id	sid	tid	token	lemma	pos
ejpam-3380	1	1	european	european	PROPN
ejpam-3380	1	2	journal	journal	PROPN
ejpam-3380	1	3	of	of	ADP
ejpam-3380	1	4	pure	pure	ADJ
ejpam-3380	1	5	and	and	CCONJ
ejpam-3380	1	6	applied	apply	VERB
ejpam-3380	1	7	mathematics	mathematic	NOUN
ejpam-3380	1	8	vol	vol	NOUN
ejpam-3380	1	9	.	.	PROPN
ejpam-3380	2	1	12	12	NUM
ejpam-3380	2	2	,	,	PUNCT
ejpam-3380	2	3	no	no	INTJ
ejpam-3380	2	4	.	.	NOUN
ejpam-3380	2	5	2	2	NUM
ejpam-3380	2	6	,	,	PUNCT
ejpam-3380	2	7	2019	2019	NUM
ejpam-3380	2	8	,	,	PUNCT
ejpam-3380	2	9	332	332	NUM
ejpam-3380	2	10	-	-	SYM
ejpam-3380	2	11	347	347	NUM
ejpam-3380	2	12	issn	issn	PROPN
ejpam-3380	2	13	1307	1307	NUM
ejpam-3380	2	14	-	-	SYM
ejpam-3380	2	15	5543	5543	NUM
ejpam-3380	2	16	–	–	PUNCT
ejpam-3380	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3380	2	18	published	publish	VERB
ejpam-3380	2	19	by	by	ADP
ejpam-3380	2	20	new	new	PROPN
ejpam-3380	2	21	york	york	PROPN
ejpam-3380	2	22	business	business	PROPN
ejpam-3380	2	23	global	global	ADJ
ejpam-3380	2	24	on	on	ADP
ejpam-3380	2	25	weak	weak	ADJ
ejpam-3380	2	26	graded	grade	VERB
ejpam-3380	2	27	rings	ring	NOUN
ejpam-3380	2	28	ii	ii	PROPN
ejpam-3380	2	29	najla	najla	PROPN
ejpam-3380	2	30	al	al	PROPN
ejpam-3380	2	31	-	-	PUNCT
ejpam-3380	2	32	subaie1	subaie1	PROPN
ejpam-3380	2	33	,	,	PUNCT
ejpam-3380	2	34	m.	m.	NOUN
ejpam-3380	2	35	m.	m.	PROPN
ejpam-3380	2	36	al	al	PROPN
ejpam-3380	2	37	-	-	PUNCT
ejpam-3380	2	38	shomrani2,∗	shomrani2,∗	PROPN
ejpam-3380	2	39	1	1	NUM
ejpam-3380	2	40	department	department	NOUN
ejpam-3380	2	41	of	of	ADP
ejpam-3380	2	42	mathematics	mathematics	PROPN
ejpam-3380	2	43	,	,	PUNCT
ejpam-3380	2	44	taif	taif	PROPN
ejpam-3380	2	45	university	university	PROPN
ejpam-3380	2	46	,	,	PUNCT
ejpam-3380	2	47	taif	taif	PROPN
ejpam-3380	2	48	,	,	PUNCT
ejpam-3380	2	49	saudi	saudi	PROPN
ejpam-3380	2	50	arabia	arabia	PROPN
ejpam-3380	2	51	2	2	NUM
ejpam-3380	2	52	king	king	PROPN
ejpam-3380	2	53	abdulaziz	abdulaziz	PROPN
ejpam-3380	2	54	university	university	PROPN
ejpam-3380	2	55	,	,	PUNCT
ejpam-3380	2	56	faculty	faculty	NOUN
ejpam-3380	2	57	of	of	ADP
ejpam-3380	2	58	science	science	NOUN
ejpam-3380	2	59	,	,	PUNCT
ejpam-3380	2	60	p.o.box	p.o.box	PROPN
ejpam-3380	2	61	80203	80203	NUM
ejpam-3380	2	62	,	,	PUNCT
ejpam-3380	2	63	jeddah	jeddah	PROPN
ejpam-3380	2	64	21589	21589	NUM
ejpam-3380	2	65	,	,	PUNCT
ejpam-3380	2	66	saudi	saudi	PROPN
ejpam-3380	2	67	arabia	arabia	PROPN
ejpam-3380	2	68	abstract	abstract	NOUN
ejpam-3380	2	69	.	.	PUNCT
ejpam-3380	3	1	the	the	DET
ejpam-3380	3	2	g	g	NOUN
ejpam-3380	3	3	-	-	PUNCT
ejpam-3380	3	4	weak	weak	ADJ
ejpam-3380	3	5	graded	grade	VERB
ejpam-3380	3	6	rings	ring	NOUN
ejpam-3380	3	7	are	be	AUX
ejpam-3380	3	8	rings	ring	NOUN
ejpam-3380	3	9	graded	grade	VERB
ejpam-3380	3	10	by	by	ADP
ejpam-3380	3	11	a	a	DET
ejpam-3380	3	12	set	set	NOUN
ejpam-3380	3	13	g	g	NOUN
ejpam-3380	3	14	of	of	ADP
ejpam-3380	3	15	left	left	ADJ
ejpam-3380	3	16	coset	coset	NOUN
ejpam-3380	3	17	representatives	representative	NOUN
ejpam-3380	3	18	for	for	ADP
ejpam-3380	3	19	the	the	DET
ejpam-3380	3	20	left	left	ADJ
ejpam-3380	3	21	action	action	NOUN
ejpam-3380	3	22	of	of	ADP
ejpam-3380	3	23	a	a	DET
ejpam-3380	3	24	subgroup	subgroup	NOUN
ejpam-3380	3	25	h	h	NOUN
ejpam-3380	3	26	of	of	ADP
ejpam-3380	3	27	a	a	DET
ejpam-3380	3	28	finite	finite	ADJ
ejpam-3380	3	29	group	group	NOUN
ejpam-3380	3	30	x.	x.	NOUN
ejpam-3380	4	1	the	the	DET
ejpam-3380	4	2	main	main	ADJ
ejpam-3380	4	3	aim	aim	NOUN
ejpam-3380	4	4	of	of	ADP
ejpam-3380	4	5	this	this	DET
ejpam-3380	4	6	article	article	NOUN
ejpam-3380	4	7	is	be	AUX
ejpam-3380	4	8	to	to	PART
ejpam-3380	4	9	study	study	VERB
ejpam-3380	4	10	the	the	DET
ejpam-3380	4	11	concept	concept	NOUN
ejpam-3380	4	12	of	of	ADP
ejpam-3380	4	13	g	g	NOUN
ejpam-3380	4	14	-	-	PUNCT
ejpam-3380	4	15	weak	weak	ADJ
ejpam-3380	4	16	graded	grade	VERB
ejpam-3380	4	17	rings	ring	NOUN
ejpam-3380	4	18	and	and	CCONJ
ejpam-3380	4	19	continue	continue	VERB
ejpam-3380	4	20	the	the	DET
ejpam-3380	4	21	investigation	investigation	NOUN
ejpam-3380	4	22	of	of	ADP
ejpam-3380	4	23	their	their	PRON
ejpam-3380	4	24	properties	property	NOUN
ejpam-3380	4	25	.	.	PUNCT
ejpam-3380	5	1	moreover	moreover	ADV
ejpam-3380	5	2	,	,	PUNCT
ejpam-3380	5	3	some	some	DET
ejpam-3380	5	4	results	result	NOUN
ejpam-3380	5	5	concerning	concern	VERB
ejpam-3380	5	6	g	g	NOUN
ejpam-3380	5	7	-	-	PUNCT
ejpam-3380	5	8	weak	weak	ADJ
ejpam-3380	5	9	graded	grade	VERB
ejpam-3380	5	10	rings	ring	NOUN
ejpam-3380	5	11	of	of	ADP
ejpam-3380	5	12	fractions	fraction	NOUN
ejpam-3380	5	13	are	be	AUX
ejpam-3380	5	14	derived	derive	VERB
ejpam-3380	5	15	.	.	PUNCT
ejpam-3380	6	1	finally	finally	ADV
ejpam-3380	6	2	,	,	PUNCT
ejpam-3380	6	3	some	some	DET
ejpam-3380	6	4	additional	additional	ADJ
ejpam-3380	6	5	examples	example	NOUN
ejpam-3380	6	6	of	of	ADP
ejpam-3380	6	7	g	g	NOUN
ejpam-3380	6	8	-	-	PUNCT
ejpam-3380	6	9	weak	weak	ADJ
ejpam-3380	6	10	graded	grade	VERB
ejpam-3380	6	11	rings	ring	NOUN
ejpam-3380	6	12	are	be	AUX
ejpam-3380	6	13	provided	provide	VERB
ejpam-3380	6	14	.	.	PUNCT
ejpam-3380	7	1	2010	2010	NUM
ejpam-3380	7	2	mathematics	mathematic	NOUN
ejpam-3380	7	3	subject	subject	NOUN
ejpam-3380	7	4	classifications	classification	NOUN
ejpam-3380	7	5	:	:	PUNCT
ejpam-3380	7	6	16w50	16w50	NUM
ejpam-3380	7	7	,	,	PUNCT
ejpam-3380	7	8	13a02	13a02	NUM
ejpam-3380	7	9	,	,	PUNCT
ejpam-3380	7	10	16d25	16d25	NUM
ejpam-3380	7	11	key	key	ADJ
ejpam-3380	7	12	words	word	NOUN
ejpam-3380	7	13	and	and	CCONJ
ejpam-3380	7	14	phrases	phrase	NOUN
ejpam-3380	7	15	:	:	PUNCT
ejpam-3380	7	16	weak	weak	ADJ
ejpam-3380	7	17	graded	grade	VERB
ejpam-3380	7	18	rings	ring	NOUN
ejpam-3380	7	19	,	,	PUNCT
ejpam-3380	7	20	fully	fully	ADV
ejpam-3380	7	21	weak	weak	ADJ
ejpam-3380	7	22	graded	grade	VERB
ejpam-3380	7	23	rings	ring	NOUN
ejpam-3380	7	24	,	,	PUNCT
ejpam-3380	7	25	left	leave	VERB
ejpam-3380	7	26	coset	coset	NOUN
ejpam-3380	7	27	representatives	representative	NOUN
ejpam-3380	7	28	,	,	PUNCT
ejpam-3380	7	29	graded	grade	VERB
ejpam-3380	7	30	rings	ring	NOUN
ejpam-3380	7	31	of	of	ADP
ejpam-3380	7	32	fractions	fraction	NOUN
ejpam-3380	7	33	,	,	PUNCT
ejpam-3380	7	34	homogeneous	homogeneous	ADJ
ejpam-3380	7	35	elements	element	NOUN
ejpam-3380	7	36	.	.	PUNCT
ejpam-3380	8	1	1	1	X
ejpam-3380	8	2	.	.	X
ejpam-3380	8	3	introduction	introduction	NOUN
ejpam-3380	8	4	recall	recall	VERB
ejpam-3380	8	5	that	that	SCONJ
ejpam-3380	8	6	,	,	PUNCT
ejpam-3380	8	7	for	for	ADP
ejpam-3380	8	8	a	a	DET
ejpam-3380	8	9	group	group	NOUN
ejpam-3380	8	10	x	x	X
ejpam-3380	8	11	and	and	CCONJ
ejpam-3380	8	12	a	a	DET
ejpam-3380	8	13	ring	ring	NOUN
ejpam-3380	8	14	r	r	NOUN
ejpam-3380	8	15	,	,	PUNCT
ejpam-3380	8	16	r	r	NOUN
ejpam-3380	8	17	is	be	AUX
ejpam-3380	8	18	called	call	VERB
ejpam-3380	8	19	x	x	ADV
ejpam-3380	8	20	-	-	VERB
ejpam-3380	8	21	graded	grade	VERB
ejpam-3380	8	22	if	if	SCONJ
ejpam-3380	8	23	,	,	PUNCT
ejpam-3380	8	24	for	for	ADP
ejpam-3380	8	25	each	each	DET
ejpam-3380	8	26	element	element	NOUN
ejpam-3380	8	27	g	g	NOUN
ejpam-3380	8	28	in	in	ADP
ejpam-3380	8	29	the	the	DET
ejpam-3380	8	30	group	group	NOUN
ejpam-3380	8	31	x	x	NOUN
ejpam-3380	8	32	,	,	PUNCT
ejpam-3380	8	33	there	there	PRON
ejpam-3380	8	34	is	be	VERB
ejpam-3380	8	35	an	an	DET
ejpam-3380	8	36	additive	additive	ADJ
ejpam-3380	8	37	subgroup	subgroup	NOUN
ejpam-3380	8	38	rg	rg	PROPN
ejpam-3380	8	39	of	of	ADP
ejpam-3380	8	40	r	r	PROPN
ejpam-3380	8	41	,	,	PUNCT
ejpam-3380	8	42	such	such	ADJ
ejpam-3380	8	43	that	that	SCONJ
ejpam-3380	8	44	r	r	NOUN
ejpam-3380	8	45	=	=	SYM
ejpam-3380	8	46	⊕	⊕	PROPN
ejpam-3380	8	47	g∈x	g∈x	VERB
ejpam-3380	8	48	rg	rg	NOUN
ejpam-3380	8	49	and	and	CCONJ
ejpam-3380	8	50	,	,	PUNCT
ejpam-3380	8	51	for	for	ADP
ejpam-3380	8	52	all	all	DET
ejpam-3380	8	53	g	g	NOUN
ejpam-3380	8	54	,	,	PUNCT
ejpam-3380	8	55	h	h	NOUN
ejpam-3380	8	56	∈	∈	PROPN
ejpam-3380	9	1	x	x	X
ejpam-3380	9	2	,	,	PUNCT
ejpam-3380	9	3	we	we	PRON
ejpam-3380	9	4	have	have	AUX
ejpam-3380	9	5	rgrh	rgrh	NOUN
ejpam-3380	9	6	⊆	⊆	NUM
ejpam-3380	9	7	rgh	rgh	PROPN
ejpam-3380	9	8	.	.	PROPN
ejpam-3380	9	9	group	group	PROPN
ejpam-3380	9	10	graded	grade	VERB
ejpam-3380	9	11	rings	ring	NOUN
ejpam-3380	9	12	as	as	ADV
ejpam-3380	9	13	well	well	ADV
ejpam-3380	9	14	as	as	ADP
ejpam-3380	9	15	clifford	clifford	PROPN
ejpam-3380	9	16	theory	theory	NOUN
ejpam-3380	9	17	for	for	ADP
ejpam-3380	9	18	group	group	NOUN
ejpam-3380	9	19	graded	grade	VERB
ejpam-3380	9	20	rings	ring	NOUN
ejpam-3380	9	21	were	be	AUX
ejpam-3380	9	22	studied	study	VERB
ejpam-3380	9	23	and	and	CCONJ
ejpam-3380	9	24	their	their	PRON
ejpam-3380	9	25	properties	property	NOUN
ejpam-3380	9	26	were	be	AUX
ejpam-3380	9	27	investigated	investigate	VERB
ejpam-3380	9	28	by	by	ADP
ejpam-3380	9	29	many	many	ADJ
ejpam-3380	9	30	mathematicians	mathematician	NOUN
ejpam-3380	9	31	,	,	PUNCT
ejpam-3380	9	32	see	see	VERB
ejpam-3380	9	33	for	for	ADP
ejpam-3380	9	34	exampl	exampl	NOUN
ejpam-3380	9	35	[	[	X
ejpam-3380	9	36	8	8	NUM
ejpam-3380	9	37	]	]	PUNCT
ejpam-3380	9	38	,	,	PUNCT
ejpam-3380	9	39	[	[	X
ejpam-3380	9	40	9],[10	9],[10	X
ejpam-3380	9	41	]	]	X
ejpam-3380	9	42	,	,	PUNCT
ejpam-3380	9	43	[	[	X
ejpam-3380	9	44	12	12	NUM
ejpam-3380	9	45	]	]	PUNCT
ejpam-3380	9	46	,	,	PUNCT
ejpam-3380	9	47	[	[	X
ejpam-3380	9	48	16	16	NUM
ejpam-3380	9	49	]	]	PUNCT
ejpam-3380	9	50	,	,	PUNCT
ejpam-3380	9	51	[	[	X
ejpam-3380	9	52	21	21	NUM
ejpam-3380	9	53	]	]	PUNCT
ejpam-3380	9	54	and	and	CCONJ
ejpam-3380	9	55	[	[	X
ejpam-3380	9	56	22	22	NUM
ejpam-3380	9	57	]	]	PUNCT
ejpam-3380	9	58	.	.	PUNCT
ejpam-3380	10	1	nevertheless	nevertheless	ADV
ejpam-3380	10	2	,	,	PUNCT
ejpam-3380	10	3	rings	ring	NOUN
ejpam-3380	10	4	and	and	CCONJ
ejpam-3380	10	5	modules	module	NOUN
ejpam-3380	10	6	can	can	AUX
ejpam-3380	10	7	be	be	AUX
ejpam-3380	10	8	graded	grade	VERB
ejpam-3380	10	9	by	by	ADP
ejpam-3380	10	10	using	use	VERB
ejpam-3380	10	11	semigroups	semigroup	NOUN
ejpam-3380	10	12	instead	instead	ADV
ejpam-3380	10	13	of	of	ADP
ejpam-3380	10	14	groups	group	NOUN
ejpam-3380	10	15	leading	lead	VERB
ejpam-3380	10	16	to	to	ADP
ejpam-3380	10	17	more	more	ADV
ejpam-3380	10	18	general	general	ADJ
ejpam-3380	10	19	results	result	NOUN
ejpam-3380	10	20	as	as	SCONJ
ejpam-3380	10	21	we	we	PRON
ejpam-3380	10	22	can	can	AUX
ejpam-3380	10	23	see	see	VERB
ejpam-3380	10	24	in	in	ADP
ejpam-3380	10	25	[	[	X
ejpam-3380	10	26	1	1	NUM
ejpam-3380	10	27	]	]	PUNCT
ejpam-3380	10	28	,	,	PUNCT
ejpam-3380	10	29	[	[	X
ejpam-3380	10	30	11	11	NUM
ejpam-3380	10	31	]	]	PUNCT
ejpam-3380	10	32	,	,	PUNCT
ejpam-3380	10	33	[	[	X
ejpam-3380	10	34	13	13	NUM
ejpam-3380	10	35	]	]	PUNCT
ejpam-3380	10	36	,	,	PUNCT
ejpam-3380	10	37	[	[	X
ejpam-3380	10	38	14	14	NUM
ejpam-3380	10	39	]	]	PUNCT
ejpam-3380	10	40	,	,	PUNCT
ejpam-3380	10	41	[	[	X
ejpam-3380	10	42	15	15	NUM
ejpam-3380	10	43	]	]	PUNCT
ejpam-3380	10	44	and	and	CCONJ
ejpam-3380	10	45	[	[	X
ejpam-3380	10	46	19	19	NUM
ejpam-3380	10	47	]	]	PUNCT
ejpam-3380	10	48	.	.	PUNCT
ejpam-3380	11	1	many	many	ADJ
ejpam-3380	11	2	ways	way	NOUN
ejpam-3380	11	3	have	have	AUX
ejpam-3380	11	4	been	be	AUX
ejpam-3380	11	5	used	use	VERB
ejpam-3380	11	6	to	to	PART
ejpam-3380	11	7	investigate	investigate	VERB
ejpam-3380	11	8	the	the	DET
ejpam-3380	11	9	properties	property	NOUN
ejpam-3380	11	10	of	of	ADP
ejpam-3380	11	11	these	these	DET
ejpam-3380	11	12	rings	ring	NOUN
ejpam-3380	11	13	.	.	PUNCT
ejpam-3380	12	1	in	in	ADP
ejpam-3380	12	2	[	[	X
ejpam-3380	12	3	7	7	NUM
ejpam-3380	12	4	]	]	PUNCT
ejpam-3380	12	5	,	,	PUNCT
ejpam-3380	12	6	cohen	cohen	PROPN
ejpam-3380	12	7	and	and	CCONJ
ejpam-3380	12	8	montgomery	montgomery	PROPN
ejpam-3380	12	9	introduced	introduce	VERB
ejpam-3380	12	10	an	an	DET
ejpam-3380	12	11	interesting	interesting	ADJ
ejpam-3380	12	12	way	way	NOUN
ejpam-3380	12	13	using	use	VERB
ejpam-3380	12	14	duality	duality	NOUN
ejpam-3380	12	15	theorems	theorem	NOUN
ejpam-3380	12	16	,	,	PUNCT
ejpam-3380	12	17	see	see	VERB
ejpam-3380	12	18	also	also	ADV
ejpam-3380	12	19	[	[	X
ejpam-3380	12	20	5	5	NUM
ejpam-3380	12	21	]	]	PUNCT
ejpam-3380	12	22	.	.	PUNCT
ejpam-3380	13	1	another	another	DET
ejpam-3380	13	2	useful	useful	ADJ
ejpam-3380	13	3	way	way	NOUN
ejpam-3380	13	4	is	be	AUX
ejpam-3380	13	5	the	the	DET
ejpam-3380	13	6	associated	associate	VERB
ejpam-3380	13	7	graded	grade	VERB
ejpam-3380	13	8	ring	ring	NOUN
ejpam-3380	13	9	construction	construction	NOUN
ejpam-3380	13	10	which	which	PRON
ejpam-3380	13	11	states	state	VERB
ejpam-3380	13	12	that	that	SCONJ
ejpam-3380	13	13	for	for	ADP
ejpam-3380	13	14	a	a	DET
ejpam-3380	13	15	valuation	valuation	NOUN
ejpam-3380	13	16	ring	ring	NOUN
ejpam-3380	13	17	r	r	NOUN
ejpam-3380	13	18	,	,	PUNCT
ejpam-3380	13	19	we	we	PRON
ejpam-3380	13	20	can	can	AUX
ejpam-3380	13	21	associate	associate	VERB
ejpam-3380	13	22	a	a	DET
ejpam-3380	13	23	ring	ring	NOUN
ejpam-3380	13	24	rg	rg	NOUN
ejpam-3380	13	25	graded	grade	VERB
ejpam-3380	13	26	by	by	ADP
ejpam-3380	13	27	the	the	DET
ejpam-3380	13	28	valuation	valuation	NOUN
ejpam-3380	13	29	group	group	PROPN
ejpam-3380	13	30	g.	g.	PROPN
ejpam-3380	14	1	this	this	DET
ejpam-3380	14	2	ring	ring	NOUN
ejpam-3380	14	3	seems	seem	VERB
ejpam-3380	14	4	to	to	PART
ejpam-3380	14	5	be	be	AUX
ejpam-3380	14	6	easier	easy	ADJ
ejpam-3380	14	7	to	to	PART
ejpam-3380	14	8	be	be	AUX
ejpam-3380	14	9	studied	study	VERB
ejpam-3380	14	10	and	and	CCONJ
ejpam-3380	14	11	the	the	DET
ejpam-3380	14	12	properties	property	NOUN
ejpam-3380	14	13	can	can	AUX
ejpam-3380	14	14	be	be	AUX
ejpam-3380	14	15	lifted	lift	VERB
ejpam-3380	14	16	back	back	ADV
ejpam-3380	14	17	from	from	ADP
ejpam-3380	14	18	rg	rg	PROPN
ejpam-3380	14	19	to	to	AUX
ejpam-3380	14	20	r.	r.	NOUN
ejpam-3380	14	21	this	this	DET
ejpam-3380	14	22	way	way	NOUN
ejpam-3380	14	23	is	be	AUX
ejpam-3380	14	24	one	one	NUM
ejpam-3380	14	25	of	of	ADP
ejpam-3380	14	26	the	the	DET
ejpam-3380	14	27	motivations	motivation	NOUN
ejpam-3380	14	28	for	for	ADP
ejpam-3380	14	29	studying	study	VERB
ejpam-3380	14	30	graded	grade	VERB
ejpam-3380	14	31	rings	ring	NOUN
ejpam-3380	14	32	,	,	PUNCT
ejpam-3380	14	33	see	see	VERB
ejpam-3380	14	34	[	[	X
ejpam-3380	14	35	16	16	NUM
ejpam-3380	14	36	]	]	PUNCT
ejpam-3380	14	37	for	for	ADP
ejpam-3380	14	38	more	more	ADJ
ejpam-3380	14	39	details	detail	NOUN
ejpam-3380	14	40	.	.	PUNCT
ejpam-3380	15	1	moreover	moreover	ADV
ejpam-3380	15	2	,	,	PUNCT
ejpam-3380	15	3	some	some	DET
ejpam-3380	15	4	mathematicians	mathematician	NOUN
ejpam-3380	15	5	introduced	introduce	VERB
ejpam-3380	15	6	categorical	categorical	ADJ
ejpam-3380	15	7	methods	method	NOUN
ejpam-3380	15	8	to	to	PART
ejpam-3380	15	9	study	study	VERB
ejpam-3380	15	10	these	these	DET
ejpam-3380	15	11	graded	grade	VERB
ejpam-3380	15	12	rings	ring	NOUN
ejpam-3380	15	13	such	such	ADJ
ejpam-3380	15	14	as	as	ADP
ejpam-3380	15	15	the	the	DET
ejpam-3380	15	16	study	study	NOUN
ejpam-3380	15	17	of	of	ADP
ejpam-3380	15	18	separable	separable	ADJ
ejpam-3380	15	19	functors	functor	NOUN
ejpam-3380	15	20	introduced	introduce	VERB
ejpam-3380	15	21	in	in	ADP
ejpam-3380	15	22	[	[	X
ejpam-3380	15	23	17	17	NUM
ejpam-3380	15	24	]	]	PUNCT
ejpam-3380	15	25	and	and	CCONJ
ejpam-3380	15	26	[	[	X
ejpam-3380	15	27	20	20	NUM
ejpam-3380	15	28	]	]	PUNCT
ejpam-3380	15	29	.	.	PUNCT
ejpam-3380	16	1	most	most	ADJ
ejpam-3380	16	2	of	of	ADP
ejpam-3380	16	3	these	these	DET
ejpam-3380	16	4	methods	method	NOUN
ejpam-3380	16	5	have	have	AUX
ejpam-3380	16	6	been	be	AUX
ejpam-3380	16	7	∗corresponding	∗corresponde	VERB
ejpam-3380	16	8	author	author	NOUN
ejpam-3380	16	9	.	.	PUNCT
ejpam-3380	17	1	doi	doi	NOUN
ejpam-3380	17	2	:	:	PUNCT
ejpam-3380	17	3	https://doi.org/10.29020/nybg.ejpam.v12i2.3380	https://doi.org/10.29020/nybg.ejpam.v12i2.3380	NOUN
ejpam-3380	17	4	email	email	NOUN
ejpam-3380	17	5	addresses	address	NOUN
ejpam-3380	17	6	:	:	PUNCT
ejpam-3380	17	7	njlalsubaie@gmail.com	njlalsubaie@gmail.com	PROPN
ejpam-3380	17	8	(	(	PUNCT
ejpam-3380	17	9	n.	n.	PROPN
ejpam-3380	17	10	al	al	PROPN
ejpam-3380	17	11	-	-	PUNCT
ejpam-3380	17	12	subaie	subaie	NOUN
ejpam-3380	17	13	)	)	PUNCT
ejpam-3380	17	14	,	,	PUNCT
ejpam-3380	17	15	malshomrani@hotmail.com	malshomrani@hotmail.com	X
ejpam-3380	17	16	(	(	PUNCT
ejpam-3380	17	17	m.	m.	PROPN
ejpam-3380	17	18	al	al	PROPN
ejpam-3380	17	19	-	-	PUNCT
ejpam-3380	17	20	shomrani	shomrani	PROPN
ejpam-3380	17	21	)	)	PUNCT
ejpam-3380	17	22	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3380	18	1	332	332	NUM
ejpam-3380	18	2	c	c	X
ejpam-3380	18	3	©	©	PROPN
ejpam-3380	18	4	2019	2019	NUM
ejpam-3380	18	5	ejpam	ejpam	NOUN
ejpam-3380	18	6	all	all	DET
ejpam-3380	18	7	rights	right	NOUN
ejpam-3380	18	8	reserved	reserve	VERB
ejpam-3380	18	9	.	.	PUNCT
ejpam-3380	19	1	najla	najla	PROPN
ejpam-3380	19	2	al	al	PROPN
ejpam-3380	19	3	-	-	PUNCT
ejpam-3380	19	4	subaie	subaie	NOUN
ejpam-3380	19	5	,	,	PUNCT
ejpam-3380	19	6	m.	m.	NOUN
ejpam-3380	19	7	m.	m.	PROPN
ejpam-3380	19	8	al	al	PROPN
ejpam-3380	19	9	-	-	PUNCT
ejpam-3380	19	10	shomrani	shomrani	PROPN
ejpam-3380	19	11	/	/	SYM
ejpam-3380	19	12	eur	eur	NOUN
ejpam-3380	19	13	.	.	PUNCT
ejpam-3380	20	1	j.	j.	PROPN
ejpam-3380	20	2	pure	pure	PROPN
ejpam-3380	20	3	appl	appl	PROPN
ejpam-3380	20	4	.	.	PROPN
ejpam-3380	20	5	math	math	PROPN
ejpam-3380	20	6	,	,	PUNCT
ejpam-3380	20	7	12	12	NUM
ejpam-3380	20	8	(	(	PUNCT
ejpam-3380	20	9	2	2	NUM
ejpam-3380	20	10	)	)	PUNCT
ejpam-3380	20	11	(	(	PUNCT
ejpam-3380	20	12	2019	2019	NUM
ejpam-3380	20	13	)	)	PUNCT
ejpam-3380	20	14	,	,	PUNCT
ejpam-3380	20	15	332	332	NUM
ejpam-3380	20	16	-	-	SYM
ejpam-3380	20	17	347	347	NUM
ejpam-3380	20	18	333	333	NUM
ejpam-3380	20	19	introduced	introduce	VERB
ejpam-3380	20	20	for	for	ADP
ejpam-3380	20	21	the	the	DET
ejpam-3380	20	22	finite	finite	PROPN
ejpam-3380	20	23	group	group	NOUN
ejpam-3380	20	24	-	-	PUNCT
ejpam-3380	20	25	grading	grade	VERB
ejpam-3380	20	26	.	.	PUNCT
ejpam-3380	21	1	however	however	ADV
ejpam-3380	21	2	,	,	PUNCT
ejpam-3380	21	3	more	more	ADV
ejpam-3380	21	4	additional	additional	ADJ
ejpam-3380	21	5	investigations	investigation	NOUN
ejpam-3380	21	6	have	have	AUX
ejpam-3380	21	7	been	be	AUX
ejpam-3380	21	8	done	do	VERB
ejpam-3380	21	9	considering	consider	VERB
ejpam-3380	21	10	the	the	DET
ejpam-3380	21	11	infinite	infinite	ADJ
ejpam-3380	21	12	case	case	NOUN
ejpam-3380	21	13	,	,	PUNCT
ejpam-3380	21	14	see	see	VERB
ejpam-3380	21	15	for	for	ADP
ejpam-3380	21	16	example	example	NOUN
ejpam-3380	21	17	[	[	X
ejpam-3380	21	18	2	2	NUM
ejpam-3380	21	19	]	]	PUNCT
ejpam-3380	21	20	.	.	PUNCT
ejpam-3380	22	1	in	in	ADP
ejpam-3380	22	2	[	[	X
ejpam-3380	22	3	6	6	NUM
ejpam-3380	22	4	]	]	PUNCT
ejpam-3380	22	5	,	,	PUNCT
ejpam-3380	22	6	a	a	DET
ejpam-3380	22	7	fixed	fix	VERB
ejpam-3380	22	8	set	set	NOUN
ejpam-3380	22	9	g	g	NOUN
ejpam-3380	22	10	of	of	ADP
ejpam-3380	22	11	left	left	ADJ
ejpam-3380	22	12	coset	coset	NOUN
ejpam-3380	22	13	representatives	representative	NOUN
ejpam-3380	22	14	for	for	ADP
ejpam-3380	22	15	the	the	DET
ejpam-3380	22	16	left	left	ADJ
ejpam-3380	22	17	action	action	NOUN
ejpam-3380	22	18	of	of	ADP
ejpam-3380	22	19	a	a	DET
ejpam-3380	22	20	subgroup	subgroup	NOUN
ejpam-3380	22	21	h	h	NOUN
ejpam-3380	22	22	on	on	ADP
ejpam-3380	22	23	a	a	DET
ejpam-3380	22	24	group	group	NOUN
ejpam-3380	22	25	x	x	PRON
ejpam-3380	22	26	was	be	AUX
ejpam-3380	22	27	constructed	construct	VERB
ejpam-3380	22	28	and	and	CCONJ
ejpam-3380	22	29	a	a	DET
ejpam-3380	22	30	binary	binary	ADJ
ejpam-3380	22	31	operation	operation	NOUN
ejpam-3380	22	32	on	on	ADP
ejpam-3380	22	33	g	g	PROPN
ejpam-3380	22	34	,	,	PUNCT
ejpam-3380	22	35	which	which	PRON
ejpam-3380	22	36	has	have	VERB
ejpam-3380	22	37	a	a	DET
ejpam-3380	22	38	left	left	ADJ
ejpam-3380	22	39	identity	identity	NOUN
ejpam-3380	22	40	and	and	CCONJ
ejpam-3380	22	41	the	the	DET
ejpam-3380	22	42	right	right	ADJ
ejpam-3380	22	43	division	division	NOUN
ejpam-3380	22	44	property	property	NOUN
ejpam-3380	22	45	,	,	PUNCT
ejpam-3380	22	46	was	be	AUX
ejpam-3380	22	47	defined	define	VERB
ejpam-3380	22	48	.	.	PUNCT
ejpam-3380	23	1	this	this	DET
ejpam-3380	23	2	binary	binary	ADJ
ejpam-3380	23	3	operation	operation	NOUN
ejpam-3380	23	4	is	be	AUX
ejpam-3380	23	5	not	not	PART
ejpam-3380	23	6	associative	associative	ADJ
ejpam-3380	23	7	in	in	ADP
ejpam-3380	23	8	a	a	DET
ejpam-3380	23	9	travail	travail	ADJ
ejpam-3380	23	10	way	way	NOUN
ejpam-3380	23	11	.	.	PUNCT
ejpam-3380	24	1	however	however	ADV
ejpam-3380	24	2	,	,	PUNCT
ejpam-3380	24	3	the	the	DET
ejpam-3380	24	4	associativity	associativity	NOUN
ejpam-3380	24	5	was	be	AUX
ejpam-3380	24	6	considered	consider	VERB
ejpam-3380	24	7	by	by	ADP
ejpam-3380	24	8	using	use	VERB
ejpam-3380	24	9	a	a	DET
ejpam-3380	24	10	”	"	PUNCT
ejpam-3380	24	11	cocycle	cocycle	NOUN
ejpam-3380	24	12	”	"	PUNCT
ejpam-3380	24	13	f	f	NOUN
ejpam-3380	24	14	:	:	PUNCT
ejpam-3380	24	15	g×g	g×g	VERB
ejpam-3380	24	16	−→	−→	NOUN
ejpam-3380	24	17	h.	h.	NOUN
ejpam-3380	24	18	the	the	DET
ejpam-3380	24	19	dependence	dependence	NOUN
ejpam-3380	24	20	on	on	ADP
ejpam-3380	24	21	the	the	DET
ejpam-3380	24	22	choice	choice	NOUN
ejpam-3380	24	23	of	of	ADP
ejpam-3380	24	24	representatives	representative	NOUN
ejpam-3380	24	25	was	be	AUX
ejpam-3380	24	26	shown	show	VERB
ejpam-3380	24	27	as	as	SCONJ
ejpam-3380	24	28	follows	follow	VERB
ejpam-3380	24	29	:	:	PUNCT
ejpam-3380	24	30	for	for	ADP
ejpam-3380	24	31	a	a	DET
ejpam-3380	24	32	given	give	VERB
ejpam-3380	24	33	subgroup	subgroup	NOUN
ejpam-3380	24	34	h	h	NOUN
ejpam-3380	24	35	of	of	ADP
ejpam-3380	24	36	a	a	DET
ejpam-3380	24	37	group	group	NOUN
ejpam-3380	24	38	x	x	NOUN
ejpam-3380	24	39	,	,	PUNCT
ejpam-3380	24	40	different	different	ADJ
ejpam-3380	24	41	sets	set	NOUN
ejpam-3380	24	42	of	of	ADP
ejpam-3380	24	43	representatives	representative	NOUN
ejpam-3380	24	44	g	g	PROPN
ejpam-3380	24	45	and	and	CCONJ
ejpam-3380	24	46	g	g	PROPN
ejpam-3380	24	47	for	for	ADP
ejpam-3380	24	48	the	the	DET
ejpam-3380	24	49	left	left	ADJ
ejpam-3380	24	50	cosets	coset	NOUN
ejpam-3380	24	51	can	can	AUX
ejpam-3380	24	52	be	be	AUX
ejpam-3380	24	53	chosen	choose	VERB
ejpam-3380	24	54	.	.	PUNCT
ejpam-3380	25	1	these	these	DET
ejpam-3380	25	2	cosets	coset	NOUN
ejpam-3380	25	3	are	be	AUX
ejpam-3380	25	4	related	relate	VERB
ejpam-3380	25	5	by	by	ADP
ejpam-3380	25	6	an	an	DET
ejpam-3380	25	7	arbitrary	arbitrary	ADJ
ejpam-3380	25	8	function	function	NOUN
ejpam-3380	25	9	γ	γ	NOUN
ejpam-3380	25	10	:	:	PUNCT
ejpam-3380	25	11	x	x	X
ejpam-3380	25	12	/	/	SYM
ejpam-3380	25	13	h	h	PROPN
ejpam-3380	25	14	−→	−→	ADJ
ejpam-3380	25	15	h	h	NOUN
ejpam-3380	25	16	,	,	PUNCT
ejpam-3380	25	17	so	so	SCONJ
ejpam-3380	25	18	that	that	SCONJ
ejpam-3380	25	19	if	if	SCONJ
ejpam-3380	25	20	s	s	VERB
ejpam-3380	25	21	∈	∈	PROPN
ejpam-3380	25	22	g	g	NOUN
ejpam-3380	25	23	then	then	ADV
ejpam-3380	25	24	γ([s])s	γ([s])s	PROPN
ejpam-3380	25	25	∈	∈	PROPN
ejpam-3380	25	26	g	g	PROPN
ejpam-3380	25	27	,	,	PUNCT
ejpam-3380	25	28	where	where	SCONJ
ejpam-3380	25	29	[	[	X
ejpam-3380	25	30	s	s	X
ejpam-3380	25	31	]	]	X
ejpam-3380	25	32	denotes	denote	VERB
ejpam-3380	25	33	the	the	DET
ejpam-3380	25	34	coset	coset	NOUN
ejpam-3380	25	35	hs	hs	PROPN
ejpam-3380	25	36	.	.	PUNCT
ejpam-3380	26	1	in	in	ADP
ejpam-3380	26	2	[	[	X
ejpam-3380	26	3	3	3	NUM
ejpam-3380	26	4	]	]	PUNCT
ejpam-3380	26	5	,	,	PUNCT
ejpam-3380	26	6	the	the	DET
ejpam-3380	26	7	concept	concept	NOUN
ejpam-3380	26	8	of	of	ADP
ejpam-3380	26	9	group	group	NOUN
ejpam-3380	26	10	graded	grade	VERB
ejpam-3380	26	11	rings	ring	NOUN
ejpam-3380	26	12	was	be	AUX
ejpam-3380	26	13	extended	extend	VERB
ejpam-3380	26	14	by	by	ADP
ejpam-3380	26	15	using	use	VERB
ejpam-3380	26	16	a	a	DET
ejpam-3380	26	17	set	set	NOUN
ejpam-3380	26	18	g	g	NOUN
ejpam-3380	26	19	of	of	ADP
ejpam-3380	26	20	left	left	ADJ
ejpam-3380	26	21	coset	coset	NOUN
ejpam-3380	26	22	representatives	representative	NOUN
ejpam-3380	26	23	with	with	ADP
ejpam-3380	26	24	specific	specific	ADJ
ejpam-3380	26	25	binary	binary	ADJ
ejpam-3380	26	26	operation	operation	NOUN
ejpam-3380	26	27	.	.	PUNCT
ejpam-3380	27	1	this	this	DET
ejpam-3380	27	2	new	new	ADJ
ejpam-3380	27	3	concept	concept	NOUN
ejpam-3380	27	4	was	be	AUX
ejpam-3380	27	5	called	call	VERB
ejpam-3380	27	6	g	g	NOUN
ejpam-3380	27	7	-	-	PUNCT
ejpam-3380	27	8	weak	weak	ADJ
ejpam-3380	27	9	graded	grade	VERB
ejpam-3380	27	10	rings	ring	NOUN
ejpam-3380	27	11	.	.	PUNCT
ejpam-3380	28	1	in	in	ADP
ejpam-3380	28	2	[	[	X
ejpam-3380	28	3	4	4	NUM
ejpam-3380	28	4	]	]	PUNCT
ejpam-3380	28	5	,	,	PUNCT
ejpam-3380	28	6	some	some	DET
ejpam-3380	28	7	properties	property	NOUN
ejpam-3380	28	8	of	of	ADP
ejpam-3380	28	9	weak	weak	ADJ
ejpam-3380	28	10	graded	grade	VERB
ejpam-3380	28	11	rings	ring	NOUN
ejpam-3380	28	12	were	be	AUX
ejpam-3380	28	13	investigated	investigate	VERB
ejpam-3380	28	14	.	.	PUNCT
ejpam-3380	29	1	moreover	moreover	ADV
ejpam-3380	29	2	,	,	PUNCT
ejpam-3380	29	3	graded	grade	VERB
ejpam-3380	29	4	rings	ring	NOUN
ejpam-3380	29	5	by	by	ADP
ejpam-3380	29	6	using	use	VERB
ejpam-3380	29	7	the	the	DET
ejpam-3380	29	8	product	product	NOUN
ejpam-3380	29	9	h	h	NOUN
ejpam-3380	29	10	×g	×g	NOUN
ejpam-3380	29	11	were	be	AUX
ejpam-3380	29	12	also	also	ADV
ejpam-3380	29	13	discussed	discuss	VERB
ejpam-3380	29	14	.	.	PUNCT
ejpam-3380	30	1	in	in	ADP
ejpam-3380	30	2	this	this	DET
ejpam-3380	30	3	article	article	NOUN
ejpam-3380	30	4	,	,	PUNCT
ejpam-3380	30	5	we	we	PRON
ejpam-3380	30	6	consider	consider	VERB
ejpam-3380	30	7	the	the	DET
ejpam-3380	30	8	g	g	NOUN
ejpam-3380	30	9	-	-	PUNCT
ejpam-3380	30	10	weak	weak	ADJ
ejpam-3380	30	11	graded	grade	VERB
ejpam-3380	30	12	rings	ring	NOUN
ejpam-3380	30	13	and	and	CCONJ
ejpam-3380	30	14	continue	continue	VERB
ejpam-3380	30	15	the	the	DET
ejpam-3380	30	16	investigation	investigation	NOUN
ejpam-3380	30	17	of	of	ADP
ejpam-3380	30	18	their	their	PRON
ejpam-3380	30	19	properties	property	NOUN
ejpam-3380	30	20	.	.	PUNCT
ejpam-3380	31	1	more	more	ADV
ejpam-3380	31	2	specifically	specifically	ADV
ejpam-3380	31	3	,	,	PUNCT
ejpam-3380	31	4	for	for	ADP
ejpam-3380	31	5	a	a	DET
ejpam-3380	31	6	finite	finite	ADJ
ejpam-3380	31	7	group	group	NOUN
ejpam-3380	31	8	x	x	PROPN
ejpam-3380	31	9	,	,	PUNCT
ejpam-3380	31	10	a	a	DET
ejpam-3380	31	11	subgroup	subgroup	NOUN
ejpam-3380	31	12	h	h	NOUN
ejpam-3380	31	13	,	,	PUNCT
ejpam-3380	31	14	a	a	DET
ejpam-3380	31	15	fixed	fix	VERB
ejpam-3380	31	16	set	set	NOUN
ejpam-3380	31	17	of	of	ADP
ejpam-3380	31	18	left	left	ADJ
ejpam-3380	31	19	coset	coset	NOUN
ejpam-3380	31	20	representatives	representative	NOUN
ejpam-3380	31	21	(	(	PUNCT
ejpam-3380	31	22	g	g	NOUN
ejpam-3380	31	23	,	,	PUNCT
ejpam-3380	31	24	∗	∗	NOUN
ejpam-3380	31	25	)	)	PUNCT
ejpam-3380	31	26	and	and	CCONJ
ejpam-3380	31	27	a	a	DET
ejpam-3380	31	28	g	g	NOUN
ejpam-3380	31	29	-	-	PUNCT
ejpam-3380	31	30	weak	weak	ADJ
ejpam-3380	31	31	graded	grade	VERB
ejpam-3380	31	32	ring	ring	NOUN
ejpam-3380	31	33	r	r	NOUN
ejpam-3380	31	34	with	with	ADP
ejpam-3380	31	35	unity	unity	NOUN
ejpam-3380	31	36	,	,	PUNCT
ejpam-3380	31	37	the	the	DET
ejpam-3380	31	38	following	follow	VERB
ejpam-3380	31	39	results	result	NOUN
ejpam-3380	31	40	are	be	AUX
ejpam-3380	31	41	proved	prove	VERB
ejpam-3380	31	42	:	:	PUNCT
ejpam-3380	31	43	(	(	PUNCT
ejpam-3380	31	44	i	i	NOUN
ejpam-3380	31	45	)	)	PUNCT
ejpam-3380	31	46	if	if	SCONJ
ejpam-3380	31	47	k	k	PROPN
ejpam-3380	31	48	is	be	AUX
ejpam-3380	31	49	a	a	DET
ejpam-3380	31	50	subring	subring	NOUN
ejpam-3380	31	51	of	of	ADP
ejpam-3380	31	52	r	r	NOUN
ejpam-3380	31	53	containing	contain	VERB
ejpam-3380	31	54	all	all	PRON
ejpam-3380	31	55	of	of	ADP
ejpam-3380	31	56	its	its	PRON
ejpam-3380	31	57	g	g	NOUN
ejpam-3380	31	58	-	-	PUNCT
ejpam-3380	31	59	homogeneous	homogeneous	ADJ
ejpam-3380	31	60	elements	element	NOUN
ejpam-3380	31	61	,	,	PUNCT
ejpam-3380	31	62	then	then	ADV
ejpam-3380	31	63	k	k	PROPN
ejpam-3380	31	64	is	be	AUX
ejpam-3380	31	65	a	a	DET
ejpam-3380	31	66	g	g	NOUN
ejpam-3380	31	67	-	-	PUNCT
ejpam-3380	31	68	weak	weak	ADJ
ejpam-3380	31	69	graded	grade	VERB
ejpam-3380	31	70	subring	subring	NOUN
ejpam-3380	31	71	.	.	PUNCT
ejpam-3380	32	1	(	(	PUNCT
ejpam-3380	32	2	ii	ii	NOUN
ejpam-3380	32	3	)	)	PUNCT
ejpam-3380	32	4	if	if	SCONJ
ejpam-3380	32	5	x	x	PRON
ejpam-3380	32	6	is	be	AUX
ejpam-3380	32	7	a	a	DET
ejpam-3380	32	8	unit	unit	NOUN
ejpam-3380	32	9	element	element	NOUN
ejpam-3380	32	10	such	such	ADJ
ejpam-3380	32	11	that	that	SCONJ
ejpam-3380	32	12	x	x	SYM
ejpam-3380	32	13	∈	∈	NOUN
ejpam-3380	32	14	rs	rs	NOUN
ejpam-3380	32	15	for	for	ADP
ejpam-3380	32	16	some	some	DET
ejpam-3380	32	17	s	s	VERB
ejpam-3380	32	18	∈	∈	PROPN
ejpam-3380	32	19	g	g	NOUN
ejpam-3380	32	20	,	,	PUNCT
ejpam-3380	32	21	then	then	ADV
ejpam-3380	32	22	x−1	x−1	PROPN
ejpam-3380	32	23	∈	∈	PROPN
ejpam-3380	32	24	rsl	rsl	NOUN
ejpam-3380	32	25	where	where	SCONJ
ejpam-3380	32	26	sl	sl	PROPN
ejpam-3380	32	27	is	be	AUX
ejpam-3380	32	28	the	the	DET
ejpam-3380	32	29	left	left	ADJ
ejpam-3380	32	30	inverse	inverse	NOUN
ejpam-3380	32	31	of	of	ADP
ejpam-3380	32	32	s.	s.	PROPN
ejpam-3380	32	33	(	(	PUNCT
ejpam-3380	32	34	iii	iii	NOUN
ejpam-3380	32	35	)	)	PUNCT
ejpam-3380	32	36	wgru	wgru	NOUN
ejpam-3380	32	37	(	(	PUNCT
ejpam-3380	32	38	r	r	NOUN
ejpam-3380	32	39	)	)	PUNCT
ejpam-3380	32	40	,	,	PUNCT
ejpam-3380	32	41	the	the	DET
ejpam-3380	32	42	set	set	NOUN
ejpam-3380	32	43	of	of	ADP
ejpam-3380	32	44	all	all	DET
ejpam-3380	32	45	weak	weak	ADJ
ejpam-3380	32	46	graded	grade	VERB
ejpam-3380	32	47	units	unit	NOUN
ejpam-3380	32	48	of	of	ADP
ejpam-3380	32	49	r	r	NOUN
ejpam-3380	32	50	,	,	PUNCT
ejpam-3380	32	51	is	be	AUX
ejpam-3380	32	52	a	a	DET
ejpam-3380	32	53	subgroup	subgroup	NOUN
ejpam-3380	32	54	of	of	ADP
ejpam-3380	32	55	u(r	u(r	NOUN
ejpam-3380	32	56	)	)	PUNCT
ejpam-3380	32	57	.	.	PUNCT
ejpam-3380	33	1	moreover	moreover	ADV
ejpam-3380	33	2	,	,	PUNCT
ejpam-3380	33	3	some	some	PRON
ejpam-3380	33	4	results	result	VERB
ejpam-3380	33	5	considering	consider	VERB
ejpam-3380	33	6	the	the	DET
ejpam-3380	33	7	g	g	NOUN
ejpam-3380	33	8	-	-	PUNCT
ejpam-3380	33	9	weak	weak	ADJ
ejpam-3380	33	10	graded	grade	VERB
ejpam-3380	33	11	rings	ring	NOUN
ejpam-3380	33	12	of	of	ADP
ejpam-3380	33	13	fraction	fraction	NOUN
ejpam-3380	33	14	are	be	AUX
ejpam-3380	33	15	proved	prove	VERB
ejpam-3380	33	16	.	.	PUNCT
ejpam-3380	34	1	finally	finally	ADV
ejpam-3380	34	2	,	,	PUNCT
ejpam-3380	34	3	some	some	DET
ejpam-3380	34	4	additional	additional	ADJ
ejpam-3380	34	5	examples	example	NOUN
ejpam-3380	34	6	of	of	ADP
ejpam-3380	34	7	g	g	NOUN
ejpam-3380	34	8	-	-	PUNCT
ejpam-3380	34	9	weak	weak	ADJ
ejpam-3380	34	10	graded	grade	VERB
ejpam-3380	34	11	rings	ring	NOUN
ejpam-3380	34	12	are	be	AUX
ejpam-3380	34	13	introduced	introduce	VERB
ejpam-3380	34	14	.	.	PUNCT
ejpam-3380	35	1	throughout	throughout	ADP
ejpam-3380	35	2	this	this	DET
ejpam-3380	35	3	article	article	NOUN
ejpam-3380	35	4	,	,	PUNCT
ejpam-3380	35	5	we	we	PRON
ejpam-3380	35	6	shall	shall	AUX
ejpam-3380	35	7	assume	assume	VERB
ejpam-3380	35	8	that	that	SCONJ
ejpam-3380	35	9	all	all	DET
ejpam-3380	35	10	groups	group	NOUN
ejpam-3380	35	11	are	be	AUX
ejpam-3380	35	12	finite	finite	ADJ
ejpam-3380	35	13	,	,	PUNCT
ejpam-3380	35	14	all	all	DET
ejpam-3380	35	15	rings	ring	NOUN
ejpam-3380	35	16	are	be	AUX
ejpam-3380	35	17	commutative	commutative	ADJ
