id	sid	tid	token	lemma	pos
iajs-3028	1	1	ihjpas	ihjpas	PROPN
iajs-3028	1	2	.	.	PUNCT
iajs-3028	2	1	36(2)2023	36(2)2023	NUM
iajs-3028	2	2	383	383	NUM
iajs-3028	2	3	this	this	DET
iajs-3028	2	4	work	work	NOUN
iajs-3028	2	5	is	be	AUX
iajs-3028	2	6	licensed	license	VERB
iajs-3028	2	7	under	under	ADP
iajs-3028	2	8	a	a	DET
iajs-3028	2	9	creative	creative	ADJ
iajs-3028	2	10	commons	common	NOUN
iajs-3028	2	11	attribution	attribution	NOUN
iajs-3028	2	12	4.0	4.0	NUM
iajs-3028	2	13	international	international	ADJ
iajs-3028	2	14	license	license	NOUN
iajs-3028	2	15	.	.	PUNCT
iajs-3028	3	1	abstract	abstract	ADJ
iajs-3028	3	2	.	.	PUNCT
iajs-3028	4	1	a	a	DET
iajs-3028	4	2	class	class	NOUN
iajs-3028	4	3	of	of	ADP
iajs-3028	4	4	hyperrings	hyperring	NOUN
iajs-3028	4	5	known	know	VERB
iajs-3028	4	6	as	as	ADP
iajs-3028	4	7	divisible	divisible	ADJ
iajs-3028	4	8	hyperrings	hyperring	NOUN
iajs-3028	4	9	will	will	AUX
iajs-3028	4	10	be	be	AUX
iajs-3028	4	11	studied	study	VERB
iajs-3028	4	12	in	in	ADP
iajs-3028	4	13	this	this	DET
iajs-3028	4	14	paper	paper	NOUN
iajs-3028	4	15	.	.	PUNCT
iajs-3028	5	1	it	it	PRON
iajs-3028	5	2	will	will	AUX
iajs-3028	5	3	be	be	AUX
iajs-3028	5	4	presented	present	VERB
iajs-3028	5	5	as	as	SCONJ
iajs-3028	5	6	each	each	DET
iajs-3028	5	7	element	element	NOUN
iajs-3028	5	8	in	in	ADP
iajs-3028	5	9	this	this	DET
iajs-3028	5	10	hyperring	hyperring	NOUN
iajs-3028	5	11	is	be	AUX
iajs-3028	5	12	a	a	DET
iajs-3028	5	13	divisible	divisible	ADJ
iajs-3028	5	14	element	element	NOUN
iajs-3028	5	15	.	.	PUNCT
iajs-3028	6	1	also	also	ADV
iajs-3028	6	2	shows	show	VERB
iajs-3028	6	3	the	the	DET
iajs-3028	6	4	relationship	relationship	NOUN
iajs-3028	6	5	between	between	ADP
iajs-3028	6	6	the	the	DET
iajs-3028	6	7	jacobsen	jacobsen	PROPN
iajs-3028	6	8	radical	radical	PROPN
iajs-3028	6	9	,	,	PUNCT
iajs-3028	6	10	and	and	CCONJ
iajs-3028	6	11	the	the	DET
iajs-3028	6	12	set	set	NOUN
iajs-3028	6	13	of	of	ADP
iajs-3028	6	14	invertible	invertible	ADJ
iajs-3028	6	15	elements	element	NOUN
iajs-3028	6	16	and	and	CCONJ
iajs-3028	6	17	gets	get	VERB
iajs-3028	6	18	some	some	DET
iajs-3028	6	19	results	result	NOUN
iajs-3028	6	20	,	,	PUNCT
iajs-3028	6	21	and	and	CCONJ
iajs-3028	6	22	linked	link	VERB
iajs-3028	6	23	these	these	DET
iajs-3028	6	24	results	result	NOUN
iajs-3028	6	25	with	with	ADP
iajs-3028	6	26	the	the	DET
iajs-3028	6	27	divisible	divisible	ADJ
iajs-3028	6	28	hyperring	hyperring	NOUN
iajs-3028	6	29	.	.	PUNCT
iajs-3028	7	1	after	after	ADP
iajs-3028	7	2	going	go	VERB
iajs-3028	7	3	through	through	ADP
iajs-3028	7	4	the	the	DET
iajs-3028	7	5	concept	concept	NOUN
iajs-3028	7	6	of	of	ADP
iajs-3028	7	7	divisible	divisible	ADJ
iajs-3028	7	8	hypermodule	hypermodule	NOUN
iajs-3028	7	9	that	that	PRON
iajs-3028	7	10	presented	present	VERB
iajs-3028	7	11	2017	2017	NUM
iajs-3028	7	12	,	,	PUNCT
iajs-3028	7	13	later	later	ADV
iajs-3028	7	14	in	in	ADP
iajs-3028	7	15	2022	2022	NUM
iajs-3028	7	16	,	,	PUNCT
iajs-3028	7	17	the	the	DET
iajs-3028	7	18	concept	concept	NOUN
iajs-3028	7	19	of	of	ADP
iajs-3028	7	20	the	the	DET
iajs-3028	7	21	divisible	divisible	ADJ
iajs-3028	7	22	hyperring	hyperring	NOUN
iajs-3028	7	23	will	will	AUX
iajs-3028	7	24	be	be	AUX
iajs-3028	7	25	related	relate	VERB
iajs-3028	7	26	to	to	ADP
iajs-3028	7	27	the	the	DET
iajs-3028	7	28	concept	concept	NOUN
iajs-3028	7	29	of	of	ADP
iajs-3028	7	30	division	division	NOUN
iajs-3028	7	31	hyperring	hyperring	NOUN
iajs-3028	7	32	,	,	PUNCT
iajs-3028	7	33	where	where	SCONJ
iajs-3028	7	34	each	each	DET
iajs-3028	7	35	division	division	NOUN
iajs-3028	7	36	hyperring	hyperring	NOUN
iajs-3028	7	37	is	be	AUX
iajs-3028	7	38	divisible	divisible	ADJ
iajs-3028	7	39	and	and	CCONJ
iajs-3028	7	40	the	the	DET
iajs-3028	7	41	converse	converse	NOUN
iajs-3028	7	42	is	be	AUX
iajs-3028	7	43	achieved	achieve	VERB
iajs-3028	7	44	under	under	ADP
iajs-3028	7	45	conditions	condition	NOUN
iajs-3028	7	46	that	that	PRON
iajs-3028	7	47	will	will	AUX
iajs-3028	7	48	be	be	AUX
iajs-3028	7	49	explained	explain	VERB
iajs-3028	7	50	in	in	ADP
iajs-3028	7	51	the	the	DET
iajs-3028	7	52	theorem	theorem	NOUN
iajs-3028	7	53	3.14	3.14	NUM
iajs-3028	7	54	.	.	PUNCT
iajs-3028	8	1	at	at	ADP
iajs-3028	8	2	the	the	DET
iajs-3028	8	3	end	end	NOUN
iajs-3028	8	4	of	of	ADP
iajs-3028	8	5	this	this	DET
iajs-3028	8	6	paper	paper	NOUN
iajs-3028	8	7	,	,	PUNCT
iajs-3028	8	8	it	it	PRON
iajs-3028	8	9	will	will	AUX
iajs-3028	8	10	be	be	AUX
iajs-3028	8	11	clear	clear	ADJ
iajs-3028	8	12	that	that	SCONJ
iajs-3028	8	13	the	the	DET
iajs-3028	8	14	goal	goal	NOUN
iajs-3028	8	15	of	of	ADP
iajs-3028	8	16	this	this	DET
iajs-3028	8	17	paper	paper	NOUN
iajs-3028	8	18	is	be	AUX
iajs-3028	8	19	to	to	PART
iajs-3028	8	20	study	study	VERB
iajs-3028	8	21	the	the	DET
iajs-3028	8	22	concept	concept	NOUN
iajs-3028	8	23	of	of	ADP
iajs-3028	8	24	divisible	divisible	ADJ
iajs-3028	8	25	hyperring	hyperring	NOUN
iajs-3028	8	26	by	by	ADP
iajs-3028	8	27	giving	give	VERB
iajs-3028	8	28	some	some	DET
iajs-3028	8	29	examples	example	NOUN
iajs-3028	8	30	,	,	PUNCT
iajs-3028	8	31	remarks	remark	VERB
iajs-3028	8	32	,	,	PUNCT
iajs-3028	8	33	and	and	CCONJ
iajs-3028	8	34	results	result	NOUN
iajs-3028	8	35	that	that	PRON
iajs-3028	8	36	are	be	AUX
iajs-3028	8	37	related	relate	VERB
iajs-3028	8	38	to	to	ADP
iajs-3028	8	39	the	the	DET
iajs-3028	8	40	concept	concept	NOUN
iajs-3028	8	41	of	of	ADP
iajs-3028	8	42	divisible	divisible	ADJ
iajs-3028	8	43	hyperrings	hyperring	NOUN
iajs-3028	8	44	.	.	PUNCT
iajs-3028	9	1	keywords	keyword	NOUN
iajs-3028	9	2	:	:	PUNCT
iajs-3028	9	3	divisible	divisible	ADJ
iajs-3028	9	4	hyperring	hyperring	NOUN
iajs-3028	9	5	,	,	PUNCT
iajs-3028	9	6	divisible	divisible	ADJ
iajs-3028	9	7	hypermodule	hypermodule	NOUN
iajs-3028	9	8	,	,	PUNCT
iajs-3028	9	9	division	division	NOUN
iajs-3028	9	10	hyperring	hyperring	NOUN
iajs-3028	9	11	,	,	PUNCT
iajs-3028	9	12	jacobson	jacobson	PROPN
iajs-3028	9	13	radical	radical	PROPN
iajs-3028	9	14	.	.	PUNCT
iajs-3028	10	1	1.introduction	1.introduction	NUM
iajs-3028	10	2	.	.	PUNCT
iajs-3028	11	1	based	base	VERB
iajs-3028	11	2	on	on	ADP
iajs-3028	11	3	the	the	DET
iajs-3028	11	4	concept	concept	NOUN
iajs-3028	11	5	of	of	ADP
iajs-3028	11	6	divisible	divisible	ADJ
iajs-3028	11	7	hypermodule	hypermodule	NOUN
iajs-3028	11	8	that	that	SCONJ
iajs-3028	11	9	sopon	sopon	ADP
iajs-3028	11	10	boriboon	boriboon	NOUN
iajs-3028	11	11	and	and	CCONJ
iajs-3028	11	12	sajee	sajee	NOUN
iajs-3028	11	13	pianskool	pianskool	NOUN
iajs-3028	11	14	introduced	introduce	VERB
iajs-3028	11	15	in	in	ADP
iajs-3028	11	16	their	their	PRON
iajs-3028	11	17	paper	paper	NOUN
iajs-3028	11	18	“	"	PUNCT
iajs-3028	11	19	baer	baer	PROPN
iajs-3028	11	20	hypermodule	hypermodule	PROPN
iajs-3028	11	21	over	over	ADP
iajs-3028	11	22	krasner	krasner	PROPN
iajs-3028	11	23	hyperring	hyperring	NOUN
iajs-3028	11	24	”	"	PUNCT
iajs-3028	12	1	[	[	X
iajs-3028	12	2	1	1	X
iajs-3028	12	3	]	]	PUNCT
iajs-3028	12	4	in	in	ADP
iajs-3028	12	5	2017	2017	NUM
iajs-3028	12	6	,	,	PUNCT
iajs-3028	12	7	and	and	CCONJ
iajs-3028	12	8	later	later	ADV
iajs-3028	12	9	by	by	ADP
iajs-3028	12	10	hashem	hashem	PROPN
iajs-3028	12	11	bordbar	bordbar	PROPN
iajs-3028	12	12	and	and	CCONJ
iajs-3028	12	13	irina	irina	PROPN
iajs-3028	12	14	cristea	cristea	PROPN
iajs-3028	12	15	in	in	ADP
iajs-3028	12	16	2022	2022	NUM
iajs-3028	12	17	in	in	ADP
iajs-3028	12	18	the	the	DET
iajs-3028	12	19	paper	paper	NOUN
iajs-3028	12	20	“	"	PUNCT
iajs-3028	12	21	divisible	divisible	ADJ
iajs-3028	12	22	hypermodule	hypermodule	NOUN
iajs-3028	12	23	”	"	PUNCT
iajs-3028	13	1	[	[	X
iajs-3028	13	2	2	2	NUM
iajs-3028	13	3	]	]	PUNCT
iajs-3028	13	4	.	.	PUNCT
iajs-3028	14	1	this	this	DET
iajs-3028	14	2	paper	paper	NOUN
iajs-3028	14	3	presents	present	VERB
iajs-3028	14	4	a	a	DET
iajs-3028	14	5	study	study	NOUN
iajs-3028	14	6	on	on	ADP
iajs-3028	14	7	the	the	DET
iajs-3028	14	8	concept	concept	NOUN
iajs-3028	14	9	of	of	ADP
iajs-3028	14	10	divisible	divisible	ADJ
iajs-3028	14	11	hyperring	hyperring	NOUN
iajs-3028	14	12	.	.	PUNCT
iajs-3028	15	1	before	before	ADP
iajs-3028	15	2	that	that	PRON
iajs-3028	15	3	,	,	PUNCT
iajs-3028	15	4	would	would	AUX
iajs-3028	15	5	like	like	VERB
iajs-3028	15	6	to	to	PART
iajs-3028	15	7	re	re	VERB
iajs-3028	15	8	-	-	VERB
iajs-3028	15	9	examine	examine	VERB
iajs-3028	15	10	the	the	DET
iajs-3028	15	11	concept	concept	NOUN
iajs-3028	15	12	of	of	ADP
iajs-3028	15	13	hyperstructure	hyperstructure	NOUN
iajs-3028	15	14	that	that	PRON
iajs-3028	15	15	was	be	AUX
iajs-3028	15	16	first	first	ADV
iajs-3028	15	17	introduced	introduce	VERB
iajs-3028	15	18	by	by	ADP
iajs-3028	15	19	the	the	DET
iajs-3028	15	20	french	french	ADJ
iajs-3028	15	21	mathematician	mathematician	NOUN
iajs-3028	15	22	marty	marty	PROPN
iajs-3028	15	23	in	in	ADP
iajs-3028	15	24	1937	1937	NUM
iajs-3028	15	25	in	in	ADP
iajs-3028	15	26	the	the	DET
iajs-3028	15	27	following	follow	VERB
iajs-3028	15	28	form	form	NOUN
iajs-3028	15	29	.	.	PUNCT
iajs-3028	16	1	the	the	DET
iajs-3028	16	2	function	function	NOUN
iajs-3028	16	3	⊚	⊚	VERB
iajs-3028	16	4	:	:	PUNCT
iajs-3028	16	5	𝒢	𝒢	PROPN
iajs-3028	16	6	×	×	NOUN
iajs-3028	16	7	𝒢	𝒢	PROPN
iajs-3028	16	8	→	→	SYM
iajs-3028	16	9	p	p	X
iajs-3028	16	10	*	*	PUNCT
iajs-3028	16	11	(	(	PUNCT
iajs-3028	16	12	𝒢	𝒢	NOUN
iajs-3028	16	13	)	)	PUNCT
iajs-3028	16	14	,	,	PUNCT
iajs-3028	16	15	which	which	PRON
iajs-3028	16	16	is	be	AUX
iajs-3028	16	17	defined	define	VERB
iajs-3028	16	18	as	as	ADP
iajs-3028	16	19	⊚	⊚	NOUN
iajs-3028	16	20	(	(	PUNCT
iajs-3028	16	21	𝓅,𝒹)=𝓅⊚𝒹	𝓅,𝒹)=𝓅⊚𝒹	NOUN
iajs-3028	16	22	,	,	PUNCT
iajs-3028	16	23	called	call	VERB
iajs-3028	16	24	“	"	PUNCT
iajs-3028	16	25	hyperoperation	hyperoperation	NOUN
iajs-3028	16	26	”	"	PUNCT
iajs-3028	17	1	[	[	X
iajs-3028	17	2	3	3	NUM
iajs-3028	17	3	]	]	PUNCT
iajs-3028	17	4	,	,	PUNCT
iajs-3028	17	5	where	where	SCONJ
iajs-3028	17	6	p*(𝒢	p*(𝒢	PROPN
iajs-3028	17	7	)	)	PUNCT
iajs-3028	17	8	is	be	AUX
iajs-3028	17	9	the	the	DET
iajs-3028	17	10	set	set	NOUN
iajs-3028	17	11	of	of	ADP
iajs-3028	17	12	all	all	DET
iajs-3028	17	13	not	not	PART
iajs-3028	17	14	empty	empty	ADJ
iajs-3028	17	15	subsets	subset	NOUN
iajs-3028	17	16	of	of	ADP
iajs-3028	17	17	𝒢.	𝒢.	PROPN
iajs-3028	17	18	an	an	DET
iajs-3028	17	19	algebraic	algebraic	ADJ
iajs-3028	17	20	hyperstructure	hyperstructure	NOUN
iajs-3028	17	21	(	(	PUNCT
iajs-3028	17	22	𝒢,⊚	𝒢,⊚	NUM
iajs-3028	17	23	)	)	PUNCT
iajs-3028	17	24	is	be	AUX
iajs-3028	17	25	referred	refer	VERB
iajs-3028	17	26	to	to	ADP
iajs-3028	17	27	as	as	ADP
iajs-3028	17	28	a	a	DET
iajs-3028	17	29	“	"	PUNCT
iajs-3028	17	30	hypergroupoid	hypergroupoid	NOUN
iajs-3028	17	31	”	"	PUNCT
iajs-3028	18	1	[	[	X
iajs-3028	18	2	3	3	NUM
iajs-3028	18	3	]	]	PUNCT
iajs-3028	18	4	.	.	PUNCT
iajs-3028	19	1	this	this	DET
iajs-3028	19	2	hypergroupaid	hypergroupaid	PROPN
iajs-3028	19	3	is	be	AUX
iajs-3028	19	4	said	say	VERB
iajs-3028	19	5	to	to	PART
iajs-3028	19	6	be	be	AUX
iajs-3028	19	7	“	"	PUNCT
iajs-3028	19	8	semihypergroup	semihypergroup	NOUN
iajs-3028	19	9	”	"	PUNCT
iajs-3028	19	10	if	if	SCONJ
iajs-3028	19	11	;	;	PUNCT
iajs-3028	19	12	𝓅⊚	𝓅⊚	X
iajs-3028	19	13	(	(	PUNCT
iajs-3028	19	14	𝒹	𝒹	NOUN
iajs-3028	19	15	⊚𝓆	⊚𝓆	NOUN
iajs-3028	19	16	)	)	PUNCT
iajs-3028	20	1	=	=	NOUN
iajs-3028	20	2	(	(	PUNCT
iajs-3028	20	3	𝓅	𝓅	PROPN
iajs-3028	20	4	⊚𝒹)⊚	⊚𝒹)⊚	NUM
iajs-3028	20	5	𝓆	𝓆	NOUN
iajs-3028	20	6	,	,	PUNCT
iajs-3028	20	7	for	for	ADP
iajs-3028	20	8	𝑎𝑙𝑙	𝑎𝑙𝑙	NOUN
iajs-3028	20	9	𝓅	𝓅	PROPN
iajs-3028	20	10	,	,	PUNCT
iajs-3028	20	11	𝒹	𝒹	PROPN
iajs-3028	20	12	,	,	PUNCT
iajs-3028	20	13	𝓆	𝓆	PROPN
iajs-3028	20	14	∈	∈	PROPN
iajs-3028	20	15	𝒢	𝒢	PROPN
iajs-3028	20	16	,	,	PUNCT
iajs-3028	20	17	i	i	PRON
iajs-3028	20	18	,	,	PUNCT
iajs-3028	20	19	e	e	NOUN
iajs-3028	20	20	:	:	PUNCT
iajs-3028	20	21	𝑈𝑢∈𝒹⊚𝓆	𝑈𝑢∈𝒹⊚𝓆	PROPN
iajs-3028	20	22	𝓅⊚	𝓅⊚	NOUN
iajs-3028	20	23	𝑢	𝑢	PART
iajs-3028	21	1	=	=	PROPN
iajs-3028	21	2	𝑈𝑣∈𝓅⊚𝒹	𝑈𝑣∈𝓅⊚𝒹	NOUN
iajs-3028	21	3	𝑣	𝑣	ADP
iajs-3028	21	4	⊚	⊚	PRON
iajs-3028	21	5	𝓆.	𝓆.	NOUN
iajs-3028	21	6	also	also	ADV
iajs-3028	21	7	,	,	PUNCT
iajs-3028	21	8	for	for	ADP
iajs-3028	21	9	any	any	DET
iajs-3028	21	10	ℰ	ℰ	PROPN
iajs-3028	21	11	≠	≠	PROPN
iajs-3028	21	12	∅	∅	NOUN
iajs-3028	21	13	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-3028	21	14	𝒞	𝒞	PROPN
iajs-3028	21	15	≠	≠	PROPN
iajs-3028	21	16	∅	∅	NOUN
iajs-3028	21	17	subsets	subset	NOUN
iajs-3028	21	18	of	of	ADP
iajs-3028	21	19	𝒢	𝒢	PROPN
iajs-3028	21	20	and	and	CCONJ
iajs-3028	21	21	𝓅	𝓅	PROPN
iajs-3028	21	22	∈	∈	PROPN
iajs-3028	21	23	𝒢	𝒢	PROPN
iajs-3028	21	24	,	,	PUNCT
iajs-3028	21	25	defined	define	VERB
iajs-3028	21	26	ℰ	ℰ	NOUN
iajs-3028	21	27	⊚	⊚	VERB
iajs-3028	21	28	𝒞	𝒞	PROPN
iajs-3028	21	29	=	=	SYM
iajs-3028	21	30	𝑈ℯ∈ℰ.𝒸∈𝒞	𝑈ℯ∈ℰ.𝒸∈𝒞	NUM
iajs-3028	21	31	ℯ	ℯ	PROPN
iajs-3028	21	32	⊚	⊚	NOUN
iajs-3028	21	33	𝒸	𝒸	X
iajs-3028	21	34	,	,	PUNCT
iajs-3028	21	35	ℰ	ℰ	PRON
iajs-3028	21	36	⊚	⊚	VERB
iajs-3028	21	37	𝓅	𝓅	NOUN
iajs-3028	21	38	=	=	SYM
iajs-3028	21	39	ℰ	ℰ	PROPN
iajs-3028	21	40	⊚	⊚	NOUN
iajs-3028	21	41	{	{	PUNCT
iajs-3028	21	42	𝓅	𝓅	NOUN
iajs-3028	21	43	}	}	PUNCT
iajs-3028	21	44	and	and	CCONJ
iajs-3028	21	45	𝓅	𝓅	NOUN
iajs-3028	21	46	⊚	⊚	NOUN
iajs-3028	22	1	𝒞	𝒞	PROPN
iajs-3028	22	2	=	=	SYM
iajs-3028	22	3	{	{	PUNCT
iajs-3028	22	4	𝓅	𝓅	NOUN
iajs-3028	22	5	}	}	PUNCT
iajs-3028	22	6	⊚	⊚	NOUN
iajs-3028	22	7	𝒞.	𝒞.	PROPN
iajs-3028	23	1	[	[	X
iajs-3028	23	2	3	3	NUM
iajs-3028	23	3	]	]	PUNCT
iajs-3028	23	4	,	,	PUNCT
iajs-3028	23	5	and	and	CCONJ
iajs-3028	23	6	(	(	PUNCT
iajs-3028	23	7	𝒢,⊚	𝒢,⊚	NUM
iajs-3028	23	8	)	)	PUNCT
iajs-3028	23	9	is	be	AUX
iajs-3028	23	10	called	call	VERB
iajs-3028	23	11	“	"	PUNCT
iajs-3028	23	12	quasihypergroup	quasihypergroup	NOUN
iajs-3028	23	13	”	"	PUNCT
iajs-3028	23	14	,	,	PUNCT
iajs-3028	23	15	if	if	SCONJ
iajs-3028	23	16	𝒢	𝒢	PROPN
iajs-3028	23	17	⊚	⊚	VERB
iajs-3028	23	18	𝑥	𝑥	NOUN
iajs-3028	23	19	=	=	SYM
iajs-3028	23	20	𝑥	𝑥	NOUN
iajs-3028	23	21	⊚	⊚	NOUN
iajs-3028	23	22	𝒢	𝒢	PROPN
iajs-3028	23	23	=	=	SYM
iajs-3028	23	24	𝒢	𝒢	PROPN
iajs-3028	23	25	,	,	PUNCT
iajs-3028	23	26	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3028	23	27	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-3028	23	28	𝑥	𝑥	PRON
iajs-3028	23	29	∈	∈	PROPN
iajs-3028	23	30	𝒢	𝒢	PROPN
iajs-3028	23	31	,	,	PUNCT
iajs-3028	23	32	[	[	X
iajs-3028	23	33	1	1	NUM
iajs-3028	23	34	]	]	PUNCT
iajs-3028	23	35	.	.	PUNCT
iajs-3028	24	1	if	if	SCONJ
iajs-3028	24	2	the	the	DET
iajs-3028	24	3	pair	pair	NOUN
iajs-3028	24	4	(	(	PUNCT
iajs-3028	24	5	𝒢,⊚	𝒢,⊚	NUM
iajs-3028	24	6	)	)	PUNCT
iajs-3028	24	7	satisfied	satisfy	VERB
iajs-3028	24	8	the	the	DET
iajs-3028	24	9	conditions	condition	NOUN
iajs-3028	24	10	of	of	ADP
iajs-3028	24	11	the	the	DET
iajs-3028	24	12	semihypergroup	semihypergroup	NOUN
iajs-3028	24	13	and	and	CCONJ
iajs-3028	24	14	the	the	DET
iajs-3028	24	15	quasihypergroup	quasihypergroup	NOUN
iajs-3028	24	16	,	,	PUNCT
