
ejpam-6690	660	2	v	v	NOUN
ejpam-6690	660	3	)	)	PUNCT
ejpam-6690	660	4	since	since	SCONJ
ejpam-6690	660	5	f̄(spec(s	f̄(spec(	VERB
ejpam-6690	660	6	)	)	PUNCT
ejpam-6690	660	7	)	)	PUNCT
ejpam-6690	661	1	=	=	SYM
ejpam-6690	661	2	f̄(v	f̄(v	NOUN
ejpam-6690	661	3	(	(	PUNCT
ejpam-6690	661	4	0	0	NUM
ejpam-6690	661	5	)	)	PUNCT
ejpam-6690	661	6	)	)	PUNCT
ejpam-6690	661	7	,	,	PUNCT
ejpam-6690	661	8	then	then	ADV
ejpam-6690	661	9	by	by	ADP
ejpam-6690	661	10	(	(	PUNCT
ejpam-6690	661	11	3	3	X
ejpam-6690	661	12	)	)	PUNCT
ejpam-6690	661	13	we	we	PRON
ejpam-6690	661	14	have	have	AUX
ejpam-6690	661	15	f̄(spec(s	f̄(spec(	VERB
ejpam-6690	661	16	)	)	PUNCT
ejpam-6690	661	17	)	)	PUNCT
ejpam-6690	662	1	=	=	SYM
ejpam-6690	662	2	v	v	X
ejpam-6690	662	3	(	(	PUNCT
ejpam-6690	662	4	ker(f	ker(f	PROPN
ejpam-6690	662	5	)	)	PUNCT
ejpam-6690	662	6	)	)	PUNCT
ejpam-6690	662	7	.	.	PUNCT
ejpam-6690	663	1	so	so	ADV
ejpam-6690	663	2	f̄(spec(s	f̄(spec(s	ADJ
ejpam-6690	663	3	)	)	PUNCT
ejpam-6690	663	4	)	)	PUNCT
ejpam-6690	664	1	is	be	AUX
ejpam-6690	664	2	dense	dense	ADJ
ejpam-6690	664	3	in	in	ADP
ejpam-6690	664	4	spec(r	spec(r	PROPN
ejpam-6690	664	5	)	)	PUNCT
ejpam-6690	664	6	if	if	SCONJ
ejpam-6690	665	1	and	and	CCONJ
ejpam-6690	665	2	only	only	ADV
ejpam-6690	665	3	if	if	SCONJ
ejpam-6690	665	4	v	v	INTJ
ejpam-6690	665	5	(	(	PUNCT
ejpam-6690	665	6	ker(f	ker(f	PROPN
ejpam-6690	665	7	)	)	PUNCT
ejpam-6690	665	8	)	)	PUNCT
ejpam-6690	666	1	=	=	SYM
ejpam-6690	666	2	spec(r	spec(r	PROPN
ejpam-6690	666	3	)	)	PUNCT
ejpam-6690	666	4	if	if	SCONJ
ejpam-6690	666	5	and	and	CCONJ
ejpam-6690	666	6	only	only	ADV
ejpam-6690	666	7	if	if	SCONJ
ejpam-6690	666	8	ker(f	ker(f	PROPN
ejpam-6690	666	9	)	)	PUNCT
ejpam-6690	666	10	⊂	⊂	PRON
ejpam-6690	667	1	√	√	NOUN
ejpam-6690	667	2	0	0	NUM
ejpam-6690	667	3	.	.	PUNCT
ejpam-6690	668	1	(	(	PUNCT
ejpam-6690	668	2	vi	vi	X
ejpam-6690	668	3	)	)	PUNCT
ejpam-6690	668	4	it	it	PRON
ejpam-6690	668	5	is	be	AUX
ejpam-6690	668	6	clear	clear	ADJ
ejpam-6690	668	7	that	that	SCONJ
ejpam-6690	668	8	g	g	PROPN
ejpam-6690	668	9	◦	◦	PROPN
ejpam-6690	668	10	f(p	f(p	PROPN
ejpam-6690	668	11	)	)	PUNCT
ejpam-6690	669	1	=	=	PRON
ejpam-6690	669	2	(	(	PUNCT
ejpam-6690	669	3	g	g	NOUN
ejpam-6690	669	4	◦	◦	NOUN
ejpam-6690	669	5	f)−1(p	f)−1(p	PROPN
ejpam-6690	669	6	)	)	PUNCT
ejpam-6690	669	7	=	=	PUNCT
ejpam-6690	670	1	(	(	PUNCT
ejpam-6690	670	2	f−1	f−1	PROPN
ejpam-6690	670	3	◦	◦	NOUN
ejpam-6690	670	4	g−1)(p	g−1)(p	NOUN
ejpam-6690	670	5	)	)	PUNCT
ejpam-6690	671	1	=	=	PUNCT
ejpam-6690	671	2	f̄(ḡ(p	f̄(ḡ(p	PROPN
ejpam-6690	671	3	)	)	PUNCT
ejpam-6690	671	4	)	)	PUNCT
ejpam-6690	671	5	.	.	PUNCT
ejpam-6690	672	1	let	let	VERB
ejpam-6690	672	2	r	r	PRON
ejpam-6690	672	3	be	be	AUX
ejpam-6690	672	4	a	a	DET
ejpam-6690	672	5	hyperring	hyperring	NOUN
ejpam-6690	672	6	.	.	PUNCT
ejpam-6690	673	1	by	by	ADP
ejpam-6690	673	2	ztop.hrg	ztop.hrg	NUM
ejpam-6690	673	3	,	,	PUNCT
ejpam-6690	673	4	we	we	PRON
ejpam-6690	673	5	mean	mean	VERB
ejpam-6690	673	6	the	the	DET
ejpam-6690	673	7	category	category	NOUN
ejpam-6690	673	8	of	of	ADP
ejpam-6690	673	9	zariski	zariski	NOUN
ejpam-6690	673	10	topology	topology	NOUN
ejpam-6690	673	11	of	of	ADP
ejpam-6690	673	12	(	(	PUNCT
ejpam-6690	673	13	krasner	krasner	NOUN
ejpam-6690	673	14	)	)	PUNCT
ejpam-6690	673	15	hyperrings	hyperring	NOUN
ejpam-6690	673	16	,	,	PUNCT
ejpam-6690	673	17	which	which	PRON
ejpam-6690	673	18	objects	object	VERB
ejpam-6690	673	19	are	be	AUX
ejpam-6690	673	20	x	x	X
ejpam-6690	673	21	=	=	SYM
ejpam-6690	673	22	spec(r	spec(r	PROPN
ejpam-6690	673	23	)	)	PUNCT
ejpam-6690	673	24	and	and	CCONJ
ejpam-6690	673	25	for	for	ADP
ejpam-6690	673	26	x	x	X
ejpam-6690	673	27	=	=	SYM
ejpam-6690	673	28	spec(r	spec(r	PROPN
ejpam-6690	673	29	)	)	PUNCT
ejpam-6690	673	30	and	and	CCONJ
ejpam-6690	673	31	y	y	PROPN
ejpam-6690	673	32	=	=	PUNCT
ejpam-6690	673	33	spec(s	spec(s	PROPN
ejpam-6690	673	34	)	)	PUNCT
ejpam-6690	673	35	,	,	PUNCT
ejpam-6690	673	36	hom(x	hom(x	PROPN
ejpam-6690	673	37	,	,	PUNCT
ejpam-6690	673	38	y	y	PROPN
ejpam-6690	673	39	)	)	PUNCT
ejpam-6690	673	40	is	be	AUX
ejpam-6690	673	41	the	the	DET
ejpam-6690	673	42	set	set	NOUN
ejpam-6690	673	43	of	of	ADP
ejpam-6690	673	44	all	all	DET
ejpam-6690	673	45	continuous	continuous	ADJ
ejpam-6690	673	46	maps	map	NOUN
ejpam-6690	673	47	induced	induce	VERB
ejpam-6690	673	48	by	by	ADP
ejpam-6690	673	49	hyperring	hyperre	VERB
ejpam-6690	673	50	homomorphisms	homomorphism	NOUN
ejpam-6690	673	51	,	,	PUNCT
ejpam-6690	673	52	with	with	ADP
ejpam-6690	673	53	usual	usual	ADJ
ejpam-6690	673	54	combinations	combination	NOUN
ejpam-6690	673	55	of	of	ADP
ejpam-6690	673	56	functions	function	NOUN
ejpam-6690	673	57	.	.	PUNCT
ejpam-6690	674	1	also	also	ADV
ejpam-6690	674	2	,	,	PUNCT
ejpam-6690	674	3	ztop.rg	ztop.rg	PROPN
ejpam-6690	674	4	denotes	denote	VERB
ejpam-6690	674	5	the	the	DET
ejpam-6690	674	6	category	category	NOUN
ejpam-6690	674	7	of	of	ADP
ejpam-6690	674	8	zariski	zariski	ADJ
ejpam-6690	674	9	topology	topology	NOUN
ejpam-6690	674	10	of	of	ADP
ejpam-6690	674	11	all	all	DET
ejpam-6690	674	12	rings	ring	NOUN
ejpam-6690	674	13	.	.	PUNCT
ejpam-6690	675	1	theorem	theorem	VERB
ejpam-6690	675	2	14	14	NUM
ejpam-6690	675	3	.	.	PUNCT
ejpam-6690	676	1	the	the	DET
ejpam-6690	676	2	mapping	mapping	NOUN
ejpam-6690	676	3	f	f	X
ejpam-6690	676	4	:	:	PUNCT
ejpam-6690	676	5	ztop.hrg	ztop.hrg	X
ejpam-6690	676	6	→	→	SYM
ejpam-6690	676	7	ztop.rg	ztop.rg	PROPN
ejpam-6690	676	8	defined	define	VERB
ejpam-6690	676	9	by	by	ADP
ejpam-6690	676	10	spec(r	spec(r	PROPN
ejpam-6690	676	11	)	)	PUNCT
ejpam-6690	676	12	7→	7→	NUM
ejpam-6690	676	13	spec(r	spec(r	PROPN
ejpam-6690	676	14	/	/	SYM
ejpam-6690	676	15	γ∗	γ∗	PROPN
ejpam-6690	676	16	)	)	PUNCT
ejpam-6690	676	17	and	and	CCONJ
ejpam-6690	676	18	(	(	PUNCT
ejpam-6690	676	19	f	f	X
ejpam-6690	676	20	:	:	PUNCT
ejpam-6690	676	21	spec(r	spec(r	PROPN
ejpam-6690	676	22	)	)	PUNCT
ejpam-6690	676	23	→	→	SYM
ejpam-6690	676	24	spec(s	spec(s	PROPN
ejpam-6690	676	25	)	)	PUNCT
ejpam-6690	676	26	)	)	PUNCT
ejpam-6690	677	1	7→	7→	PROPN
ejpam-6690	678	1	(	(	PUNCT
ejpam-6690	678	2	f∗	f∗	NOUN
ejpam-6690	678	3	:	:	PUNCT
ejpam-6690	678	4	spec(r	spec(r	PROPN
ejpam-6690	678	5	/	/	SYM
ejpam-6690	678	6	γ∗	γ∗	PROPN
ejpam-6690	678	7	)	)	PUNCT
ejpam-6690	678	8	→	→	PUNCT
ejpam-6690	678	9	spec(s	spec(s	PROPN
ejpam-6690	678	10	/	/	SYM
ejpam-6690	678	11	γ∗	γ∗	PROPN
ejpam-6690	678	12	)	)	PUNCT
ejpam-6690	678	13	)	)	PUNCT
ejpam-6690	678	14	.	.	PUNCT
ejpam-6690	679	1	then	then	ADV
ejpam-6690	679	2	the	the	DET
ejpam-6690	679	3	following	follow	VERB
ejpam-6690	679	4	statements	statement	NOUN
ejpam-6690	679	5	are	be	AUX
ejpam-6690	679	6	satisfied	satisfied	ADJ
ejpam-6690	679	7	:	:	PUNCT
ejpam-6690	679	8	(	(	PUNCT
ejpam-6690	679	9	i	i	NOUN
ejpam-6690	679	10	)	)	PUNCT
ejpam-6690	679	11	f	f	PROPN
ejpam-6690	679	12	is	be	AUX
ejpam-6690	679	13	functor	functor	PROPN
ejpam-6690	679	14	.	.	PUNCT
ejpam-6690	680	1	(	(	PUNCT
ejpam-6690	680	2	ii	ii	NOUN
ejpam-6690	680	3	)	)	PUNCT
ejpam-6690	680	4	the	the	DET
ejpam-6690	680	5	following	follow	VERB
ejpam-6690	680	6	diagram	diagram	NOUN
ejpam-6690	680	7	is	be	AUX
ejpam-6690	680	8	commutative	commutative	ADJ
ejpam-6690	680	9	(	(	PUNCT
ejpam-6690	680	10	πr	πr	NOUN
ejpam-6690	680	11	and	and	CCONJ
ejpam-6690	680	12	πs	πs	ADV
ejpam-6690	680	13	are	be	AUX
ejpam-6690	680	14	canonical	canonical	ADJ
ejpam-6690	680	15	projections	projection	NOUN
ejpam-6690	680	16	):	):	PUNCT
ejpam-6690	680	17	spec(r	spec(r	PROPN
ejpam-6690	680	18	)	)	PUNCT
ejpam-6690	680	19	spec(s	spec(s	PROPN
ejpam-6690	680	20	)	)	PUNCT
ejpam-6690	680	21	spec(r	spec(r	PROPN
ejpam-6690	680	22	/	/	SYM
ejpam-6690	680	23	γ∗	γ∗	PROPN
ejpam-6690	680	24	)	)	PUNCT
ejpam-6690	680	25	spec(s	spec(s	PROPN
ejpam-6690	680	26	/	/	SYM
ejpam-6690	680	27	γ∗	γ∗	PROPN
ejpam-6690	680	28	)	)	PUNCT
ejpam-6690	681	1	f	f	PROPN
ejpam-6690	681	2	πr	πr	PROPN
ejpam-6690	681	3	πs	πs	PROPN
ejpam-6690	681	4	f∗	f∗	NOUN
ejpam-6690	681	5	proof	proof	NOUN
ejpam-6690	681	6	.	.	PUNCT
ejpam-6690	682	1	(	(	PUNCT
ejpam-6690	682	2	i	i	NOUN
ejpam-6690	682	3	)	)	PUNCT
ejpam-6690	682	4	closed	close	VERB
ejpam-6690	682	5	subsets	subset	NOUN
ejpam-6690	682	6	of	of	ADP
ejpam-6690	682	7	spec(s	spec(s	PROPN