ejpam-3380	35	18	with	with	ADP
ejpam-3380	35	19	unities	unity	NOUN
ejpam-3380	35	20	and	and	CCONJ
ejpam-3380	35	21	all	all	DET
ejpam-3380	35	22	vector	vector	NOUN
ejpam-3380	35	23	spaces	space	NOUN
ejpam-3380	35	24	are	be	AUX
ejpam-3380	35	25	finite	finite	ADJ
ejpam-3380	35	26	dimensional	dimensional	ADJ
ejpam-3380	35	27	.	.	PUNCT
ejpam-3380	36	1	2	2	X
ejpam-3380	36	2	.	.	X
ejpam-3380	36	3	preliminaries	preliminary	NOUN
ejpam-3380	36	4	in	in	ADP
ejpam-3380	36	5	this	this	DET
ejpam-3380	36	6	section	section	NOUN
ejpam-3380	36	7	,	,	PUNCT
ejpam-3380	36	8	we	we	PRON
ejpam-3380	36	9	list	list	VERB
ejpam-3380	36	10	some	some	DET
ejpam-3380	36	11	important	important	ADJ
ejpam-3380	36	12	definitions	definition	NOUN
ejpam-3380	36	13	and	and	CCONJ
ejpam-3380	36	14	results	result	NOUN
ejpam-3380	36	15	that	that	PRON
ejpam-3380	36	16	will	will	AUX
ejpam-3380	36	17	be	be	AUX
ejpam-3380	36	18	used	use	VERB
ejpam-3380	36	19	later	later	ADV
ejpam-3380	36	20	in	in	ADP
ejpam-3380	36	21	this	this	DET
ejpam-3380	36	22	article	article	NOUN
ejpam-3380	36	23	.	.	PUNCT
ejpam-3380	37	1	definition	definition	NOUN
ejpam-3380	37	2	1	1	NUM
ejpam-3380	37	3	.	.	PUNCT
ejpam-3380	38	1	[	[	X
ejpam-3380	38	2	6	6	NUM
ejpam-3380	38	3	]	]	PUNCT
ejpam-3380	38	4	let	let	VERB
ejpam-3380	38	5	x	x	PRON
ejpam-3380	38	6	be	be	AUX
ejpam-3380	38	7	a	a	DET
ejpam-3380	38	8	group	group	NOUN
ejpam-3380	38	9	and	and	CCONJ
ejpam-3380	38	10	h	h	NOUN
ejpam-3380	38	11	be	be	AUX
ejpam-3380	38	12	a	a	DET
ejpam-3380	38	13	subgroup	subgroup	NOUN
ejpam-3380	38	14	of	of	ADP
ejpam-3380	38	15	x.	x.	NOUN
ejpam-3380	38	16	we	we	PRON
ejpam-3380	38	17	call	call	VERB
ejpam-3380	38	18	g	g	PROPN
ejpam-3380	38	19	⊂	⊂	PROPN
ejpam-3380	38	20	x	x	PUNCT
ejpam-3380	38	21	a	a	DET
ejpam-3380	38	22	set	set	NOUN
ejpam-3380	38	23	of	of	ADP
ejpam-3380	38	24	left	left	ADJ
ejpam-3380	38	25	coset	coset	NOUN
ejpam-3380	38	26	representatives	representative	NOUN
ejpam-3380	38	27	if	if	SCONJ
ejpam-3380	38	28	,	,	PUNCT
ejpam-3380	38	29	for	for	ADP
ejpam-3380	38	30	every	every	DET
ejpam-3380	38	31	x	x	SYM
ejpam-3380	38	32	∈	∈	PROPN
ejpam-3380	38	33	x	x	NOUN
ejpam-3380	38	34	,	,	PUNCT
ejpam-3380	38	35	there	there	PRON
ejpam-3380	38	36	is	be	VERB
ejpam-3380	38	37	a	a	DET
ejpam-3380	38	38	unique	unique	ADJ
ejpam-3380	38	39	s	s	X
ejpam-3380	38	40	∈	∈	NOUN
ejpam-3380	38	41	g	g	NOUN
ejpam-3380	38	42	such	such	ADJ
ejpam-3380	38	43	that	that	SCONJ
ejpam-3380	38	44	x	x	SYM
ejpam-3380	38	45	∈	∈	PROPN
ejpam-3380	38	46	hs	hs	PROPN
ejpam-3380	38	47	.	.	PROPN
ejpam-3380	39	1	in	in	ADP
ejpam-3380	39	2	addition	addition	NOUN
ejpam-3380	39	3	,	,	PUNCT
ejpam-3380	39	4	we	we	PRON
ejpam-3380	39	5	call	call	VERB
ejpam-3380	39	6	the	the	DET
ejpam-3380	39	7	decomposition	decomposition	NOUN
ejpam-3380	39	8	x	x	PUNCT
ejpam-3380	40	1	=	=	PUNCT
ejpam-3380	40	2	us	us	PROPN
ejpam-3380	40	3	,	,	PUNCT
ejpam-3380	40	4	for	for	ADP
ejpam-3380	40	5	u	u	PROPN
ejpam-3380	40	6	∈	∈	PROPN
ejpam-3380	40	7	h	h	NOUN
ejpam-3380	40	8	and	and	CCONJ
ejpam-3380	40	9	s	s	PROPN
ejpam-3380	40	10	∈	∈	PROPN
ejpam-3380	40	11	g	g	NOUN
ejpam-3380	40	12	,	,	PUNCT
ejpam-3380	40	13	the	the	DET
ejpam-3380	40	14	unique	unique	ADJ
ejpam-3380	40	15	factorization	factorization	NOUN
ejpam-3380	40	16	of	of	ADP
ejpam-3380	40	17	x.	x.	NOUN
ejpam-3380	40	18	in	in	ADP
ejpam-3380	40	19	what	what	PRON
ejpam-3380	40	20	follows	follow	VERB
ejpam-3380	40	21	,	,	PUNCT
ejpam-3380	40	22	g	g	NOUN
ejpam-3380	40	23	stands	stand	VERB
ejpam-3380	40	24	for	for	ADP
ejpam-3380	40	25	a	a	DET
ejpam-3380	40	26	fixed	fix	VERB
ejpam-3380	40	27	set	set	NOUN
ejpam-3380	40	28	of	of	ADP
ejpam-3380	40	29	left	left	ADJ
ejpam-3380	40	30	coset	coset	NOUN
ejpam-3380	40	31	representatives	representative	NOUN
ejpam-3380	40	32	for	for	ADP
ejpam-3380	40	33	the	the	DET
ejpam-3380	40	34	action	action	NOUN
ejpam-3380	40	35	of	of	ADP
ejpam-3380	40	36	the	the	DET
ejpam-3380	40	37	subgroup	subgroup	NOUN
ejpam-3380	40	38	h	h	NOUN
ejpam-3380	40	39	of	of	ADP
ejpam-3380	40	40	x	x	PUNCT
ejpam-3380	40	41	on	on	ADP
ejpam-3380	40	42	the	the	DET
ejpam-3380	40	43	group	group	NOUN
ejpam-3380	40	44	x	x	PUNCT
ejpam-3380	40	45	and	and	CCONJ
ejpam-3380	40	46	e	e	PROPN
ejpam-3380	40	47	is	be	AUX
ejpam-3380	40	48	the	the	DET
ejpam-3380	40	49	identity	identity	NOUN
ejpam-3380	40	50	element	element	NOUN
ejpam-3380	40	51	in	in	ADP
ejpam-3380	40	52	x.	x.	NOUN
ejpam-3380	40	53	definition	definition	NOUN
ejpam-3380	40	54	2	2	NUM
ejpam-3380	40	55	.	.	PUNCT
ejpam-3380	41	1	[	[	X
ejpam-3380	41	2	6	6	NUM
ejpam-3380	41	3	]	]	PUNCT
ejpam-3380	41	4	for	for	ADP
ejpam-3380	41	5	elements	element	NOUN
ejpam-3380	41	6	s	s	PART
ejpam-3380	41	7	,	,	PUNCT
ejpam-3380	41	8	t	t	PROPN
ejpam-3380	41	9	∈	∈	PROPN
ejpam-3380	41	10	g	g	NOUN
ejpam-3380	41	11	we	we	PRON
ejpam-3380	41	12	define	define	VERB
ejpam-3380	41	13	f(s	f(	NOUN
ejpam-3380	41	14	,	,	PUNCT
ejpam-3380	41	15	t	t	PROPN
ejpam-3380	41	16	)	)	PUNCT
ejpam-3380	41	17	∈	∈	PROPN
ejpam-3380	41	18	h	h	NOUN
ejpam-3380	41	19	and	and	CCONJ
ejpam-3380	41	20	s	s	NOUN
ejpam-3380	41	21	∗	∗	NOUN
ejpam-3380	41	22	t	t	NOUN
ejpam-3380	41	23	∈	∈	PROPN
ejpam-3380	41	24	g	g	NOUN
ejpam-3380	41	25	by	by	ADP
ejpam-3380	41	26	the	the	DET
ejpam-3380	41	27	unique	unique	ADJ
ejpam-3380	41	28	factorization	factorization	NOUN
ejpam-3380	41	29	st	st	NOUN
ejpam-3380	41	30	=	=	SYM
ejpam-3380	41	31	f(s	f(s	PROPN
ejpam-3380	41	32	,	,	PUNCT
ejpam-3380	41	33	t)(s	t)(s	PROPN
ejpam-3380	41	34	∗	∗	NOUN
ejpam-3380	41	35	t	t	PROPN
ejpam-3380	41	36	)	)	PUNCT
ejpam-3380	41	37	in	in	ADP
ejpam-3380	41	38	x	x	NOUN
ejpam-3380	41	39	,	,	PUNCT
ejpam-3380	41	40	where	where	SCONJ
ejpam-3380	41	41	f	f	PROPN
ejpam-3380	41	42	is	be	AUX
ejpam-3380	41	43	the	the	DET
ejpam-3380	41	44	cocycle	cocycle	NOUN
ejpam-3380	41	45	map	map	NOUN
ejpam-3380	41	46	.	.	PUNCT
ejpam-3380	42	1	moreover	moreover	ADV
ejpam-3380	42	2	,	,	PUNCT
ejpam-3380	42	3	the	the	DET
ejpam-3380	42	4	functions	function	NOUN
ejpam-3380	42	5	.	.	PUNCT
ejpam-3380	43	1	:	:	PUNCT
ejpam-3380	43	2	g×h	g×h	VERB
ejpam-3380	43	3	→	→	SYM
ejpam-3380	43	4	h	h	NOUN
ejpam-3380	43	5	and	and	CCONJ
ejpam-3380	43	6	/	/	SYM
ejpam-3380	43	7	:	:	PUNCT
ejpam-3380	43	8	g×h	g×h	PROPN
ejpam-3380	43	9	→	→	SYM
ejpam-3380	43	10	g	g	NOUN
ejpam-3380	43	11	are	be	AUX
ejpam-3380	43	12	defined	define	VERB
ejpam-3380	43	13	by	by	ADP
ejpam-3380	43	14	the	the	DET
ejpam-3380	43	15	unique	unique	ADJ
ejpam-3380	43	16	factorization	factorization	NOUN
ejpam-3380	43	17	su	su	PROPN
ejpam-3380	44	1	=	=	PUNCT
ejpam-3380	44	2	(	(	PUNCT
ejpam-3380	44	3	s.u)(s	s.u)(s	NOUN
ejpam-3380	44	4	/	/	SYM
ejpam-3380	44	5	u	u	NOUN
ejpam-3380	44	6	)	)	PUNCT
ejpam-3380	44	7	for	for	ADP
ejpam-3380	44	8	s	s	PROPN
ejpam-3380	44	9	,	,	PUNCT
ejpam-3380	44	10	s	s	PART
ejpam-3380	44	11	/	/	SYM
ejpam-3380	44	12	u	u	NOUN
ejpam-3380	44	13	∈	∈	PROPN
ejpam-3380	44	14	g	g	NOUN
ejpam-3380	44	15	and	and	CCONJ
ejpam-3380	44	16	u	u	NOUN
ejpam-3380	44	17	,	,	PUNCT
ejpam-3380	44	18	s	s	PART
ejpam-3380	44	19	.	.	PUNCT
ejpam-3380	45	1	u	u	PROPN
ejpam-3380	45	2	∈	∈	PROPN
ejpam-3380	45	3	h.	h.	PROPN
ejpam-3380	45	4	najla	najla	PROPN
ejpam-3380	45	5	al	al	PROPN
ejpam-3380	45	6	-	-	PUNCT
ejpam-3380	45	7	subaie	subaie	NOUN
ejpam-3380	45	8	,	,	PUNCT
ejpam-3380	45	9	m.	m.	NOUN
ejpam-3380	45	10	m.	m.	PROPN
ejpam-3380	45	11	al	al	PROPN
ejpam-3380	45	12	-	-	PUNCT
ejpam-3380	45	13	shomrani	shomrani	PROPN
ejpam-3380	45	14	/	/	SYM
ejpam-3380	45	15	eur	eur	NOUN
ejpam-3380	45	16	.	.	PUNCT
ejpam-3380	46	1	j.	j.	PROPN
ejpam-3380	46	2	pure	pure	PROPN
ejpam-3380	46	3	appl	appl	PROPN
ejpam-3380	46	4	.	.	PROPN
ejpam-3380	46	5	math	math	PROPN
ejpam-3380	46	6	,	,	PUNCT
ejpam-3380	46	7	12	12	NUM
ejpam-3380	46	8	(	(	PUNCT
ejpam-3380	46	9	2	2	NUM
ejpam-3380	46	10	)	)	PUNCT
ejpam-3380	46	11	(	(	PUNCT
ejpam-3380	46	12	2019	2019	NUM
ejpam-3380	46	13	)	)	PUNCT
ejpam-3380	46	14	,	,	PUNCT
ejpam-3380	46	15	332	332	NUM
ejpam-3380	46	16	-	-	SYM
ejpam-3380	46	17	347	347	NUM
ejpam-3380	46	18	334	334	NUM
ejpam-3380	46	19	the	the	DET
ejpam-3380	46	20	binary	binary	PROPN
ejpam-3380	46	21	operation	operation	NOUN
ejpam-3380	46	22	∗	∗	NOUN
ejpam-3380	46	23	on	on	ADP
ejpam-3380	46	24	g	g	PROPN
ejpam-3380	46	25	has	have	VERB
ejpam-3380	46	26	a	a	DET
ejpam-3380	46	27	unique	unique	ADJ
ejpam-3380	46	28	left	leave	VERB
ejpam-3380	46	29	identity	identity	NOUN
ejpam-3380	46	30	eg	eg	NOUN
ejpam-3380	46	31	∈	∈	PROPN
ejpam-3380	46	32	g	g	NOUN
ejpam-3380	46	33	and	and	CCONJ
ejpam-3380	46	34	satisfying	satisfy	VERB
ejpam-3380	46	35	the	the	DET
ejpam-3380	46	36	right	right	ADJ
ejpam-3380	46	37	division	division	NOUN
ejpam-3380	46	38	property	property	NOUN
ejpam-3380	46	39	,	,	PUNCT
ejpam-3380	46	40	i.e.	i.e.	X
ejpam-3380	46	41	,	,	PUNCT
ejpam-3380	46	42	for	for	ADP
ejpam-3380	46	43	all	all	DET
ejpam-3380	46	44	s	s	PROPN
ejpam-3380	46	45	,	,	PUNCT
ejpam-3380	46	46	t	t	PROPN
ejpam-3380	46	47	∈	∈	PROPN
ejpam-3380	46	48	g	g	PROPN
ejpam-3380	46	49	there	there	PRON
ejpam-3380	46	50	is	be	VERB
ejpam-3380	46	51	a	a	DET
ejpam-3380	46	52	unique	unique	ADJ
ejpam-3380	46	53	solution	solution	NOUN
ejpam-3380	47	1	p	p	X
ejpam-3380	47	2	∈	∈	PROPN
ejpam-3380	47	3	g	g	NOUN
ejpam-3380	47	4	satisfying	satisfy	VERB
ejpam-3380	47	5	the	the	DET
ejpam-3380	47	6	equation	equation	NOUN
ejpam-3380	47	7	p	p	NOUN
ejpam-3380	47	8	∗	∗	NOUN
ejpam-3380	47	9	s	s	PART
ejpam-3380	47	10	=	=	X
ejpam-3380	47	11	t	t	PROPN
ejpam-3380	48	1	[	[	X
ejpam-3380	48	2	6	6	NUM
ejpam-3380	48	3	]	]	PUNCT
ejpam-3380	48	4	.	.	PUNCT
ejpam-3380	49	1	if	if	SCONJ
ejpam-3380	49	2	e	e	PROPN
ejpam-3380	49	3	∈	∈	PROPN
ejpam-3380	49	4	g	g	PROPN
ejpam-3380	49	5	,	,	PUNCT
ejpam-3380	49	6	then	then	ADV
ejpam-3380	49	7	eg	eg	NOUN
ejpam-3380	49	8	=	=	PUNCT
ejpam-3380	49	9	e	e	PROPN
ejpam-3380	49	10	is	be	AUX
ejpam-3380	49	11	also	also	ADV
ejpam-3380	49	12	a	a	DET
ejpam-3380	49	13	right	right	ADJ
ejpam-3380	49	14	identity	identity	NOUN
ejpam-3380	49	15	.	.	PUNCT
ejpam-3380	50	1	also	also	ADV
ejpam-3380	50	2	,	,	PUNCT
ejpam-3380	50	3	there	there	PRON
ejpam-3380	50	4	is	be	VERB
ejpam-3380	50	5	a	a	DET
ejpam-3380	50	6	unique	unique	ADJ
ejpam-3380	50	7	left	left	ADJ
ejpam-3380	50	8	inverse	inverse	NOUN
ejpam-3380	50	9	sl	sl	NOUN
ejpam-3380	50	10	for	for	ADP
ejpam-3380	50	11	every	every	DET
ejpam-3380	50	12	s	s	X
ejpam-3380	50	13	∈	∈	NOUN
ejpam-3380	50	14	g	g	NOUN
ejpam-3380	50	15	satisfying	satisfy	VERB
ejpam-3380	50	16	the	the	DET
ejpam-3380	50	17	equation	equation	NOUN
ejpam-3380	50	18	sl	sl	INTJ
ejpam-3380	50	19	∗	∗	NOUN
ejpam-3380	50	20	s	s	PART
ejpam-3380	50	21	=	=	NOUN
ejpam-3380	50	22	eg	eg	NOUN
ejpam-3380	50	23	.	.	PUNCT
ejpam-3380	51	1	proposition	proposition	NOUN
ejpam-3380	51	2	1	1	NUM
ejpam-3380	51	3	.	.	PUNCT
ejpam-3380	52	1	[	[	X
ejpam-3380	52	2	6	6	NUM
ejpam-3380	52	3	]	]	PUNCT
ejpam-3380	52	4	the	the	DET
ejpam-3380	52	5	following	follow	VERB
ejpam-3380	52	6	identities	identity	NOUN
ejpam-3380	52	7	between	between	ADP
ejpam-3380	52	8	(	(	PUNCT
ejpam-3380	52	9	g	g	NOUN
ejpam-3380	52	10	,	,	PUNCT
ejpam-3380	52	11	∗	∗	NOUN
ejpam-3380	52	12	)	)	PUNCT
ejpam-3380	52	13	and	and	CCONJ
ejpam-3380	52	14	f	f	PROPN
ejpam-3380	52	15	are	be	AUX
ejpam-3380	52	16	satisfied	satisfied	ADJ
ejpam-3380	52	17	for	for	ADP
ejpam-3380	52	18	all	all	DET
ejpam-3380	52	19	s	s	PROPN
ejpam-3380	52	20	,	,	PUNCT
ejpam-3380	52	21	t	t	PROPN
ejpam-3380	52	22	,	,	PUNCT
ejpam-3380	52	23	p	p	PROPN
ejpam-3380	52	24	∈	∈	PROPN
ejpam-3380	52	25	g	g	NOUN
ejpam-3380	52	26	and	and	CCONJ
ejpam-3380	52	27	all	all	DET
ejpam-3380	52	28	u	u	NOUN
ejpam-3380	52	29	,	,	PUNCT
ejpam-3380	52	30	v	v	PROPN
ejpam-3380	52	31	∈	∈	PROPN
ejpam-3380	52	32	h	h	NOUN
ejpam-3380	52	33	:	:	PUNCT
ejpam-3380	52	34	s	s	X
ejpam-3380	52	35	.	.	PUNCT
ejpam-3380	53	1	(	(	PUNCT
ejpam-3380	53	2	t	t	PROPN
ejpam-3380	53	3	.	.	PUNCT
ejpam-3380	54	1	u	u	NOUN
ejpam-3380	54	2	)	)	PUNCT
ejpam-3380	54	3	=	=	PUNCT
ejpam-3380	54	4	f(s	f(s	PROPN
ejpam-3380	54	5	,	,	PUNCT
ejpam-3380	54	6	t	t	PROPN
ejpam-3380	54	7	)	)	PUNCT
ejpam-3380	54	8	(	(	PUNCT
ejpam-3380	54	9	(	(	PUNCT
ejpam-3380	54	10	s	s	NOUN
ejpam-3380	54	11	∗	∗	X
ejpam-3380	54	12	t	t	NOUN
ejpam-3380	54	13	)	)	PUNCT
ejpam-3380	54	14	.	.	PUNCT
ejpam-3380	55	1	u)f	u)f	PROPN
ejpam-3380	55	2	(	(	PUNCT
ejpam-3380	55	3	s	s	NOUN
ejpam-3380	55	4	/	/	SYM
ejpam-3380	55	5	(	(	PUNCT
ejpam-3380	55	6	t	t	PROPN
ejpam-3380	55	7	.	.	PUNCT
ejpam-3380	56	1	u	u	NOUN
ejpam-3380	56	2	)	)	PUNCT
ejpam-3380	56	3	,	,	PUNCT
ejpam-3380	56	4	t	t	PROPN
ejpam-3380	56	5	/	/	SYM
ejpam-3380	56	6	u	u	PROPN
ejpam-3380	56	7	)	)	PUNCT
ejpam-3380	56	8	−1	−1	NOUN
ejpam-3380	56	9	,	,	PUNCT
ejpam-3380	56	10	(	(	PUNCT
ejpam-3380	56	11	s	s	NOUN
ejpam-3380	56	12	∗	∗	X
ejpam-3380	56	13	t	t	PROPN
ejpam-3380	56	14	)	)	PUNCT
ejpam-3380	56	15	/	/	SYM
ejpam-3380	56	16	u	u	NOUN
ejpam-3380	56	17	=	=	PUNCT
ejpam-3380	56	18	(	(	PUNCT
ejpam-3380	56	19	s	s	NOUN
ejpam-3380	56	20	/	/	SYM
ejpam-3380	56	21	(	(	PUNCT
ejpam-3380	56	22	t	t	PROPN
ejpam-3380	56	23	.	.	PUNCT
ejpam-3380	57	1	u	u	NOUN
ejpam-3380	57	2	)	)	PUNCT
ejpam-3380	57	3	)	)	PUNCT
ejpam-3380	58	1	∗	∗	NOUN
ejpam-3380	58	2	(	(	PUNCT
ejpam-3380	58	3	t	t	PROPN
ejpam-3380	58	4	/	/	SYM
ejpam-3380	58	5	u	u	NOUN
ejpam-3380	58	6	)	)	PUNCT
ejpam-3380	58	7	,	,	PUNCT
ejpam-3380	58	8	s	s	X
ejpam-3380	58	9	.	.	PUNCT
ejpam-3380	59	1	uv	uv	NOUN
ejpam-3380	59	2	=	=	PUNCT
ejpam-3380	59	3	(	(	PUNCT
ejpam-3380	59	4	s	s	NOUN
ejpam-3380	59	5	.	.	PUNCT
ejpam-3380	59	6	u	u	NOUN
ejpam-3380	59	7	)	)	PUNCT
ejpam-3380	59	8	(	(	PUNCT
ejpam-3380	59	9	(	(	PUNCT
ejpam-3380	59	10	s	s	NOUN
ejpam-3380	59	11	/	/	SYM
ejpam-3380	59	12	u	u	NOUN
ejpam-3380	59	13	)	)	PUNCT
ejpam-3380	59	14	.	.	PUNCT
ejpam-3380	60	1	v	v	X
ejpam-3380	60	2	)	)	PUNCT
ejpam-3380	60	3	,	,	PUNCT
ejpam-3380	60	4	s	s	X
ejpam-3380	60	5	/	/	SYM
ejpam-3380	60	6	uv	uv	NOUN
ejpam-3380	60	7	=	=	PUNCT
ejpam-3380	60	8	(	(	PUNCT
ejpam-3380	60	9	s	s	NOUN
ejpam-3380	60	10	/	/	SYM
ejpam-3380	60	11	u	u	NOUN
ejpam-3380	60	12	)	)	PUNCT
ejpam-3380	60	13	/	/	SYM
ejpam-3380	60	14	v	v	NOUN
ejpam-3380	60	15	,	,	PUNCT
ejpam-3380	60	16	f(p	f(p	PROPN
ejpam-3380	60	17	,	,	PUNCT
ejpam-3380	60	18	s)f(p	s)f(p	PROPN
ejpam-3380	60	19	∗	∗	PROPN
ejpam-3380	60	20	s	s	PROPN
ejpam-3380	60	21	,	,	PUNCT
ejpam-3380	60	22	t	t	PROPN
ejpam-3380	60	23	)	)	PUNCT
ejpam-3380	60	24	=	=	PUNCT
ejpam-3380	61	1	(	(	PUNCT
ejpam-3380	61	2	p	p	X
ejpam-3380	61	3	.	.	PUNCT
ejpam-3380	62	1	f(s	f(s	PROPN
ejpam-3380	62	2	,	,	PUNCT
ejpam-3380	62	3	t	t	PROPN
ejpam-3380	62	4	)	)	PUNCT
ejpam-3380	62	5	)	)	PUNCT
ejpam-3380	63	1	f	f	NOUN
ejpam-3380	63	2	(	(	PUNCT
ejpam-3380	63	3	p	p	X
ejpam-3380	63	4	/	/	SYM
ejpam-3380	63	5	f(s	f(s	PROPN
ejpam-3380	63	6	,	,	PUNCT
ejpam-3380	63	7	t	t	PROPN
ejpam-3380	63	8	)	)	PUNCT
ejpam-3380	63	9	,	,	PUNCT
ejpam-3380	63	10	s	s	PART
ejpam-3380	63	11	∗	∗	NOUN
ejpam-3380	63	12	t	t	NOUN
ejpam-3380	63	13	)	)	PUNCT
ejpam-3380	63	14	and	and	CCONJ
ejpam-3380	63	15	(	(	PUNCT
ejpam-3380	63	16	p	p	X
ejpam-3380	63	17	/	/	SYM
ejpam-3380	63	18	f(s	f(s	PROPN
ejpam-3380	63	19	,	,	PUNCT
ejpam-3380	63	20	t	t	PROPN
ejpam-3380	63	21	)	)	PUNCT
ejpam-3380	63	22	)	)	PUNCT
ejpam-3380	64	1	∗	∗	NOUN
ejpam-3380	64	2	(	(	PUNCT
ejpam-3380	64	3	s	s	NOUN
ejpam-3380	64	4	∗	∗	X
ejpam-3380	64	5	t	t	NOUN
ejpam-3380	64	6	)	)	PUNCT
ejpam-3380	64	7	=	=	PUNCT
ejpam-3380	65	1	(	(	PUNCT
ejpam-3380	65	2	p	p	NOUN
ejpam-3380	65	3	∗	∗	PRON
ejpam-3380	65	4	s	s	PART
ejpam-3380	65	5	)	)	PUNCT
ejpam-3380	65	6	∗	∗	NOUN
ejpam-3380	65	7	t.	t.	NOUN
ejpam-3380	65	8	proposition	proposition	NOUN
ejpam-3380	65	9	2	2	NUM
ejpam-3380	65	10	.	.	PUNCT
ejpam-3380	66	1	[	[	X
ejpam-3380	66	2	6	6	NUM
ejpam-3380	66	3	]	]	PUNCT
ejpam-3380	66	4	the	the	DET
ejpam-3380	66	5	following	follow	VERB
ejpam-3380	66	6	identities	identity	NOUN
ejpam-3380	66	7	between	between	ADP
ejpam-3380	66	8	(	(	PUNCT
ejpam-3380	66	9	g	g	NOUN
ejpam-3380	66	10	,	,	PUNCT
ejpam-3380	66	11	∗	∗	NOUN
ejpam-3380	66	12	)	)	PUNCT
ejpam-3380	66	13	and	and	CCONJ
ejpam-3380	66	14	f	f	PROPN
ejpam-3380	66	15	are	be	AUX
ejpam-3380	66	16	satisfied	satisfied	ADJ
ejpam-3380	66	17	for	for	ADP
ejpam-3380	66	18	all	all	DET
ejpam-3380	66	19	t	t	NOUN
ejpam-3380	66	20	∈	∈	PROPN
ejpam-3380	66	21	g	g	PROPN
ejpam-3380	66	22	and	and	CCONJ
ejpam-3380	66	23	all	all	DET
ejpam-3380	66	24	v	v	ADP
ejpam-3380	66	25	∈	∈	PROPN
ejpam-3380	66	26	h	h	NOUN
ejpam-3380	66	27	:	:	PUNCT
ejpam-3380	66	28	eg	eg	PROPN
ejpam-3380	66	29	/	/	SYM
ejpam-3380	66	30	v	v	PROPN
ejpam-3380	66	31	=	=	SYM
ejpam-3380	66	32	eg	eg	NOUN
ejpam-3380	66	33	,	,	PUNCT
ejpam-3380	66	34	eg	eg	NOUN
ejpam-3380	66	35	.	.	PUNCT
ejpam-3380	67	1	v	v	X
ejpam-3380	67	2	=	=	NOUN
ejpam-3380	67	3	egve	egve	NOUN
ejpam-3380	67	4	−1	−1	NOUN
ejpam-3380	67	5	g	g	PROPN
ejpam-3380	67	6	,	,	PUNCT
ejpam-3380	67	7	t	t	PROPN
ejpam-3380	67	8	.	.	PUNCT
ejpam-3380	68	1	e	e	X
ejpam-3380	68	2	=	=	SYM
ejpam-3380	68	3	e	e	PROPN
ejpam-3380	68	4	,	,	PUNCT
ejpam-3380	68	5	t	t	PROPN
ejpam-3380	68	6	/	/	SYM
ejpam-3380	68	7	e	e	PROPN
ejpam-3380	68	8	=	=	PROPN
ejpam-3380	68	9	t	t	PROPN
ejpam-3380	68	10	,	,	PUNCT
ejpam-3380	68	11	f(eg	f(eg	NUM
ejpam-3380	68	12	,	,	PUNCT
ejpam-3380	68	13	t	t	NOUN
ejpam-3380	68	14	)	)	PUNCT
ejpam-3380	68	15	=	=	SYM
ejpam-3380	68	16	eg	eg	NOUN
ejpam-3380	68	17	,	,	PUNCT
ejpam-3380	68	18	t	t	PROPN
ejpam-3380	68	19	.	.	PUNCT
ejpam-3380	69	1	e−1	e−1	PROPN
ejpam-3380	69	2	g	g	PROPN
ejpam-3380	69	3	=	=	SYM
ejpam-3380	69	4	f	f	PROPN
ejpam-3380	69	5	(	(	PUNCT
ejpam-3380	69	6	t	t	PROPN
ejpam-3380	69	7	/	/	SYM
ejpam-3380	69	8	e−1	e−1	PROPN
ejpam-3380	69	9	g	g	PROPN
ejpam-3380	69	10	,	,	PUNCT
ejpam-3380	69	11	eg	eg	NOUN
ejpam-3380	69	12	)	)	PUNCT
ejpam-3380	69	13	−1	−1	NOUN
ejpam-3380	70	1	and	and	CCONJ
ejpam-3380	70	2	(	(	PUNCT
ejpam-3380	70	3	t	t	PROPN
ejpam-3380	70	4	/	/	SYM
ejpam-3380	70	5	e−1	e−1	PROPN
ejpam-3380	71	1	g	g	NOUN
ejpam-3380	71	2	)	)	PUNCT
ejpam-3380	71	3	∗	∗	NOUN
ejpam-3380	71	4	eg	eg	NOUN
ejpam-3380	71	5	=	=	PUNCT
ejpam-3380	71	6	t.	t.	NOUN
ejpam-3380	71	7	now	now	ADV
ejpam-3380	71	8	,	,	PUNCT
ejpam-3380	71	9	we	we	PRON
ejpam-3380	71	10	include	include	VERB
ejpam-3380	71	11	the	the	DET
ejpam-3380	71	12	definition	definition	NOUN
ejpam-3380	71	13	of	of	ADP
ejpam-3380	71	14	g	g	NOUN
ejpam-3380	71	15	-	-	PUNCT
ejpam-3380	71	16	weak	weak	ADJ
ejpam-3380	71	17	graded	grade	VERB
ejpam-3380	71	18	rings	ring	NOUN
ejpam-3380	71	19	and	and	CCONJ
ejpam-3380	71	20	some	some	DET
ejpam-3380	71	21	related	relate	VERB
ejpam-3380	71	22	results	result	NOUN
ejpam-3380	71	23	form	form	VERB
ejpam-3380	71	24	[	[	X
ejpam-3380	71	25	3	3	NUM
ejpam-3380	71	26	]	]	PUNCT
ejpam-3380	71	27	where	where	SCONJ
ejpam-3380	71	28	the	the	DET
ejpam-3380	71	29	binary	binary	PROPN
ejpam-3380	71	30	operation	operation	NOUN
ejpam-3380	71	31	∗	∗	NOUN
ejpam-3380	71	32	is	be	AUX
ejpam-3380	71	33	as	as	ADP
ejpam-3380	71	34	in	in	ADP
ejpam-3380	71	35	definition	definition	NOUN
ejpam-3380	71	36	2	2	NUM
ejpam-3380	71	37	.	.	PUNCT
ejpam-3380	71	38	definition	definition	NOUN
ejpam-3380	71	39	3	3	NUM
ejpam-3380	71	40	.	.	PUNCT
ejpam-3380	72	1	[	[	X
ejpam-3380	72	2	3	3	NUM
ejpam-3380	72	3	]	]	PUNCT
ejpam-3380	72	4	,	,	PUNCT
ejpam-3380	72	5	[	[	X
ejpam-3380	72	6	4	4	X
ejpam-3380	72	7	]	]	AUX
ejpam-3380	72	8	let	let	VERB
ejpam-3380	72	9	x	x	PRON
ejpam-3380	72	10	be	be	AUX
ejpam-3380	72	11	a	a	DET
ejpam-3380	72	12	group	group	NOUN
ejpam-3380	72	13	,	,	PUNCT
ejpam-3380	72	14	h	h	PROPN
ejpam-3380	72	15	be	be	VERB
ejpam-3380	72	16	a	a	DET
ejpam-3380	72	17	subgroup	subgroup	NOUN
ejpam-3380	72	18	of	of	ADP
ejpam-3380	72	19	x	x	PUNCT
ejpam-3380	72	20	and	and	CCONJ
ejpam-3380	72	21	(	(	PUNCT
ejpam-3380	72	22	g	g	NOUN
ejpam-3380	72	23	,	,	PUNCT
ejpam-3380	72	24	∗	∗	NOUN
ejpam-3380	72	25	)	)	PUNCT
ejpam-3380	72	26	be	be	VERB
ejpam-3380	72	27	a	a	DET
ejpam-3380	72	28	fixed	fix	VERB
ejpam-3380	72	29	set	set	NOUN
ejpam-3380	72	30	of	of	ADP
ejpam-3380	72	31	left	left	ADJ
ejpam-3380	72	32	coset	coset	NOUN
ejpam-3380	72	33	representatives	representative	NOUN
ejpam-3380	72	34	for	for	ADP
ejpam-3380	72	35	the	the	DET
ejpam-3380	72	36	left	left	ADJ
ejpam-3380	72	37	action	action	NOUN
ejpam-3380	72	38	of	of	ADP
ejpam-3380	72	39	h	h	NOUN
ejpam-3380	72	40	on	on	ADP
ejpam-3380	72	41	x.	x.	NOUN
ejpam-3380	72	42	a	a	DET
ejpam-3380	72	43	ring	ring	NOUN
ejpam-3380	72	44	r	r	NOUN
ejpam-3380	72	45	is	be	AUX
ejpam-3380	72	46	called	call	VERB
ejpam-3380	72	47	a	a	DET
ejpam-3380	72	48	g	g	NOUN
ejpam-3380	72	49	-	-	PUNCT
ejpam-3380	72	50	weak	weak	ADJ
ejpam-3380	72	51	graded	grade	VERB
ejpam-3380	72	52	ring	ring	NOUN
ejpam-3380	72	53	if	if	SCONJ
ejpam-3380	72	54	r	r	NOUN
ejpam-3380	72	55	=	=	SYM
ejpam-3380	72	56	⊕	⊕	PROPN
ejpam-3380	72	57	s∈g	s∈g	VERB
ejpam-3380	72	58	rs	rs	NOUN
ejpam-3380	72	59	(	(	PUNCT
ejpam-3380	72	60	1	1	NUM
ejpam-3380	72	61	)	)	PUNCT
ejpam-3380	72	62	and	and	CCONJ
ejpam-3380	72	63	rsrt	rsrt	VERB
ejpam-3380	72	64	⊆	⊆	NUM
ejpam-3380	72	65	rs∗t	rs∗t	NOUN
ejpam-3380	72	66	for	for	ADP
ejpam-3380	72	67	all	all	DET
ejpam-3380	72	68	s	s	PROPN
ejpam-3380	72	69	,	,	PUNCT
ejpam-3380	72	70	t	t	PROPN
ejpam-3380	72	71	∈	∈	PROPN
ejpam-3380	72	72	g	g	PROPN
ejpam-3380	72	73	,	,	PUNCT
ejpam-3380	72	74	(	(	PUNCT
ejpam-3380	72	75	2	2	X
ejpam-3380	72	76	)	)	PUNCT
ejpam-3380	72	77	where	where	SCONJ
ejpam-3380	72	78	rs	rs	NOUN
ejpam-3380	72	79	is	be	AUX
ejpam-3380	72	80	an	an	DET
ejpam-3380	72	81	additive	additive	ADJ
ejpam-3380	72	82	subgroup	subgroup	NOUN
ejpam-3380	72	83	for	for	ADP
ejpam-3380	72	84	each	each	DET
ejpam-3380	72	85	s	s	PROPN
ejpam-3380	72	86	∈	∈	PROPN
ejpam-3380	72	87	g.	g.	NOUN
ejpam-3380	73	1	if	if	SCONJ
ejpam-3380	73	2	(	(	PUNCT
ejpam-3380	73	3	2	2	X
ejpam-3380	73	4	)	)	PUNCT
ejpam-3380	73	5	is	be	AUX
ejpam-3380	73	6	replaced	replace	VERB
ejpam-3380	73	7	by	by	ADP
ejpam-3380	73	8	rsrt	rsrt	NOUN
ejpam-3380	73	9	=	=	SYM
ejpam-3380	73	10	rs∗t	rs∗t	NOUN
ejpam-3380	73	11	for	for	ADP
ejpam-3380	73	12	all	all	DET
ejpam-3380	73	13	s	s	PROPN
ejpam-3380	73	14	,	,	PUNCT
ejpam-3380	73	15	t	t	PROPN
ejpam-3380	73	16	∈	∈	PROPN
ejpam-3380	73	17	g	g	PROPN
ejpam-3380	73	18	,	,	PUNCT
ejpam-3380	73	19	(	(	PUNCT
ejpam-3380	73	20	3	3	X
ejpam-3380	73	21	)	)	PUNCT
ejpam-3380	73	22	then	then	ADV
ejpam-3380	73	23	,	,	PUNCT
ejpam-3380	73	24	r	r	NOUN
ejpam-3380	73	25	is	be	AUX
ejpam-3380	73	26	called	call	VERB
ejpam-3380	73	27	a	a	DET
ejpam-3380	73	28	fully	fully	ADV
ejpam-3380	73	29	(	(	PUNCT
ejpam-3380	73	30	or	or	CCONJ
ejpam-3380	73	31	strongly	strongly	ADV
ejpam-3380	73	32	)	)	PUNCT
ejpam-3380	73	33	g	g	NOUN
ejpam-3380	73	34	-	-	PUNCT
ejpam-3380	73	35	weak	weak	ADJ
ejpam-3380	73	36	graded	grade	VERB
ejpam-3380	73	37	ring	ring	NOUN
ejpam-3380	73	38	.	.	PUNCT
ejpam-3380	74	1	najla	najla	PROPN
ejpam-3380	74	2	al	al	PROPN
ejpam-3380	74	3	-	-	PUNCT
ejpam-3380	74	4	subaie	subaie	NOUN
ejpam-3380	74	5	,	,	PUNCT
ejpam-3380	74	6	m.	m.	NOUN
ejpam-3380	74	7	m.	m.	PROPN
ejpam-3380	74	8	al	al	PROPN
ejpam-3380	74	9	-	-	PUNCT
ejpam-3380	74	10	shomrani	shomrani	PROPN
ejpam-3380	74	11	/	/	SYM
ejpam-3380	74	12	eur	eur	NOUN
ejpam-3380	74	13	.	.	PUNCT
ejpam-3380	75	1	j.	j.	PROPN
ejpam-3380	75	2	pure	pure	PROPN
ejpam-3380	75	3	appl	appl	PROPN
ejpam-3380	75	4	.	.	PROPN
ejpam-3380	75	5	math	math	PROPN
ejpam-3380	75	6	,	,	PUNCT
ejpam-3380	75	7	12	12	NUM
ejpam-3380	75	8	(	(	PUNCT
ejpam-3380	75	9	2	2	NUM
ejpam-3380	75	10	)	)	PUNCT
ejpam-3380	75	11	(	(	PUNCT
ejpam-3380	75	12	2019	2019	NUM
ejpam-3380	75	13	)	)	PUNCT
ejpam-3380	75	14	,	,	PUNCT
ejpam-3380	75	15	332	332	NUM
ejpam-3380	75	16	-	-	SYM
ejpam-3380	75	17	347	347	NUM
ejpam-3380	75	18	335	335	NUM
ejpam-3380	75	19	it	it	PRON
ejpam-3380	75	20	can	can	AUX
ejpam-3380	75	21	be	be	AUX
ejpam-3380	75	22	noted	note	VERB
ejpam-3380	75	23	that	that	SCONJ
ejpam-3380	75	24	any	any	DET
ejpam-3380	75	25	ring	ring	NOUN
ejpam-3380	75	26	r	r	NOUN
ejpam-3380	75	27	can	can	AUX
ejpam-3380	75	28	be	be	AUX
ejpam-3380	75	29	put	put	VERB
ejpam-3380	75	30	into	into	ADP
ejpam-3380	75	31	a	a	DET
ejpam-3380	75	32	g	g	NOUN
ejpam-3380	75	33	-	-	PUNCT
ejpam-3380	75	34	weak	weak	ADJ
ejpam-3380	75	35	graded	grade	VERB
ejpam-3380	75	36	ring	ring	NOUN
ejpam-3380	75	37	by	by	ADP
ejpam-3380	75	38	placing	place	VERB
ejpam-3380	75	39	r	r	NOUN
ejpam-3380	75	40	=	=	SYM
ejpam-3380	75	41	reg	reg	NOUN
ejpam-3380	75	42	and	and	CCONJ
ejpam-3380	75	43	rs	rs	NOUN
ejpam-3380	75	44	=	=	SYM
ejpam-3380	75	45	0	0	NUM
ejpam-3380	76	1	for	for	ADP
ejpam-3380	76	2	all	all	DET
ejpam-3380	76	3	s	s	PART
ejpam-3380	76	4	∈	∈	NOUN
ejpam-3380	76	5	g	g	NOUN
ejpam-3380	76	6	with	with	ADP
ejpam-3380	76	7	s	s	PROPN
ejpam-3380	76	8	6=	6=	NUM
ejpam-3380	76	9	eg	eg	NOUN
ejpam-3380	76	10	.	.	PUNCT
ejpam-3380	77	1	this	this	PRON
ejpam-3380	77	2	is	be	AUX
ejpam-3380	77	3	called	call	VERB
ejpam-3380	77	4	the	the	DET
ejpam-3380	77	5	trivial	trivial	ADJ
ejpam-3380	77	6	g	g	NOUN
ejpam-3380	77	7	-	-	PUNCT
ejpam-3380	77	8	weak	weak	ADJ
ejpam-3380	77	9	graded	grade	VERB
ejpam-3380	77	10	ring	ring	NOUN
ejpam-3380	77	11	.	.	PUNCT
ejpam-3380	78	1	proposition	proposition	NOUN
ejpam-3380	78	2	3	3	NUM
ejpam-3380	78	3	.	.	PUNCT
ejpam-3380	79	1	[	[	X
ejpam-3380	79	2	3	3	X
ejpam-3380	79	3	]	]	PUNCT
ejpam-3380	79	4	let	let	VERB
ejpam-3380	79	5	g	g	PRON
ejpam-3380	79	6	be	be	AUX
ejpam-3380	79	7	a	a	DET
ejpam-3380	79	8	fixed	fix	VERB
ejpam-3380	79	9	set	set	NOUN
ejpam-3380	79	10	of	of	ADP
ejpam-3380	79	11	left	left	ADJ
ejpam-3380	79	12	coset	coset	NOUN
ejpam-3380	79	13	representatives	representative	NOUN
ejpam-3380	79	14	for	for	ADP
ejpam-3380	79	15	a	a	DET
ejpam-3380	79	16	subgroup	subgroup	NOUN
ejpam-3380	79	17	h	h	NOUN
ejpam-3380	79	18	of	of	ADP
ejpam-3380	79	19	a	a	DET
ejpam-3380	79	20	group	group	NOUN
ejpam-3380	79	21	x	x	NOUN
ejpam-3380	79	22	and	and	CCONJ
ejpam-3380	79	23	r	r	NOUN
ejpam-3380	79	24	be	be	VERB
ejpam-3380	79	25	a	a	DET
ejpam-3380	79	26	g	g	NOUN
ejpam-3380	79	27	-	-	PUNCT
ejpam-3380	79	28	weak	weak	ADJ
ejpam-3380	79	29	graded	grade	VERB
ejpam-3380	79	30	ring	ring	NOUN
ejpam-3380	79	31	with	with	ADP
ejpam-3380	79	32	identity	identity	NOUN
ejpam-3380	79	33	.	.	PUNCT
ejpam-3380	80	1	then	then	ADV
ejpam-3380	80	2	,	,	PUNCT
ejpam-3380	80	3	1r	1r	NUM
ejpam-3380	80	4	∈	∈	NOUN
ejpam-3380	80	5	reg	reg	NOUN
ejpam-3380	80	6	,	,	PUNCT
ejpam-3380	80	7	where	where	SCONJ
ejpam-3380	80	8	1r	1r	NOUN
ejpam-3380	80	9	is	be	AUX
ejpam-3380	80	10	the	the	DET
ejpam-3380	80	11	multiplicative	multiplicative	ADJ
ejpam-3380	80	12	identity	identity	NOUN
ejpam-3380	80	13	of	of	ADP
ejpam-3380	80	14	r.	r.	PROPN
ejpam-3380	80	15	proposition	proposition	PROPN
ejpam-3380	80	16	4	4	NUM
ejpam-3380	80	17	.	.	PUNCT
ejpam-3380	81	1	[	[	X
ejpam-3380	81	2	3	3	X
ejpam-3380	81	3	]	]	PUNCT
ejpam-3380	81	4	let	let	VERB
ejpam-3380	81	5	r	r	PRON
ejpam-3380	81	6	be	be	AUX
ejpam-3380	81	7	a	a	DET
ejpam-3380	81	8	g	g	NOUN
ejpam-3380	81	9	-	-	PUNCT
ejpam-3380	81	10	weak	weak	ADJ
ejpam-3380	81	11	graded	grade	VERB
ejpam-3380	81	12	ring	ring	NOUN
ejpam-3380	81	13	.	.	PUNCT
ejpam-3380	82	1	then	then	ADV
ejpam-3380	82	2	the	the	DET
ejpam-3380	82	3	eg	eg	NOUN
ejpam-3380	82	4	-	-	PUNCT
ejpam-3380	82	5	component	component	NOUN
ejpam-3380	82	6	reg	reg	NOUN
ejpam-3380	82	7	is	be	AUX
ejpam-3380	82	8	a	a	DET
ejpam-3380	82	9	subring	subring	NOUN
ejpam-3380	82	10	of	of	ADP
ejpam-3380	82	11	r.	r.	PROPN
ejpam-3380	82	12	3	3	NUM
ejpam-3380	82	13	.	.	PUNCT
ejpam-3380	83	1	g	g	NOUN
ejpam-3380	83	2	-	-	PUNCT
ejpam-3380	83	3	weak	weak	ADJ
ejpam-3380	83	4	graded	grade	VERB
ejpam-3380	83	5	subrings	subring	NOUN
ejpam-3380	83	6	and	and	CCONJ
ejpam-3380	83	7	related	related	ADJ
ejpam-3380	83	8	results	result	NOUN
ejpam-3380	83	9	in	in	ADP
ejpam-3380	83	10	[	[	X
ejpam-3380	83	11	3	3	NUM
ejpam-3380	83	12	]	]	PUNCT
ejpam-3380	83	13	and	and	CCONJ
ejpam-3380	83	14	[	[	X
ejpam-3380	83	15	4	4	NUM
ejpam-3380	83	16	]	]	PUNCT
ejpam-3380	83	17	,	,	PUNCT
ejpam-3380	83	18	the	the	DET
ejpam-3380	83	19	definition	definition	NOUN
ejpam-3380	83	20	of	of	ADP
ejpam-3380	83	21	weak	weak	ADJ
ejpam-3380	83	22	graded	grade	VERB
ejpam-3380	83	23	rings	ring	NOUN
ejpam-3380	83	24	was	be	AUX
ejpam-3380	83	25	given	give	VERB
ejpam-3380	83	26	and	and	CCONJ
ejpam-3380	83	27	some	some	PRON
ejpam-3380	83	28	of	of	ADP
ejpam-3380	83	29	their	their	PRON
ejpam-3380	83	30	properties	property	NOUN
ejpam-3380	83	31	were	be	AUX
ejpam-3380	83	32	derived	derive	VERB
ejpam-3380	83	33	.	.	PUNCT
ejpam-3380	84	1	in	in	ADP
ejpam-3380	84	2	this	this	DET
ejpam-3380	84	3	section	section	NOUN
ejpam-3380	84	4	,	,	PUNCT
ejpam-3380	84	5	we	we	PRON
ejpam-3380	84	6	continue	continue	VERB
ejpam-3380	84	7	the	the	DET
ejpam-3380	84	8	investigation	investigation	NOUN
ejpam-3380	84	9	of	of	ADP
ejpam-3380	84	10	the	the	DET
ejpam-3380	84	11	properties	property	NOUN
ejpam-3380	84	12	of	of	ADP
ejpam-3380	84	13	weak	weak	ADJ
ejpam-3380	84	14	graded	grade	VERB
ejpam-3380	84	15	rings	ring	NOUN
ejpam-3380	84	16	.	.	PUNCT
ejpam-3380	85	1	we	we	PRON
ejpam-3380	85	2	start	start	VERB
ejpam-3380	85	3	by	by	ADP
ejpam-3380	85	4	giving	give	VERB
ejpam-3380	85	5	a	a	DET
ejpam-3380	85	6	definition	definition	NOUN
ejpam-3380	85	7	for	for	ADP
ejpam-3380	85	8	a	a	DET
ejpam-3380	85	9	g	g	NOUN
ejpam-3380	85	10	-	-	PUNCT
ejpam-3380	85	11	homogeneous	homogeneous	ADJ
ejpam-3380	85	12	element	element	NOUN
ejpam-3380	85	13	of	of	ADP
ejpam-3380	85	14	a	a	DET
ejpam-3380	85	15	g	g	NOUN
ejpam-3380	85	16	-	-	PUNCT
ejpam-3380	85	17	weak	weak	ADJ
ejpam-3380	85	18	graded	grade	VERB
ejpam-3380	85	19	ring	ring	NOUN
ejpam-3380	85	20	r.	r.	PROPN
ejpam-3380	85	21	definition	definition	NOUN
ejpam-3380	85	22	4	4	NUM
ejpam-3380	85	23	.	.	PUNCT
ejpam-3380	86	1	let	let	VERB
ejpam-3380	86	2	r	r	PRON
ejpam-3380	86	3	be	be	AUX
ejpam-3380	86	4	a	a	DET
ejpam-3380	86	5	g	g	NOUN
ejpam-3380	86	6	-	-	PUNCT
ejpam-3380	86	7	weak	weak	ADJ
ejpam-3380	86	8	graded	grade	VERB
ejpam-3380	86	9	ring	ring	NOUN
ejpam-3380	86	10	.	.	PUNCT
ejpam-3380	87	1	then	then	ADV
ejpam-3380	87	2	,	,	PUNCT
ejpam-3380	87	3	a	a	DET
ejpam-3380	87	4	non	non	ADJ
ejpam-3380	87	5	-	-	ADJ
ejpam-3380	87	6	zero	zero	NUM
ejpam-3380	87	7	element	element	NOUN
ejpam-3380	87	8	rs	rs	NOUN
ejpam-3380	87	9	∈	∈	PROPN
ejpam-3380	87	10	r	r	NOUN
ejpam-3380	87	11	is	be	AUX
ejpam-3380	87	12	said	say	VERB
ejpam-3380	87	13	to	to	PART
ejpam-3380	87	14	be	be	AUX
ejpam-3380	87	15	a	a	DET
ejpam-3380	87	16	weak	weak	ADJ
ejpam-3380	87	17	graded	grade	VERB
ejpam-3380	87	18	or	or	CCONJ
ejpam-3380	87	19	g	g	NOUN
ejpam-3380	87	20	-	-	PUNCT
ejpam-3380	87	21	homogeneous	homogeneous	ADJ
ejpam-3380	87	22	element	element	NOUN
ejpam-3380	87	23	of	of	ADP
ejpam-3380	87	24	grade	grade	NOUN
ejpam-3380	87	25	s	s	NOUN
ejpam-3380	87	26	if	if	SCONJ
ejpam-3380	87	27	there	there	PRON
ejpam-3380	87	28	exists	exist	VERB
ejpam-3380	87	29	an	an	DET
ejpam-3380	87	30	s	s	NOUN
ejpam-3380	87	31	-	-	PUNCT
ejpam-3380	87	32	component	component	NOUN
ejpam-3380	87	33	rs	rs	NOUN
ejpam-3380	87	34	of	of	ADP
ejpam-3380	87	35	r	r	NOUN
ejpam-3380	87	36	such	such	ADJ
ejpam-3380	87	37	that	that	DET
ejpam-3380	87	38	rs	rs	NOUN
ejpam-3380	87	39	∈	∈	PROPN
ejpam-3380	87	40	rs	rs	NOUN
ejpam-3380	87	41	.	.	PUNCT
ejpam-3380	88	1	the	the	DET
ejpam-3380	88	2	grade	grade	NOUN
ejpam-3380	88	3	of	of	ADP
ejpam-3380	88	4	rs	r	NOUN
ejpam-3380	88	5	is	be	AUX