iajs-3028	24	17	then	then	ADV
iajs-3028	24	18	called	call	VERB
iajs-3028	24	19	“	"	PUNCT
iajs-3028	24	20	hypergroup	hypergroup	NOUN
iajs-3028	24	21	”	"	PUNCT
iajs-3028	25	1	[	[	X
iajs-3028	25	2	3	3	NUM
iajs-3028	25	3	]	]	PUNCT
iajs-3028	25	4	.	.	PUNCT
iajs-3028	26	1	a	a	DET
iajs-3028	26	2	set	set	ADJ
iajs-3028	26	3	doi.org/10.30526/36.2.3028	doi.org/10.30526/36.2.3028	ADJ
iajs-3028	26	4	article	article	NOUN
iajs-3028	26	5	history	history	NOUN
iajs-3028	26	6	:	:	PUNCT
iajs-3028	26	7	received	receive	VERB
iajs-3028	26	8	21	21	NUM
iajs-3028	26	9	september	september	PROPN
iajs-3028	26	10	2022	2022	NUM
iajs-3028	26	11	,	,	PUNCT
iajs-3028	26	12	accepted	accept	VERB
iajs-3028	26	13	21	21	NUM
iajs-3028	26	14	november	november	PROPN
iajs-3028	26	15	2022	2022	NUM
iajs-3028	26	16	,	,	PUNCT
iajs-3028	26	17	published	publish	VERB
iajs-3028	26	18	in	in	ADP
iajs-3028	26	19	april	april	PROPN
iajs-3028	26	20	2023	2023	NUM
iajs-3028	26	21	.	.	PUNCT
iajs-3028	27	1	ibn	ibn	PROPN
iajs-3028	27	2	al	al	PROPN
iajs-3028	27	3	-	-	PUNCT
iajs-3028	27	4	haitham	haitham	PROPN
iajs-3028	27	5	journal	journal	PROPN
iajs-3028	27	6	for	for	ADP
iajs-3028	27	7	pure	pure	ADJ
iajs-3028	27	8	and	and	CCONJ
iajs-3028	27	9	applied	applied	ADJ
iajs-3028	27	10	sciences	sciences	PROPN
iajs-3028	27	11	journal	journal	PROPN
iajs-3028	27	12	homepage	homepage	NOUN
iajs-3028	27	13	:	:	PUNCT
iajs-3028	27	14	jih.uobaghdad.edu.iq	jih.uobaghdad.edu.iq	VERB
iajs-3028	27	15	some	some	DET
iajs-3028	27	16	results	result	NOUN
iajs-3028	27	17	on	on	ADP
iajs-3028	27	18	the	the	DET
iajs-3028	27	19	divisible	divisible	ADJ
iajs-3028	27	20	hyperrings	hyperring	NOUN
iajs-3028	27	21	mayssam	mayssam	PROPN
iajs-3028	27	22	fadel	fadel	PROPN
iajs-3028	27	23	abood	abood	PROPN
iajs-3028	27	24	department	department	PROPN
iajs-3028	27	25	of	of	ADP
iajs-3028	27	26	mathematics	mathematics	PROPN
iajs-3028	27	27	,	,	PUNCT
iajs-3028	27	28	college	college	NOUN
iajs-3028	27	29	of	of	ADP
iajs-3028	27	30	science	science	NOUN
iajs-3028	27	31	for	for	ADP
iajs-3028	27	32	women	woman	NOUN
iajs-3028	27	33	,	,	PUNCT
iajs-3028	27	34	university	university	NOUN
iajs-3028	27	35	of	of	ADP
iajs-3028	27	36	baghdad	baghdad	PROPN
iajs-3028	27	37	,	,	PUNCT
iajs-3028	27	38	baghdad	baghdad	PROPN
iajs-3028	27	39	,	,	PUNCT
iajs-3028	27	40	iraq	iraq	PROPN
iajs-3028	27	41	.	.	PUNCT
iajs-3028	28	1	maisssam.fadel1203a@csw.uobaghdad.edu.iq	maisssam.fadel1203a@csw.uobaghdad.edu.iq	ADJ
iajs-3028	28	2	tamadher	tamadher	PROPN
iajs-3028	28	3	arif	arif	PROPN
iajs-3028	28	4	ibrahiem	ibrahiem	PROPN
iajs-3028	28	5	department	department	PROPN
iajs-3028	28	6	of	of	ADP
iajs-3028	28	7	mathematics	mathematics	PROPN
iajs-3028	28	8	,	,	PUNCT
iajs-3028	28	9	college	college	NOUN
iajs-3028	28	10	of	of	ADP
iajs-3028	28	11	science	science	NOUN
iajs-3028	28	12	for	for	ADP
iajs-3028	28	13	women	woman	NOUN
iajs-3028	28	14	,	,	PUNCT
iajs-3028	28	15	university	university	NOUN
iajs-3028	28	16	of	of	ADP
iajs-3028	28	17	baghdad	baghdad	PROPN
iajs-3028	28	18	,	,	PUNCT
iajs-3028	28	19	baghdad	baghdad	PROPN
iajs-3028	28	20	,	,	PUNCT
iajs-3028	28	21	iraq	iraq	PROPN
iajs-3028	28	22	.	.	PUNCT
iajs-3028	29	1	tamadherai_math@csw.uobaghdad.edu.iq	tamadherai_math@csw.uobaghdad.edu.iq	PROPN
iajs-3028	29	2	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
iajs-3028	29	3	mailto:maisssam.fadel1203a@csw.uobaghdad.edu.iq	mailto:maisssam.fadel1203a@csw.uobaghdad.edu.iq	PROPN
iajs-3028	29	4	mailto:tamadherai_math@csw.uobaghdad.edu.iq	mailto:tamadherai_math@csw.uobaghdad.edu.iq	ADJ
iajs-3028	29	5	ihjpas	ihjpa	NOUN
iajs-3028	29	6	.	.	PUNCT
iajs-3028	30	1	36(2)2023	36(2)2023	NUM
iajs-3028	30	2	384	384	NUM
iajs-3028	30	3	∅	∅	NOUN
iajs-3028	30	4	≠	≠	PROPN
iajs-3028	30	5	𝒬	𝒬	PROPN
iajs-3028	30	6	that	that	PRON
iajs-3028	30	7	contained	contain	VERB
iajs-3028	30	8	in	in	ADP
iajs-3028	30	9	(	(	PUNCT
iajs-3028	30	10	𝒢,⊚	𝒢,⊚	NOUN
iajs-3028	30	11	)	)	PUNCT
iajs-3028	30	12	is	be	AUX
iajs-3028	30	13	called	call	VERB
iajs-3028	30	14	“	"	PUNCT
iajs-3028	30	15	subhypergroup	subhypergroup	NOUN
iajs-3028	30	16	”	"	PUNCT
iajs-3028	30	17	if	if	SCONJ
iajs-3028	30	18	it	it	PRON
iajs-3028	30	19	was	be	AUX
iajs-3028	30	20	hypergroup	hypergroup	ADJ
iajs-3028	30	21	it	it	PRON
iajs-3028	30	22	-	-	PUNCT
iajs-3028	30	23	self	self	NOUN
iajs-3028	30	24	[	[	X
iajs-3028	30	25	3	3	NUM
iajs-3028	30	26	]	]	PUNCT
iajs-3028	30	27	.	.	PUNCT
iajs-3028	31	1	in	in	ADP
iajs-3028	31	2	1956	1956	NUM
iajs-3028	31	3	krasner	krasner	NOUN
iajs-3028	31	4	introduced	introduce	VERB
iajs-3028	31	5	the	the	DET
iajs-3028	31	6	concept	concept	NOUN
iajs-3028	31	7	of	of	ADP
iajs-3028	31	8	hyperring	hyperring	NOUN
iajs-3028	31	9	and	and	CCONJ
iajs-3028	31	10	hypermodule	hypermodule	NOUN
iajs-3028	31	11	which	which	PRON
iajs-3028	31	12	are	be	AUX
iajs-3028	31	13	known	know	VERB
iajs-3028	31	14	nowadays	nowadays	ADV
iajs-3028	31	15	as	as	ADP
iajs-3028	31	16	krasner	krasner	NOUN
iajs-3028	31	17	hyperring	hyperring	NOUN
iajs-3028	31	18	and	and	CCONJ
iajs-3028	31	19	krasner	krasner	PROPN
iajs-3028	31	20	hypermodule	hypermodule	PROPN
iajs-3028	31	21	respectively	respectively	ADV
iajs-3028	31	22	.	.	PUNCT
iajs-3028	32	1	after	after	SCONJ
iajs-3028	32	2	that	that	DET
iajs-3028	32	3	many	many	ADJ
iajs-3028	32	4	authors	author	NOUN
iajs-3028	32	5	introduced	introduce	VERB
iajs-3028	32	6	many	many	ADJ
iajs-3028	32	7	types	type	NOUN
iajs-3028	32	8	of	of	ADP
iajs-3028	32	9	hyperring	hyperre	VERB
iajs-3028	32	10	like	like	ADP
iajs-3028	32	11	general	general	ADJ
iajs-3028	32	12	hyperring	hyperring	NOUN
iajs-3028	32	13	and	and	CCONJ
iajs-3028	32	14	multiplicative	multiplicative	ADJ
iajs-3028	32	15	hyperring	hyperring	NOUN
iajs-3028	32	16	.	.	PUNCT
iajs-3028	33	1	in	in	ADP
iajs-3028	33	2	this	this	DET
iajs-3028	33	3	paper	paper	NOUN
iajs-3028	33	4	,	,	PUNCT
iajs-3028	33	5	the	the	DET
iajs-3028	33	6	hyperring	hyperre	VERB
iajs-3028	33	7	ℛ	ℛ	PROPN
iajs-3028	33	8	is	be	AUX
iajs-3028	33	9	a	a	DET
iajs-3028	33	10	krasner	krasner	NOUN
iajs-3028	33	11	hyperring	hyperre	VERB
iajs-3028	33	12	with	with	ADP
iajs-3028	33	13	unit	unit	NOUN
iajs-3028	33	14	element	element	NOUN
iajs-3028	33	15	1	1	NUM
iajs-3028	33	16	.	.	PUNCT
iajs-3028	34	1	𝑅𝑜	𝑅𝑜	PROPN
iajs-3028	34	2	is	be	AUX
iajs-3028	34	3	the	the	DET
iajs-3028	34	4	set	set	NOUN
iajs-3028	34	5	of	of	ADP
iajs-3028	34	6	all	all	DET
iajs-3028	34	7	non	non	ADJ
iajs-3028	34	8	-	-	ADJ
iajs-3028	34	9	zero	zero	NUM
iajs-3028	34	10	divisor	divisor	NOUN
iajs-3028	34	11	elements	element	NOUN
iajs-3028	34	12	of	of	ADP
iajs-3028	34	13	a	a	DET
iajs-3028	34	14	hyperring	hyperre	VERB
iajs-3028	34	15	ℛ	ℛ	NOUN
iajs-3028	34	16	,	,	PUNCT
iajs-3028	34	17	and	and	CCONJ
iajs-3028	34	18	a	a	DET
iajs-3028	34	19	left	left	ADJ
iajs-3028	34	20	ℛ-hypermodule	ℛ-hypermodule	PROPN
iajs-3028	34	21	will	will	AUX
iajs-3028	34	22	be	be	AUX
iajs-3028	34	23	denoted	denote	VERB
iajs-3028	34	24	by	by	ADP
iajs-3028	34	25	ℳ.	ℳ.	PROPN
iajs-3028	34	26	2	2	NUM
iajs-3028	34	27	.	.	PUNCT
iajs-3028	34	28	preliminaries	preliminary	NOUN
iajs-3028	34	29	.	.	PUNCT
iajs-3028	35	1	definition	definition	NOUN
iajs-3028	35	2	2.1	2.1	NUM
iajs-3028	36	1	[	[	X
iajs-3028	36	2	4	4	NUM
iajs-3028	36	3	]	]	PUNCT
iajs-3028	36	4	.	.	PUNCT
iajs-3028	37	1	the	the	DET
iajs-3028	37	2	hypergroup	hypergroup	NOUN
iajs-3028	37	3	(	(	PUNCT
iajs-3028	37	4	𝒢,⊚	𝒢,⊚	NUM
iajs-3028	37	5	)	)	PUNCT
iajs-3028	37	6	is	be	AUX
iajs-3028	37	7	called	call	VERB
iajs-3028	37	8	canonical	canonical	ADJ
iajs-3028	37	9	if	if	SCONJ
iajs-3028	37	10	;	;	PUNCT
iajs-3028	37	11	1	1	X
iajs-3028	37	12	.	.	NOUN
iajs-3028	37	13	⊚	⊚	PROPN
iajs-3028	37	14	is	be	AUX
iajs-3028	37	15	an	an	DET
iajs-3028	37	16	associative	associative	ADJ
iajs-3028	37	17	hyperoperation	hyperoperation	NOUN
iajs-3028	37	18	,	,	PUNCT
iajs-3028	37	19	i	i	PRON
iajs-3028	37	20	,	,	PUNCT
iajs-3028	37	21	e	e	PROPN
iajs-3028	37	22	𝓅⊚	𝓅⊚	PROPN
iajs-3028	37	23	(	(	PUNCT
iajs-3028	37	24	𝒹	𝒹	PROPN
iajs-3028	37	25	⊚	⊚	NOUN
iajs-3028	37	26	𝓆	𝓆	NOUN
iajs-3028	37	27	)	)	PUNCT
iajs-3028	38	1	=	=	SYM
iajs-3028	38	2	(	(	PUNCT
iajs-3028	38	3	𝓅⊚	𝓅⊚	X
iajs-3028	38	4	𝒹)⊚	𝒹)⊚	NOUN
iajs-3028	38	5	𝓆	𝓆	PROPN
iajs-3028	38	6	for	for	ADP
iajs-3028	38	7	every	every	DET
iajs-3028	38	8	𝓅,𝒹	𝓅,𝒹	NOUN
iajs-3028	38	9	,	,	PUNCT
iajs-3028	38	10	𝓆	𝓆	PROPN
iajs-3028	38	11	∈	∈	PROPN
iajs-3028	38	12	𝒢	𝒢	PROPN
iajs-3028	38	13	;	;	PUNCT
iajs-3028	38	14	2	2	X
iajs-3028	38	15	.	.	X
iajs-3028	38	16	there	there	PRON
iajs-3028	38	17	exist	exist	VERB
iajs-3028	38	18	an	an	DET
iajs-3028	38	19	element“0	element“0	ADJ
iajs-3028	38	20	”	"	PUNCT
iajs-3028	38	21	∈	∈	PROPN
iajs-3028	38	22	𝒢	𝒢	PROPN
iajs-3028	38	23	,	,	PUNCT
iajs-3028	38	24	such	such	ADJ
iajs-3028	38	25	that	that	DET
iajs-3028	38	26	0	0	NUM
iajs-3028	38	27	⊚	⊚	NOUN
iajs-3028	38	28	𝓅	𝓅	NOUN
iajs-3028	38	29	=	=	PUNCT
iajs-3028	38	30	𝓅⊚	𝓅⊚	NOUN
iajs-3028	38	31	0	0	NUM
iajs-3028	39	1	=	=	PUNCT
iajs-3028	39	2	{	{	PUNCT
iajs-3028	39	3	𝓅	𝓅	NOUN
iajs-3028	39	4	}	}	PUNCT
iajs-3028	39	5	.	.	PUNCT
iajs-3028	39	6	∀	∀	PUNCT
iajs-3028	40	1	𝓅	𝓅	ADP
iajs-3028	40	2	∈	∈	PROPN
iajs-3028	40	3	𝒢	𝒢	PROPN
iajs-3028	40	4	;	;	PUNCT
iajs-3028	40	5	3	3	X
iajs-3028	40	6	.	.	X
iajs-3028	40	7	there	there	PRON
iajs-3028	40	8	exist	exist	VERB
iajs-3028	40	9	a	a	DET
iajs-3028	40	10	unique	unique	ADJ
iajs-3028	40	11	−𝓅	−𝓅	ADJ
iajs-3028	40	12	∈	∈	PROPN
iajs-3028	40	13	𝒢	𝒢	PROPN
iajs-3028	40	14	,	,	PUNCT
iajs-3028	40	15	∀	∀	X
iajs-3028	40	16	𝓅	𝓅	ADP
iajs-3028	40	17	∈	∈	PROPN
iajs-3028	40	18	𝒢	𝒢	PROPN
iajs-3028	40	19	,	,	PUNCT
iajs-3028	40	20	such	such	ADJ
iajs-3028	40	21	that	that	SCONJ
iajs-3028	40	22	0	0	NUM
iajs-3028	40	23	∈	∈	PROPN
iajs-3028	40	24	𝓅⊚	𝓅⊚	NOUN
iajs-3028	40	25	(	(	PUNCT
iajs-3028	40	26	−𝓅	−𝓅	NOUN
iajs-3028	40	27	)	)	PUNCT
iajs-3028	40	28	;	;	PUNCT
iajs-3028	40	29	4	4	X
iajs-3028	40	30	.	.	X
iajs-3028	40	31	𝓅	𝓅	NOUN
iajs-3028	40	32	∈	∈	PROPN
iajs-3028	40	33	𝒹	𝒹	PROPN
iajs-3028	40	34	⊚𝓆	⊚𝓆	NOUN
iajs-3028	40	35	implies	imply	VERB
iajs-3028	40	36	𝒹	𝒹	PROPN
iajs-3028	40	37	∈	∈	PROPN
iajs-3028	40	38	𝓅⊚	𝓅⊚	X
iajs-3028	41	1	(	(	PUNCT
iajs-3028	41	2	−𝓆	−𝓆	NOUN
iajs-3028	41	3	)	)	PUNCT
iajs-3028	41	4	.	.	PUNCT
iajs-3028	42	1	5	5	X
iajs-3028	42	2	.	.	X
iajs-3028	42	3	for	for	ADP
iajs-3028	42	4	all	all	DET
iajs-3028	42	5	𝓅	𝓅	PROPN
iajs-3028	42	6	,	,	PUNCT
iajs-3028	42	7	𝒹	𝒹	PROPN
iajs-3028	42	8	∈	∈	PROPN
iajs-3028	42	9	𝒢	𝒢	PROPN
iajs-3028	42	10	,	,	PUNCT
iajs-3028	42	11	𝓅⊚𝒹	𝓅⊚𝒹	NOUN
iajs-3028	42	12	=	=	SYM
iajs-3028	42	13	𝒹⊚𝓅.	𝒹⊚𝓅.	NOUN
iajs-3028	42	14	definition	definition	NOUN
iajs-3028	42	15	2.2	2.2	NUM
iajs-3028	42	16	[	[	SYM
iajs-3028	42	17	4	4	NUM
iajs-3028	42	18	]	]	PUNCT
iajs-3028	42	19	.	.	PUNCT
iajs-3028	43	1	the	the	DET
iajs-3028	43	2	hyperstructure	hyperstructure	NOUN
iajs-3028	43	3	(	(	PUNCT
iajs-3028	43	4	ℛ⊚,⊙	ℛ⊚,⊙	PROPN
iajs-3028	43	5	)	)	PUNCT
iajs-3028	43	6	is	be	AUX
iajs-3028	43	7	said	say	VERB
iajs-3028	43	8	to	to	PART
iajs-3028	43	9	be	be	AUX
iajs-3028	43	10	krasner	krasner	NOUN
iajs-3028	43	11	hyperring	hyperring	NOUN
iajs-3028	43	12	,	,	PUNCT
iajs-3028	43	13	if	if	SCONJ
iajs-3028	43	14	:	:	PUNCT
iajs-3028	43	15	1	1	X
iajs-3028	43	16	.	.	PUNCT
iajs-3028	43	17	(	(	PUNCT
iajs-3028	43	18	ℛ,⊚	ℛ,⊚	NUM
iajs-3028	43	19	)	)	PUNCT
iajs-3028	43	20	is	be	AUX
iajs-3028	43	21	a	a	DET
iajs-3028	43	22	canonical	canonical	ADJ
iajs-3028	43	23	hypergroup	hypergroup	NOUN
iajs-3028	43	24	;	;	PUNCT
iajs-3028	43	25	2	2	X
iajs-3028	43	26	.	.	PUNCT
iajs-3028	43	27	(	(	PUNCT
iajs-3028	43	28	ℛ,⊙	ℛ,⊙	NUM
iajs-3028	43	29	)	)	PUNCT
iajs-3028	43	30	is	be	AUX
iajs-3028	43	31	a	a	DET
iajs-3028	43	32	semigroup	semigroup	NOUN
iajs-3028	43	33	,	,	PUNCT
iajs-3028	43	34	have	have	VERB
iajs-3028	43	35	𝛼⨀0	𝛼⨀0	NUM
iajs-3028	43	36	=	=	SYM
iajs-3028	43	37	0⨀𝛼	0⨀𝛼	NOUN
iajs-3028	43	38	=	=	SYM
iajs-3028	43	39	0	0	NUM
iajs-3028	43	40	,	,	PUNCT
iajs-3028	43	41	for	for	ADP
iajs-3028	43	42	all	all	DET
iajs-3028	43	43	𝛼	𝛼	PRON
iajs-3028	43	44	∈	∈	NOUN
iajs-3028	43	45	ℛ	ℛ	PROPN
iajs-3028	43	46	3	3	NUM
iajs-3028	43	47	.	.	PUNCT
iajs-3028	43	48	𝛼⨀(𝛽	𝛼⨀(𝛽	PROPN
iajs-3028	43	49	⊚	⊚	PROPN
iajs-3028	43	50	𝛾	𝛾	NOUN
iajs-3028	43	51	)	)	PUNCT
iajs-3028	43	52	=	=	SYM
iajs-3028	43	53	𝛼⨀𝛽	𝛼⨀𝛽	NOUN
iajs-3028	43	54	⊚	⊚	PROPN
iajs-3028	43	55	𝛼⨀𝛾	𝛼⨀𝛾	NOUN
iajs-3028	43	56	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-3028	43	57	(	(	PUNCT
iajs-3028	43	58	𝛽	𝛽	NOUN
iajs-3028	43	59	⊚	⊚	NOUN
iajs-3028	43	60	𝛾)⨀𝛼1	𝛾)⨀𝛼1	NOUN
iajs-3028	43	61	=	=	PUNCT
iajs-3028	43	62	𝛽⨀𝛼⊚	𝛽⨀𝛼⊚	ADV
iajs-3028	43	63	𝛾⨀𝛼.	𝛾⨀𝛼.	NOUN
iajs-3028	43	64	for	for	ADP
iajs-3028	43	65	all	all	DET
iajs-3028	43	66	𝛼	𝛼	PROPN
iajs-3028	43	67	,	,	PUNCT
iajs-3028	43	68	𝛽	𝛽	PROPN
iajs-3028	43	69	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-3028	43	70	𝛾	𝛾	PROPN
iajs-3028	43	71	∈	∈	ADP
iajs-3028	43	72	ℛ.	ℛ.	PROPN
iajs-3028	43	73	a	a	DET
iajs-3028	43	74	krasner	krasner	NOUN
iajs-3028	43	75	hyperring	hyperring	NOUN
iajs-3028	43	76	is	be	AUX
iajs-3028	43	77	commutative	commutative	ADJ
iajs-3028	43	78	if	if	SCONJ
iajs-3028	43	79	(	(	PUNCT
iajs-3028	43	80	ℛ,⨀	ℛ,⨀	PROPN
iajs-3028	43	81	)	)	PUNCT
iajs-3028	43	82	is	be	AUX
iajs-3028	43	83	a	a	DET
iajs-3028	43	84	commutative	commutative	ADJ
iajs-3028	43	85	semigroup	semigroup	NOUN
iajs-3028	43	86	.	.	PUNCT
iajs-3028	44	1	definition	definition	NOUN
iajs-3028	44	2	2.3	2.3	NUM
iajs-3028	45	1	[	[	X
iajs-3028	45	2	5	5	NUM
iajs-3028	45	3	]	]	PUNCT
iajs-3028	45	4	.	.	PUNCT
iajs-3028	46	1	a	a	DET
iajs-3028	46	2	subset	subset	ADJ
iajs-3028	46	3	𝒜	𝒜	NOUN
iajs-3028	46	4	of	of	ADP
iajs-3028	46	5	ℛ	ℛ	PROPN
iajs-3028	46	6	is	be	AUX
iajs-3028	46	7	said	say	VERB
iajs-3028	46	8	to	to	PART
iajs-3028	46	9	be	be	AUX
iajs-3028	46	10	subhyperring	subhyperre	VERB
iajs-3028	46	11	if	if	SCONJ
iajs-3028	46	12	it	it	PRON
iajs-3028	46	13	satisfy	satisfy	VERB
iajs-3028	46	14	the	the	DET
iajs-3028	46	15	conditions	condition	NOUN
iajs-3028	46	16	of	of	ADP
iajs-3028	46	17	a	a	DET
iajs-3028	46	18	hyperring	hyperring	NOUN
iajs-3028	46	19	.	.	PUNCT
iajs-3028	47	1	definition	definition	NOUN
iajs-3028	47	2	2.4	2.4	NUM
iajs-3028	47	3	[	[	X
iajs-3028	47	4	5	5	NUM
iajs-3028	47	5	]	]	PUNCT
iajs-3028	47	6	.	.	PUNCT
iajs-3028	48	1	let	let	AUX
iajs-3028	48	2	(	(	PUNCT
iajs-3028	48	3	ℛ,∔,⋅	ℛ,∔,⋅	PROPN
iajs-3028	48	4	)	)	PUNCT
iajs-3028	48	5	be	be	AUX
iajs-3028	48	6	a	a	DET
iajs-3028	48	7	commutative	commutative	ADJ
iajs-3028	48	8	hyperring	hyperring	NOUN
iajs-3028	48	9	,	,	PUNCT
iajs-3028	48	10	𝐼	𝐼	PRON
iajs-3028	48	11	be	be	VERB
iajs-3028	48	12	a	a	DET
iajs-3028	48	13	hyperideal	hyperideal	NOUN
iajs-3028	48	14	of	of	ADP
iajs-3028	48	15	ℛ	ℛ	NOUN
iajs-3028	48	16	,	,	PUNCT
iajs-3028	48	17	then	then	ADV
iajs-3028	48	18	the	the	DET
iajs-3028	48	19	set	set	ADJ
iajs-3028	48	20	ℛ/𝐼	ℛ/𝐼	NOUN
iajs-3028	48	21	=	=	SYM
iajs-3028	48	22	{	{	PUNCT
iajs-3028	48	23	𝑥	𝑥	X
iajs-3028	48	24	∔	∔	PROPN
