id	sid	tid	token	lemma	pos
ejpam-2026	1	1	european	european	PROPN
ejpam-2026	1	2	journal	journal	PROPN
ejpam-2026	1	3	of	of	ADP
ejpam-2026	1	4	pure	pure	ADJ
ejpam-2026	1	5	and	and	CCONJ
ejpam-2026	1	6	applied	apply	VERB
ejpam-2026	1	7	mathematics	mathematic	NOUN
ejpam-2026	1	8	vol	vol	NOUN
ejpam-2026	1	9	.	.	PROPN
ejpam-2026	2	1	10	10	NUM
ejpam-2026	2	2	,	,	PUNCT
ejpam-2026	2	3	no	no	INTJ
ejpam-2026	2	4	.	.	NOUN
ejpam-2026	2	5	3	3	NUM
ejpam-2026	2	6	,	,	PUNCT
ejpam-2026	2	7	2017	2017	NUM
ejpam-2026	2	8	,	,	PUNCT
ejpam-2026	2	9	516	516	NUM
ejpam-2026	2	10	-	-	SYM
ejpam-2026	2	11	520	520	NUM
ejpam-2026	2	12	issn	issn	PROPN
ejpam-2026	2	13	1307	1307	NUM
ejpam-2026	2	14	-	-	SYM
ejpam-2026	2	15	5543	5543	NUM
ejpam-2026	2	16	–	–	PUNCT
ejpam-2026	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-2026	2	18	published	publish	VERB
ejpam-2026	2	19	by	by	ADP
ejpam-2026	2	20	new	new	PROPN
ejpam-2026	2	21	york	york	PROPN
ejpam-2026	2	22	business	business	PROPN
ejpam-2026	2	23	global	global	ADJ
ejpam-2026	2	24	module	module	NOUN
ejpam-2026	2	25	over	over	ADP
ejpam-2026	2	26	pseudo	pseudo	NOUN
ejpam-2026	2	27	-	-	NOUN
ejpam-2026	2	28	valuation	valuation	NOUN
ejpam-2026	2	29	ring	ring	NOUN
ejpam-2026	2	30	and	and	CCONJ
ejpam-2026	2	31	domain	domain	NOUN
ejpam-2026	2	32	waheed	waheed	PROPN
ejpam-2026	2	33	ahmad	ahmad	PROPN
ejpam-2026	2	34	khan	khan	PROPN
ejpam-2026	2	35	department	department	PROPN
ejpam-2026	2	36	of	of	ADP
ejpam-2026	2	37	mathematics	mathematics	PROPN
ejpam-2026	2	38	,	,	PUNCT
ejpam-2026	2	39	university	university	NOUN
ejpam-2026	2	40	of	of	ADP
ejpam-2026	2	41	education	education	NOUN
ejpam-2026	2	42	lahore	lahore	NOUN
ejpam-2026	2	43	,	,	PUNCT
ejpam-2026	2	44	pakistan	pakistan	PROPN
ejpam-2026	2	45	abstract	abstract	NOUN
ejpam-2026	2	46	.	.	PUNCT
ejpam-2026	3	1	modules	module	NOUN
ejpam-2026	3	2	over	over	ADP
ejpam-2026	3	3	principal	principal	ADJ
ejpam-2026	3	4	ideal	ideal	NOUN
ejpam-2026	3	5	rings	ring	NOUN
ejpam-2026	3	6	,	,	PUNCT
ejpam-2026	3	7	dedekind	dedekind	NOUN
ejpam-2026	3	8	rings	ring	NOUN
ejpam-2026	3	9	and	and	CCONJ
ejpam-2026	3	10	valuation	valuation	NOUN
ejpam-2026	3	11	rings	ring	NOUN
ejpam-2026	3	12	have	have	AUX
ejpam-2026	3	13	been	be	AUX
ejpam-2026	3	14	discussed	discuss	VERB
ejpam-2026	3	15	in	in	ADP
ejpam-2026	3	16	the	the	DET
ejpam-2026	3	17	literature	literature	NOUN
ejpam-2026	3	18	.	.	PUNCT
ejpam-2026	4	1	in	in	ADP
ejpam-2026	4	2	this	this	DET
ejpam-2026	4	3	article	article	NOUN
ejpam-2026	4	4	,	,	PUNCT
ejpam-2026	4	5	we	we	PRON
ejpam-2026	4	6	introduce	introduce	VERB
ejpam-2026	4	7	and	and	CCONJ
ejpam-2026	4	8	discuss	discuss	VERB
ejpam-2026	4	9	the	the	DET
ejpam-2026	4	10	module	module	NOUN
ejpam-2026	4	11	over	over	ADP
ejpam-2026	4	12	pseudo	pseudo	NOUN
ejpam-2026	4	13	-	-	PUNCT
ejpam-2026	4	14	valuation	valuation	NOUN
ejpam-2026	4	15	ring(pvr	ring(pvr	NOUN
ejpam-2026	4	16	)	)	PUNCT
ejpam-2026	4	17	and	and	CCONJ
ejpam-2026	4	18	its	its	PRON
ejpam-2026	4	19	submodule	submodule	NOUN
ejpam-2026	4	20	.	.	PUNCT
ejpam-2026	5	1	also	also	ADV
ejpam-2026	5	2	as	as	ADP
ejpam-2026	5	3	a	a	DET
ejpam-2026	5	4	byproduct	byproduct	NOUN
ejpam-2026	5	5	we	we	PRON
ejpam-2026	5	6	introduce	introduce	VERB
ejpam-2026	5	7	the	the	DET
ejpam-2026	5	8	module	module	NOUN
ejpam-2026	5	9	over	over	ADP
ejpam-2026	5	10	pseudo	pseudo	NOUN
ejpam-2026	5	11	-	-	NOUN
ejpam-2026	5	12	valuation	valuation	NOUN
ejpam-2026	5	13	domain(pvd	domain(pvd	NOUN
ejpam-2026	5	14	)	)	PUNCT
ejpam-2026	5	15	.	.	PUNCT
ejpam-2026	6	1	2010	2010	NUM
ejpam-2026	6	2	mathematics	mathematic	NOUN
ejpam-2026	6	3	subject	subject	NOUN
ejpam-2026	6	4	classifications	classification	NOUN
ejpam-2026	6	5	:	:	PUNCT
ejpam-2026	6	6	13a05	13a05	NUM
ejpam-2026	6	7	,	,	PUNCT
ejpam-2026	6	8	13a18	13a18	NUM
ejpam-2026	6	9	,	,	PUNCT
ejpam-2026	6	10	12j20	12j20	NUM
ejpam-2026	6	11	key	key	ADJ
ejpam-2026	6	12	words	word	NOUN
ejpam-2026	6	13	and	and	CCONJ
ejpam-2026	6	14	phrases	phrase	NOUN
ejpam-2026	6	15	:	:	PUNCT
ejpam-2026	6	16	module	module	NOUN
ejpam-2026	6	17	,	,	PUNCT
ejpam-2026	6	18	submodule	submodule	NOUN
ejpam-2026	6	19	,	,	PUNCT
ejpam-2026	6	20	pvr	pvr	NOUN
ejpam-2026	6	21	,	,	PUNCT
ejpam-2026	6	22	pvd	pvd	PROPN
ejpam-2026	6	23	1	1	NUM
ejpam-2026	6	24	.	.	PUNCT
ejpam-2026	7	1	introduction	introduction	NOUN
ejpam-2026	7	2	and	and	CCONJ
ejpam-2026	7	3	preliminaries	preliminary	NOUN
ejpam-2026	7	4	number	number	NOUN
ejpam-2026	7	5	of	of	ADP
ejpam-2026	7	6	research	research	NOUN
ejpam-2026	7	7	articles	article	NOUN
ejpam-2026	7	8	are	be	AUX
ejpam-2026	7	9	available	available	ADJ
ejpam-2026	7	10	upon	upon	SCONJ
ejpam-2026	7	11	modules	module	NOUN
ejpam-2026	7	12	over	over	ADP
ejpam-2026	7	13	different	different	ADJ
ejpam-2026	7	14	types	type	NOUN
ejpam-2026	7	15	of	of	ADP
ejpam-2026	7	16	rings	ring	NOUN
ejpam-2026	7	17	and	and	CCONJ
ejpam-2026	7	18	domains	domain	NOUN
ejpam-2026	7	19	,	,	PUNCT
ejpam-2026	7	20	modules	module	NOUN
ejpam-2026	7	21	over	over	ADP
ejpam-2026	7	22	principal	principal	ADJ
ejpam-2026	7	23	ideal	ideal	NOUN
ejpam-2026	7	24	rings	ring	NOUN
ejpam-2026	7	25	have	have	AUX
ejpam-2026	7	26	been	be	AUX
ejpam-2026	7	27	discussed	discuss	VERB
ejpam-2026	7	28	in	in	ADP
ejpam-2026	7	29	the	the	DET
ejpam-2026	7	30	literature	literature	NOUN
ejpam-2026	7	31	,	,	PUNCT
ejpam-2026	7	32	modules	module	NOUN
ejpam-2026	7	33	over	over	ADP
ejpam-2026	7	34	dedekind	dedekind	NOUN
ejpam-2026	7	35	rings	ring	NOUN
ejpam-2026	7	36	and	and	CCONJ
ejpam-2026	7	37	valuation	valuation	NOUN
ejpam-2026	7	38	rings	ring	NOUN
ejpam-2026	7	39	have	have	AUX
ejpam-2026	7	40	been	be	AUX
ejpam-2026	7	41	introduced	introduce	VERB
ejpam-2026	7	42	by	by	ADP
ejpam-2026	7	43	i.	i.	PROPN
ejpam-2026	7	44	kaplansky	kaplansky	PROPN
ejpam-2026	7	45	in	in	ADP
ejpam-2026	7	46	[	[	X
ejpam-2026	7	47	6	6	NUM
ejpam-2026	7	48	]	]	PUNCT
ejpam-2026	7	49	.	.	PUNCT
ejpam-2026	8	1	similarly	similarly	ADV
ejpam-2026	8	2	modules	module	VERB
ejpam-2026	8	3	over	over	ADP
ejpam-2026	8	4	dedekind	dedekind	ADJ
ejpam-2026	8	5	prime	prime	NOUN
ejpam-2026	8	6	rings	ring	NOUN
ejpam-2026	8	7	introduced	introduce	VERB
ejpam-2026	8	8	by	by	ADP
ejpam-2026	8	9	david	david	PROPN
ejpam-2026	8	10	eisenbud	eisenbud	PROPN
ejpam-2026	8	11	and	and	CCONJ
ejpam-2026	8	12	j.	j.	PROPN
ejpam-2026	8	13	c.	c.	PROPN
ejpam-2026	8	14	robin	robin	PROPN
ejpam-2026	8	15	in	in	ADP
ejpam-2026	8	16	[	[	X
ejpam-2026	8	17	3	3	NUM
ejpam-2026	8	18	]	]	PUNCT
ejpam-2026	8	19	.	.	PUNCT
ejpam-2026	9	1	in	in	ADP
ejpam-2026	9	2	this	this	DET
ejpam-2026	9	3	connection	connection	NOUN