ejpam-6690	682	8	/	/	SYM
ejpam-6690	682	9	γ∗	γ∗	PROPN
ejpam-6690	682	10	)	)	PUNCT
ejpam-6690	683	1	are	be	AUX
ejpam-6690	683	2	of	of	ADP
ejpam-6690	683	3	the	the	DET
ejpam-6690	683	4	form	form	NOUN
ejpam-6690	683	5	v	v	NOUN
ejpam-6690	683	6	(	(	PUNCT
ejpam-6690	683	7	j	j	NOUN
ejpam-6690	683	8	/	/	SYM
ejpam-6690	683	9	γ∗	γ∗	PROPN
ejpam-6690	683	10	)	)	PUNCT
ejpam-6690	683	11	where	where	SCONJ
ejpam-6690	683	12	j	j	PROPN
ejpam-6690	683	13	is	be	AUX
ejpam-6690	683	14	hyperideal	hyperideal	ADJ
ejpam-6690	683	15	of	of	ADP
ejpam-6690	683	16	s	s	NOUN
ejpam-6690	683	17	,	,	PUNCT
ejpam-6690	683	18	and	and	CCONJ
ejpam-6690	683	19	v	v	X
ejpam-6690	683	20	(	(	PUNCT
ejpam-6690	683	21	j	j	NOUN
ejpam-6690	683	22	/	/	SYM
ejpam-6690	683	23	γ∗	γ∗	PROPN
ejpam-6690	683	24	)	)	PUNCT
ejpam-6690	684	1	=	=	PRON
ejpam-6690	684	2	{	{	PUNCT
ejpam-6690	684	3	p	p	X
ejpam-6690	684	4	/	/	SYM
ejpam-6690	684	5	γ∗	γ∗	NOUN
ejpam-6690	684	6	∈	∈	PROPN
ejpam-6690	684	7	spec(s	spec(s	PROPN
ejpam-6690	684	8	/	/	SYM
ejpam-6690	684	9	γ∗	γ∗	PROPN
ejpam-6690	684	10	)	)	PUNCT
ejpam-6690	684	11	;	;	PUNCT
ejpam-6690	684	12	p	p	PROPN
ejpam-6690	684	13	∈	∈	PROPN
ejpam-6690	684	14	v	v	ADP
ejpam-6690	684	15	(	(	PUNCT
ejpam-6690	684	16	j	j	NOUN
ejpam-6690	684	17	)	)	PUNCT
ejpam-6690	684	18	}	}	PUNCT
ejpam-6690	684	19	.	.	PUNCT
ejpam-6690	685	1	so	so	ADV
ejpam-6690	685	2	g∗	g∗	VERB
ejpam-6690	685	3	−1	−1	NOUN
ejpam-6690	685	4	(	(	PUNCT
ejpam-6690	685	5	v	v	NOUN
ejpam-6690	685	6	(	(	PUNCT
ejpam-6690	685	7	j	j	PROPN
ejpam-6690	685	8	/	/	SYM
ejpam-6690	685	9	γ∗	γ∗	PROPN
ejpam-6690	685	10	)	)	PUNCT
ejpam-6690	685	11	)	)	PUNCT
ejpam-6690	686	1	=	=	PRON
ejpam-6690	686	2	{	{	PUNCT
ejpam-6690	686	3	q	q	ADJ
ejpam-6690	686	4	/	/	SYM
ejpam-6690	686	5	γ∗	γ∗	NOUN
ejpam-6690	686	6	∈	∈	PROPN
ejpam-6690	686	7	spec(r	spec(r	PROPN
ejpam-6690	686	8	/	/	SYM
ejpam-6690	686	9	γ∗);q	γ∗);q	PROPN
ejpam-6690	686	10	∈	∈	NOUN
ejpam-6690	686	11	g−1(v	g−1(v	NOUN
ejpam-6690	686	12	(	(	PUNCT
ejpam-6690	686	13	j	j	NOUN
ejpam-6690	686	14	)	)	PUNCT
ejpam-6690	686	15	)	)	PUNCT
ejpam-6690	686	16	}	}	PUNCT
ejpam-6690	686	17	.	.	PUNCT
ejpam-6690	687	1	since	since	SCONJ
ejpam-6690	687	2	g	g	PROPN
ejpam-6690	687	3	is	be	AUX
ejpam-6690	687	4	continuous	continuous	ADJ
ejpam-6690	687	5	,	,	PUNCT
ejpam-6690	687	6	then	then	ADV
ejpam-6690	687	7	g−1(v	g−1(v	INTJ
ejpam-6690	687	8	(	(	PUNCT
ejpam-6690	687	9	j	j	NOUN
ejpam-6690	687	10	)	)	PUNCT
ejpam-6690	687	11	)	)	PUNCT
ejpam-6690	687	12	is	be	AUX
ejpam-6690	687	13	closed	close	VERB
ejpam-6690	687	14	subset	subset	NOUN
ejpam-6690	687	15	of	of	ADP
ejpam-6690	687	16	spec(r	spec(r	PROPN
ejpam-6690	687	17	)	)	PUNCT
ejpam-6690	687	18	and	and	CCONJ
ejpam-6690	687	19	hence	hence	ADV
ejpam-6690	687	20	there	there	PRON
ejpam-6690	687	21	is	be	VERB
ejpam-6690	687	22	hyperideal	hyperideal	ADJ
ejpam-6690	687	23	i	i	PRON
ejpam-6690	687	24	of	of	ADP
ejpam-6690	687	25	r	r	NOUN
ejpam-6690	687	26	such	such	ADJ
ejpam-6690	687	27	that	that	DET
ejpam-6690	687	28	g−1(v	g−1(v	NOUN
ejpam-6690	687	29	(	(	PUNCT
ejpam-6690	687	30	j	j	NOUN
ejpam-6690	687	31	)	)	PUNCT
ejpam-6690	687	32	)	)	PUNCT
ejpam-6690	688	1	=	=	SYM
ejpam-6690	688	2	v	v	X
ejpam-6690	688	3	(	(	PUNCT
ejpam-6690	688	4	i	i	NOUN
ejpam-6690	688	5	)	)	PUNCT
ejpam-6690	688	6	.	.	PUNCT
ejpam-6690	689	1	thus	thus	ADV
ejpam-6690	689	2	g∗	g∗	VERB
ejpam-6690	689	3	−1	−1	NOUN
ejpam-6690	689	4	(	(	PUNCT
ejpam-6690	689	5	v	v	NOUN
ejpam-6690	689	6	(	(	PUNCT
ejpam-6690	689	7	j	j	PROPN
ejpam-6690	689	8	/	/	SYM
ejpam-6690	689	9	γ∗	γ∗	PROPN
ejpam-6690	689	10	)	)	PUNCT
ejpam-6690	689	11	)	)	PUNCT
ejpam-6690	690	1	=	=	SYM
ejpam-6690	690	2	v	v	X
ejpam-6690	690	3	(	(	PUNCT
ejpam-6690	690	4	i	i	NOUN
ejpam-6690	690	5	/	/	SYM
ejpam-6690	690	6	γ∗	γ∗	PROPN
ejpam-6690	690	7	)	)	PUNCT
ejpam-6690	690	8	.	.	PUNCT
ejpam-6690	691	1	it	it	PRON
ejpam-6690	691	2	is	be	AUX
ejpam-6690	691	3	clear	clear	ADJ
ejpam-6690	692	1	that	that	SCONJ
ejpam-6690	692	2	f	f	PROPN
ejpam-6690	692	3	(	(	PUNCT
ejpam-6690	692	4	1spec(r	1spec(r	NUM
ejpam-6690	692	5	)	)	PUNCT
ejpam-6690	692	6	)	)	PUNCT
ejpam-6690	692	7	=	=	SYM
ejpam-6690	692	8	1spec(r	1spec(r	NUM
ejpam-6690	692	9	/	/	SYM
ejpam-6690	692	10	γ∗	γ∗	PROPN
ejpam-6690	692	11	)	)	PUNCT
ejpam-6690	692	12	and	and	CCONJ
ejpam-6690	692	13	(	(	PUNCT
ejpam-6690	692	14	h	h	NOUN
ejpam-6690	692	15	◦	◦	NOUN
ejpam-6690	692	16	g)∗	g)∗	NOUN
ejpam-6690	693	1	=	=	SYM
ejpam-6690	693	2	h∗	h∗	PROPN
ejpam-6690	693	3	◦	◦	NOUN
ejpam-6690	693	4	g∗.	g∗.	PROPN
ejpam-6690	693	5	(	(	PUNCT
ejpam-6690	693	6	ii	ii	NOUN
ejpam-6690	693	7	)	)	PUNCT
ejpam-6690	693	8	let	let	VERB
ejpam-6690	693	9	p	p	PROPN
ejpam-6690	693	10	∈	∈	PROPN
ejpam-6690	693	11	spec(r	spec(r	PROPN
ejpam-6690	693	12	)	)	PUNCT
ejpam-6690	693	13	,	,	PUNCT
ejpam-6690	693	14	then	then	ADV
ejpam-6690	693	15	f∗πr(p	f∗πr(p	VERB
ejpam-6690	693	16	)	)	PUNCT
ejpam-6690	693	17	=	=	PUNCT
ejpam-6690	693	18	f∗(p	f∗(p	PROPN
ejpam-6690	693	19	/	/	SYM
ejpam-6690	693	20	γ∗	γ∗	NOUN
ejpam-6690	693	21	)	)	PUNCT
ejpam-6690	694	1	=	=	SYM
ejpam-6690	694	2	f(p	f(p	PROPN
ejpam-6690	694	3	)	)	PUNCT
ejpam-6690	694	4	/γ∗	/γ∗	PUNCT
ejpam-6690	695	1	=	=	PUNCT
ejpam-6690	695	2	πsf(p	πsf(p	PROPN
ejpam-6690	695	3	)	)	PUNCT
ejpam-6690	695	4	.	.	PUNCT
ejpam-6690	696	1	theorem	theorem	VERB
ejpam-6690	696	2	15	15	NUM
ejpam-6690	696	3	.	.	PUNCT
ejpam-6690	697	1	the	the	DET
ejpam-6690	697	2	mapping	mapping	PROPN
ejpam-6690	697	3	spec(−	spec(−	PROPN
ejpam-6690	697	4	)	)	PUNCT
ejpam-6690	697	5	:	:	PUNCT
ejpam-6690	698	1	h.rg	h.rg	X
ejpam-6690	698	2	→	→	SYM
ejpam-6690	698	3	ztop.hrg	ztop.hrg	X
ejpam-6690	698	4	,	,	PUNCT
ejpam-6690	698	5	by	by	ADP
ejpam-6690	698	6	r	r	PROPN
ejpam-6690	698	7	7→	7→	NUM
ejpam-6690	698	8	spec(r	spec(r	PROPN
ejpam-6690	698	9	)	)	PUNCT
ejpam-6690	698	10	and	and	CCONJ
ejpam-6690	698	11	f	f	PROPN
ejpam-6690	698	12	7→	7→	PROPN
ejpam-6690	698	13	f̄	f̄	NOUN
ejpam-6690	698	14	is	be	AUX
ejpam-6690	698	15	a	a	DET
ejpam-6690	698	16	contravariant	contravariant	ADJ
ejpam-6690	698	17	functor	functor	NOUN
ejpam-6690	698	18	.	.	PUNCT
ejpam-6690	698	19	proof	proof	NOUN
ejpam-6690	698	20	.	.	PUNCT
ejpam-6690	699	1	by	by	ADP
ejpam-6690	699	2	theorem	theorem	NOUN
ejpam-6690	699	3	3	3	NUM
ejpam-6690	699	4	,	,	PUNCT
ejpam-6690	699	5	f̄	f̄	PROPN
ejpam-6690	699	6	is	be	AUX
ejpam-6690	699	7	continuous	continuous	ADJ
ejpam-6690	699	8	and	and	CCONJ
ejpam-6690	699	9	by	by	ADP
ejpam-6690	699	10	theorem	theorem	NOUN
ejpam-6690	699	11	13	13	NUM
ejpam-6690	699	12	,	,	PUNCT
ejpam-6690	699	13	g	g	NOUN
ejpam-6690	699	14	◦	◦	NOUN
ejpam-6690	699	15	f	f	PROPN
ejpam-6690	699	16	=	=	SYM
ejpam-6690	699	17	f̄	f̄	PROPN
ejpam-6690	699	18	◦	◦	PROPN
ejpam-6690	699	19	ḡ.	ḡ.	PROPN
ejpam-6690	699	20	b.	b.	PROPN
ejpam-6690	699	21	afshar	afshar	PROPN
ejpam-6690	699	22	,	,	PUNCT
ejpam-6690	699	23	r.	r.	PROPN
ejpam-6690	699	24	ameri	ameri	PROPN
ejpam-6690	699	25	,	,	PUNCT
ejpam-6690	699	26	m.	m.	PROPN
ejpam-6690	699	27	al	al	PROPN
ejpam-6690	699	28	-	-	PUNCT
ejpam-6690	699	29	tahan	tahan	PROPN
ejpam-6690	699	30	/	/	SYM
ejpam-6690	699	31	eur	eur	PROPN
ejpam-6690	699	32	.	.	PUNCT
ejpam-6690	700	1	j.	j.	PROPN
ejpam-6690	700	2	pure	pure	PROPN
ejpam-6690	700	3	appl	appl	PROPN
ejpam-6690	700	4	.	.	PROPN
ejpam-6690	700	5	math	math	PROPN
ejpam-6690	700	6	,	,	PUNCT
ejpam-6690	700	7	18	18	NUM
ejpam-6690	700	8	(	(	PUNCT
ejpam-6690	700	9	4	4	NUM
ejpam-6690	700	10	)	)	PUNCT
ejpam-6690	700	11	(	(	PUNCT
ejpam-6690	700	12	2025	2025	NUM
ejpam-6690	700	13	)	)	PUNCT
ejpam-6690	700	14	,	,	PUNCT
ejpam-6690	700	15	6690	6690	NUM
ejpam-6690	700	16	17	17	NUM
ejpam-6690	700	17	of	of	ADP
ejpam-6690	700	18	19	19	NUM
ejpam-6690	700	19	theorem	theorem	NOUN
ejpam-6690	700	20	16	16	NUM
ejpam-6690	700	21	.	.	PUNCT
ejpam-6690	701	1	the	the	DET
ejpam-6690	701	2	following	follow	VERB
ejpam-6690	701	3	diagram	diagram	NOUN
ejpam-6690	701	4	is	be	AUX
ejpam-6690	701	5	commutative	commutative	ADJ
ejpam-6690	701	6	:	:	PUNCT
ejpam-6690	701	7	h.rg	h.rg	PROPN
ejpam-6690	701	8	ztop.hrg	ztop.hrg	X
ejpam-6690	701	9	rg	rg	PROPN
ejpam-6690	701	10	ztop.rg	ztop.rg	PROPN
ejpam-6690	701	11	spec(−	spec(−	PROPN
ejpam-6690	701	12	)	)	PUNCT
ejpam-6690	701	13	γ∗	γ∗	PROPN
ejpam-6690	701	14	f	f	PROPN