iajs-3028	48	25	𝐼	𝐼	PROPN
iajs-3028	48	26	:	:	PUNCT
iajs-3028	48	27	𝑥	𝑥	X
iajs-3028	48	28	∈	∈	PROPN
iajs-3028	48	29	ℛ	ℛ	PROPN
iajs-3028	48	30	}	}	PUNCT
iajs-3028	48	31	is	be	AUX
iajs-3028	48	32	a	a	DET
iajs-3028	48	33	commutative	commutative	ADJ
iajs-3028	48	34	hyperring	hyperring	NOUN
iajs-3028	48	35	under	under	ADP
iajs-3028	48	36	hyperaddition	hyperaddition	NOUN
iajs-3028	48	37	(	(	PUNCT
iajs-3028	48	38	𝑥	𝑥	NOUN
iajs-3028	48	39	∔	∔	PROPN
iajs-3028	48	40	𝐼	𝐼	PROPN
iajs-3028	48	41	)	)	PUNCT
iajs-3028	48	42	∔	∔	PROPN
iajs-3028	48	43	(	(	PUNCT
iajs-3028	48	44	𝑦	𝑦	NOUN
iajs-3028	48	45	∔	∔	PROPN
iajs-3028	48	46	𝐼	𝐼	PROPN
iajs-3028	48	47	)	)	PUNCT
iajs-3028	48	48	=	=	PUNCT
iajs-3028	49	1	(	(	PUNCT
iajs-3028	49	2	𝑥	𝑥	X
iajs-3028	49	3	∔	∔	PROPN
iajs-3028	49	4	𝑦	𝑦	NOUN
iajs-3028	49	5	)	)	PUNCT
iajs-3028	49	6	∔	∔	PROPN
iajs-3028	49	7	𝐼	𝐼	PROPN
iajs-3028	49	8	and	and	CCONJ
iajs-3028	49	9	multiplication	multiplication	NOUN
iajs-3028	49	10	(	(	PUNCT
iajs-3028	49	11	𝑠	𝑠	INTJ
iajs-3028	49	12	∔	∔	PROPN
iajs-3028	49	13	𝐼)(𝑡	𝐼)(𝑡	PROPN
iajs-3028	49	14	∔	∔	PROPN
iajs-3028	49	15	𝐼	𝐼	PROPN
iajs-3028	49	16	)	)	PUNCT
iajs-3028	49	17	=	=	PUNCT
iajs-3028	49	18	𝑠𝑡	𝑠𝑡	PROPN
iajs-3028	49	19	∔	∔	PROPN
iajs-3028	49	20	𝐼.	𝐼.	PROPN
iajs-3028	50	1	and	and	CCONJ
iajs-3028	50	2	it	it	PRON
iajs-3028	50	3	is	be	AUX
iajs-3028	50	4	called	call	VERB
iajs-3028	50	5	a	a	DET
iajs-3028	50	6	quotient	quotient	NOUN
iajs-3028	50	7	hyperring	hyperring	NOUN
iajs-3028	50	8	.	.	PUNCT
iajs-3028	51	1	definition	definition	NOUN
iajs-3028	51	2	2.5	2.5	NUM
iajs-3028	52	1	[	[	X
iajs-3028	52	2	1	1	NUM
iajs-3028	52	3	]	]	PUNCT
iajs-3028	52	4	.	.	PUNCT
iajs-3028	53	1	let	let	VERB
iajs-3028	53	2	i	i	PRON
iajs-3028	53	3	be	be	AUX
iajs-3028	53	4	a	a	DET
iajs-3028	53	5	nonempty	nonempty	ADJ
iajs-3028	53	6	subset	subset	NOUN
iajs-3028	53	7	of	of	ADP
iajs-3028	53	8	a	a	DET
iajs-3028	53	9	krasner	krasner	NOUN
iajs-3028	53	10	hyperring	hyperre	VERB
iajs-3028	53	11	ℛ	ℛ	PROPN
iajs-3028	53	12	,	,	PUNCT
iajs-3028	53	13	then	then	ADV
iajs-3028	53	14	i	i	PRON
iajs-3028	53	15	is	be	AUX
iajs-3028	53	16	called	call	VERB
iajs-3028	53	17	a	a	DET
iajs-3028	53	18	“	"	PUNCT
iajs-3028	53	19	right	right	ADJ
iajs-3028	53	20	(	(	PUNCT
iajs-3028	53	21	resp	resp	NOUN
iajs-3028	53	22	.	.	PUNCT
iajs-3028	54	1	left	left	ADJ
iajs-3028	54	2	)	)	PUNCT
iajs-3028	54	3	hyperideal	hyperideal	NOUN
iajs-3028	54	4	”	"	PUNCT
iajs-3028	54	5	if	if	SCONJ
iajs-3028	54	6	for	for	ADP
iajs-3028	54	7	every	every	DET
iajs-3028	54	8	𝓅	𝓅	NOUN
iajs-3028	54	9	and	and	CCONJ
iajs-3028	54	10	𝒹	𝒹	PRON
iajs-3028	54	11	∈	∈	PROPN
iajs-3028	55	1	i	i	PRON
iajs-3028	55	2	,	,	PUNCT
iajs-3028	55	3	and	and	CCONJ
iajs-3028	55	4	𝛼	𝛼	X
iajs-3028	55	5	∈	∈	PROPN
iajs-3028	55	6	ℛ	ℛ	PROPN
iajs-3028	55	7	:	:	PUNCT
iajs-3028	55	8	(	(	PUNCT
iajs-3028	55	9	1	1	X
iajs-3028	55	10	)	)	PUNCT
iajs-3028	55	11	𝓅	𝓅	NOUN
iajs-3028	55	12	−	−	PROPN
iajs-3028	55	13	𝒹	𝒹	PROPN
iajs-3028	55	14	⊆	⊆	NUM
iajs-3028	55	15	ι	ι	NOUN
iajs-3028	55	16	;	;	PUNCT
iajs-3028	55	17	(	(	PUNCT
iajs-3028	55	18	2	2	X
iajs-3028	55	19	)	)	PUNCT
iajs-3028	55	20	𝛼⨀𝓅	𝛼⨀𝓅	NOUN
iajs-3028	55	21	∈	∈	PROPN
iajs-3028	55	22	ι	ι	PROPN
iajs-3028	55	23	(	(	PUNCT
iajs-3028	55	24	resp	resp	PROPN
iajs-3028	55	25	.	.	PUNCT
iajs-3028	56	1	𝓅⨀𝛼	𝓅⨀𝛼	PROPN
iajs-3028	56	2	∈	∈	PROPN
iajs-3028	57	1	i	i	PROPN
iajs-3028	57	2	)	)	PUNCT
iajs-3028	58	1	and	and	CCONJ
iajs-3028	58	2	called	call	VERB
iajs-3028	58	3	hyperideal	hyperideal	ADV
iajs-3028	58	4	if	if	SCONJ
iajs-3028	58	5	it	it	PRON
iajs-3028	58	6	is	be	AUX
iajs-3028	58	7	right	right	ADJ
iajs-3028	58	8	and	and	CCONJ
iajs-3028	58	9	left	leave	VERB
iajs-3028	58	10	hyperideal	hyperideal	NOUN
iajs-3028	58	11	.	.	PUNCT
iajs-3028	59	1	definition	definition	NOUN
iajs-3028	59	2	2.6	2.6	NUM
iajs-3028	60	1	[	[	X
iajs-3028	60	2	4	4	NUM
iajs-3028	60	3	]	]	PUNCT
iajs-3028	60	4	.	.	PUNCT
iajs-3028	61	1	the	the	DET
iajs-3028	61	2	hyperideal	hyperideal	NOUN
iajs-3028	61	3	i	i	PRON
iajs-3028	61	4	of	of	ADP
iajs-3028	61	5	ℛ	ℛ	PROPN
iajs-3028	61	6	is	be	AUX
iajs-3028	61	7	maximal	maximal	ADJ
iajs-3028	61	8	hyperideal	hyperideal	NOUN
iajs-3028	61	9	if	if	SCONJ
iajs-3028	61	10	every	every	DET
iajs-3028	61	11	hyperideal	hyperideal	ADJ
iajs-3028	61	12	j	j	PROPN
iajs-3028	61	13	in	in	ADP
iajs-3028	61	14	ℛ	ℛ	PROPN
iajs-3028	61	15	with	with	ADP
iajs-3028	61	16	i	i	PRON
iajs-3028	61	17	⊊	⊊	VERB
iajs-3028	61	18	j	j	PROPN
iajs-3028	61	19	⊆	⊆	NUM
iajs-3028	61	20	ℛ	ℛ	PROPN
iajs-3028	61	21	then	then	ADV
iajs-3028	61	22	j	j	PROPN
iajs-3028	61	23	=	=	SYM
iajs-3028	61	24	ℛ.	ℛ.	PROPN
iajs-3028	61	25	definition	definition	NOUN
iajs-3028	61	26	2.7	2.7	NUM
iajs-3028	61	27	[	[	X
iajs-3028	61	28	1	1	NUM
iajs-3028	61	29	]	]	PUNCT
iajs-3028	61	30	.	.	PUNCT
iajs-3028	62	1	a	a	DET
iajs-3028	62	2	canonical	canonical	ADJ
iajs-3028	62	3	hypergroup	hypergroup	NOUN
iajs-3028	62	4	(	(	PUNCT
iajs-3028	62	5	ℳ,⨁	ℳ,⨁	PROPN
iajs-3028	62	6	)	)	PUNCT
iajs-3028	62	7	is	be	AUX
iajs-3028	62	8	said	say	VERB
iajs-3028	62	9	to	to	PART
iajs-3028	62	10	be	be	AUX
iajs-3028	62	11	left	leave	VERB
iajs-3028	62	12	hypermodule	hypermodule	NOUN
iajs-3028	62	13	over	over	ADP
iajs-3028	62	14	a	a	DET
iajs-3028	62	15	hyperring	hyperring	NOUN
iajs-3028	62	16	(	(	PUNCT
iajs-3028	62	17	ℛ,⊚,⨀	ℛ,⊚,⨀	PROPN
iajs-3028	62	18	)	)	PUNCT
iajs-3028	62	19	with	with	ADP
iajs-3028	62	20	the	the	DET
iajs-3028	62	21	unit	unit	NOUN
iajs-3028	62	22	element	element	NOUN
iajs-3028	62	23	“	"	PUNCT
iajs-3028	62	24	1	1	NUM
iajs-3028	62	25	”	"	PUNCT
iajs-3028	62	26	,	,	PUNCT
iajs-3028	62	27	if	if	SCONJ
iajs-3028	62	28	the	the	DET
iajs-3028	62	29	map	map	NOUN
iajs-3028	62	30	⋅	⋅	PROPN
iajs-3028	62	31	:	:	PUNCT
iajs-3028	62	32	ℛ	ℛ	ADJ
iajs-3028	62	33	×ℳ	×ℳ	NOUN
iajs-3028	62	34	⟶ℳ	⟶ℳ	X
iajs-3028	62	35	which	which	PRON
iajs-3028	62	36	is	be	AUX
iajs-3028	62	37	defined	define	VERB
iajs-3028	62	38	as	as	ADP
iajs-3028	62	39	:	:	PUNCT
iajs-3028	62	40	⋅	⋅	X
iajs-3028	62	41	(	(	PUNCT
iajs-3028	62	42	𝑠,𝑚	𝑠,𝑚	ADP
iajs-3028	62	43	)	)	PUNCT
iajs-3028	62	44	↦	↦	PROPN
iajs-3028	62	45	𝑠.𝑚	𝑠.𝑚	PROPN
iajs-3028	62	46	=	=	SYM
iajs-3028	62	47	𝑠𝑚	𝑠𝑚	ADP
iajs-3028	62	48	∈	∈	PROPN
iajs-3028	62	49	ℳ	ℳ	PROPN
iajs-3028	62	50	,	,	PUNCT
iajs-3028	62	51	for	for	ADP
iajs-3028	62	52	𝑠	𝑠	PROPN
iajs-3028	62	53	∈	∈	PROPN
iajs-3028	62	54	ℛ,𝑚	ℛ,𝑚	NOUN
iajs-3028	62	55	∈	∈	PROPN
iajs-3028	62	56	ℳ	ℳ	PROPN
iajs-3028	62	57	satisfies	satisfy	VERB
iajs-3028	62	58	the	the	DET
iajs-3028	62	59	following	follow	VERB
iajs-3028	62	60	conditions	condition	NOUN
iajs-3028	62	61	,	,	PUNCT
iajs-3028	62	62	for	for	ADP
iajs-3028	62	63	𝓀,𝒷	𝓀,𝒷	PROPN
iajs-3028	62	64	∈	∈	PROPN
iajs-3028	62	65	ℛ	ℛ	PROPN
iajs-3028	62	66	,	,	PUNCT
iajs-3028	62	67	and	and	CCONJ
iajs-3028	62	68	𝑚ˋ	𝑚ˋ	PROPN
iajs-3028	62	69	∈	∈	PROPN
iajs-3028	62	70	ℳ	ℳ	PROPN
iajs-3028	62	71	:	:	PUNCT
iajs-3028	62	72	1	1	NUM
iajs-3028	62	73	.	.	PUNCT
iajs-3028	62	74	(	(	PUNCT
iajs-3028	62	75	𝓀⊚𝒷)𝑚	𝓀⊚𝒷)𝑚	NOUN
iajs-3028	62	76	=	=	SYM
iajs-3028	62	77	𝓀𝑚⊕𝒷𝑚	𝓀𝑚⊕𝒷𝑚	PROPN
iajs-3028	62	78	ihjpas	ihjpas	PROPN
iajs-3028	62	79	.	.	PUNCT
iajs-3028	63	1	36(2)2023	36(2)2023	NUM
iajs-3028	63	2	385	385	NUM
iajs-3028	63	3	2	2	NUM
iajs-3028	63	4	.	.	PUNCT
iajs-3028	64	1	𝓀(𝑚⨁𝑚ˋ)=𝓀𝓂⊕𝓀𝑚ˋ	𝓀(𝑚⨁𝑚ˋ)=𝓀𝓂⊕𝓀𝑚ˋ	PROPN
iajs-3028	64	2	3	3	NUM
iajs-3028	64	3	.	.	PUNCT
iajs-3028	64	4	(	(	PUNCT
iajs-3028	64	5	𝓀⨀𝒷)𝑚=𝓀(𝒷𝑚	𝓀⨀𝒷)𝑚=𝓀(𝒷𝑚	PROPN
iajs-3028	64	6	)	)	PUNCT
iajs-3028	64	7	4	4	NUM
iajs-3028	64	8	.	.	X
iajs-3028	64	9	0ℛ	0ℛ	PROPN
iajs-3028	64	10	.	.	PUNCT
iajs-3028	65	1	𝑚	𝑚	X
iajs-3028	65	2	=	=	SYM
iajs-3028	65	3	0ℳ	0ℳ	NOUN
iajs-3028	65	4	,	,	PUNCT
iajs-3028	65	5	where	where	SCONJ
iajs-3028	65	6	0ℛ	0ℛ	PROPN
iajs-3028	65	7	is	be	AUX
iajs-3028	65	8	a	a	DET
iajs-3028	65	9	zero	zero	NUM
iajs-3028	65	10	of	of	ADP
iajs-3028	65	11	ℛ	ℛ	PROPN
iajs-3028	65	12	,	,	PUNCT
iajs-3028	65	13	0ℳ	0ℳ	VERB
iajs-3028	65	14	the	the	DET
iajs-3028	65	15	secular	secular	ADJ
iajs-3028	65	16	identity	identity	NOUN
iajs-3028	65	17	of	of	ADP
iajs-3028	65	18	ℳ.	ℳ.	PROPN
iajs-3028	65	19	in	in	ADP
iajs-3028	65	20	the	the	DET
iajs-3028	65	21	same	same	ADJ
iajs-3028	65	22	way	way	NOUN
iajs-3028	65	23	,	,	PUNCT
iajs-3028	65	24	one	one	PRON
iajs-3028	65	25	can	can	AUX
iajs-3028	65	26	define	define	VERB
iajs-3028	65	27	the	the	DET
iajs-3028	65	28	right	right	NOUN
iajs-3028	65	29	ℛ-hypermodule	ℛ-hypermodule	PROPN
iajs-3028	65	30	.	.	PUNCT
iajs-3028	66	1	the	the	DET
iajs-3028	66	2	ℛhypermodule	ℛhypermodule	PROPN
iajs-3028	66	3	ℳ	ℳ	PROPN
iajs-3028	66	4	is	be	AUX
iajs-3028	66	5	said	say	VERB
iajs-3028	66	6	to	to	PART
iajs-3028	66	7	be	be	AUX
iajs-3028	66	8	unitary	unitary	ADJ
iajs-3028	66	9	if	if	SCONJ
iajs-3028	66	10	1.𝓂	1.𝓂	NUM
iajs-3028	66	11	=	=	SYM
iajs-3028	66	12	𝓂	𝓂	PROPN
iajs-3028	66	13	,	,	PUNCT
iajs-3028	66	14	where	where	SCONJ
iajs-3028	66	15	1	1	NUM
iajs-3028	66	16	is	be	AUX
iajs-3028	66	17	the	the	DET
iajs-3028	66	18	unit	unit	NOUN
iajs-3028	66	19	element	element	NOUN
iajs-3028	66	20	of	of	ADP
iajs-3028	66	21	ℛ	ℛ	PROPN
iajs-3028	66	22	and	and	CCONJ
iajs-3028	66	23	𝓂∈ℳ.	𝓂∈ℳ.	NOUN
iajs-3028	66	24	definition	definition	NOUN
iajs-3028	66	25	2.8	2.8	NUM
iajs-3028	66	26	[	[	X
iajs-3028	66	27	6	6	NUM
iajs-3028	66	28	]	]	PUNCT
iajs-3028	66	29	.	.	PUNCT
iajs-3028	67	1	if	if	SCONJ
iajs-3028	67	2	ℳ	ℳ	PROPN
iajs-3028	67	3	is	be	AUX
iajs-3028	67	4	an	an	DET
iajs-3028	67	5	ℛ-hypermodule	ℛ-hypermodule	PROPN
iajs-3028	67	6	,	,	PUNCT
iajs-3028	67	7	then	then	ADV
iajs-3028	67	8	∅	∅	NOUN
iajs-3028	67	9	≠	≠	PROPN
iajs-3028	67	10	𝒩	𝒩	PROPN
iajs-3028	67	11	⊆ℳ	⊆ℳ	PROPN
iajs-3028	67	12	is	be	AUX
iajs-3028	67	13	called	call	VERB
iajs-3028	67	14	a	a	DET
iajs-3028	67	15	“	"	PUNCT
iajs-3028	67	16	subhypermodule	subhypermodule	NOUN
iajs-3028	67	17	”	"	PUNCT
iajs-3028	67	18	if	if	SCONJ
iajs-3028	67	19	and	and	CCONJ
iajs-3028	67	20	only	only	ADV
iajs-3028	67	21	if	if	SCONJ
iajs-3028	67	22	ℴ	ℴ	NOUN
iajs-3028	67	23	−	−	NOUN
iajs-3028	67	24	𝒶	𝒶	NOUN
iajs-3028	67	25	⊆	⊆	NUM
iajs-3028	67	26	𝒩	𝒩	PROPN
iajs-3028	67	27	and	and	CCONJ
iajs-3028	67	28	ℴ𝓇	ℴ𝓇	ADP
iajs-3028	67	29	∈	∈	PROPN
iajs-3028	67	30	𝒩	𝒩	PROPN
iajs-3028	67	31	,	,	PUNCT
iajs-3028	67	32	for	for	ADP
iajs-3028	67	33	each	each	DET
iajs-3028	67	34	ℴ	ℴ	NOUN
iajs-3028	67	35	,	,	PUNCT
iajs-3028	67	36	𝒶	𝒶	PROPN
iajs-3028	67	37	∈	∈	PROPN
iajs-3028	67	38	𝒩	𝒩	PROPN
iajs-3028	67	39	,	,	PUNCT
iajs-3028	67	40	𝓇	𝓇	X
iajs-3028	67	41	∈	∈	NOUN
iajs-3028	67	42	ℛ.	ℛ.	PROPN
iajs-3028	67	43	definition	definition	NOUN
iajs-3028	67	44	2.9	2.9	NUM
iajs-3028	67	45	[	[	X
iajs-3028	67	46	2	2	NUM
iajs-3028	67	47	]	]	X
iajs-3028	67	48	let	let	VERB
iajs-3028	67	49	(	(	PUNCT
iajs-3028	67	50	ℛ,⊚,⨀	ℛ,⊚,⨀	PROPN
iajs-3028	67	51	)	)	PUNCT
iajs-3028	67	52	be	be	AUX
iajs-3028	67	53	a	a	DET
iajs-3028	67	54	hyperring	hyperring	NOUN
iajs-3028	67	55	.	.	PUNCT
iajs-3028	68	1	the	the	DET
iajs-3028	68	2	element	element	NOUN
iajs-3028	68	3	𝑟1	𝑟1	PROPN
iajs-3028	68	4	∈	∈	PROPN
iajs-3028	68	5	ℛ	ℛ	PROPN
iajs-3028	68	6	is	be	AUX
iajs-3028	68	7	named	name	VERB
iajs-3028	68	8	“	"	PUNCT
iajs-3028	68	9	right	right	ADJ
iajs-3028	68	10	(	(	PUNCT
iajs-3028	68	11	resp	resp	NOUN
iajs-3028	68	12	.	.	PUNCT
iajs-3028	69	1	left)zero	left)zero	NUM
iajs-3028	69	2	divisor	divisor	NOUN
iajs-3028	69	3	”	"	PUNCT
iajs-3028	69	4	if	if	SCONJ
iajs-3028	69	5	there	there	PRON
iajs-3028	69	6	is	be	VERB
iajs-3028	69	7	0	0	NUM
iajs-3028	69	8	≠	≠	NOUN
iajs-3028	69	9	𝑟2	𝑟2	NOUN
iajs-3028	69	10	∈	∈	NOUN
iajs-3028	69	11	ℛ	ℛ	NOUN
iajs-3028	69	12	such	such	ADJ
iajs-3028	69	13	that	that	DET
iajs-3028	69	14	𝑟1	𝑟1	NOUN
iajs-3028	69	15	.	.	PUNCT
iajs-3028	70	1	𝑟2	𝑟2	NOUN
iajs-3028	70	2	(	(	PUNCT
iajs-3028	70	3	resp	resp	NOUN
iajs-3028	70	4	.	.	PUNCT
iajs-3028	70	5	𝑟2	𝑟2	NOUN
iajs-3028	70	6	.	.	PUNCT
iajs-3028	71	1	𝑟1	𝑟1	NOUN
iajs-3028	71	2	)	)	PUNCT
iajs-3028	72	1	=	=	PRON
iajs-3028	72	2	{	{	PUNCT
iajs-3028	72	3	0	0	NUM
iajs-3028	72	4	}	}	PUNCT
iajs-3028	72	5	.	.	PUNCT
iajs-3028	73	1	and	and	CCONJ
iajs-3028	73	2	called	call	VERB
iajs-3028	73	3	zero	zero	NUM
iajs-3028	73	4	-	-	PUNCT
iajs-3028	73	5	divisor	divisor	NOUN
iajs-3028	73	6	if	if	SCONJ
iajs-3028	73	7	it	it	PRON
iajs-3028	73	8	was	be	AUX
iajs-3028	73	9	right	right	ADJ
iajs-3028	73	10	and	and	CCONJ
iajs-3028	73	11	left	leave	VERB
iajs-3028	73	12	zero	zero	NUM
iajs-3028	73	13	-	-	PUNCT
iajs-3028	73	14	divisor	divisor	NOUN
iajs-3028	73	15	.	.	PUNCT
iajs-3028	74	1	definition	definition	NOUN
iajs-3028	74	2	2.10	2.10	NUM
iajs-3028	74	3	[	[	X
iajs-3028	74	4	2	2	NUM
iajs-3028	74	5	]	]	PUNCT
iajs-3028	74	6	let	let	VERB
iajs-3028	74	7	ℳ	ℳ	PRON
iajs-3028	74	8	be	be	AUX
iajs-3028	74	9	an	an	DET
iajs-3028	74	10	ℛ-hypermodule.the	ℛ-hypermodule.the	DET
iajs-3028	74	11	element	element	NOUN
iajs-3028	74	12	𝜇	𝜇	ADP
iajs-3028	74	13	∈	∈	PROPN
iajs-3028	74	14	ℳ	ℳ	PROPN
iajs-3028	74	15	is	be	AUX
iajs-3028	74	16	called	call	VERB
iajs-3028	74	17	a	a	DET
iajs-3028	74	18	“	"	PUNCT
iajs-3028	74	19	divisible	divisible	ADJ
iajs-3028	74	20	element	element	NOUN
iajs-3028	74	21	”	"	PUNCT
iajs-3028	74	22	if	if	SCONJ
iajs-3028	74	23	for	for	ADP
iajs-3028	74	24	each	each	DET
iajs-3028	74	25	non	non	ADJ
iajs-3028	74	26	-	-	ADJ
iajs-3028	74	27	zero	zero	NUM
iajs-3028	74	28	divisor	divisor	NOUN
iajs-3028	74	29	𝜍	𝜍	ADP
iajs-3028	74	30	∈	∈	PROPN
iajs-3028	74	31	ℛ	ℛ	NOUN
iajs-3028	74	32	there	there	PRON
iajs-3028	74	33	is	be	VERB
iajs-3028	74	34	∈	∈	PROPN
iajs-3028	74	35	ℳ	ℳ	PROPN
iajs-3028	74	36	,	,	PUNCT
iajs-3028	74	37	such	such	ADJ
iajs-3028	74	38	that	that	PRON
iajs-3028	74	39	.	.	PUNCT
iajs-3028	75	1	𝜇	𝜇	X
iajs-3028	75	2	=	=	NOUN
iajs-3028	75	3	𝜍𝜂.	𝜍𝜂.	NOUN
iajs-3028	75	4	if	if	SCONJ
iajs-3028	75	5	every	every	DET
iajs-3028	75	6	element	element	NOUN
iajs-3028	75	7	in	in	ADP
iajs-3028	75	8	an	an	DET
iajs-3028	75	9	ℛ-hypermodule	ℛ-hypermodule	PROPN
iajs-3028	75	10	ℳ	ℳ	PROPN
iajs-3028	75	11	is	be	AUX
iajs-3028	75	12	a	a	DET
iajs-3028	75	13	divisible	divisible	ADJ
iajs-3028	75	14	,	,	PUNCT
iajs-3028	75	15	then	then	ADV
iajs-3028	75	16	ℳ	ℳ	PROPN
iajs-3028	75	17	is	be	AUX
iajs-3028	75	18	named	name	VERB
iajs-3028	75	19	a	a	DET
iajs-3028	75	20	“	"	PUNCT
iajs-3028	75	21	divisible	divisible	ADJ
iajs-3028	75	22	hypermodule	hypermodule	NOUN
iajs-3028	75	23	”	"	PUNCT
iajs-3028	75	24	definition	definition	NOUN
iajs-3028	75	25	2.11	2.11	NUM
iajs-3028	75	26	[	[	X
iajs-3028	75	27	4	4	NUM
iajs-3028	75	28	]	]	PUNCT
iajs-3028	75	29	.	.	PUNCT
iajs-3028	76	1	the	the	DET
iajs-3028	76	2	jacobson	jacobson	PROPN
iajs-3028	76	3	radical	radical	PROPN
iajs-3028	76	4	of	of	ADP
iajs-3028	76	5	a	a	DET
iajs-3028	76	6	hyperring	hyperre	VERB
iajs-3028	76	7	ℛ	ℛ	PROPN
iajs-3028	76	8	is	be	AUX
iajs-3028	76	9	the	the	DET
iajs-3028	76	10	intersection	intersection	NOUN
iajs-3028	76	11	of	of	ADP
iajs-3028	76	12	all	all	DET
iajs-3028	76	13	maximalhyperideals	maximalhyperideal	NOUN
iajs-3028	76	14	and	and	CCONJ
iajs-3028	76	15	it	it	PRON
iajs-3028	76	16	is	be	AUX