ejpam-3380	88	6	denoted	denote	VERB
ejpam-3380	88	7	by	by	ADP
ejpam-3380	88	8	〈	〈	PROPN
ejpam-3380	88	9	rs	rs	NOUN
ejpam-3380	88	10	〉	〉	NOUN
ejpam-3380	88	11	=	=	SYM
ejpam-3380	88	12	s.	s.	PROPN
ejpam-3380	88	13	the	the	DET
ejpam-3380	88	14	set	set	NOUN
ejpam-3380	88	15	of	of	ADP
ejpam-3380	88	16	all	all	DET
ejpam-3380	88	17	g	g	NOUN
ejpam-3380	88	18	-	-	PUNCT
ejpam-3380	88	19	homogeneous	homogeneous	ADJ
ejpam-3380	88	20	elements	element	NOUN
ejpam-3380	88	21	of	of	ADP
ejpam-3380	88	22	r	r	NOUN
ejpam-3380	88	23	is	be	AUX
ejpam-3380	88	24	defined	define	VERB
ejpam-3380	88	25	and	and	CCONJ
ejpam-3380	88	26	written	write	VERB
ejpam-3380	88	27	as	as	ADP
ejpam-3380	88	28	h(r	h(r	NOUN
ejpam-3380	88	29	)	)	PUNCT
ejpam-3380	88	30	=	=	PUNCT
ejpam-3380	88	31	∪s∈grs	∪s∈grs	PROPN
ejpam-3380	88	32	.	.	PUNCT
ejpam-3380	89	1	remark	remark	PROPN
ejpam-3380	89	2	1	1	NUM
ejpam-3380	89	3	.	.	PUNCT
ejpam-3380	90	1	by	by	ADP
ejpam-3380	90	2	definition	definition	NOUN
ejpam-3380	90	3	3	3	NUM
ejpam-3380	90	4	,	,	PUNCT
ejpam-3380	90	5	every	every	DET
ejpam-3380	90	6	element	element	NOUN
ejpam-3380	90	7	r	r	NOUN
ejpam-3380	90	8	∈	∈	NOUN
ejpam-3380	90	9	r	r	NOUN
ejpam-3380	90	10	has	have	VERB
ejpam-3380	90	11	a	a	DET
ejpam-3380	90	12	unique	unique	ADJ
ejpam-3380	90	13	decomposition	decomposition	NOUN
ejpam-3380	90	14	written	write	VERB
ejpam-3380	90	15	as	as	ADP
ejpam-3380	90	16	r	r	NOUN
ejpam-3380	90	17	=	=	SYM
ejpam-3380	90	18	∑	∑	PROPN
ejpam-3380	90	19	s∈g	s∈g	VERB
ejpam-3380	90	20	rs	rs	NOUN
ejpam-3380	90	21	with	with	ADP
ejpam-3380	90	22	rs	rs	X
ejpam-3380	90	23	∈	∈	NOUN
ejpam-3380	90	24	rs	rs	NOUN
ejpam-3380	90	25	for	for	ADP
ejpam-3380	90	26	all	all	PRON
ejpam-3380	90	27	s	s	PART
ejpam-3380	90	28	∈	∈	NOUN
ejpam-3380	90	29	g.	g.	NOUN
ejpam-3380	91	1	these	these	PRON
ejpam-3380	91	2	{	{	PUNCT
ejpam-3380	91	3	rs}s∈g	rs}s∈g	PROPN
ejpam-3380	91	4	are	be	AUX
ejpam-3380	91	5	called	call	VERB
ejpam-3380	91	6	the	the	DET
ejpam-3380	91	7	g	g	NOUN
ejpam-3380	91	8	-	-	PUNCT
ejpam-3380	91	9	homogeneous	homogeneous	ADJ
ejpam-3380	91	10	components	component	NOUN
ejpam-3380	91	11	of	of	ADP
ejpam-3380	91	12	r.	r.	PROPN
ejpam-3380	91	13	however	however	ADV
ejpam-3380	91	14	,	,	PUNCT
ejpam-3380	91	15	the	the	DET
ejpam-3380	91	16	sum	sum	NOUN
ejpam-3380	91	17	∑	∑	ADP
ejpam-3380	91	18	s∈g	s∈g	VERB
ejpam-3380	91	19	rs	rs	NOUN
ejpam-3380	91	20	is	be	AUX
ejpam-3380	91	21	finite	finite	ADJ
ejpam-3380	91	22	(	(	PUNCT
ejpam-3380	91	23	in	in	ADP
ejpam-3380	91	24	other	other	ADJ
ejpam-3380	91	25	words	word	NOUN
ejpam-3380	91	26	almost	almost	ADV
ejpam-3380	91	27	all	all	PRON
ejpam-3380	91	28	rs	r	NOUN
ejpam-3380	91	29	are	be	AUX
ejpam-3380	91	30	zero	zero	NUM
ejpam-3380	91	31	)	)	PUNCT
ejpam-3380	91	32	.	.	PUNCT
ejpam-3380	92	1	definition	definition	NOUN
ejpam-3380	92	2	5	5	NUM
ejpam-3380	92	3	.	.	PUNCT
ejpam-3380	93	1	let	let	VERB
ejpam-3380	93	2	k	k	PRON
ejpam-3380	93	3	be	be	AUX
ejpam-3380	93	4	a	a	DET
ejpam-3380	93	5	subring	subring	NOUN
ejpam-3380	93	6	of	of	ADP
ejpam-3380	93	7	a	a	DET
ejpam-3380	93	8	g	g	NOUN
ejpam-3380	93	9	-	-	PUNCT
ejpam-3380	93	10	weak	weak	ADJ
ejpam-3380	93	11	graded	grade	VERB
ejpam-3380	93	12	ring	ring	NOUN
ejpam-3380	93	13	r.	r.	PROPN
ejpam-3380	93	14	then	then	ADV
ejpam-3380	93	15	,	,	PUNCT
ejpam-3380	93	16	k	k	PROPN
ejpam-3380	93	17	is	be	AUX
ejpam-3380	93	18	said	say	VERB
ejpam-3380	93	19	to	to	PART
ejpam-3380	93	20	be	be	AUX
ejpam-3380	93	21	a	a	DET
ejpam-3380	93	22	g	g	NOUN
ejpam-3380	93	23	-	-	PUNCT
ejpam-3380	93	24	weak	weak	ADJ
ejpam-3380	93	25	graded	grade	VERB
ejpam-3380	93	26	subring	subring	NOUN
ejpam-3380	93	27	of	of	ADP
ejpam-3380	93	28	r	r	NOUN
ejpam-3380	93	29	if	if	SCONJ
ejpam-3380	93	30	k	k	PROPN
ejpam-3380	93	31	itself	itself	PRON
ejpam-3380	93	32	is	be	AUX
ejpam-3380	93	33	a	a	DET
ejpam-3380	93	34	g	g	NOUN
ejpam-3380	93	35	-	-	PUNCT
ejpam-3380	93	36	weak	weak	ADJ
ejpam-3380	93	37	graded	grade	VERB
ejpam-3380	93	38	ring	ring	NOUN
ejpam-3380	93	39	.	.	PUNCT
ejpam-3380	94	1	theorem	theorem	NOUN
ejpam-3380	94	2	1	1	NUM
ejpam-3380	94	3	.	.	PUNCT
ejpam-3380	95	1	let	let	VERB
ejpam-3380	95	2	k	k	PRON
ejpam-3380	95	3	be	be	AUX
ejpam-3380	95	4	a	a	DET
ejpam-3380	95	5	subring	subring	NOUN
ejpam-3380	95	6	of	of	ADP
ejpam-3380	95	7	a	a	DET
ejpam-3380	95	8	g	g	NOUN
ejpam-3380	95	9	-	-	PUNCT
ejpam-3380	95	10	weak	weak	ADJ
ejpam-3380	95	11	graded	grade	VERB
ejpam-3380	95	12	ring	ring	NOUN
ejpam-3380	95	13	r.	r.	PROPN
ejpam-3380	95	14	if	if	SCONJ
ejpam-3380	95	15	k	k	PROPN
ejpam-3380	95	16	contains	contain	VERB
ejpam-3380	95	17	all	all	DET
ejpam-3380	95	18	the	the	DET
ejpam-3380	95	19	ghomogeneous	ghomogeneous	ADJ
ejpam-3380	95	20	components	component	NOUN
ejpam-3380	95	21	for	for	ADP
ejpam-3380	95	22	each	each	DET
ejpam-3380	95	23	k	k	PROPN
ejpam-3380	95	24	∈	∈	PROPN
ejpam-3380	95	25	k	k	PROPN
ejpam-3380	95	26	,	,	PUNCT
ejpam-3380	95	27	then	then	ADV
ejpam-3380	95	28	k	k	PROPN
ejpam-3380	95	29	is	be	AUX
ejpam-3380	95	30	a	a	DET
ejpam-3380	95	31	g	g	NOUN
ejpam-3380	95	32	-	-	PUNCT
ejpam-3380	95	33	weak	weak	ADJ
ejpam-3380	95	34	graded	grade	VERB
ejpam-3380	95	35	subring	subring	NOUN
ejpam-3380	95	36	of	of	ADP
ejpam-3380	95	37	r.	r.	PROPN
ejpam-3380	95	38	proof	proof	NOUN
ejpam-3380	95	39	.	.	PUNCT
ejpam-3380	96	1	clearly	clearly	ADV
ejpam-3380	96	2	,	,	PUNCT
ejpam-3380	96	3	we	we	PRON
ejpam-3380	96	4	can	can	AUX
ejpam-3380	96	5	write	write	VERB
ejpam-3380	96	6	k	k	PROPN
ejpam-3380	96	7	as	as	ADP
ejpam-3380	96	8	k	k	PROPN
ejpam-3380	96	9	=	=	PUNCT
ejpam-3380	96	10	∑	∑	PROPN
ejpam-3380	96	11	s∈gks	s∈gks	PROPN
ejpam-3380	96	12	as	as	ADP
ejpam-3380	96	13	additive	additive	ADJ
ejpam-3380	96	14	subgroups	subgroup	NOUN
ejpam-3380	96	15	.	.	PUNCT
ejpam-3380	97	1	then	then	ADV
ejpam-3380	97	2	for	for	ADP
ejpam-3380	97	3	every	every	DET
ejpam-3380	97	4	k	k	PROPN
ejpam-3380	97	5	∈	∈	PROPN
ejpam-3380	97	6	k	k	NOUN
ejpam-3380	97	7	,	,	PUNCT
ejpam-3380	97	8	we	we	PRON
ejpam-3380	97	9	have	have	VERB
ejpam-3380	97	10	k	k	NOUN
ejpam-3380	97	11	=	=	PUNCT
ejpam-3380	97	12	∑	∑	PROPN
ejpam-3380	97	13	s∈g	s∈g	PROPN
ejpam-3380	97	14	ks	ks	NOUN
ejpam-3380	97	15	where	where	SCONJ
ejpam-3380	97	16	ks	ks	PROPN
ejpam-3380	97	17	∈	∈	PROPN
ejpam-3380	97	18	ks	ks	PROPN
ejpam-3380	97	19	for	for	ADP
ejpam-3380	97	20	all	all	DET
ejpam-3380	97	21	s	s	PROPN
ejpam-3380	97	22	∈	∈	NOUN
ejpam-3380	97	23	g.	g.	NOUN
ejpam-3380	97	24	we	we	PRON
ejpam-3380	97	25	need	need	VERB
ejpam-3380	97	26	to	to	PART
ejpam-3380	97	27	prove	prove	VERB
ejpam-3380	97	28	that	that	SCONJ
ejpam-3380	97	29	k	k	PROPN
ejpam-3380	97	30	=	=	PROPN
ejpam-3380	97	31	⊕	⊕	PROPN
ejpam-3380	97	32	s∈gks	s∈gks	PROPN
ejpam-3380	97	33	and	and	CCONJ
ejpam-3380	97	34	kskt	kskt	PROPN
ejpam-3380	97	35	⊆	⊆	NUM
ejpam-3380	97	36	ks∗t	ks∗t	NOUN
ejpam-3380	97	37	which	which	PRON
ejpam-3380	97	38	we	we	PRON
ejpam-3380	97	39	do	do	VERB
ejpam-3380	97	40	as	as	SCONJ
ejpam-3380	97	41	follows	follow	VERB
ejpam-3380	97	42	:	:	PUNCT
ejpam-3380	97	43	(	(	PUNCT
ejpam-3380	97	44	i	i	NOUN
ejpam-3380	97	45	)	)	PUNCT
ejpam-3380	98	1	k	k	PROPN
ejpam-3380	98	2	=	=	PROPN
ejpam-3380	98	3	⊕	⊕	PROPN
ejpam-3380	98	4	s∈gks	s∈gks	PROPN
ejpam-3380	98	5	.	.	PUNCT
ejpam-3380	99	1	since	since	SCONJ
ejpam-3380	99	2	k	k	PROPN
ejpam-3380	99	3	is	be	AUX
ejpam-3380	99	4	a	a	DET
ejpam-3380	99	5	subring	subring	NOUN
ejpam-3380	99	6	of	of	ADP
ejpam-3380	99	7	r	r	NOUN
ejpam-3380	99	8	and	and	CCONJ
ejpam-3380	99	9	r	r	NOUN
ejpam-3380	99	10	=	=	PROPN
ejpam-3380	99	11	⊕	⊕	PROPN
ejpam-3380	99	12	s∈grs	s∈grs	PROPN
ejpam-3380	99	13	,	,	PUNCT
ejpam-3380	99	14	we	we	PRON
ejpam-3380	99	15	have	have	VERB
ejpam-3380	99	16	ks	ks	NOUN
ejpam-3380	99	17	=	=	SYM
ejpam-3380	99	18	k	k	PROPN
ejpam-3380	99	19	∩	∩	NOUN
ejpam-3380	99	20	rs	rs	VERB
ejpam-3380	99	21	for	for	ADP
ejpam-3380	99	22	all	all	DET
ejpam-3380	99	23	s	s	PROPN
ejpam-3380	99	24	∈	∈	PROPN
ejpam-3380	99	25	g.	g.	NOUN
ejpam-3380	99	26	also	also	ADV
ejpam-3380	99	27	,	,	PUNCT
ejpam-3380	99	28	as	as	SCONJ
ejpam-3380	99	29	r	r	NOUN
ejpam-3380	99	30	is	be	AUX
ejpam-3380	99	31	a	a	DET
ejpam-3380	99	32	g	g	NOUN
ejpam-3380	99	33	-	-	PUNCT
ejpam-3380	99	34	weak	weak	ADJ
ejpam-3380	99	35	graded	grade	VERB
ejpam-3380	99	36	ring	ring	NOUN
ejpam-3380	99	37	,	,	PUNCT
ejpam-3380	99	38	the	the	DET
ejpam-3380	99	39	direct	direct	ADJ
ejpam-3380	99	40	sum	sum	NOUN
ejpam-3380	99	41	condition	condition	NOUN
ejpam-3380	99	42	implies	imply	VERB
ejpam-3380	99	43	that	that	SCONJ
ejpam-3380	99	44	rs	rs	INTJ
ejpam-3380	99	45	⋂	⋂	PROPN
ejpam-3380	99	46	(	(	PUNCT
ejpam-3380	99	47	∑	∑	PUNCT
ejpam-3380	99	48	t∈grt	t∈grt	PROPN
ejpam-3380	99	49	)	)	PUNCT
ejpam-3380	99	50	=	=	PRON
ejpam-3380	99	51	{	{	PUNCT
ejpam-3380	99	52	0	0	NUM
ejpam-3380	99	53	}	}	PUNCT
ejpam-3380	99	54	for	for	ADP
ejpam-3380	99	55	all	all	DET
ejpam-3380	99	56	s	s	PART
ejpam-3380	99	57	∈	∈	NOUN
ejpam-3380	99	58	g	g	NOUN
ejpam-3380	99	59	with	with	ADP
ejpam-3380	99	60	s	s	PROPN
ejpam-3380	99	61	6=	6=	PROPN
ejpam-3380	99	62	t	t	PROPN
ejpam-3380	99	63	,	,	PUNCT
ejpam-3380	99	64	which	which	PRON
ejpam-3380	99	65	consequently	consequently	ADV
ejpam-3380	99	66	implies	imply	VERB
ejpam-3380	99	67	that	that	SCONJ
ejpam-3380	99	68	ks	ks	PROPN
ejpam-3380	99	69	⋂	⋂	PROPN
ejpam-3380	99	70	(	(	PUNCT
ejpam-3380	99	71	∑	∑	PROPN
ejpam-3380	99	72	t∈gkt	t∈gkt	PROPN
ejpam-3380	99	73	)	)	PUNCT
ejpam-3380	99	74	=	=	PRON
ejpam-3380	99	75	{	{	PUNCT
ejpam-3380	99	76	0	0	NUM
ejpam-3380	99	77	}	}	PUNCT
ejpam-3380	99	78	.	.	PUNCT
ejpam-3380	100	1	thus	thus	ADV
ejpam-3380	100	2	,	,	PUNCT
ejpam-3380	100	3	k	k	PROPN
ejpam-3380	100	4	=	=	PROPN
ejpam-3380	100	5	⊕	⊕	PROPN
ejpam-3380	100	6	s∈gks	s∈gks	PROPN
ejpam-3380	100	7	.	.	PUNCT
ejpam-3380	100	8	(	(	PUNCT
ejpam-3380	100	9	ii	ii	PROPN
ejpam-3380	100	10	)	)	PUNCT
ejpam-3380	100	11	kskt	kskt	PROPN
ejpam-3380	100	12	⊆	⊆	NUM
ejpam-3380	100	13	ks∗t	ks∗t	PUNCT
ejpam-3380	100	14	.	.	PUNCT
ejpam-3380	101	1	let	let	VERB
ejpam-3380	101	2	ks	ks	NOUN
ejpam-3380	101	3	=	=	PUNCT
ejpam-3380	101	4	k	k	PROPN
ejpam-3380	101	5	∩rs	∩rs	NOUN
ejpam-3380	101	6	and	and	CCONJ
ejpam-3380	101	7	kt	kt	PROPN
ejpam-3380	101	8	=	=	SYM
ejpam-3380	101	9	k	k	PROPN
ejpam-3380	101	10	∩rt	∩rt	ADV
ejpam-3380	101	11	for	for	ADP
ejpam-3380	101	12	some	some	DET
ejpam-3380	101	13	s	s	NOUN
ejpam-3380	101	14	and	and	CCONJ
ejpam-3380	101	15	t	t	PROPN
ejpam-3380	101	16	in	in	ADP
ejpam-3380	101	17	g.	g.	PROPN
ejpam-3380	101	18	hence	hence	ADV
ejpam-3380	101	19	,	,	PUNCT
ejpam-3380	101	20	kskt	kskt	PROPN
ejpam-3380	101	21	=	=	SYM
ejpam-3380	101	22	(	(	PUNCT
ejpam-3380	101	23	k	k	PROPN
ejpam-3380	101	24	∩rs)(k	∩rs)(k	NOUN
ejpam-3380	101	25	∩rt	∩rt	NOUN
ejpam-3380	101	26	)	)	PUNCT
ejpam-3380	101	27	⊆	⊆	NUM
ejpam-3380	101	28	k	k	PROPN
ejpam-3380	101	29	∩	∩	NOUN
ejpam-3380	101	30	(	(	PUNCT
ejpam-3380	101	31	rsrt	rsrt	NOUN
ejpam-3380	101	32	)	)	PUNCT
ejpam-3380	101	33	⊆	⊆	NUM
ejpam-3380	101	34	k	k	PROPN
ejpam-3380	101	35	∩	∩	X
ejpam-3380	101	36	(	(	PUNCT
ejpam-3380	101	37	rs∗t	rs∗t	NOUN
ejpam-3380	101	38	)	)	PUNCT
ejpam-3380	101	39	=	=	SYM
ejpam-3380	101	40	ks∗t	ks∗t	PROPN
ejpam-3380	101	41	,	,	PUNCT
ejpam-3380	101	42	najla	najla	ADJ
ejpam-3380	101	43	al	al	PROPN
ejpam-3380	101	44	-	-	PUNCT
ejpam-3380	101	45	subaie	subaie	NOUN
ejpam-3380	101	46	,	,	PUNCT
ejpam-3380	101	47	m.	m.	NOUN
ejpam-3380	101	48	m.	m.	PROPN
ejpam-3380	101	49	al	al	PROPN
ejpam-3380	101	50	-	-	PUNCT
ejpam-3380	101	51	shomrani	shomrani	PROPN
ejpam-3380	101	52	/	/	SYM
ejpam-3380	101	53	eur	eur	NOUN
ejpam-3380	101	54	.	.	PUNCT
ejpam-3380	102	1	j.	j.	PROPN
ejpam-3380	102	2	pure	pure	PROPN
ejpam-3380	102	3	appl	appl	PROPN
ejpam-3380	102	4	.	.	PROPN
ejpam-3380	102	5	math	math	PROPN
ejpam-3380	102	6	,	,	PUNCT
ejpam-3380	102	7	12	12	NUM
ejpam-3380	102	8	(	(	PUNCT
ejpam-3380	102	9	2	2	NUM
ejpam-3380	102	10	)	)	PUNCT
ejpam-3380	102	11	(	(	PUNCT
ejpam-3380	102	12	2019	2019	NUM
ejpam-3380	102	13	)	)	PUNCT
ejpam-3380	102	14	,	,	PUNCT
ejpam-3380	102	15	332	332	NUM
ejpam-3380	102	16	-	-	SYM
ejpam-3380	102	17	347	347	NUM
ejpam-3380	102	18	336	336	NUM
ejpam-3380	102	19	which	which	PRON
ejpam-3380	102	20	completes	complete	VERB
ejpam-3380	102	21	the	the	DET
ejpam-3380	102	22	proof	proof	NOUN
ejpam-3380	102	23	.	.	PUNCT
ejpam-3380	103	1	it	it	PRON
ejpam-3380	103	2	can	can	AUX
ejpam-3380	103	3	be	be	AUX
ejpam-3380	103	4	noted	note	VERB
ejpam-3380	103	5	that	that	SCONJ
ejpam-3380	103	6	the	the	DET
ejpam-3380	103	7	last	last	ADJ
ejpam-3380	103	8	two	two	NUM
ejpam-3380	103	9	lines	line	NOUN
ejpam-3380	103	10	in	in	ADP
ejpam-3380	103	11	part	part	NOUN
ejpam-3380	103	12	(	(	PUNCT
ejpam-3380	103	13	i	i	NOUN
ejpam-3380	103	14	)	)	PUNCT
ejpam-3380	103	15	of	of	ADP
ejpam-3380	103	16	the	the	DET
ejpam-3380	103	17	proof	proof	NOUN
ejpam-3380	103	18	can	can	AUX
ejpam-3380	103	19	be	be	AUX
ejpam-3380	103	20	,	,	PUNCT
ejpam-3380	103	21	equivalently	equivalently	ADV
ejpam-3380	103	22	,	,	PUNCT
ejpam-3380	103	23	written	write	VERB
ejpam-3380	103	24	as	as	SCONJ
ejpam-3380	103	25	follows	follow	VERB
ejpam-3380	103	26	:	:	PUNCT
ejpam-3380	103	27	ks	ks	NOUN
ejpam-3380	103	28	∩	∩	NOUN
ejpam-3380	103	29	(	(	PUNCT
ejpam-3380	103	30	∑	∑	PROPN
ejpam-3380	103	31	t∈g	t∈g	PROPN
ejpam-3380	103	32	kt	kt	PROPN
ejpam-3380	103	33	)	)	PUNCT
ejpam-3380	103	34	=	=	PRON
ejpam-3380	103	35	(	(	PUNCT
ejpam-3380	103	36	k	k	PROPN
ejpam-3380	103	37	∩rs	∩rs	NOUN
ejpam-3380	103	38	)	)	PUNCT
ejpam-3380	103	39	∩	∩	NOUN
ejpam-3380	103	40	(	(	PUNCT
ejpam-3380	103	41	∑	∑	PROPN
ejpam-3380	103	42	t∈g	t∈g	X
ejpam-3380	103	43	k	k	X
ejpam-3380	103	44	∩rt	∩rt	NOUN
ejpam-3380	103	45	)	)	PUNCT
ejpam-3380	104	1	=	=	SYM
ejpam-3380	104	2	k	k	PROPN
ejpam-3380	104	3	∩	∩	NOUN
ejpam-3380	104	4	(	(	PUNCT
ejpam-3380	104	5	rs	rs	PROPN
ejpam-3380	104	6	∩	∩	NOUN
ejpam-3380	104	7	∑	∑	PUNCT
ejpam-3380	104	8	t∈g	t∈g	PROPN
ejpam-3380	104	9	rt	rt	PROPN
ejpam-3380	104	10	)	)	PUNCT
ejpam-3380	104	11	=	=	SYM
ejpam-3380	104	12	k	k	PROPN
ejpam-3380	104	13	∩	∩	X
ejpam-3380	104	14	{	{	PUNCT
ejpam-3380	104	15	0	0	NUM
ejpam-3380	104	16	}	}	PUNCT
ejpam-3380	104	17	=	=	PRON
ejpam-3380	104	18	{	{	PUNCT
ejpam-3380	104	19	0	0	NUM
ejpam-3380	104	20	}	}	PUNCT
ejpam-3380	104	21	,	,	PUNCT
ejpam-3380	104	22	for	for	ADP
ejpam-3380	104	23	all	all	DET
ejpam-3380	104	24	s	s	PART
ejpam-3380	104	25	∈	∈	NOUN
ejpam-3380	104	26	g	g	NOUN
ejpam-3380	104	27	with	with	ADP
ejpam-3380	104	28	s	s	PROPN
ejpam-3380	104	29	6=	6=	NUM
ejpam-3380	104	30	t.	t.	PROPN
ejpam-3380	104	31	corollary	corollary	NOUN
ejpam-3380	104	32	1	1	NUM
ejpam-3380	104	33	.	.	PUNCT
ejpam-3380	104	34	a	a	DET
ejpam-3380	104	35	subring	subre	VERB
ejpam-3380	104	36	k	k	NOUN
ejpam-3380	104	37	of	of	ADP
ejpam-3380	104	38	a	a	DET
ejpam-3380	104	39	g	g	NOUN
ejpam-3380	104	40	-	-	PUNCT
ejpam-3380	104	41	weak	weak	ADJ
ejpam-3380	104	42	graded	grade	VERB
ejpam-3380	104	43	ring	ring	NOUN
ejpam-3380	104	44	r	r	NOUN
ejpam-3380	104	45	is	be	AUX
ejpam-3380	104	46	a	a	DET
ejpam-3380	104	47	g	g	NOUN
ejpam-3380	104	48	-	-	PUNCT
ejpam-3380	104	49	weak	weak	ADJ
ejpam-3380	104	50	graded	grade	VERB
ejpam-3380	104	51	subring	subre	VERB
ejpam-3380	104	52	if	if	SCONJ
ejpam-3380	104	53	and	and	CCONJ
ejpam-3380	104	54	only	only	ADV
ejpam-3380	104	55	if	if	SCONJ
ejpam-3380	104	56	k	k	PROPN
ejpam-3380	104	57	=	=	PROPN
ejpam-3380	104	58	⊕	⊕	PROPN
ejpam-3380	104	59	s∈g(k	s∈g(k	NOUN
ejpam-3380	104	60	∩rs	∩rs	NOUN
ejpam-3380	104	61	)	)	PUNCT
ejpam-3380	104	62	.	.	PUNCT
ejpam-3380	105	1	theorem	theorem	NOUN
ejpam-3380	105	2	2	2	NUM
ejpam-3380	105	3	.	.	PUNCT
ejpam-3380	106	1	let	let	VERB
ejpam-3380	106	2	x	x	PRON
ejpam-3380	106	3	be	be	AUX
ejpam-3380	106	4	a	a	DET
ejpam-3380	106	5	group	group	NOUN
ejpam-3380	106	6	,	,	PUNCT
ejpam-3380	106	7	h	h	PROPN
ejpam-3380	106	8	be	be	VERB
ejpam-3380	106	9	a	a	DET
ejpam-3380	106	10	subgroup	subgroup	NOUN
ejpam-3380	106	11	of	of	ADP
ejpam-3380	106	12	x	x	PUNCT
ejpam-3380	106	13	and	and	CCONJ
ejpam-3380	106	14	(	(	PUNCT
ejpam-3380	106	15	g	g	NOUN
ejpam-3380	106	16	,	,	PUNCT
ejpam-3380	106	17	∗	∗	NOUN
ejpam-3380	106	18	)	)	PUNCT
ejpam-3380	106	19	be	be	VERB
ejpam-3380	106	20	a	a	DET
ejpam-3380	106	21	fixed	fix	VERB
ejpam-3380	106	22	set	set	NOUN
ejpam-3380	106	23	of	of	ADP
ejpam-3380	106	24	left	left	ADJ
ejpam-3380	106	25	coset	coset	NOUN
ejpam-3380	106	26	representatives	representative	NOUN
ejpam-3380	106	27	.	.	PUNCT
ejpam-3380	107	1	suppose	suppose	VERB
ejpam-3380	107	2	that	that	SCONJ
ejpam-3380	107	3	g	g	PROPN
ejpam-3380	107	4	has	have	VERB
ejpam-3380	107	5	a	a	DET
ejpam-3380	107	6	right	right	ADJ
ejpam-3380	107	7	inverse	inverse	NOUN
ejpam-3380	107	8	sr	sr	PROPN
ejpam-3380	107	9	for	for	ADP
ejpam-3380	107	10	each	each	DET
ejpam-3380	107	11	s	s	PROPN
ejpam-3380	107	12	∈	∈	PROPN
ejpam-3380	107	13	g	g	NOUN
ejpam-3380	107	14	and	and	CCONJ
ejpam-3380	107	15	that	that	SCONJ
ejpam-3380	107	16	sr	sr	PROPN
ejpam-3380	107	17	=	=	PUNCT
ejpam-3380	107	18	sl	sl	PROPN
ejpam-3380	107	19	=	=	SYM
ejpam-3380	107	20	s−1	s−1	PROPN
ejpam-3380	107	21	.	.	PUNCT
ejpam-3380	108	1	then	then	ADV
ejpam-3380	108	2	,	,	PUNCT
ejpam-3380	108	3	the	the	DET
ejpam-3380	108	4	g	g	NOUN
ejpam-3380	108	5	-	-	PUNCT
ejpam-3380	108	6	weak	weak	ADJ
ejpam-3380	108	7	graded	grade	VERB
ejpam-3380	108	8	ring	ring	NOUN
ejpam-3380	108	9	r	r	NOUN
ejpam-3380	108	10	is	be	AUX
ejpam-3380	108	11	fully	fully	ADV
ejpam-3380	108	12	if	if	SCONJ
ejpam-3380	109	1	and	and	CCONJ
ejpam-3380	109	2	only	only	ADV
ejpam-3380	109	3	if	if	SCONJ
ejpam-3380	109	4	1r	1r	NUM
ejpam-3380	109	5	∈	∈	PROPN
ejpam-3380	109	6	rslrs	rslrs	NOUN
ejpam-3380	109	7	for	for	ADP
ejpam-3380	109	8	all	all	DET
ejpam-3380	109	9	s	s	PROPN
ejpam-3380	109	10	∈	∈	NOUN
ejpam-3380	109	11	g.	g.	NOUN
ejpam-3380	109	12	proof	proof	NOUN
ejpam-3380	109	13	.	.	PUNCT
ejpam-3380	110	1	let	let	VERB
ejpam-3380	110	2	r	r	PRON
ejpam-3380	110	3	be	be	AUX
ejpam-3380	110	4	a	a	DET
ejpam-3380	110	5	fully	fully	ADV
ejpam-3380	110	6	g	g	NOUN
ejpam-3380	110	7	-	-	PUNCT
ejpam-3380	110	8	weak	weak	ADJ
ejpam-3380	110	9	graded	grade	VERB
ejpam-3380	110	10	ring	ring	NOUN
ejpam-3380	110	11	.	.	PUNCT
ejpam-3380	111	1	then	then	ADV
ejpam-3380	111	2	,	,	PUNCT
ejpam-3380	111	3	we	we	PRON
ejpam-3380	111	4	have	have	VERB
ejpam-3380	111	5	1r	1r	NUM
ejpam-3380	111	6	∈	∈	PROPN
ejpam-3380	111	7	reg	reg	NOUN
ejpam-3380	111	8	=	=	SYM
ejpam-3380	111	9	rsl∗s	rsl∗	NOUN
ejpam-3380	111	10	=	=	SYM
ejpam-3380	111	11	rslrs	rslrs	NOUN
ejpam-3380	111	12	.	.	PUNCT
ejpam-3380	112	1	on	on	ADP
ejpam-3380	112	2	the	the	DET
ejpam-3380	112	3	other	other	ADJ
ejpam-3380	112	4	hand	hand	NOUN
ejpam-3380	112	5	,	,	PUNCT
ejpam-3380	112	6	let	let	VERB
ejpam-3380	112	7	1r	1r	NUM
ejpam-3380	112	8	∈	∈	PROPN
ejpam-3380	112	9	rslrs	rslrs	NOUN
ejpam-3380	112	10	.	.	PUNCT
ejpam-3380	113	1	since	since	SCONJ
ejpam-3380	113	2	r	r	NOUN
ejpam-3380	113	3	is	be	AUX
ejpam-3380	113	4	g	g	NOUN
ejpam-3380	113	5	-	-	PUNCT
ejpam-3380	113	6	weak	weak	ADJ
ejpam-3380	113	7	graded	grade	VERB
ejpam-3380	113	8	ring	ring	NOUN
ejpam-3380	113	9	,	,	PUNCT
ejpam-3380	113	10	we	we	PRON
ejpam-3380	113	11	have	have	AUX
ejpam-3380	113	12	rtrs	rtr	VERB
ejpam-3380	113	13	⊆	⊆	NUM
ejpam-3380	113	14	rt∗s	rt∗s	NOUN
ejpam-3380	113	15	.	.	PUNCT
ejpam-3380	114	1	so	so	ADV
ejpam-3380	114	2	,	,	PUNCT
ejpam-3380	114	3	we	we	PRON
ejpam-3380	114	4	only	only	ADV
ejpam-3380	114	5	need	need	VERB
ejpam-3380	114	6	to	to	PART
ejpam-3380	114	7	show	show	VERB
ejpam-3380	114	8	that	that	SCONJ
ejpam-3380	114	9	rt∗s	rt∗s	NOUN
ejpam-3380	114	10	⊆	⊆	NUM
ejpam-3380	114	11	rtrs	rtrs	NOUN
ejpam-3380	114	12	which	which	PRON
ejpam-3380	114	13	we	we	PRON
ejpam-3380	114	14	do	do	VERB
ejpam-3380	114	15	as	as	SCONJ
ejpam-3380	114	16	follows	follow	VERB
ejpam-3380	114	17	:	:	PUNCT
ejpam-3380	114	18	rt∗s	rt∗s	NOUN
ejpam-3380	114	19	=	=	PUNCT
ejpam-3380	114	20	rt∗s1r	rt∗s1r	NUM
ejpam-3380	114	21	⊆	⊆	NUM
ejpam-3380	114	22	rt∗srslrs	rt∗srslrs	NOUN
ejpam-3380	114	23	⊆	⊆	NUM
ejpam-3380	114	24	r(t∗s)∗slrs	r(t∗s)∗slrs	NOUN
ejpam-3380	114	25	=	=	SYM
ejpam-3380	114	26	r	r	NOUN
ejpam-3380	114	27	(	(	PUNCT
ejpam-3380	114	28	t	t	PROPN
ejpam-3380	114	29	/	/	SYM
ejpam-3380	114	30	f(s	f(s	PROPN
ejpam-3380	114	31	,	,	PUNCT
ejpam-3380	114	32	sl	sl	NOUN
ejpam-3380	114	33	)	)	PUNCT
ejpam-3380	114	34	)	)	PUNCT
ejpam-3380	115	1	∗(s∗sl	∗(s∗sl	ADJ
ejpam-3380	115	2	)	)	PUNCT
ejpam-3380	115	3	rs	rs	NOUN
ejpam-3380	116	1	=	=	SYM
ejpam-3380	116	2	r	r	NOUN
ejpam-3380	116	3	(	(	PUNCT
ejpam-3380	116	4	t	t	PROPN
ejpam-3380	116	5	/	/	SYM
ejpam-3380	116	6	f(s	f(s	PROPN
ejpam-3380	116	7	,	,	PUNCT
ejpam-3380	116	8	sl	sl	PROPN
ejpam-3380	116	9	)	)	PUNCT
ejpam-3380	116	10	)	)	PUNCT
ejpam-3380	117	1	∗eg	∗eg	NUM
ejpam-3380	117	2	rs	rs	NOUN
ejpam-3380	117	3	=	=	SYM
ejpam-3380	117	4	r(t	r(t	NOUN
ejpam-3380	117	5	/	/	SYM
ejpam-3380	117	6	e−1	e−1	PROPN
ejpam-3380	117	7	g	g	NOUN
ejpam-3380	117	8	)	)	PUNCT
ejpam-3380	117	9	∗egrs	∗egrs	NOUN
ejpam-3380	118	1	=	=	NOUN
ejpam-3380	118	2	rtrs	rtrs	NOUN
ejpam-3380	118	3	,	,	PUNCT
ejpam-3380	118	4	as	as	SCONJ
ejpam-3380	118	5	required	require	VERB
ejpam-3380	118	6	.	.	PUNCT
ejpam-3380	119	1	theorem	theorem	NOUN
ejpam-3380	119	2	3	3	NUM
ejpam-3380	119	3	.	.	PUNCT
ejpam-3380	120	1	if	if	SCONJ
ejpam-3380	120	2	r	r	NOUN
ejpam-3380	120	3	is	be	AUX
ejpam-3380	120	4	a	a	DET
ejpam-3380	120	5	g	g	NOUN
ejpam-3380	120	6	-	-	PUNCT
ejpam-3380	120	7	weak	weak	ADJ
ejpam-3380	120	8	graded	grade	VERB
ejpam-3380	120	9	ring	ring	NOUN
ejpam-3380	120	10	and	and	CCONJ
ejpam-3380	120	11	i	i	PRON
ejpam-3380	120	12	is	be	AUX
ejpam-3380	120	13	an	an	DET
ejpam-3380	120	14	ideal	ideal	NOUN
ejpam-3380	120	15	of	of	ADP
ejpam-3380	120	16	reg	reg	NOUN
ejpam-3380	120	17	,	,	PUNCT
ejpam-3380	120	18	then	then	ADV
ejpam-3380	120	19	ir	ir	PROPN
ejpam-3380	120	20	∩reg	∩reg	PROPN
ejpam-3380	120	21	=	=	SYM
ejpam-3380	120	22	i.	i.	NOUN
ejpam-3380	120	23	proof	proof	NOUN
ejpam-3380	120	24	.	.	PUNCT
ejpam-3380	121	1	we	we	PRON
ejpam-3380	121	2	have	have	VERB
ejpam-3380	121	3	to	to	PART
ejpam-3380	121	4	show	show	VERB
ejpam-3380	121	5	that	that	SCONJ
ejpam-3380	121	6	i	i	PRON
ejpam-3380	121	7	⊆	⊆	NUM
ejpam-3380	121	8	ir	ir	PROPN
ejpam-3380	121	9	∩	∩	ADJ
ejpam-3380	121	10	reg	reg	NOUN
ejpam-3380	121	11	and	and	CCONJ
ejpam-3380	121	12	ir	ir	PROPN
ejpam-3380	121	13	∩	∩	NOUN
ejpam-3380	121	14	reg	reg	VERB
ejpam-3380	121	15	⊆	⊆	NUM
ejpam-3380	121	16	i.	i.	NOUN
ejpam-3380	121	17	it	it	PRON
ejpam-3380	121	18	is	be	AUX
ejpam-3380	121	19	obvious	obvious	ADJ
ejpam-3380	121	20	that	that	SCONJ
ejpam-3380	121	21	i	i	PRON
ejpam-3380	121	22	⊆	⊆	NUM
ejpam-3380	121	23	ir∩reg	ir∩reg	NOUN
ejpam-3380	121	24	as	as	ADP
ejpam-3380	121	25	,	,	PUNCT
ejpam-3380	121	26	for	for	ADP
ejpam-3380	121	27	any	any	DET
ejpam-3380	121	28	i	i	PRON
ejpam-3380	121	29	∈	∈	PROPN
ejpam-3380	121	30	i	i	PRON
ejpam-3380	121	31	,	,	PUNCT
ejpam-3380	121	32	we	we	PRON
ejpam-3380	121	33	can	can	AUX
ejpam-3380	121	34	write	write	VERB
ejpam-3380	121	35	i	i	PRON
ejpam-3380	121	36	=	=	SYM
ejpam-3380	121	37	i1r	i1r	PROPN
ejpam-3380	121	38	∈	∈	PROPN
ejpam-3380	121	39	ir	ir	PROPN
ejpam-3380	121	40	as	as	ADV
ejpam-3380	121	41	well	well	ADV
ejpam-3380	121	42	as	as	ADP
ejpam-3380	121	43	i	i	PRON
ejpam-3380	121	44	∈	∈	VERB
ejpam-3380	121	45	reg	reg	NOUN
ejpam-3380	121	46	since	since	SCONJ
ejpam-3380	121	47	i	i	PRON
ejpam-3380	121	48	⊆	⊆	NUM
ejpam-3380	121	49	reg	reg	NOUN
ejpam-3380	121	50	.	.	PUNCT
ejpam-3380	122	1	so	so	ADV
ejpam-3380	122	2	,	,	PUNCT
ejpam-3380	122	3	it	it	PRON
ejpam-3380	122	4	is	be	AUX
ejpam-3380	122	5	enough	enough	ADJ
ejpam-3380	122	6	to	to	PART
ejpam-3380	122	7	show	show	VERB
ejpam-3380	122	8	that	that	SCONJ
ejpam-3380	122	9	ir	ir	PROPN
ejpam-3380	122	10	∩	∩	PROPN
ejpam-3380	122	11	reg	reg	VERB
ejpam-3380	122	12	⊆	⊆	NUM
ejpam-3380	122	13	i	i	PRON
ejpam-3380	122	14	which	which	PRON
ejpam-3380	122	15	we	we	PRON
ejpam-3380	122	16	do	do	VERB
ejpam-3380	122	17	as	as	SCONJ
ejpam-3380	122	18	follows	follow	VERB
ejpam-3380	122	19	:	:	PUNCT
ejpam-3380	122	20	suppose	suppose	VERB
ejpam-3380	122	21	that	that	SCONJ
ejpam-3380	122	22	x	x	PUNCT
ejpam-3380	122	23	∈	∈	PROPN
ejpam-3380	122	24	ir	ir	NOUN
ejpam-3380	122	25	∩reg	∩reg	NOUN
ejpam-3380	122	26	which	which	PRON
ejpam-3380	122	27	implies	imply	VERB
ejpam-3380	122	28	that	that	SCONJ
ejpam-3380	122	29	x	x	PUNCT
ejpam-3380	122	30	∈	∈	PROPN
ejpam-3380	122	31	ir	ir	PROPN
ejpam-3380	122	32	and	and	CCONJ
ejpam-3380	122	33	x	x	PROPN
ejpam-3380	122	34	∈	∈	PROPN
ejpam-3380	122	35	reg	reg	NOUN
ejpam-3380	122	36	.	.	PUNCT
ejpam-3380	123	1	since	since	SCONJ
ejpam-3380	123	2	x	x	PROPN
ejpam-3380	123	3	∈	∈	PROPN
ejpam-3380	123	4	ir	ir	PROPN
ejpam-3380	123	5	,	,	PUNCT
ejpam-3380	123	6	we	we	PRON
ejpam-3380	123	7	can	can	AUX
ejpam-3380	123	8	write	write	VERB
ejpam-3380	123	9	x	x	PUNCT
ejpam-3380	123	10	as	as	ADP
ejpam-3380	123	11	x	x	X
ejpam-3380	123	12	=	=	SYM
ejpam-3380	123	13	∑	∑	PUNCT
ejpam-3380	123	14	finite	finite	PROPN
ejpam-3380	123	15	ir	ir	PROPN
ejpam-3380	123	16	,	,	PUNCT
ejpam-3380	123	17	(	(	PUNCT
ejpam-3380	123	18	4	4	NUM
ejpam-3380	123	19	)	)	PUNCT
ejpam-3380	123	20	for	for	ADP
ejpam-3380	123	21	i	i	PRON
ejpam-3380	123	22	∈	∈	PROPN
ejpam-3380	124	1	i	i	PRON
ejpam-3380	124	2	and	and	CCONJ
ejpam-3380	124	3	r	r	PROPN
ejpam-3380	124	4	∈	∈	PROPN
ejpam-3380	124	5	r.	r.	NOUN
ejpam-3380	124	6	as	as	SCONJ
ejpam-3380	124	7	r	r	NOUN
ejpam-3380	124	8	is	be	AUX
ejpam-3380	124	9	a	a	DET
ejpam-3380	124	10	g	g	NOUN
ejpam-3380	124	11	-	-	PUNCT
ejpam-3380	124	12	weak	weak	ADJ
ejpam-3380	124	13	graded	grade	VERB
ejpam-3380	124	14	ring	ring	NOUN
ejpam-3380	124	15	,	,	PUNCT
ejpam-3380	124	16	equation	equation	NOUN
ejpam-3380	124	17	(	(	PUNCT
ejpam-3380	124	18	4	4	X
ejpam-3380	124	19	)	)	PUNCT
ejpam-3380	124	20	can	can	AUX
ejpam-3380	124	21	be	be	AUX
ejpam-3380	124	22	rewritten	rewrite	VERB
ejpam-3380	124	23	as	as	ADP
ejpam-3380	124	24	x	x	X
ejpam-3380	124	25	=	=	PRON
ejpam-3380	124	26	∑	∑	PUNCT
ejpam-3380	124	27	finite	finite	VERB
ejpam-3380	124	28	irs	irs	PROPN
ejpam-3380	124	29	,	,	PUNCT
ejpam-3380	124	30	for	for	ADP
ejpam-3380	124	31	s	s	PROPN
ejpam-3380	124	32	∈	∈	PROPN
ejpam-3380	124	33	g	g	NOUN
ejpam-3380	124	34	,	,	PUNCT
ejpam-3380	124	35	where	where	SCONJ
ejpam-3380	124	36	rs	rs	NOUN
ejpam-3380	124	37	∈	∈	NOUN
ejpam-3380	124	38	rs	rs	NOUN
ejpam-3380	124	39	.	.	PUNCT
ejpam-3380	125	1	also	also	ADV
ejpam-3380	125	2	,	,	PUNCT
ejpam-3380	125	3	as	as	ADP
ejpam-3380	125	4	x	x	PROPN
ejpam-3380	125	5	∈	∈	PROPN
ejpam-3380	125	6	reg	reg	NOUN
ejpam-3380	125	7	,	,	PUNCT
ejpam-3380	125	8	hence	hence	ADV
ejpam-3380	125	9	x	x	PUNCT
ejpam-3380	125	10	can	can	AUX
ejpam-3380	125	11	be	be	AUX
ejpam-3380	125	12	written	write	VERB
ejpam-3380	125	13	as	as	ADP
ejpam-3380	125	14	x	x	X
ejpam-3380	125	15	=	=	PRON
ejpam-3380	125	16	ireg	ireg	VERB
ejpam-3380	125	17	.	.	PUNCT
ejpam-3380	126	1	thus	thus	ADV
ejpam-3380	126	2	,	,	PUNCT
ejpam-3380	126	3	x	x	PUNCT
ejpam-3380	126	4	∈	∈	PROPN
ejpam-3380	126	5	i	i	PRON
ejpam-3380	126	6	since	since	SCONJ
ejpam-3380	126	7	i	i	PRON
ejpam-3380	126	8	is	be	AUX
ejpam-3380	126	9	an	an	DET
ejpam-3380	126	10	ideal	ideal	NOUN
ejpam-3380	126	11	of	of	ADP
ejpam-3380	126	12	reg	reg	NOUN
ejpam-3380	126	13	.	.	PUNCT
ejpam-3380	127	1	therefore	therefore	ADV
ejpam-3380	127	2	,	,	PUNCT
ejpam-3380	127	3	ir	ir	PROPN
ejpam-3380	127	4	∩reg	∩reg	VERB
ejpam-3380	127	5	⊆	⊆	NUM
ejpam-3380	127	6	i	i	PRON
ejpam-3380	127	7	which	which	PRON
ejpam-3380	127	8	completes	complete	VERB
ejpam-3380	127	9	the	the	DET
ejpam-3380	127	10	proof	proof	NOUN
ejpam-3380	127	11	.	.	PUNCT
ejpam-3380	128	1	najla	najla	PROPN
ejpam-3380	128	2	al	al	PROPN
ejpam-3380	128	3	-	-	PUNCT
ejpam-3380	128	4	subaie	subaie	NOUN
ejpam-3380	128	5	,	,	PUNCT
ejpam-3380	128	6	m.	m.	NOUN
ejpam-3380	128	7	m.	m.	PROPN
ejpam-3380	128	8	al	al	PROPN
ejpam-3380	128	9	-	-	PUNCT
ejpam-3380	128	10	shomrani	shomrani	PROPN
ejpam-3380	128	11	/	/	SYM
ejpam-3380	128	12	eur	eur	NOUN
ejpam-3380	128	13	.	.	PUNCT
ejpam-3380	129	1	j.	j.	PROPN
ejpam-3380	129	2	pure	pure	PROPN
ejpam-3380	129	3	appl	appl	PROPN
ejpam-3380	129	4	.	.	PROPN
ejpam-3380	129	5	math	math	PROPN
ejpam-3380	129	6	,	,	PUNCT
ejpam-3380	129	7	12	12	NUM
ejpam-3380	129	8	(	(	PUNCT
ejpam-3380	129	9	2	2	NUM
ejpam-3380	129	10	)	)	PUNCT
ejpam-3380	129	11	(	(	PUNCT
ejpam-3380	129	12	2019	2019	NUM
ejpam-3380	129	13	)	)	PUNCT
ejpam-3380	129	14	,	,	PUNCT
ejpam-3380	129	15	332	332	NUM
ejpam-3380	129	16	-	-	SYM
ejpam-3380	129	17	347	347	NUM
ejpam-3380	129	18	337	337	NUM
ejpam-3380	129	19	definition	definition	NOUN
ejpam-3380	129	20	6	6	NUM
ejpam-3380	129	21	.	.	PUNCT
ejpam-3380	130	1	for	for	ADP
ejpam-3380	130	2	a	a	DET
ejpam-3380	130	3	g	g	NOUN
ejpam-3380	130	4	-	-	PUNCT
ejpam-3380	130	5	weak	weak	ADJ
ejpam-3380	130	6	graded	grade	VERB
ejpam-3380	130	7	ring	ring	NOUN
ejpam-3380	130	8	r	r	NOUN
ejpam-3380	130	9	,	,	PUNCT
ejpam-3380	130	10	a	a	DET
ejpam-3380	130	11	unit	unit	NOUN
ejpam-3380	130	12	x	x	SYM
ejpam-3380	130	13	∈	∈	PROPN
ejpam-3380	130	14	u(r	u(r	PROPN
ejpam-3380	130	15	)	)	PUNCT
ejpam-3380	130	16	is	be	AUX
ejpam-3380	130	17	said	say	VERB
ejpam-3380	130	18	to	to	PART
ejpam-3380	130	19	be	be	AUX
ejpam-3380	130	20	weak	weak	ADJ
ejpam-3380	130	21	graded	grade	VERB
ejpam-3380	130	22	unit	unit	NOUN
ejpam-3380	130	23	or	or	CCONJ
ejpam-3380	130	24	a	a	DET
ejpam-3380	130	25	g	g	NOUN
ejpam-3380	130	26	-	-	PUNCT
ejpam-3380	130	27	homogeneous	homogeneous	ADJ
ejpam-3380	130	28	unit	unit	NOUN
ejpam-3380	130	29	if	if	SCONJ
ejpam-3380	130	30	x	x	SYM
ejpam-3380	130	31	∈	∈	NOUN
ejpam-3380	130	32	rs	rs	NOUN
ejpam-3380	130	33	for	for	ADP
ejpam-3380	130	34	some	some	DET
ejpam-3380	130	35	s	s	VERB
ejpam-3380	130	36	∈	∈	PROPN
ejpam-3380	130	37	g	g	NOUN
ejpam-3380	130	38	,	,	PUNCT
ejpam-3380	130	39	where	where	SCONJ
ejpam-3380	130	40	u(r	u(r	NOUN
ejpam-3380	130	41	)	)	PUNCT
ejpam-3380	130	42	is	be	AUX
ejpam-3380	130	43	the	the	DET
ejpam-3380	130	44	group	group	NOUN
ejpam-3380	130	45	of	of	ADP
ejpam-3380	130	46	all	all	DET
ejpam-3380	130	47	units	unit	NOUN
ejpam-3380	130	48	in	in	ADP
ejpam-3380	130	49	r.	r.	PROPN
ejpam-3380	130	50	the	the	DET
ejpam-3380	130	51	set	set	NOUN
ejpam-3380	130	52	of	of	ADP
ejpam-3380	130	53	all	all	DET
ejpam-3380	130	54	weak	weak	ADJ
ejpam-3380	130	55	graded	grade	VERB
ejpam-3380	130	56	units	unit	NOUN
ejpam-3380	130	57	in	in	ADP
ejpam-3380	130	58	r	r	NOUN
ejpam-3380	130	59	is	be	AUX
ejpam-3380	130	60	denoted	denote	VERB
ejpam-3380	130	61	by	by	ADP
ejpam-3380	130	62	wgru	wgru	NOUN
ejpam-3380	130	63	(	(	PUNCT
ejpam-3380	130	64	r	r	NOUN
ejpam-3380	130	65	)	)	PUNCT
ejpam-3380	130	66	.	.	PUNCT
ejpam-3380	131	1	theorem	theorem	ADJ
ejpam-3380	131	2	4	4	NUM
ejpam-3380	131	3	.	.	PUNCT
ejpam-3380	132	1	let	let	VERB
ejpam-3380	132	2	r	r	PRON
ejpam-3380	132	3	be	be	AUX
ejpam-3380	132	4	a	a	DET
ejpam-3380	132	5	g	g	NOUN
ejpam-3380	132	6	-	-	PUNCT
ejpam-3380	132	7	weak	weak	ADJ
ejpam-3380	132	8	graded	grade	VERB
ejpam-3380	132	9	ring	ring	NOUN
ejpam-3380	132	10	with	with	ADP
ejpam-3380	132	11	unity	unity	NOUN
ejpam-3380	132	12	and	and	CCONJ
ejpam-3380	132	13	let	let	VERB
ejpam-3380	132	14	x	x	PRON
ejpam-3380	132	15	be	be	AUX
ejpam-3380	132	16	an	an	DET
ejpam-3380	132	17	element	element	NOUN
ejpam-3380	132	18	in	in	ADP
ejpam-3380	132	19	u(r	u(r	NOUN
ejpam-3380	132	20	)	)	PUNCT