ejpam-6690	701	15	spec(−	spec(−	PROPN
ejpam-6690	701	16	)	)	PUNCT
ejpam-6690	701	17	proof	proof	NOUN
ejpam-6690	701	18	.	.	PUNCT
ejpam-6690	702	1	it	it	PRON
ejpam-6690	702	2	is	be	AUX
ejpam-6690	702	3	clear	clear	ADJ
ejpam-6690	702	4	that	that	SCONJ
ejpam-6690	702	5	f	f	PROPN
ejpam-6690	702	6	(	(	PUNCT
ejpam-6690	702	7	spec(r	spec(r	PROPN
ejpam-6690	702	8	)	)	PUNCT
ejpam-6690	702	9	)	)	PUNCT
ejpam-6690	703	1	=	=	SYM
ejpam-6690	704	1	spec(γ∗(r	spec(γ∗(r	PROPN
ejpam-6690	704	2	)	)	PUNCT
ejpam-6690	704	3	)	)	PUNCT
ejpam-6690	704	4	.	.	PUNCT
ejpam-6690	705	1	if	if	SCONJ
ejpam-6690	705	2	f	f	PROPN
ejpam-6690	705	3	:	:	PUNCT
ejpam-6690	705	4	r	r	X
ejpam-6690	705	5	→	→	SYM
ejpam-6690	705	6	s	s	PART
ejpam-6690	705	7	is	be	AUX
ejpam-6690	705	8	a	a	DET
ejpam-6690	705	9	good	good	ADJ
ejpam-6690	705	10	homomorphism	homomorphism	NOUN
ejpam-6690	705	11	of	of	ADP
ejpam-6690	705	12	hyperrings	hyperring	NOUN
ejpam-6690	705	13	then	then	ADV
ejpam-6690	705	14	we	we	PRON
ejpam-6690	705	15	have	have	VERB
ejpam-6690	705	16	f̄	f̄	NOUN
ejpam-6690	705	17	:	:	PUNCT
ejpam-6690	705	18	spec(r	spec(r	PROPN
ejpam-6690	705	19	)	)	PUNCT
ejpam-6690	705	20	→	→	SYM
ejpam-6690	705	21	spec(s	spec(	VERB
ejpam-6690	705	22	)	)	PUNCT
ejpam-6690	705	23	by	by	ADP
ejpam-6690	705	24	p	p	PROPN
ejpam-6690	705	25	7→	7→	NUM
ejpam-6690	705	26	f−1(p	f−1(p	NOUN
ejpam-6690	705	27	)	)	PUNCT
ejpam-6690	705	28	and	and	CCONJ
ejpam-6690	705	29	f∗	f∗	NOUN
ejpam-6690	705	30	:	:	PUNCT
ejpam-6690	705	31	r	r	X
ejpam-6690	705	32	/	/	SYM
ejpam-6690	705	33	γ∗	γ∗	NOUN
ejpam-6690	705	34	→	→	SYM
ejpam-6690	705	35	s	s	X
ejpam-6690	705	36	/	/	SYM
ejpam-6690	705	37	γ∗	γ∗	NOUN
ejpam-6690	705	38	by	by	ADP
ejpam-6690	705	39	γ∗(r	γ∗(r	PROPN
ejpam-6690	705	40	)	)	PUNCT
ejpam-6690	705	41	7→	7→	NUM
ejpam-6690	705	42	γ∗(f(r	γ∗(f(r	NUM
ejpam-6690	705	43	)	)	PUNCT
ejpam-6690	705	44	)	)	PUNCT
ejpam-6690	705	45	.	.	PUNCT
ejpam-6690	706	1	so	so	ADV
ejpam-6690	706	2	f̄	f̄	PROPN
ejpam-6690	706	3	∗	∗	NOUN
ejpam-6690	706	4	=	=	PUNCT
ejpam-6690	706	5	f̄	f̄	PROPN
ejpam-6690	706	6	∗	∗	NOUN
ejpam-6690	706	7	,	,	PUNCT
ejpam-6690	706	8	because	because	SCONJ
ejpam-6690	706	9	f̄	f̄	PROPN
ejpam-6690	706	10	∗	∗	NOUN
ejpam-6690	706	11	(	(	PUNCT
ejpam-6690	706	12	p	p	X
ejpam-6690	706	13	/	/	SYM
ejpam-6690	706	14	γ∗	γ∗	NOUN
ejpam-6690	706	15	)	)	PUNCT
ejpam-6690	707	1	=	=	SYM
ejpam-6690	707	2	f−1(p	f−1(p	PROPN
ejpam-6690	707	3	)	)	PUNCT
ejpam-6690	707	4	/γ∗	/γ∗	PUNCT
ejpam-6690	708	1	=	=	PUNCT
ejpam-6690	708	2	f∗−1	f∗−1	NOUN
ejpam-6690	708	3	(	(	PUNCT
ejpam-6690	708	4	p	p	NOUN
ejpam-6690	708	5	/	/	SYM
ejpam-6690	708	6	γ∗	γ∗	NOUN
ejpam-6690	708	7	)	)	PUNCT
ejpam-6690	709	1	=	=	SYM
ejpam-6690	709	2	f̄	f̄	PROPN
ejpam-6690	709	3	∗(p	∗(p	PROPN
ejpam-6690	709	4	/	/	SYM
ejpam-6690	709	5	γ∗	γ∗	PROPN
ejpam-6690	709	6	)	)	PUNCT
ejpam-6690	709	7	.	.	PUNCT
ejpam-6690	710	1	example	example	NOUN
ejpam-6690	710	2	5	5	NUM
ejpam-6690	710	3	.	.	X
ejpam-6690	711	1	consider	consider	VERB
ejpam-6690	711	2	hyperring	hyperre	VERB
ejpam-6690	711	3	(	(	PUNCT
ejpam-6690	711	4	z̄,⊕,⊙	z̄,⊕,⊙	NUM
ejpam-6690	711	5	)	)	PUNCT
ejpam-6690	711	6	,	,	PUNCT
ejpam-6690	711	7	in	in	ADP
ejpam-6690	711	8	which	which	PRON
ejpam-6690	711	9	z̄	z̄	X
ejpam-6690	711	10	=	=	SYM
ejpam-6690	711	11	{	{	PUNCT
ejpam-6690	711	12	n̄;n	n̄;n	PROPN
ejpam-6690	711	13	∈	∈	PROPN
ejpam-6690	711	14	z	z	NOUN
ejpam-6690	711	15	}	}	PUNCT
ejpam-6690	711	16	,	,	PUNCT
ejpam-6690	711	17	n̄	n̄	NOUN
ejpam-6690	711	18	=	=	PRON
ejpam-6690	711	19	{	{	PUNCT
ejpam-6690	711	20	−n	−n	PROPN
ejpam-6690	711	21	,	,	PUNCT
ejpam-6690	711	22	n	n	CCONJ
ejpam-6690	711	23	}	}	PUNCT
ejpam-6690	711	24	,	,	PUNCT
ejpam-6690	711	25	n̄⊕	n̄⊕	ADV
ejpam-6690	711	26	m̄	m̄	NOUN
ejpam-6690	711	27	=	=	SYM
ejpam-6690	711	28	{	{	PUNCT
ejpam-6690	711	29	n+m	n+m	NUM
ejpam-6690	711	30	,	,	PUNCT
ejpam-6690	711	31	n−m	n−m	VERB
ejpam-6690	711	32	}	}	PUNCT
ejpam-6690	711	33	,	,	PUNCT
ejpam-6690	711	34	and	and	CCONJ
ejpam-6690	711	35	n̄⊙	n̄⊙	NOUN
ejpam-6690	711	36	n̄	n̄	NOUN
ejpam-6690	711	37	=	=	SYM
ejpam-6690	712	1	nm	nm	PROPN
ejpam-6690	712	2	.	.	PUNCT
ejpam-6690	713	1	analogous	analogous	ADJ
ejpam-6690	713	2	to	to	ADP
ejpam-6690	713	3	the	the	DET
ejpam-6690	713	4	classical	classical	ADJ
ejpam-6690	713	5	case	case	NOUN
ejpam-6690	713	6	,	,	PUNCT
ejpam-6690	713	7	z̄	z̄	PUNCT
ejpam-6690	713	8	is	be	AUX
ejpam-6690	713	9	a	a	DET
ejpam-6690	713	10	principal	principal	ADJ
ejpam-6690	713	11	hyperideal	hyperideal	ADJ
ejpam-6690	713	12	domain	domain	NOUN
ejpam-6690	713	13	;	;	PUNCT
ejpam-6690	713	14	which	which	PRON
ejpam-6690	713	15	is	be	AUX
ejpam-6690	713	16	,	,	PUNCT
ejpam-6690	713	17	its	its	PRON
ejpam-6690	713	18	hyperideals	hyperideal	NOUN
ejpam-6690	713	19	are	be	AUX
ejpam-6690	713	20	of	of	ADP
ejpam-6690	713	21	the	the	DET
ejpam-6690	713	22	form	form	NOUN
ejpam-6690	713	23	(	(	PUNCT
ejpam-6690	713	24	ā	ā	NOUN
ejpam-6690	713	25	)	)	PUNCT
ejpam-6690	714	1	=	=	PRON
ejpam-6690	714	2	{	{	PUNCT
ejpam-6690	714	3	na;n	na;n	PROPN
ejpam-6690	714	4	∈	∈	PROPN
ejpam-6690	714	5	z	z	X
ejpam-6690	714	6	}	}	PUNCT
ejpam-6690	714	7	,	,	PUNCT
ejpam-6690	714	8	for	for	ADP
ejpam-6690	714	9	every	every	DET
ejpam-6690	714	10	a	a	DET
ejpam-6690	714	11	∈	∈	PROPN
ejpam-6690	714	12	z.	z.	PROPN
ejpam-6690	714	13	moreover	moreover	ADV
ejpam-6690	714	14	,	,	PUNCT
ejpam-6690	714	15	the	the	DET
ejpam-6690	714	16	prime	prime	ADJ
ejpam-6690	714	17	hyperideals	hyperideal	NOUN
ejpam-6690	714	18	of	of	ADP
ejpam-6690	714	19	z̄	z̄	PROPN
ejpam-6690	714	20	are	be	AUX
ejpam-6690	714	21	of	of	ADP
ejpam-6690	714	22	the	the	DET
ejpam-6690	714	23	form	form	NOUN
ejpam-6690	714	24	(	(	PUNCT
ejpam-6690	714	25	p̄	p̄	PROPN
ejpam-6690	714	26	)	)	PUNCT
ejpam-6690	714	27	,	,	PUNCT
ejpam-6690	714	28	where	where	SCONJ
ejpam-6690	714	29	p	p	NOUN
ejpam-6690	714	30	is	be	AUX
ejpam-6690	714	31	a	a	DET
ejpam-6690	714	32	prime	prime	ADJ
ejpam-6690	714	33	number	number	NOUN
ejpam-6690	714	34	.	.	PUNCT
ejpam-6690	715	1	so	so	ADV
ejpam-6690	715	2	spec(z̄	spec(z̄	NOUN
ejpam-6690	715	3	)	)	PUNCT
ejpam-6690	716	1	=	=	PRON
ejpam-6690	716	2	{	{	PUNCT
ejpam-6690	716	3	(	(	PUNCT
ejpam-6690	716	4	p̄	p̄	PROPN
ejpam-6690	716	5	)	)	PUNCT
ejpam-6690	716	6	;	;	PUNCT
ejpam-6690	716	7	p	p	PRON
ejpam-6690	716	8	is	be	AUX
ejpam-6690	716	9	a	a	DET
ejpam-6690	716	10	prime	prime	ADJ
ejpam-6690	716	11	number	number	NOUN
ejpam-6690	716	12	}	}	PUNCT
ejpam-6690	716	13	.	.	PUNCT
ejpam-6690	717	1	since	since	SCONJ
ejpam-6690	717	2	γ∗(0̄	γ∗(0̄	ADJ
ejpam-6690	717	3	)	)	PUNCT
ejpam-6690	717	4	=	=	SYM
ejpam-6690	717	5	z̄	z̄	X
ejpam-6690	717	6	(	(	PUNCT
ejpam-6690	717	7	or	or	CCONJ
ejpam-6690	717	8	,	,	PUNCT
ejpam-6690	717	9	equivalently	equivalently	ADV
ejpam-6690	717	10	,	,	PUNCT
ejpam-6690	717	11	γ∗	γ∗	NOUN
ejpam-6690	717	12	=	=	SYM
ejpam-6690	717	13	z̄×	z̄×	NOUN
ejpam-6690	717	14	z̄	z̄	NOUN
ejpam-6690	717	15	)	)	PUNCT
ejpam-6690	717	16	,	,	PUNCT
ejpam-6690	717	17	we	we	PRON
ejpam-6690	717	18	have	have	VERB
ejpam-6690	717	19	:	:	PUNCT
ejpam-6690	717	20	spec	spec	PROPN
ejpam-6690	717	21	(	(	PUNCT
ejpam-6690	717	22	γ∗(z̄	γ∗(z̄	NOUN
ejpam-6690	717	23	)	)	PUNCT
ejpam-6690	717	24	)	)	PUNCT
ejpam-6690	718	1	=	=	PUNCT
ejpam-6690	718	2	f	f	PROPN
ejpam-6690	718	3	(	(	PUNCT
ejpam-6690	718	4	spec(z̄	spec(z̄	PROPN
ejpam-6690	718	5	)	)	PUNCT
ejpam-6690	718	6	)	)	PUNCT
ejpam-6690	718	7	∼=	∼=	VERB
ejpam-6690	718	8	z1	z1	NOUN
ejpam-6690	718	9	.	.	NOUN
ejpam-6690	718	10	5	5	NUM
ejpam-6690	718	11	.	.	X
ejpam-6690	718	12	discussion	discussion	NOUN
ejpam-6690	718	13	the	the	DET
ejpam-6690	718	14	framework	framework	NOUN
ejpam-6690	718	15	developed	develop	VERB
ejpam-6690	718	16	in	in	ADP
ejpam-6690	718	17	this	this	DET
ejpam-6690	718	18	paper	paper	NOUN
ejpam-6690	718	19	establishes	establish	VERB
ejpam-6690	718	20	a	a	DET
ejpam-6690	718	21	deep	deep	ADJ
ejpam-6690	718	22	connection	connection	NOUN
ejpam-6690	718	23	between	between	ADP
ejpam-6690	718	24	the	the	DET
ejpam-6690	718	25	algebraic	algebraic	ADJ
ejpam-6690	718	26	properties	property	NOUN
ejpam-6690	718	27	of	of	ADP
ejpam-6690	718	28	commutative	commutative	ADJ
ejpam-6690	718	29	krasner	krasner	NOUN
ejpam-6690	718	30	hyperrings	hyperring	NOUN
ejpam-6690	718	31	and	and	CCONJ