iajs-3028	76	17	denoted	denote	VERB
iajs-3028	76	18	by	by	ADP
iajs-3028	76	19	j(ℛ	j(ℛ	PROPN
iajs-3028	76	20	)	)	PUNCT
iajs-3028	76	21	.	.	PUNCT
iajs-3028	77	1	remark	remark	VERB
iajs-3028	77	2	2.12	2.12	NUM
iajs-3028	78	1	[	[	X
iajs-3028	78	2	7	7	NUM
iajs-3028	78	3	]	]	PUNCT
iajs-3028	78	4	.	.	PUNCT
iajs-3028	79	1	j(ℛ	j(ℛ	PROPN
iajs-3028	79	2	)	)	PUNCT
iajs-3028	79	3	is	be	AUX
iajs-3028	79	4	a	a	DET
iajs-3028	79	5	hyperideal	hyperideal	NOUN
iajs-3028	79	6	in	in	ADP
iajs-3028	79	7	ℛ	ℛ	PROPN
iajs-3028	79	8	notations	notation	NOUN
iajs-3028	79	9	:	:	PUNCT
iajs-3028	79	10	•	•	NUM
iajs-3028	79	11	𝑈𝑙𝑒𝑓𝑡	𝑈𝑙𝑒𝑓𝑡	PROPN
iajs-3028	79	12	(	(	PUNCT
iajs-3028	79	13	ℛ	ℛ	PROPN
iajs-3028	79	14	)	)	PUNCT
iajs-3028	79	15	=	=	NOUN
iajs-3028	79	16	{	{	PUNCT
iajs-3028	79	17	r	r	NOUN
iajs-3028	79	18	𝜖	𝜖	PROPN
iajs-3028	79	19	ℛ	ℛ	NOUN
iajs-3028	79	20	|	|	ADV
iajs-3028	79	21	ǝŕ	ǝŕ	PRON
iajs-3028	79	22	𝜖	𝜖	PROPN
iajs-3028	79	23	ℛ	ℛ	PROPN
iajs-3028	79	24	𝑠𝑢𝑐ℎ	𝑠𝑢𝑐ℎ	NOUN
iajs-3028	79	25	𝑡ℎ𝑎𝑡	𝑡ℎ𝑎𝑡	ADJ
iajs-3028	79	26	ŕ	ŕ	PROPN
iajs-3028	79	27	.	.	PUNCT
iajs-3028	80	1	𝑟	𝑟	X
iajs-3028	80	2	=	=	SYM
iajs-3028	80	3	1	1	NUM
iajs-3028	80	4	}	}	PUNCT
iajs-3028	80	5	•	•	NUM
iajs-3028	80	6	𝑈𝑟𝑖𝑔ℎ𝑡	𝑈𝑟𝑖𝑔ℎ𝑡	PROPN
iajs-3028	80	7	(	(	PUNCT
iajs-3028	80	8	ℛ	ℛ	PROPN
iajs-3028	80	9	)	)	PUNCT
iajs-3028	80	10	=	=	PRON
iajs-3028	80	11	{	{	PUNCT
iajs-3028	80	12	𝑟	𝑟	X
iajs-3028	80	13	ϵ	ϵ	SYM
iajs-3028	80	14	ℛ	ℛ	PROPN
iajs-3028	80	15	|ǝ	|ǝ	X
iajs-3028	80	16	ŕ	ŕ	X
iajs-3028	80	17	𝜖	𝜖	X
iajs-3028	80	18	ℛ	ℛ	PROPN
iajs-3028	80	19	𝑠𝑢𝑐ℎ	𝑠𝑢𝑐ℎ	NOUN
iajs-3028	80	20	𝑡ℎ𝑎𝑡	𝑡ℎ𝑎𝑡	ADJ
iajs-3028	80	21	𝑟.	𝑟.	NOUN
iajs-3028	80	22	ŕ	ŕ	X
iajs-3028	80	23	=	=	SYM
iajs-3028	80	24	1	1	NUM
iajs-3028	80	25	}	}	PUNCT
iajs-3028	80	26	•	•	ADV
iajs-3028	80	27	𝑈(ℛ	𝑈(ℛ	ADJ
iajs-3028	80	28	)	)	PUNCT
iajs-3028	80	29	=	=	NOUN
iajs-3028	80	30	{	{	PUNCT
iajs-3028	80	31	𝑟𝜖ℛ	𝑟𝜖ℛ	ADJ
iajs-3028	80	32	|ǝŕ	|ǝŕ	ADP
iajs-3028	80	33	𝜖	𝜖	X
iajs-3028	80	34	ℛ	ℛ	PROPN
iajs-3028	80	35	𝑠𝑢𝑐ℎ	𝑠𝑢𝑐ℎ	NOUN
iajs-3028	80	36	𝑡ℎ𝑎𝑡	𝑡ℎ𝑎𝑡	ADJ
iajs-3028	80	37	𝑟.	𝑟.	NOUN
iajs-3028	80	38	ŕ	ŕ	X
iajs-3028	80	39	=	=	PUNCT
iajs-3028	80	40	ŕ	ŕ	X
iajs-3028	80	41	.	.	PUNCT
iajs-3028	81	1	𝑟	𝑟	X
iajs-3028	81	2	=	=	SYM
iajs-3028	81	3	1	1	X
iajs-3028	81	4	}	}	PUNCT
iajs-3028	81	5	in	in	ADP
iajs-3028	81	6	the	the	DET
iajs-3028	81	7	following	follow	VERB
iajs-3028	81	8	proposition	proposition	NOUN
iajs-3028	81	9	,	,	PUNCT
iajs-3028	81	10	davvaz	davvaz	NOUN
iajs-3028	81	11	b	b	PROPN
iajs-3028	81	12	and	and	CCONJ
iajs-3028	81	13	salasi	salasi	NOUN
iajs-3028	81	14	a	a	PRON
iajs-3028	81	15	in	in	ADP
iajs-3028	81	16	[	[	X
iajs-3028	81	17	4	4	NUM
iajs-3028	81	18	]	]	PUNCT
iajs-3028	81	19	proved	prove	VERB
iajs-3028	81	20	the	the	DET
iajs-3028	81	21	part	part	NOUN
iajs-3028	81	22	(	(	PUNCT
iajs-3028	81	23	1	1	NUM
iajs-3028	81	24	→	→	SYM
iajs-3028	81	25	2	2	NUM
iajs-3028	81	26	)	)	PUNCT
iajs-3028	81	27	.	.	PUNCT
iajs-3028	82	1	here	here	ADV
iajs-3028	82	2	we	we	PRON
iajs-3028	82	3	add	add	VERB
iajs-3028	82	4	another	another	DET
iajs-3028	82	5	condition	condition	NOUN
iajs-3028	82	6	and	and	CCONJ
iajs-3028	82	7	give	give	VERB
iajs-3028	82	8	the	the	DET
iajs-3028	82	9	following	follow	VERB
iajs-3028	82	10	proposition	proposition	NOUN
iajs-3028	82	11	.	.	PUNCT
iajs-3028	83	1	proposition	proposition	NOUN
iajs-3028	83	2	2.13	2.13	NUM
iajs-3028	83	3	.	.	PUNCT
iajs-3028	84	1	for	for	ADP
iajs-3028	84	2	an	an	DET
iajs-3028	84	3	element	element	NOUN
iajs-3028	84	4	𝓅	𝓅	PROPN
iajs-3028	84	5	∈	∈	NOUN
iajs-3028	84	6	ℛ	ℛ	NOUN
iajs-3028	84	7	the	the	DET
iajs-3028	84	8	following	following	ADJ
iajs-3028	84	9	statements	statement	NOUN
iajs-3028	84	10	are	be	AUX
iajs-3028	84	11	equivalent	equivalent	ADJ
iajs-3028	84	12	(	(	PUNCT
iajs-3028	84	13	1	1	NUM
iajs-3028	84	14	)	)	PUNCT
iajs-3028	84	15	𝓅	𝓅	PROPN
iajs-3028	84	16	∈	∈	PROPN
iajs-3028	84	17	j(ℛ	j(ℛ	PROPN
iajs-3028	84	18	)	)	PUNCT
iajs-3028	84	19	(	(	PUNCT
iajs-3028	84	20	2	2	X
iajs-3028	84	21	)	)	PUNCT
iajs-3028	84	22	for	for	ADP
iajs-3028	84	23	any	any	DET
iajs-3028	84	24	𝒹	𝒹	PROPN
iajs-3028	84	25	∈	∈	PROPN
iajs-3028	84	26	ℛ	ℛ	PROPN
iajs-3028	84	27	,	,	PUNCT
iajs-3028	84	28	1-𝓅𝒹	1-𝓅𝒹	NUM
iajs-3028	84	29	⊆	⊆	NUM
iajs-3028	84	30	𝑈	𝑈	PROPN
iajs-3028	84	31	(	(	PUNCT
iajs-3028	84	32	ℛ	ℛ	PROPN
iajs-3028	84	33	)	)	PUNCT
iajs-3028	84	34	(	(	PUNCT
iajs-3028	84	35	3	3	X
iajs-3028	84	36	)	)	PUNCT
iajs-3028	84	37	for	for	ADP
iajs-3028	84	38	any	any	DET
iajs-3028	84	39	𝒹,𝓆	𝒹,𝓆	PROPN
iajs-3028	84	40	∈	∈	PROPN
iajs-3028	84	41	ℛ	ℛ	PROPN
iajs-3028	84	42	,	,	PUNCT
iajs-3028	84	43	1-𝓅𝒹𝓆	1-𝓅𝒹𝓆	NUM
iajs-3028	84	44	⊆	⊆	NUM
iajs-3028	84	45	𝑈(ℛ	𝑈(ℛ	ADJ
iajs-3028	84	46	)	)	PUNCT
iajs-3028	84	47	.	.	PUNCT
iajs-3028	85	1	proof	proof	NOUN
iajs-3028	85	2	.	.	PUNCT
iajs-3028	86	1	(	(	PUNCT
iajs-3028	86	2	1	1	NUM
iajs-3028	86	3	↔	↔	PROPN
iajs-3028	86	4	2	2	NUM
iajs-3028	86	5	proved	prove	VERB
iajs-3028	86	6	in	in	ADP
iajs-3028	86	7	(	(	PUNCT
iajs-3028	86	8	4	4	NUM
iajs-3028	86	9	,	,	PUNCT
iajs-3028	86	10	prop	prop	NOUN
iajs-3028	86	11	.	.	PUNCT
iajs-3028	86	12	2.14	2.14	NUM
iajs-3028	86	13	)	)	PUNCT
iajs-3028	86	14	)	)	PUNCT
iajs-3028	86	15	.	.	PUNCT
iajs-3028	87	1	now	now	ADV
iajs-3028	87	2	,	,	PUNCT
iajs-3028	87	3	will	will	AUX
iajs-3028	87	4	prove	prove	VERB
iajs-3028	87	5	(	(	PUNCT
iajs-3028	87	6	1	1	NUM
iajs-3028	87	7	↔	↔	PROPN
iajs-3028	87	8	3	3	NUM
iajs-3028	87	9	)	)	PUNCT
iajs-3028	87	10	.	.	PUNCT
iajs-3028	88	1	to	to	PART
iajs-3028	88	2	prove	prove	VERB
iajs-3028	88	3	that	that	SCONJ
iajs-3028	88	4	,	,	PUNCT
iajs-3028	88	5	for	for	ADP
iajs-3028	88	6	𝓅	𝓅	PROPN
iajs-3028	88	7	∈	∈	PROPN
iajs-3028	88	8	j(ℛ	j(ℛ	PROPN
iajs-3028	88	9	)	)	PUNCT
iajs-3028	88	10	,	,	PUNCT
iajs-3028	88	11	and	and	CCONJ
iajs-3028	88	12	for	for	ADP
iajs-3028	88	13	all	all	DET
iajs-3028	88	14	𝒹,𝓆	𝒹,𝓆	PROPN
iajs-3028	88	15	∈	∈	PROPN
iajs-3028	88	16	ℛ	ℛ	PROPN
iajs-3028	88	17	,	,	PUNCT
iajs-3028	88	18	the	the	DET
iajs-3028	88	19	set	set	NOUN
iajs-3028	88	20	1𝓅𝒹𝓆	1𝓅𝒹𝓆	PROPN
iajs-3028	88	21	⊆	⊆	NUM
iajs-3028	88	22	𝑈(ℛ	𝑈(ℛ	ADJ
iajs-3028	88	23	)	)	PUNCT
iajs-3028	88	24	.	.	PUNCT
iajs-3028	89	1	suppose	suppose	VERB
iajs-3028	89	2	that	that	SCONJ
iajs-3028	89	3	∃𝒹𝑜	∃𝒹𝑜	NOUN
iajs-3028	89	4	,	,	PUNCT
iajs-3028	89	5	𝓆𝑜	𝓆𝑜	ADP
iajs-3028	89	6	∈	∈	PROPN
iajs-3028	89	7	ℛ	ℛ	PROPN
iajs-3028	89	8	such	such	ADJ
iajs-3028	89	9	that	that	SCONJ
iajs-3028	89	10	𝓈	𝓈	PROPN
iajs-3028	89	11	∉	∉	X
iajs-3028	89	12	𝑈(ℛ	𝑈(ℛ	ADJ
iajs-3028	89	13	)	)	PUNCT
iajs-3028	89	14	,	,	PUNCT
iajs-3028	89	15	for	for	ADP
iajs-3028	89	16	some	some	DET
iajs-3028	89	17	𝓈	𝓈	NOUN
iajs-3028	89	18	∈	∈	PROPN
iajs-3028	89	19	1	1	NUM
iajs-3028	89	20	(	(	PUNCT
iajs-3028	89	21	𝒹𝑜𝓆𝑜	𝒹𝑜𝓆𝑜	NOUN
iajs-3028	89	22	)	)	PUNCT
iajs-3028	89	23	𝓅.	𝓅.	NOUN
iajs-3028	89	24	but	but	CCONJ
iajs-3028	89	25	𝓈	𝓈	X
iajs-3028	89	26	∉	∉	X
iajs-3028	89	27	𝑈(ℛ	𝑈(ℛ	ADJ
iajs-3028	89	28	)	)	PUNCT
iajs-3028	89	29	this	this	DET
iajs-3028	89	30	lead	lead	NOUN
iajs-3028	89	31	to	to	ADP
iajs-3028	89	32	ℛ𝓈	ℛ𝓈	PROPN
iajs-3028	89	33	≠	≠	PROPN
iajs-3028	89	34	ℛ	ℛ	PROPN
iajs-3028	89	35	then	then	ADV
iajs-3028	89	36	there	there	PRON
iajs-3028	89	37	is	be	VERB
iajs-3028	89	38	a	a	DET
iajs-3028	89	39	maximal	maximal	ADJ
iajs-3028	89	40	hyperideal	hyperideal	NOUN
iajs-3028	89	41	𝐼	𝐼	ADP
iajs-3028	89	42	such	such	ADJ
iajs-3028	89	43	that	that	SCONJ
iajs-3028	89	44	ℛ𝓈	ℛ𝓈	PROPN
iajs-3028	89	45	⊆	⊆	NUM
iajs-3028	89	46	𝐼	𝐼	PROPN
iajs-3028	89	47	(	(	PUNCT
iajs-3028	89	48	4	4	NUM
iajs-3028	89	49	,	,	PUNCT
iajs-3028	89	50	prop.2.12	prop.2.12	NOUN
iajs-3028	89	51	)	)	PUNCT
iajs-3028	89	52	,	,	PUNCT
iajs-3028	89	53	since	since	SCONJ
iajs-3028	89	54	𝓈	𝓈	PROPN
iajs-3028	89	55	∈	∈	PROPN
iajs-3028	89	56	1	1	NUM
iajs-3028	89	57	(	(	PUNCT
iajs-3028	89	58	𝒹𝑜𝓆𝑜	𝒹𝑜𝓆𝑜	NOUN
iajs-3028	89	59	)	)	PUNCT
iajs-3028	89	60	𝓅	𝓅	PROPN
iajs-3028	89	61	,	,	PUNCT
iajs-3028	89	62	one	one	NUM
iajs-3028	89	63	obtain	obtain	VERB
iajs-3028	89	64	that	that	DET
iajs-3028	89	65	1∈	1∈	PROPN
iajs-3028	89	66	𝓈	𝓈	X
iajs-3028	89	67	+	+	CCONJ
iajs-3028	89	68	(	(	PUNCT
iajs-3028	89	69	𝒹𝑜𝓆𝑜	𝒹𝑜𝓆𝑜	NOUN
iajs-3028	89	70	)	)	PUNCT
iajs-3028	89	71	𝓅	𝓅	ADP
iajs-3028	89	72	⊆	⊆	NUM
iajs-3028	89	73	𝐼	𝐼	PROPN
iajs-3028	89	74	+	+	CCONJ
iajs-3028	89	75	j(ℛ	j(ℛ	NOUN
iajs-3028	89	76	)	)	PUNCT
iajs-3028	89	77	⊆	⊆	NUM
iajs-3028	89	78	𝐼	𝐼	PROPN
iajs-3028	89	79	(	(	PUNCT
iajs-3028	89	80	by	by	ADP
iajs-3028	89	81	definition	definition	NOUN
iajs-3028	89	82	2.1	2.1	NUM
iajs-3028	89	83	)	)	PUNCT
iajs-3028	89	84	that	that	PRON
iajs-3028	89	85	is	be	AUX
iajs-3028	89	86	1∈	1∈	PROPN
iajs-3028	89	87	𝐼	𝐼	PROPN
iajs-3028	89	88	and	and	CCONJ
iajs-3028	89	89	this	this	DET
iajs-3028	89	90	contradiction	contradiction	NOUN
iajs-3028	89	91	.	.	PUNCT
iajs-3028	90	1	conversely	conversely	ADV
iajs-3028	90	2	,	,	PUNCT
iajs-3028	90	3	for	for	ADP
iajs-3028	90	4	any	any	DET
iajs-3028	90	5	𝒹,𝓆	𝒹,𝓆	PROPN
iajs-3028	90	6	∈	∈	PROPN
iajs-3028	90	7	ℛ	ℛ	PROPN
iajs-3028	90	8	,	,	PUNCT
iajs-3028	90	9	let	let	VERB
iajs-3028	90	10	1-𝓅𝒹𝓆	1-𝓅𝒹𝓆	NUM
iajs-3028	90	11	⊆	⊆	PUNCT
iajs-3028	90	12	𝑈(ℛ	𝑈(ℛ	ADJ
iajs-3028	90	13	)	)	PUNCT
iajs-3028	90	14	.	.	PUNCT
iajs-3028	91	1	to	to	PART
iajs-3028	91	2	prove	prove	VERB
iajs-3028	91	3	that	that	SCONJ
iajs-3028	91	4	𝓅	𝓅	PROPN
iajs-3028	91	5	∈	∈	PROPN
iajs-3028	91	6	j(ℛ	j(ℛ	PROPN
iajs-3028	91	7	)	)	PUNCT
iajs-3028	91	8	,	,	PUNCT
iajs-3028	91	9	assume	assume	VERB
iajs-3028	91	10	𝓅	𝓅	PROPN
iajs-3028	91	11	∉	∉	PROPN
iajs-3028	91	12	j(ℛ	j(ℛ	PROPN
iajs-3028	91	13	)	)	PUNCT
iajs-3028	91	14	,	,	PUNCT
iajs-3028	91	15	so	so	ADV
iajs-3028	91	16	∃	∃	PROPN
iajs-3028	91	17	𝐼	𝐼	PROPN
iajs-3028	91	18	is	be	AUX
iajs-3028	91	19	a	a	DET
iajs-3028	91	20	maximal	maximal	ADJ
iajs-3028	91	21	hyperideal	hyperideal	NOUN
iajs-3028	91	22	of	of	ADP
iajs-3028	91	23	ℛ	ℛ	NOUN
iajs-3028	91	24	such	such	ADJ
iajs-3028	91	25	that	that	SCONJ
iajs-3028	91	26	𝓅	𝓅	PROPN
iajs-3028	91	27	∉	∉	PROPN
iajs-3028	91	28	𝐼.	𝐼.	PROPN
iajs-3028	91	29	thus	thus	ADV
iajs-3028	91	30	<	<	X
iajs-3028	91	31	𝐼	𝐼	PROPN
iajs-3028	91	32	,	,	PUNCT
iajs-3028	91	33	𝓅	𝓅	PROPN
iajs-3028	91	34	>	>	X
iajs-3028	91	35	≠	≠	PROPN
iajs-3028	91	36	𝐼	𝐼	PROPN
iajs-3028	91	37	,	,	PUNCT
iajs-3028	91	38	and	and	CCONJ
iajs-3028	91	39	so	so	ADV
iajs-3028	91	40	<	<	X
iajs-3028	91	41	𝐼	𝐼	PROPN
iajs-3028	91	42	,	,	PUNCT
iajs-3028	91	43	𝓅	𝓅	PROPN
iajs-3028	91	44	>	>	X
iajs-3028	91	45	=	=	PUNCT
iajs-3028	91	46	ℛ.	ℛ.	PROPN
iajs-3028	91	47	since	since	SCONJ
iajs-3028	91	48	1	1	NUM
iajs-3028	91	49	ihjpas	ihjpa	NOUN
iajs-3028	91	50	.	.	PUNCT
iajs-3028	92	1	36(2)2023	36(2)2023	NUM
iajs-3028	92	2	386	386	NUM
iajs-3028	92	3	∈	∈	PROPN
iajs-3028	92	4	ℛ	ℛ	PROPN
iajs-3028	92	5	,	,	PUNCT
iajs-3028	92	6	then	then	ADV
iajs-3028	92	7	there	there	PRON
iajs-3028	92	8	is	be	VERB
iajs-3028	92	9	𝑎	𝑎	DET
iajs-3028	92	10	∈	∈	PROPN
iajs-3028	92	11	𝐼	𝐼	PROPN
iajs-3028	92	12	,	,	PUNCT
iajs-3028	92	13	and	and	CCONJ
iajs-3028	92	14	𝒹,𝓆	𝒹,𝓆	NOUN
iajs-3028	92	15	∈	∈	PROPN
iajs-3028	92	16	ℛ	ℛ	PROPN
iajs-3028	92	17	such	such	ADJ
iajs-3028	92	18	that	that	DET
iajs-3028	92	19	1∈	1∈	PROPN
iajs-3028	92	20	𝑎	𝑎	X
iajs-3028	92	21	+	+	X
iajs-3028	92	22	𝓅(𝒹𝓆	𝓅(𝒹𝓆	NOUN
iajs-3028	92	23	)	)	PUNCT
iajs-3028	92	24	.	.	PUNCT
iajs-3028	93	1	hence	hence	ADV
iajs-3028	93	2	𝑎	𝑎	VERB
iajs-3028	93	3	∈1	∈1	ADJ
iajs-3028	93	4	𝓅(𝒹𝓆	𝓅(𝒹𝓆	NOUN
iajs-3028	93	5	)	)	PUNCT
iajs-3028	93	6	⊆	⊆	NUM
iajs-3028	93	7	𝑈(ℛ	𝑈(ℛ	ADJ
iajs-3028	93	8	)	)	PUNCT
iajs-3028	93	9	,	,	PUNCT
iajs-3028	93	10	this	this	PRON
iajs-3028	93	11	implies	imply	VERB
iajs-3028	93	12	that	that	SCONJ
iajs-3028	93	13	1	1	NUM
iajs-3028	93	14	∈	∈	PROPN
iajs-3028	93	15	𝐼	𝐼	PROPN
iajs-3028	93	16	,	,	PUNCT
iajs-3028	93	17	and	and	CCONJ
iajs-3028	93	18	this	this	DET
iajs-3028	93	19	contradiction	contradiction	NOUN
iajs-3028	93	20	,	,	PUNCT
iajs-3028	93	21	therefore	therefore	ADV
iajs-3028	93	22	𝓅	𝓅	PROPN
iajs-3028	93	23	∈	∈	PROPN
iajs-3028	93	24	j(ℛ).∎	j(ℛ).∎	X
iajs-3028	93	25	3	3	X
iajs-3028	93	26	.	.	X
iajs-3028	93	27	main	main	ADJ
iajs-3028	93	28	results	result	NOUN
iajs-3028	93	29	the	the	DET
iajs-3028	93	30	concept	concept	NOUN
iajs-3028	93	31	of	of	ADP
iajs-3028	93	32	a	a	DET
iajs-3028	93	33	divisible	divisible	ADJ
iajs-3028	93	34	hyperring	hyperring	NOUN
iajs-3028	93	35	will	will	AUX
iajs-3028	93	36	be	be	AUX
iajs-3028	93	37	discussed	discuss	VERB
iajs-3028	93	38	in	in	ADP
iajs-3028	93	39	this	this	DET
iajs-3028	93	40	section	section	NOUN
iajs-3028	93	41	.	.	PUNCT
iajs-3028	94	1	definition	definition	NOUN
iajs-3028	94	2	3.1	3.1	NUM
iajs-3028	94	3	.	.	PUNCT
iajs-3028	95	1	the	the	DET
iajs-3028	95	2	family	family	NOUN
iajs-3028	95	3	of	of	ADP
iajs-3028	95	4	all	all	DET
iajs-3028	95	5	divisible	divisible	ADJ
iajs-3028	95	6	elements	element	NOUN
iajs-3028	95	7	of	of	ADP
iajs-3028	95	8	an	an	DET
iajs-3028	95	9	ℛ-hypermodule	ℛ-hypermodule	PROPN
iajs-3028	95	10	ℳ	ℳ	PROPN
iajs-3028	95	11	are	be	AUX
iajs-3028	95	12	defined	define	VERB
iajs-3028	95	13	as	as	ADP
iajs-3028	95	14	𝑑(ℳ	𝑑(ℳ	NOUN
iajs-3028	95	15	)	)	PUNCT
iajs-3028	96	1	=	=	NOUN
iajs-3028	96	2	{	{	PUNCT
iajs-3028	96	3	𝑦	𝑦	NOUN
iajs-3028	96	4	∈	∈	PROPN
iajs-3028	96	5	ℳ|	ℳ|	PROPN
iajs-3028	96	6	for	for	ADP
iajs-3028	96	7	each	each	DET
iajs-3028	96	8	𝑟	𝑟	DET
iajs-3028	96	9	∈	∈	PROPN
iajs-3028	96	10	𝑅𝑂	𝑅𝑂	PROPN
iajs-3028	96	11	,	,	PUNCT
iajs-3028	96	12	there	there	PRON
iajs-3028	96	13	is	be	VERB
iajs-3028	96	14	𝑥	𝑥	DET
iajs-3028	96	15	∈	∈	PROPN
iajs-3028	96	16	ℳ	ℳ	PROPN
iajs-3028	96	17	,	,	PUNCT
iajs-3028	96	18	such	such	ADJ
iajs-3028	96	19	that	that	SCONJ
iajs-3028	96	20	𝑦	𝑦	NOUN
iajs-3028	96	21	=	=	PUNCT
iajs-3028	96	22	𝑟𝑥	𝑟𝑥	PART
iajs-3028	96	23	}	}	PUNCT
iajs-3028	96	24	.	.	PUNCT
iajs-3028	97	1	remark	remark	PROPN
iajs-3028	97	2	3.2	3.2	NUM
iajs-3028	97	3	.	.	PUNCT
iajs-3028	98	1	𝑑(ℳ	𝑑(ℳ	NOUN
iajs-3028	98	2	)	)	PUNCT
iajs-3028	98	3	is	be	AUX
iajs-3028	98	4	a	a	DET
iajs-3028	98	5	divisible	divisible	ADJ
iajs-3028	98	6	hypergroup	hypergroup	NOUN
iajs-3028	98	7	of	of	ADP
iajs-3028	98	8	(	(	PUNCT
iajs-3028	98	9	ℳ,+	ℳ,+	PROPN
iajs-3028	98	10	)	)	PUNCT
iajs-3028	98	11	.	.	PUNCT
iajs-3028	99	1	proof	proof	NOUN
iajs-3028	99	2	.	.	PUNCT
iajs-3028	100	1	let	let	VERB
iajs-3028	100	2	𝑚1	𝑚1	NOUN
iajs-3028	100	3	,	,	PUNCT
iajs-3028	100	4	𝑚2	𝑚2	NOUN
iajs-3028	100	5	,	,	PUNCT
iajs-3028	100	6	𝑚3	𝑚3	NOUN
iajs-3028	100	7	∈	∈	PROPN
iajs-3028	100	8	𝑑(ℳ	𝑑(ℳ	PROPN
iajs-3028	100	9	)	)	PUNCT
iajs-3028	100	10	.	.	PUNCT
iajs-3028	101	1	the	the	DET
iajs-3028	101	2	associative	associative	NOUN
iajs-3028	101	3	of	of	ADP
iajs-3028	101	4	“	"	PUNCT
iajs-3028	101	5	+	+	PROPN
iajs-3028	101	6	”	"	PUNCT
iajs-3028	101	7	is	be	AUX
iajs-3028	101	8	an	an	DET
iajs-3028	101	9	obvious	obvious	ADJ
iajs-3028	101	10	.	.	PUNCT
iajs-3028	102	1	now	now	ADV
iajs-3028	102	2	,	,	PUNCT
iajs-3028	102	3	to	to	PART
iajs-3028	102	4	prove	prove	VERB
iajs-3028	102	5	𝑚	𝑚	ADP
iajs-3028	102	6	∘	∘	NUM
iajs-3028	102	7	𝑑(ℳ	𝑑(ℳ	NOUN
iajs-3028	102	8	)	)	PUNCT
iajs-3028	102	9	=	=	SYM