ejpam-3380	132	21	.	.	PUNCT
ejpam-3380	133	1	if	if	SCONJ
ejpam-3380	133	2	x	x	SYM
ejpam-3380	133	3	∈	∈	NOUN
ejpam-3380	133	4	rs	rs	NOUN
ejpam-3380	133	5	,	,	PUNCT
ejpam-3380	133	6	for	for	ADP
ejpam-3380	133	7	some	some	DET
ejpam-3380	133	8	s	s	PART
ejpam-3380	133	9	∈	∈	PROPN
ejpam-3380	133	10	g	g	NOUN
ejpam-3380	133	11	,	,	PUNCT
ejpam-3380	133	12	then	then	ADV
ejpam-3380	133	13	x−1	x−1	PROPN
ejpam-3380	133	14	∈	∈	PROPN
ejpam-3380	133	15	rsl	rsl	NOUN
ejpam-3380	133	16	.	.	PUNCT
ejpam-3380	134	1	proof	proof	NOUN
ejpam-3380	134	2	.	.	PUNCT
ejpam-3380	135	1	let	let	VERB
ejpam-3380	135	2	x	x	SYM
ejpam-3380	135	3	∈	∈	NOUN
ejpam-3380	135	4	rs	rs	NOUN
ejpam-3380	135	5	for	for	ADP
ejpam-3380	135	6	some	some	DET
ejpam-3380	135	7	s	s	PART
ejpam-3380	135	8	∈	∈	PROPN
ejpam-3380	135	9	g	g	NOUN
ejpam-3380	135	10	and	and	CCONJ
ejpam-3380	135	11	let	let	VERB
ejpam-3380	135	12	x−1	x−1	PROPN
ejpam-3380	135	13	=	=	PUNCT
ejpam-3380	135	14	∑	∑	PUNCT
ejpam-3380	135	15	t∈g	t∈g	X
ejpam-3380	135	16	rt	rt	PROPN
ejpam-3380	135	17	where	where	SCONJ
ejpam-3380	135	18	rt	rt	PROPN
ejpam-3380	135	19	∈	∈	PROPN
ejpam-3380	135	20	rt	rt	PROPN
ejpam-3380	135	21	such	such	ADJ
ejpam-3380	135	22	that	that	SCONJ
ejpam-3380	135	23	all	all	PRON
ejpam-3380	135	24	but	but	SCONJ
ejpam-3380	135	25	a	a	DET
ejpam-3380	135	26	finite	finite	ADJ
ejpam-3380	135	27	number	number	NOUN
ejpam-3380	135	28	of	of	ADP
ejpam-3380	135	29	them	they	PRON
ejpam-3380	135	30	are	be	AUX
ejpam-3380	135	31	zero	zero	NUM
ejpam-3380	135	32	.	.	PUNCT
ejpam-3380	136	1	hence	hence	ADV
ejpam-3380	136	2	,	,	PUNCT
ejpam-3380	136	3	rtx	rtx	PROPN
ejpam-3380	136	4	∈	∈	PROPN
ejpam-3380	136	5	rtrs	rtrs	VERB
ejpam-3380	136	6	⊆	⊆	NUM
ejpam-3380	136	7	rt∗s	rt∗s	NOUN
ejpam-3380	136	8	for	for	ADP
ejpam-3380	136	9	any	any	DET
ejpam-3380	136	10	t	t	PROPN
ejpam-3380	136	11	∈	∈	PROPN
ejpam-3380	136	12	g.	g.	PROPN
ejpam-3380	136	13	thus	thus	ADV
ejpam-3380	136	14	,	,	PUNCT
ejpam-3380	136	15	x−1x	x−1x	X
ejpam-3380	136	16	=	=	SYM
ejpam-3380	136	17	∑	∑	PUNCT
ejpam-3380	136	18	t∈g	t∈g	NOUN
ejpam-3380	136	19	rtx	rtx	PROPN
ejpam-3380	136	20	is	be	AUX
ejpam-3380	136	21	the	the	DET
ejpam-3380	136	22	unique	unique	ADJ
ejpam-3380	136	23	expansion	expansion	NOUN
ejpam-3380	136	24	for	for	ADP
ejpam-3380	136	25	x−1x	x−1x	NOUN
ejpam-3380	136	26	in	in	ADP
ejpam-3380	136	27	the	the	DET
ejpam-3380	136	28	direct	direct	ADJ
ejpam-3380	136	29	sumr	sumr	NOUN
ejpam-3380	136	30	=	=	SYM
ejpam-3380	136	31	⊕	⊕	PROPN
ejpam-3380	136	32	t∈grt∗s	t∈grt∗s	PROPN
ejpam-3380	136	33	which	which	PRON
ejpam-3380	136	34	is	be	AUX
ejpam-3380	136	35	equivalent	equivalent	ADJ
ejpam-3380	136	36	to	to	ADP
ejpam-3380	136	37	r	r	NOUN
ejpam-3380	136	38	=	=	SYM
ejpam-3380	136	39	⊕	⊕	PROPN
ejpam-3380	136	40	s∈grs	s∈grs	PROPN
ejpam-3380	136	41	since	since	SCONJ
ejpam-3380	136	42	g	g	PROPN
ejpam-3380	136	43	is	be	AUX
ejpam-3380	136	44	closed	close	VERB
ejpam-3380	136	45	under	under	ADP
ejpam-3380	136	46	the	the	DET
ejpam-3380	136	47	binary	binary	ADJ
ejpam-3380	136	48	operation	operation	NOUN
ejpam-3380	136	49	∗.	∗.	PROPN
ejpam-3380	136	50	but	but	CCONJ
ejpam-3380	136	51	,	,	PUNCT
ejpam-3380	136	52	on	on	ADP
ejpam-3380	136	53	the	the	DET
ejpam-3380	136	54	other	other	ADJ
ejpam-3380	136	55	side	side	NOUN
ejpam-3380	136	56	,	,	PUNCT
ejpam-3380	136	57	we	we	PRON
ejpam-3380	136	58	have	have	VERB
ejpam-3380	136	59	x−1x	x−1x	NOUN
ejpam-3380	136	60	=	=	SYM
ejpam-3380	136	61	1r	1r	NUM
ejpam-3380	136	62	∈	∈	PROPN
ejpam-3380	136	63	reg	reg	NOUN
ejpam-3380	136	64	=	=	SYM
ejpam-3380	136	65	rsl∗s	rsl∗	NOUN
ejpam-3380	136	66	.	.	PUNCT
ejpam-3380	137	1	consequently	consequently	ADV
ejpam-3380	137	2	,	,	PUNCT
ejpam-3380	137	3	∑	∑	PROPN
ejpam-3380	137	4	t∈g	t∈g	X
ejpam-3380	137	5	rtx	rtx	PROPN
ejpam-3380	137	6	=	=	SYM
ejpam-3380	137	7	1r	1r	NUM
ejpam-3380	137	8	which	which	PRON
ejpam-3380	137	9	implies	imply	VERB
ejpam-3380	137	10	that	that	PRON
ejpam-3380	137	11	rt	rt	PROPN
ejpam-3380	137	12	=	=	NOUN
ejpam-3380	137	13	0	0	PROPN
ejpam-3380	137	14	for	for	ADP
ejpam-3380	137	15	all	all	DET
ejpam-3380	137	16	t	t	PROPN
ejpam-3380	137	17	6=	6=	PROPN
ejpam-3380	137	18	sl	sl	PROPN
ejpam-3380	137	19	.	.	PUNCT
ejpam-3380	138	1	so	so	ADV
ejpam-3380	138	2	,	,	PUNCT
ejpam-3380	138	3	if	if	SCONJ
ejpam-3380	138	4	we	we	PRON
ejpam-3380	138	5	put	put	VERB
ejpam-3380	138	6	t	t	NOUN
ejpam-3380	138	7	=	=	PUNCT
ejpam-3380	138	8	sl	sl	INTJ
ejpam-3380	138	9	we	we	PRON
ejpam-3380	138	10	get	get	VERB
ejpam-3380	138	11	rslx	rslx	NOUN
ejpam-3380	138	12	=	=	NOUN
ejpam-3380	138	13	1r	1r	NUM
ejpam-3380	138	14	.	.	PUNCT
ejpam-3380	139	1	therefore	therefore	ADV
ejpam-3380	139	2	,	,	PUNCT
ejpam-3380	139	3	rsl	rsl	NOUN
ejpam-3380	139	4	=	=	PROPN
ejpam-3380	139	5	x−1	x−1	PROPN
ejpam-3380	139	6	∈	∈	PROPN
ejpam-3380	139	7	rsl	rsl	NOUN
ejpam-3380	139	8	as	as	SCONJ
ejpam-3380	139	9	required	require	VERB
ejpam-3380	139	10	.	.	PUNCT
ejpam-3380	140	1	theorem	theorem	ADJ
ejpam-3380	140	2	5	5	NUM
ejpam-3380	140	3	.	.	PUNCT
ejpam-3380	141	1	if	if	SCONJ
ejpam-3380	141	2	r	r	NOUN
ejpam-3380	141	3	is	be	AUX
ejpam-3380	141	4	a	a	DET
ejpam-3380	141	5	g	g	NOUN
ejpam-3380	141	6	-	-	PUNCT
ejpam-3380	141	7	weak	weak	ADJ
ejpam-3380	141	8	graded	grade	VERB
ejpam-3380	141	9	ring	ring	NOUN
ejpam-3380	141	10	,	,	PUNCT
ejpam-3380	141	11	then	then	ADV
ejpam-3380	141	12	the	the	DET
ejpam-3380	141	13	set	set	NOUN
ejpam-3380	141	14	of	of	ADP
ejpam-3380	141	15	all	all	DET
ejpam-3380	141	16	its	its	PRON
ejpam-3380	141	17	weak	weak	ADJ
ejpam-3380	141	18	graded	grade	VERB
ejpam-3380	141	19	units	unit	NOUN
ejpam-3380	141	20	,	,	PUNCT
ejpam-3380	141	21	wgru	wgru	NOUN
ejpam-3380	141	22	(	(	PUNCT
ejpam-3380	141	23	r	r	NOUN
ejpam-3380	141	24	)	)	PUNCT
ejpam-3380	141	25	,	,	PUNCT
ejpam-3380	141	26	is	be	AUX
ejpam-3380	141	27	a	a	DET
ejpam-3380	141	28	subgroup	subgroup	NOUN
ejpam-3380	141	29	of	of	ADP
ejpam-3380	141	30	u(r	u(r	PROPN
ejpam-3380	141	31	)	)	PUNCT
ejpam-3380	141	32	and	and	CCONJ
ejpam-3380	141	33	the	the	DET
ejpam-3380	141	34	map	map	NOUN
ejpam-3380	141	35	〈	〈	PROPN
ejpam-3380	141	36	−	−	PROPN
ejpam-3380	141	37	〉	〉	NOUN
ejpam-3380	141	38	:	:	PUNCT
ejpam-3380	141	39	wgru	wgru	NOUN
ejpam-3380	141	40	(	(	PUNCT
ejpam-3380	141	41	r	r	NOUN
ejpam-3380	141	42	)	)	PUNCT
ejpam-3380	141	43	−→	−→	NOUN
ejpam-3380	141	44	g	g	NOUN
ejpam-3380	141	45	,	,	PUNCT
ejpam-3380	141	46	satisfies	satisfy	VERB
ejpam-3380	141	47	the	the	DET
ejpam-3380	141	48	homomorphism	homomorphism	PROPN
ejpam-3380	141	49	property	property	NOUN
ejpam-3380	141	50	of	of	ADP
ejpam-3380	141	51	groups	group	NOUN
ejpam-3380	141	52	with	with	ADP
ejpam-3380	141	53	〈	〈	PROPN
ejpam-3380	141	54	−〉−1(eg	−〉−1(eg	NUM
ejpam-3380	141	55	)	)	PUNCT
ejpam-3380	141	56	=	=	SYM
ejpam-3380	141	57	wgru(reg	wgru(reg	NOUN
ejpam-3380	141	58	)	)	PUNCT
ejpam-3380	141	59	,	,	PUNCT
ejpam-3380	141	60	where	where	SCONJ
ejpam-3380	141	61	〈	〈	PROPN
ejpam-3380	141	62	−	−	PROPN
ejpam-3380	141	63	〉	〉	PROPN
ejpam-3380	141	64	is	be	AUX
ejpam-3380	141	65	the	the	DET
ejpam-3380	141	66	ggrade	ggrade	NOUN
ejpam-3380	141	67	.	.	PUNCT
ejpam-3380	142	1	proof	proof	NOUN
ejpam-3380	142	2	.	.	PUNCT
ejpam-3380	143	1	first	first	ADV
ejpam-3380	143	2	,	,	PUNCT
ejpam-3380	143	3	to	to	PART
ejpam-3380	143	4	show	show	VERB
ejpam-3380	143	5	that	that	SCONJ
ejpam-3380	143	6	the	the	DET
ejpam-3380	143	7	set	set	ADJ
ejpam-3380	143	8	wgru	wgru	NOUN
ejpam-3380	143	9	(	(	PUNCT
ejpam-3380	143	10	r	r	NOUN
ejpam-3380	143	11	)	)	PUNCT
ejpam-3380	143	12	is	be	AUX
ejpam-3380	143	13	a	a	DET
ejpam-3380	143	14	subgroup	subgroup	NOUN
ejpam-3380	143	15	of	of	ADP
ejpam-3380	143	16	u(r	u(r	PROPN
ejpam-3380	143	17	)	)	PUNCT
ejpam-3380	143	18	,	,	PUNCT
ejpam-3380	143	19	let	let	VERB
ejpam-3380	143	20	x	x	PRON
ejpam-3380	143	21	,	,	PUNCT
ejpam-3380	143	22	y	y	PROPN
ejpam-3380	143	23	∈	∈	PROPN
ejpam-3380	143	24	wgru	wgru	NOUN
ejpam-3380	143	25	(	(	PUNCT
ejpam-3380	143	26	r	r	NOUN
ejpam-3380	143	27	)	)	PUNCT
ejpam-3380	143	28	.	.	PUNCT
ejpam-3380	144	1	then	then	ADV
ejpam-3380	144	2	x	x	SYM
ejpam-3380	144	3	∈	∈	NOUN
ejpam-3380	144	4	rs	rs	NOUN
ejpam-3380	144	5	and	and	CCONJ
ejpam-3380	144	6	y	y	PROPN
ejpam-3380	144	7	∈	∈	PROPN
ejpam-3380	144	8	rt	rt	PROPN
ejpam-3380	144	9	for	for	ADP
ejpam-3380	144	10	some	some	DET
ejpam-3380	144	11	s	s	NOUN
ejpam-3380	144	12	,	,	PUNCT
ejpam-3380	144	13	t	t	PROPN
ejpam-3380	144	14	∈	∈	PROPN
ejpam-3380	144	15	g.	g.	PROPN
ejpam-3380	144	16	now	now	ADV
ejpam-3380	144	17	,	,	PUNCT
ejpam-3380	144	18	since	since	SCONJ
ejpam-3380	144	19	r	r	NOUN
ejpam-3380	144	20	is	be	AUX
ejpam-3380	144	21	a	a	DET
ejpam-3380	144	22	g	g	NOUN
ejpam-3380	144	23	-	-	PUNCT
ejpam-3380	144	24	weak	weak	ADJ
ejpam-3380	144	25	graded	grade	VERB
ejpam-3380	144	26	ring	ring	NOUN
ejpam-3380	144	27	,	,	PUNCT
ejpam-3380	144	28	we	we	PRON
ejpam-3380	144	29	have	have	AUX
ejpam-3380	144	30	xy	xy	PROPN
ejpam-3380	144	31	∈	∈	PROPN
ejpam-3380	144	32	rsrt	rsrt	VERB
ejpam-3380	144	33	⊆	⊆	NUM
ejpam-3380	144	34	rs∗t	rs∗t	NOUN
ejpam-3380	144	35	.	.	PUNCT
ejpam-3380	145	1	also	also	ADV
ejpam-3380	145	2	,	,	PUNCT
ejpam-3380	145	3	as	as	SCONJ
ejpam-3380	145	4	g	g	PROPN
ejpam-3380	145	5	is	be	AUX
ejpam-3380	145	6	closed	close	VERB
ejpam-3380	145	7	under	under	ADP
ejpam-3380	145	8	the	the	DET
ejpam-3380	145	9	operation	operation	NOUN
ejpam-3380	145	10	∗	∗	NOUN
ejpam-3380	145	11	,	,	PUNCT
ejpam-3380	145	12	we	we	PRON
ejpam-3380	145	13	can	can	AUX
ejpam-3380	145	14	put	put	VERB
ejpam-3380	145	15	s	s	PRON
ejpam-3380	145	16	∗	∗	NOUN
ejpam-3380	145	17	t	t	NOUN
ejpam-3380	146	1	=	=	SYM
ejpam-3380	146	2	p	p	NOUN
ejpam-3380	146	3	for	for	ADP
ejpam-3380	146	4	some	some	DET
ejpam-3380	146	5	p	p	PROPN
ejpam-3380	146	6	∈	∈	PROPN
ejpam-3380	146	7	g.	g.	NOUN
ejpam-3380	146	8	hence	hence	ADV
ejpam-3380	146	9	,	,	PUNCT
ejpam-3380	146	10	xy	xy	PROPN
ejpam-3380	146	11	is	be	AUX
ejpam-3380	146	12	weak	weak	ADJ
ejpam-3380	146	13	graded	grade	VERB
ejpam-3380	146	14	for	for	ADP
ejpam-3380	146	15	all	all	DET
ejpam-3380	146	16	x	x	NOUN
ejpam-3380	146	17	,	,	PUNCT
ejpam-3380	146	18	y	y	PROPN
ejpam-3380	146	19	∈wgru	∈wgru	INTJ
ejpam-3380	146	20	(	(	PUNCT
ejpam-3380	146	21	r	r	NOUN
ejpam-3380	146	22	)	)	PUNCT
ejpam-3380	146	23	.	.	PUNCT
ejpam-3380	147	1	in	in	ADP
ejpam-3380	147	2	addition	addition	NOUN
ejpam-3380	147	3	,	,	PUNCT
ejpam-3380	147	4	as	as	SCONJ
ejpam-3380	147	5	x	x	X
ejpam-3380	147	6	,	,	PUNCT
ejpam-3380	147	7	y	y	PROPN
ejpam-3380	147	8	∈wgru	∈wgru	INTJ
ejpam-3380	147	9	(	(	PUNCT
ejpam-3380	147	10	r	r	NOUN
ejpam-3380	147	11	)	)	PUNCT
ejpam-3380	147	12	⊆	⊆	NUM
ejpam-3380	147	13	u(r	u(r	NOUN
ejpam-3380	147	14	)	)	PUNCT
ejpam-3380	147	15	and	and	CCONJ
ejpam-3380	147	16	that	that	SCONJ
ejpam-3380	147	17	u(r	u(r	NOUN
ejpam-3380	147	18	)	)	PUNCT
ejpam-3380	147	19	is	be	AUX
ejpam-3380	147	20	a	a	DET
ejpam-3380	147	21	group	group	NOUN
ejpam-3380	147	22	under	under	ADP
ejpam-3380	147	23	the	the	DET
ejpam-3380	147	24	ring	ring	NOUN
ejpam-3380	147	25	multiplication	multiplication	NOUN
ejpam-3380	147	26	,	,	PUNCT
ejpam-3380	147	27	then	then	ADV
ejpam-3380	147	28	xy	xy	PROPN
ejpam-3380	147	29	∈	∈	PROPN
ejpam-3380	147	30	u(r	u(r	PROPN
ejpam-3380	147	31	)	)	PUNCT
ejpam-3380	147	32	.	.	PUNCT
ejpam-3380	148	1	thus	thus	ADV
ejpam-3380	148	2	xy	xy	PROPN
ejpam-3380	148	3	is	be	AUX
ejpam-3380	148	4	a	a	DET
ejpam-3380	148	5	weak	weak	ADJ
ejpam-3380	148	6	graded	grade	VERB
ejpam-3380	148	7	unit	unit	NOUN
ejpam-3380	148	8	for	for	ADP
ejpam-3380	148	9	any	any	DET
ejpam-3380	148	10	x	x	NOUN
ejpam-3380	148	11	,	,	PUNCT
ejpam-3380	148	12	y	y	PROPN
ejpam-3380	148	13	∈wgru	∈wgru	INTJ
ejpam-3380	148	14	(	(	PUNCT
ejpam-3380	148	15	r	r	NOUN
ejpam-3380	148	16	)	)	PUNCT
ejpam-3380	148	17	,	,	PUNCT
ejpam-3380	148	18	i.e.	i.e.	X
ejpam-3380	148	19	,	,	PUNCT
ejpam-3380	148	20	xy	xy	ADJ
ejpam-3380	148	21	∈wgru	∈wgru	INTJ
ejpam-3380	148	22	(	(	PUNCT
ejpam-3380	148	23	r	r	NOUN
ejpam-3380	148	24	)	)	PUNCT
ejpam-3380	148	25	.	.	PUNCT
ejpam-3380	149	1	moreover	moreover	ADV
ejpam-3380	149	2	,	,	PUNCT
ejpam-3380	149	3	if	if	SCONJ
ejpam-3380	149	4	x	x	PRON
ejpam-3380	149	5	∈wgru	∈wgru	INTJ
ejpam-3380	149	6	(	(	PUNCT
ejpam-3380	149	7	r	r	NOUN
ejpam-3380	149	8	)	)	PUNCT
ejpam-3380	149	9	,	,	PUNCT
ejpam-3380	149	10	then	then	ADV
ejpam-3380	149	11	,	,	PUNCT
ejpam-3380	149	12	by	by	ADP
ejpam-3380	149	13	theorem	theorem	NOUN
ejpam-3380	149	14	4	4	NUM
ejpam-3380	149	15	,	,	PUNCT
ejpam-3380	149	16	x−1	x−1	PROPN
ejpam-3380	149	17	∈wgru	∈wgru	PROPN
ejpam-3380	149	18	(	(	PUNCT
ejpam-3380	149	19	r	r	NOUN
ejpam-3380	149	20	)	)	PUNCT
ejpam-3380	149	21	.	.	PUNCT
ejpam-3380	150	1	next	next	ADV
ejpam-3380	150	2	,	,	PUNCT
ejpam-3380	150	3	to	to	PART
ejpam-3380	150	4	show	show	VERB
ejpam-3380	150	5	that	that	SCONJ
ejpam-3380	150	6	the	the	DET
ejpam-3380	150	7	map	map	NOUN
ejpam-3380	150	8	〈	〈	PROPN
ejpam-3380	150	9	−	−	PROPN
ejpam-3380	150	10	〉	〉	PROPN
ejpam-3380	150	11	satisfies	satisfy	VERB
ejpam-3380	150	12	the	the	DET
ejpam-3380	150	13	homomorphism	homomorphism	PROPN
ejpam-3380	150	14	property	property	NOUN
ejpam-3380	150	15	of	of	ADP
ejpam-3380	150	16	groups	group	NOUN
ejpam-3380	150	17	,	,	PUNCT
ejpam-3380	150	18	let	let	VERB
ejpam-3380	150	19	x	x	PRON
ejpam-3380	150	20	,	,	PUNCT
ejpam-3380	150	21	y	y	PROPN
ejpam-3380	150	22	∈wgru	∈wgru	INTJ
ejpam-3380	150	23	(	(	PUNCT
ejpam-3380	150	24	r	r	NOUN
ejpam-3380	150	25	)	)	PUNCT
ejpam-3380	150	26	and	and	CCONJ
ejpam-3380	150	27	suppose	suppose	VERB
ejpam-3380	150	28	that	that	SCONJ
ejpam-3380	150	29	x	x	PUNCT
ejpam-3380	150	30	∈	∈	NOUN
ejpam-3380	150	31	rs	rs	NOUN
ejpam-3380	150	32	and	and	CCONJ
ejpam-3380	150	33	y	y	PROPN
ejpam-3380	150	34	∈	∈	PROPN
ejpam-3380	150	35	rt	rt	PROPN
ejpam-3380	150	36	for	for	ADP
ejpam-3380	150	37	some	some	DET
ejpam-3380	150	38	s	s	NOUN
ejpam-3380	150	39	,	,	PUNCT
ejpam-3380	150	40	t	t	PROPN
ejpam-3380	150	41	∈	∈	PROPN
ejpam-3380	150	42	g	g	PROPN
ejpam-3380	150	43	,	,	PUNCT
ejpam-3380	150	44	i.e.	i.e.	X
ejpam-3380	150	45	〈	〈	PROPN
ejpam-3380	150	46	x	x	SYM
ejpam-3380	150	47	〉	〉	NOUN
ejpam-3380	150	48	=	=	SYM
ejpam-3380	150	49	s	s	PROPN
ejpam-3380	150	50	and	and	CCONJ
ejpam-3380	150	51	〈	〈	PROPN
ejpam-3380	150	52	y	y	PROPN
ejpam-3380	150	53	〉	〉	NOUN
ejpam-3380	150	54	=	=	PUNCT
ejpam-3380	150	55	t.	t.	PROPN
ejpam-3380	150	56	thus	thus	ADV
ejpam-3380	150	57	,	,	PUNCT
ejpam-3380	150	58	xy	xy	PROPN
ejpam-3380	150	59	∈	∈	PROPN
ejpam-3380	150	60	rsrt	rsrt	VERB
ejpam-3380	150	61	⊆	⊆	NUM
ejpam-3380	150	62	rs∗t	rs∗t	NOUN
ejpam-3380	150	63	=	=	SYM
ejpam-3380	150	64	rp	rp	NOUN
ejpam-3380	150	65	,	,	PUNCT
ejpam-3380	150	66	for	for	ADP
ejpam-3380	150	67	some	some	DET
ejpam-3380	150	68	p	p	PROPN
ejpam-3380	150	69	∈	∈	PROPN
ejpam-3380	150	70	g.	g.	NOUN
ejpam-3380	150	71	consequently	consequently	ADV
ejpam-3380	150	72	,	,	PUNCT
ejpam-3380	150	73	〈	〈	PROPN
ejpam-3380	150	74	xy	xy	PROPN
ejpam-3380	150	75	〉	〉	NOUN
ejpam-3380	150	76	=	=	SYM
ejpam-3380	151	1	p	p	NOUN
ejpam-3380	151	2	=	=	PUNCT
ejpam-3380	151	3	s	s	NOUN
ejpam-3380	151	4	∗	∗	NOUN
ejpam-3380	151	5	t	t	NOUN
ejpam-3380	151	6	=	=	SYM
ejpam-3380	151	7	〈	〈	PROPN
ejpam-3380	151	8	x	x	PROPN
ejpam-3380	151	9	〉	〉	PROPN
ejpam-3380	151	10	∗	∗	NOUN
ejpam-3380	151	11	〈	〈	PROPN
ejpam-3380	151	12	y	y	PROPN
ejpam-3380	151	13	〉	〉	PROPN
ejpam-3380	151	14	.	.	PUNCT
ejpam-3380	152	1	najla	najla	PROPN
ejpam-3380	152	2	al	al	PROPN
ejpam-3380	152	3	-	-	PUNCT
ejpam-3380	152	4	subaie	subaie	NOUN
ejpam-3380	152	5	,	,	PUNCT
ejpam-3380	152	6	m.	m.	NOUN
ejpam-3380	152	7	m.	m.	PROPN
ejpam-3380	152	8	al	al	PROPN
ejpam-3380	152	9	-	-	PUNCT
ejpam-3380	152	10	shomrani	shomrani	PROPN
ejpam-3380	152	11	/	/	SYM
ejpam-3380	152	12	eur	eur	NOUN
ejpam-3380	152	13	.	.	PUNCT
ejpam-3380	153	1	j.	j.	PROPN
ejpam-3380	153	2	pure	pure	PROPN
ejpam-3380	153	3	appl	appl	PROPN
ejpam-3380	153	4	.	.	PROPN
ejpam-3380	153	5	math	math	PROPN
ejpam-3380	153	6	,	,	PUNCT
ejpam-3380	153	7	12	12	NUM
ejpam-3380	153	8	(	(	PUNCT
ejpam-3380	153	9	2	2	NUM
ejpam-3380	153	10	)	)	PUNCT
ejpam-3380	153	11	(	(	PUNCT
ejpam-3380	153	12	2019	2019	NUM
ejpam-3380	153	13	)	)	PUNCT
ejpam-3380	153	14	,	,	PUNCT
ejpam-3380	153	15	332	332	NUM
ejpam-3380	153	16	-	-	SYM
ejpam-3380	153	17	347	347	NUM
ejpam-3380	153	18	338	338	NUM
ejpam-3380	153	19	finally	finally	ADV
ejpam-3380	153	20	,	,	PUNCT
ejpam-3380	153	21	〈	〈	PROPN
ejpam-3380	153	22	−〉−1(eg	−〉−1(eg	X
ejpam-3380	153	23	)	)	PUNCT
ejpam-3380	153	24	=	=	SYM
ejpam-3380	153	25	{	{	PUNCT
ejpam-3380	153	26	x	x	PUNCT
ejpam-3380	153	27	∈wgru	∈wgru	INTJ
ejpam-3380	153	28	(	(	PUNCT
ejpam-3380	153	29	r	r	NOUN
ejpam-3380	153	30	)	)	PUNCT
ejpam-3380	153	31	:	:	PUNCT
ejpam-3380	154	1	〈	〈	PROPN
ejpam-3380	154	2	x	x	SYM
ejpam-3380	154	3	〉	〉	NOUN
ejpam-3380	154	4	=	=	SYM
ejpam-3380	154	5	eg	eg	NOUN
ejpam-3380	154	6	}	}	PUNCT
ejpam-3380	154	7	=	=	SYM
ejpam-3380	154	8	{	{	PUNCT
ejpam-3380	154	9	x	x	PUNCT
ejpam-3380	154	10	∈wgru	∈wgru	INTJ
ejpam-3380	154	11	(	(	PUNCT
ejpam-3380	154	12	r	r	NOUN
ejpam-3380	154	13	)	)	PUNCT
ejpam-3380	154	14	:	:	PUNCT
ejpam-3380	154	15	x	x	X
ejpam-3380	154	16	∈	∈	NOUN
ejpam-3380	154	17	reg	reg	NOUN
ejpam-3380	154	18	}	}	PUNCT
ejpam-3380	154	19	=	=	NOUN
ejpam-3380	154	20	wgru(reg	wgru(reg	NOUN
ejpam-3380	154	21	)	)	PUNCT
ejpam-3380	154	22	,	,	PUNCT
ejpam-3380	154	23	as	as	SCONJ
ejpam-3380	154	24	required	require	VERB
ejpam-3380	154	25	.	.	PUNCT
ejpam-3380	155	1	it	it	PRON
ejpam-3380	155	2	should	should	AUX
ejpam-3380	155	3	be	be	AUX
ejpam-3380	155	4	noted	note	VERB
ejpam-3380	155	5	that	that	SCONJ
ejpam-3380	155	6	g	g	PROPN
ejpam-3380	155	7	is	be	AUX
ejpam-3380	155	8	,	,	PUNCT
ejpam-3380	155	9	in	in	ADP
ejpam-3380	155	10	general	general	ADJ
ejpam-3380	155	11	,	,	PUNCT
ejpam-3380	155	12	neither	neither	CCONJ
ejpam-3380	155	13	a	a	DET
ejpam-3380	155	14	group	group	NOUN
ejpam-3380	155	15	nor	nor	CCONJ
ejpam-3380	155	16	even	even	ADV
ejpam-3380	155	17	a	a	DET
ejpam-3380	155	18	monoid	monoid	NOUN
ejpam-3380	155	19	.	.	PUNCT
ejpam-3380	155	20	proposition	proposition	NOUN
ejpam-3380	155	21	5	5	NUM
ejpam-3380	155	22	.	.	PUNCT
ejpam-3380	156	1	if	if	SCONJ
ejpam-3380	156	2	a	a	DET
ejpam-3380	156	3	ring	ring	NOUN
ejpam-3380	156	4	r	r	NOUN
ejpam-3380	156	5	is	be	AUX
ejpam-3380	156	6	a	a	DET
ejpam-3380	156	7	g	g	NOUN
ejpam-3380	156	8	-	-	PUNCT
ejpam-3380	156	9	weak	weak	ADJ
ejpam-3380	156	10	graded	grade	VERB
ejpam-3380	156	11	ring	ring	NOUN
ejpam-3380	156	12	,	,	PUNCT
ejpam-3380	156	13	then	then	ADV
ejpam-3380	156	14	the	the	DET
ejpam-3380	156	15	conjugation	conjugation	NOUN
ejpam-3380	156	16	in	in	ADP
ejpam-3380	156	17	r	r	NOUN
ejpam-3380	156	18	defines	define	VERB
ejpam-3380	156	19	an	an	DET
ejpam-3380	156	20	action	action	NOUN
ejpam-3380	156	21	given	give	VERB
ejpam-3380	156	22	by	by	ADP
ejpam-3380	156	23	ϕr	ϕr	NOUN
ejpam-3380	156	24	:	:	PUNCT
ejpam-3380	156	25	reg	reg	NOUN
ejpam-3380	156	26	,	,	PUNCT
ejpam-3380	156	27	x	x	PUNCT
ejpam-3380	156	28	−→	−→	ADJ
ejpam-3380	156	29	rxeg	rxeg	NOUN
ejpam-3380	156	30	=	=	SYM
ejpam-3380	156	31	x−1regx	x−1regx	NOUN
ejpam-3380	156	32	of	of	ADP
ejpam-3380	156	33	the	the	DET
ejpam-3380	156	34	group	group	NOUN
ejpam-3380	156	35	wgru	wgru	NOUN
ejpam-3380	156	36	(	(	PUNCT
ejpam-3380	156	37	r	r	NOUN
ejpam-3380	156	38	)	)	PUNCT
ejpam-3380	156	39	as	as	ADP
ejpam-3380	156	40	automorphisms	automorphism	NOUN
ejpam-3380	156	41	of	of	ADP
ejpam-3380	156	42	the	the	DET
ejpam-3380	156	43	subring	subre	VERB
ejpam-3380	156	44	reg	reg	NOUN
ejpam-3380	156	45	,	,	PUNCT
ejpam-3380	156	46	where	where	SCONJ
ejpam-3380	156	47	reg	reg	PROPN
ejpam-3380	156	48	∈	∈	PROPN
ejpam-3380	156	49	reg	reg	NOUN
ejpam-3380	156	50	and	and	CCONJ
ejpam-3380	156	51	x	x	PART
ejpam-3380	156	52	∈wgru	∈wgru	INTJ
ejpam-3380	156	53	(	(	PUNCT
ejpam-3380	156	54	r	r	NOUN
ejpam-3380	156	55	)	)	PUNCT
ejpam-3380	156	56	.	.	PUNCT
ejpam-3380	157	1	proof	proof	NOUN
ejpam-3380	157	2	.	.	PUNCT
ejpam-3380	158	1	the	the	DET
ejpam-3380	158	2	conjugation	conjugation	NOUN
ejpam-3380	158	3	by	by	ADP
ejpam-3380	158	4	a	a	DET
ejpam-3380	158	5	weak	weak	ADJ
ejpam-3380	158	6	graded	grade	VERB
ejpam-3380	158	7	unit	unit	NOUN
ejpam-3380	158	8	x	x	PUNCT
ejpam-3380	158	9	in	in	ADP
ejpam-3380	158	10	rs	rs	NOUN
ejpam-3380	158	11	for	for	ADP
ejpam-3380	158	12	some	some	DET
ejpam-3380	158	13	s	s	NOUN
ejpam-3380	158	14	∈	∈	PROPN
ejpam-3380	158	15	g	g	NOUN
ejpam-3380	158	16	is	be	AUX
ejpam-3380	158	17	an	an	DET
ejpam-3380	158	18	automorphism	automorphism	NOUN
ejpam-3380	158	19	of	of	ADP
ejpam-3380	158	20	the	the	DET
ejpam-3380	158	21	ring	ring	NOUN
ejpam-3380	158	22	r	r	NOUN
ejpam-3380	158	23	satisfying	satisfying	NOUN
ejpam-3380	158	24	:	:	PUNCT
ejpam-3380	158	25	rx	rx	VERB
ejpam-3380	158	26	eg	eg	NOUN
ejpam-3380	158	27	=	=	PUNCT
ejpam-3380	158	28	x−1regx	x−1regx	PROPN
ejpam-3380	159	1	⊆	⊆	NUM
ejpam-3380	159	2	rslregrs	rslregrs	NOUN
ejpam-3380	159	3	⊆	⊆	NUM
ejpam-3380	159	4	rslreg∗s	rslreg∗s	NOUN
ejpam-3380	159	5	=	=	SYM
ejpam-3380	159	6	rslrs	rslrs	NOUN
ejpam-3380	159	7	⊆	⊆	NUM
ejpam-3380	159	8	reg	reg	NOUN
ejpam-3380	159	9	.	.	PUNCT
ejpam-3380	160	1	since	since	SCONJ
ejpam-3380	160	2	wgru	wgru	NOUN
ejpam-3380	160	3	(	(	PUNCT
ejpam-3380	160	4	r	r	NOUN
ejpam-3380	160	5	)	)	PUNCT
ejpam-3380	160	6	is	be	AUX
ejpam-3380	160	7	a	a	DET
ejpam-3380	160	8	group	group	NOUN
ejpam-3380	160	9	,	,	PUNCT
ejpam-3380	160	10	the	the	DET
ejpam-3380	160	11	proof	proof	NOUN
ejpam-3380	160	12	is	be	AUX
ejpam-3380	160	13	completed	complete	VERB
ejpam-3380	160	14	.	.	PUNCT
ejpam-3380	161	1	4	4	X
ejpam-3380	161	2	.	.	X
ejpam-3380	161	3	g	g	NOUN
ejpam-3380	161	4	-	-	PUNCT
ejpam-3380	161	5	weak	weak	ADJ
ejpam-3380	161	6	graded	grade	VERB
ejpam-3380	161	7	rings	ring	NOUN
ejpam-3380	161	8	of	of	ADP
ejpam-3380	161	9	fractions	fraction	NOUN
ejpam-3380	161	10	we	we	PRON
ejpam-3380	161	11	start	start	VERB
ejpam-3380	161	12	this	this	DET
ejpam-3380	161	13	section	section	NOUN
ejpam-3380	161	14	by	by	ADP
ejpam-3380	161	15	recalling	recall	VERB
ejpam-3380	161	16	the	the	DET
ejpam-3380	161	17	following	follow	VERB
ejpam-3380	161	18	well	well	ADV
ejpam-3380	161	19	known	know	VERB
ejpam-3380	161	20	results	result	NOUN
ejpam-3380	162	1	[	[	X
ejpam-3380	162	2	18	18	NUM
ejpam-3380	162	3	]	]	SYM
ejpam-3380	162	4	:	:	PUNCT
ejpam-3380	162	5	(	(	PUNCT
ejpam-3380	162	6	a	a	X
ejpam-3380	162	7	)	)	PUNCT
ejpam-3380	162	8	let	let	VERB
ejpam-3380	162	9	r	r	PRON
ejpam-3380	162	10	be	be	AUX
ejpam-3380	162	11	a	a	DET
ejpam-3380	162	12	ring	ring	NOUN
ejpam-3380	162	13	and	and	CCONJ
ejpam-3380	162	14	k	k	PROPN
ejpam-3380	162	15	be	be	AUX
ejpam-3380	162	16	a	a	DET
ejpam-3380	162	17	multiplicatively	multiplicatively	ADV
ejpam-3380	162	18	closed	close	VERB
ejpam-3380	162	19	subset	subset	NOUN
ejpam-3380	162	20	of	of	ADP
ejpam-3380	162	21	r	r	NOUN
ejpam-3380	162	22	such	such	ADJ
ejpam-3380	162	23	that	that	DET
ejpam-3380	162	24	1r	1r	NUM
ejpam-3380	162	25	∈	∈	PROPN
ejpam-3380	162	26	k	k	X
ejpam-3380	162	27	,	,	PUNCT
ejpam-3380	162	28	0	0	NUM
ejpam-3380	162	29	/∈	/∈	PUNCT
ejpam-3380	163	1	k.	k.	PROPN
ejpam-3380	164	1	then	then	ADV
ejpam-3380	164	2	the	the	DET
ejpam-3380	164	3	left	left	ADJ
ejpam-3380	164	4	ring	ring	NOUN
ejpam-3380	164	5	of	of	ADP
ejpam-3380	164	6	fractions	fraction	NOUN
ejpam-3380	164	7	with	with	ADP
ejpam-3380	164	8	respect	respect	NOUN
ejpam-3380	164	9	to	to	ADP
ejpam-3380	164	10	k	k	PROPN
ejpam-3380	164	11	,	,	PUNCT
ejpam-3380	164	12	k−1r	k−1r	PROPN
ejpam-3380	164	13	,	,	PUNCT
ejpam-3380	164	14	exists	exist	VERB
ejpam-3380	164	15	if	if	SCONJ
ejpam-3380	164	16	and	and	CCONJ
ejpam-3380	164	17	only	only	ADV
ejpam-3380	164	18	if	if	SCONJ
ejpam-3380	164	19	r	r	NOUN
ejpam-3380	164	20	satisfies	satisfy	VERB
ejpam-3380	164	21	the	the	DET
ejpam-3380	164	22	left	left	ADJ
ejpam-3380	164	23	ore	ore	NOUN
ejpam-3380	164	24	conditions	condition	NOUN
ejpam-3380	164	25	with	with	ADP
ejpam-3380	164	26	respect	respect	NOUN
ejpam-3380	164	27	to	to	ADP
ejpam-3380	164	28	k	k	NOUN
ejpam-3380	164	29	,	,	PUNCT
ejpam-3380	164	30	i.e.	i.e.	X
ejpam-3380	164	31	:	:	PUNCT
ejpam-3380	164	32	(	(	PUNCT
ejpam-3380	164	33	i	i	NOUN
ejpam-3380	164	34	)	)	PUNCT
ejpam-3380	164	35	if	if	SCONJ
ejpam-3380	164	36	rk	rk	NOUN
ejpam-3380	164	37	=	=	SYM
ejpam-3380	164	38	0	0	NUM
ejpam-3380	164	39	,	,	PUNCT
ejpam-3380	164	40	for	for	ADP
ejpam-3380	164	41	some	some	DET
ejpam-3380	164	42	k	k	PROPN
ejpam-3380	164	43	∈	∈	PROPN
ejpam-3380	164	44	k	k	PROPN
ejpam-3380	164	45	and	and	CCONJ
ejpam-3380	164	46	r	r	NOUN
ejpam-3380	164	47	∈	∈	PROPN
ejpam-3380	164	48	r	r	NOUN
ejpam-3380	164	49	,	,	PUNCT
ejpam-3380	164	50	then	then	ADV
ejpam-3380	164	51	there	there	PRON
ejpam-3380	164	52	is	be	VERB
ejpam-3380	164	53	an	an	DET
ejpam-3380	164	54	element	element	NOUN
ejpam-3380	164	55	k′	k′	PROPN
ejpam-3380	164	56	∈	∈	PROPN
ejpam-3380	165	1	k	k	ADP
ejpam-3380	165	2	such	such	ADJ
ejpam-3380	165	3	that	that	DET
ejpam-3380	165	4	k′r	k′r	NOUN
ejpam-3380	165	5	=	=	SYM
ejpam-3380	165	6	0	0	X
ejpam-3380	165	7	.	.	PUNCT
ejpam-3380	165	8	(	(	PUNCT
ejpam-3380	165	9	ii	ii	NOUN
ejpam-3380	165	10	)	)	PUNCT
ejpam-3380	165	11	for	for	ADP
ejpam-3380	165	12	r	r	NOUN
ejpam-3380	165	13	∈	∈	PROPN
ejpam-3380	165	14	r	r	NOUN
ejpam-3380	165	15	and	and	CCONJ
ejpam-3380	165	16	k	k	PROPN
ejpam-3380	165	17	∈	∈	PROPN
ejpam-3380	166	1	k	k	NOUN
ejpam-3380	166	2	,	,	PUNCT
ejpam-3380	166	3	there	there	PRON
ejpam-3380	166	4	are	be	VERB
ejpam-3380	166	5	elements	element	NOUN
ejpam-3380	166	6	r′	r′	PROPN
ejpam-3380	166	7	∈	∈	PROPN
ejpam-3380	166	8	r	r	NOUN
ejpam-3380	166	9	and	and	CCONJ
ejpam-3380	166	10	k′	k′	PROPN
ejpam-3380	166	11	∈	∈	PROPN
ejpam-3380	166	12	k	k	NOUN
ejpam-3380	166	13	such	such	ADJ
ejpam-3380	166	14	that	that	DET
ejpam-3380	166	15	k′r	k′r	NOUN
ejpam-3380	166	16	=	=	NOUN
ejpam-3380	166	17	r′k	r′k	NOUN
ejpam-3380	166	18	.	.	PUNCT
ejpam-3380	167	1	if	if	SCONJ
ejpam-3380	167	2	ore	ore	NOUN
ejpam-3380	167	3	conditions	condition	NOUN
ejpam-3380	167	4	with	with	ADP
ejpam-3380	167	5	respect	respect	NOUN
ejpam-3380	167	6	to	to	ADP
ejpam-3380	167	7	k	k	PROPN
ejpam-3380	167	8	are	be	AUX
ejpam-3380	167	9	satisfied	satisfied	ADJ
ejpam-3380	167	10	,	,	PUNCT
ejpam-3380	167	11	then	then	ADV
ejpam-3380	167	12	k−1r	k−1r	PROPN
ejpam-3380	168	1	=	=	PUNCT
ejpam-3380	168	2	rk	rk	NOUN
ejpam-3380	168	3	=	=	NOUN
ejpam-3380	168	4	{	{	PUNCT
ejpam-3380	168	5	r	r	NOUN
ejpam-3380	168	6	k	k	NOUN
ejpam-3380	168	7	:	:	PUNCT
ejpam-3380	168	8	r	r	NOUN
ejpam-3380	168	9	∈	∈	PROPN
ejpam-3380	168	10	r	r	NOUN
ejpam-3380	168	11	,	,	PUNCT
ejpam-3380	168	12	k	k	PROPN
ejpam-3380	168	13	∈	∈	PROPN
ejpam-3380	168	14	k	k	X
ejpam-3380	168	15	}	}	PUNCT
ejpam-3380	168	16	.	.	PUNCT
ejpam-3380	169	1	the	the	DET
ejpam-3380	169	2	addition	addition	NOUN
ejpam-3380	169	3	and	and	CCONJ
ejpam-3380	169	4	multiplication	multiplication	NOUN
ejpam-3380	169	5	operations	operation	NOUN
ejpam-3380	169	6	on	on	ADP
ejpam-3380	169	7	rk	rk	PRON
ejpam-3380	169	8	are	be	AUX
ejpam-3380	169	9	defined	define	VERB
ejpam-3380	169	10	,	,	PUNCT
ejpam-3380	169	11	respectively	respectively	ADV
ejpam-3380	169	12	,	,	PUNCT
ejpam-3380	169	13	by	by	ADP
ejpam-3380	169	14	r	r	PROPN
ejpam-3380	169	15	k	k	PROPN
ejpam-3380	170	1	+	+	CCONJ
ejpam-3380	170	2	r′	r′	PROPN
ejpam-3380	170	3	k′	k′	PROPN
ejpam-3380	170	4	=	=	SYM
ejpam-3380	170	5	k′r+kr′	k′r+kr′	PROPN
ejpam-3380	170	6	kk′	kk′	PROPN
ejpam-3380	170	7	and	and	CCONJ
ejpam-3380	170	8	r	r	PROPN
ejpam-3380	170	9	k	k	PROPN
ejpam-3380	170	10	·	·	PUNCT
ejpam-3380	170	11	r′	r′	PROPN
ejpam-3380	170	12	k′	k′	PROPN
ejpam-3380	170	13	=	=	SYM
ejpam-3380	170	14	r1r′	r1r′	PROPN
ejpam-3380	170	15	k′1k	k′1k	ADJ
ejpam-3380	170	16	for	for	ADP
ejpam-3380	170	17	k′1	k′1	NOUN
ejpam-3380	170	18	∈	∈	PROPN
ejpam-3380	170	19	k	k	PROPN
ejpam-3380	170	20	,	,	PUNCT
ejpam-3380	170	21	r1	r1	PROPN
ejpam-3380	170	22	∈	∈	PROPN
ejpam-3380	170	23	r	r	NOUN
ejpam-3380	170	24	with	with	ADP
ejpam-3380	170	25	k′1r	k′1r	X
ejpam-3380	170	26	=	=	SYM
ejpam-3380	170	27	r1k	r1k	PROPN
ejpam-3380	170	28	′.	′.	NOUN
ejpam-3380	170	29	(	(	PUNCT
ejpam-3380	170	30	b	b	NOUN
ejpam-3380	170	31	)	)	PUNCT
ejpam-3380	170	32	for	for	ADP
ejpam-3380	170	33	every	every	DET
ejpam-3380	170	34	m	m	NOUN
ejpam-3380	170	35	in	in	ADP
ejpam-3380	170	36	r	r	NOUN
ejpam-3380	170	37	-	-	PUNCT
ejpam-3380	170	38	mod	mod	NOUN
ejpam-3380	170	39	,	,	PUNCT
ejpam-3380	170	40	we	we	PRON
ejpam-3380	170	41	can	can	AUX
ejpam-3380	170	42	construct	construct	VERB
ejpam-3380	170	43	a	a	DET
ejpam-3380	170	44	fraction	fraction	NOUN
ejpam-3380	170	45	k−1	k−1	PROPN
ejpam-3380	170	46	m	m	VERB
ejpam-3380	170	47	which	which	PRON
ejpam-3380	170	48	is	be	AUX
ejpam-3380	170	49	a	a	DET
ejpam-3380	170	50	left	left	ADJ
ejpam-3380	170	51	k−1rmodule	k−1rmodule	NOUN
ejpam-3380	170	52	.	.	PUNCT
ejpam-3380	171	1	moreover	moreover	ADV
ejpam-3380	171	2	,	,	PUNCT
ejpam-3380	171	3	k−1	k−1	PROPN
ejpam-3380	171	4	m	m	VERB
ejpam-3380	171	5	∼=	∼=	NOUN
ejpam-3380	171	6	k−1r⊗r	k−1r⊗r	NOUN
ejpam-3380	171	7	m	m	PROPN
ejpam-3380	171	8	.	.	PUNCT
ejpam-3380	172	1	now	now	ADV
ejpam-3380	172	2	,	,	PUNCT
ejpam-3380	172	3	we	we	PRON
ejpam-3380	172	4	prove	prove	VERB
ejpam-3380	172	5	the	the	DET
ejpam-3380	172	6	following	follow	VERB
ejpam-3380	172	7	lemma	lemma	PROPN
ejpam-3380	172	8	which	which	PRON
ejpam-3380	172	9	will	will	AUX
ejpam-3380	172	10	be	be	AUX
ejpam-3380	172	11	used	use	VERB
ejpam-3380	172	12	to	to	PART
ejpam-3380	172	13	prove	prove	VERB
ejpam-3380	172	14	the	the	DET
ejpam-3380	172	15	the	the	DET
ejpam-3380	172	16	next	next	ADJ
ejpam-3380	172	17	theorem	theorem	NOUN
ejpam-3380	172	18	.	.	PUNCT
ejpam-3380	173	1	najla	najla	PROPN
ejpam-3380	173	2	al	al	PROPN
ejpam-3380	173	3	-	-	PUNCT
ejpam-3380	173	4	subaie	subaie	NOUN
ejpam-3380	173	5	,	,	PUNCT
ejpam-3380	173	6	m.	m.	NOUN
ejpam-3380	173	7	m.	m.	PROPN
ejpam-3380	173	8	al	al	PROPN
ejpam-3380	173	9	-	-	PUNCT
ejpam-3380	173	10	shomrani	shomrani	PROPN
ejpam-3380	173	11	/	/	SYM
ejpam-3380	173	12	eur	eur	NOUN
ejpam-3380	173	13	.	.	PUNCT
ejpam-3380	174	1	j.	j.	PROPN
ejpam-3380	174	2	pure	pure	PROPN
ejpam-3380	174	3	appl	appl	PROPN
ejpam-3380	174	4	.	.	PROPN
ejpam-3380	174	5	math	math	PROPN
ejpam-3380	174	6	,	,	PUNCT
ejpam-3380	174	7	12	12	NUM
ejpam-3380	174	8	(	(	PUNCT
ejpam-3380	174	9	2	2	NUM
ejpam-3380	174	10	)	)	PUNCT
ejpam-3380	174	11	(	(	PUNCT
ejpam-3380	174	12	2019	2019	NUM
ejpam-3380	174	13	)	)	PUNCT
ejpam-3380	174	14	,	,	PUNCT
ejpam-3380	174	15	332	332	NUM
ejpam-3380	174	16	-	-	SYM
ejpam-3380	174	17	347	347	NUM
ejpam-3380	174	18	339	339	NUM
ejpam-3380	174	19	lemma	lemma	PROPN
ejpam-3380	174	20	1	1	NUM
ejpam-3380	174	21	.	.	PUNCT
ejpam-3380	175	1	let	let	VERB
ejpam-3380	175	2	x	x	PRON
ejpam-3380	175	3	be	be	AUX
ejpam-3380	175	4	a	a	DET
ejpam-3380	175	5	group	group	NOUN
ejpam-3380	175	6	,	,	PUNCT
ejpam-3380	175	7	h	h	PROPN
ejpam-3380	175	8	be	be	VERB
ejpam-3380	175	9	a	a	DET
ejpam-3380	175	10	subgroup	subgroup	NOUN
ejpam-3380	175	11	of	of	ADP
ejpam-3380	175	12	x	x	PROPN
ejpam-3380	175	13	and	and	CCONJ
ejpam-3380	175	14	g	g	PROPN
ejpam-3380	175	15	⊂	⊂	PROPN
ejpam-3380	175	16	x	x	X
ejpam-3380	175	17	be	be	AUX
ejpam-3380	175	18	a	a	DET
ejpam-3380	175	19	set	set	NOUN
ejpam-3380	175	20	of	of	ADP
ejpam-3380	175	21	left	left	ADJ
ejpam-3380	175	22	coset	coset	NOUN
ejpam-3380	175	23	representatives	representative	NOUN