ejpam-6690	718	32	the	the	DET
ejpam-6690	718	33	topological	topological	ADJ
ejpam-6690	718	34	structure	structure	NOUN
ejpam-6690	718	35	of	of	ADP
ejpam-6690	718	36	their	their	PRON
ejpam-6690	718	37	spectra	spectra	NOUN
ejpam-6690	718	38	under	under	ADP
ejpam-6690	718	39	the	the	DET
ejpam-6690	718	40	zariski	zariski	NOUN
ejpam-6690	718	41	topology	topology	NOUN
ejpam-6690	718	42	.	.	PUNCT
ejpam-6690	719	1	by	by	ADP
ejpam-6690	719	2	characterizing	characterize	VERB
ejpam-6690	719	3	connectedness	connectedness	NOUN
ejpam-6690	719	4	,	,	PUNCT
ejpam-6690	719	5	irreducibility	irreducibility	NOUN
ejpam-6690	719	6	,	,	PUNCT
ejpam-6690	719	7	compactness	compactness	NOUN
ejpam-6690	719	8	,	,	PUNCT
ejpam-6690	719	9	and	and	CCONJ
ejpam-6690	719	10	separation	separation	NOUN
ejpam-6690	719	11	in	in	ADP
ejpam-6690	719	12	terms	term	NOUN
ejpam-6690	719	13	of	of	ADP
ejpam-6690	719	14	properties	property	NOUN
ejpam-6690	719	15	like	like	ADP
ejpam-6690	719	16	the	the	DET
ejpam-6690	719	17	presence	presence	NOUN
ejpam-6690	719	18	of	of	ADP
ejpam-6690	719	19	nontrivial	nontrivial	ADJ
ejpam-6690	719	20	idempotents	idempotent	NOUN
ejpam-6690	719	21	or	or	CCONJ
ejpam-6690	719	22	the	the	DET
ejpam-6690	719	23	nature	nature	NOUN
ejpam-6690	719	24	of	of	ADP
ejpam-6690	719	25	the	the	DET
ejpam-6690	719	26	nilradical	nilradical	ADJ
ejpam-6690	719	27	,	,	PUNCT
ejpam-6690	719	28	we	we	PRON
ejpam-6690	719	29	provide	provide	VERB
ejpam-6690	719	30	a	a	DET
ejpam-6690	719	31	bridge	bridge	NOUN
ejpam-6690	719	32	between	between	ADP
ejpam-6690	719	33	hyperstructure	hyperstructure	NOUN
ejpam-6690	719	34	theory	theory	NOUN
ejpam-6690	719	35	and	and	CCONJ
ejpam-6690	719	36	topological	topological	ADJ
ejpam-6690	719	37	algebra	algebra	NOUN
ejpam-6690	719	38	.	.	PUNCT
ejpam-6690	720	1	a	a	DET
ejpam-6690	720	2	central	central	ADJ
ejpam-6690	720	3	contribution	contribution	NOUN
ejpam-6690	720	4	of	of	ADP
ejpam-6690	720	5	this	this	DET
ejpam-6690	720	6	work	work	NOUN
ejpam-6690	720	7	is	be	AUX
ejpam-6690	720	8	the	the	DET
ejpam-6690	720	9	introduction	introduction	NOUN
ejpam-6690	720	10	of	of	ADP
ejpam-6690	720	11	a	a	DET
ejpam-6690	720	12	topological	topological	ADJ
ejpam-6690	720	13	structure	structure	NOUN
ejpam-6690	720	14	on	on	ADP
ejpam-6690	720	15	the	the	DET
ejpam-6690	720	16	set	set	NOUN
ejpam-6690	720	17	of	of	ADP
ejpam-6690	720	18	prime	prime	ADJ
ejpam-6690	720	19	strongly	strongly	ADV
ejpam-6690	720	20	regular	regular	ADJ
ejpam-6690	720	21	relations	relation	NOUN
ejpam-6690	720	22	,	,	PUNCT
ejpam-6690	720	23	which	which	PRON
ejpam-6690	720	24	reflects	reflect	VERB
ejpam-6690	720	25	the	the	DET
ejpam-6690	720	26	hyperring	hyperring	NOUN
ejpam-6690	720	27	’s	’s	PART
ejpam-6690	720	28	internal	internal	ADJ
ejpam-6690	720	29	congruence	congruence	NOUN
ejpam-6690	720	30	geometry	geometry	NOUN
ejpam-6690	720	31	.	.	PUNCT
ejpam-6690	721	1	the	the	DET
ejpam-6690	721	2	demonstration	demonstration	NOUN
ejpam-6690	721	3	of	of	ADP
ejpam-6690	721	4	a	a	DET
ejpam-6690	721	5	homeomorphism	homeomorphism	NOUN
ejpam-6690	721	6	between	between	ADP
ejpam-6690	721	7	this	this	DET
ejpam-6690	721	8	space	space	NOUN
ejpam-6690	721	9	and	and	CCONJ
ejpam-6690	721	10	a	a	DET
ejpam-6690	721	11	subspace	subspace	NOUN
ejpam-6690	721	12	of	of	ADP
ejpam-6690	721	13	the	the	DET
ejpam-6690	721	14	classical	classical	ADJ
ejpam-6690	721	15	spectrum	spectrum	NOUN
ejpam-6690	721	16	via	via	ADP
ejpam-6690	721	17	the	the	DET
ejpam-6690	721	18	fundamental	fundamental	ADJ
ejpam-6690	721	19	relation	relation	NOUN
ejpam-6690	721	20	γ∗	γ∗	PROPN
ejpam-6690	721	21	offers	offer	VERB
ejpam-6690	721	22	a	a	DET
ejpam-6690	721	23	novel	novel	ADJ
ejpam-6690	721	24	perspective	perspective	NOUN
ejpam-6690	721	25	,	,	PUNCT
ejpam-6690	721	26	enabling	enable	VERB
ejpam-6690	721	27	a	a	DET
ejpam-6690	721	28	reinterpretation	reinterpretation	NOUN
ejpam-6690	721	29	of	of	ADP
ejpam-6690	721	30	classical	classical	ADJ
ejpam-6690	721	31	results	result	NOUN
ejpam-6690	721	32	through	through	ADP
ejpam-6690	721	33	hyperalgebraic	hyperalgebraic	ADJ
ejpam-6690	721	34	lenses	lense	NOUN
ejpam-6690	721	35	.	.	PUNCT
ejpam-6690	722	1	this	this	DET
ejpam-6690	722	2	correspondence	correspondence	NOUN
ejpam-6690	722	3	not	not	PART
ejpam-6690	722	4	only	only	ADV
ejpam-6690	722	5	enriches	enrich	VERB
ejpam-6690	722	6	the	the	DET
ejpam-6690	722	7	conceptual	conceptual	ADJ
ejpam-6690	722	8	understanding	understanding	NOUN
ejpam-6690	722	9	of	of	ADP
ejpam-6690	722	10	hyperrings	hyperring	NOUN
ejpam-6690	722	11	but	but	CCONJ
ejpam-6690	722	12	also	also	ADV
ejpam-6690	722	13	sets	set	VERB
ejpam-6690	722	14	the	the	DET
ejpam-6690	722	15	stage	stage	NOUN
ejpam-6690	722	16	for	for	ADP
ejpam-6690	722	17	potential	potential	ADJ
ejpam-6690	722	18	generalizations	generalization	NOUN
ejpam-6690	722	19	of	of	ADP
ejpam-6690	722	20	geometric	geometric	ADJ
ejpam-6690	722	21	tools	tool	NOUN
ejpam-6690	722	22	,	,	PUNCT
ejpam-6690	722	23	such	such	ADJ
ejpam-6690	722	24	as	as	ADP
ejpam-6690	722	25	schemes	scheme	NOUN
ejpam-6690	722	26	and	and	CCONJ
ejpam-6690	722	27	sheaves	sheaf	NOUN
ejpam-6690	722	28	,	,	PUNCT
ejpam-6690	722	29	to	to	ADP
ejpam-6690	722	30	hyperring	hyperre	VERB
ejpam-6690	722	31	contexts	contexts	NOUN
ejpam-6690	722	32	.	.	PUNCT
ejpam-6690	723	1	additionally	additionally	ADV
ejpam-6690	723	2	,	,	PUNCT
ejpam-6690	723	3	our	our	PRON
ejpam-6690	723	4	categorical	categorical	ADJ
ejpam-6690	723	5	treatment	treatment	NOUN
ejpam-6690	723	6	of	of	ADP
ejpam-6690	723	7	zariski	zariski	ADJ
ejpam-6690	723	8	topology	topology	NOUN
ejpam-6690	723	9	via	via	ADP
ejpam-6690	723	10	the	the	DET
ejpam-6690	723	11	spec(−	spec(−	PROPN
ejpam-6690	723	12	)	)	PUNCT
ejpam-6690	723	13	functor	functor	PROPN
ejpam-6690	723	14	reveals	reveal	VERB
ejpam-6690	723	15	the	the	DET
ejpam-6690	723	16	functorial	functorial	ADJ
ejpam-6690	723	17	behavior	behavior	NOUN
ejpam-6690	723	18	of	of	ADP
ejpam-6690	723	19	spectra	spectra	NOUN
ejpam-6690	723	20	under	under	ADP
ejpam-6690	723	21	good	good	ADJ
ejpam-6690	723	22	homomorphisms	homomorphism	NOUN
ejpam-6690	723	23	.	.	PUNCT
ejpam-6690	724	1	we	we	PRON
ejpam-6690	724	2	demonstrated	demonstrate	VERB
ejpam-6690	724	3	how	how	SCONJ
ejpam-6690	724	4	the	the	DET
ejpam-6690	724	5	interplay	interplay	NOUN
ejpam-6690	724	6	between	between	ADP
ejpam-6690	724	7	strongly	strongly	ADV
ejpam-6690	724	8	regular	regular	ADJ
ejpam-6690	724	9	relations	relation	NOUN
ejpam-6690	724	10	and	and	CCONJ
ejpam-6690	724	11	the	the	DET
ejpam-6690	724	12	fundamental	fundamental	ADJ
ejpam-6690	724	13	relation	relation	NOUN
ejpam-6690	724	14	γ∗	γ∗	NOUN
ejpam-6690	724	15	enables	enable	VERB
ejpam-6690	724	16	the	the	DET
ejpam-6690	724	17	transfer	transfer	NOUN
ejpam-6690	724	18	of	of	ADP
ejpam-6690	724	19	structural	structural	ADJ
ejpam-6690	724	20	information	information	NOUN
ejpam-6690	724	21	from	from	ADP
ejpam-6690	724	22	hyperrings	hyperring	NOUN
ejpam-6690	724	23	to	to	ADP
ejpam-6690	724	24	their	their	PRON
ejpam-6690	724	25	associated	associated	ADJ
ejpam-6690	724	26	fundamental	fundamental	ADJ
ejpam-6690	724	27	rings	ring	NOUN
ejpam-6690	724	28	.	.	PUNCT
ejpam-6690	725	1	this	this	PRON
ejpam-6690	725	2	opens	open	VERB
ejpam-6690	725	3	the	the	DET
ejpam-6690	725	4	door	door	NOUN
ejpam-6690	725	5	to	to	ADP
ejpam-6690	725	6	developing	develop	VERB
ejpam-6690	725	7	new	new	ADJ
ejpam-6690	725	8	categorical	categorical	ADJ
ejpam-6690	725	9	frameworks	framework	NOUN
ejpam-6690	725	10	for	for	ADP
ejpam-6690	725	11	hyperrings	hyperring	NOUN
ejpam-6690	725	12	that	that	PRON
ejpam-6690	725	13	mirror	mirror	VERB
ejpam-6690	725	14	the	the	DET
ejpam-6690	725	15	structural	structural	ADJ
ejpam-6690	725	16	richness	richness	NOUN
ejpam-6690	725	17	found	find	VERB
ejpam-6690	725	18	in	in	ADP
ejpam-6690	725	19	classical	classical	ADJ
ejpam-6690	725	20	algebraic	algebraic	ADJ
ejpam-6690	725	21	geometry	geometry	NOUN
ejpam-6690	725	22	.	.	PUNCT
ejpam-6690	726	1	b.	b.	PROPN
ejpam-6690	726	2	afshar	afshar	PROPN
ejpam-6690	726	3	,	,	PUNCT
ejpam-6690	726	4	r.	r.	PROPN
ejpam-6690	726	5	ameri	ameri	PROPN
ejpam-6690	726	6	,	,	PUNCT
ejpam-6690	726	7	m.	m.	PROPN