ejpam-2026	9	4	we	we	PRON
ejpam-2026	9	5	introduced	introduce	VERB
ejpam-2026	9	6	module	module	NOUN
ejpam-2026	9	7	over	over	ADP
ejpam-2026	9	8	pseudovaluation	pseudovaluation	NOUN
ejpam-2026	9	9	ring	ring	NOUN
ejpam-2026	9	10	and	and	CCONJ
ejpam-2026	9	11	pseudovaluation	pseudovaluation	NOUN
ejpam-2026	9	12	domain	domain	NOUN
ejpam-2026	9	13	,	,	PUNCT
ejpam-2026	9	14	and	and	CCONJ
ejpam-2026	9	15	also	also	ADV
ejpam-2026	9	16	their	their	PRON
ejpam-2026	9	17	submodules	submodule	NOUN
ejpam-2026	9	18	.	.	PUNCT
ejpam-2026	10	1	we	we	PRON
ejpam-2026	10	2	begin	begin	VERB
ejpam-2026	10	3	with	with	ADP
ejpam-2026	10	4	the	the	DET
ejpam-2026	10	5	basics	basic	NOUN
ejpam-2026	10	6	of	of	ADP
ejpam-2026	10	7	module	module	NOUN
ejpam-2026	10	8	.	.	PUNCT
ejpam-2026	11	1	r	r	NOUN
ejpam-2026	11	2	-	-	PUNCT
ejpam-2026	11	3	module	module	NOUN
ejpam-2026	11	4	m	m	NOUN
ejpam-2026	11	5	over	over	ADP
ejpam-2026	11	6	the	the	DET
ejpam-2026	11	7	ring	ring	NOUN
ejpam-2026	11	8	r	r	NOUN
ejpam-2026	11	9	consists	consist	VERB
ejpam-2026	11	10	of	of	ADP
ejpam-2026	11	11	an	an	DET
ejpam-2026	11	12	abelian	abelian	ADJ
ejpam-2026	11	13	group	group	NOUN
ejpam-2026	11	14	(	(	PUNCT
ejpam-2026	11	15	m	m	PROPN
ejpam-2026	11	16	,	,	PUNCT
ejpam-2026	11	17	+	+	ADJ
ejpam-2026	11	18	)	)	PUNCT
ejpam-2026	11	19	and	and	CCONJ
ejpam-2026	11	20	an	an	DET
ejpam-2026	11	21	operation	operation	NOUN
ejpam-2026	11	22	r	r	NOUN
ejpam-2026	11	23	×	×	NOUN
ejpam-2026	11	24	m	m	INTJ
ejpam-2026	11	25	→	→	SYM
ejpam-2026	11	26	m	m	PROPN
ejpam-2026	11	27	(	(	PUNCT
ejpam-2026	11	28	called	call	VERB
ejpam-2026	11	29	scalar	scalar	ADJ
ejpam-2026	11	30	multiplication	multiplication	NOUN
ejpam-2026	11	31	,	,	PUNCT
ejpam-2026	11	32	usually	usually	ADV
ejpam-2026	11	33	just	just	ADV
ejpam-2026	11	34	written	write	VERB
ejpam-2026	11	35	by	by	ADP
ejpam-2026	11	36	juxtaposition	juxtaposition	NOUN
ejpam-2026	11	37	,	,	PUNCT
ejpam-2026	11	38	i.e.	i.e.	X
ejpam-2026	11	39	as	as	ADP
ejpam-2026	11	40	rx	rx	ADJ
ejpam-2026	11	41	for	for	ADP
ejpam-2026	11	42	r	r	NOUN
ejpam-2026	11	43	∈	∈	NOUN
ejpam-2026	11	44	r	r	NOUN
ejpam-2026	11	45	and	and	CCONJ
ejpam-2026	11	46	x	x	SYM
ejpam-2026	11	47	∈	∈	PROPN
ejpam-2026	11	48	m	m	PROPN
ejpam-2026	11	49	)	)	PUNCT
ejpam-2026	11	50	such	such	ADJ
ejpam-2026	11	51	that	that	PRON
ejpam-2026	11	52	for	for	ADP
ejpam-2026	11	53	all	all	DET
ejpam-2026	11	54	r	r	NOUN
ejpam-2026	11	55	,	,	PUNCT
ejpam-2026	11	56	s	s	NOUN
ejpam-2026	11	57	in	in	ADP
ejpam-2026	11	58	r	r	NOUN
ejpam-2026	11	59	,	,	PUNCT
ejpam-2026	11	60	x	x	PRON
ejpam-2026	11	61	,	,	PUNCT
ejpam-2026	11	62	y	y	PROPN
ejpam-2026	11	63	in	in	ADP
ejpam-2026	11	64	m	m	PROPN
ejpam-2026	11	65	,	,	PUNCT
ejpam-2026	11	66	we	we	PRON
ejpam-2026	11	67	have	have	VERB
ejpam-2026	11	68	r(x+y	r(x+y	NUM
ejpam-2026	11	69	)	)	PUNCT
ejpam-2026	12	1	=	=	SYM
ejpam-2026	12	2	rx+ry	rx+ry	NOUN
ejpam-2026	12	3	,	,	PUNCT
ejpam-2026	12	4	(	(	PUNCT
ejpam-2026	12	5	r+s)x	r+s)x	NOUN
ejpam-2026	12	6	=	=	SYM
ejpam-2026	12	7	rx+sx	rx+sx	NOUN
ejpam-2026	12	8	,	,	PUNCT
ejpam-2026	12	9	(	(	PUNCT
ejpam-2026	12	10	rs)x	rs)x	NOUN
ejpam-2026	12	11	=	=	SYM
ejpam-2026	12	12	r(sx	r(sx	PROPN
ejpam-2026	12	13	)	)	PUNCT
ejpam-2026	12	14	,	,	PUNCT
ejpam-2026	12	15	1rx	1rx	NOUN
ejpam-2026	13	1	=	=	PUNCT
ejpam-2026	13	2	x	x	X
ejpam-2026	13	3	if	if	SCONJ
ejpam-2026	13	4	r	r	NOUN
ejpam-2026	13	5	has	have	VERB
ejpam-2026	13	6	identity	identity	NOUN
ejpam-2026	13	7	1r	1r	NUM
ejpam-2026	13	8	.	.	PUNCT
ejpam-2026	14	1	suppose	suppose	VERB
ejpam-2026	14	2	m	m	PRON
ejpam-2026	14	3	is	be	AUX
ejpam-2026	14	4	a	a	DET
ejpam-2026	14	5	left	left	ADJ
ejpam-2026	14	6	r	r	NOUN
ejpam-2026	14	7	-	-	PUNCT
ejpam-2026	14	8	module	module	NOUN
ejpam-2026	14	9	and	and	CCONJ
ejpam-2026	14	10	n	n	NOUN
ejpam-2026	14	11	is	be	AUX
ejpam-2026	14	12	a	a	DET
ejpam-2026	14	13	subgroup	subgroup	NOUN
ejpam-2026	14	14	of	of	ADP
ejpam-2026	14	15	m	m	PROPN
ejpam-2026	14	16	.	.	PUNCT
ejpam-2026	15	1	then	then	ADV
ejpam-2026	15	2	n	n	PRON
ejpam-2026	15	3	is	be	AUX
ejpam-2026	15	4	a	a	DET
ejpam-2026	15	5	submodule	submodule	NOUN
ejpam-2026	15	6	(	(	PUNCT
ejpam-2026	15	7	or	or	CCONJ
ejpam-2026	15	8	r	r	NOUN
ejpam-2026	15	9	-	-	PUNCT
ejpam-2026	15	10	submodule	submodule	NOUN
ejpam-2026	15	11	,	,	PUNCT
ejpam-2026	15	12	to	to	PART
ejpam-2026	15	13	be	be	AUX
ejpam-2026	15	14	more	more	ADV
ejpam-2026	15	15	explicit	explicit	ADJ
ejpam-2026	15	16	)	)	PUNCT
ejpam-2026	15	17	if	if	SCONJ
ejpam-2026	15	18	for	for	ADP
ejpam-2026	15	19	any	any	DET
ejpam-2026	15	20	n	n	PRON
ejpam-2026	15	21	∈	∈	NOUN
ejpam-2026	15	22	n	n	NOUN
ejpam-2026	15	23	and	and	CCONJ
ejpam-2026	15	24	any	any	DET
ejpam-2026	15	25	r	r	NOUN
ejpam-2026	15	26	∈	∈	NOUN
ejpam-2026	15	27	r	r	NOUN
ejpam-2026	15	28	,	,	PUNCT
ejpam-2026	15	29	the	the	DET
ejpam-2026	15	30	product	product	NOUN
ejpam-2026	15	31	rn	rn	PROPN
ejpam-2026	15	32	is	be	AUX
ejpam-2026	15	33	in	in	ADP
ejpam-2026	15	34	n	n	PROPN
ejpam-2026	15	35	(	(	PUNCT
ejpam-2026	15	36	or	or	CCONJ
ejpam-2026	15	37	nr	nr	PROPN
ejpam-2026	15	38	for	for	ADP
ejpam-2026	15	39	a	a	DET
ejpam-2026	15	40	right	right	ADJ
ejpam-2026	15	41	module	module	NOUN
ejpam-2026	15	42	)	)	PUNCT
ejpam-2026	15	43	.	.	PUNCT
ejpam-2026	16	1	let	let	VERB
ejpam-2026	16	2	r	r	PRON
ejpam-2026	16	3	be	be	AUX
ejpam-2026	16	4	a	a	DET
ejpam-2026	16	5	ring	ring	NOUN
ejpam-2026	16	6	.	.	PUNCT
ejpam-2026	17	1	let	let	VERB
ejpam-2026	17	2	m	m	PRON
ejpam-2026	17	3	be	be	AUX
ejpam-2026	17	4	a	a	DET
ejpam-2026	17	5	r−module	r−module	PROPN
ejpam-2026	17	6	.	.	PUNCT
ejpam-2026	18	1	a	a	DET
ejpam-2026	18	2	subset	subset	NOUN
ejpam-2026	18	3	n	n	NOUN
ejpam-2026	18	4	of	of	ADP
ejpam-2026	18	5	m	m	PROPN
ejpam-2026	18	6	is	be	AUX
ejpam-2026	18	7	a	a	DET
ejpam-2026	18	8	submodule	submodule	NOUN
ejpam-2026	18	9	of	of	ADP
ejpam-2026	18	10	m	m	PROPN
ejpam-2026	18	11	⇐	⇐	ADJ
ejpam-2026	18	12	⇒	⇒	NOUN
ejpam-2026	18	13	(	(	PUNCT
ejpam-2026	18	14	1	1	NUM
ejpam-2026	18	15	)	)	PUNCT
ejpam-2026	18	16	n	n	PROPN
ejpam-2026	18	17	6=	6=	ADP
ejpam-2026	18	18	φ	φ	PROPN
ejpam-2026	18	19	.	.	PUNCT
ejpam-2026	19	1	(	(	PUNCT
ejpam-2026	19	2	2	2	X
ejpam-2026	19	3	)	)	PUNCT
ejpam-2026	19	4	x+	x+	PUNCT
ejpam-2026	19	5	ry	ry	NOUN
ejpam-2026	19	6	∈	∈	PROPN
ejpam-2026	19	7	n	n	PRON
ejpam-2026	19	8	∀	∀	NOUN
ejpam-2026	19	9	r	r	NOUN
ejpam-2026	19	10	∈	∈	NOUN
ejpam-2026	19	11	r	r	NOUN
ejpam-2026	19	12	and	and	CCONJ
ejpam-2026	19	13	∀x	∀x	NUM
ejpam-2026	19	14	,	,	PUNCT
ejpam-2026	19	15	y	y	PROPN
ejpam-2026	19	16	∈	∈	PROPN
ejpam-2026	19	17	n	n	CCONJ
ejpam-2026	19	18	[	[	X
ejpam-2026	19	19	2	2	NUM