ejpam-3380	175	24	.	.	PUNCT
ejpam-3380	176	1	then	then	ADV
ejpam-3380	176	2	,	,	PUNCT
ejpam-3380	176	3	for	for	ADP
ejpam-3380	176	4	all	all	DET
ejpam-3380	176	5	s	s	PROPN
ejpam-3380	176	6	,	,	PUNCT
ejpam-3380	176	7	t	t	PROPN
ejpam-3380	176	8	∈	∈	PROPN
ejpam-3380	176	9	g	g	PROPN
ejpam-3380	176	10	,	,	PUNCT
ejpam-3380	176	11	the	the	DET
ejpam-3380	176	12	multiplication	multiplication	NOUN
ejpam-3380	176	13	s∗	s∗	PROPN
ejpam-3380	176	14	t	t	PROPN
ejpam-3380	176	15	in	in	ADP
ejpam-3380	176	16	g	g	PROPN
ejpam-3380	176	17	has	have	VERB
ejpam-3380	176	18	a	a	DET
ejpam-3380	176	19	left	left	ADJ
ejpam-3380	176	20	inverse	inverse	NOUN
ejpam-3380	176	21	given	give	VERB
ejpam-3380	176	22	by	by	ADP
ejpam-3380	176	23	:	:	PUNCT
ejpam-3380	176	24	(	(	PUNCT
ejpam-3380	176	25	s	s	NOUN
ejpam-3380	176	26	∗	∗	NOUN
ejpam-3380	176	27	t)l	t)l	NOUN
ejpam-3380	177	1	=	=	PUNCT
ejpam-3380	177	2	(	(	PUNCT
ejpam-3380	177	3	tl	tl	PROPN
ejpam-3380	177	4	/	/	SYM
ejpam-3380	177	5	f	f	PROPN
ejpam-3380	177	6	(	(	PUNCT
ejpam-3380	177	7	sl	sl	PROPN
ejpam-3380	177	8	/	/	SYM
ejpam-3380	177	9	f(s	f(s	PROPN
ejpam-3380	177	10	,	,	PUNCT
ejpam-3380	177	11	t	t	PROPN
ejpam-3380	177	12	)	)	PUNCT
ejpam-3380	177	13	,	,	PUNCT
ejpam-3380	177	14	s	s	PART
ejpam-3380	177	15	∗	∗	NOUN
ejpam-3380	177	16	t	t	NOUN
ejpam-3380	177	17	)	)	PUNCT
ejpam-3380	177	18	−1	−1	NOUN
ejpam-3380	177	19	)	)	PUNCT
ejpam-3380	177	20	∗	∗	NOUN
ejpam-3380	177	21	(	(	PUNCT
ejpam-3380	177	22	sl	sl	NOUN
ejpam-3380	177	23	/	/	SYM
ejpam-3380	177	24	f(s	f(s	PROPN
ejpam-3380	177	25	,	,	PUNCT
ejpam-3380	177	26	t	t	PROPN
ejpam-3380	177	27	)	)	PUNCT
ejpam-3380	177	28	)	)	PUNCT
ejpam-3380	177	29	.	.	PUNCT
ejpam-3380	178	1	proof	proof	NOUN
ejpam-3380	178	2	.	.	PUNCT
ejpam-3380	179	1	we	we	PRON
ejpam-3380	179	2	have	have	VERB
ejpam-3380	179	3	to	to	PART
ejpam-3380	179	4	show	show	VERB
ejpam-3380	179	5	that	that	SCONJ
ejpam-3380	179	6	(	(	PUNCT
ejpam-3380	179	7	(	(	PUNCT
ejpam-3380	179	8	tl	tl	PROPN
ejpam-3380	179	9	/	/	SYM
ejpam-3380	179	10	f	f	PROPN
ejpam-3380	179	11	(	(	PUNCT
ejpam-3380	179	12	sl	sl	PROPN
ejpam-3380	179	13	/	/	SYM
ejpam-3380	179	14	f(s	f(s	PROPN
ejpam-3380	179	15	,	,	PUNCT
ejpam-3380	179	16	t	t	PROPN
ejpam-3380	179	17	)	)	PUNCT
ejpam-3380	179	18	,	,	PUNCT
ejpam-3380	179	19	s	s	PART
ejpam-3380	179	20	∗	∗	NOUN
ejpam-3380	179	21	t	t	NOUN
ejpam-3380	179	22	)	)	PUNCT
ejpam-3380	179	23	−1	−1	NOUN
ejpam-3380	179	24	)	)	PUNCT
ejpam-3380	179	25	∗	∗	NOUN
ejpam-3380	179	26	(	(	PUNCT
ejpam-3380	179	27	sl	sl	NOUN
ejpam-3380	179	28	/	/	SYM
ejpam-3380	179	29	f(s	f(s	PROPN
ejpam-3380	179	30	,	,	PUNCT
ejpam-3380	179	31	t	t	PROPN
ejpam-3380	179	32	)	)	PUNCT
ejpam-3380	179	33	)	)	PUNCT
ejpam-3380	179	34	)	)	PUNCT
ejpam-3380	180	1	∗	∗	NOUN
ejpam-3380	180	2	(	(	PUNCT
ejpam-3380	180	3	s	s	NOUN
ejpam-3380	180	4	∗	∗	X
ejpam-3380	180	5	t	t	NOUN
ejpam-3380	180	6	)	)	PUNCT
ejpam-3380	180	7	=	=	SYM
ejpam-3380	181	1	eg	eg	NOUN
ejpam-3380	181	2	.	.	PUNCT
ejpam-3380	182	1	(	(	PUNCT
ejpam-3380	182	2	5	5	NUM
ejpam-3380	182	3	)	)	PUNCT
ejpam-3380	182	4	to	to	PART
ejpam-3380	182	5	do	do	VERB
ejpam-3380	182	6	so	so	ADV
ejpam-3380	182	7	,	,	PUNCT
ejpam-3380	182	8	we	we	PRON
ejpam-3380	182	9	start	start	VERB
ejpam-3380	182	10	with	with	ADP
ejpam-3380	182	11	the	the	DET
ejpam-3380	182	12	left	left	ADJ
ejpam-3380	182	13	hand	hand	NOUN
ejpam-3380	182	14	side	side	NOUN
ejpam-3380	182	15	of	of	ADP
ejpam-3380	182	16	equation	equation	NOUN
ejpam-3380	182	17	(	(	PUNCT
ejpam-3380	182	18	5	5	NUM
ejpam-3380	182	19	)	)	PUNCT
ejpam-3380	182	20	as	as	SCONJ
ejpam-3380	182	21	follows	follow	VERB
ejpam-3380	182	22	:((	:((	PUNCT
ejpam-3380	183	1	tl	tl	PROPN
ejpam-3380	183	2	/	/	SYM
ejpam-3380	183	3	f	f	PROPN
ejpam-3380	183	4	(	(	PUNCT
ejpam-3380	183	5	sl	sl	PROPN
ejpam-3380	183	6	/	/	SYM
ejpam-3380	183	7	f(s	f(s	PROPN
ejpam-3380	183	8	,	,	PUNCT
ejpam-3380	183	9	t	t	PROPN
ejpam-3380	183	10	)	)	PUNCT
ejpam-3380	183	11	,	,	PUNCT
ejpam-3380	183	12	s	s	PART
ejpam-3380	183	13	∗	∗	NOUN
ejpam-3380	183	14	t	t	NOUN
ejpam-3380	183	15	)	)	PUNCT
ejpam-3380	183	16	−1	−1	NOUN
ejpam-3380	183	17	)	)	PUNCT
ejpam-3380	184	1	∗	∗	NOUN
ejpam-3380	184	2	(	(	PUNCT
ejpam-3380	184	3	sl	sl	NOUN
ejpam-3380	184	4	/	/	SYM
ejpam-3380	184	5	f(s	f(s	PROPN
ejpam-3380	184	6	,	,	PUNCT
ejpam-3380	184	7	t	t	PROPN
ejpam-3380	184	8	)	)	PUNCT
ejpam-3380	184	9	)	)	PUNCT
ejpam-3380	184	10	)	)	PUNCT
ejpam-3380	185	1	∗	∗	NOUN
ejpam-3380	185	2	(	(	PUNCT
ejpam-3380	185	3	s	s	NOUN
ejpam-3380	185	4	∗	∗	X
ejpam-3380	185	5	t	t	PROPN
ejpam-3380	185	6	)	)	PUNCT
ejpam-3380	185	7	=(	=(	NOUN
ejpam-3380	185	8	(	(	PUNCT
ejpam-3380	185	9	tl	tl	PROPN
ejpam-3380	185	10	/	/	SYM
ejpam-3380	185	11	f	f	PROPN
ejpam-3380	185	12	(	(	PUNCT
ejpam-3380	185	13	sl	sl	PROPN
ejpam-3380	185	14	/	/	SYM
ejpam-3380	185	15	f(s	f(s	PROPN
ejpam-3380	185	16	,	,	PUNCT
ejpam-3380	185	17	t	t	PROPN
ejpam-3380	185	18	)	)	PUNCT
ejpam-3380	185	19	,	,	PUNCT
ejpam-3380	185	20	s	s	PART
ejpam-3380	185	21	∗	∗	NOUN
ejpam-3380	185	22	t	t	NOUN
ejpam-3380	185	23	)	)	PUNCT
ejpam-3380	185	24	−1	−1	NOUN
ejpam-3380	185	25	)	)	PUNCT
ejpam-3380	185	26	/	/	SYM
ejpam-3380	186	1	f	f	PROPN
ejpam-3380	186	2	(	(	PUNCT
ejpam-3380	186	3	sl	sl	PROPN
ejpam-3380	186	4	/	/	SYM
ejpam-3380	186	5	f(s	f(s	PROPN
ejpam-3380	186	6	,	,	PUNCT
ejpam-3380	186	7	t	t	PROPN
ejpam-3380	186	8	)	)	PUNCT
ejpam-3380	186	9	,	,	PUNCT
ejpam-3380	186	10	s	s	NOUN
ejpam-3380	186	11	∗	∗	NOUN
ejpam-3380	186	12	t	t	NOUN
ejpam-3380	186	13	)	)	PUNCT
ejpam-3380	186	14	)	)	PUNCT
ejpam-3380	187	1	∗	∗	NOUN
ejpam-3380	187	2	(	(	PUNCT
ejpam-3380	187	3	(	(	PUNCT
ejpam-3380	187	4	sl	sl	INTJ
ejpam-3380	187	5	/	/	SYM
ejpam-3380	187	6	f(s	f(s	PROPN
ejpam-3380	187	7	,	,	PUNCT
ejpam-3380	187	8	t	t	PROPN
ejpam-3380	187	9	)	)	PUNCT
ejpam-3380	187	10	)	)	PUNCT
ejpam-3380	187	11	∗	∗	NOUN
ejpam-3380	187	12	(	(	PUNCT
ejpam-3380	187	13	s	s	NOUN
ejpam-3380	187	14	∗	∗	X
ejpam-3380	187	15	t	t	PROPN
ejpam-3380	187	16	)	)	PUNCT
ejpam-3380	187	17	)	)	PUNCT
ejpam-3380	188	1	=(	=(	NOUN
ejpam-3380	188	2	tl	tl	PROPN
ejpam-3380	188	3	/	/	SYM
ejpam-3380	188	4	f	f	PROPN
ejpam-3380	188	5	(	(	PUNCT
ejpam-3380	188	6	sl	sl	PROPN
ejpam-3380	188	7	/	/	SYM
ejpam-3380	188	8	f(s	f(s	PROPN
ejpam-3380	188	9	,	,	PUNCT
ejpam-3380	188	10	t	t	PROPN
ejpam-3380	188	11	)	)	PUNCT
ejpam-3380	188	12	,	,	PUNCT
ejpam-3380	188	13	s	s	PART
ejpam-3380	188	14	∗	∗	NOUN
ejpam-3380	188	15	t	t	NOUN
ejpam-3380	188	16	)	)	PUNCT
ejpam-3380	189	1	−1	−1	NOUN
ejpam-3380	189	2	f	f	PROPN
ejpam-3380	189	3	(	(	PUNCT
ejpam-3380	189	4	sl	sl	PROPN
ejpam-3380	189	5	/	/	SYM
ejpam-3380	189	6	f(s	f(s	PROPN
ejpam-3380	189	7	,	,	PUNCT
ejpam-3380	189	8	t	t	PROPN
ejpam-3380	189	9	)	)	PUNCT
ejpam-3380	189	10	,	,	PUNCT
ejpam-3380	189	11	s	s	NOUN
ejpam-3380	189	12	∗	∗	NOUN
ejpam-3380	189	13	t	t	NOUN
ejpam-3380	189	14	)	)	PUNCT
ejpam-3380	189	15	)	)	PUNCT
ejpam-3380	190	1	∗	∗	NOUN
ejpam-3380	190	2	(	(	PUNCT
ejpam-3380	190	3	(	(	PUNCT
ejpam-3380	190	4	sl	sl	INTJ
ejpam-3380	190	5	∗	∗	NOUN
ejpam-3380	190	6	s	s	PART
ejpam-3380	190	7	)	)	PUNCT
ejpam-3380	190	8	∗	∗	NOUN
ejpam-3380	190	9	t	t	NOUN
ejpam-3380	190	10	)	)	PUNCT
ejpam-3380	191	1	=	=	SYM
ejpam-3380	191	2	tl	tl	PROPN
ejpam-3380	191	3	∗	∗	X
ejpam-3380	191	4	t	t	NOUN
ejpam-3380	191	5	=	=	SYM
ejpam-3380	191	6	eg	eg	NOUN
ejpam-3380	191	7	.	.	PUNCT
ejpam-3380	192	1	the	the	DET
ejpam-3380	192	2	right	right	ADJ
ejpam-3380	192	3	division	division	NOUN
ejpam-3380	192	4	property	property	NOUN
ejpam-3380	192	5	yields	yield	NOUN
ejpam-3380	192	6	(	(	PUNCT
ejpam-3380	192	7	tl	tl	PROPN
ejpam-3380	192	8	/	/	SYM
ejpam-3380	192	9	f	f	PROPN
ejpam-3380	192	10	(	(	PUNCT
ejpam-3380	192	11	sl	sl	PROPN
ejpam-3380	192	12	/	/	SYM
ejpam-3380	192	13	f(s	f(s	PROPN
ejpam-3380	192	14	,	,	PUNCT
ejpam-3380	192	15	t	t	PROPN
ejpam-3380	192	16	)	)	PUNCT
ejpam-3380	192	17	,	,	PUNCT
ejpam-3380	192	18	s	s	PART
ejpam-3380	192	19	∗	∗	NOUN
ejpam-3380	192	20	t	t	NOUN
ejpam-3380	192	21	)	)	PUNCT
ejpam-3380	192	22	−1	−1	NOUN
ejpam-3380	192	23	)	)	PUNCT
ejpam-3380	193	1	∗	∗	NOUN
ejpam-3380	193	2	(	(	PUNCT
ejpam-3380	193	3	sl	sl	NOUN
ejpam-3380	193	4	/	/	SYM
ejpam-3380	193	5	f(s	f(s	PROPN
ejpam-3380	193	6	,	,	PUNCT
ejpam-3380	193	7	t	t	PROPN
ejpam-3380	193	8	)	)	PUNCT
ejpam-3380	193	9	)	)	PUNCT
ejpam-3380	194	1	=	=	PUNCT
ejpam-3380	194	2	(	(	PUNCT
ejpam-3380	194	3	s	s	NOUN
ejpam-3380	194	4	∗	∗	NOUN
ejpam-3380	194	5	t)l	t)l	NOUN
ejpam-3380	194	6	as	as	SCONJ
ejpam-3380	194	7	required	require	VERB
ejpam-3380	194	8	.	.	PUNCT
ejpam-3380	195	1	theorem	theorem	ADJ
ejpam-3380	195	2	6	6	NUM
ejpam-3380	195	3	.	.	PUNCT
ejpam-3380	196	1	let	let	AUX
ejpam-3380	196	2	r	r	PRON
ejpam-3380	196	3	be	be	AUX
ejpam-3380	196	4	a	a	DET
ejpam-3380	196	5	g	g	NOUN
ejpam-3380	196	6	-	-	PUNCT
ejpam-3380	196	7	weak	weak	ADJ
ejpam-3380	196	8	graded	grade	VERB
ejpam-3380	196	9	ring	ring	NOUN
ejpam-3380	196	10	and	and	CCONJ
ejpam-3380	196	11	k	k	PROPN
ejpam-3380	196	12	be	be	AUX
ejpam-3380	196	13	a	a	DET
ejpam-3380	196	14	multiplicatively	multiplicatively	ADV
ejpam-3380	196	15	closed	close	VERB
ejpam-3380	196	16	set	set	NOUN
ejpam-3380	196	17	of	of	ADP
ejpam-3380	196	18	g	g	NOUN
ejpam-3380	196	19	-	-	PUNCT
ejpam-3380	196	20	homogeneous	homogeneous	ADJ
ejpam-3380	196	21	elements	element	NOUN
ejpam-3380	196	22	not	not	PART
ejpam-3380	196	23	containing	contain	VERB
ejpam-3380	196	24	0	0	NUM
ejpam-3380	196	25	.	.	PUNCT
ejpam-3380	197	1	then	then	ADV
ejpam-3380	197	2	the	the	DET
ejpam-3380	197	3	localization	localization	NOUN
ejpam-3380	197	4	rk	rk	NOUN
ejpam-3380	197	5	can	can	AUX
ejpam-3380	197	6	be	be	AUX
ejpam-3380	197	7	written	write	VERB
ejpam-3380	197	8	as	as	ADP
ejpam-3380	197	9	the	the	DET
ejpam-3380	197	10	direct	direct	ADJ
ejpam-3380	197	11	sum	sum	NOUN
ejpam-3380	197	12	of	of	ADP
ejpam-3380	197	13	its	its	PRON
ejpam-3380	197	14	g	g	NOUN
ejpam-3380	197	15	-	-	PUNCT
ejpam-3380	197	16	components	component	NOUN
ejpam-3380	197	17	as	as	SCONJ
ejpam-3380	197	18	follows	follow	VERB
ejpam-3380	197	19	:	:	PUNCT
ejpam-3380	197	20	rk	rk	PROPN
ejpam-3380	197	21	=	=	PROPN
ejpam-3380	197	22	⊕	⊕	PROPN
ejpam-3380	197	23	s∈g	s∈g	NOUN
ejpam-3380	197	24	(	(	PUNCT
ejpam-3380	197	25	rk)s	rk)s	PROPN
ejpam-3380	197	26	,	,	PUNCT
ejpam-3380	197	27	(	(	PUNCT
ejpam-3380	197	28	as	as	ADP
ejpam-3380	197	29	additive	additive	ADJ
ejpam-3380	197	30	subgroups	subgroup	NOUN
ejpam-3380	197	31	)	)	PUNCT
ejpam-3380	197	32	with	with	ADP
ejpam-3380	197	33	,	,	PUNCT
ejpam-3380	197	34	rk	rk	NOUN
ejpam-3380	197	35	=	=	PUNCT
ejpam-3380	197	36	{	{	PUNCT
ejpam-3380	197	37	r	r	NOUN
ejpam-3380	197	38	k	k	NOUN
ejpam-3380	197	39	:	:	PUNCT
ejpam-3380	197	40	r	r	NOUN
ejpam-3380	197	41	∈	∈	PROPN
ejpam-3380	197	42	r	r	NOUN
ejpam-3380	197	43	,	,	PUNCT
ejpam-3380	197	44	k	k	PROPN
ejpam-3380	197	45	∈	∈	PROPN
ejpam-3380	197	46	k	k	X
ejpam-3380	197	47	}	}	PUNCT
ejpam-3380	197	48	and	and	CCONJ
ejpam-3380	197	49	(	(	PUNCT
ejpam-3380	197	50	rk)s	rk)s	NOUN
ejpam-3380	197	51	=	=	PUNCT
ejpam-3380	197	52	{	{	PUNCT
ejpam-3380	197	53	r	r	NOUN
ejpam-3380	197	54	k	k	PROPN
ejpam-3380	197	55	∈	∈	PROPN
ejpam-3380	197	56	rk	rk	NOUN
ejpam-3380	197	57	:	:	PUNCT
ejpam-3380	198	1	r	r	NOUN
ejpam-3380	198	2	and	and	CCONJ
ejpam-3380	198	3	k	k	PROPN
ejpam-3380	198	4	are	be	AUX
ejpam-3380	198	5	g	g	NOUN
ejpam-3380	198	6	-	-	PUNCT
ejpam-3380	198	7	homogeneous	homogeneous	ADJ
ejpam-3380	198	8	and	and	CCONJ
ejpam-3380	198	9	〈	〈	PROPN
ejpam-3380	198	10	k〉l	k〉l	PROPN
ejpam-3380	198	11	∗	∗	NOUN
ejpam-3380	198	12	〈	〈	PROPN
ejpam-3380	198	13	r	r	PROPN
ejpam-3380	198	14	〉	〉	NOUN
ejpam-3380	198	15	=	=	SYM
ejpam-3380	198	16	s	s	PART
ejpam-3380	198	17	}	}	PUNCT
ejpam-3380	198	18	such	such	ADJ
ejpam-3380	198	19	that	that	SCONJ
ejpam-3380	198	20	f(s	f(s	PROPN
ejpam-3380	198	21	,	,	PUNCT
ejpam-3380	198	22	t	t	PROPN
ejpam-3380	198	23	)	)	PUNCT
ejpam-3380	198	24	=	=	PUNCT
ejpam-3380	199	1	f(s	f(s	PROPN
ejpam-3380	199	2	,	,	PUNCT
ejpam-3380	199	3	p	p	X
ejpam-3380	199	4	)	)	PUNCT
ejpam-3380	199	5	for	for	ADP
ejpam-3380	199	6	all	all	DET
ejpam-3380	199	7	s	s	PROPN
ejpam-3380	199	8	,	,	PUNCT
ejpam-3380	199	9	t	t	PROPN
ejpam-3380	199	10	,	,	PUNCT
ejpam-3380	199	11	p	p	PROPN
ejpam-3380	199	12	∈	∈	PROPN
ejpam-3380	199	13	g	g	NOUN
ejpam-3380	199	14	,	,	PUNCT
ejpam-3380	199	15	where	where	SCONJ
ejpam-3380	199	16	〈	〈	PROPN
ejpam-3380	199	17	k	k	PROPN
ejpam-3380	199	18	〉	〉	PROPN
ejpam-3380	199	19	,	,	PUNCT
ejpam-3380	199	20	〈	〈	PROPN
ejpam-3380	199	21	r	r	PROPN
ejpam-3380	199	22	〉	〉	PROPN
ejpam-3380	199	23	are	be	AUX
ejpam-3380	199	24	the	the	DET
ejpam-3380	199	25	g	g	NOUN
ejpam-3380	199	26	-	-	PUNCT
ejpam-3380	199	27	grades	grade	NOUN
ejpam-3380	199	28	of	of	ADP
ejpam-3380	199	29	k	k	PROPN
ejpam-3380	199	30	and	and	CCONJ
ejpam-3380	199	31	r	r	NOUN
ejpam-3380	199	32	respectively	respectively	ADV
ejpam-3380	199	33	.	.	PUNCT
ejpam-3380	200	1	proof	proof	NOUN
ejpam-3380	200	2	.	.	PUNCT
ejpam-3380	201	1	we	we	PRON
ejpam-3380	201	2	begin	begin	VERB
ejpam-3380	201	3	the	the	DET
ejpam-3380	201	4	proof	proof	NOUN
ejpam-3380	201	5	by	by	ADP
ejpam-3380	201	6	showing	show	VERB
ejpam-3380	201	7	that	that	SCONJ
ejpam-3380	201	8	(	(	PUNCT
ejpam-3380	201	9	rk)s	rk)s	PROPN
ejpam-3380	201	10	is	be	AUX
ejpam-3380	201	11	an	an	DET
ejpam-3380	201	12	additive	additive	ADJ
ejpam-3380	201	13	subgroup	subgroup	NOUN
ejpam-3380	201	14	of	of	ADP
ejpam-3380	201	15	rk	rk	PROPN
ejpam-3380	201	16	as	as	SCONJ
ejpam-3380	201	17	follows	follow	VERB
ejpam-3380	201	18	:	:	PUNCT
ejpam-3380	201	19	if	if	SCONJ
ejpam-3380	201	20	r	r	PROPN
ejpam-3380	201	21	k	k	X
ejpam-3380	201	22	,	,	PUNCT
ejpam-3380	201	23	r′	r′	PROPN
ejpam-3380	201	24	k′	k′	PROPN
ejpam-3380	201	25	∈	∈	PROPN
ejpam-3380	201	26	(	(	PUNCT
ejpam-3380	201	27	rk)s	rk)s	PROPN
ejpam-3380	201	28	,	,	PUNCT
ejpam-3380	201	29	then	then	ADV
ejpam-3380	201	30	〈	〈	PROPN
ejpam-3380	201	31	k〉l	k〉l	PROPN
ejpam-3380	201	32	∗	∗	NOUN
ejpam-3380	201	33	〈	〈	PROPN
ejpam-3380	201	34	r	r	NOUN
ejpam-3380	201	35	〉	〉	NOUN
ejpam-3380	201	36	=	=	SYM
ejpam-3380	201	37	s	s	NOUN
ejpam-3380	201	38	=	=	PUNCT
ejpam-3380	201	39	〈	〈	PROPN
ejpam-3380	201	40	k′〉l	k′〉l	PROPN
ejpam-3380	201	41	∗	∗	NOUN
ejpam-3380	201	42	〈	〈	PROPN
ejpam-3380	201	43	r′	r′	PROPN
ejpam-3380	201	44	〉	〉	PROPN
ejpam-3380	201	45	.	.	PUNCT
ejpam-3380	202	1	hence	hence	ADV
ejpam-3380	202	2	,	,	PUNCT
ejpam-3380	202	3	r	r	PROPN
ejpam-3380	202	4	k	k	PROPN
ejpam-3380	202	5	+	+	CCONJ
ejpam-3380	202	6	r′	r′	PROPN
ejpam-3380	202	7	k′	k′	PROPN
ejpam-3380	202	8	=	=	SYM
ejpam-3380	202	9	k′r+kr′	k′r+kr′	PROPN
ejpam-3380	202	10	kk′	kk′	PROPN
ejpam-3380	202	11	.	.	PUNCT
ejpam-3380	203	1	najla	najla	PROPN
ejpam-3380	203	2	al	al	PROPN
ejpam-3380	203	3	-	-	PUNCT
ejpam-3380	203	4	subaie	subaie	NOUN
ejpam-3380	203	5	,	,	PUNCT
ejpam-3380	203	6	m.	m.	NOUN
ejpam-3380	203	7	m.	m.	PROPN
ejpam-3380	203	8	al	al	PROPN
ejpam-3380	203	9	-	-	PUNCT
ejpam-3380	203	10	shomrani	shomrani	PROPN
ejpam-3380	203	11	/	/	SYM
ejpam-3380	203	12	eur	eur	NOUN
ejpam-3380	203	13	.	.	PUNCT
ejpam-3380	204	1	j.	j.	PROPN
ejpam-3380	204	2	pure	pure	PROPN
ejpam-3380	204	3	appl	appl	PROPN
ejpam-3380	204	4	.	.	PROPN
ejpam-3380	204	5	math	math	PROPN
ejpam-3380	204	6	,	,	PUNCT
ejpam-3380	204	7	12	12	NUM
ejpam-3380	204	8	(	(	PUNCT
ejpam-3380	204	9	2	2	NUM
ejpam-3380	204	10	)	)	PUNCT
ejpam-3380	204	11	(	(	PUNCT
ejpam-3380	204	12	2019	2019	NUM
ejpam-3380	204	13	)	)	PUNCT
ejpam-3380	204	14	,	,	PUNCT
ejpam-3380	204	15	332	332	NUM
ejpam-3380	204	16	-	-	SYM
ejpam-3380	204	17	347	347	NUM
ejpam-3380	204	18	340	340	NUM
ejpam-3380	204	19	consequently	consequently	ADV
ejpam-3380	204	20	,	,	PUNCT
ejpam-3380	204	21	〈	〈	PROPN
ejpam-3380	204	22	k	k	X
ejpam-3380	205	1	′r	′r	PROPN
ejpam-3380	205	2	+	+	NUM
ejpam-3380	205	3	kr′	kr′	PROPN
ejpam-3380	205	4	kk′	kk′	PROPN
ejpam-3380	205	5	〉	〉	NUM
ejpam-3380	205	6	=	=	SYM
ejpam-3380	205	7	〈	〈	PROPN
ejpam-3380	205	8	kk′〉l	kk′〉l	PROPN
ejpam-3380	205	9	∗	∗	NOUN
ejpam-3380	205	10	〈	〈	NOUN
ejpam-3380	205	11	kr′	kr′	PROPN
ejpam-3380	205	12	〉	〉	NOUN
ejpam-3380	205	13	=	=	SYM
ejpam-3380	205	14	(	(	PUNCT
ejpam-3380	205	15	〈	〈	PROPN
ejpam-3380	205	16	k	k	PROPN
ejpam-3380	205	17	〉	〉	PROPN
ejpam-3380	205	18	∗	∗	NOUN
ejpam-3380	205	19	〈	〈	PROPN
ejpam-3380	205	20	k′	k′	PROPN
ejpam-3380	205	21	〉	〉	PROPN
ejpam-3380	205	22	)	)	PUNCT
ejpam-3380	205	23	l	l	NOUN
ejpam-3380	205	24	∗	∗	NOUN
ejpam-3380	205	25	(	(	PUNCT
ejpam-3380	205	26	〈	〈	PROPN
ejpam-3380	205	27	k	k	PROPN
ejpam-3380	205	28	〉	〉	PROPN
ejpam-3380	205	29	∗	∗	NOUN
ejpam-3380	205	30	〈	〈	PROPN
ejpam-3380	205	31	r′	r′	PROPN
ejpam-3380	205	32	〉	〉	NUM
ejpam-3380	205	33	)	)	PUNCT
ejpam-3380	205	34	=	=	PUNCT
ejpam-3380	205	35	(	(	PUNCT
ejpam-3380	205	36	(	(	PUNCT
ejpam-3380	205	37	〈	〈	PROPN
ejpam-3380	205	38	k′〉l	k′〉l	PROPN
ejpam-3380	205	39	/	/	SYM
ejpam-3380	205	40	f	f	PROPN
ejpam-3380	205	41	(	(	PUNCT
ejpam-3380	205	42	〈	〈	PROPN
ejpam-3380	205	43	k〉l	k〉l	PROPN
ejpam-3380	205	44	/	/	SYM
ejpam-3380	205	45	f(〈k	f(〈k	NOUN
ejpam-3380	205	46	〉	〉	PROPN
ejpam-3380	205	47	,	,	PUNCT
ejpam-3380	205	48	〈	〈	PROPN
ejpam-3380	205	49	k′	k′	PROPN
ejpam-3380	205	50	〉	〉	PROPN
ejpam-3380	205	51	)	)	PUNCT
ejpam-3380	205	52	,	,	PUNCT
ejpam-3380	205	53	〈	〈	PROPN
ejpam-3380	205	54	k	k	PROPN
ejpam-3380	205	55	〉	〉	PROPN
ejpam-3380	205	56	∗	∗	NOUN
ejpam-3380	205	57	〈	〈	PROPN
ejpam-3380	205	58	k′	k′	PROPN
ejpam-3380	205	59	〉	〉	PROPN
ejpam-3380	205	60	)	)	PUNCT
ejpam-3380	205	61	−1	−1	NOUN
ejpam-3380	205	62	)	)	PUNCT
ejpam-3380	205	63	∗	∗	NOUN
ejpam-3380	205	64	(	(	PUNCT
ejpam-3380	205	65	〈	〈	PROPN
ejpam-3380	205	66	k〉l	k〉l	PROPN
ejpam-3380	205	67	/	/	SYM
ejpam-3380	205	68	f(〈k	f(〈k	NOUN
ejpam-3380	205	69	〉	〉	PROPN
ejpam-3380	205	70	,	,	PUNCT
ejpam-3380	205	71	〈	〈	PROPN
ejpam-3380	205	72	k′	k′	PROPN
ejpam-3380	205	73	〉	〉	NUM
ejpam-3380	205	74	)	)	PUNCT
ejpam-3380	205	75	)	)	PUNCT
ejpam-3380	205	76	)	)	PUNCT
ejpam-3380	206	1	∗	∗	NOUN
ejpam-3380	206	2	(	(	PUNCT
ejpam-3380	206	3	〈	〈	PROPN
ejpam-3380	206	4	k	k	PROPN
ejpam-3380	206	5	〉	〉	PROPN
ejpam-3380	206	6	∗	∗	NOUN
ejpam-3380	206	7	〈	〈	PROPN
ejpam-3380	206	8	r′	r′	PROPN
ejpam-3380	206	9	〉	〉	NOUN
ejpam-3380	206	10	)	)	PUNCT
ejpam-3380	206	11	=	=	PUNCT
ejpam-3380	207	1	(	(	PUNCT
ejpam-3380	207	2	(	(	PUNCT
ejpam-3380	207	3	〈	〈	PROPN
ejpam-3380	207	4	k′〉l	k′〉l	PROPN
ejpam-3380	207	5	/	/	SYM
ejpam-3380	207	6	f	f	PROPN
ejpam-3380	207	7	(	(	PUNCT
ejpam-3380	207	8	〈	〈	PROPN
ejpam-3380	207	9	k〉l	k〉l	PROPN
ejpam-3380	207	10	/	/	SYM
ejpam-3380	207	11	f(〈k	f(〈k	NOUN
ejpam-3380	207	12	〉	〉	PROPN
ejpam-3380	207	13	,	,	PUNCT
ejpam-3380	207	14	〈	〈	PROPN
ejpam-3380	207	15	k′	k′	PROPN
ejpam-3380	207	16	〉	〉	PROPN
ejpam-3380	207	17	)	)	PUNCT
ejpam-3380	207	18	,	,	PUNCT
ejpam-3380	207	19	〈	〈	PROPN
ejpam-3380	207	20	k	k	PROPN
ejpam-3380	207	21	〉	〉	PROPN
ejpam-3380	207	22	∗	∗	NOUN
ejpam-3380	207	23	〈	〈	PROPN
ejpam-3380	207	24	k′	k′	PROPN
ejpam-3380	207	25	〉	〉	PROPN
ejpam-3380	207	26	)	)	PUNCT
ejpam-3380	207	27	−1	−1	NOUN
ejpam-3380	207	28	)	)	PUNCT
ejpam-3380	207	29	/	/	SYM
ejpam-3380	208	1	f	f	PROPN
ejpam-3380	208	2	(	(	PUNCT
ejpam-3380	208	3	〈	〈	PROPN
ejpam-3380	208	4	k〉l	k〉l	PROPN
ejpam-3380	208	5	/	/	SYM
ejpam-3380	208	6	f(〈k	f(〈k	NOUN
ejpam-3380	208	7	〉	〉	PROPN
ejpam-3380	208	8	,	,	PUNCT
ejpam-3380	208	9	〈	〈	PROPN
ejpam-3380	208	10	k′	k′	PROPN
ejpam-3380	208	11	〉	〉	PROPN
ejpam-3380	208	12	)	)	PUNCT
ejpam-3380	208	13	,	,	PUNCT
ejpam-3380	208	14	〈	〈	PROPN
ejpam-3380	208	15	k	k	PROPN
ejpam-3380	208	16	〉	〉	PROPN
ejpam-3380	208	17	∗	∗	NOUN
ejpam-3380	208	18	〈	〈	PROPN
ejpam-3380	208	19	r′	r′	PROPN
ejpam-3380	208	20	〉	〉	NOUN
ejpam-3380	208	21	)	)	PUNCT
ejpam-3380	208	22	)	)	PUNCT
ejpam-3380	209	1	∗	∗	NOUN
ejpam-3380	209	2	(	(	PUNCT
ejpam-3380	209	3	〈	〈	PROPN
ejpam-3380	209	4	k〉l	k〉l	PROPN
ejpam-3380	209	5	/	/	SYM
ejpam-3380	209	6	f(〈k	f(〈k	NOUN
ejpam-3380	209	7	〉	〉	PROPN
ejpam-3380	209	8	,	,	PUNCT
ejpam-3380	209	9	〈	〈	PROPN
ejpam-3380	209	10	k′	k′	PROPN
ejpam-3380	209	11	〉	〉	PROPN
ejpam-3380	209	12	)	)	PUNCT
ejpam-3380	209	13	∗	∗	NOUN
ejpam-3380	209	14	(	(	PUNCT
ejpam-3380	209	15	〈	〈	PROPN
ejpam-3380	209	16	k	k	PROPN
ejpam-3380	209	17	〉	〉	PROPN
ejpam-3380	209	18	∗	∗	NOUN
ejpam-3380	209	19	〈	〈	PROPN
ejpam-3380	209	20	r′	r′	PROPN
ejpam-3380	209	21	〉	〉	NOUN
ejpam-3380	209	22	)	)	PUNCT
ejpam-3380	209	23	)	)	PUNCT
ejpam-3380	210	1	=	=	PRON
ejpam-3380	210	2	(	(	PUNCT
ejpam-3380	210	3	〈	〈	PROPN
ejpam-3380	210	4	k′〉l	k′〉l	PROPN
ejpam-3380	210	5	/	/	SYM
ejpam-3380	210	6	f	f	PROPN
ejpam-3380	210	7	(	(	PUNCT
ejpam-3380	210	8	〈	〈	PROPN
ejpam-3380	210	9	k〉l	k〉l	PROPN
ejpam-3380	210	10	/	/	SYM
ejpam-3380	210	11	f(〈k	f(〈k	NOUN
ejpam-3380	210	12	〉	〉	PROPN
ejpam-3380	210	13	,	,	PUNCT
ejpam-3380	210	14	〈	〈	PROPN
ejpam-3380	210	15	k′	k′	PROPN
ejpam-3380	210	16	〉	〉	PROPN
ejpam-3380	210	17	)	)	PUNCT
ejpam-3380	210	18	,	,	PUNCT
ejpam-3380	210	19	〈	〈	PROPN
ejpam-3380	210	20	k	k	PROPN
ejpam-3380	210	21	〉	〉	PROPN
ejpam-3380	210	22	∗	∗	NOUN
ejpam-3380	210	23	〈	〈	PROPN
ejpam-3380	210	24	k′	k′	PROPN
ejpam-3380	210	25	〉	〉	PROPN
ejpam-3380	210	26	)	)	PUNCT
ejpam-3380	210	27	−1	−1	NOUN
ejpam-3380	211	1	f	f	NOUN
ejpam-3380	211	2	(	(	PUNCT
ejpam-3380	211	3	〈	〈	PROPN
ejpam-3380	211	4	k〉l	k〉l	PROPN
ejpam-3380	211	5	/	/	SYM
ejpam-3380	211	6	f(〈k	f(〈k	NOUN
ejpam-3380	211	7	〉	〉	PROPN
ejpam-3380	211	8	,	,	PUNCT
ejpam-3380	211	9	〈	〈	PROPN
ejpam-3380	211	10	k′	k′	PROPN
ejpam-3380	211	11	〉	〉	PROPN
ejpam-3380	211	12	)	)	PUNCT
ejpam-3380	211	13	,	,	PUNCT
ejpam-3380	211	14	〈	〈	PROPN
ejpam-3380	211	15	k	k	PROPN
ejpam-3380	211	16	〉	〉	PROPN
ejpam-3380	211	17	∗	∗	NOUN
ejpam-3380	211	18	〈	〈	PROPN
ejpam-3380	211	19	r′	r′	PROPN
ejpam-3380	211	20	〉	〉	NOUN
ejpam-3380	211	21	)	)	PUNCT
ejpam-3380	211	22	∗	∗	NOUN
ejpam-3380	211	23	(	(	PUNCT
ejpam-3380	211	24	(	(	PUNCT
ejpam-3380	211	25	〈	〈	PROPN
ejpam-3380	211	26	k〉l	k〉l	PROPN
ejpam-3380	211	27	∗	∗	PROPN
ejpam-3380	211	28	〈	〈	PROPN
ejpam-3380	211	29	k	k	PROPN
ejpam-3380	211	30	〉	〉	PROPN
ejpam-3380	211	31	)	)	PUNCT
ejpam-3380	211	32	∗	∗	NOUN
ejpam-3380	211	33	〈	〈	PROPN
ejpam-3380	211	34	r′	r′	PROPN
ejpam-3380	211	35	〉	〉	NOUN
ejpam-3380	211	36	)	)	PUNCT
ejpam-3380	211	37	)	)	PUNCT
ejpam-3380	212	1	=	=	PUNCT
ejpam-3380	213	1	〈	〈	PROPN
ejpam-3380	213	2	k′〉l	k′〉l	PROPN
ejpam-3380	213	3	∗	∗	NOUN
ejpam-3380	213	4	(	(	PUNCT
ejpam-3380	213	5	eg	eg	NOUN
ejpam-3380	213	6	∗	∗	PROPN
ejpam-3380	213	7	〈	〈	PROPN
ejpam-3380	213	8	r′	r′	PROPN
ejpam-3380	213	9	〉	〉	NOUN
ejpam-3380	213	10	)	)	PUNCT
ejpam-3380	213	11	=	=	PUNCT
ejpam-3380	214	1	〈	〈	PROPN
ejpam-3380	214	2	k′〉l	k′〉l	PROPN
ejpam-3380	214	3	∗	∗	NOUN
ejpam-3380	214	4	〈	〈	PROPN
ejpam-3380	214	5	r′	r′	PROPN
ejpam-3380	214	6	〉	〉	PROPN
ejpam-3380	214	7	=	=	SYM
ejpam-3380	214	8	s.	s.	PROPN
ejpam-3380	214	9	also	also	ADV
ejpam-3380	214	10	,	,	PUNCT
ejpam-3380	214	11	if	if	SCONJ
ejpam-3380	214	12	r	r	NOUN
ejpam-3380	214	13	k	k	PROPN
ejpam-3380	214	14	∈	∈	PROPN
ejpam-3380	214	15	(	(	PUNCT
ejpam-3380	214	16	rk)s	rk)s	PROPN
ejpam-3380	214	17	,	,	PUNCT
ejpam-3380	214	18	then	then	ADV
ejpam-3380	214	19	−	−	PROPN
ejpam-3380	214	20	(	(	PUNCT
ejpam-3380	214	21	r	r	NOUN
ejpam-3380	214	22	k	k	PROPN
ejpam-3380	214	23	)	)	PUNCT
ejpam-3380	215	1	=	=	SYM
ejpam-3380	215	2	−r	−r	PROPN
ejpam-3380	215	3	k	k	PROPN
ejpam-3380	215	4	∈	∈	PROPN
ejpam-3380	215	5	(	(	PUNCT
ejpam-3380	215	6	rk)s	rk)s	PROPN
ejpam-3380	215	7	.	.	PUNCT
ejpam-3380	216	1	indeed	indeed	ADV
ejpam-3380	216	2	,	,	PUNCT
ejpam-3380	216	3	if	if	SCONJ
ejpam-3380	216	4	r	r	NOUN
ejpam-3380	216	5	k	k	PROPN
ejpam-3380	216	6	∈	∈	PROPN
ejpam-3380	216	7	(	(	PUNCT
ejpam-3380	216	8	rk)s	rk)s	PROPN
ejpam-3380	216	9	,	,	PUNCT
ejpam-3380	216	10	then	then	ADV
ejpam-3380	216	11	〈	〈	PROPN
ejpam-3380	216	12	k〉l	k〉l	PROPN
ejpam-3380	216	13	∗	∗	NOUN
ejpam-3380	216	14	〈	〈	PROPN
ejpam-3380	216	15	r	r	NOUN
ejpam-3380	216	16	〉	〉	NOUN
ejpam-3380	216	17	=	=	SYM
ejpam-3380	216	18	s	s	NOUN
ejpam-3380	216	19	=	=	PUNCT
ejpam-3380	216	20	〈	〈	PROPN
ejpam-3380	216	21	k〉l	k〉l	PROPN
ejpam-3380	216	22	∗	∗	PROPN
ejpam-3380	216	23	〈	〈	PROPN
ejpam-3380	216	24	−r	−r	NOUN
ejpam-3380	216	25	〉	〉	PROPN
ejpam-3380	216	26	as	as	SCONJ
ejpam-3380	216	27	r	r	NOUN
ejpam-3380	216	28	is	be	AUX
ejpam-3380	216	29	a	a	DET
ejpam-3380	216	30	g	g	NOUN
ejpam-3380	216	31	-	-	PUNCT
ejpam-3380	216	32	homogeneous	homogeneous	ADJ
ejpam-3380	216	33	element	element	NOUN
ejpam-3380	216	34	,	,	PUNCT
ejpam-3380	216	35	i.e.	i.e.	X
ejpam-3380	216	36	r	r	NOUN
ejpam-3380	216	37	is	be	AUX
ejpam-3380	216	38	contained	contain	VERB
ejpam-3380	216	39	in	in	ADP
ejpam-3380	216	40	fixed	fix	VERB
ejpam-3380	216	41	component	component	NOUN
ejpam-3380	216	42	and	and	CCONJ
ejpam-3380	216	43	since	since	SCONJ
ejpam-3380	216	44	each	each	DET
ejpam-3380	216	45	component	component	NOUN
ejpam-3380	216	46	are	be	AUX
ejpam-3380	216	47	additive	additive	ADJ
ejpam-3380	216	48	subgroup	subgroup	NOUN
ejpam-3380	216	49	of	of	ADP
ejpam-3380	216	50	r	r	NOUN
ejpam-3380	216	51	yields	yield	NOUN
ejpam-3380	216	52	−r	−r	PROPN
ejpam-3380	216	53	has	have	VERB
ejpam-3380	216	54	the	the	DET
ejpam-3380	216	55	same	same	ADJ
ejpam-3380	216	56	g	g	NOUN
ejpam-3380	216	57	-	-	PUNCT
ejpam-3380	216	58	grade	grade	NOUN
ejpam-3380	216	59	.	.	PUNCT
ejpam-3380	217	1	hence	hence	ADV
ejpam-3380	217	2	,	,	PUNCT
ejpam-3380	217	3	(	(	PUNCT
ejpam-3380	217	4	rk)s	rk)s	PROPN
ejpam-3380	217	5	is	be	AUX
ejpam-3380	217	6	an	an	DET
ejpam-3380	217	7	additive	additive	ADJ
ejpam-3380	217	8	subgroup	subgroup	NOUN
ejpam-3380	217	9	of	of	ADP
ejpam-3380	217	10	rk	rk	NOUN
ejpam-3380	217	11	for	for	ADP
ejpam-3380	217	12	all	all	DET
ejpam-3380	217	13	s	s	PROPN
ejpam-3380	217	14	∈	∈	PROPN
ejpam-3380	217	15	g.	g.	NOUN
ejpam-3380	217	16	next	next	ADV
ejpam-3380	217	17	,	,	PUNCT
ejpam-3380	217	18	we	we	PRON
ejpam-3380	217	19	have	have	VERB
ejpam-3380	217	20	to	to	PART
ejpam-3380	217	21	show	show	VERB
ejpam-3380	217	22	that	that	SCONJ
ejpam-3380	217	23	rk	rk	NOUN
ejpam-3380	217	24	=	=	SYM
ejpam-3380	217	25	⊕	⊕	PROPN
ejpam-3380	217	26	s∈g(rk)s	s∈g(rk)s	PROPN
ejpam-3380	217	27	.	.	PUNCT
ejpam-3380	218	1	it	it	PRON
ejpam-3380	218	2	is	be	AUX
ejpam-3380	218	3	obvious	obvious	ADJ
ejpam-3380	218	4	that	that	SCONJ
ejpam-3380	218	5	rk	rk	NOUN
ejpam-3380	218	6	=	=	SYM
ejpam-3380	218	7	∑	∑	PROPN
ejpam-3380	218	8	s∈g(rk)s	s∈g(rk)s	PROPN
ejpam-3380	218	9	.	.	PUNCT
ejpam-3380	219	1	so	so	ADV
ejpam-3380	219	2	,	,	PUNCT
ejpam-3380	219	3	let	let	VERB
ejpam-3380	219	4	r	r	NOUN
ejpam-3380	219	5	k	k	X
ejpam-3380	219	6	∈	∈	PROPN
ejpam-3380	219	7	(	(	PUNCT
ejpam-3380	219	8	rk)sj	rk)sj	PROPN
ejpam-3380	219	9	⋂	⋂	PROPN
ejpam-3380	219	10	{	{	PUNCT
ejpam-3380	219	11	(	(	PUNCT
ejpam-3380	219	12	rk)s1	rk)s1	PROPN
ejpam-3380	219	13	+	+	X
ejpam-3380	219	14	.	.	PUNCT
ejpam-3380	219	15	.	.	PUNCT
ejpam-3380	219	16	.	.	PUNCT
ejpam-3380	220	1	+	+	CCONJ
ejpam-3380	220	2	(	(	PUNCT
ejpam-3380	220	3	rk)sj−1	rk)sj−1	NOUN
ejpam-3380	220	4	+	+	NUM
ejpam-3380	220	5	(	(	PUNCT
ejpam-3380	220	6	rk)sj+1	rk)sj+1	NOUN
ejpam-3380	220	7	+	+	CCONJ
ejpam-3380	220	8	.	.	PUNCT
ejpam-3380	220	9	.	.	PUNCT
ejpam-3380	220	10	.	.	PUNCT
ejpam-3380	221	1	+	+	CCONJ
ejpam-3380	221	2	(	(	PUNCT
ejpam-3380	221	3	rk)sn	rk)sn	NOUN
ejpam-3380	221	4	}	}	PUNCT
ejpam-3380	221	5	for	for	ADP
ejpam-3380	221	6	all	all	DET
ejpam-3380	221	7	sj	sj	PROPN
ejpam-3380	221	8	6=	6=	PROPN
ejpam-3380	221	9	s1	s1	NOUN
ejpam-3380	221	10	,	,	PUNCT
ejpam-3380	221	11	.	.	PUNCT
ejpam-3380	221	12	.	.	PUNCT
ejpam-3380	221	13	.	.	PUNCT
ejpam-3380	222	1	,	,	PUNCT
ejpam-3380	222	2	sj−1	sj−1	NOUN
ejpam-3380	222	3	,	,	PUNCT
ejpam-3380	222	4	sj+1	sj+1	X
ejpam-3380	222	5	,	,	PUNCT
ejpam-3380	222	6	.	.	PUNCT
ejpam-3380	222	7	.	.	PUNCT
ejpam-3380	223	1	.	.	PUNCT
ejpam-3380	224	1	,	,	PUNCT
ejpam-3380	224	2	sn	sn	PROPN
ejpam-3380	224	3	.	.	PUNCT
ejpam-3380	225	1	thus	thus	ADV
ejpam-3380	225	2	,	,	PUNCT
ejpam-3380	225	3	〈	〈	PROPN
ejpam-3380	225	4	k〉l	k〉l	PROPN
ejpam-3380	225	5	∗	∗	NOUN
ejpam-3380	225	6	〈	〈	PROPN
ejpam-3380	225	7	r	r	NOUN
ejpam-3380	225	8	〉	〉	NOUN
ejpam-3380	225	9	=	=	SYM
ejpam-3380	225	10	sj	sj	NOUN
ejpam-3380	225	11	and	and	CCONJ
ejpam-3380	225	12	〈	〈	PROPN
ejpam-3380	225	13	k〉l	k〉l	PROPN
ejpam-3380	225	14	∗	∗	NOUN
ejpam-3380	225	15	〈	〈	PROPN
ejpam-3380	225	16	r	r	PROPN
ejpam-3380	225	17	〉	〉	NOUN
ejpam-3380	225	18	=	=	SYM
ejpam-3380	225	19	si	si	NOUN
ejpam-3380	225	20	,	,	PUNCT
ejpam-3380	225	21	where	where	SCONJ
ejpam-3380	225	22	i	i	PRON
ejpam-3380	225	23	=	=	NOUN
ejpam-3380	225	24	1	1	NUM
ejpam-3380	225	25	,	,	PUNCT
ejpam-3380	225	26	...	...	PUNCT
ejpam-3380	225	27	,	,	PUNCT
ejpam-3380	225	28	j−1	j−1	PROPN
ejpam-3380	225	29	,	,	PUNCT
ejpam-3380	225	30	j+1	j+1	PROPN
ejpam-3380	225	31	,	,	PUNCT
ejpam-3380	225	32	...	...	PUNCT
ejpam-3380	225	33	,	,	PUNCT
ejpam-3380	225	34	n.	n.	NOUN
ejpam-3380	225	35	this	this	PRON
ejpam-3380	225	36	means	mean	VERB
ejpam-3380	225	37	that	that	SCONJ
ejpam-3380	225	38	,	,	PUNCT
ejpam-3380	225	39	either	either	CCONJ
ejpam-3380	225	40	(	(	PUNCT
ejpam-3380	225	41	rk)sj	rk)sj	SYM
ejpam-3380	225	42	=	=	SYM
ejpam-3380	225	43	(	(	PUNCT
ejpam-3380	225	44	rk)si	rk)si	X
ejpam-3380	225	45	which	which	PRON
ejpam-3380	225	46	is	be	AUX
ejpam-3380	225	47	a	a	DET
ejpam-3380	225	48	contradiction	contradiction	NOUN
ejpam-3380	225	49	or	or	CCONJ
ejpam-3380	225	50	r	r	NOUN
ejpam-3380	225	51	k	k	NOUN
ejpam-3380	225	52	=	=	PUNCT
ejpam-3380	225	53	0rk	0rk	NOUN
ejpam-3380	225	54	which	which	PRON
ejpam-3380	225	55	implies	imply	VERB
ejpam-3380	225	56	rk	rk	PROPN
ejpam-3380	225	57	=	=	SYM
ejpam-3380	225	58	⊕	⊕	PROPN
ejpam-3380	225	59	s∈g(rk)s	s∈g(rk)s	PROPN
ejpam-3380	225	60	as	as	SCONJ
ejpam-3380	225	61	required	require	VERB
ejpam-3380	225	62	.	.	PUNCT
ejpam-3380	226	1	5	5	X
ejpam-3380	226	2	.	.	PUNCT
ejpam-3380	226	3	additional	additional	ADJ
ejpam-3380	226	4	examples	example	NOUN
ejpam-3380	226	5	of	of	ADP
ejpam-3380	226	6	g	g	NOUN
ejpam-3380	226	7	-	-	PUNCT
ejpam-3380	226	8	weak	weak	ADJ
ejpam-3380	226	9	graded	grade	VERB
ejpam-3380	226	10	rings	ring	NOUN
ejpam-3380	226	11	in	in	ADP
ejpam-3380	226	12	this	this	DET
ejpam-3380	226	13	section	section	NOUN
ejpam-3380	226	14	,	,	PUNCT
ejpam-3380	226	15	we	we	PRON
ejpam-3380	226	16	give	give	VERB
ejpam-3380	226	17	additional	additional	ADJ
ejpam-3380	226	18	examples	example	NOUN
ejpam-3380	226	19	of	of	ADP
ejpam-3380	226	20	g	g	NOUN
ejpam-3380	226	21	-	-	PUNCT
ejpam-3380	226	22	weak	weak	ADJ
ejpam-3380	226	23	and	and	CCONJ
ejpam-3380	226	24	fully	fully	ADV
ejpam-3380	226	25	g	g	NOUN
ejpam-3380	226	26	-	-	PUNCT
ejpam-3380	226	27	weak	weak	ADJ
ejpam-3380	226	28	graded	grade	VERB
ejpam-3380	226	29	rings	ring	NOUN
ejpam-3380	226	30	that	that	PRON
ejpam-3380	226	31	are	be	AUX
ejpam-3380	226	32	not	not	PART
ejpam-3380	226	33	trivially	trivially	ADV
ejpam-3380	226	34	constructed	construct	VERB
ejpam-3380	226	35	.	.	PUNCT
ejpam-3380	227	1	example	example	NOUN
ejpam-3380	227	2	1	1	NUM
ejpam-3380	227	3	.	.	X
ejpam-3380	228	1	consider	consider	VERB
ejpam-3380	228	2	a	a	DET
ejpam-3380	228	3	ring	ring	NOUN
ejpam-3380	228	4	r	r	NOUN
ejpam-3380	228	5	to	to	PART
ejpam-3380	228	6	be	be	AUX
ejpam-3380	228	7	the	the	DET
ejpam-3380	228	8	ring	ring	NOUN
ejpam-3380	228	9	of	of	ADP
ejpam-3380	228	10	all	all	DET
ejpam-3380	228	11	2×	2×	NUM
ejpam-3380	228	12	2	2	NUM
ejpam-3380	228	13	matrices	matrix	NOUN
ejpam-3380	228	14	over	over	ADP
ejpam-3380	228	15	the	the	DET