ejpam-6690	726	8	al	al	PROPN
ejpam-6690	726	9	-	-	PUNCT
ejpam-6690	726	10	tahan	tahan	PROPN
ejpam-6690	726	11	/	/	SYM
ejpam-6690	726	12	eur	eur	PROPN
ejpam-6690	726	13	.	.	PUNCT
ejpam-6690	727	1	j.	j.	PROPN
ejpam-6690	727	2	pure	pure	PROPN
ejpam-6690	727	3	appl	appl	PROPN
ejpam-6690	727	4	.	.	PROPN
ejpam-6690	727	5	math	math	PROPN
ejpam-6690	727	6	,	,	PUNCT
ejpam-6690	727	7	18	18	NUM
ejpam-6690	727	8	(	(	PUNCT
ejpam-6690	727	9	4	4	NUM
ejpam-6690	727	10	)	)	PUNCT
ejpam-6690	727	11	(	(	PUNCT
ejpam-6690	727	12	2025	2025	NUM
ejpam-6690	727	13	)	)	PUNCT
ejpam-6690	727	14	,	,	PUNCT
ejpam-6690	727	15	6690	6690	NUM
ejpam-6690	727	16	18	18	NUM
ejpam-6690	727	17	of	of	ADP
ejpam-6690	727	18	19	19	NUM
ejpam-6690	727	19	an	an	DET
ejpam-6690	727	20	open	open	ADJ
ejpam-6690	727	21	question	question	NOUN
ejpam-6690	727	22	addressed	address	VERB
ejpam-6690	727	23	in	in	ADP
ejpam-6690	727	24	this	this	DET
ejpam-6690	727	25	study	study	NOUN
ejpam-6690	727	26	concerns	concern	VERB
ejpam-6690	727	27	the	the	DET
ejpam-6690	727	28	existence	existence	NOUN
ejpam-6690	727	29	of	of	ADP
ejpam-6690	727	30	functors	functor	NOUN
ejpam-6690	727	31	from	from	ADP
ejpam-6690	727	32	hyperrings	hyperring	NOUN
ejpam-6690	727	33	to	to	ADP
ejpam-6690	727	34	integral	integral	ADJ
ejpam-6690	727	35	domains	domain	NOUN
ejpam-6690	727	36	via	via	ADP
ejpam-6690	727	37	families	family	NOUN
ejpam-6690	727	38	of	of	ADP
ejpam-6690	727	39	strongly	strongly	ADV
ejpam-6690	727	40	regular	regular	ADJ
ejpam-6690	727	41	relations	relation	NOUN
ejpam-6690	727	42	.	.	PUNCT
ejpam-6690	728	1	although	although	SCONJ
ejpam-6690	728	2	such	such	ADJ
ejpam-6690	728	3	sequences	sequence	NOUN
ejpam-6690	728	4	are	be	AUX
ejpam-6690	728	5	not	not	PART
ejpam-6690	728	6	functorial	functorial	ADJ
ejpam-6690	728	7	in	in	ADP
ejpam-6690	728	8	general	general	ADJ
ejpam-6690	728	9	,	,	PUNCT
ejpam-6690	728	10	we	we	PRON
ejpam-6690	728	11	propose	propose	VERB
ejpam-6690	728	12	sufficient	sufficient	ADJ
ejpam-6690	728	13	conditions	condition	NOUN
ejpam-6690	728	14	under	under	ADP
ejpam-6690	728	15	which	which	PRON
ejpam-6690	728	16	they	they	PRON
ejpam-6690	728	17	may	may	AUX
ejpam-6690	728	18	define	define	VERB
ejpam-6690	728	19	a	a	DET
ejpam-6690	728	20	contravariant	contravariant	ADJ
ejpam-6690	728	21	functor	functor	NOUN
ejpam-6690	728	22	,	,	PUNCT
ejpam-6690	728	23	potentially	potentially	ADV
ejpam-6690	728	24	leading	lead	VERB
ejpam-6690	728	25	to	to	ADP
ejpam-6690	728	26	a	a	DET
ejpam-6690	728	27	functorial	functorial	NOUN
ejpam-6690	728	28	theory	theory	NOUN
ejpam-6690	728	29	of	of	ADP
ejpam-6690	728	30	hyperring	hyperre	VERB
ejpam-6690	728	31	localization	localization	NOUN
ejpam-6690	728	32	.	.	PUNCT
ejpam-6690	729	1	this	this	DET
ejpam-6690	729	2	exploratory	exploratory	ADJ
ejpam-6690	729	3	work	work	NOUN
ejpam-6690	729	4	highlights	highlight	VERB
ejpam-6690	729	5	the	the	DET
ejpam-6690	729	6	untapped	untapped	ADJ
ejpam-6690	729	7	potential	potential	NOUN
ejpam-6690	729	8	of	of	ADP
ejpam-6690	729	9	combining	combine	VERB
ejpam-6690	729	10	topological	topological	ADJ
ejpam-6690	729	11	,	,	PUNCT
ejpam-6690	729	12	algebraic	algebraic	ADJ
ejpam-6690	729	13	,	,	PUNCT
ejpam-6690	729	14	and	and	CCONJ
ejpam-6690	729	15	categorical	categorical	ADJ
ejpam-6690	729	16	methods	method	NOUN
ejpam-6690	729	17	in	in	ADP
ejpam-6690	729	18	the	the	DET
ejpam-6690	729	19	study	study	NOUN
ejpam-6690	729	20	of	of	ADP
ejpam-6690	729	21	hyperstructures	hyperstructure	NOUN
ejpam-6690	729	22	,	,	PUNCT
ejpam-6690	729	23	and	and	CCONJ
ejpam-6690	729	24	invites	invite	VERB
ejpam-6690	729	25	further	further	ADJ
ejpam-6690	729	26	developments	development	NOUN
ejpam-6690	729	27	in	in	ADP
ejpam-6690	729	28	hyperalgebraic	hyperalgebraic	PROPN
ejpam-6690	729	29	geometry	geometry	NOUN
ejpam-6690	729	30	and	and	CCONJ
ejpam-6690	729	31	its	its	PRON
ejpam-6690	729	32	applications	application	NOUN
ejpam-6690	729	33	.	.	PUNCT
ejpam-6690	730	1	6	6	NUM
ejpam-6690	730	2	.	.	X
ejpam-6690	730	3	conclusion	conclusion	NOUN
ejpam-6690	730	4	in	in	ADP
ejpam-6690	730	5	this	this	DET
ejpam-6690	730	6	paper	paper	NOUN
ejpam-6690	730	7	,	,	PUNCT
ejpam-6690	730	8	we	we	PRON
ejpam-6690	730	9	presented	present	VERB
ejpam-6690	730	10	a	a	DET
ejpam-6690	730	11	detailed	detailed	ADJ
ejpam-6690	730	12	study	study	NOUN
ejpam-6690	730	13	of	of	ADP
ejpam-6690	730	14	the	the	DET
ejpam-6690	730	15	zariski	zariski	ADJ
ejpam-6690	730	16	topology	topology	NOUN
ejpam-6690	730	17	on	on	ADP
ejpam-6690	730	18	the	the	DET
ejpam-6690	730	19	spectrum	spectrum	NOUN
ejpam-6690	730	20	of	of	ADP
ejpam-6690	730	21	prime	prime	ADJ
ejpam-6690	730	22	hyperideals	hyperideal	NOUN
ejpam-6690	730	23	in	in	ADP
ejpam-6690	730	24	commutative	commutative	ADJ
ejpam-6690	730	25	krasner	krasner	NOUN
ejpam-6690	730	26	hyperrings	hyperring	NOUN
ejpam-6690	730	27	.	.	PUNCT
ejpam-6690	731	1	we	we	PRON
ejpam-6690	731	2	established	establish	VERB
ejpam-6690	731	3	precise	precise	ADJ
ejpam-6690	731	4	algebraic	algebraic	ADJ
ejpam-6690	731	5	conditions	condition	NOUN
ejpam-6690	731	6	under	under	ADP
ejpam-6690	731	7	which	which	PRON
ejpam-6690	731	8	the	the	DET
ejpam-6690	731	9	spectrum	spectrum	NOUN
ejpam-6690	731	10	exhibits	exhibit	VERB
ejpam-6690	731	11	topological	topological	ADJ
ejpam-6690	731	12	properties	property	NOUN
ejpam-6690	731	13	such	such	ADJ
ejpam-6690	731	14	as	as	ADP
ejpam-6690	731	15	connectedness	connectedness	NOUN
ejpam-6690	731	16	,	,	PUNCT
ejpam-6690	731	17	irreducibility	irreducibility	NOUN
ejpam-6690	731	18	,	,	PUNCT
ejpam-6690	731	19	compactness	compactness	NOUN
ejpam-6690	731	20	,	,	PUNCT
ejpam-6690	731	21	and	and	CCONJ
ejpam-6690	731	22	various	various	ADJ
ejpam-6690	731	23	separation	separation	NOUN
ejpam-6690	731	24	axioms	axiom	VERB
ejpam-6690	731	25	.	.	PUNCT
ejpam-6690	732	1	our	our	PRON
ejpam-6690	732	2	results	result	NOUN
ejpam-6690	732	3	reveal	reveal	VERB
ejpam-6690	732	4	that	that	SCONJ
ejpam-6690	732	5	these	these	DET
ejpam-6690	732	6	properties	property	NOUN
ejpam-6690	732	7	are	be	AUX
ejpam-6690	732	8	intrinsically	intrinsically	ADV
ejpam-6690	732	9	tied	tie	VERB
ejpam-6690	732	10	to	to	ADP
ejpam-6690	732	11	the	the	DET
ejpam-6690	732	12	structure	structure	NOUN
ejpam-6690	732	13	of	of	ADP
ejpam-6690	732	14	the	the	DET
ejpam-6690	732	15	hyperring	hyperring	NOUN
ejpam-6690	732	16	—	—	PUNCT
ejpam-6690	732	17	specifically	specifically	ADV
ejpam-6690	732	18	the	the	DET
ejpam-6690	732	19	presence	presence	NOUN
ejpam-6690	732	20	of	of	ADP
ejpam-6690	732	21	nontrivial	nontrivial	ADJ
ejpam-6690	732	22	idempotents	idempotent	NOUN
ejpam-6690	732	23	,	,	PUNCT
ejpam-6690	732	24	the	the	DET
ejpam-6690	732	25	nature	nature	NOUN
ejpam-6690	732	26	of	of	ADP
ejpam-6690	732	27	the	the	DET
ejpam-6690	732	28	nilradical	nilradical	ADJ
ejpam-6690	732	29	,	,	PUNCT
ejpam-6690	732	30	and	and	CCONJ
ejpam-6690	732	31	the	the	DET
ejpam-6690	732	32	behavior	behavior	NOUN
ejpam-6690	732	33	of	of	ADP
ejpam-6690	732	34	hyperideals	hyperideal	NOUN
ejpam-6690	732	35	under	under	ADP
ejpam-6690	732	36	multiplication	multiplication	NOUN
ejpam-6690	732	37	.	.	PUNCT
ejpam-6690	733	1	beyond	beyond	ADP
ejpam-6690	733	2	the	the	DET
ejpam-6690	733	3	classical	classical	ADJ
ejpam-6690	733	4	spectrum	spectrum	NOUN
ejpam-6690	733	5	,	,	PUNCT
ejpam-6690	733	6	we	we	PRON
ejpam-6690	733	7	introduced	introduce	VERB
ejpam-6690	733	8	a	a	DET
ejpam-6690	733	9	topology	topology	NOUN
ejpam-6690	733	10	on	on	ADP
ejpam-6690	733	11	the	the	DET
ejpam-6690	733	12	set	set	NOUN
ejpam-6690	733	13	of	of	ADP
ejpam-6690	733	14	prime	prime	ADJ
ejpam-6690	733	15	strongly	strongly	ADV
ejpam-6690	733	16	regular	regular	ADJ
ejpam-6690	733	17	relations	relation	NOUN
ejpam-6690	733	18	and	and	CCONJ
ejpam-6690	733	19	demonstrated	demonstrate	VERB
ejpam-6690	733	20	its	its	PRON
ejpam-6690	733	21	homeomorphism	homeomorphism	NOUN
ejpam-6690	733	22	with	with	ADP
ejpam-6690	733	23	a	a	DET
ejpam-6690	733	24	subspace	subspace	NOUN
ejpam-6690	733	25	of	of	ADP
ejpam-6690	733	26	the	the	DET
ejpam-6690	733	27	spectrum	spectrum	NOUN
ejpam-6690	733	28	determined	determine	VERB
ejpam-6690	733	29	by	by	ADP