ejpam-2026	19	20	,	,	PUNCT
ejpam-2026	19	21	proposition	proposition	NOUN
ejpam-2026	19	22	1	1	NUM
ejpam-2026	19	23	]	]	PUNCT
ejpam-2026	19	24	.	.	PUNCT
ejpam-2026	20	1	similarly	similarly	ADV
ejpam-2026	20	2	,	,	PUNCT
ejpam-2026	20	3	it	it	PRON
ejpam-2026	20	4	is	be	AUX
ejpam-2026	20	5	well	well	ADV
ejpam-2026	20	6	known	know	VERB
ejpam-2026	20	7	that	that	SCONJ
ejpam-2026	20	8	the	the	DET
ejpam-2026	20	9	left	left	ADJ
ejpam-2026	20	10	r	r	NOUN
ejpam-2026	20	11	-	-	PUNCT
ejpam-2026	20	12	module	module	NOUN
ejpam-2026	20	13	m	m	NOUN
ejpam-2026	20	14	is	be	AUX
ejpam-2026	20	15	said	say	VERB
ejpam-2026	20	16	to	to	PART
ejpam-2026	20	17	be	be	AUX
ejpam-2026	20	18	finitely	finitely	ADV
ejpam-2026	20	19	generated	generate	VERB
ejpam-2026	20	20	if	if	SCONJ
ejpam-2026	20	21	there	there	PRON
ejpam-2026	20	22	exist	exist	VERB
ejpam-2026	20	23	m1,m2	m1,m2	PROPN
ejpam-2026	20	24	,	,	PUNCT
ejpam-2026	20	25	...	...	PUNCT
ejpam-2026	20	26	,	,	PUNCT
ejpam-2026	20	27	mn	mn	PROPN
ejpam-2026	20	28	∈	∈	PROPN
ejpam-2026	20	29	m	m	VERB
ejpam-2026	20	30	such	such	ADJ
ejpam-2026	20	31	that	that	SCONJ
ejpam-2026	20	32	in	in	ADP
ejpam-2026	20	33	this	this	DET
ejpam-2026	20	34	case	case	NOUN
ejpam-2026	20	35	,	,	PUNCT
ejpam-2026	20	36	we	we	PRON
ejpam-2026	20	37	say	say	VERB
ejpam-2026	20	38	that	that	SCONJ
ejpam-2026	20	39	{	{	PUNCT
ejpam-2026	20	40	m1,m2	m1,m2	PROPN
ejpam-2026	20	41	,	,	PUNCT
ejpam-2026	20	42	...	...	PUNCT
ejpam-2026	20	43	,	,	PUNCT
ejpam-2026	20	44	mn	mn	PROPN
ejpam-2026	20	45	}	}	PUNCT
ejpam-2026	20	46	is	be	AUX
ejpam-2026	20	47	a	a	DET
ejpam-2026	20	48	set	set	NOUN
ejpam-2026	20	49	of	of	ADP
ejpam-2026	20	50	generators	generator	NOUN
ejpam-2026	20	51	for	for	ADP
ejpam-2026	20	52	m	m	PROPN
ejpam-2026	20	53	.	.	PUNCT
ejpam-2026	21	1	the	the	DET
ejpam-2026	21	2	module	module	NOUN
ejpam-2026	21	3	m	m	VERB
ejpam-2026	21	4	is	be	AUX
ejpam-2026	21	5	called	call	VERB
ejpam-2026	21	6	cyclic	cyclic	ADJ
ejpam-2026	21	7	if	if	SCONJ
ejpam-2026	21	8	there	there	PRON
ejpam-2026	21	9	exists	exist	VERB
ejpam-2026	21	10	m	m	VERB
ejpam-2026	21	11	∈	∈	PROPN
ejpam-2026	21	12	m	m	VERB
ejpam-2026	21	13	such	such	ADJ
ejpam-2026	21	14	that	that	SCONJ
ejpam-2026	21	15	m	m	PROPN
ejpam-2026	21	16	=	=	ADJ
ejpam-2026	21	17	rm	rm	PROPN
ejpam-2026	21	18	.	.	PUNCT
ejpam-2026	22	1	a	a	DET
ejpam-2026	22	2	free	free	ADJ
ejpam-2026	22	3	module	module	NOUN
ejpam-2026	22	4	is	be	AUX
ejpam-2026	22	5	a	a	DET
ejpam-2026	22	6	module	module	NOUN
ejpam-2026	22	7	with	with	ADP
ejpam-2026	22	8	a	a	DET
ejpam-2026	22	9	free	free	ADJ
ejpam-2026	22	10	basis	basis	NOUN
ejpam-2026	22	11	,	,	PUNCT
ejpam-2026	22	12	a	a	DET
ejpam-2026	22	13	linearly	linearly	ADV
ejpam-2026	22	14	independent	independent	ADJ
ejpam-2026	22	15	generating	generating	NOUN
ejpam-2026	22	16	set	set	NOUN
ejpam-2026	22	17	.	.	PUNCT
ejpam-2026	23	1	for	for	ADP
ejpam-2026	23	2	an	an	DET
ejpam-2026	23	3	r	r	NOUN
ejpam-2026	23	4	-	-	PUNCT
ejpam-2026	23	5	module	module	NOUN
ejpam-2026	23	6	m	m	NOUN
ejpam-2026	23	7	,	,	PUNCT
ejpam-2026	23	8	the	the	DET
ejpam-2026	23	9	set	set	NOUN
ejpam-2026	23	10	e	e	NOUN
ejpam-2026	23	11	=	=	SYM
ejpam-2026	23	12	{	{	PUNCT
ejpam-2026	23	13	e1	e1	PROPN
ejpam-2026	23	14	,	,	PUNCT
ejpam-2026	23	15	e2	e2	PROPN
ejpam-2026	23	16	,	,	PUNCT
ejpam-2026	23	17	...	...	PUNCT
ejpam-2026	23	18	en	en	X
ejpam-2026	23	19	}	}	PUNCT
ejpam-2026	23	20	is	be	AUX
ejpam-2026	23	21	a	a	DET
ejpam-2026	23	22	free	free	ADJ
ejpam-2026	23	23	basis	basis	NOUN
ejpam-2026	23	24	for	for	ADP
ejpam-2026	23	25	m	m	PRON
ejpam-2026	23	26	such	such	ADJ
ejpam-2026	23	27	that	that	SCONJ
ejpam-2026	23	28	(	(	PUNCT
ejpam-2026	23	29	1	1	X
ejpam-2026	23	30	)	)	PUNCT
ejpam-2026	23	31	if	if	SCONJ
ejpam-2026	23	32	e	e	PRON
ejpam-2026	23	33	is	be	AUX
ejpam-2026	23	34	a	a	DET
ejpam-2026	23	35	generating	generate	VERB
ejpam-2026	23	36	set	set	NOUN
ejpam-2026	23	37	for	for	ADP
ejpam-2026	23	38	m	m	PRON
ejpam-2026	23	39	;	;	PUNCT
ejpam-2026	23	40	that	that	PRON
ejpam-2026	23	41	is	be	AUX
ejpam-2026	23	42	to	to	PART
ejpam-2026	23	43	say	say	VERB
ejpam-2026	23	44	,	,	PUNCT
ejpam-2026	23	45	every	every	DET
ejpam-2026	23	46	element	element	NOUN
ejpam-2026	23	47	of	of	ADP
ejpam-2026	23	48	m	m	PROPN
ejpam-2026	23	49	is	be	AUX
ejpam-2026	23	50	a	a	DET
ejpam-2026	23	51	finite	finite	ADJ
ejpam-2026	23	52	sum	sum	NOUN
ejpam-2026	23	53	of	of	ADP
ejpam-2026	23	54	elements	element	NOUN
ejpam-2026	23	55	of	of	ADP
ejpam-2026	23	56	email	email	NOUN
ejpam-2026	23	57	addresses	address	NOUN
ejpam-2026	23	58	:	:	PUNCT
ejpam-2026	23	59	sirwak2003@yahoo.com	sirwak2003@yahoo.com	X
ejpam-2026	24	1	(	(	PUNCT
ejpam-2026	24	2	w.	w.	PROPN
ejpam-2026	24	3	a.	a.	PROPN
ejpam-2026	24	4	khan	khan	PROPN
ejpam-2026	24	5	)	)	PUNCT
ejpam-2026	24	6	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-2026	25	1	516	516	NUM
ejpam-2026	25	2	c	c	NOUN
ejpam-2026	25	3	©	©	PROPN
ejpam-2026	25	4	2017	2017	NUM
ejpam-2026	25	5	ejpam	ejpam	NOUN
ejpam-2026	25	6	all	all	DET
ejpam-2026	25	7	rights	right	NOUN
ejpam-2026	25	8	reserved	reserve	VERB
ejpam-2026	25	9	.	.	PUNCT
ejpam-2026	26	1	w.	w.	PROPN
ejpam-2026	26	2	a.	a.	PROPN
ejpam-2026	26	3	khan	khan	PROPN
ejpam-2026	26	4	/	/	SYM
ejpam-2026	26	5	eur	eur	PROPN
ejpam-2026	26	6	.	.	PUNCT
ejpam-2026	27	1	j.	j.	PROPN
ejpam-2026	27	2	pure	pure	PROPN
ejpam-2026	27	3	appl	appl	PROPN
ejpam-2026	27	4	.	.	PROPN
ejpam-2026	27	5	math	math	PROPN
ejpam-2026	27	6	,	,	PUNCT
ejpam-2026	27	7	10	10	NUM
ejpam-2026	27	8	(	(	PUNCT
ejpam-2026	27	9	3	3	NUM
ejpam-2026	27	10	)	)	PUNCT
ejpam-2026	27	11	(	(	PUNCT
ejpam-2026	27	12	2017	2017	NUM
ejpam-2026	27	13	)	)	PUNCT
ejpam-2026	27	14	,	,	PUNCT
ejpam-2026	27	15	516	516	NUM
ejpam-2026	27	16	-	-	SYM
ejpam-2026	27	17	520	520	NUM
ejpam-2026	27	18	517	517	NUM
ejpam-2026	27	19	e	e	NOUN
ejpam-2026	27	20	multiplied	multiply	VERB
ejpam-2026	27	21	by	by	ADP
ejpam-2026	27	22	coefficients	coefficient	NOUN
ejpam-2026	27	23	in	in	ADP
ejpam-2026	27	24	r	r	NOUN
ejpam-2026	27	25	;	;	PUNCT
ejpam-2026	27	26	(	(	PUNCT
ejpam-2026	27	27	2	2	X
ejpam-2026	27	28	)	)	PUNCT
ejpam-2026	27	29	e	e	NOUN
ejpam-2026	27	30	is	be	AUX
ejpam-2026	27	31	a	a	DET
ejpam-2026	27	32	free	free	ADJ
ejpam-2026	27	33	set	set	NOUN
ejpam-2026	27	34	,	,	PUNCT
ejpam-2026	27	35	that	that	ADV
ejpam-2026	27	36	is	is	ADV
ejpam-2026	27	37	,	,	PUNCT
ejpam-2026	27	38	if	if	SCONJ
ejpam-2026	27	39	r1e1	r1e1	NOUN
ejpam-2026	27	40	+	+	X
ejpam-2026	27	41	r2e2	r2e2	PROPN
ejpam-2026	28	1	+	+	CCONJ
ejpam-2026	28	2	...	...	PUNCT
ejpam-2026	28	3	+	+	NUM
ejpam-2026	28	4	rnen	rnen	X