ejpam-3380	228	16	field	field	NOUN
ejpam-3380	228	17	r	r	NOUN
ejpam-3380	228	18	,	,	PUNCT
ejpam-3380	228	19	i.e.	i.e.	X
ejpam-3380	228	20	,	,	PUNCT
ejpam-3380	228	21	r	r	NOUN
ejpam-3380	228	22	=	=	SYM
ejpam-3380	228	23	m2(r	m2(r	PROPN
ejpam-3380	228	24	)	)	PUNCT
ejpam-3380	228	25	=	=	SYM
ejpam-3380	228	26	{	{	PUNCT
ejpam-3380	228	27	(	(	PUNCT
ejpam-3380	228	28	a	a	DET
ejpam-3380	228	29	b	b	NOUN
ejpam-3380	228	30	c	c	NOUN
ejpam-3380	228	31	d	d	PROPN
ejpam-3380	228	32	)	)	PUNCT
ejpam-3380	228	33	:	:	PUNCT
ejpam-3380	228	34	a	a	DET
ejpam-3380	228	35	,	,	PUNCT
ejpam-3380	228	36	b	b	NOUN
ejpam-3380	228	37	,	,	PUNCT
ejpam-3380	228	38	c	c	NOUN
ejpam-3380	228	39	,	,	PUNCT
ejpam-3380	228	40	d	d	PROPN
ejpam-3380	228	41	∈	∈	PROPN
ejpam-3380	228	42	r	r	NOUN
ejpam-3380	228	43	}	}	PUNCT
ejpam-3380	228	44	.	.	PUNCT
ejpam-3380	229	1	let	let	VERB
ejpam-3380	229	2	x	x	PRON
ejpam-3380	229	3	be	be	AUX
ejpam-3380	229	4	the	the	DET
ejpam-3380	229	5	dihedral	dihedral	ADJ
ejpam-3380	229	6	group	group	NOUN
ejpam-3380	229	7	d6	d6	NOUN
ejpam-3380	229	8	=	=	PUNCT
ejpam-3380	229	9	<	<	X
ejpam-3380	229	10	x	x	X
ejpam-3380	229	11	,	,	PUNCT
ejpam-3380	229	12	y	y	NOUN
ejpam-3380	229	13	:	:	PUNCT
ejpam-3380	229	14	x6	x6	PROPN
ejpam-3380	229	15	=	=	SYM
ejpam-3380	229	16	y2	y2	NOUN
ejpam-3380	230	1	=	=	SYM
ejpam-3380	231	1	1	1	NUM
ejpam-3380	231	2	,	,	PUNCT
ejpam-3380	231	3	xy	xy	PROPN
ejpam-3380	232	1	=	=	PUNCT
ejpam-3380	232	2	yx5	yx5	NOUN
ejpam-3380	232	3	>	>	X
ejpam-3380	232	4	and	and	CCONJ
ejpam-3380	232	5	h	h	NOUN
ejpam-3380	232	6	be	be	AUX
ejpam-3380	232	7	the	the	DET
ejpam-3380	232	8	non	non	ADJ
ejpam-3380	232	9	-	-	ADJ
ejpam-3380	232	10	normal	normal	ADJ
ejpam-3380	232	11	subgroup	subgroup	NOUN
ejpam-3380	232	12	{	{	PUNCT
ejpam-3380	232	13	1	1	NUM
ejpam-3380	232	14	,	,	PUNCT
ejpam-3380	232	15	x3	x3	ADJ
ejpam-3380	232	16	,	,	PUNCT
ejpam-3380	232	17	y	y	NOUN
ejpam-3380	232	18	,	,	PUNCT
ejpam-3380	232	19	x3y	x3y	NUM
ejpam-3380	232	20	}	}	PUNCT
ejpam-3380	232	21	.	.	PUNCT
ejpam-3380	233	1	choose	choose	VERB
ejpam-3380	233	2	g	g	NOUN
ejpam-3380	233	3	=	=	PUNCT
ejpam-3380	233	4	{	{	PUNCT
ejpam-3380	233	5	1	1	NUM
ejpam-3380	233	6	,	,	PUNCT
ejpam-3380	233	7	x	x	NOUN
ejpam-3380	233	8	,	,	PUNCT
ejpam-3380	233	9	x5	x5	PROPN
ejpam-3380	233	10	}	}	PUNCT
ejpam-3380	233	11	to	to	PART
ejpam-3380	233	12	be	be	AUX
ejpam-3380	233	13	the	the	DET
ejpam-3380	233	14	set	set	NOUN
ejpam-3380	233	15	of	of	ADP
ejpam-3380	233	16	left	left	ADJ
ejpam-3380	233	17	coset	coset	NOUN
ejpam-3380	233	18	representatives	representative	NOUN
ejpam-3380	233	19	.	.	PUNCT
ejpam-3380	234	1	then	then	ADV
ejpam-3380	234	2	the	the	DET
ejpam-3380	234	3	∗	∗	NOUN
ejpam-3380	234	4	and	and	CCONJ
ejpam-3380	234	5	f	f	PROPN
ejpam-3380	234	6	operations	operation	NOUN
ejpam-3380	234	7	as	as	ADV
ejpam-3380	234	8	well	well	ADV
ejpam-3380	234	9	as	as	ADP
ejpam-3380	234	10	the	the	DET
ejpam-3380	234	11	actions	action	NOUN
ejpam-3380	234	12	/	/	PUNCT
ejpam-3380	234	13	and	and	CCONJ
ejpam-3380	234	14	.	.	PUNCT
ejpam-3380	234	15	are	be	AUX
ejpam-3380	234	16	given	give	VERB
ejpam-3380	234	17	by	by	ADP
ejpam-3380	234	18	the	the	DET
ejpam-3380	234	19	following	following	ADJ
ejpam-3380	234	20	tables	table	NOUN
ejpam-3380	234	21	:	:	PUNCT
ejpam-3380	234	22	thus	thus	ADV
ejpam-3380	234	23	,	,	PUNCT
ejpam-3380	234	24	r	r	NOUN
ejpam-3380	234	25	=	=	SYM
ejpam-3380	234	26	r1	r1	PROPN
ejpam-3380	234	27	⊕rx	⊕rx	NOUN
ejpam-3380	234	28	⊕rx5	⊕rx5	ADP
ejpam-3380	234	29	,	,	PUNCT
ejpam-3380	234	30	where	where	SCONJ
ejpam-3380	234	31	r1	r1	PROPN
ejpam-3380	234	32	=	=	SYM
ejpam-3380	234	33	{	{	PUNCT
ejpam-3380	234	34	(	(	PUNCT
ejpam-3380	234	35	a	a	DET
ejpam-3380	234	36	0	0	NUM
ejpam-3380	234	37	0	0	NUM
ejpam-3380	234	38	d	d	NOUN
ejpam-3380	234	39	)	)	PUNCT
ejpam-3380	234	40	:	:	PUNCT
ejpam-3380	234	41	a	a	X
ejpam-3380	234	42	,	,	PUNCT
ejpam-3380	234	43	d	d	X
ejpam-3380	234	44	∈	∈	PROPN
ejpam-3380	234	45	r	r	NOUN
ejpam-3380	234	46	}	}	PUNCT
ejpam-3380	234	47	,	,	PUNCT
ejpam-3380	234	48	najla	najla	PROPN
ejpam-3380	234	49	al	al	PROPN
ejpam-3380	234	50	-	-	PUNCT
ejpam-3380	234	51	subaie	subaie	NOUN
ejpam-3380	234	52	,	,	PUNCT
ejpam-3380	234	53	m.	m.	NOUN
ejpam-3380	234	54	m.	m.	PROPN
ejpam-3380	234	55	al	al	PROPN
ejpam-3380	234	56	-	-	PUNCT
ejpam-3380	234	57	shomrani	shomrani	PROPN
ejpam-3380	234	58	/	/	SYM
ejpam-3380	234	59	eur	eur	NOUN
ejpam-3380	234	60	.	.	PUNCT
ejpam-3380	235	1	j.	j.	PROPN
ejpam-3380	235	2	pure	pure	PROPN
ejpam-3380	235	3	appl	appl	PROPN
ejpam-3380	235	4	.	.	PROPN
ejpam-3380	235	5	math	math	PROPN
ejpam-3380	235	6	,	,	PUNCT
ejpam-3380	235	7	12	12	NUM
ejpam-3380	235	8	(	(	PUNCT
ejpam-3380	235	9	2	2	NUM
ejpam-3380	235	10	)	)	PUNCT
ejpam-3380	235	11	(	(	PUNCT
ejpam-3380	235	12	2019	2019	NUM
ejpam-3380	235	13	)	)	PUNCT
ejpam-3380	235	14	,	,	PUNCT
ejpam-3380	235	15	332	332	NUM
ejpam-3380	235	16	-	-	SYM
ejpam-3380	235	17	347	347	NUM
ejpam-3380	235	18	341	341	NUM
ejpam-3380	235	19	table	table	NOUN
ejpam-3380	235	20	1	1	NUM
ejpam-3380	235	21	:	:	PUNCT
ejpam-3380	235	22	∗	∗	NOUN
ejpam-3380	235	23	and	and	CCONJ
ejpam-3380	235	24	f	f	PROPN
ejpam-3380	235	25	operations	operation	NOUN
ejpam-3380	235	26	.	.	PUNCT
ejpam-3380	236	1	∗	∗	NOUN
ejpam-3380	236	2	1	1	NUM
ejpam-3380	236	3	x	x	SYM
ejpam-3380	236	4	x5	x5	NOUN
ejpam-3380	236	5	1	1	NUM
ejpam-3380	236	6	1	1	NUM
ejpam-3380	236	7	x	x	SYM
ejpam-3380	236	8	x5	x5	NOUN
ejpam-3380	236	9	x	x	SYM
ejpam-3380	236	10	x	x	SYM
ejpam-3380	236	11	x5	x5	NOUN
ejpam-3380	236	12	1	1	NUM
ejpam-3380	236	13	x5	x5	NOUN
ejpam-3380	236	14	x5	x5	PROPN
ejpam-3380	236	15	1	1	NUM
ejpam-3380	236	16	x	x	SYM
ejpam-3380	236	17	f	f	PROPN
ejpam-3380	236	18	1	1	NUM
ejpam-3380	236	19	x	x	SYM
ejpam-3380	236	20	x5	x5	NOUN
ejpam-3380	236	21	1	1	NUM
ejpam-3380	236	22	1	1	NUM
ejpam-3380	236	23	1	1	NUM
ejpam-3380	236	24	1	1	NUM
ejpam-3380	236	25	x	x	SYM
ejpam-3380	236	26	1	1	NUM
ejpam-3380	236	27	x3	x3	ADJ
ejpam-3380	236	28	1	1	NUM
ejpam-3380	236	29	x5	x5	NOUN
ejpam-3380	236	30	1	1	NUM
ejpam-3380	236	31	1	1	NUM
ejpam-3380	236	32	x3	x3	ADJ
ejpam-3380	236	33	table	table	NOUN
ejpam-3380	236	34	2	2	NUM
ejpam-3380	236	35	:	:	PUNCT
ejpam-3380	236	36	.	.	PUNCT
ejpam-3380	237	1	and	and	CCONJ
ejpam-3380	237	2	/	/	SYM
ejpam-3380	237	3	actions	action	NOUN
ejpam-3380	237	4	.	.	PUNCT
ejpam-3380	238	1	s.u	s.u	NOUN
ejpam-3380	238	2	1	1	NUM
ejpam-3380	238	3	x3	x3	PROPN
ejpam-3380	238	4	y	y	PROPN
ejpam-3380	238	5	x3y	x3y	NUM
ejpam-3380	238	6	1	1	NUM
ejpam-3380	238	7	1	1	NUM
ejpam-3380	238	8	x3	x3	NOUN
ejpam-3380	238	9	y	y	PROPN
ejpam-3380	238	10	x3y	x3y	PROPN
ejpam-3380	238	11	x	x	SYM
ejpam-3380	238	12	1	1	NUM
ejpam-3380	238	13	x3	x3	PROPN
ejpam-3380	238	14	y	y	PROPN
ejpam-3380	238	15	x3y	x3y	PROPN
ejpam-3380	238	16	x5	x5	PROPN
ejpam-3380	238	17	1	1	NUM
ejpam-3380	238	18	x3	x3	PROPN
ejpam-3380	238	19	y	y	PROPN
ejpam-3380	238	20	x3y	x3y	PROPN
ejpam-3380	238	21	s	s	PROPN
ejpam-3380	238	22	/	/	SYM
ejpam-3380	238	23	u	u	NOUN
ejpam-3380	238	24	1	1	NUM
ejpam-3380	238	25	x3	x3	NOUN
ejpam-3380	238	26	y	y	PROPN
ejpam-3380	238	27	x3y	x3y	NUM
ejpam-3380	238	28	1	1	NUM
ejpam-3380	238	29	1	1	NUM
ejpam-3380	238	30	1	1	NUM
ejpam-3380	238	31	1	1	NUM
ejpam-3380	238	32	1	1	NUM
ejpam-3380	238	33	x	x	SYM
ejpam-3380	238	34	x	x	SYM
ejpam-3380	238	35	x	x	SYM
ejpam-3380	238	36	x5	x5	NUM
ejpam-3380	238	37	x5	x5	PROPN
ejpam-3380	238	38	x5	x5	PROPN
ejpam-3380	238	39	x5	x5	PROPN
ejpam-3380	238	40	x5	x5	PROPN
ejpam-3380	238	41	x	x	PUNCT
ejpam-3380	238	42	x	x	X
ejpam-3380	238	43	rx	rx	VERB
ejpam-3380	238	44	=	=	PUNCT
ejpam-3380	238	45	{	{	PUNCT
ejpam-3380	238	46	(	(	PUNCT
ejpam-3380	238	47	0	0	NUM
ejpam-3380	238	48	0	0	NUM
ejpam-3380	238	49	c	c	NOUN
ejpam-3380	238	50	0	0	NUM
ejpam-3380	238	51	)	)	PUNCT
ejpam-3380	238	52	:	:	PUNCT
ejpam-3380	239	1	c	c	X
ejpam-3380	239	2	∈	∈	PROPN
ejpam-3380	239	3	r	r	NOUN
ejpam-3380	239	4	}	}	PUNCT
ejpam-3380	239	5	and	and	CCONJ
ejpam-3380	239	6	rx5	rx5	NOUN
ejpam-3380	239	7	=	=	SYM
ejpam-3380	239	8	{	{	PUNCT
ejpam-3380	239	9	(	(	PUNCT
ejpam-3380	239	10	0	0	NUM
ejpam-3380	239	11	b	b	NOUN
ejpam-3380	239	12	0	0	NUM
ejpam-3380	239	13	0	0	NUM
ejpam-3380	239	14	)	)	PUNCT
ejpam-3380	239	15	:	:	PUNCT
ejpam-3380	239	16	b	b	X
ejpam-3380	239	17	∈	∈	NOUN
ejpam-3380	239	18	r	r	NOUN
ejpam-3380	239	19	}	}	PUNCT
ejpam-3380	239	20	.	.	PUNCT
ejpam-3380	240	1	moreover	moreover	ADV
ejpam-3380	240	2	,	,	PUNCT
ejpam-3380	240	3	the	the	DET
ejpam-3380	240	4	inclusion	inclusion	NOUN
ejpam-3380	240	5	property	property	NOUN
ejpam-3380	240	6	rsrt	rsrt	NOUN
ejpam-3380	240	7	⊆	⊆	NUM
ejpam-3380	240	8	rs∗t	rs∗t	NOUN
ejpam-3380	240	9	is	be	AUX
ejpam-3380	240	10	satisfied	satisfied	ADJ
ejpam-3380	240	11	for	for	ADP
ejpam-3380	240	12	all	all	DET
ejpam-3380	240	13	s	s	PROPN
ejpam-3380	240	14	,	,	PUNCT
ejpam-3380	240	15	t	t	PROPN
ejpam-3380	240	16	∈	∈	PROPN
ejpam-3380	240	17	g.	g.	PROPN
ejpam-3380	240	18	this	this	PRON
ejpam-3380	240	19	can	can	AUX
ejpam-3380	240	20	be	be	AUX
ejpam-3380	240	21	detailed	detail	VERB
ejpam-3380	240	22	as	as	SCONJ
ejpam-3380	240	23	follows	follow	VERB
ejpam-3380	240	24	:	:	PUNCT
ejpam-3380	240	25	(	(	PUNCT
ejpam-3380	240	26	i	i	NOUN
ejpam-3380	240	27	)	)	PUNCT
ejpam-3380	240	28	r1r1	r1r1	VERB
ejpam-3380	240	29	⊆	⊆	NUM
ejpam-3380	240	30	r1∗1	r1∗1	NOUN
ejpam-3380	240	31	=	=	SYM
ejpam-3380	240	32	r1	r1	PROPN
ejpam-3380	240	33	,	,	PUNCT
ejpam-3380	240	34	as	as	ADP
ejpam-3380	240	35	for	for	ADP
ejpam-3380	240	36	all	all	DET
ejpam-3380	240	37	(	(	PUNCT
ejpam-3380	240	38	a1	a1	NOUN
ejpam-3380	240	39	0	0	NUM
ejpam-3380	240	40	0	0	NUM
ejpam-3380	240	41	d1	d1	PROPN
ejpam-3380	240	42	)	)	PUNCT
ejpam-3380	240	43	,	,	PUNCT
ejpam-3380	240	44	(	(	PUNCT
ejpam-3380	240	45	a2	a2	PROPN
ejpam-3380	240	46	0	0	NUM
ejpam-3380	240	47	0	0	NUM
ejpam-3380	240	48	d2	d2	PROPN
ejpam-3380	240	49	)	)	PUNCT
ejpam-3380	240	50	∈	∈	PROPN
ejpam-3380	240	51	r1	r1	PROPN
ejpam-3380	240	52	,	,	PUNCT
ejpam-3380	240	53	we	we	PRON
ejpam-3380	240	54	have	have	VERB
ejpam-3380	240	55	(	(	PUNCT
ejpam-3380	240	56	a1	a1	NOUN
ejpam-3380	240	57	0	0	NUM
ejpam-3380	240	58	0	0	NUM
ejpam-3380	240	59	d1	d1	NOUN
ejpam-3380	240	60	)	)	PUNCT
ejpam-3380	240	61	(	(	PUNCT
ejpam-3380	240	62	a2	a2	PROPN
ejpam-3380	240	63	0	0	NUM
ejpam-3380	240	64	0	0	NUM
ejpam-3380	240	65	d2	d2	PROPN
ejpam-3380	240	66	)	)	PUNCT
ejpam-3380	240	67	=	=	PRON
ejpam-3380	241	1	(	(	PUNCT
ejpam-3380	241	2	a1a2	a1a2	INTJ
ejpam-3380	241	3	0	0	NUM
ejpam-3380	241	4	0	0	NUM
ejpam-3380	241	5	d1d2	d1d2	PRON
ejpam-3380	241	6	)	)	PUNCT
ejpam-3380	241	7	∈	∈	PROPN
ejpam-3380	241	8	r1	r1	NOUN
ejpam-3380	241	9	=	=	PUNCT
ejpam-3380	241	10	r1∗1	r1∗1	NOUN
ejpam-3380	241	11	.	.	PUNCT
ejpam-3380	241	12	(	(	PUNCT
ejpam-3380	241	13	ii	ii	NOUN
ejpam-3380	241	14	)	)	PUNCT
ejpam-3380	241	15	r1rx	r1rx	PUNCT
ejpam-3380	242	1	⊆	⊆	NUM
ejpam-3380	242	2	r1∗x	r1∗x	NOUN
ejpam-3380	242	3	=	=	PUNCT
ejpam-3380	242	4	rx	rx	VERB
ejpam-3380	242	5	,	,	PUNCT
ejpam-3380	242	6	as	as	ADP
ejpam-3380	242	7	for	for	ADP
ejpam-3380	242	8	all	all	DET
ejpam-3380	242	9	(	(	PUNCT
ejpam-3380	242	10	a	a	DET
ejpam-3380	242	11	0	0	NUM
ejpam-3380	242	12	0	0	NUM
ejpam-3380	242	13	d	d	NOUN
ejpam-3380	242	14	)	)	PUNCT
ejpam-3380	242	15	∈	∈	PROPN
ejpam-3380	242	16	r1	r1	NOUN
ejpam-3380	242	17	and	and	CCONJ
ejpam-3380	242	18	(	(	PUNCT
ejpam-3380	242	19	0	0	NUM
ejpam-3380	242	20	0	0	NUM
ejpam-3380	242	21	c	c	NOUN
ejpam-3380	242	22	0	0	NUM
ejpam-3380	242	23	)	)	PUNCT
ejpam-3380	242	24	∈	∈	PROPN
ejpam-3380	242	25	rx	rx	VERB
ejpam-3380	242	26	,	,	PUNCT
ejpam-3380	242	27	we	we	PRON
ejpam-3380	242	28	have	have	VERB
ejpam-3380	242	29	(	(	PUNCT
ejpam-3380	242	30	a	a	DET
ejpam-3380	242	31	0	0	NUM
ejpam-3380	242	32	0	0	NUM
ejpam-3380	242	33	d	d	NOUN
ejpam-3380	242	34	)	)	PUNCT
ejpam-3380	242	35	(	(	PUNCT
ejpam-3380	242	36	0	0	NUM
ejpam-3380	242	37	0	0	NUM
ejpam-3380	242	38	c	c	NOUN
ejpam-3380	242	39	0	0	NUM
ejpam-3380	242	40	)	)	PUNCT
ejpam-3380	242	41	=	=	PRON
ejpam-3380	243	1	(	(	PUNCT
ejpam-3380	243	2	0	0	NUM
ejpam-3380	243	3	0	0	NUM
ejpam-3380	243	4	dc	dc	PROPN
ejpam-3380	243	5	0	0	NUM
ejpam-3380	243	6	)	)	PUNCT
ejpam-3380	243	7	∈	∈	PROPN
ejpam-3380	243	8	rx	rx	NOUN
ejpam-3380	243	9	=	=	SYM
ejpam-3380	243	10	r1∗x	r1∗x	PROPN
ejpam-3380	243	11	.	.	PUNCT
ejpam-3380	243	12	(	(	PUNCT
ejpam-3380	243	13	iii	iii	X
ejpam-3380	243	14	)	)	PUNCT
ejpam-3380	243	15	r1rx5	r1rx5	NOUN
ejpam-3380	243	16	⊆	⊆	NUM
ejpam-3380	243	17	r1∗x5	r1∗x5	X
ejpam-3380	243	18	=	=	SYM
ejpam-3380	243	19	rx5	rx5	PROPN
ejpam-3380	243	20	,	,	PUNCT
ejpam-3380	243	21	as	as	ADP
ejpam-3380	243	22	for	for	ADP
ejpam-3380	243	23	all	all	DET
ejpam-3380	243	24	(	(	PUNCT
ejpam-3380	243	25	a	a	DET
ejpam-3380	243	26	0	0	NUM
ejpam-3380	243	27	0	0	NUM
ejpam-3380	243	28	d	d	NOUN
ejpam-3380	243	29	)	)	PUNCT
ejpam-3380	243	30	∈	∈	PROPN
ejpam-3380	243	31	r1	r1	NOUN
ejpam-3380	243	32	and	and	CCONJ
ejpam-3380	243	33	(	(	PUNCT
ejpam-3380	243	34	0	0	NUM
ejpam-3380	243	35	b	b	NOUN
ejpam-3380	243	36	0	0	NUM
ejpam-3380	243	37	0	0	NUM
ejpam-3380	243	38	)	)	PUNCT
ejpam-3380	244	1	∈	∈	PROPN
ejpam-3380	244	2	rx5	rx5	PROPN
ejpam-3380	244	3	,	,	PUNCT
ejpam-3380	244	4	we	we	PRON
ejpam-3380	244	5	have	have	VERB
ejpam-3380	244	6	(	(	PUNCT
ejpam-3380	244	7	a	a	DET
ejpam-3380	244	8	0	0	NUM
ejpam-3380	244	9	0	0	NUM
ejpam-3380	244	10	d	d	NOUN
ejpam-3380	244	11	)	)	PUNCT
ejpam-3380	244	12	(	(	PUNCT
ejpam-3380	244	13	0	0	NUM
ejpam-3380	244	14	b	b	NOUN
ejpam-3380	244	15	0	0	NUM
ejpam-3380	244	16	0	0	NUM
ejpam-3380	244	17	)	)	PUNCT
ejpam-3380	245	1	=	=	PRON
ejpam-3380	246	1	(	(	PUNCT
ejpam-3380	246	2	0	0	NUM
ejpam-3380	246	3	ab	ab	NOUN
ejpam-3380	246	4	0	0	NUM
ejpam-3380	246	5	0	0	NUM
ejpam-3380	246	6	)	)	PUNCT
ejpam-3380	246	7	∈	∈	PROPN
ejpam-3380	246	8	rx5	rx5	NOUN
ejpam-3380	246	9	=	=	X
ejpam-3380	246	10	r1∗x5	r1∗x5	X
ejpam-3380	246	11	.	.	PUNCT
ejpam-3380	247	1	(	(	PUNCT
ejpam-3380	247	2	iv	iv	X
ejpam-3380	247	3	)	)	PUNCT
ejpam-3380	247	4	rxr1	rxr1	NOUN
ejpam-3380	247	5	⊆	⊆	NUM
ejpam-3380	247	6	rx∗1	rx∗1	NOUN
ejpam-3380	247	7	=	=	SYM
ejpam-3380	247	8	rx	rx	NOUN
ejpam-3380	247	9	,	,	PUNCT
ejpam-3380	247	10	as	as	ADP
ejpam-3380	247	11	for	for	ADP
ejpam-3380	247	12	all	all	PRON
ejpam-3380	247	13	(	(	PUNCT
ejpam-3380	247	14	0	0	NUM
ejpam-3380	247	15	0	0	NUM
ejpam-3380	247	16	c	c	NOUN
ejpam-3380	247	17	0	0	NUM
ejpam-3380	247	18	)	)	PUNCT
ejpam-3380	247	19	∈	∈	PROPN
ejpam-3380	247	20	rx	rx	NOUN
ejpam-3380	247	21	and	and	CCONJ
ejpam-3380	247	22	(	(	PUNCT
ejpam-3380	247	23	a	a	DET
ejpam-3380	247	24	0	0	NUM
ejpam-3380	247	25	0	0	NUM
ejpam-3380	247	26	d	d	NOUN
ejpam-3380	247	27	)	)	PUNCT
ejpam-3380	247	28	∈	∈	PROPN
ejpam-3380	247	29	r1	r1	NOUN
ejpam-3380	247	30	,	,	PUNCT
ejpam-3380	247	31	we	we	PRON
ejpam-3380	247	32	have	have	VERB
ejpam-3380	247	33	(	(	PUNCT
ejpam-3380	247	34	0	0	NUM
ejpam-3380	247	35	0	0	NUM
ejpam-3380	247	36	c	c	NOUN
ejpam-3380	247	37	0	0	NUM
ejpam-3380	247	38	)	)	PUNCT
ejpam-3380	247	39	(	(	PUNCT
ejpam-3380	247	40	a	a	DET
ejpam-3380	247	41	0	0	NUM
ejpam-3380	247	42	0	0	NUM
ejpam-3380	247	43	d	d	NOUN
ejpam-3380	247	44	)	)	PUNCT
ejpam-3380	248	1	=	=	PUNCT
ejpam-3380	249	1	(	(	PUNCT
ejpam-3380	249	2	0	0	NUM
ejpam-3380	249	3	0	0	NUM
ejpam-3380	249	4	ca	ca	NOUN
ejpam-3380	249	5	0	0	NUM
ejpam-3380	249	6	)	)	PUNCT
ejpam-3380	249	7	∈	∈	NOUN
ejpam-3380	249	8	rx	rx	NOUN
ejpam-3380	249	9	=	=	SYM
ejpam-3380	249	10	rx∗1	rx∗1	NOUN
ejpam-3380	249	11	.	.	PUNCT
ejpam-3380	250	1	najla	najla	PROPN
ejpam-3380	250	2	al	al	PROPN
ejpam-3380	250	3	-	-	PUNCT
ejpam-3380	250	4	subaie	subaie	NOUN
ejpam-3380	250	5	,	,	PUNCT
ejpam-3380	250	6	m.	m.	NOUN
ejpam-3380	250	7	m.	m.	PROPN
ejpam-3380	250	8	al	al	PROPN
ejpam-3380	250	9	-	-	PUNCT
ejpam-3380	250	10	shomrani	shomrani	PROPN
ejpam-3380	250	11	/	/	SYM
ejpam-3380	250	12	eur	eur	NOUN
ejpam-3380	250	13	.	.	PUNCT
ejpam-3380	251	1	j.	j.	PROPN
ejpam-3380	251	2	pure	pure	PROPN
ejpam-3380	251	3	appl	appl	PROPN
ejpam-3380	251	4	.	.	PROPN
ejpam-3380	251	5	math	math	PROPN
ejpam-3380	251	6	,	,	PUNCT
ejpam-3380	251	7	12	12	NUM
ejpam-3380	251	8	(	(	PUNCT
ejpam-3380	251	9	2	2	NUM
ejpam-3380	251	10	)	)	PUNCT
ejpam-3380	251	11	(	(	PUNCT
ejpam-3380	251	12	2019	2019	NUM
ejpam-3380	251	13	)	)	PUNCT
ejpam-3380	251	14	,	,	PUNCT
ejpam-3380	251	15	332	332	NUM
ejpam-3380	251	16	-	-	SYM
ejpam-3380	251	17	347	347	NUM
ejpam-3380	251	18	342	342	NUM
ejpam-3380	251	19	(	(	PUNCT
ejpam-3380	251	20	v	v	NOUN
ejpam-3380	251	21	)	)	PUNCT
ejpam-3380	251	22	rxrx	rxrx	NOUN
ejpam-3380	251	23	⊆	⊆	NUM
ejpam-3380	251	24	rx∗x	rx∗x	PROPN
ejpam-3380	251	25	=	=	SYM
ejpam-3380	251	26	rx5	rx5	NOUN
ejpam-3380	251	27	,	,	PUNCT
ejpam-3380	251	28	as	as	ADP
ejpam-3380	251	29	for	for	ADP
ejpam-3380	251	30	all	all	DET
ejpam-3380	251	31	(	(	PUNCT
ejpam-3380	251	32	0	0	NUM
ejpam-3380	251	33	0	0	NUM
ejpam-3380	251	34	c1	c1	NOUN
ejpam-3380	251	35	0	0	NUM
ejpam-3380	251	36	)	)	PUNCT
ejpam-3380	251	37	,	,	PUNCT
ejpam-3380	251	38	(	(	PUNCT
ejpam-3380	251	39	0	0	NUM
ejpam-3380	251	40	0	0	NUM
ejpam-3380	251	41	c2	c2	PROPN
ejpam-3380	251	42	0	0	NUM
ejpam-3380	251	43	)	)	PUNCT
ejpam-3380	251	44	∈	∈	PROPN
ejpam-3380	251	45	rx	rx	VERB
ejpam-3380	251	46	,	,	PUNCT
ejpam-3380	251	47	we	we	PRON
ejpam-3380	251	48	have	have	AUX
ejpam-3380	251	49	(	(	PUNCT
ejpam-3380	251	50	0	0	NUM
ejpam-3380	251	51	0	0	NUM
ejpam-3380	251	52	c1	c1	NOUN
ejpam-3380	251	53	0	0	NUM
ejpam-3380	251	54	)	)	PUNCT
ejpam-3380	251	55	(	(	PUNCT
ejpam-3380	251	56	0	0	NUM
ejpam-3380	251	57	0	0	NUM
ejpam-3380	251	58	c2	c2	PROPN
ejpam-3380	251	59	0	0	NUM
ejpam-3380	251	60	)	)	PUNCT
ejpam-3380	251	61	=	=	PUNCT
ejpam-3380	252	1	(	(	PUNCT
ejpam-3380	252	2	0	0	NUM
ejpam-3380	252	3	0	0	NUM
ejpam-3380	252	4	0	0	NUM
ejpam-3380	252	5	0	0	NUM
ejpam-3380	252	6	)	)	PUNCT
ejpam-3380	252	7	∈	∈	PROPN
ejpam-3380	252	8	rx5	rx5	NOUN
ejpam-3380	253	1	=	=	PUNCT
ejpam-3380	254	1	rx∗x	rx∗x	PROPN
ejpam-3380	254	2	.	.	PUNCT
ejpam-3380	255	1	(	(	PUNCT
ejpam-3380	255	2	vi	vi	NOUN
ejpam-3380	255	3	)	)	PUNCT
ejpam-3380	255	4	rxrx5	rxrx5	NOUN
ejpam-3380	255	5	⊆	⊆	NUM
ejpam-3380	255	6	rx∗x5	rx∗x5	PROPN
ejpam-3380	255	7	=	=	SYM
ejpam-3380	255	8	r1	r1	PROPN
ejpam-3380	255	9	,	,	PUNCT
ejpam-3380	255	10	as	as	ADP
ejpam-3380	255	11	for	for	ADP
ejpam-3380	255	12	all	all	PRON
ejpam-3380	255	13	(	(	PUNCT
ejpam-3380	255	14	0	0	NUM
ejpam-3380	255	15	0	0	NUM
ejpam-3380	255	16	c	c	NOUN
ejpam-3380	255	17	0	0	NUM
ejpam-3380	255	18	)	)	PUNCT
ejpam-3380	255	19	∈	∈	PROPN
ejpam-3380	255	20	rx	rx	NOUN
ejpam-3380	255	21	and	and	CCONJ
ejpam-3380	255	22	(	(	PUNCT
ejpam-3380	255	23	0	0	NUM
ejpam-3380	255	24	b	b	NOUN
ejpam-3380	255	25	0	0	NUM
ejpam-3380	255	26	0	0	NUM
ejpam-3380	255	27	)	)	PUNCT
ejpam-3380	256	1	∈	∈	PROPN
ejpam-3380	256	2	rx5	rx5	PROPN
ejpam-3380	256	3	,	,	PUNCT
ejpam-3380	256	4	we	we	PRON
ejpam-3380	256	5	have	have	VERB
ejpam-3380	256	6	(	(	PUNCT
ejpam-3380	256	7	0	0	NUM
ejpam-3380	256	8	0	0	NUM
ejpam-3380	256	9	c	c	NOUN
ejpam-3380	256	10	0	0	NUM
ejpam-3380	256	11	)	)	PUNCT
ejpam-3380	256	12	(	(	PUNCT
ejpam-3380	256	13	0	0	NUM
ejpam-3380	256	14	b	b	NOUN
ejpam-3380	256	15	0	0	NUM
ejpam-3380	256	16	0	0	NUM
ejpam-3380	256	17	)	)	PUNCT
ejpam-3380	256	18	=	=	PRON
ejpam-3380	257	1	(	(	PUNCT
ejpam-3380	257	2	0	0	NUM
ejpam-3380	257	3	0	0	NUM
ejpam-3380	257	4	0	0	NUM
ejpam-3380	257	5	cb	cb	PROPN
ejpam-3380	257	6	)	)	PUNCT
ejpam-3380	257	7	∈	∈	PROPN
ejpam-3380	257	8	r1	r1	PROPN
ejpam-3380	257	9	=	=	PUNCT
ejpam-3380	257	10	rx∗x5	rx∗x5	PROPN
ejpam-3380	257	11	.	.	PUNCT
ejpam-3380	258	1	(	(	PUNCT
ejpam-3380	258	2	vii	vii	PROPN
ejpam-3380	258	3	)	)	PUNCT
ejpam-3380	258	4	rx5r1	rx5r1	PROPN
ejpam-3380	258	5	⊆	⊆	NUM
ejpam-3380	258	6	rx5∗1	rx5∗1	NOUN
ejpam-3380	258	7	=	=	SYM
ejpam-3380	259	1	rx5	rx5	PROPN
ejpam-3380	259	2	,	,	PUNCT
ejpam-3380	259	3	as	as	ADP
ejpam-3380	259	4	for	for	ADP
ejpam-3380	259	5	all	all	PRON
ejpam-3380	259	6	(	(	PUNCT
ejpam-3380	259	7	0	0	NUM
ejpam-3380	259	8	b	b	NOUN
ejpam-3380	259	9	0	0	NUM
ejpam-3380	259	10	0	0	NUM
ejpam-3380	259	11	)	)	PUNCT
ejpam-3380	259	12	∈	∈	PROPN
ejpam-3380	259	13	rx5	rx5	NOUN
ejpam-3380	259	14	and	and	CCONJ
ejpam-3380	259	15	(	(	PUNCT
ejpam-3380	259	16	a	a	DET
ejpam-3380	259	17	0	0	NUM
ejpam-3380	259	18	0	0	NUM
ejpam-3380	259	19	d	d	NOUN
ejpam-3380	259	20	)	)	PUNCT
ejpam-3380	259	21	∈	∈	PROPN
ejpam-3380	259	22	r1	r1	NOUN
ejpam-3380	259	23	,	,	PUNCT
ejpam-3380	259	24	we	we	PRON
ejpam-3380	259	25	have	have	VERB
ejpam-3380	259	26	(	(	PUNCT
ejpam-3380	259	27	0	0	NUM
ejpam-3380	259	28	b	b	NOUN
ejpam-3380	259	29	0	0	NUM
ejpam-3380	259	30	0	0	NUM
ejpam-3380	259	31	)	)	PUNCT
ejpam-3380	259	32	(	(	PUNCT
ejpam-3380	259	33	a	a	DET
ejpam-3380	259	34	0	0	NUM
ejpam-3380	259	35	0	0	NUM
ejpam-3380	259	36	d	d	NOUN
ejpam-3380	259	37	)	)	PUNCT
ejpam-3380	260	1	=	=	PUNCT
ejpam-3380	261	1	(	(	PUNCT
ejpam-3380	261	2	0	0	NUM
ejpam-3380	261	3	bd	bd	PROPN
ejpam-3380	261	4	0	0	NUM
ejpam-3380	261	5	0	0	NUM
ejpam-3380	261	6	)	)	PUNCT
ejpam-3380	261	7	∈	∈	PROPN
ejpam-3380	261	8	rx5	rx5	NOUN
ejpam-3380	262	1	=	=	SYM
ejpam-3380	263	1	rx5∗1	rx5∗1	PROPN
ejpam-3380	263	2	.	.	PUNCT
ejpam-3380	264	1	(	(	PUNCT
ejpam-3380	264	2	viii	viii	NOUN
ejpam-3380	264	3	)	)	PUNCT
ejpam-3380	264	4	rx5rx	rx5rx	VERB
ejpam-3380	264	5	⊆	⊆	NUM
ejpam-3380	264	6	rx5∗x	rx5∗x	NOUN
ejpam-3380	264	7	=	=	SYM
ejpam-3380	264	8	r1	r1	PROPN
ejpam-3380	264	9	,	,	PUNCT
ejpam-3380	264	10	as	as	ADP
ejpam-3380	264	11	for	for	ADP
ejpam-3380	264	12	all	all	PRON
ejpam-3380	264	13	(	(	PUNCT
ejpam-3380	264	14	0	0	NUM
ejpam-3380	264	15	b	b	NOUN
ejpam-3380	264	16	0	0	NUM
ejpam-3380	264	17	0	0	NUM
ejpam-3380	264	18	)	)	PUNCT
ejpam-3380	264	19	∈	∈	PROPN
ejpam-3380	264	20	rx5	rx5	NOUN
ejpam-3380	265	1	and	and	CCONJ
ejpam-3380	265	2	(	(	PUNCT
ejpam-3380	265	3	0	0	NUM
ejpam-3380	265	4	0	0	NUM
ejpam-3380	265	5	c	c	NOUN
ejpam-3380	265	6	0	0	NUM
ejpam-3380	265	7	)	)	PUNCT
ejpam-3380	265	8	∈	∈	PROPN
ejpam-3380	265	9	rx	rx	VERB
ejpam-3380	265	10	,	,	PUNCT
ejpam-3380	265	11	we	we	PRON
ejpam-3380	265	12	have	have	AUX
ejpam-3380	265	13	(	(	PUNCT
ejpam-3380	265	14	0	0	NUM
ejpam-3380	265	15	b	b	NOUN
ejpam-3380	265	16	0	0	NUM
ejpam-3380	265	17	0	0	NUM
ejpam-3380	265	18	)	)	PUNCT
ejpam-3380	265	19	(	(	PUNCT
ejpam-3380	265	20	0	0	NUM
ejpam-3380	265	21	0	0	NUM
ejpam-3380	265	22	c	c	NOUN
ejpam-3380	265	23	0	0	NUM
ejpam-3380	265	24	)	)	PUNCT
ejpam-3380	265	25	=	=	PRON
ejpam-3380	265	26	(	(	PUNCT
ejpam-3380	265	27	bc	bc	PROPN
ejpam-3380	265	28	0	0	PROPN
ejpam-3380	265	29	0	0	NUM
ejpam-3380	265	30	0	0	NUM
ejpam-3380	265	31	)	)	PUNCT
ejpam-3380	265	32	∈	∈	PROPN
ejpam-3380	265	33	r1	r1	NOUN
ejpam-3380	265	34	=	=	PUNCT
ejpam-3380	265	35	rx5∗x	rx5∗x	PROPN
ejpam-3380	265	36	.	.	PUNCT
ejpam-3380	266	1	(	(	PUNCT
ejpam-3380	266	2	ix	ix	ADJ
ejpam-3380	266	3	)	)	PUNCT
ejpam-3380	266	4	rx5rx5	rx5rx5	NOUN
ejpam-3380	266	5	⊆	⊆	NUM
ejpam-3380	266	6	rx5∗x5	rx5∗x5	X
ejpam-3380	266	7	=	=	PUNCT
ejpam-3380	266	8	rx	rx	PROPN
ejpam-3380	266	9	,	,	PUNCT
ejpam-3380	266	10	as	as	ADP
ejpam-3380	266	11	for	for	ADP
ejpam-3380	266	12	all	all	PRON
ejpam-3380	266	13	(	(	PUNCT
ejpam-3380	266	14	0	0	NUM
ejpam-3380	266	15	b1	b1	NOUN
ejpam-3380	266	16	0	0	NUM
ejpam-3380	266	17	0	0	NUM
ejpam-3380	266	18	)	)	PUNCT
ejpam-3380	266	19	,	,	PUNCT
ejpam-3380	266	20	(	(	PUNCT
ejpam-3380	266	21	0	0	NUM
ejpam-3380	266	22	b2	b2	NOUN
ejpam-3380	266	23	0	0	NUM
ejpam-3380	266	24	0	0	NUM
ejpam-3380	266	25	)	)	PUNCT
ejpam-3380	266	26	∈	∈	PROPN
ejpam-3380	266	27	rx5	rx5	PROPN
ejpam-3380	266	28	,	,	PUNCT
ejpam-3380	266	29	we	we	PRON
ejpam-3380	266	30	have	have	VERB
ejpam-3380	266	31	(	(	PUNCT
ejpam-3380	266	32	0	0	NUM
ejpam-3380	266	33	b1	b1	NOUN
ejpam-3380	266	34	0	0	NUM
ejpam-3380	266	35	0	0	NUM
ejpam-3380	266	36	)	)	PUNCT
ejpam-3380	266	37	(	(	PUNCT
ejpam-3380	266	38	0	0	NUM
ejpam-3380	266	39	b2	b2	NOUN
ejpam-3380	266	40	0	0	NUM
ejpam-3380	266	41	0	0	NUM
ejpam-3380	266	42	)	)	PUNCT
ejpam-3380	266	43	=	=	PRON
ejpam-3380	267	1	(	(	PUNCT
ejpam-3380	267	2	0	0	NUM
ejpam-3380	267	3	0	0	NUM
ejpam-3380	267	4	0	0	NUM
ejpam-3380	267	5	0	0	NUM
ejpam-3380	267	6	)	)	PUNCT
ejpam-3380	267	7	∈	∈	NOUN
ejpam-3380	267	8	rx	rx	NOUN
ejpam-3380	267	9	=	=	NOUN
ejpam-3380	267	10	rx5∗x5	rx5∗x5	X
ejpam-3380	267	11	.	.	PUNCT
ejpam-3380	268	1	therefore	therefore	ADV
ejpam-3380	268	2	,	,	PUNCT
ejpam-3380	268	3	r	r	NOUN
ejpam-3380	268	4	is	be	AUX
ejpam-3380	268	5	a	a	DET
ejpam-3380	268	6	g	g	NOUN
ejpam-3380	268	7	-	-	PUNCT
ejpam-3380	268	8	weak	weak	ADJ
ejpam-3380	268	9	graded	grade	VERB
ejpam-3380	268	10	ring	ring	NOUN
ejpam-3380	268	11	.	.	PUNCT
ejpam-3380	269	1	however	however	ADV
ejpam-3380	269	2	,	,	PUNCT
ejpam-3380	269	3	it	it	PRON
ejpam-3380	269	4	is	be	AUX
ejpam-3380	269	5	not	not	PART
ejpam-3380	269	6	a	a	DET
ejpam-3380	269	7	fully	fully	ADV
ejpam-3380	269	8	g	g	NOUN
ejpam-3380	269	9	-	-	PUNCT
ejpam-3380	269	10	weak	weak	ADJ
ejpam-3380	269	11	graded	grade	VERB
ejpam-3380	269	12	ring	ring	NOUN
ejpam-3380	269	13	.	.	PUNCT
ejpam-3380	270	1	for	for	ADP
ejpam-3380	270	2	instance	instance	NOUN
ejpam-3380	270	3	,	,	PUNCT
ejpam-3380	270	4	rx5rx5	rx5rx5	NOUN
ejpam-3380	270	5	6=	6=	NUM
ejpam-3380	270	6	rx5∗x5	rx5∗x5	NOUN
ejpam-3380	270	7	since	since	SCONJ
ejpam-3380	270	8	rx	rx	VERB
ejpam-3380	270	9	=	=	SYM
ejpam-3380	270	10	rx5∗x5	rx5∗x5	X
ejpam-3380	270	11	*	*	PUNCT
ejpam-3380	270	12	rx5rx5	rx5rx5	NOUN
ejpam-3380	270	13	.	.	PUNCT
ejpam-3380	270	14	example	example	NOUN
ejpam-3380	271	1	2	2	NUM
ejpam-3380	271	2	.	.	PUNCT
ejpam-3380	271	3	let	let	VERB
ejpam-3380	271	4	x	x	PUNCT
ejpam-3380	271	5	=	=	PRON
ejpam-3380	271	6	(	(	PUNCT
ejpam-3380	271	7	z6,+	z6,+	NOUN
ejpam-3380	271	8	)	)	PUNCT
ejpam-3380	271	9	and	and	CCONJ
ejpam-3380	271	10	h	h	NOUN
ejpam-3380	272	1	=	=	NOUN
ejpam-3380	272	2	<	<	X
ejpam-3380	272	3	3	3	NUM
ejpam-3380	272	4	>	>	PUNCT
ejpam-3380	272	5	=	=	SYM
ejpam-3380	272	6	{	{	PUNCT
ejpam-3380	272	7	0	0	NUM
ejpam-3380	272	8	,	,	PUNCT
ejpam-3380	272	9	3	3	NUM
ejpam-3380	272	10	}	}	PUNCT
ejpam-3380	272	11	.	.	PUNCT
ejpam-3380	273	1	choose	choose	VERB
ejpam-3380	273	2	the	the	DET
ejpam-3380	273	3	set	set	NOUN
ejpam-3380	273	4	of	of	ADP
ejpam-3380	273	5	left	left	ADJ
ejpam-3380	273	6	coset	coset	NOUN
ejpam-3380	273	7	representatives	representative	NOUN
ejpam-3380	273	8	to	to	PART
ejpam-3380	273	9	be	be	AUX
ejpam-3380	273	10	g	g	NOUN
ejpam-3380	273	11	=	=	PUNCT
ejpam-3380	273	12	{	{	PUNCT
ejpam-3380	273	13	1	1	NUM
ejpam-3380	273	14	,	,	PUNCT
ejpam-3380	273	15	3	3	NUM
ejpam-3380	273	16	,	,	PUNCT
ejpam-3380	273	17	5	5	NUM
ejpam-3380	273	18	}	}	PUNCT
ejpam-3380	273	19	.	.	PUNCT
ejpam-3380	274	1	then	then	ADV
ejpam-3380	274	2	the	the	DET
ejpam-3380	274	3	∗	∗	NOUN
ejpam-3380	274	4	and	and	CCONJ
ejpam-3380	274	5	f	f	PROPN
ejpam-3380	274	6	operations	operation	NOUN
ejpam-3380	274	7	as	as	ADV
ejpam-3380	274	8	well	well	ADV
ejpam-3380	274	9	as	as	ADP
ejpam-3380	274	10	the	the	DET
ejpam-3380	274	11	actions	action	NOUN
ejpam-3380	274	12	/	/	PUNCT
ejpam-3380	274	13	,	,	PUNCT
ejpam-3380	274	14	.	.	PUNCT
ejpam-3380	275	1	are	be	AUX
ejpam-3380	275	2	given	give	VERB
ejpam-3380	275	3	by	by	ADP
ejpam-3380	275	4	the	the	DET
ejpam-3380	275	5	following	following	ADJ
ejpam-3380	275	6	tables	table	NOUN
ejpam-3380	275	7	:	:	PUNCT
ejpam-3380	275	8	if	if	SCONJ
ejpam-3380	275	9	we	we	PRON
ejpam-3380	275	10	consider	consider	VERB
ejpam-3380	275	11	the	the	DET
ejpam-3380	275	12	morita	morita	PROPN
ejpam-3380	275	13	ring	ring	PROPN
ejpam-3380	275	14	t	t	PROPN
ejpam-3380	275	15	=	=	PUNCT
ejpam-3380	275	16	(	(	PUNCT
ejpam-3380	275	17	r	r	NOUN
ejpam-3380	275	18	m	m	VERB
ejpam-3380	275	19	n	n	NUM
ejpam-3380	275	20	s	s	PART
ejpam-3380	275	21	)	)	PUNCT
ejpam-3380	275	22	,	,	PUNCT
ejpam-3380	275	23	then	then	ADV
ejpam-3380	275	24	we	we	PRON
ejpam-3380	275	25	have	have	VERB
ejpam-3380	275	26	:	:	PUNCT
ejpam-3380	275	27	t	t	PROPN
ejpam-3380	275	28	=	=	SYM
ejpam-3380	275	29	t1	t1	PROPN
ejpam-3380	275	30	⊕	⊕	PROPN
ejpam-3380	275	31	t3	t3	PROPN
ejpam-3380	275	32	⊕	⊕	PROPN
ejpam-3380	275	33	t5	t5	PROPN
ejpam-3380	275	34	,	,	PUNCT
ejpam-3380	275	35	where	where	SCONJ
ejpam-3380	275	36	t1	t1	NOUN
ejpam-3380	275	37	=	=	PUNCT
ejpam-3380	275	38	(	(	PUNCT
ejpam-3380	275	39	0	0	NUM
ejpam-3380	275	40	m	m	NOUN
ejpam-3380	275	41	0	0	NUM
ejpam-3380	275	42	0	0	NUM
ejpam-3380	275	43	)	)	PUNCT
ejpam-3380	275	44	,	,	PUNCT
ejpam-3380	275	45	t3	t3	NOUN
ejpam-3380	275	46	=	=	PUNCT
ejpam-3380	276	1	(	(	PUNCT
ejpam-3380	276	2	r	r	NOUN
ejpam-3380	276	3	0	0	NUM
ejpam-3380	276	4	0	0	NUM
ejpam-3380	276	5	s	s	NOUN
ejpam-3380	276	6	)	)	PUNCT
ejpam-3380	276	7	and	and	CCONJ
ejpam-3380	276	8	t5	t5	PROPN
ejpam-3380	276	9	=	=	PUNCT
ejpam-3380	276	10	(	(	PUNCT
ejpam-3380	276	11	0	0	NUM
ejpam-3380	276	12	0	0	NUM
ejpam-3380	276	13	n	n	NOUN
ejpam-3380	276	14	0	0	NUM
ejpam-3380	276	15	)	)	PUNCT
ejpam-3380	276	16	.	.	PUNCT
ejpam-3380	277	1	najla	najla	PROPN
ejpam-3380	277	2	al	al	PROPN
ejpam-3380	277	3	-	-	PUNCT
ejpam-3380	277	4	subaie	subaie	NOUN
ejpam-3380	277	5	,	,	PUNCT
ejpam-3380	277	6	m.	m.	NOUN
ejpam-3380	277	7	m.	m.	PROPN
ejpam-3380	277	8	al	al	PROPN
ejpam-3380	277	9	-	-	PUNCT
ejpam-3380	277	10	shomrani	shomrani	PROPN
ejpam-3380	277	11	/	/	SYM
ejpam-3380	277	12	eur	eur	NOUN
ejpam-3380	277	13	.	.	PUNCT
ejpam-3380	278	1	j.	j.	PROPN
ejpam-3380	278	2	pure	pure	PROPN
ejpam-3380	278	3	appl	appl	PROPN
ejpam-3380	278	4	.	.	PROPN
ejpam-3380	278	5	math	math	PROPN
ejpam-3380	278	6	,	,	PUNCT
ejpam-3380	278	7	12	12	NUM
ejpam-3380	278	8	(	(	PUNCT
ejpam-3380	278	9	2	2	NUM
ejpam-3380	278	10	)	)	PUNCT
ejpam-3380	278	11	(	(	PUNCT
ejpam-3380	278	12	2019	2019	NUM
ejpam-3380	278	13	)	)	PUNCT
ejpam-3380	278	14	,	,	PUNCT
ejpam-3380	278	15	332	332	NUM
ejpam-3380	278	16	-	-	SYM
ejpam-3380	278	17	347	347	NUM
ejpam-3380	278	18	343	343	NUM
ejpam-3380	278	19	table	table	NOUN
ejpam-3380	278	20	3	3	NUM
ejpam-3380	278	21	:	:	PUNCT
ejpam-3380	278	22	∗	∗	NOUN
ejpam-3380	278	23	and	and	CCONJ
ejpam-3380	278	24	f	f	PROPN
ejpam-3380	278	25	operations	operation	NOUN
ejpam-3380	278	26	.	.	PUNCT
ejpam-3380	279	1	∗	∗	NOUN
ejpam-3380	279	2	3	3	NUM
ejpam-3380	279	3	1	1	NUM
ejpam-3380	279	4	5	5	NUM
ejpam-3380	279	5	3	3	NUM
ejpam-3380	279	6	3	3	NUM
ejpam-3380	279	7	1	1	NUM
ejpam-3380	279	8	5	5	NUM
ejpam-3380	279	9	1	1	NUM
ejpam-3380	279	10	1	1	NUM
ejpam-3380	279	11	5	5	NUM
ejpam-3380	279	12	3	3	NUM
ejpam-3380	279	13	5	5	NUM
ejpam-3380	279	14	5	5	NUM
ejpam-3380	279	15	3	3	NUM
ejpam-3380	279	16	1	1	NUM
ejpam-3380	279	17	f	f	NOUN
ejpam-3380	279	18	3	3	NUM
ejpam-3380	279	19	1	1	NUM
ejpam-3380	279	20	5	5	NUM
ejpam-3380	279	21	3	3	NUM
ejpam-3380	279	22	3	3	NUM
ejpam-3380	279	23	3	3	NUM
ejpam-3380	279	24	3	3	NUM
ejpam-3380	279	25	1	1	NUM
ejpam-3380	279	26	3	3	NUM
ejpam-3380	279	27	3	3	NUM
ejpam-3380	279	28	3	3	NUM
ejpam-3380	279	29	5	5	NUM
ejpam-3380	279	30	3	3	NUM
ejpam-3380	279	31	3	3	NUM
ejpam-3380	279	32	3	3	NUM
ejpam-3380	279	33	table	table	NOUN
ejpam-3380	279	34	4	4	NUM
ejpam-3380	279	35	:	:	PUNCT
ejpam-3380	279	36	.	.	PUNCT
ejpam-3380	280	1	and	and	CCONJ
ejpam-3380	280	2	/	/	SYM
ejpam-3380	280	3	actions	action	NOUN
ejpam-3380	280	4	.	.	PUNCT
ejpam-3380	281	1	s	s	PART
ejpam-3380	281	2	.	.	PUNCT
ejpam-3380	282	1	u	u	NOUN
ejpam-3380	282	2	0	0	NUM
ejpam-3380	282	3	3	3	NUM
ejpam-3380	282	4	3	3	NUM
ejpam-3380	282	5	0	0	NUM
ejpam-3380	282	6	3	3	NUM
ejpam-3380	282	7	1	1	NUM
ejpam-3380	282	8	0	0	NUM
ejpam-3380	282	9	3	3	NUM