ejpam-6690	733	30	the	the	DET
ejpam-6690	733	31	fundamental	fundamental	ADJ
ejpam-6690	733	32	relation	relation	NOUN
ejpam-6690	733	33	γ∗.	γ∗.	ADP
ejpam-6690	733	34	this	this	DET
ejpam-6690	733	35	dual	dual	ADJ
ejpam-6690	733	36	topological	topological	ADJ
ejpam-6690	733	37	viewpoint	viewpoint	NOUN
ejpam-6690	733	38	offers	offer	VERB
ejpam-6690	733	39	a	a	DET
ejpam-6690	733	40	new	new	ADJ
ejpam-6690	733	41	route	route	NOUN
ejpam-6690	733	42	for	for	ADP
ejpam-6690	733	43	developing	develop	VERB
ejpam-6690	733	44	geometric	geometric	ADJ
ejpam-6690	733	45	theories	theory	NOUN
ejpam-6690	733	46	over	over	ADP
ejpam-6690	733	47	hyperstructures	hyperstructure	NOUN
ejpam-6690	733	48	,	,	PUNCT
ejpam-6690	733	49	particularly	particularly	ADV
ejpam-6690	733	50	in	in	ADP
ejpam-6690	733	51	connection	connection	NOUN
ejpam-6690	733	52	with	with	ADP
ejpam-6690	733	53	sheaves	sheaf	NOUN
ejpam-6690	733	54	and	and	CCONJ
ejpam-6690	733	55	categorical	categorical	ADJ
ejpam-6690	733	56	constructions	construction	NOUN
ejpam-6690	733	57	.	.	PUNCT
ejpam-6690	734	1	by	by	ADP
ejpam-6690	734	2	incorporating	incorporate	VERB
ejpam-6690	734	3	functorial	functorial	NOUN
ejpam-6690	734	4	methods	method	NOUN
ejpam-6690	734	5	and	and	CCONJ
ejpam-6690	734	6	exploring	explore	VERB
ejpam-6690	734	7	the	the	DET
ejpam-6690	734	8	behavior	behavior	NOUN
ejpam-6690	734	9	of	of	ADP
ejpam-6690	734	10	spectra	spectra	NOUN
ejpam-6690	734	11	under	under	ADP
ejpam-6690	734	12	good	good	ADJ
ejpam-6690	734	13	homomorphisms	homomorphism	NOUN
ejpam-6690	734	14	,	,	PUNCT
ejpam-6690	734	15	we	we	PRON
ejpam-6690	734	16	laid	lay	VERB
ejpam-6690	734	17	a	a	DET
ejpam-6690	734	18	foundational	foundational	ADJ
ejpam-6690	734	19	platform	platform	NOUN
ejpam-6690	734	20	for	for	ADP
ejpam-6690	734	21	extending	extend	VERB
ejpam-6690	734	22	algebraic	algebraic	ADJ
ejpam-6690	734	23	geometry	geometry	NOUN
ejpam-6690	734	24	tools	tool	NOUN
ejpam-6690	734	25	to	to	ADP
ejpam-6690	734	26	hyperrings	hyperring	NOUN
ejpam-6690	734	27	.	.	PUNCT
ejpam-6690	735	1	the	the	DET
ejpam-6690	735	2	categorical	categorical	ADJ
ejpam-6690	735	3	insights	insight	NOUN
ejpam-6690	735	4	and	and	CCONJ
ejpam-6690	735	5	topological	topological	ADJ
ejpam-6690	735	6	frameworks	framework	NOUN
ejpam-6690	735	7	developed	develop	VERB
ejpam-6690	735	8	here	here	ADV
ejpam-6690	735	9	are	be	AUX
ejpam-6690	735	10	expected	expect	VERB
ejpam-6690	735	11	to	to	PART
ejpam-6690	735	12	contribute	contribute	VERB
ejpam-6690	735	13	to	to	ADP
ejpam-6690	735	14	ongoing	ongoing	ADJ
ejpam-6690	735	15	research	research	NOUN
ejpam-6690	735	16	in	in	ADP
ejpam-6690	735	17	the	the	DET
ejpam-6690	735	18	theory	theory	NOUN
ejpam-6690	735	19	of	of	ADP
ejpam-6690	735	20	hypermodules	hypermodule	NOUN
ejpam-6690	735	21	,	,	PUNCT
ejpam-6690	735	22	spectral	spectral	ADJ
ejpam-6690	735	23	constructions	construction	NOUN
ejpam-6690	735	24	,	,	PUNCT
ejpam-6690	735	25	and	and	CCONJ
ejpam-6690	735	26	algebraic	algebraic	ADJ
ejpam-6690	735	27	geometry	geometry	NOUN
ejpam-6690	735	28	beyond	beyond	ADP
ejpam-6690	735	29	the	the	DET
ejpam-6690	735	30	confines	confine	NOUN
ejpam-6690	735	31	of	of	ADP
ejpam-6690	735	32	traditional	traditional	ADJ
ejpam-6690	735	33	ring	ring	NOUN
ejpam-6690	735	34	theory	theory	NOUN
ejpam-6690	735	35	.	.	PUNCT
ejpam-6690	736	1	the	the	DET
ejpam-6690	736	2	primary	primary	ADJ
ejpam-6690	736	3	motivation	motivation	NOUN
ejpam-6690	736	4	of	of	ADP
ejpam-6690	736	5	this	this	DET
ejpam-6690	736	6	paper	paper	NOUN
ejpam-6690	736	7	was	be	AUX
ejpam-6690	736	8	to	to	PART
ejpam-6690	736	9	establish	establish	VERB
ejpam-6690	736	10	a	a	DET
ejpam-6690	736	11	framework	framework	NOUN
ejpam-6690	736	12	for	for	ADP
ejpam-6690	736	13	the	the	DET
ejpam-6690	736	14	study	study	NOUN
ejpam-6690	736	15	of	of	ADP
ejpam-6690	736	16	sheaves	sheaf	NOUN
ejpam-6690	736	17	on	on	ADP
ejpam-6690	736	18	the	the	DET
ejpam-6690	736	19	zariski	zariski	ADJ
ejpam-6690	736	20	topology	topology	NOUN
ejpam-6690	736	21	of	of	ADP
ejpam-6690	736	22	krasner	krasner	NOUN
ejpam-6690	736	23	hyperrings	hyperring	NOUN
ejpam-6690	736	24	and	and	CCONJ
ejpam-6690	736	25	hypermodules	hypermodule	NOUN
ejpam-6690	736	26	,	,	PUNCT
ejpam-6690	736	27	and	and	CCONJ
ejpam-6690	736	28	to	to	PART
ejpam-6690	736	29	examine	examine	VERB
ejpam-6690	736	30	the	the	DET
ejpam-6690	736	31	properties	property	NOUN
ejpam-6690	736	32	of	of	ADP
ejpam-6690	736	33	the	the	DET
ejpam-6690	736	34	associated	associate	VERB
ejpam-6690	736	35	hyperringed	hyperringe	VERB
ejpam-6690	736	36	spaces	space	NOUN
ejpam-6690	736	37	and	and	CCONJ
ejpam-6690	736	38	hyperschemes	hyperscheme	NOUN
ejpam-6690	736	39	,	,	PUNCT
ejpam-6690	736	40	with	with	ADP
ejpam-6690	736	41	the	the	DET
ejpam-6690	736	42	aim	aim	NOUN
ejpam-6690	736	43	of	of	ADP
ejpam-6690	736	44	conducting	conduct	VERB
ejpam-6690	736	45	investigations	investigation	NOUN
ejpam-6690	736	46	in	in	ADP
ejpam-6690	736	47	hyperalgebraic	hyperalgebraic	PROPN
ejpam-6690	736	48	geometry	geometry	NOUN
ejpam-6690	736	49	.	.	PUNCT
ejpam-6690	737	1	in	in	ADP
ejpam-6690	737	2	addition	addition	NOUN
ejpam-6690	737	3	,	,	PUNCT
ejpam-6690	737	4	extending	extend	VERB
ejpam-6690	737	5	the	the	DET
ejpam-6690	737	6	obtained	obtain	VERB
ejpam-6690	737	7	results	result	NOUN
ejpam-6690	737	8	to	to	ADP
ejpam-6690	737	9	other	other	ADJ
ejpam-6690	737	10	types	type	NOUN
ejpam-6690	737	11	of	of	ADP
ejpam-6690	737	12	hyperrings	hyperring	NOUN
ejpam-6690	737	13	and	and	CCONJ
ejpam-6690	737	14	hypermodules	hypermodule	NOUN
ejpam-6690	737	15	,	,	PUNCT
ejpam-6690	737	16	as	as	ADV
ejpam-6690	737	17	well	well	ADV
ejpam-6690	737	18	as	as	ADP
ejpam-6690	737	19	exploring	explore	VERB
ejpam-6690	737	20	the	the	DET
ejpam-6690	737	21	role	role	NOUN
ejpam-6690	737	22	of	of	ADP
ejpam-6690	737	23	regular	regular	ADJ
ejpam-6690	737	24	relations	relation	NOUN
ejpam-6690	737	25	in	in	ADP
ejpam-6690	737	26	this	this	DET
ejpam-6690	737	27	context	context	NOUN
ejpam-6690	737	28	,	,	PUNCT
ejpam-6690	737	29	will	will	AUX
ejpam-6690	737	30	be	be	AUX
ejpam-6690	737	31	among	among	ADP
ejpam-6690	737	32	our	our	PRON
ejpam-6690	737	33	future	future	ADJ
ejpam-6690	737	34	works	work	NOUN
ejpam-6690	737	35	.	.	PUNCT
ejpam-6690	738	1	acknowledgements	acknowledgement	NOUN
ejpam-6690	738	2	this	this	DET
ejpam-6690	738	3	research	research	NOUN
ejpam-6690	738	4	was	be	AUX
ejpam-6690	738	5	supported	support	VERB
ejpam-6690	738	6	by	by	ADP
ejpam-6690	738	7	abu	abu	PROPN
ejpam-6690	738	8	dhabi	dhabi	PROPN
ejpam-6690	738	9	university	university	PROPN
ejpam-6690	738	10	with	with	ADP
ejpam-6690	738	11	grant	grant	NOUN
ejpam-6690	738	12	number	number	NOUN
ejpam-6690	738	13	19300884	19300884	NUM
ejpam-6690	738	14	.	.	PUNCT
ejpam-6690	739	1	references	reference	NOUN
ejpam-6690	739	2	[	[	X
ejpam-6690	739	3	1	1	NUM
ejpam-6690	739	4	]	]	X
ejpam-6690	739	5	f	f	PROPN
ejpam-6690	739	6	marty	marty	PROPN
ejpam-6690	739	7	.	.	PUNCT
ejpam-6690	740	1	sur	sur	PROPN
ejpam-6690	740	2	une	une	PROPN
ejpam-6690	740	3	generalization	generalization	PROPN
ejpam-6690	740	4	de	de	X
ejpam-6690	740	5	la	la	PROPN
ejpam-6690	740	6	notion	notion	NOUN
ejpam-6690	740	7	de	de	ADP
ejpam-6690	740	8	groups	group	NOUN
ejpam-6690	740	9	.	.	PUNCT
ejpam-6690	741	1	in	in	ADP
ejpam-6690	741	2	8th	8th	ADJ
ejpam-6690	741	3	congress	congress	PROPN
ejpam-6690	741	4	math	math	NOUN
ejpam-6690	741	5	.	.	PUNCT
ejpam-6690	742	1	scandinaves	scandinave	NOUN
ejpam-6690	742	2	,	,	PUNCT
ejpam-6690	742	3	stockholm,(1934	stockholm,(1934	NOUN
ejpam-6690	742	4	)	)	PUNCT
ejpam-6690	742	5	.	.	PUNCT
ejpam-6690	742	6	,	,	PUNCT
ejpam-6690	742	7	1934	1934	NUM
ejpam-6690	742	8	.	.	PUNCT
ejpam-6690	743	1	b.	b.	PROPN
ejpam-6690	743	2	afshar	afshar	PROPN
ejpam-6690	743	3	,	,	PUNCT
ejpam-6690	743	4	r.	r.	PROPN
ejpam-6690	743	5	ameri	ameri	PROPN
ejpam-6690	743	6	,	,	PUNCT
ejpam-6690	743	7	m.	m.	PROPN
ejpam-6690	743	8	al	al	PROPN
ejpam-6690	743	9	-	-	PUNCT
ejpam-6690	743	10	tahan	tahan	PROPN
ejpam-6690	743	11	/	/	SYM
ejpam-6690	743	12	eur	eur	PROPN
ejpam-6690	743	13	.	.	PUNCT
ejpam-6690	744	1	j.	j.	PROPN
ejpam-6690	744	2	pure	pure	PROPN
ejpam-6690	744	3	appl	appl	PROPN
ejpam-6690	744	4	.	.	PROPN