ejpam-2026	28	5	=	=	SYM
ejpam-2026	28	6	0	0	PROPN
ejpam-2026	28	7	,	,	PUNCT
ejpam-2026	28	8	then	then	ADV
ejpam-2026	28	9	r1	r1	PROPN
ejpam-2026	28	10	=	=	PUNCT
ejpam-2026	28	11	r2	r2	PROPN
ejpam-2026	28	12	=	=	PUNCT
ejpam-2026	28	13	...	...	PUNCT
ejpam-2026	28	14	=	=	PUNCT
ejpam-2026	28	15	rn	rn	PROPN
ejpam-2026	28	16	=	=	PROPN
ejpam-2026	28	17	0	0	PROPN
ejpam-2026	28	18	(	(	PUNCT
ejpam-2026	28	19	where	where	SCONJ
ejpam-2026	28	20	0	0	NUM
ejpam-2026	28	21	is	be	AUX
ejpam-2026	28	22	the	the	DET
ejpam-2026	28	23	zero	zero	NUM
ejpam-2026	28	24	element	element	NOUN
ejpam-2026	28	25	of	of	ADP
ejpam-2026	28	26	m	m	PROPN
ejpam-2026	28	27	and	and	CCONJ
ejpam-2026	28	28	0	0	NUM
ejpam-2026	28	29	is	be	AUX
ejpam-2026	28	30	the	the	DET
ejpam-2026	28	31	zero	zero	NUM
ejpam-2026	28	32	element	element	NOUN
ejpam-2026	28	33	of	of	ADP
ejpam-2026	28	34	r	r	NOUN
ejpam-2026	28	35	)	)	PUNCT
ejpam-2026	28	36	.	.	PUNCT
ejpam-2026	29	1	let	let	VERB
ejpam-2026	29	2	r	r	PRON
ejpam-2026	29	3	be	be	AUX
ejpam-2026	29	4	an	an	DET
ejpam-2026	29	5	integral	integral	ADJ
ejpam-2026	29	6	domain	domain	NOUN
ejpam-2026	29	7	with	with	ADP
ejpam-2026	29	8	quotient	quotient	NOUN
ejpam-2026	29	9	field	field	PROPN
ejpam-2026	29	10	k.	k.	PROPN
ejpam-2026	30	1	a	a	DET
ejpam-2026	30	2	prime	prime	ADJ
ejpam-2026	30	3	ideal	ideal	NOUN
ejpam-2026	30	4	p	p	NOUN
ejpam-2026	30	5	of	of	ADP
ejpam-2026	30	6	r	r	NOUN
ejpam-2026	30	7	is	be	AUX
ejpam-2026	30	8	called	call	VERB
ejpam-2026	30	9	strongly	strongly	ADV
ejpam-2026	30	10	prime	prime	ADJ
ejpam-2026	30	11	if	if	SCONJ
ejpam-2026	30	12	x	x	NOUN
ejpam-2026	30	13	,	,	PUNCT
ejpam-2026	30	14	y	y	PROPN
ejpam-2026	30	15	∈	∈	PROPN
ejpam-2026	30	16	k	k	PROPN
ejpam-2026	30	17	and	and	CCONJ
ejpam-2026	30	18	xy	xy	PROPN
ejpam-2026	30	19	∈	∈	PROPN
ejpam-2026	30	20	p	p	NOUN
ejpam-2026	30	21	imply	imply	VERB
ejpam-2026	30	22	that	that	SCONJ
ejpam-2026	30	23	x	x	PUNCT
ejpam-2026	30	24	∈	∈	PROPN
ejpam-2026	30	25	p	p	NOUN
ejpam-2026	30	26	or	or	CCONJ
ejpam-2026	30	27	y	y	PROPN
ejpam-2026	30	28	∈	∈	PROPN
ejpam-2026	30	29	p	p	NOUN
ejpam-2026	30	30	(	(	PUNCT
ejpam-2026	30	31	alternatively	alternatively	ADV
ejpam-2026	30	32	p	p	NOUN
ejpam-2026	30	33	is	be	AUX
ejpam-2026	30	34	strongly	strongly	ADV
ejpam-2026	30	35	prime	prime	ADJ
ejpam-2026	30	36	if	if	SCONJ
ejpam-2026	30	37	and	and	CCONJ
ejpam-2026	30	38	only	only	ADV
ejpam-2026	30	39	if	if	SCONJ
ejpam-2026	30	40	x−1p	x−1p	PROPN
ejpam-2026	31	1	⊂	⊂	PROPN
ejpam-2026	31	2	p	p	X
ejpam-2026	31	3	whenever	whenever	SCONJ
ejpam-2026	31	4	x	x	SYM
ejpam-2026	31	5	∈	∈	PROPN
ejpam-2026	31	6	k\r	k\r	PROPN
ejpam-2026	31	7	)	)	PUNCT
ejpam-2026	32	1	[	[	X
ejpam-2026	32	2	5	5	NUM
ejpam-2026	32	3	,	,	PUNCT
ejpam-2026	32	4	definition	definition	NOUN
ejpam-2026	32	5	,	,	PUNCT
ejpam-2026	32	6	page2	page2	PROPN
ejpam-2026	32	7	]	]	X
ejpam-2026	32	8	.	.	PUNCT
ejpam-2026	33	1	a	a	DET
ejpam-2026	33	2	domain	domain	NOUN
ejpam-2026	33	3	r	r	NOUN
ejpam-2026	33	4	is	be	AUX
ejpam-2026	33	5	called	call	VERB
ejpam-2026	33	6	a	a	DET
ejpam-2026	33	7	pseudo	pseudo	NOUN
ejpam-2026	33	8	-	-	ADJ
ejpam-2026	33	9	valuation	valuation	NOUN
ejpam-2026	33	10	domain	domain	NOUN
ejpam-2026	33	11	if	if	SCONJ
ejpam-2026	33	12	every	every	DET
ejpam-2026	33	13	prime	prime	ADJ
ejpam-2026	33	14	ideal	ideal	NOUN
ejpam-2026	33	15	of	of	ADP
ejpam-2026	33	16	r	r	NOUN
ejpam-2026	33	17	is	be	AUX
ejpam-2026	33	18	a	a	DET
ejpam-2026	33	19	strongly	strongly	ADV
ejpam-2026	33	20	prime	prime	ADJ
ejpam-2026	33	21	[	[	X
ejpam-2026	33	22	5	5	NUM
ejpam-2026	33	23	,	,	PUNCT
ejpam-2026	33	24	definition	definition	NOUN
ejpam-2026	33	25	,	,	PUNCT
ejpam-2026	33	26	page2	page2	PROPN
ejpam-2026	33	27	]	]	X
ejpam-2026	33	28	.	.	PUNCT
ejpam-2026	34	1	an	an	DET
ejpam-2026	34	2	integral	integral	ADJ
ejpam-2026	34	3	domain	domain	NOUN
ejpam-2026	34	4	r	r	NOUN
ejpam-2026	34	5	is	be	AUX
ejpam-2026	34	6	a	a	DET
ejpam-2026	34	7	pseudo	pseudo	NOUN
ejpam-2026	34	8	-	-	ADJ
ejpam-2026	34	9	valuation	valuation	NOUN
ejpam-2026	34	10	domain	domain	NOUN
ejpam-2026	34	11	if	if	SCONJ
ejpam-2026	34	12	and	and	CCONJ
ejpam-2026	34	13	only	only	ADV
ejpam-2026	34	14	if	if	SCONJ
ejpam-2026	34	15	for	for	ADP
ejpam-2026	34	16	every	every	DET
ejpam-2026	34	17	nonzero	nonzero	NOUN
ejpam-2026	34	18	x	x	SYM
ejpam-2026	34	19	∈	∈	PROPN
ejpam-2026	34	20	k	k	NOUN
ejpam-2026	34	21	,	,	PUNCT
ejpam-2026	34	22	either	either	CCONJ
ejpam-2026	34	23	x	x	SYM
ejpam-2026	34	24	∈	∈	NOUN
ejpam-2026	34	25	r	r	NOUN
ejpam-2026	34	26	or	or	CCONJ
ejpam-2026	34	27	ax−1	ax−1	VERB
ejpam-2026	34	28	∈	∈	NOUN
ejpam-2026	34	29	r	r	NOUN
ejpam-2026	34	30	for	for	ADP
ejpam-2026	34	31	every	every	DET
ejpam-2026	34	32	nonunit	nonunit	NOUN
ejpam-2026	34	33	a	a	DET
ejpam-2026	34	34	∈	∈	NOUN
ejpam-2026	34	35	r	r	NOUN
ejpam-2026	35	1	[	[	X
ejpam-2026	35	2	5	5	NUM
ejpam-2026	35	3	,	,	PUNCT
ejpam-2026	35	4	theorem	theorem	VERB
ejpam-2026	35	5	1.5(3	1.5(3	NUM
ejpam-2026	35	6	)	)	PUNCT
ejpam-2026	35	7	]	]	PUNCT
ejpam-2026	35	8	.	.	PUNCT
ejpam-2026	36	1	every	every	DET
ejpam-2026	36	2	valuation	valuation	NOUN
ejpam-2026	36	3	domain	domain	NOUN
ejpam-2026	36	4	is	be	AUX
ejpam-2026	36	5	a	a	DET
ejpam-2026	36	6	pseudo	pseudo	NOUN
ejpam-2026	36	7	-	-	ADJ
ejpam-2026	36	8	valuation	valuation	NOUN
ejpam-2026	36	9	domain	domain	NOUN
ejpam-2026	36	10	[	[	X
ejpam-2026	36	11	5	5	NUM
ejpam-2026	36	12	,	,	PUNCT
ejpam-2026	36	13	proposition	proposition	NOUN
ejpam-2026	36	14	.	.	PUNCT
ejpam-2026	37	1	1.1	1.1	NUM
ejpam-2026	37	2	]	]	PUNCT
ejpam-2026	37	3	but	but	CCONJ
ejpam-2026	37	4	converse	converse	NOUN
ejpam-2026	37	5	is	be	AUX
ejpam-2026	37	6	not	not	PART
ejpam-2026	37	7	true	true	ADJ
ejpam-2026	37	8	for	for	ADP
ejpam-2026	37	9	example	example	NOUN
ejpam-2026	37	10	the	the	DET
ejpam-2026	37	11	valuation	valuation	NOUN
ejpam-2026	37	12	domain	domain	NOUN
ejpam-2026	37	13	v	v	NOUN
ejpam-2026	37	14	of	of	ADP
ejpam-2026	37	15	the	the	DET
ejpam-2026	37	16	form	form	NOUN
ejpam-2026	37	17	k	k	PROPN
ejpam-2026	38	1	+	+	NOUN
ejpam-2026	38	2	m	m	X
ejpam-2026	38	3	,	,	PUNCT
ejpam-2026	38	4	where	where	SCONJ
ejpam-2026	38	5	k	k	PROPN
ejpam-2026	38	6	is	be	AUX
ejpam-2026	38	7	a	a	DET
ejpam-2026	38	8	field	field	NOUN
ejpam-2026	38	9	and	and	CCONJ
ejpam-2026	38	10	m	m	NOUN
ejpam-2026	38	11	is	be	AUX
ejpam-2026	38	12	the	the	DET
ejpam-2026	38	13	maximal	maximal	ADJ
ejpam-2026	38	14	ideal	ideal	NOUN
ejpam-2026	38	15	of	of	ADP
ejpam-2026	38	16	v	v	NOUN
ejpam-2026	38	17	.	.	PUNCT
ejpam-2026	39	1	if	if	SCONJ
ejpam-2026	39	2	f	f	PROPN
ejpam-2026	39	3	is	be	AUX
ejpam-2026	39	4	a	a	DET