iajs-3028	102	10	𝑑(ℳ	𝑑(ℳ	NOUN
iajs-3028	102	11	)	)	PUNCT
iajs-3028	102	12	∘	∘	NOUN
iajs-3028	102	13	𝑚	𝑚	NOUN
iajs-3028	102	14	=	=	SYM
iajs-3028	102	15	𝑑(ℳ	𝑑(ℳ	PROPN
iajs-3028	102	16	)	)	PUNCT
iajs-3028	102	17	,	,	PUNCT
iajs-3028	102	18	let	let	VERB
iajs-3028	102	19	𝑥	𝑥	DET
iajs-3028	102	20	∈	∈	VERB
iajs-3028	102	21	𝑚	𝑚	ADP
iajs-3028	102	22	∘	∘	PROPN
iajs-3028	102	23	𝑑(ℳ	𝑑(ℳ	PROPN
iajs-3028	102	24	)	)	PUNCT
iajs-3028	102	25	.	.	PUNCT
iajs-3028	103	1	it	it	PRON
iajs-3028	103	2	is	be	AUX
iajs-3028	103	3	mean	mean	VERB
iajs-3028	103	4	𝑥	𝑥	DET
iajs-3028	103	5	∈	∈	PROPN
iajs-3028	103	6	𝑚	𝑚	ADP
iajs-3028	103	7	∘	∘	PROPN
iajs-3028	103	8	𝑦	𝑦	NUM
iajs-3028	103	9	;	;	PUNCT
iajs-3028	103	10	𝑦	𝑦	NUM
iajs-3028	103	11	∈	∈	NOUN
iajs-3028	103	12	d(ℳ	d(ℳ	NOUN
iajs-3028	103	13	)	)	PUNCT
iajs-3028	103	14	.	.	PUNCT
iajs-3028	104	1	for	for	ADP
iajs-3028	104	2	each	each	DET
iajs-3028	104	3	𝑟	𝑟	PRON
iajs-3028	104	4	∈	∈	PROPN
iajs-3028	104	5	𝑅𝑜	𝑅𝑜	PROPN
iajs-3028	104	6	there	there	PRON
iajs-3028	104	7	is	be	VERB
iajs-3028	104	8	𝑧	𝑧	DET
iajs-3028	104	9	∈	∈	PROPN
iajs-3028	104	10	ℳ	ℳ	NOUN
iajs-3028	104	11	such	such	ADJ
iajs-3028	104	12	that	that	PRON
iajs-3028	104	13	𝑦	𝑦	NOUN
iajs-3028	104	14	=	=	SYM
iajs-3028	104	15	𝑟𝑧	𝑟𝑧	ADJ
iajs-3028	104	16	;	;	PUNCT
iajs-3028	104	17	𝑚	𝑚	PROPN
iajs-3028	104	18	∈	∈	PROPN
iajs-3028	104	19	𝑑(ℳ	𝑑(ℳ	PROPN
iajs-3028	104	20	)	)	PUNCT
iajs-3028	104	21	𝑚𝑒𝑎𝑛	𝑚𝑒𝑎𝑛	NOUN
iajs-3028	104	22	𝑡ℎ𝑎𝑡	𝑡ℎ𝑎𝑡	VERB
iajs-3028	104	23	𝑚	𝑚	X
iajs-3028	104	24	=	=	SYM
iajs-3028	104	25	𝑟𝑤,𝑤	𝑟𝑤,𝑤	PUNCT
iajs-3028	104	26	∈	∈	PROPN
iajs-3028	104	27	ℳ.	ℳ.	PROPN
iajs-3028	104	28	follows	follow	VERB
iajs-3028	104	29	𝑥	𝑥	PRON
iajs-3028	104	30	∈	∈	NOUN
iajs-3028	104	31	(	(	PUNCT
iajs-3028	104	32	𝑟𝑤	𝑟𝑤	NOUN
iajs-3028	104	33	)	)	PUNCT
iajs-3028	104	34	∘	∘	PROPN
iajs-3028	104	35	(	(	PUNCT
iajs-3028	104	36	𝑟𝑧	𝑟𝑧	PROPN
iajs-3028	104	37	)	)	PUNCT
iajs-3028	104	38	,	,	PUNCT
iajs-3028	104	39	lead	lead	VERB
iajs-3028	104	40	to	to	ADP
iajs-3028	104	41	𝑥	𝑥	DET
iajs-3028	104	42	∈	∈	PROPN
iajs-3028	104	43	𝑟(𝑤	𝑟(𝑤	PROPN
iajs-3028	104	44	∘	∘	NOUN
iajs-3028	104	45	𝑧	𝑧	PART
iajs-3028	104	46	)	)	PUNCT
iajs-3028	104	47	∈	∈	PROPN
iajs-3028	104	48	𝑑(ℳ	𝑑(ℳ	PROPN
iajs-3028	104	49	)	)	PUNCT
iajs-3028	104	50	.	.	PUNCT
iajs-3028	105	1	in	in	ADP
iajs-3028	105	2	the	the	DET
iajs-3028	105	3	same	same	ADJ
iajs-3028	105	4	way	way	NOUN
iajs-3028	105	5	𝑑(ℳ	𝑑(ℳ	NOUN
iajs-3028	105	6	)	)	PUNCT
iajs-3028	105	7	∘	∘	PART
iajs-3028	105	8	𝑚	𝑚	NOUN
iajs-3028	105	9	=	=	SYM
iajs-3028	105	10	𝑑(ℳ	𝑑(ℳ	PROPN
iajs-3028	105	11	)	)	PUNCT
iajs-3028	105	12	can	can	AUX
iajs-3028	105	13	be	be	AUX
iajs-3028	105	14	proved	prove	VERB
iajs-3028	105	15	.	.	PUNCT
iajs-3028	106	1	proposition	proposition	NOUN
iajs-3028	106	2	3.3	3.3	NUM
iajs-3028	106	3	.	.	PUNCT
iajs-3028	107	1	if	if	SCONJ
iajs-3028	107	2	the	the	DET
iajs-3028	107	3	hyperstructure	hyperstructure	NOUN
iajs-3028	107	4	(	(	PUNCT
iajs-3028	107	5	ℛ,⊚,⊙	ℛ,⊚,⊙	PROPN
iajs-3028	107	6	)	)	PUNCT
iajs-3028	107	7	is	be	AUX
iajs-3028	107	8	commutative	commutative	ADJ
iajs-3028	107	9	hyperring	hyperring	NOUN
iajs-3028	107	10	then	then	ADV
iajs-3028	107	11	;	;	PUNCT
iajs-3028	107	12	1	1	X
iajs-3028	107	13	.	.	NUM
iajs-3028	107	14	d(ℳ	d(ℳ	NOUN
iajs-3028	107	15	)	)	PUNCT
iajs-3028	107	16	is	be	AUX
iajs-3028	107	17	divisible	divisible	ADJ
iajs-3028	107	18	ℛ-subhypermodule	ℛ-subhypermodule	PROPN
iajs-3028	107	19	of	of	ADP
iajs-3028	107	20	an	an	DET
iajs-3028	107	21	ℛ-hypermodule	ℛ-hypermodule	PROPN
iajs-3028	107	22	ℳ	ℳ	PROPN
iajs-3028	107	23	2	2	NUM
iajs-3028	107	24	.	.	NOUN
iajs-3028	107	25	d(ℳ	d(ℳ	NOUN
iajs-3028	107	26	∕d(ℳ	∕d(ℳ	NUM
iajs-3028	107	27	)	)	PUNCT
iajs-3028	107	28	)	)	PUNCT
iajs-3028	108	1	=	=	PUNCT
iajs-3028	108	2	0	0	X
iajs-3028	108	3	.	.	PUNCT
iajs-3028	109	1	proof	proof	NOUN
iajs-3028	109	2	.	.	PUNCT
iajs-3028	110	1	1	1	X
iajs-3028	110	2	.	.	X
iajs-3028	110	3	let	let	VERB
iajs-3028	110	4	𝑚	𝑚	NOUN
iajs-3028	110	5	,	,	PUNCT
iajs-3028	110	6	𝑛	𝑛	DET
iajs-3028	110	7	∈	∈	PROPN
iajs-3028	110	8	d(ℳ	d(ℳ	NOUN
iajs-3028	110	9	)	)	PUNCT
iajs-3028	110	10	,	,	PUNCT
iajs-3028	110	11	for	for	ADP
iajs-3028	110	12	any	any	DET
iajs-3028	110	13	𝑟	𝑟	PRON
iajs-3028	110	14	∈	∈	PROPN
iajs-3028	110	15	𝑅𝑂	𝑅𝑂	PROPN
iajs-3028	110	16	,	,	PUNCT
iajs-3028	110	17	𝑚	𝑚	X
iajs-3028	110	18	=	=	SYM
iajs-3028	110	19	𝑟𝑚ˊ	𝑟𝑚ˊ	X
iajs-3028	110	20	and	and	CCONJ
iajs-3028	110	21	𝑛	𝑛	NOUN
iajs-3028	110	22	=	=	NOUN
iajs-3028	110	23	𝑟𝑛ˊ.	𝑟𝑛ˊ.	NUM
iajs-3028	110	24	so	so	ADV
iajs-3028	110	25	𝑚	𝑚	ADP
iajs-3028	110	26	−	−	PROPN
iajs-3028	111	1	𝑛	𝑛	PROPN
iajs-3028	111	2	=	=	PUNCT
iajs-3028	111	3	𝑟𝑚ˊ	𝑟𝑚ˊ	NOUN
iajs-3028	111	4	−	−	PROPN
iajs-3028	111	5	𝑟𝑛ˊ	𝑟𝑛ˊ	PROPN
iajs-3028	111	6	=	=	SYM
iajs-3028	111	7	𝑟(𝑚ˊ	𝑟(𝑚ˊ	PROPN
iajs-3028	111	8	−	−	PROPN
iajs-3028	111	9	𝑛ˊ	𝑛ˊ	PROPN
iajs-3028	111	10	)	)	PUNCT
iajs-3028	111	11	∈	∈	PROPN
iajs-3028	111	12	d(ℳ	d(ℳ	NOUN
iajs-3028	111	13	)	)	PUNCT
iajs-3028	111	14	.	.	PUNCT
iajs-3028	112	1	now	now	ADV
iajs-3028	112	2	,	,	PUNCT
iajs-3028	112	3	if	if	SCONJ
iajs-3028	112	4	𝑦	𝑦	NOUN
iajs-3028	112	5	∈	∈	NOUN
iajs-3028	112	6	d(ℳ	d(ℳ	NOUN
iajs-3028	112	7	)	)	PUNCT
iajs-3028	112	8	,	,	PUNCT
iajs-3028	112	9	0	0	NUM
iajs-3028	112	10	≠	≠	PROPN
iajs-3028	112	11	𝑎	𝑎	PRON
iajs-3028	112	12	∈	∈	NOUN
iajs-3028	112	13	ℛ	ℛ	NOUN
iajs-3028	112	14	and	and	CCONJ
iajs-3028	112	15	for	for	ADP
iajs-3028	112	16	any	any	DET
iajs-3028	112	17	r∈	r∈	PROPN
iajs-3028	112	18	𝑅𝑂	𝑅𝑂	PROPN
iajs-3028	112	19	,	,	PUNCT
iajs-3028	112	20	∃𝑥	∃𝑥	PROPN
iajs-3028	112	21	∈	∈	PROPN
iajs-3028	112	22	ℳ	ℳ	VERB
iajs-3028	112	23	such	such	ADJ
iajs-3028	112	24	that	that	SCONJ
iajs-3028	112	25	,	,	PUNCT
iajs-3028	112	26	𝑦	𝑦	NOUN
iajs-3028	112	27	=	=	X
iajs-3028	112	28	𝑟𝑥.	𝑟𝑥.	NOUN
iajs-3028	112	29	therefore	therefore	ADV
iajs-3028	112	30	,	,	PUNCT
iajs-3028	112	31	𝑎𝑦	𝑎𝑦	PROPN
iajs-3028	112	32	=	=	PUNCT
iajs-3028	112	33	𝑎(𝑟𝑥	𝑎(𝑟𝑥	NOUN
iajs-3028	112	34	)	)	PUNCT
iajs-3028	112	35	=	=	PUNCT
iajs-3028	112	36	(	(	PUNCT
iajs-3028	112	37	𝑎𝑟)𝑥	𝑎𝑟)𝑥	PROPN
iajs-3028	112	38	=	=	SYM
iajs-3028	112	39	(	(	PUNCT
iajs-3028	112	40	𝑟𝑎)𝑥	𝑟𝑎)𝑥	NOUN
iajs-3028	112	41	=	=	SYM
iajs-3028	112	42	𝑟(𝑎𝑥	𝑟(𝑎𝑥	NOUN
iajs-3028	112	43	)	)	PUNCT
iajs-3028	112	44	.	.	PUNCT
iajs-3028	113	1	this	this	PRON
iajs-3028	113	2	implies	imply	VERB
iajs-3028	113	3	that	that	SCONJ
iajs-3028	113	4	𝑎𝑦	𝑎𝑦	PROPN
iajs-3028	113	5	∈	∈	PROPN
iajs-3028	113	6	𝑑(ℳ	𝑑(ℳ	PROPN
iajs-3028	113	7	)	)	PUNCT
iajs-3028	113	8	.	.	PUNCT
iajs-3028	114	1	2	2	X
iajs-3028	114	2	.	.	X
iajs-3028	115	1	if	if	SCONJ
iajs-3028	115	2	y	y	PROPN
iajs-3028	115	3	+	+	NOUN
iajs-3028	115	4	d(ℳ	d(ℳ	NOUN
iajs-3028	115	5	)	)	PUNCT
iajs-3028	115	6	∈	∈	PROPN
iajs-3028	115	7	d(ℳ	d(ℳ	NOUN
iajs-3028	115	8	∕d(ℳ	∕d(ℳ	NOUN
iajs-3028	115	9	)	)	PUNCT
iajs-3028	115	10	)	)	PUNCT
iajs-3028	115	11	for	for	ADP
iajs-3028	115	12	y∈d(ℳ	y∈d(ℳ	NUM
iajs-3028	115	13	)	)	PUNCT
iajs-3028	115	14	.	.	PUNCT
iajs-3028	116	1	then	then	ADV
iajs-3028	116	2	∀	∀	VERB
iajs-3028	116	3	𝑟	𝑟	X
iajs-3028	116	4	∈	∈	PROPN
iajs-3028	116	5	𝑅𝑂	𝑅𝑂	PROPN
iajs-3028	116	6	,	,	PUNCT
iajs-3028	116	7	∃	∃	PROPN
iajs-3028	116	8	x+d(ℳ	x+d(ℳ	ADV
iajs-3028	116	9	)	)	PUNCT
iajs-3028	116	10	∈	∈	PROPN
iajs-3028	116	11	ℳ	ℳ	PROPN
iajs-3028	116	12	∕d(ℳ	∕d(ℳ	NOUN
iajs-3028	116	13	)	)	PUNCT
iajs-3028	116	14	such	such	ADJ
iajs-3028	116	15	that	that	SCONJ
iajs-3028	116	16	,	,	PUNCT
iajs-3028	116	17	y+d(ℳ	y+d(ℳ	ADV
iajs-3028	116	18	)	)	PUNCT
iajs-3028	116	19	=	=	PUNCT
iajs-3028	116	20	r(x+d(ℳ	r(x+d(ℳ	NUM
iajs-3028	116	21	)	)	PUNCT
iajs-3028	116	22	)	)	PUNCT
iajs-3028	116	23	.	.	PUNCT
iajs-3028	117	1	thus	thus	ADV
iajs-3028	117	2	y	y	X
iajs-3028	117	3	-	-	PUNCT
iajs-3028	117	4	rx	rx	VERB
iajs-3028	117	5	∈	∈	NOUN
iajs-3028	117	6	d(ℳ	d(ℳ	NOUN
iajs-3028	117	7	)	)	PUNCT
iajs-3028	117	8	.	.	PUNCT
iajs-3028	118	1	this	this	PRON
iajs-3028	118	2	implies	imply	VERB
iajs-3028	118	3	that	that	SCONJ
iajs-3028	118	4	∃𝑥ˊ	∃𝑥ˊ	PROPN
iajs-3028	118	5	∈	∈	PROPN
iajs-3028	118	6	ℳ	ℳ	PROPN
iajs-3028	118	7	such	such	ADJ
iajs-3028	118	8	that	that	PRON
iajs-3028	118	9	,	,	PUNCT
iajs-3028	118	10	(	(	PUNCT
iajs-3028	118	11	y	y	NOUN
iajs-3028	118	12	-	-	PUNCT
iajs-3028	118	13	rx	rx	NOUN
iajs-3028	118	14	)	)	PUNCT
iajs-3028	118	15	=	=	PUNCT
iajs-3028	118	16	r𝑥ˊ.	r𝑥ˊ.	NUM
iajs-3028	118	17	it	it	PRON
iajs-3028	118	18	follows	follow	VERB
iajs-3028	118	19	that	that	SCONJ
iajs-3028	118	20	y	y	PROPN
iajs-3028	118	21	=	=	NOUN
iajs-3028	118	22	r(x+𝑥ˊ	r(x+𝑥ˊ	PROPN
iajs-3028	118	23	)	)	PUNCT
iajs-3028	118	24	and	and	CCONJ
iajs-3028	118	25	y	y	PROPN
iajs-3028	118	26	∈	∈	PROPN
iajs-3028	118	27	d(ℳ	d(ℳ	PROPN
iajs-3028	118	28	)	)	PUNCT
iajs-3028	118	29	or	or	CCONJ
iajs-3028	118	30	equivalently	equivalently	ADV
iajs-3028	118	31	y	y	PROPN
iajs-3028	118	32	+	+	NOUN
iajs-3028	118	33	d(ℳ	d(ℳ	NOUN
iajs-3028	118	34	)	)	PUNCT
iajs-3028	118	35	=	=	SYM
iajs-3028	118	36	d(ℳ	d(ℳ	NOUN
iajs-3028	118	37	)	)	PUNCT
iajs-3028	118	38	,	,	PUNCT
iajs-3028	118	39	thus	thus	ADV
iajs-3028	118	40	d(ℳ	d(ℳ	NOUN
iajs-3028	118	41	∕d(ℳ	∕d(ℳ	NUM
iajs-3028	118	42	)	)	PUNCT
iajs-3028	118	43	)	)	PUNCT
iajs-3028	119	1	=	=	PUNCT
iajs-3028	119	2	0	0	X
iajs-3028	119	3	.	.	PUNCT
iajs-3028	119	4	remark	remark	PROPN
iajs-3028	119	5	3.4	3.4	NUM
iajs-3028	119	6	.	.	PUNCT
iajs-3028	120	1	the	the	DET
iajs-3028	120	2	set	set	NOUN
iajs-3028	120	3	d(ℛ	d(ℛ	X
iajs-3028	120	4	)	)	PUNCT
iajs-3028	121	1	=	=	PRON
iajs-3028	121	2	{	{	PUNCT
iajs-3028	121	3	a	a	DET
iajs-3028	121	4	∈	∈	NOUN
iajs-3028	121	5	ℛ|	ℛ|	NOUN
iajs-3028	121	6	for	for	ADP
iajs-3028	121	7	all	all	DET
iajs-3028	121	8	r	r	NOUN
iajs-3028	121	9	∈	∈	PROPN
iajs-3028	121	10	𝑅𝑂	𝑅𝑂	PROPN
iajs-3028	121	11	,	,	PUNCT
iajs-3028	121	12	there	there	PRON
iajs-3028	121	13	is	be	VERB
iajs-3028	121	14	b	b	NUM
iajs-3028	121	15	∈	∈	NOUN
iajs-3028	121	16	ℛ	ℛ	NOUN
iajs-3028	121	17	such	such	ADJ
iajs-3028	121	18	that	that	PRON
iajs-3028	121	19	.	.	PUNCT
iajs-3028	122	1	a	a	DET
iajs-3028	122	2	=	=	NOUN
iajs-3028	122	3	rb	rb	NOUN
iajs-3028	122	4	}	}	PUNCT
iajs-3028	122	5	.	.	PUNCT
iajs-3028	123	1	remark	remark	NOUN
iajs-3028	123	2	3.5	3.5	NUM
iajs-3028	123	3	.	.	PUNCT
iajs-3028	124	1	the	the	DET
iajs-3028	124	2	set	set	NOUN
iajs-3028	124	3	d(ℛ	d(ℛ	PROPN
iajs-3028	124	4	)	)	PUNCT
iajs-3028	124	5	is	be	AUX
iajs-3028	124	6	a	a	DET
iajs-3028	124	7	right	right	ADJ
iajs-3028	124	8	hyperideal	hyperideal	NOUN
iajs-3028	124	9	of	of	ADP
iajs-3028	124	10	ℛ.	ℛ.	PROPN
iajs-3028	124	11	indeed	indeed	ADV
iajs-3028	124	12	,	,	PUNCT
iajs-3028	124	13	for	for	ADP
iajs-3028	124	14	any	any	DET
iajs-3028	124	15	y	y	PROPN
iajs-3028	124	16	and	and	CCONJ
iajs-3028	124	17	x	x	NOUN
iajs-3028	124	18	in	in	ADP
iajs-3028	124	19	d(ℛ	d(ℛ	PROPN
iajs-3028	124	20	)	)	PUNCT
iajs-3028	124	21	,	,	PUNCT
iajs-3028	124	22	y	y	PROPN
iajs-3028	124	23	=	=	NOUN
iajs-3028	124	24	rs	rs	X
iajs-3028	124	25	and	and	CCONJ
iajs-3028	124	26	x	x	X
iajs-3028	124	27	=	=	NOUN
iajs-3028	124	28	ra	ra	PROPN
iajs-3028	124	29	,	,	PUNCT
iajs-3028	124	30	for	for	ADP
iajs-3028	124	31	all	all	DET
iajs-3028	124	32	s	s	NOUN
iajs-3028	124	33	and	and	CCONJ
iajs-3028	124	34	a	a	DET
iajs-3028	124	35	belong	belong	NOUN
iajs-3028	124	36	to	to	ADP
iajs-3028	124	37	ℛ	ℛ	PROPN
iajs-3028	124	38	,	,	PUNCT
iajs-3028	124	39	y	y	PROPN
iajs-3028	124	40	-	-	PUNCT
iajs-3028	124	41	x	x	NOUN
iajs-3028	124	42	⊆d(ℛ	⊆d(ℛ	NOUN
iajs-3028	124	43	)	)	PUNCT
iajs-3028	124	44	.	.	PUNCT
iajs-3028	125	1	also	also	ADV
iajs-3028	125	2	for	for	ADP
iajs-3028	125	3	any	any	DET
iajs-3028	125	4	r	r	NOUN
iajs-3028	125	5	∈	∈	NOUN
iajs-3028	125	6	𝑅𝑜	𝑅𝑜	ADP
iajs-3028	125	7	there	there	PRON
iajs-3028	125	8	is	be	VERB
iajs-3028	125	9	s	s	PROPN
iajs-3028	125	10	∈	∈	NOUN
iajs-3028	125	11	ℛ	ℛ	NOUN
iajs-3028	125	12	such	such	ADJ
iajs-3028	125	13	that	that	SCONJ
iajs-3028	125	14	,	,	PUNCT
iajs-3028	125	15	y	y	PROPN
iajs-3028	125	16	=	=	NOUN
iajs-3028	125	17	rs	rs	ADJ
iajs-3028	125	18	,	,	PUNCT
iajs-3028	125	19	thus	thus	ADV
iajs-3028	125	20	yt	yt	VERB
iajs-3028	125	21	=	=	SYM
iajs-3028	125	22	(	(	PUNCT
iajs-3028	125	23	rs)t	rs)t	NOUN
iajs-3028	125	24	=	=	SYM
iajs-3028	125	25	r(st	r(st	NOUN
iajs-3028	125	26	)	)	PUNCT
iajs-3028	125	27	,	,	PUNCT
iajs-3028	125	28	this	this	PRON
iajs-3028	125	29	implies	imply	VERB
iajs-3028	125	30	yt	yt	PROPN
iajs-3028	125	31	∈	∈	PROPN
iajs-3028	125	32	d(ℛ	d(ℛ	PROPN
iajs-3028	125	33	)	)	PUNCT
iajs-3028	125	34	,	,	PUNCT
iajs-3028	125	35	t∈	t∈	VERB
iajs-3028	125	36	ℛ.	ℛ.	PROPN
iajs-3028	125	37	definition	definition	NOUN
iajs-3028	125	38	3.6	3.6	NUM
iajs-3028	125	39	[	[	X
iajs-3028	125	40	3	3	NUM
iajs-3028	125	41	]	]	PUNCT
iajs-3028	125	42	.	.	PUNCT
iajs-3028	126	1	a	a	DET
iajs-3028	126	2	hyperring	hyperring	NOUN
iajs-3028	126	3	(	(	PUNCT
iajs-3028	126	4	ℛ,⊚,⊙	ℛ,⊚,⊙	PROPN
iajs-3028	126	5	)	)	PUNCT
iajs-3028	126	6	is	be	AUX
iajs-3028	126	7	a	a	DET
iajs-3028	126	8	division	division	NOUN
iajs-3028	126	9	hyperring	hyperre	VERB
iajs-3028	126	10	if	if	SCONJ
iajs-3028	126	11	(	(	PUNCT
iajs-3028	126	12	ℛ\{0},⨀	ℛ\{0},⨀	NOUN
iajs-3028	126	13	)	)	PUNCT
iajs-3028	126	14	is	be	AUX
iajs-3028	126	15	a	a	DET
iajs-3028	126	16	hypergroup	hypergroup	NOUN
iajs-3028	126	17	with	with	ADP
iajs-3028	126	18	unit	unit	NOUN
iajs-3028	126	19	element	element	NOUN
iajs-3028	126	20	1	1	NUM
iajs-3028	126	21	.	.	PUNCT
iajs-3028	126	22	proposition	proposition	NOUN
iajs-3028	126	23	3.7	3.7	NUM
iajs-3028	126	24	.	.	PUNCT
iajs-3028	127	1	if	if	SCONJ
iajs-3028	127	2	ℛ	ℛ	PROPN
iajs-3028	127	3	is	be	AUX
iajs-3028	127	4	a	a	DET
iajs-3028	127	5	division	division	NOUN
iajs-3028	127	6	hyperring	hyperring	NOUN
iajs-3028	127	7	,	,	PUNCT
iajs-3028	127	8	then	then	ADV
iajs-3028	127	9	d(ℛ	d(ℛ	X
iajs-3028	127	10	)	)	PUNCT
iajs-3028	128	1	=	=	SYM
iajs-3028	128	2	ℛ.	ℛ.	NOUN
iajs-3028	128	3	proof	proof	NOUN
iajs-3028	128	4	.	.	PUNCT
iajs-3028	129	1	it	it	PRON
iajs-3028	129	2	is	be	AUX
iajs-3028	129	3	clear	clear	ADJ
iajs-3028	129	4	that	that	SCONJ
iajs-3028	129	5	d(ℛ)⊆	d(ℛ)⊆	NUM
iajs-3028	129	6	ℛ.	ℛ.	PROPN
iajs-3028	129	7	now	now	ADV
iajs-3028	129	8	,	,	PUNCT
iajs-3028	129	9	for	for	ADP
iajs-3028	129	10	any	any	DET
iajs-3028	129	11	a	a	DET
iajs-3028	129	12	∈	∈	PROPN
iajs-3028	129	13	ℛ	ℛ	NOUN
iajs-3028	129	14	and	and	CCONJ
iajs-3028	129	15	any	any	DET
iajs-3028	129	16	r	r	NOUN
iajs-3028	129	17	∈	∈	PROPN
iajs-3028	129	18	𝑅𝑂=ℛ∗	𝑅𝑂=ℛ∗	NOUN
iajs-3028	129	19	,	,	PUNCT
iajs-3028	129	20	∃	∃	PROPN
iajs-3028	129	21	b	b	PROPN
iajs-3028	129	22	=	=	X
iajs-3028	129	23	𝑟−1a	𝑟−1a	PUNCT
iajs-3028	129	24	∈	∈	PROPN
iajs-3028	129	25	ℛ	ℛ	NOUN
iajs-3028	129	26	such	such	ADJ
iajs-3028	129	27	that	that	SCONJ
iajs-3028	129	28	a	a	DET
iajs-3028	129	29	=	=	SYM
iajs-3028	129	30	rb	rb	NOUN
iajs-3028	129	31	,	,	PUNCT
iajs-3028	129	32	therefore	therefore	ADV
iajs-3028	129	33	a	a	DET
iajs-3028	129	34	∈	∈	NOUN
iajs-3028	129	35	d(ℛ	d(ℛ	X
iajs-3028	129	36	)	)	PUNCT
iajs-3028	129	37	.	.	PUNCT
iajs-3028	130	1	proposition	proposition	NOUN
iajs-3028	130	2	3.8	3.8	NUM