ejpam-3380	282	10	5	5	NUM
ejpam-3380	282	11	0	0	NUM
ejpam-3380	282	12	3	3	NUM
ejpam-3380	282	13	s	s	NOUN
ejpam-3380	282	14	/	/	SYM
ejpam-3380	282	15	u	u	NOUN
ejpam-3380	282	16	0	0	NUM
ejpam-3380	282	17	3	3	NUM
ejpam-3380	282	18	3	3	NUM
ejpam-3380	282	19	3	3	NUM
ejpam-3380	282	20	3	3	NUM
ejpam-3380	282	21	1	1	NUM
ejpam-3380	282	22	1	1	NUM
ejpam-3380	282	23	1	1	NUM
ejpam-3380	282	24	5	5	NUM
ejpam-3380	282	25	5	5	NUM
ejpam-3380	282	26	5	5	NUM
ejpam-3380	282	27	moreover	moreover	ADV
ejpam-3380	282	28	,	,	PUNCT
ejpam-3380	282	29	the	the	DET
ejpam-3380	282	30	inclusion	inclusion	NOUN
ejpam-3380	282	31	property	property	NOUN
ejpam-3380	282	32	tstt	tstt	NOUN
ejpam-3380	282	33	⊆	⊆	NUM
ejpam-3380	282	34	ts∗t	ts∗t	PROPN
ejpam-3380	282	35	is	be	AUX
ejpam-3380	282	36	satisfied	satisfied	ADJ
ejpam-3380	282	37	for	for	ADP
ejpam-3380	282	38	all	all	DET
ejpam-3380	282	39	s	s	PROPN
ejpam-3380	282	40	,	,	PUNCT
ejpam-3380	282	41	t	t	PROPN
ejpam-3380	282	42	∈	∈	PROPN
ejpam-3380	282	43	g	g	PROPN
ejpam-3380	282	44	which	which	PRON
ejpam-3380	282	45	can	can	AUX
ejpam-3380	282	46	illustrated	illustrate	VERB
ejpam-3380	282	47	as	as	SCONJ
ejpam-3380	282	48	follows	follow	VERB
ejpam-3380	282	49	:	:	PUNCT
ejpam-3380	282	50	(	(	PUNCT
ejpam-3380	282	51	i	i	NOUN
ejpam-3380	282	52	)	)	PUNCT
ejpam-3380	282	53	t3t3	t3t3	NOUN
ejpam-3380	283	1	⊆	⊆	NUM
ejpam-3380	283	2	t3∗3	t3∗3	NOUN
ejpam-3380	283	3	=	=	SYM
ejpam-3380	283	4	t3	t3	PROPN
ejpam-3380	283	5	,	,	PUNCT
ejpam-3380	283	6	as	as	ADP
ejpam-3380	283	7	for	for	ADP
ejpam-3380	283	8	all	all	PRON
ejpam-3380	283	9	(	(	PUNCT
ejpam-3380	283	10	r1	r1	PROPN
ejpam-3380	283	11	0	0	NUM
ejpam-3380	283	12	0	0	NUM
ejpam-3380	283	13	s1	s1	NOUN
ejpam-3380	283	14	)	)	PUNCT
ejpam-3380	283	15	and	and	CCONJ
ejpam-3380	283	16	(	(	PUNCT
ejpam-3380	283	17	r2	r2	PROPN
ejpam-3380	283	18	0	0	NUM
ejpam-3380	283	19	0	0	NUM
ejpam-3380	283	20	s2	s2	PROPN
ejpam-3380	283	21	)	)	PUNCT
ejpam-3380	283	22	∈	∈	PROPN
ejpam-3380	283	23	t3	t3	PROPN
ejpam-3380	283	24	,	,	PUNCT
ejpam-3380	283	25	we	we	PRON
ejpam-3380	283	26	have	have	VERB
ejpam-3380	283	27	(	(	PUNCT
ejpam-3380	283	28	r1	r1	VERB
ejpam-3380	283	29	0	0	NUM
ejpam-3380	283	30	0	0	NUM
ejpam-3380	283	31	s1	s1	NOUN
ejpam-3380	283	32	)	)	PUNCT
ejpam-3380	283	33	(	(	PUNCT
ejpam-3380	283	34	r2	r2	PROPN
ejpam-3380	283	35	0	0	NUM
ejpam-3380	283	36	0	0	NUM
ejpam-3380	283	37	s2	s2	NOUN
ejpam-3380	283	38	)	)	PUNCT
ejpam-3380	284	1	=	=	PUNCT
ejpam-3380	284	2	(	(	PUNCT
ejpam-3380	284	3	r1r2	r1r2	X
ejpam-3380	284	4	0	0	NUM
ejpam-3380	284	5	0	0	NUM
ejpam-3380	284	6	s1s2	s1s2	PROPN
ejpam-3380	284	7	)	)	PUNCT
ejpam-3380	284	8	∈	∈	PROPN
ejpam-3380	284	9	t3	t3	NOUN
ejpam-3380	284	10	=	=	PUNCT
ejpam-3380	284	11	t3∗3	t3∗3	X
ejpam-3380	284	12	.	.	PUNCT
ejpam-3380	285	1	(	(	PUNCT
ejpam-3380	285	2	ii	ii	NOUN
ejpam-3380	285	3	)	)	PUNCT
ejpam-3380	285	4	t3t1	t3t1	PUNCT
ejpam-3380	286	1	⊆	⊆	NUM
ejpam-3380	286	2	t3∗1	t3∗1	PROPN
ejpam-3380	286	3	=	=	SYM
ejpam-3380	286	4	t1	t1	PROPN
ejpam-3380	286	5	,	,	PUNCT
ejpam-3380	286	6	as	as	ADP
ejpam-3380	286	7	for	for	ADP
ejpam-3380	286	8	all	all	DET
ejpam-3380	286	9	(	(	PUNCT
ejpam-3380	286	10	r	r	NOUN
ejpam-3380	286	11	0	0	NUM
ejpam-3380	286	12	0	0	NUM
ejpam-3380	286	13	s	s	PART
ejpam-3380	286	14	)	)	PUNCT
ejpam-3380	286	15	∈	∈	PROPN
ejpam-3380	286	16	t3	t3	NOUN
ejpam-3380	286	17	and	and	CCONJ
ejpam-3380	286	18	(	(	PUNCT
ejpam-3380	286	19	0	0	NUM
ejpam-3380	286	20	m	m	NOUN
ejpam-3380	286	21	0	0	NUM
ejpam-3380	286	22	0	0	NUM
ejpam-3380	286	23	)	)	PUNCT
ejpam-3380	286	24	∈	∈	PROPN
ejpam-3380	286	25	t1	t1	NOUN
ejpam-3380	286	26	,	,	PUNCT
ejpam-3380	286	27	we	we	PRON
ejpam-3380	286	28	have	have	VERB
ejpam-3380	286	29	(	(	PUNCT
ejpam-3380	286	30	r	r	NOUN
ejpam-3380	286	31	0	0	NUM
ejpam-3380	286	32	0	0	NUM
ejpam-3380	286	33	s	s	NOUN
ejpam-3380	286	34	)	)	PUNCT
ejpam-3380	286	35	(	(	PUNCT
ejpam-3380	286	36	0	0	NUM
ejpam-3380	286	37	m	m	NOUN
ejpam-3380	286	38	0	0	NUM
ejpam-3380	286	39	0	0	NUM
ejpam-3380	286	40	)	)	PUNCT
ejpam-3380	287	1	=	=	PRON
ejpam-3380	287	2	(	(	PUNCT
ejpam-3380	287	3	0	0	NUM
ejpam-3380	287	4	rm	rm	NOUN
ejpam-3380	287	5	0	0	NUM
ejpam-3380	287	6	0	0	NUM
ejpam-3380	287	7	)	)	PUNCT
ejpam-3380	287	8	∈	∈	PROPN
ejpam-3380	287	9	t1	t1	NOUN
ejpam-3380	287	10	=	=	SYM
ejpam-3380	288	1	t3∗1	t3∗1	PROPN
ejpam-3380	288	2	.	.	PUNCT
ejpam-3380	289	1	(	(	PUNCT
ejpam-3380	289	2	iii	iii	X
ejpam-3380	289	3	)	)	PUNCT
ejpam-3380	289	4	t3t5	t3t5	NOUN
ejpam-3380	289	5	⊆	⊆	NUM
ejpam-3380	289	6	t3∗5	t3∗5	NOUN
ejpam-3380	289	7	=	=	PUNCT
ejpam-3380	289	8	t5	t5	PROPN
ejpam-3380	289	9	,	,	PUNCT
ejpam-3380	289	10	as	as	ADP
ejpam-3380	289	11	for	for	ADP
ejpam-3380	289	12	all	all	DET
ejpam-3380	289	13	(	(	PUNCT
ejpam-3380	289	14	r	r	NOUN
ejpam-3380	289	15	0	0	NUM
ejpam-3380	289	16	0	0	NUM
ejpam-3380	289	17	s	s	PART
ejpam-3380	289	18	)	)	PUNCT
ejpam-3380	289	19	∈	∈	PROPN
ejpam-3380	289	20	t3	t3	NOUN
ejpam-3380	289	21	and	and	CCONJ
ejpam-3380	289	22	(	(	PUNCT
ejpam-3380	289	23	0	0	NUM
ejpam-3380	289	24	0	0	NUM
ejpam-3380	289	25	s	s	NOUN
ejpam-3380	289	26	0	0	NUM
ejpam-3380	289	27	)	)	PUNCT
ejpam-3380	289	28	∈	∈	PROPN
ejpam-3380	289	29	t5	t5	PROPN
ejpam-3380	289	30	,	,	PUNCT
ejpam-3380	289	31	we	we	PRON
ejpam-3380	289	32	have	have	VERB
ejpam-3380	289	33	(	(	PUNCT
ejpam-3380	289	34	r	r	NOUN
ejpam-3380	289	35	0	0	NUM
ejpam-3380	289	36	0	0	NUM
ejpam-3380	289	37	s	s	NOUN
ejpam-3380	289	38	)	)	PUNCT
ejpam-3380	289	39	(	(	PUNCT
ejpam-3380	289	40	0	0	NUM
ejpam-3380	289	41	0	0	NUM
ejpam-3380	289	42	s	s	NOUN
ejpam-3380	289	43	0	0	NUM
ejpam-3380	289	44	)	)	PUNCT
ejpam-3380	289	45	=	=	PRON
ejpam-3380	290	1	(	(	PUNCT
ejpam-3380	290	2	0	0	NUM
ejpam-3380	290	3	0	0	NUM
ejpam-3380	290	4	sn	sn	NOUN
ejpam-3380	290	5	0	0	NUM
ejpam-3380	290	6	)	)	PUNCT
ejpam-3380	290	7	∈	∈	PROPN
ejpam-3380	290	8	t5	t5	PROPN
ejpam-3380	290	9	=	=	SYM
ejpam-3380	290	10	t3∗5	t3∗5	PROPN
ejpam-3380	290	11	.	.	PUNCT
ejpam-3380	291	1	(	(	PUNCT
ejpam-3380	291	2	iv	iv	X
ejpam-3380	291	3	)	)	PUNCT
ejpam-3380	291	4	t1t3	t1t3	NOUN
ejpam-3380	291	5	⊆	⊆	NUM
ejpam-3380	291	6	t1∗3	t1∗3	ADJ
ejpam-3380	292	1	=	=	SYM
ejpam-3380	292	2	t1	t1	NOUN
ejpam-3380	292	3	,	,	PUNCT
ejpam-3380	292	4	as	as	ADP
ejpam-3380	292	5	for	for	ADP
ejpam-3380	292	6	all	all	PRON
ejpam-3380	292	7	(	(	PUNCT
ejpam-3380	292	8	0	0	NUM
ejpam-3380	292	9	m	m	NOUN
ejpam-3380	292	10	0	0	NUM
ejpam-3380	292	11	0	0	NUM
ejpam-3380	292	12	)	)	PUNCT
ejpam-3380	292	13	∈	∈	PROPN
ejpam-3380	292	14	t1	t1	NOUN
ejpam-3380	292	15	and	and	CCONJ
ejpam-3380	292	16	(	(	PUNCT
ejpam-3380	292	17	r	r	NOUN
ejpam-3380	292	18	0	0	NUM
ejpam-3380	292	19	0	0	NUM
ejpam-3380	292	20	s	s	PART
ejpam-3380	292	21	)	)	PUNCT
ejpam-3380	292	22	∈	∈	PROPN
ejpam-3380	292	23	t3	t3	PROPN
ejpam-3380	292	24	,	,	PUNCT
ejpam-3380	292	25	we	we	PRON
ejpam-3380	292	26	have	have	VERB
ejpam-3380	292	27	(	(	PUNCT
ejpam-3380	292	28	0	0	NUM
ejpam-3380	292	29	m	m	NOUN
ejpam-3380	292	30	0	0	NUM
ejpam-3380	292	31	0	0	NUM
ejpam-3380	292	32	)	)	PUNCT
ejpam-3380	293	1	(	(	PUNCT
ejpam-3380	293	2	r	r	NOUN
ejpam-3380	293	3	0	0	NUM
ejpam-3380	293	4	0	0	NUM
ejpam-3380	293	5	s	s	PART
ejpam-3380	293	6	)	)	PUNCT
ejpam-3380	293	7	=	=	SYM
ejpam-3380	294	1	(	(	PUNCT
ejpam-3380	294	2	0	0	NUM
ejpam-3380	294	3	ms	ms	NOUN
ejpam-3380	294	4	0	0	NUM
ejpam-3380	294	5	0	0	NUM
ejpam-3380	294	6	)	)	PUNCT
ejpam-3380	294	7	∈	∈	PROPN
ejpam-3380	294	8	t1	t1	NOUN
ejpam-3380	294	9	=	=	PUNCT
ejpam-3380	295	1	t1∗3	t1∗3	ADJ
ejpam-3380	295	2	.	.	PUNCT
ejpam-3380	296	1	(	(	PUNCT
ejpam-3380	296	2	v	v	NOUN
ejpam-3380	296	3	)	)	PUNCT
ejpam-3380	296	4	t1t1	t1t1	CCONJ
ejpam-3380	297	1	⊆	⊆	NUM
ejpam-3380	297	2	t1∗1	t1∗1	NOUN
ejpam-3380	297	3	=	=	SYM
ejpam-3380	297	4	t5	t5	PROPN
ejpam-3380	297	5	,	,	PUNCT
ejpam-3380	297	6	as	as	ADP
ejpam-3380	297	7	for	for	ADP
ejpam-3380	297	8	all	all	PRON
ejpam-3380	297	9	(	(	PUNCT
ejpam-3380	297	10	0	0	NUM
ejpam-3380	297	11	m1	m1	NOUN
ejpam-3380	297	12	0	0	NUM
ejpam-3380	297	13	0	0	NUM
ejpam-3380	297	14	)	)	PUNCT
ejpam-3380	297	15	and	and	CCONJ
ejpam-3380	297	16	(	(	PUNCT
ejpam-3380	297	17	0	0	NUM
ejpam-3380	297	18	m2	m2	PROPN
ejpam-3380	297	19	0	0	NUM
ejpam-3380	297	20	0	0	NUM
ejpam-3380	297	21	)	)	PUNCT
ejpam-3380	297	22	∈	∈	PROPN
ejpam-3380	297	23	t1	t1	NOUN
ejpam-3380	297	24	,	,	PUNCT
ejpam-3380	297	25	we	we	PRON
ejpam-3380	297	26	have	have	VERB
ejpam-3380	297	27	(	(	PUNCT
ejpam-3380	297	28	0	0	NUM
ejpam-3380	297	29	m1	m1	NOUN
ejpam-3380	297	30	0	0	NUM
ejpam-3380	297	31	0	0	NUM
ejpam-3380	297	32	)	)	PUNCT
ejpam-3380	297	33	(	(	PUNCT
ejpam-3380	297	34	0	0	NUM
ejpam-3380	297	35	m2	m2	PROPN
ejpam-3380	297	36	0	0	NUM
ejpam-3380	297	37	0	0	NUM
ejpam-3380	297	38	)	)	PUNCT
ejpam-3380	297	39	=	=	PRON
ejpam-3380	298	1	(	(	PUNCT
ejpam-3380	298	2	0	0	NUM
ejpam-3380	298	3	0	0	NUM
ejpam-3380	298	4	0	0	NUM
ejpam-3380	298	5	0	0	NUM
ejpam-3380	298	6	)	)	PUNCT
ejpam-3380	298	7	∈	∈	PROPN
ejpam-3380	298	8	t5	t5	PROPN
ejpam-3380	298	9	=	=	SYM
ejpam-3380	298	10	t1∗1	t1∗1	PROPN
ejpam-3380	298	11	.	.	PUNCT
ejpam-3380	299	1	najla	najla	PROPN
ejpam-3380	299	2	al	al	PROPN
ejpam-3380	299	3	-	-	PUNCT
ejpam-3380	299	4	subaie	subaie	NOUN
ejpam-3380	299	5	,	,	PUNCT
ejpam-3380	299	6	m.	m.	NOUN
ejpam-3380	299	7	m.	m.	PROPN
ejpam-3380	299	8	al	al	PROPN
ejpam-3380	299	9	-	-	PUNCT
ejpam-3380	299	10	shomrani	shomrani	PROPN
ejpam-3380	299	11	/	/	SYM
ejpam-3380	299	12	eur	eur	NOUN
ejpam-3380	299	13	.	.	PUNCT
ejpam-3380	300	1	j.	j.	PROPN
ejpam-3380	300	2	pure	pure	PROPN
ejpam-3380	300	3	appl	appl	PROPN
ejpam-3380	300	4	.	.	PROPN
ejpam-3380	300	5	math	math	PROPN
ejpam-3380	300	6	,	,	PUNCT
ejpam-3380	300	7	12	12	NUM
ejpam-3380	300	8	(	(	PUNCT
ejpam-3380	300	9	2	2	NUM
ejpam-3380	300	10	)	)	PUNCT
ejpam-3380	300	11	(	(	PUNCT
ejpam-3380	300	12	2019	2019	NUM
ejpam-3380	300	13	)	)	PUNCT
ejpam-3380	300	14	,	,	PUNCT
ejpam-3380	300	15	332	332	NUM
ejpam-3380	300	16	-	-	SYM
ejpam-3380	300	17	347	347	NUM
ejpam-3380	300	18	344	344	NUM
ejpam-3380	300	19	(	(	PUNCT
ejpam-3380	300	20	vi	vi	NOUN
ejpam-3380	300	21	)	)	PUNCT
ejpam-3380	300	22	t1t5	t1t5	VERB
ejpam-3380	300	23	⊆	⊆	NUM
ejpam-3380	300	24	t1∗5	t1∗5	NOUN
ejpam-3380	300	25	=	=	SYM
ejpam-3380	300	26	t3	t3	PROPN
ejpam-3380	300	27	,	,	PUNCT
ejpam-3380	300	28	as	as	ADP
ejpam-3380	300	29	for	for	ADP
ejpam-3380	300	30	all	all	PRON
ejpam-3380	300	31	(	(	PUNCT
ejpam-3380	300	32	0	0	NUM
ejpam-3380	300	33	m	m	NOUN
ejpam-3380	300	34	0	0	NUM
ejpam-3380	300	35	0	0	NUM
ejpam-3380	300	36	)	)	PUNCT
ejpam-3380	300	37	∈	∈	PROPN
ejpam-3380	300	38	t1	t1	NOUN
ejpam-3380	300	39	and	and	CCONJ
ejpam-3380	300	40	(	(	PUNCT
ejpam-3380	300	41	0	0	NUM
ejpam-3380	300	42	0	0	NUM
ejpam-3380	300	43	n	n	CCONJ
ejpam-3380	300	44	0	0	NUM
ejpam-3380	300	45	)	)	PUNCT
ejpam-3380	300	46	∈	∈	PROPN
ejpam-3380	300	47	t5	t5	PROPN
ejpam-3380	300	48	,	,	PUNCT
ejpam-3380	300	49	we	we	PRON
ejpam-3380	300	50	have	have	VERB
ejpam-3380	300	51	(	(	PUNCT
ejpam-3380	300	52	0	0	NUM
ejpam-3380	300	53	m	m	NOUN
ejpam-3380	300	54	0	0	NUM
ejpam-3380	300	55	0	0	NUM
ejpam-3380	300	56	)	)	PUNCT
ejpam-3380	300	57	(	(	PUNCT
ejpam-3380	300	58	0	0	NUM
ejpam-3380	300	59	0	0	NUM
ejpam-3380	300	60	n	n	NOUN
ejpam-3380	300	61	0	0	NUM
ejpam-3380	300	62	)	)	PUNCT
ejpam-3380	301	1	=	=	PRON
ejpam-3380	301	2	(	(	PUNCT
ejpam-3380	301	3	mn	mn	PROPN
ejpam-3380	301	4	0	0	NUM
ejpam-3380	301	5	0	0	NUM
ejpam-3380	301	6	0	0	NUM
ejpam-3380	301	7	)	)	PUNCT
ejpam-3380	302	1	∈	∈	PROPN
ejpam-3380	302	2	t3	t3	PROPN
ejpam-3380	302	3	=	=	SYM
ejpam-3380	302	4	t1∗5	t1∗5	PROPN
ejpam-3380	302	5	.	.	PUNCT
ejpam-3380	303	1	(	(	PUNCT
ejpam-3380	303	2	vii	vii	PROPN
ejpam-3380	303	3	)	)	PUNCT
ejpam-3380	303	4	t5t3	t5t3	PRON
ejpam-3380	304	1	⊆	⊆	NUM
ejpam-3380	304	2	t5∗3	t5∗3	NUM
ejpam-3380	304	3	=	=	SYM
ejpam-3380	304	4	t5	t5	PROPN
ejpam-3380	304	5	,	,	PUNCT
ejpam-3380	304	6	as	as	ADP
ejpam-3380	304	7	for	for	ADP
ejpam-3380	304	8	all	all	DET
ejpam-3380	304	9	(	(	PUNCT
ejpam-3380	304	10	0	0	NUM
ejpam-3380	304	11	0	0	NUM
ejpam-3380	304	12	n	n	CCONJ
ejpam-3380	304	13	0	0	NUM
ejpam-3380	304	14	)	)	PUNCT
ejpam-3380	304	15	∈	∈	PROPN
ejpam-3380	304	16	t5	t5	PROPN
ejpam-3380	304	17	and	and	CCONJ
ejpam-3380	304	18	(	(	PUNCT
ejpam-3380	304	19	r	r	NOUN
ejpam-3380	304	20	0	0	NUM
ejpam-3380	304	21	0	0	NUM
ejpam-3380	304	22	s	s	PART
ejpam-3380	304	23	)	)	PUNCT
ejpam-3380	304	24	∈	∈	PROPN
ejpam-3380	304	25	t3	t3	PROPN
ejpam-3380	304	26	,	,	PUNCT
ejpam-3380	304	27	we	we	PRON
ejpam-3380	304	28	have	have	VERB
ejpam-3380	304	29	(	(	PUNCT
ejpam-3380	304	30	0	0	NUM
ejpam-3380	304	31	0	0	NUM
ejpam-3380	304	32	n	n	NOUN
ejpam-3380	304	33	0	0	NUM
ejpam-3380	304	34	)	)	PUNCT
ejpam-3380	305	1	(	(	PUNCT
ejpam-3380	305	2	r	r	NOUN
ejpam-3380	305	3	0	0	NUM
ejpam-3380	305	4	0	0	NUM
ejpam-3380	305	5	s	s	PART
ejpam-3380	305	6	)	)	PUNCT
ejpam-3380	305	7	=	=	SYM
ejpam-3380	306	1	(	(	PUNCT
ejpam-3380	306	2	0	0	NUM
ejpam-3380	306	3	0	0	NUM
ejpam-3380	306	4	nr	nr	NOUN
ejpam-3380	306	5	0	0	NUM
ejpam-3380	306	6	)	)	PUNCT
ejpam-3380	306	7	∈	∈	PROPN
ejpam-3380	306	8	t5	t5	PROPN
ejpam-3380	306	9	=	=	PUNCT
ejpam-3380	306	10	t5∗3	t5∗3	X
ejpam-3380	306	11	.	.	PUNCT
ejpam-3380	307	1	(	(	PUNCT
ejpam-3380	307	2	viii	viii	NOUN
ejpam-3380	307	3	)	)	PUNCT
ejpam-3380	307	4	t5t1	t5t1	NOUN
ejpam-3380	307	5	⊆	⊆	NUM
ejpam-3380	307	6	t5∗1	t5∗1	NUM
ejpam-3380	307	7	=	=	SYM
ejpam-3380	307	8	t3	t3	PROPN
ejpam-3380	307	9	,	,	PUNCT
ejpam-3380	307	10	as	as	ADP
ejpam-3380	307	11	for	for	ADP
ejpam-3380	307	12	all	all	DET
ejpam-3380	307	13	(	(	PUNCT
ejpam-3380	307	14	0	0	NUM
ejpam-3380	307	15	0	0	NUM
ejpam-3380	307	16	n	n	CCONJ
ejpam-3380	307	17	0	0	NUM
ejpam-3380	307	18	)	)	PUNCT
ejpam-3380	307	19	∈	∈	PROPN
ejpam-3380	307	20	t5	t5	PROPN
ejpam-3380	307	21	and	and	CCONJ
ejpam-3380	307	22	(	(	PUNCT
ejpam-3380	307	23	0	0	NUM
ejpam-3380	307	24	m	m	NOUN
ejpam-3380	307	25	0	0	NUM
ejpam-3380	307	26	0	0	NUM
ejpam-3380	307	27	)	)	PUNCT
ejpam-3380	307	28	∈	∈	PROPN
ejpam-3380	307	29	t1	t1	NOUN
ejpam-3380	307	30	,	,	PUNCT
ejpam-3380	307	31	we	we	PRON
ejpam-3380	307	32	have	have	VERB
ejpam-3380	307	33	(	(	PUNCT
ejpam-3380	307	34	0	0	NUM
ejpam-3380	307	35	0	0	NUM
ejpam-3380	307	36	n	n	NOUN
ejpam-3380	307	37	0	0	NUM
ejpam-3380	307	38	)	)	PUNCT
ejpam-3380	307	39	(	(	PUNCT
ejpam-3380	307	40	0	0	NUM
ejpam-3380	307	41	m	m	NOUN
ejpam-3380	307	42	0	0	NUM
ejpam-3380	307	43	0	0	NUM
ejpam-3380	307	44	)	)	PUNCT
ejpam-3380	308	1	=	=	PRON
ejpam-3380	308	2	(	(	PUNCT
ejpam-3380	308	3	0	0	NUM
ejpam-3380	308	4	0	0	NUM
ejpam-3380	308	5	0	0	NUM
ejpam-3380	308	6	mn	mn	PROPN
ejpam-3380	308	7	)	)	PUNCT
ejpam-3380	308	8	∈	∈	PROPN
ejpam-3380	308	9	t3	t3	NOUN
ejpam-3380	308	10	=	=	SYM
ejpam-3380	308	11	t5∗1	t5∗1	PROPN
ejpam-3380	308	12	.	.	PUNCT
ejpam-3380	309	1	(	(	PUNCT
ejpam-3380	309	2	ix	ix	ADP
ejpam-3380	309	3	)	)	PUNCT
ejpam-3380	309	4	t5t5	t5t5	PROPN
ejpam-3380	310	1	⊆	⊆	NUM
ejpam-3380	310	2	t5∗5	t5∗5	NOUN
ejpam-3380	310	3	=	=	SYM
ejpam-3380	310	4	t1	t1	PROPN
ejpam-3380	310	5	,	,	PUNCT
ejpam-3380	310	6	as	as	ADP
ejpam-3380	310	7	for	for	ADP
ejpam-3380	310	8	all	all	DET
ejpam-3380	310	9	(	(	PUNCT
ejpam-3380	310	10	0	0	NUM
ejpam-3380	310	11	0	0	NUM
ejpam-3380	310	12	n1	n1	NOUN
ejpam-3380	310	13	0	0	NUM
ejpam-3380	310	14	)	)	PUNCT
ejpam-3380	310	15	and	and	CCONJ
ejpam-3380	310	16	(	(	PUNCT
ejpam-3380	310	17	0	0	NUM
ejpam-3380	310	18	0	0	NUM
ejpam-3380	310	19	n2	n2	NOUN
ejpam-3380	310	20	0	0	NUM
ejpam-3380	310	21	)	)	PUNCT
ejpam-3380	310	22	∈	∈	PROPN
ejpam-3380	310	23	t5	t5	PROPN
ejpam-3380	310	24	,	,	PUNCT
ejpam-3380	310	25	we	we	PRON
ejpam-3380	310	26	have	have	VERB
ejpam-3380	310	27	(	(	PUNCT
ejpam-3380	310	28	0	0	NUM
ejpam-3380	310	29	0	0	NUM
ejpam-3380	310	30	n1	n1	NOUN
ejpam-3380	310	31	0	0	NUM
ejpam-3380	310	32	)	)	PUNCT
ejpam-3380	310	33	(	(	PUNCT
ejpam-3380	310	34	0	0	NUM
ejpam-3380	310	35	0	0	NUM
ejpam-3380	310	36	n2	n2	NOUN
ejpam-3380	310	37	0	0	NUM
ejpam-3380	310	38	)	)	PUNCT
ejpam-3380	310	39	=	=	PUNCT
ejpam-3380	311	1	(	(	PUNCT
ejpam-3380	311	2	0	0	NUM
ejpam-3380	311	3	0	0	NUM
ejpam-3380	311	4	0	0	NUM
ejpam-3380	311	5	0	0	NUM
ejpam-3380	311	6	)	)	PUNCT
ejpam-3380	311	7	∈	∈	PROPN
ejpam-3380	311	8	t1	t1	NOUN
ejpam-3380	311	9	=	=	PUNCT
ejpam-3380	312	1	t5∗5	t5∗5	PROPN
ejpam-3380	312	2	.	.	PUNCT
ejpam-3380	313	1	thus	thus	ADV
ejpam-3380	313	2	,	,	PUNCT
ejpam-3380	313	3	t	t	PROPN
ejpam-3380	313	4	is	be	AUX
ejpam-3380	313	5	a	a	DET
ejpam-3380	313	6	g	g	NOUN
ejpam-3380	313	7	-	-	PUNCT
ejpam-3380	313	8	weak	weak	ADJ
ejpam-3380	313	9	graded	grade	VERB
ejpam-3380	313	10	ring	ring	NOUN
ejpam-3380	313	11	.	.	PUNCT
ejpam-3380	314	1	however	however	ADV
ejpam-3380	314	2	,	,	PUNCT
ejpam-3380	314	3	it	it	PRON
ejpam-3380	314	4	is	be	AUX
ejpam-3380	314	5	not	not	PART
ejpam-3380	314	6	a	a	DET
ejpam-3380	314	7	fully	fully	ADV
ejpam-3380	314	8	g	g	NOUN
ejpam-3380	314	9	-	-	PUNCT
ejpam-3380	314	10	weak	weak	ADJ
ejpam-3380	314	11	graded	grade	VERB
ejpam-3380	314	12	ring	ring	NOUN
ejpam-3380	314	13	.	.	PUNCT
ejpam-3380	315	1	for	for	ADP
ejpam-3380	315	2	instance	instance	NOUN
ejpam-3380	315	3	,	,	PUNCT
ejpam-3380	315	4	t5t5	t5t5	X
ejpam-3380	315	5	6=	6=	NUM
ejpam-3380	315	6	t5∗5	t5∗5	PROPN
ejpam-3380	315	7	since	since	SCONJ
ejpam-3380	315	8	t1	t1	NOUN
ejpam-3380	315	9	=	=	NOUN
ejpam-3380	316	1	t5∗5	t5∗5	NOUN
ejpam-3380	317	1	*	*	PUNCT
ejpam-3380	317	2	t5t5	t5t5	X
ejpam-3380	317	3	.	.	PUNCT
ejpam-3380	317	4	example	example	NOUN
ejpam-3380	317	5	3	3	X
ejpam-3380	317	6	.	.	X
ejpam-3380	317	7	consider	consider	VERB
ejpam-3380	317	8	the	the	DET
ejpam-3380	317	9	ring	ring	NOUN
ejpam-3380	317	10	of	of	ADP
ejpam-3380	317	11	real	real	ADJ
ejpam-3380	317	12	quaternions	quaternion	NOUN
ejpam-3380	317	13	(	(	PUNCT
ejpam-3380	317	14	h,+	h,+	ADJ
ejpam-3380	317	15	,	,	PUNCT
ejpam-3380	317	16	·	·	PUNCT
ejpam-3380	317	17	)	)	PUNCT
ejpam-3380	317	18	.	.	PUNCT
ejpam-3380	318	1	let	let	VERB
ejpam-3380	318	2	x	x	PUNCT
ejpam-3380	318	3	=	=	PUNCT
ejpam-3380	318	4	d6	d6	NOUN
ejpam-3380	318	5	=	=	PUNCT
ejpam-3380	318	6	{	{	PUNCT
ejpam-3380	318	7	1	1	NUM
ejpam-3380	318	8	,	,	PUNCT
ejpam-3380	318	9	x	x	NOUN
ejpam-3380	318	10	,	,	PUNCT
ejpam-3380	318	11	x2	x2	PROPN
ejpam-3380	318	12	,	,	PUNCT
ejpam-3380	318	13	x3	x3	PROPN
ejpam-3380	318	14	,	,	PUNCT
ejpam-3380	318	15	x4	x4	PROPN
ejpam-3380	318	16	,	,	PUNCT
ejpam-3380	318	17	x5	x5	PROPN
ejpam-3380	318	18	,	,	PUNCT
ejpam-3380	318	19	y	y	PROPN
ejpam-3380	318	20	,	,	PUNCT
ejpam-3380	318	21	xy	xy	PROPN
ejpam-3380	318	22	,	,	PUNCT
ejpam-3380	318	23	x2y	x2y	PROPN
ejpam-3380	318	24	,	,	PUNCT
ejpam-3380	318	25	x3y	x3y	ADJ
ejpam-3380	318	26	,	,	PUNCT
ejpam-3380	318	27	x4y	x4y	PRON
ejpam-3380	318	28	,	,	PUNCT
ejpam-3380	318	29	x5y	x5y	PROPN
ejpam-3380	318	30	}	}	PUNCT
ejpam-3380	318	31	and	and	CCONJ
ejpam-3380	318	32	h	h	NOUN
ejpam-3380	318	33	=	=	SYM
ejpam-3380	318	34	{	{	PUNCT
ejpam-3380	318	35	1	1	NUM
ejpam-3380	318	36	,	,	PUNCT
ejpam-3380	318	37	x2	x2	PROPN
ejpam-3380	318	38	,	,	PUNCT
ejpam-3380	318	39	x4	x4	PROPN
ejpam-3380	318	40	}	}	PUNCT
ejpam-3380	318	41	be	be	VERB
ejpam-3380	318	42	an	an	DET
ejpam-3380	318	43	additive	additive	ADJ
ejpam-3380	318	44	subgroup	subgroup	NOUN
ejpam-3380	318	45	of	of	ADP
ejpam-3380	318	46	the	the	DET
ejpam-3380	318	47	group	group	NOUN
ejpam-3380	318	48	x.	x.	NOUN
ejpam-3380	318	49	take	take	VERB
ejpam-3380	318	50	the	the	DET
ejpam-3380	318	51	set	set	NOUN
ejpam-3380	318	52	of	of	ADP
ejpam-3380	318	53	left	left	ADJ
ejpam-3380	318	54	coset	coset	NOUN
ejpam-3380	318	55	representatives	representative	NOUN
ejpam-3380	318	56	to	to	PART
ejpam-3380	318	57	be	be	AUX
ejpam-3380	318	58	g	g	NOUN
ejpam-3380	318	59	=	=	PUNCT
ejpam-3380	318	60	{	{	PUNCT
ejpam-3380	318	61	1	1	NUM
ejpam-3380	318	62	,	,	PUNCT
ejpam-3380	318	63	y	y	PROPN
ejpam-3380	318	64	,	,	PUNCT
ejpam-3380	318	65	x5	x5	PROPN
ejpam-3380	318	66	,	,	PUNCT
ejpam-3380	318	67	xy	xy	NOUN
ejpam-3380	318	68	}	}	PUNCT
ejpam-3380	318	69	.	.	PUNCT
ejpam-3380	319	1	then	then	ADV
ejpam-3380	319	2	the	the	DET
ejpam-3380	319	3	∗	∗	NOUN
ejpam-3380	319	4	and	and	CCONJ
ejpam-3380	319	5	f	f	PROPN
ejpam-3380	319	6	operations	operation	NOUN
ejpam-3380	319	7	as	as	ADV
ejpam-3380	319	8	well	well	ADV
ejpam-3380	319	9	as	as	ADP
ejpam-3380	319	10	the	the	DET
ejpam-3380	319	11	actions	action	NOUN
ejpam-3380	319	12	/	/	PUNCT
ejpam-3380	319	13	,	,	PUNCT
ejpam-3380	319	14	.	.	PUNCT
ejpam-3380	320	1	are	be	AUX
ejpam-3380	320	2	given	give	VERB
ejpam-3380	320	3	by	by	ADP
ejpam-3380	320	4	the	the	DET
ejpam-3380	320	5	following	following	ADJ
ejpam-3380	320	6	tables	table	NOUN
ejpam-3380	320	7	:	:	PUNCT
ejpam-3380	320	8	table	table	NOUN
ejpam-3380	320	9	5	5	NUM
ejpam-3380	320	10	:	:	PUNCT
ejpam-3380	320	11	∗	∗	NOUN
ejpam-3380	320	12	and	and	CCONJ
ejpam-3380	320	13	f	f	PROPN
ejpam-3380	320	14	operations	operation	NOUN
ejpam-3380	320	15	.	.	PUNCT
ejpam-3380	321	1	∗	∗	NOUN
ejpam-3380	321	2	1	1	NUM
ejpam-3380	321	3	y	y	PROPN
ejpam-3380	321	4	x5	x5	PROPN
ejpam-3380	321	5	xy	xy	NOUN
ejpam-3380	321	6	1	1	NUM
ejpam-3380	321	7	1	1	NUM
ejpam-3380	321	8	y	y	PROPN
ejpam-3380	321	9	x5	x5	PROPN
ejpam-3380	321	10	xy	xy	PROPN
ejpam-3380	321	11	y	y	PROPN
ejpam-3380	322	1	y	y	PROPN
ejpam-3380	322	2	1	1	NUM
ejpam-3380	322	3	xy	xy	NOUN
ejpam-3380	322	4	x5	x5	PROPN
ejpam-3380	322	5	x5	x5	PROPN
ejpam-3380	322	6	x5	x5	PROPN
ejpam-3380	322	7	xy	xy	PROPN
ejpam-3380	322	8	1	1	NUM
ejpam-3380	322	9	y	y	INTJ
ejpam-3380	322	10	xy	xy	INTJ
ejpam-3380	322	11	xy	xy	PROPN
ejpam-3380	323	1	x5	x5	PROPN
ejpam-3380	323	2	y	y	PROPN
ejpam-3380	323	3	1	1	NUM
ejpam-3380	323	4	f	f	NOUN
ejpam-3380	323	5	1	1	NUM
ejpam-3380	323	6	y	y	PROPN
ejpam-3380	323	7	x5	x5	PROPN
ejpam-3380	323	8	xy	xy	NOUN
ejpam-3380	323	9	1	1	NUM
ejpam-3380	323	10	1	1	NUM
ejpam-3380	323	11	1	1	NUM
ejpam-3380	323	12	1	1	NUM
ejpam-3380	323	13	1	1	NUM
ejpam-3380	323	14	y	y	SYM
ejpam-3380	323	15	1	1	NUM
ejpam-3380	323	16	1	1	NUM
ejpam-3380	323	17	1	1	NUM
ejpam-3380	323	18	1	1	NUM
ejpam-3380	323	19	x5	x5	NOUN
ejpam-3380	323	20	1	1	NUM
ejpam-3380	323	21	x4	x4	NOUN
ejpam-3380	323	22	x4	x4	PROPN
ejpam-3380	323	23	1	1	NUM
ejpam-3380	324	1	xy	xy	ADP
ejpam-3380	324	2	1	1	NUM
ejpam-3380	324	3	x2	x2	NOUN
ejpam-3380	324	4	x2	x2	NOUN
ejpam-3380	324	5	1	1	NUM
ejpam-3380	324	6	najla	najla	PROPN
ejpam-3380	324	7	al	al	PROPN
ejpam-3380	324	8	-	-	PUNCT
ejpam-3380	324	9	subaie	subaie	NOUN
ejpam-3380	324	10	,	,	PUNCT
ejpam-3380	324	11	m.	m.	NOUN
ejpam-3380	324	12	m.	m.	PROPN
ejpam-3380	324	13	al	al	PROPN
ejpam-3380	324	14	-	-	PUNCT
ejpam-3380	324	15	shomrani	shomrani	PROPN
ejpam-3380	324	16	/	/	SYM
ejpam-3380	324	17	eur	eur	NOUN
ejpam-3380	324	18	.	.	PUNCT
ejpam-3380	325	1	j.	j.	PROPN
ejpam-3380	325	2	pure	pure	PROPN
ejpam-3380	325	3	appl	appl	PROPN
ejpam-3380	325	4	.	.	PROPN
ejpam-3380	325	5	math	math	PROPN
ejpam-3380	325	6	,	,	PUNCT
ejpam-3380	325	7	12	12	NUM
ejpam-3380	325	8	(	(	PUNCT
ejpam-3380	325	9	2	2	NUM
ejpam-3380	325	10	)	)	PUNCT
ejpam-3380	325	11	(	(	PUNCT
ejpam-3380	325	12	2019	2019	NUM
ejpam-3380	325	13	)	)	PUNCT
ejpam-3380	325	14	,	,	PUNCT
ejpam-3380	325	15	332	332	NUM
ejpam-3380	325	16	-	-	SYM
ejpam-3380	325	17	347	347	NUM
ejpam-3380	325	18	345	345	NUM
ejpam-3380	325	19	table	table	NOUN
ejpam-3380	325	20	6	6	NUM
ejpam-3380	325	21	:	:	PUNCT
ejpam-3380	325	22	.	.	PUNCT
ejpam-3380	326	1	and	and	CCONJ
ejpam-3380	326	2	/	/	SYM
ejpam-3380	326	3	actions	action	NOUN
ejpam-3380	326	4	.	.	PUNCT
ejpam-3380	327	1	s	s	PART
ejpam-3380	327	2	.	.	PUNCT
ejpam-3380	328	1	u	u	NOUN
ejpam-3380	328	2	1	1	NUM
ejpam-3380	328	3	x2	x2	NOUN
ejpam-3380	328	4	x4	x4	PROPN
ejpam-3380	328	5	1	1	NUM
ejpam-3380	328	6	1	1	NUM
ejpam-3380	328	7	x2	x2	NOUN
ejpam-3380	328	8	x4	x4	PROPN
ejpam-3380	328	9	y	y	PROPN
ejpam-3380	328	10	1	1	NUM
ejpam-3380	328	11	x4	x4	PROPN
ejpam-3380	328	12	x2	x2	PROPN
ejpam-3380	328	13	x5	x5	PROPN
ejpam-3380	328	14	1	1	NUM
ejpam-3380	328	15	x2	x2	NOUN
ejpam-3380	328	16	x4	x4	PROPN
ejpam-3380	328	17	xy	xy	PROPN
ejpam-3380	328	18	1	1	NUM
ejpam-3380	329	1	x4	x4	PROPN
ejpam-3380	329	2	x2	x2	PROPN
ejpam-3380	329	3	s	s	PART
ejpam-3380	329	4	/	/	SYM
ejpam-3380	329	5	u	u	PROPN
ejpam-3380	329	6	1	1	NUM
ejpam-3380	329	7	x2	x2	NOUN
ejpam-3380	329	8	x4	x4	PROPN
ejpam-3380	329	9	1	1	NUM
ejpam-3380	329	10	1	1	NUM
ejpam-3380	329	11	1	1	NUM
ejpam-3380	329	12	1	1	NUM
ejpam-3380	329	13	y	y	NOUN
ejpam-3380	329	14	y	y	PROPN
ejpam-3380	329	15	y	y	PROPN
ejpam-3380	329	16	y	y	PROPN
ejpam-3380	329	17	x5	x5	PROPN
ejpam-3380	329	18	x5	x5	PROPN
ejpam-3380	329	19	x5	x5	PROPN
ejpam-3380	329	20	x5	x5	PROPN
ejpam-3380	329	21	xy	xy	X
ejpam-3380	329	22	xy	xy	PROPN
ejpam-3380	329	23	xy	xy	PROPN
ejpam-3380	329	24	xy	xy	PROPN
ejpam-3380	330	1	thus	thus	ADV
ejpam-3380	330	2	,	,	PUNCT
ejpam-3380	330	3	h	h	PROPN
ejpam-3380	330	4	=	=	PROPN
ejpam-3380	330	5	r1	r1	PROPN
ejpam-3380	330	6	⊕	⊕	PROPN
ejpam-3380	330	7	ry	ry	PROPN
ejpam-3380	330	8	⊕	⊕	PROPN
ejpam-3380	330	9	rx5	rx5	PROPN
ejpam-3380	330	10	⊕	⊕	PROPN
ejpam-3380	330	11	rxy	rxy	ADJ
ejpam-3380	330	12	where	where	SCONJ
ejpam-3380	330	13	,	,	PUNCT
ejpam-3380	330	14	r1	r1	PROPN
ejpam-3380	330	15	=	=	SYM
ejpam-3380	330	16	r	r	NOUN
ejpam-3380	330	17	,	,	PUNCT
ejpam-3380	330	18	ry	ry	NOUN
ejpam-3380	330	19	=	=	PROPN
ejpam-3380	330	20	ri	ri	PROPN
ejpam-3380	330	21	,	,	PUNCT
ejpam-3380	330	22	rx5	rx5	PROPN
ejpam-3380	330	23	=	=	SYM
ejpam-3380	330	24	rj	rj	PROPN
ejpam-3380	330	25	and	and	CCONJ
ejpam-3380	330	26	rxy	rxy	ADJ
ejpam-3380	330	27	=	=	SYM
ejpam-3380	330	28	rk	rk	NOUN
ejpam-3380	330	29	.	.	PUNCT
ejpam-3380	331	1	the	the	DET
ejpam-3380	331	2	inclusion	inclusion	NOUN
ejpam-3380	331	3	property	property	NOUN
ejpam-3380	331	4	can	can	AUX
ejpam-3380	331	5	be	be	AUX
ejpam-3380	331	6	checked	check	VERB
ejpam-3380	331	7	as	as	SCONJ
ejpam-3380	331	8	follows	follow	VERB
ejpam-3380	331	9	for	for	ADP
ejpam-3380	331	10	any	any	DET
ejpam-3380	331	11	r	r	NOUN
ejpam-3380	331	12	,	,	PUNCT
ejpam-3380	331	13	r′	r′	NOUN
ejpam-3380	331	14	∈	∈	PROPN
ejpam-3380	331	15	r	r	NOUN
ejpam-3380	331	16	:	:	PUNCT
ejpam-3380	331	17	(	(	PUNCT
ejpam-3380	331	18	i	i	NOUN
ejpam-3380	331	19	)	)	PUNCT
ejpam-3380	331	20	r1r1	r1r1	VERB
ejpam-3380	331	21	⊆	⊆	NUM
ejpam-3380	331	22	r1∗1	r1∗1	NOUN
ejpam-3380	331	23	=	=	SYM
ejpam-3380	331	24	r1	r1	PROPN
ejpam-3380	331	25	,	,	PUNCT
ejpam-3380	331	26	as	as	ADP
ejpam-3380	331	27	for	for	ADP
ejpam-3380	331	28	all	all	DET
ejpam-3380	331	29	r	r	NOUN
ejpam-3380	331	30	,	,	PUNCT
ejpam-3380	331	31	r′	r′	PROPN
ejpam-3380	331	32	∈	∈	PROPN
ejpam-3380	331	33	r1	r1	NOUN
ejpam-3380	331	34	,	,	PUNCT
ejpam-3380	331	35	we	we	PRON
ejpam-3380	331	36	have	have	VERB
ejpam-3380	331	37	(	(	PUNCT
ejpam-3380	331	38	r)(r′	r)(r′	ADJ
ejpam-3380	331	39	)	)	PUNCT
ejpam-3380	331	40	=	=	PUNCT
ejpam-3380	331	41	rr′	rr′	NOUN
ejpam-3380	331	42	∈	∈	PROPN
ejpam-3380	331	43	r1	r1	NOUN
ejpam-3380	331	44	=	=	PUNCT
ejpam-3380	331	45	r1∗1	r1∗1	NOUN
ejpam-3380	331	46	.	.	PUNCT
ejpam-3380	331	47	(	(	PUNCT
ejpam-3380	331	48	ii	ii	NOUN
ejpam-3380	331	49	)	)	PUNCT
ejpam-3380	331	50	r1ry	r1ry	X
ejpam-3380	332	1	⊆	⊆	NUM
ejpam-3380	332	2	r1∗y	r1∗y	X
ejpam-3380	332	3	=	=	PUNCT
ejpam-3380	332	4	ry	ry	NOUN
ejpam-3380	332	5	,	,	PUNCT
ejpam-3380	332	6	as	as	ADP
ejpam-3380	332	7	for	for	ADP
ejpam-3380	332	8	all	all	DET
ejpam-3380	332	9	r	r	NOUN
ejpam-3380	332	10	∈	∈	PROPN
ejpam-3380	332	11	r1	r1	NOUN
ejpam-3380	332	12	,	,	PUNCT
ejpam-3380	332	13	r	r	NOUN
ejpam-3380	332	14	′i	′i	PROPN
ejpam-3380	332	15	∈	∈	PROPN
ejpam-3380	332	16	ry	ry	PROPN
ejpam-3380	332	17	,	,	PUNCT
ejpam-3380	332	18	we	we	PRON
ejpam-3380	332	19	have	have	AUX
ejpam-3380	332	20	(	(	PUNCT
ejpam-3380	332	21	r)(r′i	r)(r′i	PROPN
ejpam-3380	332	22	)	)	PUNCT
ejpam-3380	332	23	=	=	NOUN
ejpam-3380	332	24	(	(	PUNCT
ejpam-3380	332	25	rr′)i	rr′)i	X
ejpam-3380	332	26	∈	∈	NOUN
ejpam-3380	332	27	ry	ry	NOUN
ejpam-3380	332	28	=	=	SYM
ejpam-3380	332	29	r1∗y	r1∗y	PROPN
ejpam-3380	332	30	.	.	PUNCT
ejpam-3380	332	31	(	(	PUNCT
ejpam-3380	332	32	iii	iii	X
ejpam-3380	332	33	)	)	PUNCT
ejpam-3380	332	34	r1rx5	r1rx5	NOUN
ejpam-3380	332	35	⊆	⊆	NUM
ejpam-3380	332	36	r1∗x5	r1∗x5	X
ejpam-3380	332	37	=	=	SYM
ejpam-3380	332	38	rx5	rx5	PROPN
ejpam-3380	332	39	,	,	PUNCT
ejpam-3380	332	40	as	as	ADP
ejpam-3380	332	41	for	for	ADP
ejpam-3380	332	42	all	all	DET
ejpam-3380	332	43	r	r	NOUN
ejpam-3380	332	44	∈	∈	PROPN
ejpam-3380	332	45	r1	r1	NOUN
ejpam-3380	332	46	,	,	PUNCT
ejpam-3380	332	47	r	r	NOUN
ejpam-3380	332	48	′j	′j	PROPN
ejpam-3380	332	49	∈	∈	PROPN
ejpam-3380	332	50	rx5	rx5	PROPN
ejpam-3380	332	51	,	,	PUNCT
ejpam-3380	332	52	we	we	PRON
ejpam-3380	332	53	have	have	VERB
ejpam-3380	332	54	(	(	PUNCT
ejpam-3380	332	55	r)(r′j	r)(r′j	PROPN
ejpam-3380	332	56	)	)	PUNCT
ejpam-3380	332	57	=	=	SYM
ejpam-3380	332	58	(	(	PUNCT
ejpam-3380	332	59	rr′)j	rr′)j	NOUN
ejpam-3380	332	60	∈	∈	PROPN
ejpam-3380	332	61	rx5	rx5	NOUN
ejpam-3380	332	62	=	=	X
ejpam-3380	332	63	r1∗x5	r1∗x5	X
ejpam-3380	332	64	.	.	PUNCT
ejpam-3380	333	1	(	(	PUNCT
ejpam-3380	333	2	iv	iv	X
ejpam-3380	333	3	)	)	PUNCT
ejpam-3380	333	4	r1rxy	r1rxy	NOUN
ejpam-3380	333	5	⊆	⊆	NUM
ejpam-3380	333	6	r1∗xy	r1∗xy	X
ejpam-3380	333	7	=	=	SYM
ejpam-3380	333	8	rxy	rxy	ADJ
ejpam-3380	333	9	,	,	PUNCT
ejpam-3380	333	10	as	as	ADP
ejpam-3380	333	11	for	for	ADP
ejpam-3380	333	12	all	all	DET
ejpam-3380	333	13	r	r	NOUN
ejpam-3380	333	14	∈	∈	PROPN
ejpam-3380	333	15	r1	r1	NOUN
ejpam-3380	333	16	,	,	PUNCT
ejpam-3380	333	17	r	r	NOUN
ejpam-3380	333	18	′k	′k	NOUN
ejpam-3380	333	19	∈	∈	PROPN
ejpam-3380	333	20	rxy	rxy	PROPN
ejpam-3380	333	21	,	,	PUNCT
ejpam-3380	333	22	we	we	PRON
ejpam-3380	333	23	have	have	VERB
ejpam-3380	333	24	(	(	PUNCT
ejpam-3380	333	25	r)(r′k	r)(r′k	NOUN
ejpam-3380	333	26	)	)	PUNCT
ejpam-3380	333	27	=	=	SYM
ejpam-3380	333	28	(	(	PUNCT
ejpam-3380	333	29	rr′)k	rr′)k	X
ejpam-3380	333	30	∈	∈	PROPN
ejpam-3380	333	31	rxy	rxy	NOUN
ejpam-3380	333	32	=	=	SYM
ejpam-3380	333	33	r1∗xy	r1∗xy	PROPN
ejpam-3380	333	34	.	.	PUNCT
ejpam-3380	334	1	(	(	PUNCT
ejpam-3380	334	2	v	v	NOUN
ejpam-3380	334	3	)	)	PUNCT
ejpam-3380	334	4	ryr1	ryr1	NOUN
ejpam-3380	334	5	⊆	⊆	NUM
ejpam-3380	334	6	ry∗1	ry∗1	NOUN
ejpam-3380	334	7	=	=	SYM
ejpam-3380	334	8	ry	ry	PROPN
ejpam-3380	334	9	,	,	PUNCT
ejpam-3380	334	10	as	as	ADP
ejpam-3380	334	11	for	for	ADP
ejpam-3380	334	12	all	all	DET
ejpam-3380	334	13	ri	ri	NOUN
ejpam-3380	334	14	∈	∈	PROPN
ejpam-3380	334	15	ry	ry	NOUN
ejpam-3380	334	16	,	,	PUNCT
ejpam-3380	334	17	r	r	NOUN
ejpam-3380	334	18	′	′	NOUN
ejpam-3380	334	19	∈	∈	PROPN
ejpam-3380	334	20	r1	r1	NOUN
ejpam-3380	334	21	,	,	PUNCT
ejpam-3380	334	22	we	we	PRON
ejpam-3380	334	23	have	have	VERB
ejpam-3380	334	24	(	(	PUNCT
ejpam-3380	334	25	ri)(r′	ri)(r′	PROPN
ejpam-3380	334	26	)	)	PUNCT
ejpam-3380	334	27	=	=	SYM
ejpam-3380	334	28	(	(	PUNCT
ejpam-3380	334	29	rr′)i	rr′)i	X
ejpam-3380	334	30	∈	∈	NOUN
ejpam-3380	334	31	ry	ry	NOUN
ejpam-3380	334	32	=	=	NOUN
ejpam-3380	334	33	ry∗1	ry∗1	NOUN
ejpam-3380	334	34	.	.	PUNCT
ejpam-3380	335	1	(	(	PUNCT
ejpam-3380	335	2	vi	vi	NOUN
ejpam-3380	335	3	)	)	PUNCT
ejpam-3380	335	4	ryry	ryry	VERB
ejpam-3380	335	5	⊆	⊆	NUM
ejpam-3380	335	6	ry∗y	ry∗y	PROPN
ejpam-3380	335	7	=	=	PUNCT
ejpam-3380	335	8	r1	r1	PROPN
ejpam-3380	335	9	,	,	PUNCT
ejpam-3380	335	10	as	as	ADP
ejpam-3380	335	11	for	for	ADP
ejpam-3380	335	12	all	all	DET
ejpam-3380	335	13	ri	ri	NOUN
ejpam-3380	335	14	,	,	PUNCT
ejpam-3380	335	15	r′i	r′i	PROPN
ejpam-3380	335	16	∈	∈	PROPN