ejpam-2026	39	5	proper	proper	ADJ
ejpam-2026	39	6	subfield	subfield	NOUN
ejpam-2026	39	7	of	of	ADP
ejpam-2026	39	8	k	k	NOUN
ejpam-2026	39	9	,	,	PUNCT
ejpam-2026	39	10	then	then	ADV
ejpam-2026	39	11	r	r	NOUN
ejpam-2026	39	12	=	=	PUNCT
ejpam-2026	39	13	f	f	X
ejpam-2026	40	1	+	+	NOUN
ejpam-2026	40	2	m	m	VERB
ejpam-2026	40	3	is	be	AUX
ejpam-2026	40	4	a	a	DET
ejpam-2026	40	5	pseudo	pseudo	NOUN
ejpam-2026	40	6	-	-	ADJ
ejpam-2026	40	7	valuation	valuation	NOUN
ejpam-2026	40	8	domain	domain	NOUN
ejpam-2026	40	9	which	which	PRON
ejpam-2026	40	10	is	be	AUX
ejpam-2026	40	11	not	not	PART
ejpam-2026	40	12	a	a	DET
ejpam-2026	40	13	valuation	valuation	NOUN
ejpam-2026	40	14	domain	domain	NOUN
ejpam-2026	40	15	.	.	PUNCT
ejpam-2026	41	1	whereas	whereas	SCONJ
ejpam-2026	41	2	r	r	NOUN
ejpam-2026	41	3	and	and	CCONJ
ejpam-2026	41	4	v	v	NOUN
ejpam-2026	41	5	have	have	VERB
ejpam-2026	41	6	the	the	DET
ejpam-2026	41	7	same	same	ADJ
ejpam-2026	41	8	quotient	quotient	NOUN
ejpam-2026	41	9	field	field	NOUN
ejpam-2026	41	10	l	l	NOUN
ejpam-2026	41	11	and	and	CCONJ
ejpam-2026	41	12	that	that	SCONJ
ejpam-2026	41	13	m	m	NOUN
ejpam-2026	41	14	is	be	AUX
ejpam-2026	41	15	the	the	DET
ejpam-2026	41	16	maximal	maximal	ADJ
ejpam-2026	41	17	ideal	ideal	NOUN
ejpam-2026	41	18	of	of	ADP
ejpam-2026	41	19	r	r	NOUN
ejpam-2026	41	20	[	[	X
ejpam-2026	41	21	4	4	NUM
ejpam-2026	41	22	,	,	PUNCT
ejpam-2026	41	23	theorem	theorem	VERB
ejpam-2026	41	24	a	a	DET
ejpam-2026	41	25	,	,	PUNCT
ejpam-2026	41	26	page	page	NOUN
ejpam-2026	41	27	560	560	NUM
ejpam-2026	41	28	]	]	PUNCT
ejpam-2026	41	29	.	.	PUNCT
ejpam-2026	42	1	a	a	DET
ejpam-2026	42	2	quasi	quasi	ADJ
ejpam-2026	42	3	-	-	ADJ
ejpam-2026	42	4	local	local	ADJ
ejpam-2026	42	5	domain	domain	NOUN
ejpam-2026	42	6	(	(	PUNCT
ejpam-2026	42	7	r	r	NOUN
ejpam-2026	42	8	,	,	PUNCT
ejpam-2026	42	9	m	m	NOUN
ejpam-2026	42	10	)	)	PUNCT
ejpam-2026	42	11	is	be	AUX
ejpam-2026	42	12	a	a	DET
ejpam-2026	42	13	pseudo	pseudo	NOUN
ejpam-2026	42	14	-	-	ADJ
ejpam-2026	42	15	valuation	valuation	NOUN
ejpam-2026	42	16	domain	domain	NOUN
ejpam-2026	42	17	if	if	SCONJ
ejpam-2026	42	18	and	and	CCONJ
ejpam-2026	42	19	only	only	ADV
ejpam-2026	42	20	if	if	SCONJ
ejpam-2026	42	21	x−1	x−1	PROPN
ejpam-2026	42	22	m	m	VERB
ejpam-2026	42	23	⊂m	⊂m	ADJ
ejpam-2026	42	24	whenever	whenever	SCONJ
ejpam-2026	42	25	x	x	SYM
ejpam-2026	42	26	∈	∈	PROPN
ejpam-2026	42	27	k\r	k\r	PROPN
ejpam-2026	42	28	[	[	X
ejpam-2026	42	29	5	5	NUM
ejpam-2026	42	30	,	,	PUNCT
ejpam-2026	42	31	theorem	theorem	ADJ
ejpam-2026	42	32	.	.	PROPN
ejpam-2026	42	33	1.4	1.4	NUM
ejpam-2026	42	34	]	]	PUNCT
ejpam-2026	42	35	.	.	PUNCT
ejpam-2026	43	1	throughout	throughout	ADP
ejpam-2026	43	2	we	we	PRON
ejpam-2026	43	3	always	always	ADV
ejpam-2026	43	4	mean	mean	VERB
ejpam-2026	43	5	by	by	ADP
ejpam-2026	43	6	a	a	DET
ejpam-2026	43	7	ring	ring	NOUN
ejpam-2026	43	8	r	r	NOUN
ejpam-2026	43	9	,	,	PUNCT
ejpam-2026	43	10	a	a	DET
ejpam-2026	43	11	commutative	commutative	ADJ
ejpam-2026	43	12	ring	ring	NOUN
ejpam-2026	43	13	having	have	VERB
ejpam-2026	43	14	unity	unity	NOUN
ejpam-2026	43	15	1	1	NUM
ejpam-2026	43	16	.	.	NOUN
ejpam-2026	43	17	2	2	NUM
ejpam-2026	43	18	.	.	X
ejpam-2026	43	19	module	module	NOUN
ejpam-2026	43	20	over	over	ADP
ejpam-2026	43	21	pseudo	pseudo	NOUN
ejpam-2026	43	22	-	-	NOUN
ejpam-2026	43	23	valuation	valuation	NOUN
ejpam-2026	43	24	ring	ring	NOUN
ejpam-2026	43	25	a	a	DET
ejpam-2026	43	26	commutative	commutative	ADJ
ejpam-2026	43	27	ring	ring	NOUN
ejpam-2026	43	28	r	r	NOUN
ejpam-2026	43	29	with	with	ADP
ejpam-2026	43	30	1	1	NUM
ejpam-2026	43	31	is	be	AUX
ejpam-2026	43	32	called	call	VERB
ejpam-2026	43	33	a	a	DET
ejpam-2026	43	34	pseudo	pseudo	NOUN
ejpam-2026	43	35	-	-	NOUN
ejpam-2026	43	36	valuation	valuation	NOUN
ejpam-2026	43	37	ring	ring	NOUN
ejpam-2026	43	38	(	(	PUNCT
ejpam-2026	43	39	pv	pv	INTJ
ejpam-2026	43	40	r	r	NOUN
ejpam-2026	43	41	)	)	PUNCT
ejpam-2026	43	42	if	if	SCONJ
ejpam-2026	43	43	for	for	ADP
ejpam-2026	43	44	every	every	DET
ejpam-2026	43	45	a	a	PRON
ejpam-2026	43	46	,	,	PUNCT
ejpam-2026	43	47	b	b	NOUN
ejpam-2026	43	48	in	in	ADP
ejpam-2026	43	49	r	r	NOUN
ejpam-2026	43	50	,	,	PUNCT
ejpam-2026	43	51	either	either	CCONJ
ejpam-2026	43	52	a	a	DET
ejpam-2026	43	53	divides	divide	NOUN
ejpam-2026	43	54	b	b	NOUN
ejpam-2026	43	55	or	or	CCONJ
ejpam-2026	43	56	b	b	PROPN
ejpam-2026	43	57	divides	divide	NOUN
ejpam-2026	43	58	ac	ac	ADV
ejpam-2026	43	59	for	for	ADP
ejpam-2026	43	60	each	each	DET
ejpam-2026	43	61	nonunit	nonunit	NOUN
ejpam-2026	43	62	c	c	PROPN
ejpam-2026	43	63	in	in	ADP
ejpam-2026	43	64	r.	r.	PROPN
ejpam-2026	43	65	we	we	PRON
ejpam-2026	43	66	begin	begin	VERB
ejpam-2026	43	67	with	with	ADP
ejpam-2026	43	68	the	the	DET
ejpam-2026	43	69	following	follow	VERB
ejpam-2026	43	70	definition	definition	NOUN
ejpam-2026	43	71	.	.	PUNCT
ejpam-2026	44	1	definition	definition	NOUN
ejpam-2026	44	2	1	1	NUM
ejpam-2026	44	3	.	.	PUNCT
ejpam-2026	45	1	if	if	SCONJ
ejpam-2026	45	2	r	r	NOUN
ejpam-2026	45	3	is	be	AUX
ejpam-2026	45	4	a	a	DET
ejpam-2026	45	5	pseudo	pseudo	NOUN
ejpam-2026	45	6	-	-	NOUN
ejpam-2026	45	7	valuation	valuation	NOUN
ejpam-2026	45	8	ring	ring	NOUN
ejpam-2026	45	9	and	and	CCONJ
ejpam-2026	45	10	m	m	AUX
ejpam-2026	45	11	be	be	AUX
ejpam-2026	45	12	an	an	DET
ejpam-2026	45	13	additive	additive	ADJ
ejpam-2026	45	14	abelian	abelian	NOUN
ejpam-2026	45	15	group	group	NOUN
ejpam-2026	45	16	then	then	ADV
ejpam-2026	45	17	m	m	VERB
ejpam-2026	45	18	is	be	AUX
ejpam-2026	45	19	said	say	VERB
ejpam-2026	45	20	to	to	PART
ejpam-2026	45	21	be	be	AUX
ejpam-2026	45	22	r	r	NOUN
ejpam-2026	45	23	-	-	PUNCT
ejpam-2026	45	24	module	module	NOUN
ejpam-2026	45	25	if	if	SCONJ
ejpam-2026	45	26	for	for	ADP
ejpam-2026	45	27	all	all	DET
ejpam-2026	45	28	r	r	NOUN
ejpam-2026	45	29	,	,	PUNCT
ejpam-2026	45	30	s	s	PROPN
ejpam-2026	45	31	,	,	PUNCT
ejpam-2026	45	32	t	t	PROPN
ejpam-2026	45	33	∈	∈	PROPN
ejpam-2026	45	34	r	r	NOUN
ejpam-2026	45	35	,	,	PUNCT
ejpam-2026	45	36	where	where	SCONJ
ejpam-2026	45	37	t	t	PROPN
ejpam-2026	45	38	is	be	AUX
ejpam-2026	45	39	a	a	DET
ejpam-2026	45	40	nonunit	nonunit	NOUN
ejpam-2026	45	41	of	of	ADP
ejpam-2026	45	42	r	r	NOUN
ejpam-2026	45	43	and	and	CCONJ
ejpam-2026	45	44	m	m	PROPN
ejpam-2026	45	45	,	,	PUNCT
ejpam-2026	45	46	n	n	PRON
ejpam-2026	45	47	∈	∈	NOUN
ejpam-2026	45	48	m	m	VERB
ejpam-2026	45	49	satisfies	satisfie	NOUN
ejpam-2026	45	50	:	:	PUNCT
ejpam-2026	45	51	(	(	PUNCT
ejpam-2026	45	52	1	1	X
ejpam-2026	45	53	)	)	PUNCT
ejpam-2026	45	54	either	either	CCONJ
ejpam-2026	45	55	r	r	X
ejpam-2026	45	56	/	/	SYM
ejpam-2026	45	57	s	s	PART