ejpam-6690	744	5	math	math	PROPN
ejpam-6690	744	6	,	,	PUNCT
ejpam-6690	744	7	18	18	NUM
ejpam-6690	744	8	(	(	PUNCT
ejpam-6690	744	9	4	4	NUM
ejpam-6690	744	10	)	)	PUNCT
ejpam-6690	744	11	(	(	PUNCT
ejpam-6690	744	12	2025	2025	NUM
ejpam-6690	744	13	)	)	PUNCT
ejpam-6690	744	14	,	,	PUNCT
ejpam-6690	744	15	6690	6690	NUM
ejpam-6690	744	16	19	19	NUM
ejpam-6690	744	17	of	of	ADP
ejpam-6690	744	18	19	19	NUM
ejpam-6690	745	1	[	[	X
ejpam-6690	745	2	2	2	NUM
ejpam-6690	745	3	]	]	PUNCT
ejpam-6690	745	4	a	a	DET
ejpam-6690	745	5	connes	conne	NOUN
ejpam-6690	745	6	and	and	CCONJ
ejpam-6690	745	7	c	c	PROPN
ejpam-6690	745	8	consani	consani	NOUN
ejpam-6690	745	9	.	.	PUNCT
ejpam-6690	746	1	from	from	ADP
ejpam-6690	746	2	monoids	monoid	NOUN
ejpam-6690	746	3	to	to	ADP
ejpam-6690	746	4	hyperstructures	hyperstructure	NOUN
ejpam-6690	746	5	:	:	PUNCT
ejpam-6690	746	6	in	in	ADP
ejpam-6690	746	7	search	search	NOUN
ejpam-6690	746	8	of	of	ADP
ejpam-6690	746	9	an	an	DET
ejpam-6690	746	10	absolute	absolute	ADJ
ejpam-6690	746	11	arithmetic	arithmetic	NOUN
ejpam-6690	746	12	.	.	PUNCT
ejpam-6690	746	13	casimir	casimir	PROPN
ejpam-6690	746	14	force	force	PROPN
ejpam-6690	746	15	,	,	PUNCT
ejpam-6690	746	16	casimir	casimir	NOUN
ejpam-6690	746	17	operators	operator	NOUN
ejpam-6690	746	18	and	and	CCONJ
ejpam-6690	746	19	the	the	DET
ejpam-6690	746	20	riemann	riemann	PROPN
ejpam-6690	746	21	hypothesis	hypothesis	NOUN
ejpam-6690	746	22	,	,	PUNCT
ejpam-6690	746	23	de	de	NOUN
ejpam-6690	746	24	gruyter	gruyter	NOUN
ejpam-6690	746	25	,	,	PUNCT
ejpam-6690	746	26	pages	page	NOUN
ejpam-6690	746	27	147–198	147–198	NUM
ejpam-6690	746	28	,	,	PUNCT
ejpam-6690	746	29	2010	2010	NUM
ejpam-6690	746	30	.	.	PUNCT
ejpam-6690	747	1	[	[	X
ejpam-6690	747	2	3	3	X
ejpam-6690	747	3	]	]	PUNCT
ejpam-6690	747	4	a	a	DET
ejpam-6690	747	5	connes	conne	NOUN
ejpam-6690	747	6	and	and	CCONJ
ejpam-6690	747	7	c	c	PROPN
ejpam-6690	747	8	consani	consani	NOUN
ejpam-6690	747	9	.	.	PUNCT
ejpam-6690	748	1	the	the	DET
ejpam-6690	748	2	hyperring	hyperring	NOUN
ejpam-6690	748	3	of	of	ADP
ejpam-6690	748	4	adele	adele	PROPN
ejpam-6690	748	5	classes	class	NOUN
ejpam-6690	748	6	.	.	PUNCT
ejpam-6690	749	1	journal	journal	NOUN
ejpam-6690	749	2	of	of	ADP
ejpam-6690	749	3	number	number	NOUN
ejpam-6690	749	4	theory	theory	NOUN
ejpam-6690	749	5	,	,	PUNCT
ejpam-6690	749	6	131(2):159–194	131(2):159–194	PROPN
ejpam-6690	749	7	,	,	PUNCT
ejpam-6690	749	8	2011	2011	NUM
ejpam-6690	749	9	.	.	PUNCT
ejpam-6690	750	1	[	[	X
ejpam-6690	750	2	4	4	X
ejpam-6690	750	3	]	]	PUNCT
ejpam-6690	750	4	a	a	DET
ejpam-6690	750	5	connes	conne	NOUN
ejpam-6690	750	6	and	and	CCONJ
ejpam-6690	750	7	c	c	PROPN
ejpam-6690	750	8	consani	consani	NOUN
ejpam-6690	750	9	.	.	PUNCT
ejpam-6690	751	1	universal	universal	ADJ
ejpam-6690	751	2	thickening	thickening	NOUN
ejpam-6690	751	3	of	of	ADP
ejpam-6690	751	4	the	the	DET
ejpam-6690	751	5	field	field	NOUN
ejpam-6690	751	6	of	of	ADP
ejpam-6690	751	7	real	real	ADJ
ejpam-6690	751	8	numbers	number	NOUN
ejpam-6690	751	9	.	.	PUNCT
ejpam-6690	752	1	advances	advance	NOUN
ejpam-6690	752	2	in	in	ADP
ejpam-6690	752	3	the	the	DET
ejpam-6690	752	4	theory	theory	NOUN
ejpam-6690	752	5	of	of	ADP
ejpam-6690	752	6	numbers	number	NOUN
ejpam-6690	752	7	:	:	PUNCT
ejpam-6690	752	8	proceedings	proceeding	NOUN
ejpam-6690	752	9	of	of	ADP
ejpam-6690	752	10	the	the	DET
ejpam-6690	752	11	thirteenth	thirteenth	ADJ
ejpam-6690	752	12	conference	conference	NOUN
ejpam-6690	752	13	of	of	ADP
ejpam-6690	752	14	the	the	DET
ejpam-6690	752	15	canadian	canadian	ADJ
ejpam-6690	752	16	number	number	NOUN
ejpam-6690	752	17	theory	theory	NOUN
ejpam-6690	752	18	association	association	NOUN
ejpam-6690	752	19	,	,	PUNCT
ejpam-6690	752	20	pages	page	NOUN
ejpam-6690	752	21	11–74	11–74	NUM
ejpam-6690	752	22	,	,	PUNCT
ejpam-6690	752	23	2015	2015	NUM
ejpam-6690	752	24	.	.	PUNCT
ejpam-6690	753	1	[	[	X
ejpam-6690	753	2	5	5	NUM
ejpam-6690	753	3	]	]	PUNCT
ejpam-6690	753	4	m	m	NOUN
ejpam-6690	753	5	krasner	krasner	NOUN
ejpam-6690	753	6	.	.	PUNCT
ejpam-6690	754	1	a	a	DET
ejpam-6690	754	2	class	class	NOUN
ejpam-6690	754	3	of	of	ADP
ejpam-6690	754	4	hyperrings	hyperring	NOUN
ejpam-6690	754	5	and	and	CCONJ
ejpam-6690	754	6	hyperfields	hyperfield	NOUN
ejpam-6690	754	7	.	.	PUNCT
ejpam-6690	755	1	international	international	ADJ
ejpam-6690	755	2	journal	journal	PROPN
ejpam-6690	755	3	of	of	ADP
ejpam-6690	755	4	mathematics	mathematics	PROPN
ejpam-6690	755	5	and	and	CCONJ
ejpam-6690	755	6	mathematical	mathematical	ADJ
ejpam-6690	755	7	sciences	science	NOUN
ejpam-6690	755	8	,	,	PUNCT
ejpam-6690	755	9	6(2):307–311	6(2):307–311	NUM
ejpam-6690	755	10	,	,	PUNCT
ejpam-6690	755	11	1983	1983	NUM
ejpam-6690	755	12	.	.	PUNCT
ejpam-6690	756	1	[	[	X
ejpam-6690	756	2	6	6	NUM
ejpam-6690	756	3	]	]	X
ejpam-6690	756	4	p	p	X
ejpam-6690	756	5	corsini	corsini	PROPN
ejpam-6690	756	6	.	.	PUNCT
ejpam-6690	757	1	prolegomena	prolegomenon	NOUN
ejpam-6690	757	2	of	of	ADP
ejpam-6690	757	3	hypergroup	hypergroup	PROPN
ejpam-6690	757	4	theory	theory	PROPN
ejpam-6690	757	5	.	.	PUNCT
ejpam-6690	758	1	aviani	aviani	PROPN
ejpam-6690	758	2	editore	editore	PROPN
ejpam-6690	758	3	,	,	PUNCT
ejpam-6690	758	4	1993	1993	NUM
ejpam-6690	758	5	.	.	PUNCT
ejpam-6690	759	1	[	[	X
ejpam-6690	759	2	7	7	NUM
ejpam-6690	759	3	]	]	X
ejpam-6690	759	4	m	m	VERB
ejpam-6690	759	5	koskas	koskas	PROPN
ejpam-6690	759	6	.	.	PUNCT
ejpam-6690	760	1	groupoides	groupoide	NOUN
ejpam-6690	760	2	,	,	PUNCT
ejpam-6690	760	3	demi	demi	NOUN
ejpam-6690	760	4	-	-	PUNCT
ejpam-6690	760	5	hypergroupes	hypergroupe	NOUN
ejpam-6690	760	6	et	et	NOUN
ejpam-6690	760	7	hypergroupes	hypergroupe	NOUN
ejpam-6690	760	8	.	.	PUNCT
ejpam-6690	761	1	j.	j.	PROPN
ejpam-6690	761	2	math	math	PROPN
ejpam-6690	761	3	.	.	PUNCT
ejpam-6690	762	1	pures	pure	NOUN
ejpam-6690	762	2	appl	appl	PROPN
ejpam-6690	762	3	,	,	PUNCT
ejpam-6690	762	4	49(9):155–192	49(9):155–192	NOUN
ejpam-6690	762	5	,	,	PUNCT
ejpam-6690	762	6	1970	1970	NUM
ejpam-6690	762	7	.	.	PUNCT
ejpam-6690	763	1	[	[	X
ejpam-6690	763	2	8	8	NUM
ejpam-6690	763	3	]	]	X
ejpam-6690	763	4	d	d	X
ejpam-6690	763	5	freni	freni	PROPN
ejpam-6690	763	6	.	.	PUNCT
ejpam-6690	764	1	a	a	DET
ejpam-6690	764	2	new	new	ADJ
ejpam-6690	764	3	characterization	characterization	NOUN
ejpam-6690	764	4	of	of	ADP
ejpam-6690	764	5	the	the	DET
ejpam-6690	764	6	derived	derive	VERB
ejpam-6690	764	7	hypergroup	hypergroup	NOUN
ejpam-6690	764	8	via	via	ADP
ejpam-6690	764	9	strongly	strongly	ADV
ejpam-6690	764	10	regular	regular	ADJ
ejpam-6690	764	11	equivalences	equivalence	NOUN
ejpam-6690	764	12	.	.	PUNCT
ejpam-6690	765	1	communications	communication	NOUN
ejpam-6690	765	2	in	in	ADP
ejpam-6690	765	3	algebra	algebra	NOUN
ejpam-6690	765	4	,	,	PUNCT
ejpam-6690	765	5	30(8):3977–3989	30(8):3977–3989	NUM
ejpam-6690	765	6	,	,	PUNCT
ejpam-6690	765	7	2002	2002	NUM
ejpam-6690	765	8	.	.	PUNCT
ejpam-6690	766	1	[	[	X
ejpam-6690	766	2	9	9	NUM
ejpam-6690	766	3	]	]	PUNCT
ejpam-6690	766	4	t	t	PROPN
ejpam-6690	766	5	vougiouklis	vougioukli	VERB
ejpam-6690	766	6	.	.	PUNCT
ejpam-6690	767	1	the	the	DET
ejpam-6690	767	2	fundamental	fundamental	ADJ
ejpam-6690	767	3	relation	relation	NOUN
ejpam-6690	767	4	in	in	ADP
ejpam-6690	767	5	hyperrings	hyperring	NOUN
ejpam-6690	767	6	.	.	PUNCT
ejpam-6690	768	1	the	the	DET
ejpam-6690	768	2	general	general	ADJ
ejpam-6690	768	3	hyperfield	hyperfield	NOUN
ejpam-6690	768	4	.	.	PUNCT
ejpam-6690	769	1	in	in	ADP
ejpam-6690	769	2	proc	proc	PROPN
ejpam-6690	769	3	.	.	PUNCT
ejpam-6690	770	1	fourth	fourth	ADJ
ejpam-6690	770	2	int	int	NOUN
ejpam-6690	770	3	.	.	PUNCT
ejpam-6690	771	1	congress	congress	PROPN
ejpam-6690	771	2	on	on	ADP
ejpam-6690	771	3	algebraic	algebraic	PROPN
ejpam-6690	771	4	hyperstructures	hyperstructure	NOUN
ejpam-6690	771	5	and	and	CCONJ
ejpam-6690	771	6	applications	application	NOUN
ejpam-6690	771	7	(	(	PUNCT
ejpam-6690	771	8	aha	aha	INTJ
ejpam-6690	771	9	1990	1990	NUM
ejpam-6690	771	10	)	)	PUNCT
ejpam-6690	771	11	,	,	PUNCT
ejpam-6690	771	12	world	world	NOUN
ejpam-6690	771	13	scientific	scientific	ADJ
ejpam-6690	771	14	.	.	PUNCT
ejpam-6690	771	15	,	,	PUNCT
ejpam-6690	771	16	pages	page	NOUN
ejpam-6690	771	17	203–211	203–211	NUM