ejpam-3380	335	17	ry	ry	NOUN
ejpam-3380	335	18	,	,	PUNCT
ejpam-3380	335	19	we	we	PRON
ejpam-3380	335	20	have	have	AUX
ejpam-3380	335	21	(	(	PUNCT
ejpam-3380	335	22	ri)(r′i	ri)(r′i	ADJ
ejpam-3380	335	23	)	)	PUNCT
ejpam-3380	336	1	=	=	PUNCT
ejpam-3380	336	2	(	(	PUNCT
ejpam-3380	336	3	−rr′	−rr′	PROPN
ejpam-3380	336	4	)	)	PUNCT
ejpam-3380	336	5	∈	∈	PROPN
ejpam-3380	336	6	r1	r1	NOUN
ejpam-3380	336	7	=	=	SYM
ejpam-3380	336	8	ry∗y	ry∗y	PROPN
ejpam-3380	336	9	.	.	PROPN
ejpam-3380	336	10	(	(	PUNCT
ejpam-3380	336	11	vii	vii	PROPN
ejpam-3380	336	12	)	)	PUNCT
ejpam-3380	336	13	ryrx5	ryrx5	NOUN
ejpam-3380	336	14	⊆	⊆	NUM
ejpam-3380	336	15	ry∗x5	ry∗x5	NUM
ejpam-3380	336	16	=	=	SYM
ejpam-3380	336	17	rxy	rxy	PROPN
ejpam-3380	336	18	,	,	PUNCT
ejpam-3380	336	19	as	as	ADP
ejpam-3380	336	20	for	for	ADP
ejpam-3380	336	21	all	all	DET
ejpam-3380	336	22	ri	ri	NOUN
ejpam-3380	336	23	∈	∈	PROPN
ejpam-3380	336	24	ry	ry	NOUN
ejpam-3380	336	25	,	,	PUNCT
ejpam-3380	336	26	r	r	NOUN
ejpam-3380	336	27	′j	′j	PROPN
ejpam-3380	336	28	∈	∈	PROPN
ejpam-3380	336	29	rx5	rx5	PROPN
ejpam-3380	336	30	,	,	PUNCT
ejpam-3380	336	31	we	we	PRON
ejpam-3380	336	32	have	have	VERB
ejpam-3380	336	33	(	(	PUNCT
ejpam-3380	336	34	ri)(r′j	ri)(r′j	ADJ
ejpam-3380	336	35	)	)	PUNCT
ejpam-3380	336	36	=	=	SYM
ejpam-3380	336	37	(	(	PUNCT
ejpam-3380	336	38	rr′)k	rr′)k	X
ejpam-3380	336	39	∈	∈	PROPN
ejpam-3380	336	40	rxy	rxy	NOUN
ejpam-3380	336	41	=	=	SYM
ejpam-3380	336	42	ry∗x5	ry∗x5	PROPN
ejpam-3380	336	43	.	.	PUNCT
ejpam-3380	337	1	(	(	PUNCT
ejpam-3380	337	2	viii	viii	NOUN
ejpam-3380	337	3	)	)	PUNCT
ejpam-3380	337	4	ryrxy	ryrxy	NOUN
ejpam-3380	337	5	⊆	⊆	NUM
ejpam-3380	337	6	ry∗xy	ry∗xy	PROPN
ejpam-3380	337	7	=	=	SYM
ejpam-3380	338	1	rx5	rx5	PROPN
ejpam-3380	338	2	,	,	PUNCT
ejpam-3380	338	3	as	as	ADP
ejpam-3380	338	4	for	for	ADP
ejpam-3380	338	5	all	all	DET
ejpam-3380	338	6	ri	ri	NOUN
ejpam-3380	338	7	∈	∈	PROPN
ejpam-3380	338	8	ry	ry	NOUN
ejpam-3380	338	9	,	,	PUNCT
ejpam-3380	338	10	r	r	NOUN
ejpam-3380	338	11	′k	′k	NOUN
ejpam-3380	338	12	∈	∈	PROPN
ejpam-3380	338	13	rxy	rxy	PROPN
ejpam-3380	338	14	,	,	PUNCT
ejpam-3380	338	15	we	we	PRON
ejpam-3380	338	16	have	have	VERB
ejpam-3380	338	17	(	(	PUNCT
ejpam-3380	338	18	ri)(r′k	ri)(r′k	NOUN
ejpam-3380	338	19	)	)	PUNCT
ejpam-3380	338	20	=	=	PUNCT
ejpam-3380	339	1	(	(	PUNCT
ejpam-3380	339	2	−rr′)j	−rr′)j	NOUN
ejpam-3380	339	3	∈	∈	PROPN
ejpam-3380	339	4	rx5	rx5	NOUN
ejpam-3380	339	5	=	=	SYM
ejpam-3380	339	6	ry∗xy	ry∗xy	PROPN
ejpam-3380	339	7	.	.	PUNCT
ejpam-3380	340	1	(	(	PUNCT
ejpam-3380	340	2	ix	ix	PROPN
ejpam-3380	340	3	)	)	PUNCT
ejpam-3380	340	4	rx5r1	rx5r1	PROPN
ejpam-3380	340	5	⊆	⊆	NUM
ejpam-3380	340	6	rx5∗1	rx5∗1	NOUN
ejpam-3380	340	7	=	=	SYM
ejpam-3380	341	1	rx5	rx5	PROPN
ejpam-3380	341	2	,	,	PUNCT
ejpam-3380	341	3	as	as	ADP
ejpam-3380	341	4	for	for	ADP
ejpam-3380	341	5	all	all	DET
ejpam-3380	341	6	rj	rj	PROPN
ejpam-3380	341	7	∈	∈	PROPN
ejpam-3380	341	8	rx5	rx5	NOUN
ejpam-3380	341	9	,	,	PUNCT
ejpam-3380	341	10	r′	r′	PROPN
ejpam-3380	341	11	∈	∈	PROPN
ejpam-3380	341	12	r1	r1	NOUN
ejpam-3380	341	13	,	,	PUNCT
ejpam-3380	341	14	we	we	PRON
ejpam-3380	341	15	have	have	AUX
ejpam-3380	341	16	(	(	PUNCT
ejpam-3380	341	17	rj)(r′	rj)(r′	PROPN
ejpam-3380	341	18	)	)	PUNCT
ejpam-3380	341	19	=	=	SYM
ejpam-3380	342	1	(	(	PUNCT
ejpam-3380	342	2	rr′)j	rr′)j	NOUN
ejpam-3380	342	3	∈	∈	PROPN
ejpam-3380	342	4	rx5	rx5	NOUN
ejpam-3380	342	5	=	=	NOUN
ejpam-3380	342	6	rx5∗1	rx5∗1	PROPN
ejpam-3380	342	7	.	.	PUNCT
ejpam-3380	343	1	(	(	PUNCT
ejpam-3380	343	2	x	x	X
ejpam-3380	343	3	)	)	PUNCT
ejpam-3380	343	4	rx5ry	rx5ry	ADP
ejpam-3380	343	5	⊆	⊆	NUM
ejpam-3380	343	6	rx5∗y	rx5∗y	PROPN
ejpam-3380	343	7	=	=	PUNCT
ejpam-3380	343	8	rxy	rxy	PROPN
ejpam-3380	343	9	,	,	PUNCT
ejpam-3380	343	10	as	as	ADP
ejpam-3380	343	11	for	for	ADP
ejpam-3380	343	12	all	all	DET
ejpam-3380	343	13	rj	rj	PROPN
ejpam-3380	343	14	∈	∈	PROPN
ejpam-3380	343	15	rx5	rx5	NOUN
ejpam-3380	343	16	,	,	PUNCT
ejpam-3380	343	17	r′i	r′i	PROPN
ejpam-3380	343	18	∈	∈	PROPN
ejpam-3380	343	19	ry	ry	NOUN
ejpam-3380	343	20	,	,	PUNCT
ejpam-3380	343	21	we	we	PRON
ejpam-3380	343	22	have	have	VERB
ejpam-3380	343	23	(	(	PUNCT
ejpam-3380	343	24	rj)(r′i	rj)(r′i	NOUN
ejpam-3380	343	25	)	)	PUNCT
ejpam-3380	343	26	=	=	PRON
ejpam-3380	344	1	(	(	PUNCT
ejpam-3380	344	2	−rr′)k	−rr′)k	PROPN
ejpam-3380	344	3	∈	∈	PROPN
ejpam-3380	344	4	rxy	rxy	NOUN
ejpam-3380	344	5	=	=	SYM
ejpam-3380	344	6	rx5∗y	rx5∗y	PROPN
ejpam-3380	344	7	.	.	PUNCT
ejpam-3380	345	1	(	(	PUNCT
ejpam-3380	345	2	xi	xi	NOUN
ejpam-3380	345	3	)	)	PUNCT
ejpam-3380	345	4	rx5rx5	rx5rx5	NOUN
ejpam-3380	345	5	⊆	⊆	NUM
ejpam-3380	345	6	rx5∗x5	rx5∗x5	X
ejpam-3380	345	7	=	=	SYM
ejpam-3380	345	8	r1	r1	PROPN
ejpam-3380	345	9	,	,	PUNCT
ejpam-3380	345	10	as	as	ADP
ejpam-3380	345	11	for	for	ADP
ejpam-3380	345	12	all	all	DET
ejpam-3380	345	13	rj	rj	NOUN
ejpam-3380	345	14	,	,	PUNCT
ejpam-3380	345	15	r′j	r′j	PROPN
ejpam-3380	345	16	∈	∈	PROPN
ejpam-3380	345	17	rx5	rx5	PROPN
ejpam-3380	345	18	,	,	PUNCT
ejpam-3380	345	19	we	we	PRON
ejpam-3380	345	20	have	have	VERB
ejpam-3380	345	21	(	(	PUNCT
ejpam-3380	345	22	rj)(r′j	rj)(r′j	ADJ
ejpam-3380	345	23	)	)	PUNCT
ejpam-3380	345	24	=	=	PUNCT
ejpam-3380	345	25	(	(	PUNCT
ejpam-3380	345	26	−rr′	−rr′	PROPN
ejpam-3380	345	27	)	)	PUNCT
ejpam-3380	345	28	∈	∈	PROPN
ejpam-3380	345	29	r1	r1	NOUN
ejpam-3380	345	30	=	=	SYM
ejpam-3380	345	31	rx5∗x5	rx5∗x5	X
ejpam-3380	345	32	.	.	PUNCT
ejpam-3380	346	1	(	(	PUNCT
ejpam-3380	346	2	xii	xii	NOUN
ejpam-3380	346	3	)	)	PUNCT
ejpam-3380	346	4	rx5rxy	rx5rxy	NOUN
ejpam-3380	346	5	⊆	⊆	NUM
ejpam-3380	346	6	rx5∗xy	rx5∗xy	PROPN
ejpam-3380	346	7	=	=	SYM
ejpam-3380	346	8	ry	ry	NOUN
ejpam-3380	346	9	,	,	PUNCT
ejpam-3380	346	10	as	as	ADP
ejpam-3380	346	11	for	for	ADP
ejpam-3380	346	12	all	all	DET
ejpam-3380	346	13	rj	rj	PROPN
ejpam-3380	346	14	∈	∈	PROPN
ejpam-3380	346	15	rx5	rx5	NOUN
ejpam-3380	346	16	,	,	PUNCT
ejpam-3380	346	17	r′k	r′k	PROPN
ejpam-3380	346	18	∈	∈	PROPN
ejpam-3380	346	19	rxy	rxy	PROPN
ejpam-3380	346	20	,	,	PUNCT
ejpam-3380	346	21	we	we	PRON
ejpam-3380	346	22	have	have	VERB
ejpam-3380	346	23	(	(	PUNCT
ejpam-3380	346	24	rj)(r′k	rj)(r′k	NOUN
ejpam-3380	346	25	)	)	PUNCT
ejpam-3380	347	1	=	=	PUNCT
ejpam-3380	347	2	(	(	PUNCT
ejpam-3380	347	3	rr′)i	rr′)i	X
ejpam-3380	347	4	∈	∈	NOUN
ejpam-3380	347	5	ry	ry	NOUN
ejpam-3380	347	6	=	=	PUNCT
ejpam-3380	347	7	rx5∗xy	rx5∗xy	PROPN
ejpam-3380	347	8	.	.	PUNCT
ejpam-3380	348	1	(	(	PUNCT
ejpam-3380	348	2	xiii	xiii	PROPN
ejpam-3380	348	3	)	)	PUNCT
ejpam-3380	348	4	rxyr1	rxyr1	NOUN
ejpam-3380	349	1	⊆	⊆	NUM
ejpam-3380	349	2	rxy∗1	rxy∗1	NOUN
ejpam-3380	349	3	=	=	SYM
ejpam-3380	349	4	rxy	rxy	ADJ
ejpam-3380	349	5	,	,	PUNCT
ejpam-3380	349	6	as	as	ADP
ejpam-3380	349	7	for	for	ADP
ejpam-3380	349	8	all	all	DET
ejpam-3380	349	9	rk	rk	NOUN
ejpam-3380	349	10	∈	∈	PROPN
ejpam-3380	349	11	rxy	rxy	NOUN
ejpam-3380	349	12	,	,	PUNCT
ejpam-3380	349	13	r	r	NOUN
ejpam-3380	349	14	′	′	NOUN
ejpam-3380	349	15	∈	∈	PROPN
ejpam-3380	349	16	r1	r1	NOUN
ejpam-3380	349	17	,	,	PUNCT
ejpam-3380	349	18	we	we	PRON
ejpam-3380	349	19	have	have	VERB
ejpam-3380	349	20	(	(	PUNCT
ejpam-3380	349	21	rk)(r′	rk)(r′	PROPN
ejpam-3380	349	22	)	)	PUNCT
ejpam-3380	349	23	=	=	SYM
ejpam-3380	349	24	(	(	PUNCT
ejpam-3380	349	25	rr′)k	rr′)k	X
ejpam-3380	349	26	∈	∈	PROPN
ejpam-3380	349	27	rxy	rxy	ADJ
ejpam-3380	349	28	=	=	SYM
ejpam-3380	349	29	rxy∗1	rxy∗1	NOUN
ejpam-3380	349	30	.	.	PUNCT
ejpam-3380	350	1	references	reference	NOUN
ejpam-3380	350	2	346	346	NUM
ejpam-3380	350	3	(	(	PUNCT
ejpam-3380	350	4	xiv	xiv	NOUN
ejpam-3380	350	5	)	)	PUNCT
ejpam-3380	350	6	rxyry	rxyry	NOUN
ejpam-3380	350	7	⊆	⊆	NUM
ejpam-3380	350	8	rxy∗y	rxy∗y	NOUN
ejpam-3380	350	9	=	=	SYM
ejpam-3380	350	10	rx5	rx5	NOUN
ejpam-3380	350	11	,	,	PUNCT
ejpam-3380	350	12	as	as	ADP
ejpam-3380	350	13	for	for	ADP
ejpam-3380	350	14	all	all	DET
ejpam-3380	350	15	rk	rk	NOUN
ejpam-3380	350	16	∈	∈	PROPN
ejpam-3380	350	17	rxy	rxy	NOUN
ejpam-3380	350	18	,	,	PUNCT
ejpam-3380	350	19	r	r	NOUN
ejpam-3380	350	20	′i	′i	PROPN
ejpam-3380	350	21	∈	∈	PROPN
ejpam-3380	350	22	ry	ry	PROPN
ejpam-3380	350	23	,	,	PUNCT
ejpam-3380	350	24	we	we	PRON
ejpam-3380	350	25	have	have	VERB
ejpam-3380	350	26	(	(	PUNCT
ejpam-3380	350	27	rk)(r′i	rk)(r′i	NOUN
ejpam-3380	350	28	)	)	PUNCT
ejpam-3380	351	1	=	=	PUNCT
ejpam-3380	351	2	(	(	PUNCT
ejpam-3380	351	3	rr′)j	rr′)j	NOUN
ejpam-3380	351	4	∈	∈	PROPN
ejpam-3380	351	5	rx5	rx5	NOUN
ejpam-3380	351	6	=	=	NOUN
ejpam-3380	351	7	rxy∗y	rxy∗y	NOUN
ejpam-3380	351	8	.	.	PUNCT
ejpam-3380	352	1	(	(	PUNCT
ejpam-3380	352	2	xv	xv	PROPN
ejpam-3380	352	3	)	)	PUNCT
ejpam-3380	352	4	rxyrx5	rxyrx5	VERB
ejpam-3380	352	5	⊆	⊆	NUM
ejpam-3380	352	6	rxy∗x5	rxy∗x5	NOUN
ejpam-3380	352	7	=	=	SYM
ejpam-3380	352	8	ry	ry	NOUN
ejpam-3380	352	9	,	,	PUNCT
ejpam-3380	352	10	as	as	ADP
ejpam-3380	352	11	for	for	ADP
ejpam-3380	352	12	all	all	DET
ejpam-3380	352	13	rk	rk	NOUN
ejpam-3380	352	14	∈	∈	PROPN
ejpam-3380	352	15	rxy	rxy	NOUN
ejpam-3380	352	16	,	,	PUNCT
ejpam-3380	352	17	r	r	NOUN
ejpam-3380	352	18	′j	′j	PROPN
ejpam-3380	352	19	∈	∈	PROPN
ejpam-3380	352	20	rx5	rx5	PROPN
ejpam-3380	352	21	,	,	PUNCT
ejpam-3380	352	22	we	we	PRON
ejpam-3380	352	23	have	have	VERB
ejpam-3380	352	24	(	(	PUNCT
ejpam-3380	352	25	rk)(r′j	rk)(r′j	PROPN
ejpam-3380	352	26	)	)	PUNCT
ejpam-3380	353	1	=	=	PUNCT
ejpam-3380	354	1	(	(	PUNCT
ejpam-3380	354	2	−rr′)i	−rr′)i	PROPN
ejpam-3380	354	3	∈	∈	PROPN
ejpam-3380	354	4	ry	ry	NOUN
ejpam-3380	354	5	=	=	PROPN
ejpam-3380	354	6	rxy∗x5	rxy∗x5	PROPN
ejpam-3380	354	7	.	.	PUNCT
ejpam-3380	355	1	(	(	PUNCT
ejpam-3380	355	2	xvi	xvi	NOUN
ejpam-3380	355	3	)	)	PUNCT
ejpam-3380	355	4	rxyrxy	rxyrxy	VERB
ejpam-3380	355	5	⊆	⊆	NUM
ejpam-3380	355	6	rxy∗xy	rxy∗xy	NOUN
ejpam-3380	355	7	=	=	SYM
ejpam-3380	355	8	r1	r1	PROPN
ejpam-3380	355	9	,	,	PUNCT
ejpam-3380	355	10	as	as	ADP
ejpam-3380	355	11	for	for	ADP
ejpam-3380	355	12	all	all	DET
ejpam-3380	355	13	rk	rk	NOUN
ejpam-3380	355	14	,	,	PUNCT
ejpam-3380	355	15	r′k	r′k	NOUN
ejpam-3380	355	16	∈	∈	PROPN
ejpam-3380	355	17	rxy	rxy	PROPN
ejpam-3380	355	18	,	,	PUNCT
ejpam-3380	355	19	we	we	PRON
ejpam-3380	355	20	have	have	VERB
ejpam-3380	355	21	(	(	PUNCT
ejpam-3380	355	22	rk)(r′k	rk)(r′k	NOUN
ejpam-3380	355	23	)	)	PUNCT
ejpam-3380	355	24	=	=	PUNCT
ejpam-3380	355	25	(	(	PUNCT
ejpam-3380	355	26	−rr′	−rr′	PROPN
ejpam-3380	355	27	)	)	PUNCT
ejpam-3380	355	28	∈	∈	PROPN
ejpam-3380	355	29	r1	r1	NOUN
ejpam-3380	355	30	=	=	SYM
ejpam-3380	355	31	rxy∗xy	rxy∗xy	PROPN
ejpam-3380	355	32	.	.	PUNCT
ejpam-3380	356	1	therefore	therefore	ADV
ejpam-3380	356	2	,	,	PUNCT
ejpam-3380	356	3	h	h	NOUN
ejpam-3380	356	4	is	be	AUX
ejpam-3380	356	5	a	a	DET
ejpam-3380	356	6	fully	fully	ADV
ejpam-3380	356	7	g	g	NOUN
ejpam-3380	356	8	-	-	PUNCT
ejpam-3380	356	9	weak	weak	ADJ
ejpam-3380	356	10	graded	grade	VERB
ejpam-3380	356	11	ring	ring	NOUN
ejpam-3380	356	12	.	.	PUNCT
ejpam-3380	357	1	references	reference	NOUN
ejpam-3380	357	2	[	[	X
ejpam-3380	357	3	1	1	NUM
ejpam-3380	357	4	]	]	X
ejpam-3380	357	5	g.	g.	PROPN
ejpam-3380	357	6	abrams	abrams	PROPN
ejpam-3380	357	7	and	and	CCONJ
ejpam-3380	357	8	c.	c.	PROPN
ejpam-3380	357	9	menini	menini	PROPN
ejpam-3380	357	10	.	.	PUNCT
ejpam-3380	358	1	rings	ring	NOUN
ejpam-3380	358	2	of	of	ADP
ejpam-3380	358	3	endomorphisms	endomorphism	NOUN
ejpam-3380	358	4	of	of	ADP
ejpam-3380	358	5	semigroup	semigroup	NOUN
ejpam-3380	358	6	-	-	PUNCT
ejpam-3380	358	7	graded	grade	VERB
ejpam-3380	358	8	modules	module	NOUN
ejpam-3380	358	9	.	.	PUNCT
ejpam-3380	359	1	rocky	rocky	ADJ
ejpam-3380	359	2	mountain	mountain	PROPN
ejpam-3380	359	3	j.	j.	PROPN
ejpam-3380	359	4	math	math	PROPN
ejpam-3380	359	5	.	.	PUNCT
ejpam-3380	359	6	,	,	PUNCT
ejpam-3380	359	7	26(2):375	26(2):375	NUM
ejpam-3380	359	8	-	-	SYM
ejpam-3380	359	9	406	406	NUM
ejpam-3380	359	10	,	,	PUNCT
ejpam-3380	359	11	(	(	PUNCT
ejpam-3380	359	12	1996	1996	NUM
ejpam-3380	359	13	)	)	PUNCT
ejpam-3380	359	14	.	.	PUNCT
ejpam-3380	360	1	[	[	X
ejpam-3380	360	2	2	2	X
ejpam-3380	360	3	]	]	PUNCT
ejpam-3380	360	4	t.	t.	PROPN
ejpam-3380	360	5	albu	albu	PROPN
ejpam-3380	360	6	,	,	PUNCT
ejpam-3380	360	7	c.	c.	PROPN
ejpam-3380	360	8	nǎstǎsescu	nǎstǎsescu	PROPN
ejpam-3380	360	9	,	,	PUNCT
ejpam-3380	360	10	infinite	infinite	ADJ
ejpam-3380	360	11	group	group	NOUN
ejpam-3380	360	12	-	-	PUNCT
ejpam-3380	360	13	graded	grade	VERB
ejpam-3380	360	14	rings	ring	NOUN
ejpam-3380	360	15	,	,	PUNCT
ejpam-3380	360	16	rings	ring	NOUN
ejpam-3380	360	17	of	of	ADP
ejpam-3380	360	18	endomorphisms	endomorphism	NOUN
ejpam-3380	360	19	and	and	CCONJ
ejpam-3380	360	20	localization	localization	NOUN
ejpam-3380	360	21	.	.	PUNCT
ejpam-3380	361	1	j.	j.	PROPN
ejpam-3380	361	2	pure	pure	PROPN
ejpam-3380	361	3	appl	appl	PROPN
ejpam-3380	361	4	.	.	PUNCT
ejpam-3380	362	1	algebra	algebra	PROPN
ejpam-3380	362	2	,	,	PUNCT
ejpam-3380	362	3	59:125	59:125	PROPN
ejpam-3380	362	4	-	-	SYM
ejpam-3380	362	5	150	150	NUM
ejpam-3380	362	6	,	,	PUNCT
ejpam-3380	362	7	(	(	PUNCT
ejpam-3380	362	8	1989	1989	NUM
ejpam-3380	362	9	)	)	PUNCT
ejpam-3380	362	10	.	.	PUNCT
ejpam-3380	363	1	[	[	X
ejpam-3380	363	2	3	3	X
ejpam-3380	363	3	]	]	X
ejpam-3380	363	4	m.	m.	NOUN
ejpam-3380	363	5	m.	m.	NOUN
ejpam-3380	363	6	al	al	PROPN
ejpam-3380	363	7	-	-	PUNCT
ejpam-3380	363	8	shomrani	shomrani	PROPN
ejpam-3380	363	9	.	.	PUNCT
ejpam-3380	364	1	a	a	DET
ejpam-3380	364	2	construction	construction	NOUN
ejpam-3380	364	3	of	of	ADP
ejpam-3380	364	4	graded	grade	VERB
ejpam-3380	364	5	rings	ring	NOUN
ejpam-3380	364	6	using	use	VERB
ejpam-3380	364	7	a	a	DET
ejpam-3380	364	8	set	set	NOUN
ejpam-3380	364	9	of	of	ADP
ejpam-3380	364	10	left	left	ADJ
ejpam-3380	364	11	coset	coset	NOUN
ejpam-3380	364	12	representatives	representative	NOUN
ejpam-3380	364	13	.	.	PUNCT
ejpam-3380	365	1	jp	jp	PROPN
ejpam-3380	365	2	journal	journal	PROPN
ejpam-3380	365	3	of	of	ADP
ejpam-3380	365	4	algebra	algebra	PROPN
ejpam-3380	365	5	,	,	PUNCT
ejpam-3380	365	6	number	number	NOUN
ejpam-3380	365	7	theory	theory	NOUN
ejpam-3380	365	8	and	and	CCONJ
ejpam-3380	365	9	applications	application	NOUN
ejpam-3380	365	10	,	,	PUNCT
ejpam-3380	365	11	25(2):133	25(2):133	NUM
ejpam-3380	365	12	-	-	SYM
ejpam-3380	365	13	144	144	NUM
ejpam-3380	365	14	,	,	PUNCT
ejpam-3380	365	15	(	(	PUNCT
ejpam-3380	365	16	2012	2012	NUM
ejpam-3380	365	17	)	)	PUNCT
ejpam-3380	365	18	.	.	PUNCT
ejpam-3380	366	1	[	[	X
ejpam-3380	366	2	4	4	X
ejpam-3380	366	3	]	]	X
ejpam-3380	366	4	n.	n.	PROPN
ejpam-3380	366	5	al	al	PROPN
ejpam-3380	366	6	-	-	PUNCT
ejpam-3380	366	7	subaie	subaie	PROPN
ejpam-3380	366	8	,	,	PUNCT
ejpam-3380	366	9	m.	m.	NOUN
ejpam-3380	366	10	al	al	PROPN
ejpam-3380	366	11	-	-	PUNCT
ejpam-3380	366	12	shomrani	shomrani	PROPN
ejpam-3380	366	13	,	,	PUNCT
ejpam-3380	366	14	on	on	ADP
ejpam-3380	366	15	weak	weak	ADJ
ejpam-3380	366	16	graded	grade	VERB
ejpam-3380	366	17	rings	ring	NOUN
ejpam-3380	366	18	.	.	PUNCT
ejpam-3380	367	1	eur	eur	PROPN
ejpam-3380	367	2	.	.	PUNCT
ejpam-3380	368	1	j.	j.	PROPN
ejpam-3380	368	2	pure	pure	PROPN
ejpam-3380	368	3	appl	appl	PROPN
ejpam-3380	368	4	.	.	PUNCT
ejpam-3380	368	5	math	math	PROPN
ejpam-3380	368	6	.	.	PUNCT
ejpam-3380	368	7	,	,	PUNCT
ejpam-3380	368	8	10:967	10:967	NUM
ejpam-3380	368	9	-	-	SYM
ejpam-3380	368	10	980	980	NUM
ejpam-3380	368	11	,	,	PUNCT
ejpam-3380	368	12	(	(	PUNCT
ejpam-3380	368	13	2017	2017	NUM
ejpam-3380	368	14	)	)	PUNCT
ejpam-3380	368	15	.	.	PUNCT
ejpam-3380	369	1	[	[	X
ejpam-3380	369	2	5	5	NUM
ejpam-3380	369	3	]	]	PUNCT
ejpam-3380	369	4	m.	m.	NOUN
ejpam-3380	369	5	beattie	beattie	PROPN
ejpam-3380	369	6	,	,	PUNCT
ejpam-3380	369	7	duality	duality	NOUN
ejpam-3380	369	8	theorems	theorem	NOUN
ejpam-3380	369	9	for	for	ADP
ejpam-3380	369	10	rings	ring	NOUN
ejpam-3380	369	11	with	with	ADP
ejpam-3380	369	12	actions	action	NOUN
ejpam-3380	369	13	and	and	CCONJ
ejpam-3380	369	14	coactions	coaction	NOUN
ejpam-3380	369	15	.	.	PUNCT
ejpam-3380	370	1	j.	j.	PROPN
ejpam-3380	370	2	algebra	algebra	PROPN
ejpam-3380	370	3	,	,	PUNCT
ejpam-3380	370	4	115:302	115:302	PROPN
ejpam-3380	370	5	-	-	PUNCT
ejpam-3380	370	6	312	312	NUM
ejpam-3380	370	7	,	,	PUNCT
ejpam-3380	370	8	(	(	PUNCT
ejpam-3380	370	9	1988	1988	NUM
ejpam-3380	370	10	)	)	PUNCT
ejpam-3380	370	11	.	.	PUNCT
ejpam-3380	371	1	[	[	X
ejpam-3380	371	2	6	6	NUM
ejpam-3380	371	3	]	]	PUNCT
ejpam-3380	371	4	e.	e.	PROPN
ejpam-3380	371	5	j.	j.	PROPN
ejpam-3380	371	6	beggs	beggs	PROPN
ejpam-3380	371	7	.	.	PUNCT
ejpam-3380	372	1	making	make	VERB
ejpam-3380	372	2	non	non	ADJ
ejpam-3380	372	3	-	-	ADJ
ejpam-3380	372	4	trivially	trivially	ADV
ejpam-3380	372	5	associated	associated	ADJ
ejpam-3380	372	6	tensor	tensor	NOUN
ejpam-3380	372	7	categories	category	NOUN
ejpam-3380	372	8	from	from	ADP
ejpam-3380	372	9	left	left	ADJ
ejpam-3380	372	10	coset	coset	NOUN
ejpam-3380	372	11	representatives	representative	NOUN
ejpam-3380	372	12	.	.	PUNCT
ejpam-3380	373	1	j.	j.	PROPN
ejpam-3380	373	2	pure	pure	PROPN
ejpam-3380	373	3	appl	appl	PROPN
ejpam-3380	373	4	.	.	PUNCT
ejpam-3380	374	1	algebra	algebra	NOUN
ejpam-3380	374	2	,	,	PUNCT
ejpam-3380	374	3	177(1):5	177(1):5	PROPN
ejpam-3380	374	4	-	-	PUNCT
ejpam-3380	374	5	41	41	NUM
ejpam-3380	374	6	,	,	PUNCT
ejpam-3380	374	7	(	(	PUNCT
ejpam-3380	374	8	2003	2003	NUM
ejpam-3380	374	9	)	)	PUNCT
ejpam-3380	374	10	.	.	PUNCT
ejpam-3380	375	1	[	[	X
ejpam-3380	375	2	7	7	X
ejpam-3380	375	3	]	]	X
ejpam-3380	375	4	m.	m.	NOUN
ejpam-3380	375	5	cohen	cohen	PROPN
ejpam-3380	375	6	and	and	CCONJ
ejpam-3380	375	7	s.	s.	PROPN
ejpam-3380	375	8	montgomery	montgomery	PROPN
ejpam-3380	375	9	.	.	PUNCT
ejpam-3380	376	1	group	group	NOUN
ejpam-3380	376	2	-	-	PUNCT
ejpam-3380	376	3	graded	grade	VERB
ejpam-3380	376	4	rings	ring	NOUN
ejpam-3380	376	5	,	,	PUNCT
ejpam-3380	376	6	smash	smash	VERB
ejpam-3380	376	7	product	product	NOUN
ejpam-3380	376	8	and	and	CCONJ
ejpam-3380	376	9	group	group	NOUN
ejpam-3380	376	10	actions	action	NOUN
ejpam-3380	376	11	.	.	PUNCT
ejpam-3380	377	1	trans	trans	PROPN
ejpam-3380	377	2	.	.	PUNCT
ejpam-3380	378	1	amer	amer	PROPN
ejpam-3380	378	2	.	.	PUNCT
ejpam-3380	378	3	math	math	PROPN
ejpam-3380	378	4	.	.	PUNCT
ejpam-3380	379	1	soc	soc	PROPN
ejpam-3380	379	2	.	.	PUNCT
ejpam-3380	379	3	,	,	PUNCT
ejpam-3380	379	4	282(1):237	282(1):237	NUM
ejpam-3380	379	5	-	-	SYM
ejpam-3380	379	6	258	258	NUM
ejpam-3380	379	7	,	,	PUNCT
ejpam-3380	379	8	(	(	PUNCT
ejpam-3380	379	9	1984	1984	NUM
ejpam-3380	379	10	)	)	PUNCT
ejpam-3380	379	11	.	.	PUNCT
ejpam-3380	380	1	[	[	X
ejpam-3380	380	2	8	8	NUM
ejpam-3380	380	3	]	]	X
ejpam-3380	380	4	e.	e.	PROPN
ejpam-3380	380	5	c.	c.	PROPN
ejpam-3380	380	6	dade	dade	PROPN
ejpam-3380	380	7	.	.	PUNCT
ejpam-3380	381	1	group	group	PROPN
ejpam-3380	381	2	graded	grade	VERB
ejpam-3380	381	3	rings	ring	NOUN
ejpam-3380	381	4	and	and	CCONJ
ejpam-3380	381	5	modules	module	NOUN
ejpam-3380	381	6	.	.	PUNCT
ejpam-3380	382	1	math	math	NOUN
ejpam-3380	382	2	.	.	PUNCT
ejpam-3380	383	1	z.	z.	PROPN
ejpam-3380	383	2	,	,	PUNCT
ejpam-3380	383	3	174:241	174:241	PROPN
ejpam-3380	383	4	-	-	SYM
ejpam-3380	383	5	262	262	NUM
ejpam-3380	383	6	,	,	PUNCT
ejpam-3380	383	7	(	(	PUNCT
ejpam-3380	383	8	1980	1980	NUM
ejpam-3380	383	9	)	)	PUNCT
ejpam-3380	383	10	.	.	PUNCT
ejpam-3380	384	1	[	[	X
ejpam-3380	384	2	9	9	NUM
ejpam-3380	384	3	]	]	X
ejpam-3380	384	4	e.	e.	PROPN
ejpam-3380	384	5	dade	dade	PROPN
ejpam-3380	384	6	,	,	PUNCT
ejpam-3380	384	7	clifford	clifford	PROPN
ejpam-3380	384	8	theory	theory	NOUN
ejpam-3380	384	9	for	for	ADP
ejpam-3380	384	10	group	group	NOUN
ejpam-3380	384	11	graded	grade	VERB
ejpam-3380	384	12	rings	ring	NOUN
ejpam-3380	384	13	.	.	PUNCT
ejpam-3380	385	1	j.	j.	PROPN
ejpam-3380	385	2	reine	reine	PROPN
ejpam-3380	385	3	angew	angew	PROPN
ejpam-3380	385	4	.	.	PUNCT
ejpam-3380	386	1	math	math	NOUN
ejpam-3380	386	2	.	.	PUNCT
ejpam-3380	386	3	,	,	PUNCT
ejpam-3380	386	4	369:40	369:40	NUM
ejpam-3380	386	5	-	-	SYM
ejpam-3380	386	6	86	86	NUM
ejpam-3380	386	7	,	,	PUNCT
ejpam-3380	386	8	(	(	PUNCT
ejpam-3380	386	9	1986	1986	NUM
ejpam-3380	386	10	)	)	PUNCT
ejpam-3380	386	11	.	.	PUNCT
ejpam-3380	387	1	[	[	X
ejpam-3380	387	2	10	10	NUM
ejpam-3380	387	3	]	]	X
ejpam-3380	387	4	e.	e.	PROPN
ejpam-3380	387	5	dade	dade	PROPN
ejpam-3380	387	6	,	,	PUNCT
ejpam-3380	387	7	clifford	clifford	PROPN
ejpam-3380	387	8	theory	theory	NOUN
ejpam-3380	387	9	for	for	ADP
ejpam-3380	387	10	group	group	NOUN
ejpam-3380	387	11	graded	grade	VERB
ejpam-3380	387	12	rings	ring	NOUN
ejpam-3380	387	13	ii	ii	PROPN
ejpam-3380	387	14	.	.	PUNCT
ejpam-3380	388	1	j.	j.	PROPN
ejpam-3380	388	2	reine	reine	PROPN
ejpam-3380	388	3	angew	angew	PROPN
ejpam-3380	388	4	.	.	PUNCT
ejpam-3380	389	1	math	math	NOUN
ejpam-3380	389	2	.	.	PUNCT
ejpam-3380	389	3	,	,	PUNCT
ejpam-3380	389	4	387:148181	387:148181	NUM
ejpam-3380	389	5	,	,	PUNCT
ejpam-3380	389	6	(	(	PUNCT
ejpam-3380	389	7	1988	1988	NUM
ejpam-3380	389	8	)	)	PUNCT
ejpam-3380	389	9	.	.	PUNCT
ejpam-3380	390	1	[	[	X
ejpam-3380	390	2	11	11	NUM
ejpam-3380	390	3	]	]	PUNCT
ejpam-3380	390	4	s.	s.	PROPN
ejpam-3380	390	5	dascalescu	dascalescu	PROPN
ejpam-3380	390	6	,	,	PUNCT
ejpam-3380	390	7	a.	a.	PROPN
ejpam-3380	390	8	v.	v.	PROPN
ejpam-3380	390	9	kelarev	kelarev	PROPN
ejpam-3380	390	10	and	and	CCONJ
ejpam-3380	390	11	l.	l.	PROPN
ejpam-3380	390	12	van	van	PROPN
ejpam-3380	390	13	wyk	wyk	PROPN
ejpam-3380	390	14	.	.	PUNCT
ejpam-3380	391	1	semigroup	semigroup	PROPN
ejpam-3380	391	2	gradings	grading	NOUN
ejpam-3380	391	3	of	of	ADP
ejpam-3380	391	4	full	full	ADJ
ejpam-3380	391	5	matrix	matrix	NOUN
ejpam-3380	391	6	rings	ring	NOUN
ejpam-3380	391	7	.	.	PUNCT
ejpam-3380	392	1	comm	comm	NOUN
ejpam-3380	392	2	.	.	PUNCT
ejpam-3380	393	1	algebra	algebra	NOUN
ejpam-3380	393	2	,	,	PUNCT
ejpam-3380	393	3	29(11):5023	29(11):5023	NUM
ejpam-3380	393	4	-	-	SYM
ejpam-3380	393	5	5031	5031	NUM
ejpam-3380	393	6	,	,	PUNCT
ejpam-3380	393	7	(	(	PUNCT
ejpam-3380	393	8	2001	2001	NUM
ejpam-3380	393	9	)	)	PUNCT
ejpam-3380	393	10	.	.	PUNCT
ejpam-3380	394	1	[	[	X
ejpam-3380	394	2	12	12	NUM
ejpam-3380	394	3	]	]	PUNCT
ejpam-3380	394	4	j.	j.	PROPN
ejpam-3380	394	5	l.	l.	PROPN
ejpam-3380	394	6	gómez	gómez	PROPN
ejpam-3380	394	7	pardo	pardo	PROPN
ejpam-3380	394	8	and	and	CCONJ
ejpam-3380	394	9	c.	c.	PROPN
ejpam-3380	394	10	nǎstǎsescu	nǎstǎsescu	PROPN
ejpam-3380	394	11	.	.	PUNCT
ejpam-3380	395	1	relative	relative	ADJ
ejpam-3380	395	2	projectivity	projectivity	NOUN
ejpam-3380	395	3	,	,	PUNCT
ejpam-3380	395	4	graded	grade	VERB
ejpam-3380	395	5	cliffored	cliffore	VERB
ejpam-3380	395	6	theory	theory	NOUN
ejpam-3380	395	7	and	and	CCONJ
ejpam-3380	395	8	applications	application	NOUN
ejpam-3380	395	9	.	.	PUNCT
ejpam-3380	396	1	j.	j.	PROPN
ejpam-3380	396	2	algebra	algebra	PROPN
ejpam-3380	396	3	,	,	PUNCT
ejpam-3380	396	4	141(2):484	141(2):484	NOUN
ejpam-3380	396	5	-	-	SYM
ejpam-3380	396	6	504	504	NUM
ejpam-3380	396	7	,	,	PUNCT
ejpam-3380	396	8	(	(	PUNCT
ejpam-3380	396	9	1991	1991	NUM
ejpam-3380	396	10	)	)	PUNCT
ejpam-3380	396	11	.	.	PUNCT
ejpam-3380	397	1	references	reference	NOUN
ejpam-3380	397	2	347	347	NUM
ejpam-3380	397	3	[	[	X
ejpam-3380	397	4	13	13	NUM
ejpam-3380	397	5	]	]	PUNCT
ejpam-3380	397	6	g.	g.	PROPN
ejpam-3380	397	7	karpilovsky	karpilovsky	PROPN
ejpam-3380	397	8	.	.	PUNCT
ejpam-3380	398	1	the	the	DET
ejpam-3380	398	2	jacobson	jacobson	PROPN
ejpam-3380	398	3	radical	radical	PROPN
ejpam-3380	398	4	of	of	ADP
ejpam-3380	398	5	monoid	monoid	NOUN
ejpam-3380	398	6	-	-	PUNCT
ejpam-3380	398	7	graded	grade	VERB
ejpam-3380	398	8	algebras	algebra	NOUN
ejpam-3380	398	9	.	.	PUNCT
ejpam-3380	399	1	tsukuba	tsukuba	PROPN
ejpam-3380	399	2	j.	j.	PROPN
ejpam-3380	399	3	math	math	PROPN
ejpam-3380	399	4	,	,	PUNCT
ejpam-3380	399	5	16(1):19	16(1):19	NUM
ejpam-3380	399	6	-	-	SYM
ejpam-3380	399	7	52	52	NUM
ejpam-3380	399	8	,	,	PUNCT
ejpam-3380	399	9	(	(	PUNCT
ejpam-3380	399	10	1992	1992	NUM
ejpam-3380	399	11	)	)	PUNCT
ejpam-3380	399	12	.	.	PUNCT
ejpam-3380	400	1	[	[	X
ejpam-3380	400	2	14	14	NUM
ejpam-3380	400	3	]	]	X
ejpam-3380	400	4	a.	a.	NOUN
ejpam-3380	400	5	v.	v.	PROPN
ejpam-3380	400	6	kelarev	kelarev	PROPN
ejpam-3380	400	7	.	.	PUNCT
ejpam-3380	401	1	applications	application	NOUN
ejpam-3380	401	2	of	of	ADP
ejpam-3380	401	3	epigroups	epigroup	NOUN
ejpam-3380	401	4	to	to	PART
ejpam-3380	401	5	graded	grade	VERB
ejpam-3380	401	6	ring	ring	NOUN
ejpam-3380	401	7	theory	theory	NOUN
ejpam-3380	401	8	.	.	PUNCT
ejpam-3380	402	1	semigroup	semigroup	PROPN
ejpam-3380	402	2	forum	forum	PROPN
ejpam-3380	402	3	,	,	PUNCT
ejpam-3380	402	4	50(3):327	50(3):327	PROPN
ejpam-3380	402	5	-	-	SYM
ejpam-3380	402	6	350	350	NUM
ejpam-3380	402	7	,	,	PUNCT
ejpam-3380	402	8	(	(	PUNCT
ejpam-3380	402	9	1995	1995	NUM
ejpam-3380	402	10	)	)	PUNCT
ejpam-3380	402	11	.	.	PUNCT
ejpam-3380	403	1	[	[	X
ejpam-3380	403	2	15	15	NUM
ejpam-3380	403	3	]	]	X
ejpam-3380	403	4	a.	a.	NOUN
ejpam-3380	403	5	v.	v.	PROPN
ejpam-3380	403	6	kelarev	kelarev	PROPN
ejpam-3380	403	7	.	.	PUNCT
ejpam-3380	404	1	semisimple	semisimple	PROPN
ejpam-3380	404	2	rings	ring	NOUN
ejpam-3380	404	3	graded	grade	VERB
ejpam-3380	404	4	by	by	ADP
ejpam-3380	404	5	inverse	inverse	NOUN
ejpam-3380	404	6	semigroups	semigroup	NOUN
ejpam-3380	404	7	.	.	PUNCT
ejpam-3380	405	1	j.	j.	PROPN
ejpam-3380	405	2	algebra	algebra	PROPN
ejpam-3380	405	3	,	,	PUNCT
ejpam-3380	405	4	205:451459	205:451459	NUM
ejpam-3380	405	5	,	,	PUNCT
ejpam-3380	405	6	(	(	PUNCT
ejpam-3380	405	7	1998	1998	NUM
ejpam-3380	405	8	)	)	PUNCT
ejpam-3380	405	9	.	.	PUNCT
ejpam-3380	406	1	[	[	X
ejpam-3380	406	2	16	16	NUM
ejpam-3380	406	3	]	]	X
ejpam-3380	406	4	c.	c.	PROPN
ejpam-3380	406	5	nǎstǎsescu	nǎstǎsescu	PROPN
ejpam-3380	406	6	,	,	PUNCT
ejpam-3380	406	7	f.	f.	PROPN
ejpam-3380	406	8	oystaeyen	oystaeyen	PROPN
ejpam-3380	406	9	,	,	PUNCT
ejpam-3380	406	10	graded	grade	VERB
ejpam-3380	406	11	ring	ring	NOUN
ejpam-3380	406	12	theory	theory	NOUN
ejpam-3380	406	13	.	.	PUNCT
ejpam-3380	407	1	north	north	NOUN
ejpam-3380	407	2	-	-	PUNCT
ejpam-3380	407	3	holland	holland	PROPN
ejpam-3380	407	4	publishing	publishing	PROPN
ejpam-3380	407	5	company	company	NOUN
ejpam-3380	407	6	,	,	PUNCT
ejpam-3380	407	7	(	(	PUNCT
ejpam-3380	407	8	1982	1982	NUM
ejpam-3380	407	9	)	)	PUNCT
ejpam-3380	407	10	.	.	PUNCT
ejpam-3380	408	1	[	[	X
ejpam-3380	408	2	17	17	NUM
ejpam-3380	408	3	]	]	X
ejpam-3380	408	4	c.	c.	PROPN
ejpam-3380	408	5	nǎstǎsescu	nǎstǎsescu	PROPN
ejpam-3380	408	6	,	,	PUNCT
ejpam-3380	408	7	m.	m.	PROPN
ejpam-3380	408	8	bergh	bergh	PROPN
ejpam-3380	408	9	,	,	PUNCT
ejpam-3380	408	10	f.	f.	PROPN
ejpam-3380	408	11	oystaeyen	oystaeyen	PROPN
ejpam-3380	408	12	,	,	PUNCT
ejpam-3380	408	13	separable	separable	ADJ
ejpam-3380	408	14	functor	functor	PROPN
ejpam-3380	408	15	,	,	PUNCT
ejpam-3380	408	16	applications	application	NOUN
ejpam-3380	408	17	to	to	PART
ejpam-3380	408	18	graded	grade	VERB
ejpam-3380	408	19	rings	ring	NOUN
ejpam-3380	408	20	and	and	CCONJ
ejpam-3380	408	21	modules	module	NOUN
ejpam-3380	408	22	.	.	PUNCT
ejpam-3380	409	1	j.	j.	PROPN
ejpam-3380	409	2	algebra	algebra	PROPN
ejpam-3380	409	3	,	,	PUNCT
ejpam-3380	409	4	123:397	123:397	NOUN
ejpam-3380	409	5	-	-	PUNCT
ejpam-3380	409	6	413	413	NUM
ejpam-3380	409	7	,	,	PUNCT
ejpam-3380	409	8	(	(	PUNCT
ejpam-3380	409	9	1989	1989	NUM
ejpam-3380	409	10	)	)	PUNCT
ejpam-3380	409	11	.	.	PUNCT
ejpam-3380	410	1	[	[	X
ejpam-3380	410	2	18	18	NUM
ejpam-3380	410	3	]	]	X
ejpam-3380	410	4	c.	c.	PROPN
ejpam-3380	410	5	nǎstǎsescu	nǎstǎsescu	PROPN
ejpam-3380	410	6	and	and	CCONJ
ejpam-3380	410	7	f.	f.	PROPN
ejpam-3380	410	8	v.	v.	PROPN
ejpam-3380	410	9	oystaeyen	oystaeyen	PROPN
ejpam-3380	410	10	.	.	PUNCT
ejpam-3380	411	1	methods	method	NOUN
ejpam-3380	411	2	of	of	ADP
ejpam-3380	411	3	graded	grade	VERB
ejpam-3380	411	4	rings	ring	NOUN
ejpam-3380	411	5	.	.	PUNCT
ejpam-3380	412	1	springer	springer	NOUN
ejpam-3380	412	2	-	-	PUNCT
ejpam-3380	412	3	verlag	verlag	PROPN
ejpam-3380	412	4	berlin	berlin	PROPN
ejpam-3380	412	5	heidelberg	heidelberg	PROPN
ejpam-3380	412	6	,	,	PUNCT
ejpam-3380	412	7	new	new	PROPN
ejpam-3380	412	8	york	york	PROPN
ejpam-3380	412	9	,	,	PUNCT
ejpam-3380	412	10	(	(	PUNCT
ejpam-3380	412	11	2004	2004	NUM
ejpam-3380	412	12	)	)	PUNCT
ejpam-3380	412	13	.	.	PUNCT
ejpam-3380	413	1	[	[	X
ejpam-3380	413	2	19	19	NUM
ejpam-3380	413	3	]	]	X
ejpam-3380	413	4	p.	p.	NOUN
ejpam-3380	413	5	nystedt	nystedt	NOUN
ejpam-3380	413	6	and	and	CCONJ
ejpam-3380	413	7	j.	j.	PROPN
ejpam-3380	413	8	oinert	oinert	PROPN
ejpam-3380	413	9	.	.	PUNCT
ejpam-3380	414	1	simple	simple	ADJ
ejpam-3380	414	2	semigroup	semigroup	PROPN
ejpam-3380	414	3	graded	grade	VERB
ejpam-3380	414	4	rings	ring	NOUN
ejpam-3380	414	5	.	.	PUNCT
ejpam-3380	415	1	j.	j.	PROPN
ejpam-3380	415	2	algebra	algebra	PROPN
ejpam-3380	415	3	appl	appl	PROPN
ejpam-3380	415	4	.	.	PROPN
ejpam-3380	415	5	,	,	PUNCT
ejpam-3380	415	6	14(7):110	14(7):110	PROPN
ejpam-3380	415	7	,	,	PUNCT
ejpam-3380	415	8	(	(	PUNCT
ejpam-3380	415	9	2015	2015	NUM
ejpam-3380	415	10	)	)	PUNCT
ejpam-3380	415	11	.	.	PUNCT
ejpam-3380	416	1	[	[	X
ejpam-3380	416	2	20	20	NUM
ejpam-3380	416	3	]	]	PUNCT
ejpam-3380	416	4	m.	m.	PROPN
ejpam-3380	416	5	rafael	rafael	PROPN
ejpam-3380	416	6	,	,	PUNCT
ejpam-3380	416	7	j.	j.	PROPN
ejpam-3380	416	8	oinert	oinert	PROPN
ejpam-3380	416	9	,	,	PUNCT
ejpam-3380	416	10	separable	separable	PROPN
ejpam-3380	416	11	functors	functors	PROPN
ejpam-3380	416	12	revisited	revisit	VERB
ejpam-3380	416	13	.	.	PUNCT
ejpam-3380	417	1	comm	comm	NOUN
ejpam-3380	417	2	.	.	PUNCT
ejpam-3380	418	1	algebra	algebra	NOUN
ejpam-3380	418	2	,	,	PUNCT
ejpam-3380	418	3	18:144	18:144	NUM
ejpam-3380	418	4	–	–	PUNCT
ejpam-3380	418	5	1459,(1990	1459,(1990	NUM
ejpam-3380	418	6	)	)	PUNCT
ejpam-3380	418	7	.	.	PUNCT
ejpam-3380	419	1	[	[	X
ejpam-3380	419	2	21	21	NUM
ejpam-3380	419	3	]	]	PUNCT
ejpam-3380	419	4	a.	a.	NOUN
ejpam-3380	419	5	turull	turull	PROPN
ejpam-3380	419	6	,	,	PUNCT
ejpam-3380	419	7	clifford	clifford	PROPN
ejpam-3380	419	8	theory	theory	NOUN
ejpam-3380	419	9	with	with	ADP
ejpam-3380	419	10	schur	schur	PROPN
ejpam-3380	419	11	indices	index	NOUN
ejpam-3380	419	12	.	.	PUNCT
ejpam-3380	420	1	j.	j.	PROPN
ejpam-3380	420	2	algebra	algebra	PROPN
ejpam-3380	420	3	,	,	PUNCT
ejpam-3380	420	4	170:661	170:661	PROPN
ejpam-3380	420	5	-	-	SYM
ejpam-3380	420	6	677	677	NUM
ejpam-3380	420	7	,	,	PUNCT
ejpam-3380	420	8	(	(	PUNCT
ejpam-3380	420	9	1994	1994	NUM
ejpam-3380	420	10	)	)	PUNCT
ejpam-3380	420	11	.	.	PUNCT
ejpam-3380	421	1	[	[	X
ejpam-3380	421	2	22	22	NUM
ejpam-3380	421	3	]	]	PUNCT
ejpam-3380	421	4	b.	b.	PROPN
ejpam-3380	421	5	zhou	zhou	PROPN
ejpam-3380	421	6	,	,	PUNCT
ejpam-3380	421	7	two	two	NUM
ejpam-3380	421	8	clifford	clifford	PROPN
ejpam-3380	421	9	’s	’s	PART
ejpam-3380	421	10	theorems	theorem	NOUN
ejpam-3380	421	11	for	for	ADP
ejpam-3380	421	12	strongly	strongly	ADV
ejpam-3380	421	13	group	group	NOUN
ejpam-3380	421	14	-	-	PUNCT
ejpam-3380	421	15	graded	grade	VERB
ejpam-3380	421	16	rings	ring	NOUN
ejpam-3380	421	17	.	.	PUNCT
ejpam-3380	422	1	j.	j.	PROPN
ejpam-3380	422	2	algebra	algebra	PROPN
ejpam-3380	422	3	,	,	PUNCT
ejpam-3380	422	4	139:172	139:172	PROPN
ejpam-3380	422	5	-	-	PUNCT
ejpam-3380	422	6	189	189	NUM
ejpam-3380	422	7	,	,	PUNCT
ejpam-3380	422	8	(	(	PUNCT
ejpam-3380	422	9	1991	1991	NUM
ejpam-3380	422	10	)	)	PUNCT
ejpam-3380	422	11	.	.	PUNCT