ejpam-2026	45	58	m	m	NOUN
ejpam-2026	45	59	∈	∈	ADJ
ejpam-2026	45	60	m	m	ADJ
ejpam-2026	45	61	or	or	CCONJ
ejpam-2026	45	62	s	s	PROPN
ejpam-2026	45	63	/	/	SYM
ejpam-2026	45	64	rt	rt	PROPN
ejpam-2026	45	65	m	m	PROPN
ejpam-2026	45	66	∈	∈	PROPN
ejpam-2026	45	67	m	m	VERB
ejpam-2026	45	68	,	,	PUNCT
ejpam-2026	45	69	for	for	ADP
ejpam-2026	45	70	all	all	DET
ejpam-2026	45	71	r	r	NOUN
ejpam-2026	45	72	,	,	PUNCT
ejpam-2026	45	73	s	s	PROPN
ejpam-2026	45	74	,	,	PUNCT
ejpam-2026	45	75	t	t	PROPN
ejpam-2026	45	76	∈	∈	PROPN
ejpam-2026	45	77	r	r	NOUN
ejpam-2026	45	78	,	,	PUNCT
ejpam-2026	45	79	where	where	SCONJ
ejpam-2026	45	80	t	t	PROPN
ejpam-2026	45	81	is	be	AUX
ejpam-2026	45	82	a	a	DET
ejpam-2026	45	83	nonunit	nonunit	NOUN
ejpam-2026	45	84	of	of	ADP
ejpam-2026	45	85	r	r	NOUN
ejpam-2026	45	86	and	and	CCONJ
ejpam-2026	45	87	m	m	NOUN
ejpam-2026	45	88	∈m	∈m	NOUN
ejpam-2026	45	89	.	.	PUNCT
ejpam-2026	46	1	(	(	PUNCT
ejpam-2026	46	2	2	2	NUM
ejpam-2026	46	3	)	)	PUNCT
ejpam-2026	46	4	(	(	PUNCT
ejpam-2026	46	5	r	r	NOUN
ejpam-2026	46	6	+	+	NOUN
ejpam-2026	46	7	s)m	s)m	ADJ
ejpam-2026	46	8	=	=	PUNCT
ejpam-2026	47	1	rm+	rm+	NOUN
ejpam-2026	47	2	sm	sm	INTJ
ejpam-2026	47	3	(	(	PUNCT
ejpam-2026	47	4	3	3	NUM
ejpam-2026	47	5	)	)	PUNCT
ejpam-2026	47	6	r(m+	r(m+	NOUN
ejpam-2026	47	7	n	n	CCONJ
ejpam-2026	47	8	)	)	PUNCT
ejpam-2026	47	9	=	=	SYM
ejpam-2026	47	10	rm+	rm+	PROPN
ejpam-2026	47	11	rn	rn	PROPN
ejpam-2026	47	12	(	(	PUNCT
ejpam-2026	47	13	4	4	NUM
ejpam-2026	47	14	)	)	PUNCT
ejpam-2026	47	15	(	(	PUNCT
ejpam-2026	47	16	rs)m	rs)m	PROPN
ejpam-2026	47	17	=	=	SYM
ejpam-2026	47	18	r(sm	r(sm	PROPN
ejpam-2026	47	19	)	)	PUNCT
ejpam-2026	47	20	if	if	SCONJ
ejpam-2026	47	21	one	one	PRON
ejpam-2026	47	22	writes	write	VERB
ejpam-2026	47	23	the	the	DET
ejpam-2026	47	24	scalar	scalar	ADJ
ejpam-2026	47	25	action	action	NOUN
ejpam-2026	47	26	as	as	ADP
ejpam-2026	47	27	fr	fr	ADV
ejpam-2026	47	28	so	so	SCONJ
ejpam-2026	47	29	that	that	SCONJ
ejpam-2026	47	30	fr(x	fr(x	X
ejpam-2026	47	31	)	)	PUNCT
ejpam-2026	47	32	=	=	SYM
ejpam-2026	48	1	r	r	X
ejpam-2026	48	2	/	/	SYM
ejpam-2026	48	3	sm	sm	PROPN
ejpam-2026	48	4	,	,	PUNCT
ejpam-2026	48	5	and	and	CCONJ
ejpam-2026	48	6	f	f	PROPN
ejpam-2026	48	7	for	for	ADP
ejpam-2026	48	8	the	the	DET
ejpam-2026	48	9	map	map	NOUN
ejpam-2026	48	10	which	which	PRON
ejpam-2026	48	11	takes	take	VERB
ejpam-2026	48	12	each	each	DET
ejpam-2026	48	13	r	r	NOUN
ejpam-2026	48	14	to	to	ADP
ejpam-2026	48	15	its	its	PRON
ejpam-2026	48	16	corresponding	corresponding	ADJ
ejpam-2026	48	17	map	map	NOUN
ejpam-2026	48	18	fr	fr	ADV
ejpam-2026	48	19	,	,	PUNCT
ejpam-2026	48	20	then	then	ADV
ejpam-2026	48	21	the	the	DET
ejpam-2026	48	22	second	second	ADJ
ejpam-2026	48	23	axiom	axiom	NOUN
ejpam-2026	48	24	states	state	VERB
ejpam-2026	48	25	that	that	SCONJ
ejpam-2026	48	26	every	every	DET
ejpam-2026	48	27	fr	fr	NOUN
ejpam-2026	48	28	is	be	AUX
ejpam-2026	48	29	a	a	DET
ejpam-2026	48	30	group	group	NOUN
ejpam-2026	48	31	homomorphism	homomorphism	NOUN
ejpam-2026	48	32	of	of	ADP
ejpam-2026	48	33	m	m	PROPN
ejpam-2026	48	34	,	,	PUNCT
ejpam-2026	48	35	and	and	CCONJ
ejpam-2026	48	36	the	the	DET
ejpam-2026	48	37	other	other	ADJ
ejpam-2026	48	38	third	third	ADJ
ejpam-2026	48	39	and	and	CCONJ
ejpam-2026	48	40	fourth	fourth	ADJ
ejpam-2026	48	41	axioms	axiom	NOUN
ejpam-2026	48	42	asserts	assert	VERB
ejpam-2026	48	43	that	that	SCONJ
ejpam-2026	48	44	f	f	PROPN
ejpam-2026	48	45	is	be	AUX
ejpam-2026	48	46	a	a	DET
ejpam-2026	48	47	ring	ring	NOUN
ejpam-2026	48	48	homomorphism	homomorphism	NOUN
ejpam-2026	48	49	from	from	ADP
ejpam-2026	48	50	r	r	NOUN
ejpam-2026	48	51	to	to	ADP
ejpam-2026	48	52	the	the	DET
ejpam-2026	48	53	endomorphism	endomorphism	PROPN
ejpam-2026	48	54	ring	ring	PROPN
ejpam-2026	48	55	end(m	end(m	PROPN
ejpam-2026	48	56	)	)	PUNCT
ejpam-2026	48	57	.	.	PUNCT
ejpam-2026	49	1	remark	remark	PROPN
ejpam-2026	49	2	1	1	NUM
ejpam-2026	49	3	.	.	PUNCT
ejpam-2026	50	1	following	follow	VERB
ejpam-2026	50	2	[	[	X
ejpam-2026	50	3	1	1	NUM
ejpam-2026	50	4	,	,	PUNCT
ejpam-2026	50	5	proposition	proposition	NOUN
ejpam-2026	50	6	3(4	3(4	NUM
ejpam-2026	50	7	)	)	PUNCT
ejpam-2026	50	8	]	]	PUNCT
ejpam-2026	50	9	,	,	PUNCT
ejpam-2026	50	10	an	an	DET
ejpam-2026	50	11	integral	integral	ADJ
ejpam-2026	50	12	domain	domain	NOUN
ejpam-2026	50	13	is	be	AUX
ejpam-2026	50	14	a	a	DET
ejpam-2026	50	15	pv	pv	NOUN
ejpam-2026	50	16	d	d	X
ejpam-2026	50	17	⇔	⇔	X
ejpam-2026	50	18	for	for	ADP
ejpam-2026	50	19	every	every	DET
ejpam-2026	50	20	a	a	PROPN
ejpam-2026	50	21	,	,	PUNCT
ejpam-2026	50	22	b	b	X
ejpam-2026	50	23	∈	∈	NOUN
ejpam-2026	50	24	r	r	NOUN
ejpam-2026	50	25	either	either	CCONJ
ejpam-2026	50	26	a	a	DET
ejpam-2026	50	27	/	/	SYM
ejpam-2026	50	28	b	b	NOUN
ejpam-2026	50	29	or	or	CCONJ
ejpam-2026	50	30	b	b	X
ejpam-2026	50	31	/	/	SYM
ejpam-2026	50	32	ac	ac	PROPN
ejpam-2026	50	33	for	for	ADP
ejpam-2026	50	34	every	every	DET
ejpam-2026	50	35	non	non	ADJ
ejpam-2026	50	36	unit	unit	NOUN
ejpam-2026	50	37	c	c	PROPN
ejpam-2026	50	38	∈	∈	PROPN
ejpam-2026	50	39	r.	r.	PROPN
ejpam-2026	50	40	an	an	DET
ejpam-2026	50	41	integral	integral	ADJ
ejpam-2026	50	42	domain	domain	NOUN
ejpam-2026	50	43	is	be	AUX
ejpam-2026	50	44	a	a	DET
ejpam-2026	50	45	pv	pv	NOUN
ejpam-2026	50	46	r	r	NOUN
ejpam-2026	50	47	⇔	⇔	NOUN
ejpam-2026	50	48	it	it	PRON
ejpam-2026	50	49	is	be	AUX
ejpam-2026	50	50	a	a	DET
ejpam-2026	50	51	pv	pv	NOUN
ejpam-2026	50	52	d	d	NOUN
ejpam-2026	50	53	theorem	theorem	NOUN
ejpam-2026	50	54	1	1	NUM
ejpam-2026	50	55	.	.	PUNCT
ejpam-2026	51	1	if	if	SCONJ
ejpam-2026	51	2	r	r	NOUN
ejpam-2026	51	3	is	be	AUX
ejpam-2026	51	4	a	a	DET
ejpam-2026	51	5	pv	pv	NOUN
ejpam-2026	51	6	r	r	NOUN
ejpam-2026	51	7	,	,	PUNCT
ejpam-2026	51	8	and	and	CCONJ
ejpam-2026	51	9	m	m	PROPN
ejpam-2026	51	10	is	be	AUX
ejpam-2026	51	11	an	an	DET
ejpam-2026	51	12	abelian	abelian	ADJ
ejpam-2026	51	13	group	group	NOUN
ejpam-2026	51	14	.	.	PUNCT
ejpam-2026	52	1	then	then	ADV
ejpam-2026	52	2	m	m	PROPN
ejpam-2026	52	3	is	be	AUX
ejpam-2026	52	4	a	a	DET
ejpam-2026	52	5	module	module	NOUN
ejpam-2026	52	6	over	over	ADP
ejpam-2026	52	7	pv	pv	NOUN
ejpam-2026	52	8	r	r	NOUN
ejpam-2026	52	9	⇔	⇔	NOUN
ejpam-2026	52	10	if	if	SCONJ
ejpam-2026	52	11	∀	∀	NOUN
ejpam-2026	52	12	r	r	NOUN
ejpam-2026	52	13	,	,	PUNCT
ejpam-2026	52	14	s	s	PROPN
ejpam-2026	52	15	,	,	PUNCT
ejpam-2026	52	16	t	t	PROPN
ejpam-2026	52	17	∈	∈	PROPN
ejpam-2026	52	18	r	r	NOUN
ejpam-2026	52	19	where	where	SCONJ
ejpam-2026	52	20	t	t	PROPN
ejpam-2026	52	21	is	be	AUX
ejpam-2026	52	22	a	a	DET
ejpam-2026	52	23	nonunit	nonunit	NOUN
ejpam-2026	52	24	and	and	CCONJ