iajs-3028	130	3	.	.	PUNCT
iajs-3028	131	1	let	let	VERB
iajs-3028	131	2	ℛ	ℛ	NOUN
iajs-3028	131	3	be	be	AUX
iajs-3028	131	4	a	a	DET
iajs-3028	131	5	hyperring	hyperring	NOUN
iajs-3028	131	6	that	that	PRON
iajs-3028	131	7	satisfies	satisfy	VERB
iajs-3028	131	8	the	the	DET
iajs-3028	131	9	property	property	NOUN
iajs-3028	131	10	d(ℛ)∩	d(ℛ)∩	PROPN
iajs-3028	131	11	𝑅𝑂	𝑅𝑂	PROPN
iajs-3028	131	12	≠	≠	PROPN
iajs-3028	131	13	∅	∅	NOUN
iajs-3028	131	14	,	,	PUNCT
iajs-3028	131	15	then	then	ADV
iajs-3028	131	16	:	:	PUNCT
iajs-3028	131	17	d(ℛ)=	d(ℛ)=	NOUN
iajs-3028	131	18	∩	∩	NOUN
iajs-3028	131	19	{	{	PUNCT
iajs-3028	131	20	𝑅ˆ|	𝑅ˆ|	NOUN
iajs-3028	131	21	𝑅ˆ	𝑅ˆ	PROPN
iajs-3028	131	22	is	be	AUX
iajs-3028	131	23	a	a	DET
iajs-3028	131	24	left	left	ADJ
iajs-3028	131	25	hyperideal	hyperideal	NOUN
iajs-3028	131	26	of	of	ADP
iajs-3028	131	27	hyperring	hyperre	VERB
iajs-3028	131	28	ℛ	ℛ	PROPN
iajs-3028	131	29	such	such	ADJ
iajs-3028	131	30	that	that	PRON
iajs-3028	131	31	.	.	PUNCT
iajs-3028	132	1	𝑅ˆ	𝑅ˆ	ADJ
iajs-3028	132	2	∩	∩	ADJ
iajs-3028	132	3	𝑅𝑂	𝑅𝑂	PROPN
iajs-3028	132	4	≠	≠	PROPN
iajs-3028	132	5	∅	∅	NOUN
iajs-3028	132	6	}	}	PUNCT
iajs-3028	132	7	.	.	PUNCT
iajs-3028	133	1	proof	proof	NOUN
iajs-3028	133	2	.	.	PUNCT
iajs-3028	134	1	it	it	PRON
iajs-3028	134	2	’s	’	VERB
iajs-3028	134	3	clear	clear	ADJ
iajs-3028	134	4	that	that	SCONJ
iajs-3028	134	5	∩{rˆ|	∩{rˆ|	ADP
iajs-3028	134	6	rˆ	rˆ	PROPN
iajs-3028	134	7	is	be	AUX
iajs-3028	134	8	a	a	DET
iajs-3028	134	9	left	left	ADJ
iajs-3028	134	10	hyperideal	hyperideal	NOUN
iajs-3028	134	11	in	in	ADP
iajs-3028	134	12	a	a	DET
iajs-3028	134	13	hyperring	hyperre	VERB
iajs-3028	134	14	ℛ	ℛ	NOUN
iajs-3028	134	15	,	,	PUNCT
iajs-3028	134	16	such	such	ADJ
iajs-3028	134	17	that	that	SCONJ
iajs-3028	134	18	rˆ∩𝑅𝑂≠∅	rˆ∩𝑅𝑂≠∅	NOUN
iajs-3028	134	19	}	}	PUNCT
iajs-3028	134	20	⊆	⊆	NUM
iajs-3028	134	21	d(ℛ	d(ℛ	NUM
iajs-3028	134	22	)	)	PUNCT
iajs-3028	134	23	.	.	PUNCT
iajs-3028	135	1	now	now	ADV
iajs-3028	135	2	to	to	PART
iajs-3028	135	3	prove	prove	VERB
iajs-3028	135	4	the	the	DET
iajs-3028	135	5	converse	converse	NOUN
iajs-3028	135	6	inclusion	inclusion	NOUN
iajs-3028	135	7	,	,	PUNCT
iajs-3028	135	8	let	let	VERB
iajs-3028	135	9	y	y	PROPN
iajs-3028	135	10	∈	∈	PROPN
iajs-3028	135	11	d(ℛ	d(ℛ	PROPN
iajs-3028	135	12	)	)	PUNCT
iajs-3028	135	13	and	and	CCONJ
iajs-3028	135	14	𝑅ˆ	𝑅ˆ	PROPN
iajs-3028	135	15	be	be	VERB
iajs-3028	135	16	the	the	DET
iajs-3028	135	17	left	left	ADJ
iajs-3028	135	18	hyperideal	hyperideal	NOUN
iajs-3028	135	19	of	of	ADP
iajs-3028	135	20	a	a	DET
iajs-3028	135	21	hyperring	hyperre	VERB
iajs-3028	135	22	ℛ	ℛ	NOUN
iajs-3028	135	23	such	such	ADJ
iajs-3028	135	24	that	that	PRON
iajs-3028	135	25	,	,	PUNCT
iajs-3028	135	26	𝑅ˆ	𝑅ˆ	PROPN
iajs-3028	135	27	∩	∩	ADJ
iajs-3028	135	28	𝑅𝑂	𝑅𝑂	PROPN
iajs-3028	135	29	≠	≠	PROPN
iajs-3028	135	30	∅	∅	NOUN
iajs-3028	135	31	,	,	PUNCT
iajs-3028	135	32	let	let	VERB
iajs-3028	135	33	𝑦𝑜	𝑦𝑜	ADP
iajs-3028	135	34	∈	∈	PROPN
iajs-3028	135	35	𝑅	𝑅	PROPN
iajs-3028	135	36	ˆ	ˆ	PROPN
iajs-3028	135	37	∩	∩	ADJ
iajs-3028	135	38	𝑅𝑂	𝑅𝑂	PROPN
iajs-3028	135	39	,	,	PUNCT
iajs-3028	135	40	then	then	ADV
iajs-3028	135	41	for	for	ADP
iajs-3028	135	42	any	any	DET
iajs-3028	135	43	y∈	y∈	NOUN
iajs-3028	135	44	d(ℛ	d(ℛ	PROPN
iajs-3028	135	45	)	)	PUNCT
iajs-3028	135	46	,	,	PUNCT
iajs-3028	135	47	there	there	PRON
iajs-3028	135	48	is	be	VERB
iajs-3028	135	49	s	s	PROPN
iajs-3028	135	50	∈	∈	NOUN
iajs-3028	135	51	ℛ	ℛ	NOUN
iajs-3028	135	52	such	such	ADJ
iajs-3028	135	53	that	that	SCONJ
iajs-3028	135	54	,	,	PUNCT
iajs-3028	136	1	y	y	PROPN
iajs-3028	136	2	=	=	NOUN
iajs-3028	136	3	𝑦𝑜s	𝑦𝑜s	NOUN
iajs-3028	136	4	∈	∈	PROPN
iajs-3028	136	5	𝑅ˆ	𝑅ˆ	PROPN
iajs-3028	136	6	this	this	PRON
iajs-3028	136	7	gives	give	VERB
iajs-3028	136	8	d(ℛ	d(ℛ	PROPN
iajs-3028	136	9	)	)	PUNCT
iajs-3028	137	1	⊆	⊆	X
iajs-3028	137	2	𝑅ˆ.	𝑅ˆ.	ADJ
iajs-3028	137	3	since	since	SCONJ
iajs-3028	137	4	𝑅ˆ	𝑅ˆ	PROPN
iajs-3028	137	5	is	be	AUX
iajs-3028	137	6	an	an	DET
iajs-3028	137	7	arbitrary	arbitrary	ADJ
iajs-3028	137	8	hyperideal	hyperideal	NOUN
iajs-3028	137	9	,	,	PUNCT
iajs-3028	137	10	so	so	ADV
iajs-3028	137	11	y	y	PROPN
iajs-3028	137	12	∈	∈	PROPN
iajs-3028	137	13	∩{rˆ	∩{rˆ	PROPN
iajs-3028	137	14	}	}	PUNCT
iajs-3028	137	15	.	.	PUNCT
iajs-3028	138	1	ihjpas	ihjpas	PROPN
iajs-3028	138	2	.	.	PUNCT
iajs-3028	139	1	36(2)2023	36(2)2023	NUM
iajs-3028	139	2	387	387	NUM
iajs-3028	139	3	proposition	proposition	NOUN
iajs-3028	139	4	3.9	3.9	NUM
iajs-3028	139	5	.	.	PUNCT
iajs-3028	140	1	if	if	SCONJ
iajs-3028	140	2	ℛ	ℛ	PROPN
iajs-3028	140	3	is	be	AUX
iajs-3028	140	4	commutative	commutative	ADJ
iajs-3028	140	5	hyperring	hyperring	NOUN
iajs-3028	140	6	and	and	CCONJ
iajs-3028	140	7	𝑅𝑂	𝑅𝑂	PROPN
iajs-3028	140	8	∩	∩	NOUN
iajs-3028	140	9	𝐽(ℛ	𝐽(ℛ	NUM
iajs-3028	140	10	)	)	PUNCT
iajs-3028	140	11	≠	≠	PROPN
iajs-3028	140	12	∅	∅	NOUN
iajs-3028	140	13	,	,	PUNCT
iajs-3028	140	14	then	then	ADV
iajs-3028	140	15	1+d(ℛ	1+d(ℛ	NUM
iajs-3028	140	16	)	)	PUNCT
iajs-3028	140	17	⊆	⊆	NUM
iajs-3028	140	18	𝑈(ℛ	𝑈(ℛ	ADJ
iajs-3028	140	19	)	)	PUNCT
iajs-3028	140	20	.	.	PUNCT
iajs-3028	141	1	proof	proof	NOUN
iajs-3028	141	2	.	.	PUNCT
iajs-3028	142	1	let	let	VERB
iajs-3028	142	2	𝑟𝑜	𝑟𝑜	PRON
iajs-3028	142	3	∈	∈	PROPN
iajs-3028	142	4	𝑅	𝑅	PROPN
iajs-3028	142	5	𝑂	𝑂	PROPN
iajs-3028	142	6	∩	∩	NOUN
iajs-3028	142	7	𝐽(ℛ	𝐽(ℛ	NUM
iajs-3028	142	8	)	)	PUNCT
iajs-3028	142	9	and	and	CCONJ
iajs-3028	142	10	for	for	ADP
iajs-3028	142	11	all	all	DET
iajs-3028	142	12	𝑟	𝑟	DET
iajs-3028	142	13	∈	∈	PROPN
iajs-3028	142	14	d(ℛ	d(ℛ	PROPN
iajs-3028	142	15	)	)	PUNCT
iajs-3028	142	16	,	,	PUNCT
iajs-3028	142	17	∃𝑎	∃𝑎	PROPN
iajs-3028	142	18	∈	∈	PROPN
iajs-3028	142	19	ℛ	ℛ	NOUN
iajs-3028	142	20	such	such	ADJ
iajs-3028	142	21	that	that	PRON
iajs-3028	142	22	,	,	PUNCT
iajs-3028	142	23	𝑟	𝑟	X
iajs-3028	142	24	=	=	SYM
iajs-3028	142	25	𝑟𝑜𝑎.	𝑟𝑜𝑎.	NUM
iajs-3028	142	26	by	by	ADP
iajs-3028	142	27	(	(	PUNCT
iajs-3028	142	28	prop.2.13	prop.2.13	NOUN
iajs-3028	142	29	-	-	PUNCT
iajs-3028	142	30	3	3	NUM
iajs-3028	142	31	)	)	PUNCT
iajs-3028	142	32	,	,	PUNCT
iajs-3028	142	33	1+𝑟	1+𝑟	NUM
iajs-3028	142	34	=	=	SYM
iajs-3028	142	35	1	1	NUM
iajs-3028	142	36	+	+	CCONJ
iajs-3028	142	37	𝑟𝑜𝑎	𝑟𝑜𝑎	NOUN
iajs-3028	142	38	=	=	SYM
iajs-3028	142	39	1	1	NUM
iajs-3028	142	40	−	−	NUM
iajs-3028	142	41	1	1	NUM
iajs-3028	142	42	∙	∙	PROPN
iajs-3028	142	43	𝑟𝑜(−𝑎	𝑟𝑜(−𝑎	NOUN
iajs-3028	142	44	)	)	PUNCT
iajs-3028	142	45	∈	∈	PROPN
iajs-3028	142	46	𝑈(ℛ	𝑈(ℛ	ADJ
iajs-3028	142	47	)	)	PUNCT
iajs-3028	142	48	implies	imply	VERB
iajs-3028	142	49	1	1	NUM
iajs-3028	142	50	+	+	SYM
iajs-3028	142	51	𝑑(ℛ	𝑑(ℛ	ADJ
iajs-3028	142	52	)	)	PUNCT
iajs-3028	142	53	⊆	⊆	NUM
iajs-3028	142	54	𝑈(ℛ	𝑈(ℛ	ADJ
iajs-3028	142	55	)	)	PUNCT
iajs-3028	142	56	.	.	PUNCT
iajs-3028	143	1	proposition	proposition	NOUN
iajs-3028	143	2	3.10	3.10	NUM
iajs-3028	143	3	.	.	PUNCT
iajs-3028	144	1	if	if	SCONJ
iajs-3028	144	2	ℛ	ℛ	PROPN
iajs-3028	144	3	is	be	AUX
iajs-3028	144	4	a	a	DET
iajs-3028	144	5	hyperring	hyperring	NOUN
iajs-3028	144	6	satisfy	satisfy	NOUN
iajs-3028	144	7	𝑅𝑂	𝑅𝑂	PROPN
iajs-3028	144	8	=	=	SYM
iajs-3028	144	9	𝑅	𝑅	PROPN
iajs-3028	144	10	*	*	PROPN
iajs-3028	144	11	,	,	PUNCT
iajs-3028	144	12	then	then	ADV
iajs-3028	144	13	either	either	CCONJ
iajs-3028	144	14	d(ℛ)=0	d(ℛ)=0	ADJ
iajs-3028	144	15	or	or	CCONJ
iajs-3028	144	16	d(ℛ)=ℛ.	d(ℛ)=ℛ.	NOUN
iajs-3028	144	17	proof	proof	NOUN
iajs-3028	144	18	.	.	PUNCT
iajs-3028	145	1	if	if	SCONJ
iajs-3028	145	2	d(ℛ	d(ℛ	NOUN
iajs-3028	145	3	)	)	PUNCT
iajs-3028	145	4	≠	≠	PROPN
iajs-3028	145	5	0	0	NUM
iajs-3028	145	6	,	,	PUNCT
iajs-3028	145	7	then	then	ADV
iajs-3028	145	8	for	for	ADP
iajs-3028	145	9	any	any	DET
iajs-3028	145	10	𝑦	𝑦	NOUN
iajs-3028	145	11	in	in	ADP
iajs-3028	145	12	d(ℛ)\{0	d(ℛ)\{0	NOUN
iajs-3028	145	13	}	}	PUNCT
iajs-3028	145	14	,	,	PUNCT
iajs-3028	145	15	have	have	VERB
iajs-3028	145	16	𝑦2	𝑦2	NOUN
iajs-3028	145	17	≠	≠	PROPN
iajs-3028	145	18	0	0	NUM
iajs-3028	145	19	and	and	CCONJ
iajs-3028	145	20	∃	∃	PROPN
iajs-3028	145	21	𝑥	𝑥	X
iajs-3028	145	22	∈	∈	PROPN
iajs-3028	145	23	ℛ	ℛ	PROPN
iajs-3028	145	24	such	such	ADJ
iajs-3028	145	25	that	that	PRON
iajs-3028	145	26	𝑦	𝑦	PROPN
iajs-3028	145	27	=	=	SYM
iajs-3028	145	28	𝑦2𝑥	𝑦2𝑥	NOUN
iajs-3028	145	29	,	,	PUNCT
iajs-3028	145	30	thus	thus	ADV
iajs-3028	145	31	𝑦(1	𝑦(1	VERB
iajs-3028	145	32	−	−	PROPN
iajs-3028	145	33	𝑦𝑥	𝑦𝑥	NOUN
iajs-3028	145	34	)	)	PUNCT
iajs-3028	145	35	=	=	SYM
iajs-3028	146	1	0	0	NUM
iajs-3028	146	2	,	,	PUNCT
iajs-3028	146	3	this	this	PRON
iajs-3028	146	4	implies	imply	VERB
iajs-3028	146	5	𝑦𝑥	𝑦𝑥	NOUN
iajs-3028	146	6	=	=	SYM
iajs-3028	146	7	1	1	NUM
iajs-3028	146	8	so	so	ADV
iajs-3028	146	9	,	,	PUNCT
iajs-3028	146	10	𝑦	𝑦	PROPN
iajs-3028	146	11	∈	∈	PROPN
iajs-3028	146	12	𝑈𝑟𝑖𝑔ℎ𝑡(ℛ	𝑈𝑟𝑖𝑔ℎ𝑡(ℛ	NOUN
iajs-3028	146	13	)	)	PUNCT
iajs-3028	146	14	(	(	PUNCT
iajs-3028	146	15	1	1	X
iajs-3028	146	16	)	)	PUNCT
iajs-3028	146	17	also	also	ADV
iajs-3028	146	18	,	,	PUNCT
iajs-3028	146	19	1	1	NUM
iajs-3028	146	20	−	−	NOUN
iajs-3028	146	21	𝑦𝑥	𝑦𝑥	NOUN
iajs-3028	146	22	=	=	SYM
iajs-3028	146	23	0	0	NUM
iajs-3028	146	24	leads	lead	VERB
iajs-3028	146	25	to	to	ADP
iajs-3028	146	26	𝑦	𝑦	NOUN
iajs-3028	146	27	−	−	NOUN
iajs-3028	146	28	𝑦𝑥𝑦	𝑦𝑥𝑦	NOUN
iajs-3028	146	29	=	=	SYM
iajs-3028	146	30	0	0	NUM
iajs-3028	146	31	coming	come	VERB
iajs-3028	146	32	after	after	ADP
iajs-3028	146	33	𝑦(1	𝑦(1	PROPN
iajs-3028	146	34	−	−	PROPN
iajs-3028	146	35	𝑥𝑦	𝑥𝑦	NOUN
iajs-3028	146	36	)	)	PUNCT
iajs-3028	146	37	=	=	SYM
iajs-3028	146	38	0	0	X
iajs-3028	146	39	.	.	PUNCT
iajs-3028	147	1	but	but	CCONJ
iajs-3028	147	2	𝑅𝑂	𝑅𝑂	PROPN
iajs-3028	147	3	=	=	SYM
iajs-3028	147	4	𝑅	𝑅	PROPN
iajs-3028	147	5	*	*	PROPN
iajs-3028	147	6	,	,	PUNCT
iajs-3028	147	7	then	then	ADV
iajs-3028	147	8	𝑥𝑦	𝑥𝑦	NOUN
iajs-3028	147	9	=	=	SYM
iajs-3028	147	10	1	1	NUM
iajs-3028	147	11	,	,	PUNCT
iajs-3028	147	12	thus	thus	ADV
iajs-3028	147	13	𝑦	𝑦	NOUN
iajs-3028	147	14	∈	∈	PROPN
iajs-3028	147	15	𝑈𝑙𝑒𝑓𝑡(ℛ	𝑈𝑙𝑒𝑓𝑡(ℛ	NOUN
iajs-3028	147	16	)	)	PUNCT
iajs-3028	147	17	(	(	PUNCT
iajs-3028	147	18	2	2	X
iajs-3028	147	19	)	)	PUNCT
iajs-3028	147	20	now	now	ADV
iajs-3028	147	21	by	by	ADP
iajs-3028	147	22	(	(	PUNCT
iajs-3028	147	23	1	1	NUM
iajs-3028	147	24	)	)	PUNCT
iajs-3028	147	25	and	and	CCONJ
iajs-3028	147	26	(	(	PUNCT
iajs-3028	147	27	2	2	X
iajs-3028	147	28	)	)	PUNCT
iajs-3028	147	29	get	get	VERB
iajs-3028	147	30	𝑦	𝑦	NOUN
iajs-3028	147	31	∈	∈	NOUN
iajs-3028	147	32	𝑈(ℛ	𝑈(ℛ	ADJ
iajs-3028	147	33	)	)	PUNCT
iajs-3028	147	34	.	.	PUNCT
iajs-3028	148	1	since	since	SCONJ
iajs-3028	148	2	𝑦	𝑦	NOUN
iajs-3028	148	3	∈	∈	NOUN
iajs-3028	148	4	𝑈(ℛ	𝑈(ℛ	ADJ
iajs-3028	148	5	)	)	PUNCT
iajs-3028	148	6	∩	∩	ADJ
iajs-3028	148	7	𝑑(ℛ	𝑑(ℛ	PROPN
iajs-3028	148	8	)	)	PUNCT
iajs-3028	148	9	𝑖𝑚𝑝𝑙𝑖𝑒𝑠	𝑖𝑚𝑝𝑙𝑖𝑒𝑠	NOUN
iajs-3028	148	10	𝑑(ℛ	𝑑(ℛ	PROPN
iajs-3028	148	11	)	)	PUNCT
iajs-3028	148	12	=	=	SYM
iajs-3028	148	13	ℛ.	ℛ.	PROPN
iajs-3028	148	14	definition	definition	NOUN
iajs-3028	148	15	3.11	3.11	NUM
iajs-3028	148	16	.	.	PUNCT
iajs-3028	149	1	the	the	DET
iajs-3028	149	2	hyperring	hyperre	VERB
iajs-3028	149	3	ℛ	ℛ	PROPN
iajs-3028	149	4	is	be	AUX
iajs-3028	149	5	called	call	VERB
iajs-3028	149	6	“	"	PUNCT
iajs-3028	149	7	divisible	divisible	ADJ
iajs-3028	149	8	hyperring	hyperring	NOUN
iajs-3028	149	9	”	"	PUNCT
iajs-3028	149	10	if	if	SCONJ
iajs-3028	149	11	any	any	DET
iajs-3028	149	12	element	element	NOUN
iajs-3028	149	13	belongs	belong	VERB
iajs-3028	149	14	to	to	ADP
iajs-3028	149	15	ℛ	ℛ	PROPN
iajs-3028	149	16	is	be	AUX
iajs-3028	149	17	a	a	DET
iajs-3028	149	18	divisible	divisible	ADJ
iajs-3028	149	19	element	element	NOUN
iajs-3028	149	20	.	.	PUNCT
iajs-3028	150	1	the	the	DET
iajs-3028	150	2	following	follow	VERB
iajs-3028	150	3	corollary	corollary	NOUN
iajs-3028	150	4	comes	come	VERB
iajs-3028	150	5	straightaway	straightaway	ADV
iajs-3028	150	6	from	from	ADP
iajs-3028	150	7	definition	definition	NOUN
iajs-3028	150	8	3.11	3.11	NUM
iajs-3028	150	9	and	and	CCONJ
iajs-3028	150	10	remark	remark	VERB
iajs-3028	150	11	3.4	3.4	NUM
iajs-3028	150	12	corollary	corollary	NOUN
iajs-3028	150	13	3.12	3.12	NUM
iajs-3028	150	14	.	.	PUNCT
iajs-3028	151	1	ℛ	ℛ	PROPN
iajs-3028	151	2	is	be	AUX
iajs-3028	151	3	divisible	divisible	ADJ
iajs-3028	151	4	hyperring	hyperring	NOUN
iajs-3028	151	5	if	if	SCONJ
iajs-3028	152	1	and	and	CCONJ
iajs-3028	152	2	only	only	ADV
iajs-3028	152	3	if	if	SCONJ
iajs-3028	152	4	d(ℛ)=ℛ.	d(ℛ)=ℛ.	PROPN
iajs-3028	152	5	corollary	corollary	VERB
iajs-3028	152	6	3.13	3.13	NUM
iajs-3028	152	7	.	.	PUNCT
iajs-3028	153	1	every	every	DET
iajs-3028	153	2	division	division	NOUN
iajs-3028	153	3	hyperring	hyperring	NOUN
iajs-3028	153	4	is	be	AUX
iajs-3028	153	5	divisible	divisible	ADJ
iajs-3028	153	6	hyperring	hyperring	NOUN
iajs-3028	153	7	next	next	ADV
iajs-3028	153	8	,	,	PUNCT
iajs-3028	153	9	will	will	AUX
iajs-3028	153	10	refer	refer	VERB
iajs-3028	153	11	to	to	ADP
iajs-3028	153	12	the	the	DET
iajs-3028	153	13	class	class	NOUN
iajs-3028	153	14	�	�	PROPN
iajs-3028	153	15	̌	̌	PRON
iajs-3028	153	16	�	�	PROPN
iajs-3028	153	17	of	of	ADP
iajs-3028	153	18	𝓡	𝓡	PROPN
iajs-3028	153	19	having	have	VERB
iajs-3028	153	20	the	the	DET
iajs-3028	153	21	properties	property	NOUN
iajs-3028	153	22	:	:	PUNCT
iajs-3028	153	23	•	•	NUM
iajs-3028	153	24	𝑑(ℛ	𝑑(ℛ	PROPN
iajs-3028	153	25	)	)	PUNCT
iajs-3028	154	1	≠	≠	PROPN
iajs-3028	154	2	0	0	NUM
iajs-3028	154	3	•	•	NOUN
iajs-3028	154	4	if	if	SCONJ
iajs-3028	154	5	{	{	PUNCT
iajs-3028	154	6	𝐼𝑗}𝑗∈𝐽	𝐼𝑗}𝑗∈𝐽	NOUN
iajs-3028	154	7	is	be	AUX
iajs-3028	154	8	a	a	DET
iajs-3028	154	9	set	set	NOUN
iajs-3028	154	10	of	of	ADP
iajs-3028	154	11	all	all	DET
iajs-3028	154	12	maximal	maximal	ADJ
iajs-3028	154	13	right	right	ADJ
iajs-3028	154	14	hyperideals	hyperideal	NOUN
iajs-3028	154	15	of	of	ADP
iajs-3028	154	16	a	a	DET
iajs-3028	154	17	hyperring	hyperre	VERB
iajs-3028	154	18	ℛ	ℛ	NOUN
iajs-3028	154	19	,	,	PUNCT
iajs-3028	154	20	then	then	ADV
iajs-3028	154	21	have	have	VERB
iajs-3028	154	22	𝑑(ℛ	𝑑(ℛ	PROPN
iajs-3028	154	23	)	)	PUNCT
iajs-3028	154	24	∩	∩	NOUN
iajs-3028	154	25	𝐼𝑗	𝐼𝑗	NOUN
iajs-3028	154	26	=	=	SYM
iajs-3028	154	27	0	0	NUM
iajs-3028	154	28	,	,	PUNCT
iajs-3028	154	29	or	or	CCONJ
iajs-3028	154	30	𝑑(𝑅	𝑑(𝑅	NOUN
iajs-3028	154	31	)	)	PUNCT
iajs-3028	154	32	∩	∩	NOUN
iajs-3028	154	33	𝐼𝑗	𝐼𝑗	NOUN
iajs-3028	154	34	∩	∩	ADJ
iajs-3028	154	35	𝑅	𝑅	NOUN
iajs-3028	154	36	𝑂	𝑂	PROPN
iajs-3028	154	37	≠	≠	PROPN
iajs-3028	154	38	∅	∅	NOUN
iajs-3028	154	39	,	,	PUNCT
iajs-3028	154	40	∀𝑗	∀𝑗	NOUN
iajs-3028	154	41	∈	∈	PROPN
iajs-3028	154	42	𝐽.	𝐽.	PROPN
iajs-3028	154	43	theorem	theorem	VERB
iajs-3028	154	44	3.14	3.14	NUM
iajs-3028	154	45	.	.	PUNCT
iajs-3028	155	1	a	a	DET
iajs-3028	155	2	hyperring	hyperre	VERB
iajs-3028	155	3	ℛ	ℛ	PROPN
iajs-3028	155	4	is	be	AUX
iajs-3028	155	5	a	a	DET
iajs-3028	155	6	division	division	NOUN
iajs-3028	155	7	if	if	SCONJ
iajs-3028	155	8	and	and	CCONJ
iajs-3028	155	9	only	only	ADV
iajs-3028	155	10	if	if	SCONJ
iajs-3028	155	11	a	a	PRON