ejpam-6690	771	18	.	.	PUNCT
ejpam-6690	772	1	world	world	NOUN
ejpam-6690	772	2	scientific	scientific	ADJ
ejpam-6690	772	3	,	,	PUNCT
ejpam-6690	772	4	1991	1991	NUM
ejpam-6690	772	5	.	.	PUNCT
ejpam-6690	773	1	[	[	X
ejpam-6690	773	2	10	10	NUM
ejpam-6690	773	3	]	]	SYM
ejpam-6690	773	4	b	b	NOUN
ejpam-6690	773	5	afshar	afshar	ADJ
ejpam-6690	773	6	and	and	CCONJ
ejpam-6690	773	7	r	r	PROPN
ejpam-6690	773	8	ameri	ameri	PROPN
ejpam-6690	773	9	.	.	PUNCT
ejpam-6690	774	1	strongly	strongly	ADV
ejpam-6690	774	2	regular	regular	ADJ
ejpam-6690	774	3	relations	relation	NOUN
ejpam-6690	774	4	derived	derive	VERB
ejpam-6690	774	5	from	from	ADP
ejpam-6690	774	6	fundamental	fundamental	ADJ
ejpam-6690	774	7	relation	relation	NOUN
ejpam-6690	774	8	.	.	PUNCT
ejpam-6690	775	1	journal	journal	PROPN
ejpam-6690	775	2	of	of	ADP
ejpam-6690	775	3	algebraic	algebraic	PROPN
ejpam-6690	775	4	hyperstructures	hyperstructure	NOUN
ejpam-6690	775	5	and	and	CCONJ
ejpam-6690	775	6	logical	logical	ADJ
ejpam-6690	775	7	algebras	algebra	NOUN
ejpam-6690	775	8	,	,	PUNCT
ejpam-6690	775	9	4(2):123–130	4(2):123–130	NOUN
ejpam-6690	775	10	,	,	PUNCT
ejpam-6690	775	11	2023	2023	NUM
ejpam-6690	775	12	.	.	PUNCT
ejpam-6690	776	1	[	[	X
ejpam-6690	776	2	11	11	NUM
ejpam-6690	776	3	]	]	X
ejpam-6690	776	4	r	r	NOUN
ejpam-6690	776	5	ameri	ameri	PROPN
ejpam-6690	776	6	and	and	CCONJ
ejpam-6690	776	7	b	b	NOUN
ejpam-6690	776	8	afshar	afshar	ADJ
ejpam-6690	776	9	.	.	PUNCT
ejpam-6690	777	1	strongly	strongly	ADV
ejpam-6690	777	2	regular	regular	ADJ
ejpam-6690	777	3	relations	relation	NOUN
ejpam-6690	777	4	on	on	ADP
ejpam-6690	777	5	regular	regular	ADJ
ejpam-6690	777	6	hypergroups	hypergroup	NOUN
ejpam-6690	777	7	.	.	PUNCT
ejpam-6690	778	1	journal	journal	PROPN
ejpam-6690	778	2	of	of	ADP
ejpam-6690	778	3	mahani	mahani	PROPN
ejpam-6690	778	4	mathematical	mathematical	PROPN
ejpam-6690	778	5	research	research	NOUN
ejpam-6690	778	6	,	,	PUNCT
ejpam-6690	778	7	14(1):73–83	14(1):73–83	NUM
ejpam-6690	778	8	,	,	PUNCT
ejpam-6690	778	9	2025	2025	NUM
ejpam-6690	778	10	.	.	PUNCT
ejpam-6690	779	1	[	[	X
ejpam-6690	779	2	12	12	NUM
ejpam-6690	779	3	]	]	X
ejpam-6690	779	4	m	m	VERB
ejpam-6690	779	5	al	al	PROPN
ejpam-6690	779	6	-	-	PUNCT
ejpam-6690	779	7	tahan	tahan	PROPN
ejpam-6690	779	8	and	and	CCONJ
ejpam-6690	779	9	b	b	PROPN
ejpam-6690	779	10	davvaz	davvaz	NOUN
ejpam-6690	779	11	.	.	PUNCT
ejpam-6690	780	1	hv	hv	NOUN
ejpam-6690	780	2	-	-	PUNCT
ejpam-6690	780	3	module	module	NOUN
ejpam-6690	780	4	of	of	ADP
ejpam-6690	780	5	functions	function	NOUN
ejpam-6690	780	6	over	over	ADP
ejpam-6690	780	7	hv	hv	NOUN
ejpam-6690	780	8	-	-	PUNCT
ejpam-6690	780	9	ring	ring	NOUN
ejpam-6690	780	10	of	of	ADP
ejpam-6690	780	11	arithmetics	arithmetic	NOUN
ejpam-6690	780	12	and	and	CCONJ
ejpam-6690	780	13	it	it	PRON
ejpam-6690	780	14	’s	’	VERB
ejpam-6690	780	15	fundamental	fundamental	ADJ
ejpam-6690	780	16	module	module	NOUN
ejpam-6690	780	17	.	.	PUNCT
ejpam-6690	781	1	analele	analele	ADP
ejpam-6690	781	2	stiintifice	stiintifice	PROPN
ejpam-6690	781	3	ale	ale	PROPN
ejpam-6690	781	4	universitatii	universitatii	PROPN
ejpam-6690	781	5	ovidius	ovidius	PROPN
ejpam-6690	781	6	constanta	constanta	PROPN
ejpam-6690	781	7	:	:	PUNCT
ejpam-6690	781	8	seria	seria	PROPN
ejpam-6690	781	9	matematica	matematica	PROPN
ejpam-6690	781	10	,	,	PUNCT
ejpam-6690	781	11	32(1):5–18	32(1):5–18	NUM
ejpam-6690	781	12	,	,	PUNCT
ejpam-6690	781	13	2024	2024	NUM
ejpam-6690	781	14	.	.	PUNCT
ejpam-6690	782	1	[	[	X
ejpam-6690	782	2	13	13	NUM
ejpam-6690	782	3	]	]	SYM
ejpam-6690	782	4	m	m	VERB
ejpam-6690	782	5	al	al	PROPN
ejpam-6690	782	6	-	-	PUNCT
ejpam-6690	782	7	tahan	tahan	PROPN
ejpam-6690	782	8	and	and	CCONJ
ejpam-6690	782	9	b	b	PROPN
ejpam-6690	782	10	davvaz	davvaz	NOUN
ejpam-6690	782	11	.	.	PUNCT
ejpam-6690	783	1	chemical	chemical	NOUN
ejpam-6690	783	2	hyperstructures	hyperstructure	NOUN
ejpam-6690	783	3	for	for	ADP
ejpam-6690	783	4	elements	element	NOUN
ejpam-6690	783	5	with	with	ADP
ejpam-6690	783	6	four	four	NUM
ejpam-6690	783	7	oxidation	oxidation	ADJ
ejpam-6690	783	8	states	state	NOUN
ejpam-6690	783	9	.	.	PUNCT
ejpam-6690	784	1	iranian	iranian	ADJ
ejpam-6690	784	2	journal	journal	PROPN
ejpam-6690	784	3	of	of	ADP
ejpam-6690	784	4	mathematical	mathematical	ADJ
ejpam-6690	784	5	chemistry	chemistry	NOUN
ejpam-6690	784	6	,	,	PUNCT
ejpam-6690	784	7	13(2):85–97	13(2):85–97	NUM
ejpam-6690	784	8	,	,	PUNCT
ejpam-6690	784	9	2022	2022	NUM
ejpam-6690	784	10	.	.	PUNCT
ejpam-6690	785	1	[	[	X
ejpam-6690	785	2	14	14	NUM
ejpam-6690	785	3	]	]	X
ejpam-6690	785	4	r	r	NOUN
ejpam-6690	785	5	ameri	ameri	PROPN
ejpam-6690	785	6	and	and	CCONJ
ejpam-6690	785	7	a	a	DET
ejpam-6690	785	8	kordi	kordi	NOUN
ejpam-6690	785	9	.	.	PUNCT
ejpam-6690	786	1	clean	clean	PROPN
ejpam-6690	786	2	multiplicative	multiplicative	ADJ
ejpam-6690	786	3	hyperrings	hyperring	NOUN
ejpam-6690	786	4	.	.	PUNCT
ejpam-6690	787	1	italian	italian	ADJ
ejpam-6690	787	2	journal	journal	NOUN
ejpam-6690	787	3	of	of	ADP
ejpam-6690	787	4	pure	pure	ADJ
ejpam-6690	787	5	and	and	CCONJ
ejpam-6690	787	6	applied	applied	ADJ
ejpam-6690	787	7	mathematics	mathematic	NOUN
ejpam-6690	787	8	,	,	PUNCT
ejpam-6690	787	9	35:625–636	35:625–636	NUM
ejpam-6690	787	10	,	,	PUNCT
ejpam-6690	787	11	2015	2015	NUM
ejpam-6690	787	12	.	.	PUNCT
ejpam-6690	788	1	[	[	X
ejpam-6690	788	2	15	15	NUM
ejpam-6690	788	3	]	]	SYM
ejpam-6690	788	4	b	b	NOUN
ejpam-6690	788	5	davvaz	davvaz	NOUN
ejpam-6690	788	6	and	and	CCONJ
ejpam-6690	788	7	v	v	ADP
ejpam-6690	788	8	leoreanu	leoreanu	NOUN
ejpam-6690	788	9	-	-	PUNCT
ejpam-6690	788	10	fotea	fotea	NOUN
ejpam-6690	788	11	.	.	PUNCT
ejpam-6690	789	1	hyperring	hyperre	VERB
ejpam-6690	789	2	theory	theory	NOUN
ejpam-6690	789	3	and	and	CCONJ
ejpam-6690	789	4	applications	application	NOUN
ejpam-6690	789	5	.	.	PUNCT
ejpam-6690	790	1	international	international	ADJ
ejpam-6690	790	2	academic	academic	ADJ
ejpam-6690	790	3	press	press	NOUN
ejpam-6690	790	4	,	,	PUNCT
ejpam-6690	790	5	usa	usa	PROPN
ejpam-6690	790	6	,	,	PUNCT
ejpam-6690	790	7	2007	2007	NUM
ejpam-6690	790	8	.	.	PUNCT
ejpam-6690	791	1	[	[	X
ejpam-6690	791	2	16	16	NUM
ejpam-6690	791	3	]	]	X
ejpam-6690	791	4	j	j	PROPN
ejpam-6690	791	5	rmunkres	rmunkre	VERB
ejpam-6690	791	6	.	.	PUNCT
ejpam-6690	792	1	topology	topology	NOUN
ejpam-6690	792	2	.	.	PUNCT
ejpam-6690	793	1	featured	feature	VERB
ejpam-6690	793	2	titles	title	NOUN
ejpam-6690	793	3	for	for	ADP
ejpam-6690	793	4	topology	topology	NOUN
ejpam-6690	793	5	series	series	NOUN
ejpam-6690	793	6	.	.	PUNCT
ejpam-6690	794	1	prentice	prentice	PROPN
ejpam-6690	794	2	hall	hall	PROPN
ejpam-6690	794	3	,	,	PUNCT
ejpam-6690	794	4	incorporated	incorporate	VERB
ejpam-6690	794	5	,	,	PUNCT
ejpam-6690	794	6	812:813	812:813	NOUN
ejpam-6690	794	7	,	,	PUNCT
ejpam-6690	794	8	2000	2000	NUM
ejpam-6690	794	9	.	.	PUNCT
ejpam-6690	795	1	[	[	X
ejpam-6690	795	2	17	17	NUM
ejpam-6690	795	3	]	]	X
ejpam-6690	795	4	w	w	PROPN
ejpam-6690	795	5	jonsson	jonsson	PROPN
ejpam-6690	795	6	.	.	PUNCT
ejpam-6690	795	7	introduction	introduction	NOUN
ejpam-6690	795	8	to	to	ADP
ejpam-6690	795	9	commutative	commutative	ADJ
ejpam-6690	795	10	algebra	algebra	PROPN
ejpam-6690	795	11	.	.	PUNCT
ejpam-6690	796	1	cambridge	cambridge	PROPN
ejpam-6690	796	2	university	university	PROPN
ejpam-6690	796	3	press	press	NOUN
ejpam-6690	796	4	,	,	PUNCT
ejpam-6690	796	5	canadian	canadian	PROPN
ejpam-6690	796	6	mathematical	mathematical	ADJ
ejpam-6690	796	7	bulletin	bulletin	NOUN
ejpam-6690	796	8	,	,	PUNCT
ejpam-6690	796	9	1970	1970	NUM
ejpam-6690	796	10	.	.	PUNCT
ejpam-6690	797	1	[	[	X
ejpam-6690	797	2	18	18	NUM
ejpam-6690	797	3	]	]	SYM
ejpam-6690	797	4	b	b	NOUN
ejpam-6690	797	5	davvaz	davvaz	NOUN
ejpam-6690	797	6	and	and	CCONJ
ejpam-6690	797	7	v	v	ADP
ejpam-6690	797	8	leoreanu	leoreanu	NOUN
ejpam-6690	797	9	-	-	PUNCT
ejpam-6690	797	10	fotea	fotea	NOUN
ejpam-6690	797	11	.	.	PUNCT
ejpam-6690	798	1	krasner	krasner	PROPN
ejpam-6690	798	2	hyperring	hyperring	PROPN
ejpam-6690	798	3	theory	theory	NOUN
ejpam-6690	798	4	.	.	PUNCT
ejpam-6690	799	1	world	world	PROPN
ejpam-6690	799	2	scientific	scientific	ADJ
ejpam-6690	799	3	,	,	PUNCT
ejpam-6690	799	4	2024	2024	NUM
ejpam-6690	799	5	.	.	PUNCT