ejpam-2026	52	25	m	m	PROPN
ejpam-2026	52	26	∈	∈	PROPN
ejpam-2026	52	27	m	m	PROPN
ejpam-2026	52	28	,	,	PUNCT
ejpam-2026	52	29	either	either	CCONJ
ejpam-2026	52	30	r	r	X
ejpam-2026	52	31	/	/	SYM
ejpam-2026	52	32	s	s	PART
ejpam-2026	52	33	m	m	NOUN
ejpam-2026	52	34	∈	∈	ADJ
ejpam-2026	52	35	m	m	ADJ
ejpam-2026	52	36	or	or	CCONJ
ejpam-2026	52	37	s	s	PROPN
ejpam-2026	52	38	/	/	SYM
ejpam-2026	52	39	rt	rt	PROPN
ejpam-2026	52	40	m	m	PROPN
ejpam-2026	52	41	∈m	∈m	NOUN
ejpam-2026	52	42	.	.	PUNCT
ejpam-2026	53	1	w.	w.	PROPN
ejpam-2026	53	2	a.	a.	PROPN
ejpam-2026	53	3	khan	khan	PROPN
ejpam-2026	53	4	/	/	SYM
ejpam-2026	53	5	eur	eur	PROPN
ejpam-2026	53	6	.	.	PUNCT
ejpam-2026	54	1	j.	j.	PROPN
ejpam-2026	54	2	pure	pure	PROPN
ejpam-2026	54	3	appl	appl	PROPN
ejpam-2026	54	4	.	.	PROPN
ejpam-2026	54	5	math	math	PROPN
ejpam-2026	54	6	,	,	PUNCT
ejpam-2026	54	7	10	10	NUM
ejpam-2026	54	8	(	(	PUNCT
ejpam-2026	54	9	3	3	NUM
ejpam-2026	54	10	)	)	PUNCT
ejpam-2026	54	11	(	(	PUNCT
ejpam-2026	54	12	2017	2017	NUM
ejpam-2026	54	13	)	)	PUNCT
ejpam-2026	54	14	,	,	PUNCT
ejpam-2026	54	15	516	516	NUM
ejpam-2026	54	16	-	-	SYM
ejpam-2026	54	17	520	520	NUM
ejpam-2026	54	18	518	518	NUM
ejpam-2026	54	19	proof	proof	NOUN
ejpam-2026	54	20	.	.	PUNCT
ejpam-2026	55	1	let	let	VERB
ejpam-2026	55	2	r	r	NOUN
ejpam-2026	55	3	is	be	AUX
ejpam-2026	55	4	a	a	DET
ejpam-2026	55	5	pv	pv	NOUN
ejpam-2026	55	6	r	r	NOUN
ejpam-2026	55	7	and	and	CCONJ
ejpam-2026	55	8	m	m	PROPN
ejpam-2026	55	9	is	be	AUX
ejpam-2026	55	10	a	a	DET
ejpam-2026	55	11	module	module	NOUN
ejpam-2026	55	12	then	then	ADV
ejpam-2026	55	13	by	by	ADP
ejpam-2026	55	14	definition	definition	NOUN
ejpam-2026	55	15	for	for	ADP
ejpam-2026	55	16	all	all	DET
ejpam-2026	55	17	r	r	NOUN
ejpam-2026	55	18	,	,	PUNCT
ejpam-2026	55	19	s	s	PROPN
ejpam-2026	55	20	,	,	PUNCT
ejpam-2026	55	21	t	t	PROPN
ejpam-2026	55	22	∈	∈	PROPN
ejpam-2026	55	23	r	r	NOUN
ejpam-2026	55	24	and	and	CCONJ
ejpam-2026	55	25	m	m	PROPN
ejpam-2026	55	26	∈	∈	PROPN
ejpam-2026	55	27	m	m	NOUN
ejpam-2026	55	28	,	,	PUNCT
ejpam-2026	55	29	clearly	clearly	ADV
ejpam-2026	55	30	either	either	CCONJ
ejpam-2026	55	31	r	r	X
ejpam-2026	55	32	/	/	SYM
ejpam-2026	55	33	s	s	PART
ejpam-2026	55	34	m	m	NOUN
ejpam-2026	55	35	∈	∈	ADJ
ejpam-2026	55	36	m	m	ADJ
ejpam-2026	55	37	or	or	CCONJ
ejpam-2026	55	38	s	s	PROPN
ejpam-2026	55	39	/	/	SYM
ejpam-2026	55	40	rt	rt	PROPN
ejpam-2026	55	41	m	m	PROPN
ejpam-2026	55	42	∈	∈	PROPN
ejpam-2026	55	43	m	m	VERB
ejpam-2026	55	44	.	.	PUNCT
ejpam-2026	56	1	conversely	conversely	ADV
ejpam-2026	56	2	suppose	suppose	VERB
ejpam-2026	56	3	m	m	NOUN
ejpam-2026	56	4	is	be	AUX
ejpam-2026	56	5	an	an	DET
ejpam-2026	56	6	abelian	abelian	ADJ
ejpam-2026	56	7	group	group	NOUN
ejpam-2026	56	8	and	and	CCONJ
ejpam-2026	56	9	r	r	NOUN
ejpam-2026	56	10	is	be	AUX
ejpam-2026	56	11	a	a	DET
ejpam-2026	56	12	pv	pv	NOUN
ejpam-2026	56	13	r	r	NOUN
ejpam-2026	56	14	,	,	PUNCT
ejpam-2026	56	15	and	and	CCONJ
ejpam-2026	56	16	either	either	CCONJ
ejpam-2026	56	17	r	r	X
ejpam-2026	56	18	/	/	SYM
ejpam-2026	56	19	s	s	PART
ejpam-2026	56	20	m	m	NOUN
ejpam-2026	56	21	∈	∈	ADJ
ejpam-2026	56	22	m	m	ADJ
ejpam-2026	56	23	or	or	CCONJ
ejpam-2026	56	24	s	s	PROPN
ejpam-2026	56	25	/	/	SYM
ejpam-2026	56	26	rt	rt	PROPN
ejpam-2026	56	27	m	m	PROPN
ejpam-2026	56	28	∈	∈	PROPN
ejpam-2026	56	29	m	m	NOUN
ejpam-2026	56	30	,	,	PUNCT
ejpam-2026	56	31	where	where	SCONJ
ejpam-2026	56	32	r	r	NOUN
ejpam-2026	56	33	,	,	PUNCT
ejpam-2026	56	34	s	s	PROPN
ejpam-2026	56	35	,	,	PUNCT
ejpam-2026	56	36	t	t	PROPN
ejpam-2026	56	37	∈	∈	PROPN
ejpam-2026	56	38	r	r	NOUN
ejpam-2026	56	39	and	and	CCONJ
ejpam-2026	56	40	m	m	NOUN
ejpam-2026	56	41	∈m	∈m	NOUN
ejpam-2026	56	42	.	.	PUNCT
ejpam-2026	57	1	we	we	PRON
ejpam-2026	57	2	have	have	VERB
ejpam-2026	57	3	;	;	PUNCT
ejpam-2026	57	4	(	(	PUNCT
ejpam-2026	57	5	1	1	X
ejpam-2026	57	6	)	)	PUNCT
ejpam-2026	57	7	either	either	CCONJ
ejpam-2026	57	8	(	(	PUNCT
ejpam-2026	57	9	r	r	X
ejpam-2026	57	10	/	/	SYM
ejpam-2026	57	11	s	s	NOUN
ejpam-2026	57	12	)	)	PUNCT
ejpam-2026	57	13	m	m	VERB
ejpam-2026	57	14	∈	∈	NOUN
ejpam-2026	57	15	m	m	ADJ
ejpam-2026	57	16	or	or	CCONJ
ejpam-2026	57	17	s	s	PROPN
ejpam-2026	57	18	/	/	SYM
ejpam-2026	57	19	rt	rt	PROPN
ejpam-2026	57	20	m	m	PROPN
ejpam-2026	57	21	∈	∈	PROPN
ejpam-2026	57	22	m	m	NOUN
ejpam-2026	57	23	.	.	PUNCT
ejpam-2026	58	1	if	if	SCONJ
ejpam-2026	58	2	s	s	PRON
ejpam-2026	58	3	=	=	NOUN
ejpam-2026	58	4	1	1	NUM
ejpam-2026	58	5	then	then	ADV
ejpam-2026	58	6	rm	rm	PROPN
ejpam-2026	58	7	∈	∈	PROPN
ejpam-2026	58	8	m	m	PROPN
ejpam-2026	58	9	,	,	PUNCT
ejpam-2026	58	10	if	if	SCONJ
ejpam-2026	58	11	s	s	X
ejpam-2026	58	12	6=	6=	NUM
ejpam-2026	58	13	1	1	NUM
ejpam-2026	58	14	then	then	ADV
ejpam-2026	58	15	r	r	X
ejpam-2026	58	16	/	/	SYM
ejpam-2026	58	17	s	s	PART
ejpam-2026	58	18	∈	∈	NOUN
ejpam-2026	58	19	r⇒	r⇒	NOUN
ejpam-2026	58	20	r	r	NOUN
ejpam-2026	58	21	is	be	AUX
ejpam-2026	58	22	a	a	DET
ejpam-2026	58	23	pvr	pvr	NOUN
ejpam-2026	58	24	and	and	CCONJ
ejpam-2026	58	25	let	let	VERB
ejpam-2026	58	26	r	r	VERB
ejpam-2026	58	27	/	/	SYM
ejpam-2026	58	28	s	s	PART
ejpam-2026	58	29	=	=	NOUN
ejpam-2026	58	30	r1	r1	PROPN
ejpam-2026	58	31	⇒	⇒	PROPN
ejpam-2026	58	32	r1	r1	PROPN
ejpam-2026	58	33	m	m	PROPN
ejpam-2026	58	34	∈m	∈m	NOUN
ejpam-2026	58	35	.	.	PUNCT
ejpam-2026	59	1	(	(	PUNCT
ejpam-2026	59	2	2	2	X
ejpam-2026	59	3	)	)	PUNCT
ejpam-2026	59	4	it	it	PRON
ejpam-2026	59	5	is	be	AUX
ejpam-2026	59	6	followed	follow	VERB
ejpam-2026	59	7	by	by	ADP
ejpam-2026	59	8	(	(	PUNCT
ejpam-2026	59	9	1	1	X
ejpam-2026	59	10	)	)	PUNCT
ejpam-2026	59	11	that	that	SCONJ
ejpam-2026	59	12	either	either	ADV
ejpam-2026	59	13	(	(	PUNCT
ejpam-2026	59	14	r	r	X
ejpam-2026	59	15	/	/	SYM
ejpam-2026	59	16	s	s	PART
ejpam-2026	59	17	+	+	NOUN
ejpam-2026	59	18	a	a	X
ejpam-2026	59	19	/	/	SYM
ejpam-2026	59	20	b)m	b)m	NOUN
ejpam-2026	59	21	=	=	SYM
ejpam-2026	59	22	(	(	PUNCT
ejpam-2026	59	23	r	r	X
ejpam-2026	59	24	/	/	SYM
ejpam-2026	59	25	s	s	NOUN
ejpam-2026	59	26	m	m	VERB
ejpam-2026	60	1	+	+	NOUN
ejpam-2026	60	2	a	a	PRON
ejpam-2026	60	3	/	/	SYM
ejpam-2026	60	4	b	b	NOUN
ejpam-2026	60	5	m	m	NOUN
ejpam-2026	60	6	)	)	PUNCT
ejpam-2026	60	7	∈	∈	PROPN
ejpam-2026	60	8	m	m	VERB
ejpam-2026	60	9	or	or	CCONJ
ejpam-2026	60	10	(	(	PUNCT
ejpam-2026	60	11	r	r	NOUN
ejpam-2026	60	12	/	/	SYM
ejpam-2026	60	13	st	st	PROPN
ejpam-2026	61	1	+	+	CCONJ
ejpam-2026	61	2	a	a	DET
ejpam-2026	61	3	/	/	SYM
ejpam-2026	61	4	bc)m	bc)m	PROPN
ejpam-2026	61	5	=	=	SYM
ejpam-2026	61	6	(	(	PUNCT
ejpam-2026	61	7	r	r	X
ejpam-2026	61	8	/	/	SYM
ejpam-2026	61	9	st	st	PROPN
ejpam-2026	61	10	m+	m+	NUM