iajs-3028	155	12	)	)	PUNCT
iajs-3028	155	13	ℛ	ℛ	PROPN
iajs-3028	155	14	is	be	AUX
iajs-3028	155	15	divisible	divisible	ADJ
iajs-3028	155	16	hyperring	hyperring	NOUN
iajs-3028	155	17	b	b	NOUN
iajs-3028	155	18	)	)	PUNCT
iajs-3028	155	19	ℛ	ℛ	PROPN
iajs-3028	155	20	∈	∈	PROPN
iajs-3028	155	21	�	�	PROPN
iajs-3028	155	22	̆	̆	NOUN
iajs-3028	155	23	�	�	PROPN
iajs-3028	155	24	c	c	NOUN
iajs-3028	155	25	)	)	PUNCT
iajs-3028	155	26	𝑅0	𝑅0	NOUN
iajs-3028	155	27	=	=	SYM
iajs-3028	155	28	𝑈(ℛ	𝑈(ℛ	ADJ
iajs-3028	155	29	)	)	PUNCT
iajs-3028	155	30	proof	proof	NOUN
iajs-3028	155	31	.	.	PUNCT
iajs-3028	156	1	let	let	VERB
iajs-3028	156	2	𝐼𝑗	𝐼𝑗	NOUN
iajs-3028	156	3	be	be	AUX
iajs-3028	156	4	a	a	DET
iajs-3028	156	5	maximal	maximal	ADJ
iajs-3028	156	6	right	right	ADJ
iajs-3028	156	7	hyperideal	hyperideal	NOUN
iajs-3028	156	8	of	of	ADP
iajs-3028	156	9	a	a	DET
iajs-3028	156	10	hyperring	hyperre	VERB
iajs-3028	156	11	ℛ	ℛ	NOUN
iajs-3028	156	12	,	,	PUNCT
iajs-3028	156	13	by	by	ADP
iajs-3028	156	14	(	(	PUNCT
iajs-3028	156	15	a	a	X
iajs-3028	156	16	)	)	PUNCT
iajs-3028	156	17	,	,	PUNCT
iajs-3028	156	18	ℛ	ℛ	PROPN
iajs-3028	156	19	is	be	AUX
iajs-3028	156	20	divisible	divisible	ADJ
iajs-3028	156	21	.	.	PUNCT
iajs-3028	157	1	this	this	PRON
iajs-3028	157	2	implies	imply	VERB
iajs-3028	157	3	that	that	SCONJ
iajs-3028	157	4	d(ℛ)=ℛ.	d(ℛ)=ℛ.	VERB
iajs-3028	157	5	the	the	DET
iajs-3028	157	6	condition	condition	NOUN
iajs-3028	157	7	(	(	PUNCT
iajs-3028	157	8	b	b	NOUN
iajs-3028	157	9	)	)	PUNCT
iajs-3028	157	10	gives	give	VERB
iajs-3028	157	11	𝐼𝑗=0	𝐼𝑗=0	NUM
iajs-3028	157	12	or	or	CCONJ
iajs-3028	157	13	𝐼𝑗	𝐼𝑗	NOUN
iajs-3028	157	14	∩	∩	ADJ
iajs-3028	157	15	𝑅	𝑅	NOUN
iajs-3028	157	16	𝑂	𝑂	NOUN
iajs-3028	157	17	≠	≠	PROPN
iajs-3028	157	18	∅	∅	NOUN
iajs-3028	157	19	for	for	ADP
iajs-3028	157	20	maximal	maximal	ADJ
iajs-3028	157	21	right	right	ADJ
iajs-3028	157	22	hyperideal	hyperideal	NOUN
iajs-3028	157	23	𝐼𝑗	𝐼𝑗	NOUN
iajs-3028	157	24	of	of	ADP
iajs-3028	157	25	ℛ.	ℛ.	PROPN
iajs-3028	157	26	finally	finally	ADV
iajs-3028	157	27	,	,	PUNCT
iajs-3028	157	28	by	by	ADP
iajs-3028	157	29	(	(	PUNCT
iajs-3028	157	30	c	c	NOUN
iajs-3028	157	31	)	)	PUNCT
iajs-3028	157	32	,	,	PUNCT
iajs-3028	157	33	either	either	CCONJ
iajs-3028	157	34	𝐼𝑗	𝐼𝑗	PROPN
iajs-3028	157	35	=	=	PUNCT
iajs-3028	157	36	0	0	NUM
iajs-3028	157	37	or	or	CCONJ
iajs-3028	157	38	𝐼𝑗	𝐼𝑗	PROPN
iajs-3028	157	39	=	=	SYM
iajs-3028	157	40	ℛ	ℛ	PROPN
iajs-3028	157	41	,	,	PUNCT
iajs-3028	157	42	but	but	CCONJ
iajs-3028	157	43	𝐼𝑗	𝐼𝑗	PROPN
iajs-3028	157	44	≠	≠	PROPN
iajs-3028	157	45	ℛ	ℛ	PROPN
iajs-3028	157	46	,	,	PUNCT
iajs-3028	157	47	so	so	ADV
iajs-3028	157	48	𝐼𝑗	𝐼𝑗	NOUN
iajs-3028	157	49	=	=	SYM
iajs-3028	157	50	0	0	NUM
iajs-3028	157	51	.	.	PUNCT
iajs-3028	158	1	for	for	ADP
iajs-3028	158	2	the	the	DET
iajs-3028	158	3	other	other	ADJ
iajs-3028	158	4	side	side	NOUN
iajs-3028	158	5	,	,	PUNCT
iajs-3028	158	6	since	since	SCONJ
iajs-3028	158	7	ℛ	ℛ	PROPN
iajs-3028	158	8	is	be	AUX
iajs-3028	158	9	a	a	DET
iajs-3028	158	10	division	division	NOUN
iajs-3028	158	11	then	then	ADV
iajs-3028	158	12	by	by	ADP
iajs-3028	158	13	corollary	corollary	ADJ
iajs-3028	158	14	3.13	3.13	NUM
iajs-3028	158	15	,	,	PUNCT
iajs-3028	158	16	ℛ	ℛ	PROPN
iajs-3028	158	17	is	be	AUX
iajs-3028	158	18	divisible	divisible	ADJ
iajs-3028	158	19	hyperring	hyperring	NOUN
iajs-3028	158	20	.	.	PUNCT
iajs-3028	159	1	now	now	ADV
iajs-3028	159	2	to	to	PART
iajs-3028	159	3	prove	prove	VERB
iajs-3028	159	4	(	(	PUNCT
iajs-3028	159	5	b	b	NOUN
iajs-3028	159	6	)	)	PUNCT
iajs-3028	159	7	,	,	PUNCT
iajs-3028	159	8	since	since	SCONJ
iajs-3028	159	9	d(ℛ)=ℛ	d(ℛ)=ℛ	PRON
iajs-3028	159	10	therefore	therefore	ADV
iajs-3028	159	11	𝑑(ℛ	𝑑(ℛ	PROPN
iajs-3028	159	12	)	)	PUNCT
iajs-3028	159	13	≠	≠	PROPN
iajs-3028	159	14	0	0	NUM
iajs-3028	159	15	,	,	PUNCT
iajs-3028	159	16	and	and	CCONJ
iajs-3028	159	17	if	if	SCONJ
iajs-3028	159	18	{	{	PUNCT
iajs-3028	159	19	𝐼𝑗}𝑗∈𝐽	𝐼𝑗}𝑗∈𝐽	NOUN
iajs-3028	159	20	is	be	AUX
iajs-3028	159	21	a	a	DET
iajs-3028	159	22	set	set	NOUN
iajs-3028	159	23	of	of	ADP
iajs-3028	159	24	all	all	DET
iajs-3028	159	25	maximal	maximal	ADJ
iajs-3028	159	26	right	right	ADJ
iajs-3028	159	27	hyperideals	hyperideal	NOUN
iajs-3028	159	28	of	of	ADP
iajs-3028	159	29	a	a	DET
iajs-3028	159	30	hyperring	hyperre	VERB
iajs-3028	159	31	ℛ	ℛ	NOUN
iajs-3028	159	32	,	,	PUNCT
iajs-3028	159	33	then	then	ADV
iajs-3028	159	34	𝑑(𝑅	𝑑(𝑅	NOUN
iajs-3028	159	35	)	)	PUNCT
iajs-3028	159	36	∩	∩	NOUN
iajs-3028	159	37	𝐼𝑗	𝐼𝑗	NOUN
iajs-3028	159	38	∩	∩	ADJ
iajs-3028	159	39	𝑅	𝑅	NOUN
iajs-3028	159	40	𝑂	𝑂	PROPN
iajs-3028	159	41	≠	≠	PROPN
iajs-3028	159	42	∅.	∅.	VERB
iajs-3028	159	43	finally	finally	ADV
iajs-3028	159	44	,	,	PUNCT
iajs-3028	159	45	since	since	SCONJ
iajs-3028	159	46	ℛ	ℛ	PROPN
iajs-3028	159	47	is	be	AUX
iajs-3028	159	48	a	a	DET
iajs-3028	159	49	division	division	NOUN
iajs-3028	159	50	then	then	ADV
iajs-3028	159	51	every	every	DET
iajs-3028	159	52	element	element	NOUN
iajs-3028	159	53	is	be	AUX
iajs-3028	159	54	unite	unite	ADJ
iajs-3028	159	55	and	and	CCONJ
iajs-3028	159	56	so	so	ADV
iajs-3028	159	57	ℛ	ℛ	PROPN
iajs-3028	159	58	has	have	VERB
iajs-3028	159	59	no	no	DET
iajs-3028	159	60	zero	zero	NUM
iajs-3028	159	61	divisor	divisor	NOUN
iajs-3028	159	62	hence	hence	ADV
iajs-3028	159	63	𝑅0	𝑅0	NOUN
iajs-3028	159	64	=	=	SYM
iajs-3028	159	65	𝑈(ℛ	𝑈(ℛ	ADJ
iajs-3028	159	66	)	)	PUNCT
iajs-3028	159	67	,	,	PUNCT
iajs-3028	159	68	then	then	ADV
iajs-3028	159	69	(	(	PUNCT
iajs-3028	159	70	c	c	X
iajs-3028	159	71	)	)	PUNCT
iajs-3028	159	72	holds	hold	VERB
iajs-3028	159	73	.	.	PUNCT
iajs-3028	160	1	∎	∎	PROPN
iajs-3028	160	2	theorem	theorem	VERB
iajs-3028	160	3	3.15	3.15	NUM
iajs-3028	160	4	.	.	PUNCT
iajs-3028	161	1	a	a	DET
iajs-3028	161	2	divisible	divisible	ADJ
iajs-3028	161	3	hyperring	hyperring	NOUN
iajs-3028	161	4	can	can	AUX
iajs-3028	161	5	be	be	AUX
iajs-3028	161	6	written	write	VERB
iajs-3028	161	7	as	as	ADP
iajs-3028	161	8	ℛ=xℛ	ℛ=xℛ	PROPN
iajs-3028	161	9	for	for	ADP
iajs-3028	161	10	any	any	DET
iajs-3028	161	11	x	x	PROPN
iajs-3028	161	12	∈	∈	PROPN
iajs-3028	161	13	𝑅𝑂	𝑅𝑂	PROPN
iajs-3028	161	14	.	.	PUNCT
iajs-3028	162	1	proof	proof	NOUN
iajs-3028	162	2	.	.	PUNCT
iajs-3028	163	1	it	it	PRON
iajs-3028	163	2	is	be	AUX
iajs-3028	163	3	enough	enough	ADJ
iajs-3028	163	4	to	to	PART
iajs-3028	163	5	show	show	VERB
iajs-3028	163	6	that	that	SCONJ
iajs-3028	163	7	ℛ	ℛ	PROPN
iajs-3028	163	8	⊆	⊆	NUM
iajs-3028	163	9	xℛ.	xℛ.	NOUN
iajs-3028	163	10	since	since	SCONJ
iajs-3028	163	11	ℛ	ℛ	PROPN
iajs-3028	163	12	is	be	AUX
iajs-3028	163	13	divisible	divisible	ADJ
iajs-3028	163	14	hyperring	hyperring	NOUN
iajs-3028	163	15	,	,	PUNCT
iajs-3028	163	16	then	then	ADV
iajs-3028	163	17	for	for	ADP
iajs-3028	163	18	any	any	DET
iajs-3028	163	19	𝑎	𝑎	PROPN
iajs-3028	163	20	∈	∈	NOUN
iajs-3028	163	21	ℛ	ℛ	NOUN
iajs-3028	163	22	there	there	PRON
iajs-3028	163	23	is	be	VERB
iajs-3028	163	24	𝑏	𝑏	DET
iajs-3028	163	25	∈	∈	PROPN
iajs-3028	163	26	ℛ	ℛ	NOUN
iajs-3028	163	27	such	such	ADJ
iajs-3028	163	28	that	that	SCONJ
iajs-3028	163	29	𝑎	𝑎	PROPN
iajs-3028	163	30	=	=	SYM
iajs-3028	163	31	x𝑏	x𝑏	NOUN
iajs-3028	164	1	and	and	CCONJ
iajs-3028	164	2	so	so	ADV
iajs-3028	164	3	𝑎	𝑎	PRON
iajs-3028	164	4	∈	∈	PROPN
iajs-3028	164	5	xℛ	xℛ	NOUN
iajs-3028	164	6	,	,	PUNCT
iajs-3028	164	7	thus	thus	ADV
iajs-3028	164	8	ℛ	ℛ	ADJ
iajs-3028	164	9	⊆	⊆	NUM
iajs-3028	164	10	xℛ.	xℛ.	PROPN
iajs-3028	164	11	ihjpas	ihjpa	NOUN
iajs-3028	164	12	.	.	PUNCT
iajs-3028	165	1	36(2)2023	36(2)2023	NUM
iajs-3028	165	2	388	388	NUM
iajs-3028	165	3	as	as	ADP
iajs-3028	165	4	the	the	DET
iajs-3028	165	5	definition	definition	NOUN
iajs-3028	165	6	of	of	ADP
iajs-3028	165	7	the	the	DET
iajs-3028	165	8	direct	direct	ADJ
iajs-3028	165	9	sum	sum	NOUN
iajs-3028	165	10	of	of	ADP
iajs-3028	165	11	two	two	NUM
iajs-3028	165	12	hyperideals	hyperideal	NOUN
iajs-3028	165	13	in	in	ADP
iajs-3028	165	14	[	[	X
iajs-3028	165	15	8	8	NUM
iajs-3028	165	16	]	]	PUNCT
iajs-3028	165	17	and	and	CCONJ
iajs-3028	165	18	the	the	DET
iajs-3028	165	19	direct	direct	ADJ
iajs-3028	165	20	sum	sum	NOUN
iajs-3028	165	21	of	of	ADP
iajs-3028	165	22	subhypermodules	subhypermodule	NOUN
iajs-3028	165	23	in	in	ADP
iajs-3028	165	24	[	[	X
iajs-3028	165	25	1	1	NUM
iajs-3028	165	26	]	]	PUNCT
iajs-3028	165	27	,	,	PUNCT
iajs-3028	165	28	we	we	PRON
iajs-3028	165	29	will	will	AUX
iajs-3028	165	30	define	define	VERB
iajs-3028	165	31	the	the	DET
iajs-3028	165	32	direct	direct	ADJ
iajs-3028	165	33	sum	sum	NOUN
iajs-3028	165	34	of	of	ADP
iajs-3028	165	35	two	two	NUM
iajs-3028	165	36	subhyperrings	subhyperring	NOUN
iajs-3028	165	37	as	as	SCONJ
iajs-3028	165	38	follows	follow	VERB
iajs-3028	165	39	;	;	PUNCT
iajs-3028	165	40	definition	definition	NOUN
iajs-3028	165	41	3.16	3.16	NUM
iajs-3028	165	42	.	.	PUNCT
iajs-3028	166	1	the	the	DET
iajs-3028	166	2	direct	direct	ADJ
iajs-3028	166	3	sum	sum	NOUN
iajs-3028	166	4	of	of	ADP
iajs-3028	166	5	two	two	NUM
iajs-3028	166	6	subhyperrings	subhyperring	NOUN
iajs-3028	166	7	𝒮	𝒮	PROPN
iajs-3028	166	8	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-3028	166	9	𝒯	𝒯	PROPN
iajs-3028	166	10	,	,	PUNCT
iajs-3028	166	11	is	be	AUX
iajs-3028	166	12	denoted	denote	VERB
iajs-3028	166	13	by	by	ADP
iajs-3028	166	14	𝒮⊕𝒯	𝒮⊕𝒯	PROPN
iajs-3028	166	15	such	such	ADJ
iajs-3028	166	16	that	that	SCONJ
iajs-3028	166	17	for	for	SCONJ
iajs-3028	166	18	each	each	DET
iajs-3028	166	19	element	element	NOUN
iajs-3028	166	20	𝓀	𝓀	PROPN
iajs-3028	166	21	∈	∈	PROPN
iajs-3028	166	22	𝒮⨁𝒯	𝒮⨁𝒯	VERB
iajs-3028	166	23	,	,	PUNCT
iajs-3028	166	24	there	there	PRON
iajs-3028	166	25	is	be	VERB
iajs-3028	166	26	unique	unique	ADJ
iajs-3028	166	27	elements	element	NOUN
iajs-3028	166	28	𝓈	𝓈	ADP
iajs-3028	166	29	∈	∈	PROPN
iajs-3028	166	30	𝒮	𝒮	PROPN
iajs-3028	166	31	,	,	PUNCT
iajs-3028	166	32	𝓉	𝓉	PROPN
iajs-3028	166	33	∈	∈	PROPN
iajs-3028	166	34	𝒯	𝒯	PROPN
iajs-3028	166	35	,	,	PUNCT
iajs-3028	166	36	𝓀	𝓀	PROPN
iajs-3028	166	37	=	=	SYM
iajs-3028	166	38	𝓈	𝓈	AUX
iajs-3028	166	39	+	+	CCONJ
iajs-3028	166	40	𝓉.	𝓉.	NOUN
iajs-3028	166	41	theorem	theorem	NOUN
iajs-3028	166	42	3.17	3.17	NUM
iajs-3028	166	43	.	.	PUNCT
iajs-3028	167	1	if	if	SCONJ
iajs-3028	167	2	𝒮	𝒮	PROPN
iajs-3028	167	3	and	and	CCONJ
iajs-3028	167	4	𝒯	𝒯	PROPN
iajs-3028	167	5	be	be	VERB
iajs-3028	167	6	subhyperrings	subhyperring	NOUN
iajs-3028	167	7	of	of	ADP
iajs-3028	167	8	a	a	DET
iajs-3028	167	9	hyperring	hyperre	VERB
iajs-3028	167	10	ℛ	ℛ	NOUN
iajs-3028	167	11	,	,	PUNCT
iajs-3028	167	12	and	and	CCONJ
iajs-3028	167	13	𝒮	𝒮	PROPN
iajs-3028	167	14	,	,	PUNCT
iajs-3028	167	15	𝒯	𝒯	PROPN
iajs-3028	167	16	are	be	AUX
iajs-3028	167	17	divisible	divisible	ADJ
iajs-3028	167	18	,	,	PUNCT
iajs-3028	167	19	then	then	ADV
iajs-3028	167	20	𝒮⨁𝒯	𝒮⨁𝒯	VERB
iajs-3028	167	21	is	be	AUX
iajs-3028	167	22	divisible	divisible	ADJ
iajs-3028	167	23	.	.	PUNCT
iajs-3028	168	1	proof	proof	NOUN
iajs-3028	168	2	.	.	PUNCT
iajs-3028	169	1	let	let	VERB
iajs-3028	169	2	𝑘	𝑘	PRON
iajs-3028	169	3	∈	∈	PROPN
iajs-3028	169	4	𝑆⨁𝒯	𝑆⨁𝒯	NOUN
iajs-3028	169	5	and	and	CCONJ
iajs-3028	169	6	𝑟	𝑟	PRON
iajs-3028	169	7	∈	∈	PROPN
iajs-3028	169	8	𝑅𝑂	𝑅𝑂	PROPN
iajs-3028	169	9	,	,	PUNCT
iajs-3028	169	10	∃𝓈	∃𝓈	PROPN
iajs-3028	169	11	∈	∈	PROPN
iajs-3028	169	12	𝒮	𝒮	PROPN
iajs-3028	169	13	and	and	CCONJ
iajs-3028	169	14	𝓉	𝓉	PROPN
iajs-3028	169	15	∈	∈	PROPN
iajs-3028	169	16	𝒯	𝒯	PROPN
iajs-3028	169	17	such	such	ADJ
iajs-3028	169	18	that	that	SCONJ
iajs-3028	169	19	𝑘	𝑘	PRON
iajs-3028	169	20	=	=	PUNCT
iajs-3028	169	21	𝓈	𝓈	X
iajs-3028	169	22	+	+	X
iajs-3028	169	23	𝓉.	𝓉.	NOUN
iajs-3028	169	24	now	now	ADV
iajs-3028	169	25	,	,	PUNCT
iajs-3028	169	26	𝒯	𝒯	PROPN
iajs-3028	169	27	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-3028	169	28	𝒮	𝒮	PROPN
iajs-3028	169	29	are	be	AUX
iajs-3028	169	30	divisible	divisible	ADJ
iajs-3028	169	31	,	,	PUNCT
iajs-3028	169	32	so	so	ADV
iajs-3028	169	33	𝒮	𝒮	PROPN
iajs-3028	169	34	=	=	PUNCT
iajs-3028	169	35	𝑥𝒮	𝑥𝒮	PROPN
iajs-3028	169	36	and	and	CCONJ
iajs-3028	169	37	𝒯	𝒯	PROPN
iajs-3028	169	38	=	=	PUNCT
iajs-3028	169	39	𝑥𝒯	𝑥𝒯	PROPN
iajs-3028	169	40	,	,	PUNCT
iajs-3028	169	41	for	for	ADP
iajs-3028	169	42	𝑥	𝑥	PROPN
iajs-3028	169	43	∈	∈	PROPN
iajs-3028	169	44	𝑅𝑂.	𝑅𝑂.	PUNCT
iajs-3028	169	45	hence	hence	ADV
iajs-3028	169	46	𝓈	𝓈	X
iajs-3028	169	47	=	=	X
iajs-3028	169	48	𝑥𝓈1	𝑥𝓈1	NOUN
iajs-3028	169	49	and	and	CCONJ
iajs-3028	169	50	𝓉	𝓉	PROPN
iajs-3028	169	51	=	=	PUNCT
iajs-3028	169	52	𝑥𝓉1	𝑥𝓉1	NOUN
iajs-3028	169	53	for	for	ADP
iajs-3028	169	54	some	some	DET
iajs-3028	169	55	𝓈1	𝓈1	NOUN
iajs-3028	169	56	∈	∈	PROPN
iajs-3028	169	57	𝒮	𝒮	NOUN
iajs-3028	169	58	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-3028	169	59	𝓉1	𝓉1	PROPN
iajs-3028	169	60	∈	∈	PROPN
iajs-3028	169	61	𝒯.	𝒯.	PROPN
iajs-3028	169	62	thus	thus	ADV
iajs-3028	169	63	𝓀	𝓀	X
iajs-3028	170	1	=	=	SYM
iajs-3028	170	2	𝓈	𝓈	PROPN
iajs-3028	170	3	+	+	X
iajs-3028	170	4	𝓉	𝓉	PROPN
iajs-3028	170	5	=	=	PUNCT
iajs-3028	170	6	𝑥𝓈1	𝑥𝓈1	NOUN
iajs-3028	170	7	+	+	X
iajs-3028	170	8	𝑥𝓉1	𝑥𝓉1	NOUN
iajs-3028	170	9	=	=	SYM
iajs-3028	170	10	𝑥(𝓈1	𝑥(𝓈1	NOUN
iajs-3028	170	11	+	+	CCONJ
iajs-3028	170	12	𝓉1	𝓉1	NOUN
iajs-3028	170	13	)	)	PUNCT
iajs-3028	170	14	.	.	PUNCT
iajs-3028	171	1	if	if	SCONJ
iajs-3028	171	2	𝑢	𝑢	PRON
iajs-3028	171	3	∈	∈	VERB
iajs-3028	171	4	𝓈1+𝓉1	𝓈1+𝓉1	ADP
iajs-3028	171	5	⊆	⊆	NUM
iajs-3028	171	6	𝒮	𝒮	PROPN
iajs-3028	171	7	+	+	CCONJ
iajs-3028	171	8	𝒯.	𝒯.	PROPN
iajs-3028	171	9	then	then	ADV
iajs-3028	171	10	,	,	PUNCT
iajs-3028	171	11	𝑘	𝑘	X
iajs-3028	171	12	=	=	PUNCT
iajs-3028	171	13	𝑥𝑢.	𝑥𝑢.	NOUN
iajs-3028	171	14	definition	definition	NOUN
iajs-3028	171	15	3.18	3.18	NUM
iajs-3028	171	16	[	[	X
iajs-3028	171	17	2	2	NUM
iajs-3028	171	18	]	]	PUNCT
iajs-3028	171	19	.	.	PUNCT
iajs-3028	172	1	the	the	DET
iajs-3028	172	2	function	function	NOUN
iajs-3028	172	3	ℓ	ℓ	PROPN
iajs-3028	172	4	from	from	ADP
iajs-3028	172	5	the	the	DET
iajs-3028	172	6	hyperring	hyperring	NOUN
iajs-3028	172	7	(	(	PUNCT
iajs-3028	172	8	ℛ⊚,⊙	ℛ⊚,⊙	PROPN
iajs-3028	172	9	)	)	PUNCT
iajs-3028	172	10	with	with	ADP
iajs-3028	172	11	1ℛ	1ℛ	PROPN
iajs-3028	172	12	into	into	ADP
iajs-3028	172	13	the	the	DET
iajs-3028	172	14	hyperring	hyperring	NOUN
iajs-3028	172	15	(	(	PUNCT
iajs-3028	172	16	𝒮,∔ˈ,⋅ˈ	𝒮,∔ˈ,⋅ˈ	NUM
iajs-3028	172	17	)	)	PUNCT
iajs-3028	172	18	,	,	PUNCT
iajs-3028	172	19	with	with	ADP
iajs-3028	172	20	1𝒮	1𝒮	NOUN
iajs-3028	172	21	,	,	PUNCT
iajs-3028	172	22	is	be	AUX
iajs-3028	172	23	a	a	DET
iajs-3028	172	24	hyperring	hyperre	VERB
iajs-3028	172	25	homomorphism	homomorphism	NOUN
iajs-3028	172	26	if	if	SCONJ
iajs-3028	172	27	for	for	ADP
iajs-3028	172	28	each	each	DET
iajs-3028	172	29	𝓀,𝒷	𝓀,𝒷	PROPN
iajs-3028	172	30	∈	∈	PROPN
iajs-3028	172	31	ℛ	ℛ	PROPN
iajs-3028	172	32	,	,	PUNCT
iajs-3028	172	33	1	1	NUM
iajs-3028	172	34	.	.	PUNCT
iajs-3028	173	1	ℓ(𝓀⊚	ℓ(𝓀⊚	PROPN
iajs-3028	173	2	𝒷	𝒷	PROPN
iajs-3028	173	3	)	)	PUNCT
iajs-3028	173	4	=	=	SYM
iajs-3028	173	5	ℓ(𝓀	ℓ(𝓀	NOUN
iajs-3028	173	6	)	)	PUNCT
iajs-3028	173	7	∔ˈ	∔ˈ	ADP
iajs-3028	173	8	ℓ(𝒷	ℓ(𝒷	ADJ
iajs-3028	173	9	)	)	PUNCT
iajs-3028	173	10	2	2	NUM
iajs-3028	173	11	.	.	NOUN
iajs-3028	173	12	ℓ(𝓀⊙𝒷)=	ℓ(𝓀⊙𝒷)=	PROPN
iajs-3028	173	13	ℓ(𝓀	ℓ(𝓀	NOUN
iajs-3028	173	14	)	)	PUNCT
iajs-3028	173	15	⋅ˈ	⋅ˈ	X
iajs-3028	173	16	ℓ(𝒷	ℓ(𝒷	ADJ
iajs-3028	173	17	)	)	PUNCT
iajs-3028	173	18	3	3	NUM
iajs-3028	173	19	.	.	PUNCT
iajs-3028	174	1	ℓ(1ℛ)=	ℓ(1ℛ)=	X
iajs-3028	174	2	1𝒮.	1𝒮.	NUM
iajs-3028	174	3	remark	remark	NOUN
iajs-3028	174	4	3.19	3.19	NUM
iajs-3028	174	5	.	.	PUNCT
iajs-3028	175	1	a	a	DET
iajs-3028	175	2	function	function	NOUN
iajs-3028	175	3	ℓ	ℓ	PROPN
iajs-3028	175	4	is	be	AUX
iajs-3028	175	5	named	name	VERB
iajs-3028	175	6	as	as	ADP
iajs-3028	175	7	surjective	surjective	PROPN
iajs-3028	175	8	ℛ-homomorphism	ℛ-homomorphism	PROPN
iajs-3028	175	9	if	if	SCONJ
iajs-3028	175	10	im(ℓ	im(ℓ	VERB
iajs-3028	175	11	)	)	PUNCT
iajs-3028	175	12	=	=	SYM
iajs-3028	175	13	𝒮.	𝒮.	NOUN
iajs-3028	175	14	proposition	proposition	NOUN
iajs-3028	175	15	3.20	3.20	NUM
iajs-3028	175	16	.	.	PUNCT
iajs-3028	176	1	let	let	VERB
iajs-3028	176	2	𝒮	𝒮	PRON
iajs-3028	176	3	be	be	AUX
iajs-3028	176	4	a	a	DET
iajs-3028	176	5	subhyperring	subhyperring	NOUN
iajs-3028	176	6	of	of	ADP
iajs-3028	176	7	a	a	DET
iajs-3028	176	8	divisible	divisible	ADJ
iajs-3028	176	9	hyperring	hyperring	NOUN
iajs-3028	176	10	ℛ	ℛ	NOUN
iajs-3028	176	11	,	,	PUNCT
iajs-3028	176	12	then	then	ADV
iajs-3028	176	13	the	the	DET
iajs-3028	176	14	quotient	quotient	NOUN
iajs-3028	176	15	hyperring	hyperre	VERB
iajs-3028	176	16	ℛ	ℛ	PROPN
iajs-3028	176	17	𝒮	𝒮	PROPN
iajs-3028	176	18	is	be	AUX
iajs-3028	176	19	a	a	DET
iajs-3028	176	20	divisible	divisible	ADJ
iajs-3028	176	21	hyperring	hyperring	NOUN
iajs-3028	176	22	proof	proof	NOUN
iajs-3028	176	23	.	.	PUNCT
iajs-3028	177	1	let	let	VERB
iajs-3028	177	2	𝑎	𝑎	NOUN
iajs-3028	177	3	+	+	NUM
iajs-3028	177	4	𝒮	𝒮	NOUN
iajs-3028	177	5	∈	∈	NOUN
iajs-3028	177	6	𝑑	𝑑	NOUN
iajs-3028	177	7	(	(	PUNCT
iajs-3028	177	8	ℛ	ℛ	PROPN
iajs-3028	177	9	𝒮	𝒮	PROPN
iajs-3028	177	10	)	)	PUNCT
iajs-3028	177	11	,	,	PUNCT
iajs-3028	177	12	so	so	ADV
iajs-3028	177	13	∃𝑎ˊ	∃𝑎ˊ	ADV
iajs-3028	177	14	∈	∈	PROPN
iajs-3028	177	15	ℛ	ℛ	PROPN
iajs-3028	177	16	such	such	ADJ
iajs-3028	177	17	that	that	SCONJ
iajs-3028	177	18	𝑎	𝑎	PROPN
iajs-3028	177	19	=	=	SYM
iajs-3028	177	20	𝑟𝑎ˊ	𝑟𝑎ˊ	NOUN
iajs-3028	177	21	for	for	ADP
iajs-3028	177	22	𝑟	𝑟	NOUN
iajs-3028	177	23	∈	∈	PROPN
iajs-3028	177	24	𝑅𝑂.	𝑅𝑂.	PUNCT
iajs-3028	177	25	thus	thus	ADV
iajs-3028	177	26	𝑎	𝑎	X
iajs-3028	177	27	+	+	NUM
iajs-3028	177	28	𝒮	𝒮	NOUN
iajs-3028	177	29	=	=	SYM
iajs-3028	177	30	𝑟𝑎ˊ	𝑟𝑎ˊ	PROPN
iajs-3028	177	31	+	+	CCONJ
iajs-3028	177	32	𝒮	𝒮	NOUN
iajs-3028	177	33	=	=	SYM
iajs-3028	177	34	𝑟(𝑎ˊ	𝑟(𝑎ˊ	NUM
iajs-3028	177	35	+	+	SYM
iajs-3028	177	36	𝒮	𝒮	PROPN
iajs-3028	177	37	)	)	PUNCT
iajs-3028	177	38	,	,	PUNCT
iajs-3028	177	39	which	which	PRON
iajs-3028	177	40	implies	imply	VERB
iajs-3028	177	41	ℛ	ℛ	PROPN
iajs-3028	177	42	𝒮	𝒮	PROPN
iajs-3028	177	43	is	be	AUX
iajs-3028	177	44	divisible	divisible	ADJ
iajs-3028	177	45	hyperring	hyperring	NOUN
iajs-3028	177	46	.	.	PUNCT
iajs-3028	178	1	corollary	corollary	ADJ
iajs-3028	178	2	3.21	3.21	NUM
iajs-3028	178	3	.	.	PUNCT
iajs-3028	179	1	let	let	VERB
iajs-3028	179	2	𝑓	𝑓	PRON
iajs-3028	179	3	be	be	AUX
iajs-3028	179	4	a	a	DET
iajs-3028	179	5	surjective	surjective	ADJ
iajs-3028	179	6	r	r	NOUN
iajs-3028	179	7	-	-	PUNCT
iajs-3028	179	8	homomorphism	homomorphism	NOUN
iajs-3028	179	9	where	where	SCONJ
iajs-3028	179	10	ℛ	ℛ	PROPN
iajs-3028	179	11	and	and	CCONJ
iajs-3028	179	12	𝒮	𝒮	PROPN
iajs-3028	179	13	are	be	AUX
iajs-3028	179	14	hyperrings	hyperring	NOUN
iajs-3028	179	15	.	.	PUNCT
iajs-3028	180	1	if	if	SCONJ
iajs-3028	180	2	ℛ	ℛ	PROPN
iajs-3028	180	3	is	be	AUX
iajs-3028	180	4	a	a	DET
iajs-3028	180	5	divisible	divisible	ADJ
iajs-3028	180	6	hyperring	hyperring	NOUN
iajs-3028	180	7	,	,	PUNCT
iajs-3028	180	8	then	then	ADV
iajs-3028	180	9	so	so	ADV
iajs-3028	180	10	is	be	AUX
iajs-3028	180	11	𝒮.	𝒮.	PROPN
iajs-3028	180	12	examples	example	NOUN
iajs-3028	180	13	3.22	3.22	NUM
iajs-3028	180	14	.	.	PUNCT
iajs-3028	181	1	[	[	X
iajs-3028	181	2	2	2	NUM
iajs-3028	181	3	]	]	PUNCT
iajs-3028	181	4	.	.	PUNCT
iajs-3028	182	1	the	the	DET
iajs-3028	182	2	hyperoperation	hyperoperation	NOUN
iajs-3028	182	3	“	"	PUNCT
iajs-3028	182	4	⊚	⊚	PROPN
iajs-3028	182	5	”	"	PUNCT
iajs-3028	182	6	and	and	CCONJ
iajs-3028	182	7	the	the	DET
iajs-3028	182	8	multiplication	multiplication	NOUN
iajs-3028	182	9	“	"	PUNCT
iajs-3028	182	10	⨀	⨀	NOUN
iajs-3028	182	11	”	"	PUNCT
iajs-3028	182	12	are	be	AUX
iajs-3028	182	13	defined	define	VERB
iajs-3028	182	14	by	by	ADP
iajs-3028	182	15	the	the	DET
iajs-3028	182	16	following	following	NOUN
iajs-3028	182	17	in	in	ADP
iajs-3028	182	18	tables	table	NOUN
iajs-3028	182	19	1	1	NUM
iajs-3028	182	20	and	and	CCONJ
iajs-3028	182	21	2	2	NUM
iajs-3028	182	22	on	on	ADP
iajs-3028	182	23	ℛ	ℛ	PROPN
iajs-3028	182	24	=	=	PUNCT
iajs-3028	182	25	{	{	PUNCT
iajs-3028	182	26	0	0	NUM
iajs-3028	182	27	,	,	PUNCT
iajs-3028	182	28	1	1	NUM
iajs-3028	182	29	,	,	PUNCT
iajs-3028	182	30	𝓅,𝒹	𝓅,𝒹	VERB
iajs-3028	182	31	}	}	PUNCT
iajs-3028	182	32	.	.	PUNCT
iajs-3028	183	1	table	table	NOUN
iajs-3028	183	2	1	1	NUM
iajs-3028	183	3	:	:	PUNCT
iajs-3028	183	4	additive	additive	ADJ
iajs-3028	183	5	hyperoperayion	hyperoperayion	NOUN
iajs-3028	183	6	⊚	⊚	PROPN
iajs-3028	183	7	0	0	NUM
iajs-3028	183	8	1	1	NUM
iajs-3028	183	9	𝓅	𝓅	PROPN
iajs-3028	183	10	𝒹	𝒹	PROPN
iajs-3028	183	11	0	0	NUM
iajs-3028	183	12	{	{	PUNCT
iajs-3028	183	13	0	0	NUM
iajs-3028	183	14	}	}	SYM
iajs-3028	183	15	1	1	NUM
iajs-3028	183	16	𝓅	𝓅	PROPN
iajs-3028	183	17	𝒹	𝒹	PROPN
iajs-3028	183	18	1	1	NUM
iajs-3028	183	19	1	1	NUM
iajs-3028	183	20	ℛ	ℛ	PROPN
iajs-3028	183	21	{	{	PUNCT
iajs-3028	183	22	1.𝓅.𝒹	1.𝓅.𝒹	NUM
iajs-3028	183	23	}	}	PUNCT
iajs-3028	183	24	{	{	PUNCT
iajs-3028	183	25	1,𝓅,𝒹	1,𝓅,𝒹	NUM
iajs-3028	183	26	}	}	PUNCT
iajs-3028	183	27	𝓅	𝓅	X
iajs-3028	183	28	𝓅	𝓅	X
iajs-3028	183	29	{	{	PUNCT
iajs-3028	183	30	1,𝓅,𝒹	1,𝓅,𝒹	NUM
iajs-3028	183	31	}	}	SYM
iajs-3028	183	32	ℛ	ℛ	PROPN
iajs-3028	183	33	{	{	PUNCT
iajs-3028	183	34	1,𝓅,𝒹	1,𝓅,𝒹	NUM
iajs-3028	183	35	}	}	PUNCT
iajs-3028	183	36	𝒹	𝒹	PROPN
iajs-3028	183	37	𝒹	𝒹	X
iajs-3028	183	38	{	{	PUNCT
iajs-3028	183	39	1,𝓅,𝒹	1,𝓅,𝒹	NUM
iajs-3028	183	40	}	}	PUNCT
iajs-3028	183	41	{	{	PUNCT
iajs-3028	183	42	1,𝓅,𝒹	1,𝓅,𝒹	NUM
iajs-3028	183	43	}	}	PUNCT
iajs-3028	183	44	ℛ	ℛ	NOUN
iajs-3028	183	45	ihjpas	ihjpas	NOUN
iajs-3028	183	46	.	.	PUNCT
iajs-3028	184	1	36(2)2023	36(2)2023	NUM
iajs-3028	184	2	389	389	NUM
iajs-3028	184	3	table	table	NOUN
iajs-3028	184	4	2	2	NUM
iajs-3028	184	5	:	:	PUNCT
iajs-3028	184	6	multiplication	multiplication	NOUN
iajs-3028	184	7	then	then	ADV
iajs-3028	184	8	ℛ	ℛ	PROPN
iajs-3028	184	9	is	be	AUX
iajs-3028	184	10	a	a	DET
iajs-3028	184	11	hyperring	hyperring	NOUN
iajs-3028	184	12	and	and	CCONJ
iajs-3028	184	13	every	every	DET
iajs-3028	184	14	nonzero	nonzero	PROPN
iajs-3028	184	15	element	element	NOUN
iajs-3028	184	16	is	be	AUX
iajs-3028	184	17	a	a	DET
iajs-3028	184	18	divisible	divisible	ADJ
iajs-3028	184	19	,	,	PUNCT
iajs-3028	184	20	thus	thus	ADV
iajs-3028	184	21	ℛ	ℛ	PROPN
iajs-3028	184	22	is	be	AUX
iajs-3028	184	23	divisible	divisible	ADJ
iajs-3028	184	24	hyperring	hyperring	NOUN
iajs-3028	184	25	conclusion	conclusion	NOUN
iajs-3028	184	26	.	.	PUNCT
iajs-3028	185	1	in	in	ADP
iajs-3028	185	2	this	this	DET
iajs-3028	185	3	research	research	NOUN
iajs-3028	185	4	,	,	PUNCT
iajs-3028	185	5	we	we	PRON
iajs-3028	185	6	discussed	discuss	VERB
iajs-3028	185	7	the	the	DET
iajs-3028	185	8	concept	concept	NOUN
iajs-3028	185	9	of	of	ADP
iajs-3028	185	10	divisible	divisible	ADJ
iajs-3028	185	11	hyperring	hyperring	NOUN
iajs-3028	185	12	.	.	PUNCT
iajs-3028	186	1	some	some	PRON
iajs-3028	186	2	of	of	ADP
iajs-3028	186	3	the	the	DET
iajs-3028	186	4	properties	property	NOUN
iajs-3028	186	5	of	of	ADP
iajs-3028	186	6	divisible	divisible	ADJ
iajs-3028	186	7	hyperring	hyperring	NOUN
iajs-3028	186	8	were	be	AUX
iajs-3028	186	9	studied	study	VERB
iajs-3028	186	10	.	.	PUNCT
iajs-3028	187	1	clarify	clarify	VERB
iajs-3028	187	2	some	some	DET
iajs-3028	187	3	concepts	concept	NOUN
iajs-3028	187	4	related	relate	VERB
iajs-3028	187	5	to	to	ADP
iajs-3028	187	6	this	this	DET
iajs-3028	187	7	concept	concept	NOUN
iajs-3028	187	8	.	.	PUNCT
iajs-3028	188	1	references	reference	NOUN
iajs-3028	188	2	.	.	PUNCT
iajs-3028	189	1	1	1	X
iajs-3028	189	2	.	.	X
iajs-3028	189	3	boriboon	boriboon	NOUN
iajs-3028	189	4	,	,	PUNCT
iajs-3028	189	5	s.	s.	PROPN
iajs-3028	189	6	;	;	PUNCT
iajs-3028	189	7	pianskool	pianskool	NOUN
iajs-3028	189	8	,	,	PUNCT
iajs-3028	189	9	s.	s.	PROPN
iajs-3028	189	10	,	,	PUNCT
iajs-3028	189	11	baer	baer	PROPN
iajs-3028	189	12	hypermodules	hypermodule	VERB
iajs-3028	189	13	over	over	ADP
iajs-3028	189	14	krasner	krasner	PROPN
iajs-3028	189	15	hyperrings	hyperring	NOUN
iajs-3028	189	16	.	.	PUNCT
iajs-3028	190	1	22ndannu	22ndannu	X
iajs-3028	190	2	.	.	PUNCT
iajs-3028	191	1	meet	meet	VERB
iajs-3028	191	2	math	math	NOUN
iajs-3028	191	3	.	.	PUNCT
iajs-3028	192	1	2	2	NUM
iajs-3028	192	2	jun	jun	PROPN
iajs-3028	192	3	2017	2017	NUM
iajs-3028	192	4	,	,	PUNCT
iajs-3028	192	5	1	1	NUM
iajs-3028	192	6	-	-	SYM
iajs-3028	192	7	9	9	NUM
iajs-3028	192	8	2	2	NUM
iajs-3028	192	9	.	.	PUNCT
iajs-3028	192	10	bordbar	bordbar	PROPN
iajs-3028	192	11	,	,	PUNCT
iajs-3028	192	12	h.	h.	PROPN
iajs-3028	192	13	;	;	PUNCT
iajs-3028	192	14	cristea	cristea	PROPN
iajs-3028	192	15	i.	i.	NOUN
iajs-3028	192	16	,	,	PUNCT
iajs-3028	192	17	divisible	divisible	ADJ
iajs-3028	192	18	hypermodules	hypermodule	NOUN
iajs-3028	192	19	.	.	PUNCT
iajs-3028	193	1	analele	analele	ADP
iajs-3028	193	2	univ	univ	ADJ
iajs-3028	193	3	“	"	PUNCT
iajs-3028	193	4	ovidius	ovidius	PROPN
iajs-3028	193	5	”	"	PUNCT
iajs-3028	193	6	,	,	PUNCT
iajs-3028	193	7	constanta	constanta	PROPN
iajs-3028	193	8	ser	ser	PROPN
iajs-3028	193	9	mat	mat	PROPN
iajs-3028	193	10	.	.	PROPN
iajs-3028	193	11	2022	2022	NUM
iajs-3028	193	12	,	,	PUNCT
iajs-3028	193	13	30	30	NUM
iajs-3028	193	14	,	,	PUNCT
iajs-3028	193	15	1	1	NUM
iajs-3028	193	16	,	,	PUNCT
iajs-3028	193	17	57	57	NUM
iajs-3028	193	18	-	-	SYM
iajs-3028	193	19	74	74	NUM
iajs-3028	193	20	.	.	PUNCT
iajs-3028	194	1	3	3	X
iajs-3028	194	2	.	.	X
iajs-3028	194	3	harijani	harijani	PROPN
iajs-3028	194	4	,	,	PUNCT
iajs-3028	194	5	km	km	PROPN
iajs-3028	194	6	.	.	PUNCT
iajs-3028	194	7	;	;	PUNCT
iajs-3028	195	1	anvariyeh	anvariyeh	PROPN
iajs-3028	195	2	,	,	PUNCT
iajs-3028	195	3	sm	sm	PROPN
iajs-3028	195	4	.	.	PROPN
iajs-3028	195	5	,	,	PUNCT
iajs-3028	195	6	non	non	ADJ
iajs-3028	195	7	-	-	ADJ
iajs-3028	195	8	commutative	commutative	ADJ
iajs-3028	195	9	hypervaluation	hypervaluation	NOUN
iajs-3028	195	10	on	on	ADP
iajs-3028	195	11	division	division	NOUN
iajs-3028	195	12	hyperrings	hyperring	NOUN
iajs-3028	195	13	.	.	PUNCT
iajs-3028	196	1	j	j	PROPN
iajs-3028	196	2	math	math	PROPN
iajs-3028	196	3	ext	ext	PROPN
iajs-3028	196	4	.	.	PROPN
iajs-3028	197	1	2019	2019	NUM
iajs-3028	197	2	,	,	PUNCT
iajs-3028	197	3	13	13	NUM
iajs-3028	197	4	,	,	PUNCT
iajs-3028	197	5	2	2	NUM
iajs-3028	197	6	,	,	PUNCT
iajs-3028	197	7	93–110	93–110	PROPN
iajs-3028	197	8	.	.	PROPN
iajs-3028	198	1	4	4	NUM
iajs-3028	198	2	.	.	X
iajs-3028	198	3	davvaz	davvaz	PROPN
iajs-3028	198	4	,	,	PUNCT
iajs-3028	198	5	b.	b.	PROPN
iajs-3028	198	6	;	;	PUNCT
iajs-3028	198	7	salasi	salasi	PROPN
iajs-3028	198	8	,	,	PUNCT
iajs-3028	198	9	a.	a.	PROPN
iajs-3028	198	10	,	,	PUNCT
iajs-3028	198	11	a	a	DET
iajs-3028	198	12	realization	realization	NOUN
iajs-3028	198	13	of	of	ADP
iajs-3028	198	14	hyperrings	hyperring	NOUN
iajs-3028	198	15	.	.	PUNCT
iajs-3028	199	1	commun	commun	PROPN
iajs-3028	199	2	algebra	algebra	PROPN
iajs-3028	199	3	2006	2006	NUM
iajs-3028	199	4	,	,	PUNCT
iajs-3028	199	5	34,12	34,12	NUM
iajs-3028	199	6	,	,	PUNCT
iajs-3028	199	7	4389–400	4389–400	NOUN
iajs-3028	199	8	.	.	PUNCT
iajs-3028	200	1	5	5	NUM
iajs-3028	200	2	.	.	X
iajs-3028	200	3	abumghaiseeb	abumghaiseeb	PROPN
iajs-3028	200	4	,	,	PUNCT
iajs-3028	200	5	a.	a.	NOUN
iajs-3028	200	6	on	on	ADP
iajs-3028	200	7	𝜹-primary	𝜹-primary	PROPN
iajs-3028	200	8	hyperideals	hyperideal	NOUN
iajs-3028	200	9	and	and	CCONJ
iajs-3028	200	10	fuzzy	fuzzy	ADJ
iajs-3028	200	11	hyperideals	hyperideal	NOUN
iajs-3028	200	12	expansions	expansion	NOUN
iajs-3028	200	13	,	,	PUNCT
iajs-3028	200	14	republic	republic	NOUN
iajs-3028	200	15	of	of	ADP
iajs-3028	200	16	turkey	turkey	PROPN
iajs-3028	200	17	yildiz	yildiz	PROPN
iajs-3028	200	18	technical	technical	PROPN
iajs-3028	200	19	university	university	PROPN
iajs-3028	200	20	graduate	graduate	NOUN
iajs-3028	200	21	school	school	NOUN
iajs-3028	200	22	of	of	ADP
iajs-3028	200	23	natural	natural	ADJ
iajs-3028	200	24	and	and	CCONJ
iajs-3028	200	25	applied	applied	ADJ
iajs-3028	200	26	sciences	science	NOUN
iajs-3028	200	27	.	.	PUNCT
iajs-3028	200	28	2018	2018	NUM
iajs-3028	200	29	,	,	PUNCT
iajs-3028	200	30	phd	phd	NOUN
iajs-3028	200	31	thesis	thesis	NOUN
iajs-3028	200	32	.	.	PUNCT
iajs-3028	201	1	6	6	X
iajs-3028	201	2	.	.	X
iajs-3028	201	3	bordbar	bordbar	PROPN
iajs-3028	201	4	,	,	PUNCT
iajs-3028	201	5	h.	h.	PROPN
iajs-3028	201	6	;	;	PUNCT
iajs-3028	201	7	novák	novák	NOUN
iajs-3028	201	8	,	,	PUNCT
iajs-3028	201	9	m.	m.	NOUN
iajs-3028	201	10	;	;	PUNCT
iajs-3028	201	11	cristea	cristea	PROPN
iajs-3028	201	12	,	,	PUNCT
iajs-3028	201	13	i.	i.	PROPN
iajs-3028	201	14	,	,	PUNCT
iajs-3028	201	15	a	a	DET
iajs-3028	201	16	note	note	NOUN
iajs-3028	201	17	on	on	ADP
iajs-3028	201	18	the	the	DET
iajs-3028	201	19	support	support	NOUN
iajs-3028	201	20	of	of	ADP
iajs-3028	201	21	a	a	DET
iajs-3028	201	22	hypermodule	hypermodule	NOUN
iajs-3028	201	23	.	.	PUNCT
iajs-3028	202	1	journal	journal	PROPN
iajs-3028	202	2	of	of	ADP
iajs-3028	202	3	algebra	algebra	PROPN
iajs-3028	202	4	and	and	CCONJ
iajs-3028	202	5	its	its	PRON
iajs-3028	202	6	applications	application	NOUN
iajs-3028	202	7	.	.	PUNCT
iajs-3028	203	1	2020	2020	NUM
iajs-3028	203	2	,	,	PUNCT
iajs-3028	203	3	19,01	19,01	NUM
iajs-3028	203	4	,	,	PUNCT
iajs-3028	203	5	2050019	2050019	NUM
iajs-3028	203	6	.	.	PUNCT
iajs-3028	204	1	7	7	X
iajs-3028	204	2	.	.	X
iajs-3028	204	3	m	m	PROPN
iajs-3028	204	4	,	,	PUNCT
iajs-3028	204	5	anbarloei	anbarloei	ADV
iajs-3028	204	6	.	.	PUNCT
iajs-3028	204	7	,	,	PUNCT
iajs-3028	204	8	j	j	PROPN
iajs-3028	204	9	-	-	ADJ
iajs-3028	204	10	prime	prime	ADJ
iajs-3028	204	11	hyperideals	hyperideal	NOUN
iajs-3028	204	12	and	and	CCONJ
iajs-3028	204	13	their	their	PRON
iajs-3028	204	14	generalizations	generalization	NOUN
iajs-3028	204	15	.	.	PUNCT
iajs-3028	205	1	arxiv:2110.07073	arxiv:2110.07073	NOUN
iajs-3028	205	2	.	.	PUNCT
iajs-3028	206	1	[	[	X
iajs-3028	206	2	math.ac	math.ac	X
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