ejpam-2026	61	11	a	a	PRON
ejpam-2026	61	12	/	/	SYM
ejpam-2026	61	13	bc	bc	PROPN
ejpam-2026	61	14	m	m	PROPN
ejpam-2026	61	15	)	)	PUNCT
ejpam-2026	61	16	∈m	∈m	NOUN
ejpam-2026	61	17	.	.	PUNCT
ejpam-2026	62	1	(	(	PUNCT
ejpam-2026	62	2	3	3	X
ejpam-2026	62	3	)	)	PUNCT
ejpam-2026	62	4	either	either	CCONJ
ejpam-2026	62	5	r	r	NOUN
ejpam-2026	62	6	/	/	SYM
ejpam-2026	62	7	s(m+n	s(m+n	NUM
ejpam-2026	62	8	)	)	PUNCT
ejpam-2026	62	9	=	=	PRON
ejpam-2026	63	1	(	(	PUNCT
ejpam-2026	63	2	r	r	X
ejpam-2026	63	3	/	/	SYM
ejpam-2026	63	4	s	s	AUX
ejpam-2026	63	5	m+r	m+r	PROPN
ejpam-2026	63	6	/	/	SYM
ejpam-2026	63	7	s	s	PROPN
ejpam-2026	63	8	n	n	CCONJ
ejpam-2026	63	9	)	)	PUNCT
ejpam-2026	63	10	∈m	∈m	NOUN
ejpam-2026	63	11	or	or	CCONJ
ejpam-2026	63	12	r	r	NOUN
ejpam-2026	63	13	/	/	SYM
ejpam-2026	63	14	st	st	PROPN
ejpam-2026	63	15	(	(	PUNCT
ejpam-2026	63	16	m+n	m+n	PROPN
ejpam-2026	63	17	)	)	PUNCT
ejpam-2026	64	1	=	=	PRON
ejpam-2026	64	2	(	(	PUNCT
ejpam-2026	64	3	r	r	X
ejpam-2026	64	4	/	/	SYM
ejpam-2026	64	5	stm+r	stm+r	PROPN
ejpam-2026	64	6	/	/	SYM
ejpam-2026	64	7	stn	stn	PROPN
ejpam-2026	64	8	)	)	PUNCT
ejpam-2026	64	9	=	=	SYM
ejpam-2026	64	10	m	m	PROPN
ejpam-2026	64	11	,	,	PUNCT
ejpam-2026	64	12	which	which	PRON
ejpam-2026	64	13	is	be	AUX
ejpam-2026	64	14	clear	clear	ADJ
ejpam-2026	64	15	by	by	ADP
ejpam-2026	64	16	(	(	PUNCT
ejpam-2026	64	17	2	2	NUM
ejpam-2026	64	18	)	)	PUNCT
ejpam-2026	64	19	.	.	PUNCT
ejpam-2026	65	1	(	(	PUNCT
ejpam-2026	65	2	4	4	X
ejpam-2026	65	3	)	)	PUNCT
ejpam-2026	65	4	either	either	CCONJ
ejpam-2026	65	5	(	(	PUNCT
ejpam-2026	65	6	r	r	X
ejpam-2026	65	7	/	/	SYM
ejpam-2026	65	8	s	s	PART
ejpam-2026	65	9	·	·	PUNCT
ejpam-2026	65	10	a	a	DET
ejpam-2026	65	11	/	/	SYM
ejpam-2026	65	12	b)m	b)m	NOUN
ejpam-2026	65	13	∈m	∈m	NOUN
ejpam-2026	65	14	or	or	CCONJ
ejpam-2026	65	15	(	(	PUNCT
ejpam-2026	65	16	r	r	PROPN
ejpam-2026	65	17	/	/	SYM
ejpam-2026	65	18	st	st	NOUN
ejpam-2026	65	19	·	·	PUNCT
ejpam-2026	65	20	a	a	DET
ejpam-2026	65	21	/	/	SYM
ejpam-2026	65	22	bc)m	bc)m	PROPN
ejpam-2026	65	23	∈m	∈m	NOUN
ejpam-2026	65	24	by	by	ADP
ejpam-2026	65	25	definition	definition	NOUN
ejpam-2026	65	26	∀	∀	NOUN
ejpam-2026	65	27	r	r	NOUN
ejpam-2026	65	28	∈	∈	NOUN
ejpam-2026	65	29	r	r	NOUN
ejpam-2026	65	30	and	and	CCONJ
ejpam-2026	65	31	either	either	CCONJ
ejpam-2026	65	32	r	r	X
ejpam-2026	65	33	/	/	SYM
ejpam-2026	65	34	s	s	PART
ejpam-2026	65	35	m	m	NOUN
ejpam-2026	65	36	∈m	∈m	NOUN
ejpam-2026	65	37	or	or	CCONJ
ejpam-2026	65	38	s	s	NOUN
ejpam-2026	65	39	/	/	SYM
ejpam-2026	65	40	rt	rt	PROPN
ejpam-2026	65	41	m	m	NOUN
ejpam-2026	65	42	∈m	∈m	NOUN
ejpam-2026	65	43	.	.	PUNCT
ejpam-2026	66	1	the	the	DET
ejpam-2026	66	2	above	above	ADJ
ejpam-2026	66	3	theorem	theorem	NOUN
ejpam-2026	66	4	reflects	reflect	VERB
ejpam-2026	66	5	that	that	SCONJ
ejpam-2026	66	6	m	m	PROPN
ejpam-2026	66	7	is	be	AUX
ejpam-2026	66	8	a	a	DET
ejpam-2026	66	9	module	module	NOUN
ejpam-2026	66	10	.	.	PUNCT
ejpam-2026	67	1	thus	thus	ADV
ejpam-2026	67	2	we	we	PRON
ejpam-2026	67	3	can	can	AUX
ejpam-2026	67	4	conclude	conclude	VERB
ejpam-2026	67	5	that	that	SCONJ
ejpam-2026	67	6	the	the	DET
ejpam-2026	67	7	condition	condition	NOUN
ejpam-2026	67	8	we	we	PRON
ejpam-2026	67	9	put	put	VERB
ejpam-2026	67	10	in	in	ADP
ejpam-2026	67	11	definition1	definition1	NOUN
ejpam-2026	67	12	is	be	AUX
ejpam-2026	67	13	necessary	necessary	ADJ
ejpam-2026	67	14	and	and	CCONJ
ejpam-2026	67	15	sufficient	sufficient	ADJ
ejpam-2026	67	16	.	.	PUNCT
ejpam-2026	67	17	example	example	NOUN
ejpam-2026	68	1	1	1	NUM
ejpam-2026	68	2	.	.	PUNCT
ejpam-2026	69	1	let	let	VERB
ejpam-2026	69	2	r	r	NOUN
ejpam-2026	69	3	=	=	SYM
ejpam-2026	69	4	q+xq(y)[[x	q+xq(y)[[x	PROPN
ejpam-2026	69	5	]	]	X
ejpam-2026	69	6	]	]	X
ejpam-2026	69	7	,	,	PUNCT
ejpam-2026	69	8	r	r	NOUN
ejpam-2026	69	9	is	be	AUX
ejpam-2026	69	10	a	a	DET
ejpam-2026	69	11	pv	pv	PROPN
ejpam-2026	69	12	r.	r.	NOUN
ejpam-2026	69	13	consider	consider	VERB
ejpam-2026	69	14	r[x	r[x	PROPN
ejpam-2026	69	15	]	]	PUNCT
ejpam-2026	69	16	is	be	AUX
ejpam-2026	69	17	an	an	DET
ejpam-2026	69	18	abelian	abelian	ADJ
ejpam-2026	69	19	group	group	NOUN
ejpam-2026	69	20	and	and	CCONJ
ejpam-2026	69	21	define	define	VERB
ejpam-2026	69	22	mapping	mapping	NOUN
ejpam-2026	69	23	r×r[x]→	r×r[x]→	PROPN
ejpam-2026	69	24	r[x	r[x	NOUN
ejpam-2026	69	25	]	]	X
ejpam-2026	69	26	i.e.	i.e.	X
ejpam-2026	69	27	,	,	PUNCT
ejpam-2026	69	28	(	(	PUNCT
ejpam-2026	69	29	a+	a+	X
ejpam-2026	69	30	xb(y))r	xb(y))r	PUNCT
ejpam-2026	69	31	∈	∈	PROPN
ejpam-2026	69	32	r[x	r[x	NOUN
ejpam-2026	69	33	]	]	PUNCT
ejpam-2026	69	34	where	where	SCONJ
ejpam-2026	69	35	a	a	DET
ejpam-2026	69	36	∈	∈	PROPN
ejpam-2026	69	37	q	q	NOUN
ejpam-2026	69	38	,	,	PUNCT
ejpam-2026	69	39	b(y	b(y	PROPN
ejpam-2026	69	40	)	)	PUNCT
ejpam-2026	69	41	∈	∈	PROPN
ejpam-2026	70	1	q(y)[[x	q(y)[[x	NUM
ejpam-2026	70	2	]	]	X
ejpam-2026	70	3	]	]	PUNCT
ejpam-2026	70	4	and	and	CCONJ
ejpam-2026	70	5	r	r	NOUN
ejpam-2026	70	6	=	=	SYM
ejpam-2026	70	7	f(x	f(x	PROPN
ejpam-2026	70	8	)	)	PUNCT
ejpam-2026	70	9	∈	∈	PROPN
ejpam-2026	70	10	r[x	r[x	NOUN
ejpam-2026	70	11	]	]	X
ejpam-2026	70	12	.	.	PUNCT
ejpam-2026	71	1	for	for	ADP
ejpam-2026	71	2	all	all	PRON
ejpam-2026	71	3	a+	a+	PRON
ejpam-2026	71	4	xb(y	xb(y	NUM
ejpam-2026	71	5	)	)	PUNCT
ejpam-2026	71	6	,	,	PUNCT
ejpam-2026	71	7	c+	c+	X
ejpam-2026	71	8	xd(y	xd(y	PUNCT
ejpam-2026	71	9	)	)	PUNCT
ejpam-2026	71	10	in	in	ADP
ejpam-2026	71	11	r	r	NOUN
ejpam-2026	71	12	and	and	CCONJ
ejpam-2026	71	13	r	r	NOUN
ejpam-2026	71	14	,	,	PUNCT
ejpam-2026	71	15	s	s	PROPN
ejpam-2026	71	16	in	in	ADP
ejpam-2026	71	17	r[x	r[x	NOUN
ejpam-2026	71	18	]	]	PUNCT
ejpam-2026	71	19	we	we	PRON
ejpam-2026	71	20	have	have	VERB
ejpam-2026	71	21	;	;	PUNCT
ejpam-2026	71	22	(	(	PUNCT
ejpam-2026	71	23	1	1	X
ejpam-2026	71	24	)	)	PUNCT
ejpam-2026	72	1	[	[	X
ejpam-2026	72	2	(	(	PUNCT
ejpam-2026	72	3	a+	a+	PUNCT
ejpam-2026	72	4	xb(y))(c+	xb(y))(c+	NOUN
ejpam-2026	72	5	xd(y))]r	xd(y))]r	PROPN
ejpam-2026	72	6	=	=	SYM
ejpam-2026	72	7	a+	a+	PUNCT
ejpam-2026	72	8	xb(y)((c+	xb(y)((c+	PROPN
ejpam-2026	72	9	xd(y))r	xd(y))r	PROPN
ejpam-2026	72	10	)	)	PUNCT
ejpam-2026	72	11	.	.	PUNCT
ejpam-2026	73	1	(	(	PUNCT
ejpam-2026	73	2	2	2	X
ejpam-2026	73	3	)	)	PUNCT
ejpam-2026	74	1	[	[	X
ejpam-2026	74	2	(	(	PUNCT
ejpam-2026	74	3	a+	a+	X
ejpam-2026	74	4	xb(y	xb(y	NUM
ejpam-2026	74	5	)	)	PUNCT
ejpam-2026	75	1	+	+	CCONJ
ejpam-2026	75	2	c+	c+	VERB
ejpam-2026	75	3	xd(y)]r	xd(y)]r	PUNCT
ejpam-2026	75	4	=	=	PUNCT
ejpam-2026	76	1	[	[	X
ejpam-2026	76	2	a+	a+	X
ejpam-2026	76	3	xb(y)]r	xb(y)]r	PUNCT
ejpam-2026	76	4	+	+	CCONJ
ejpam-2026	77	1	[	[	X
ejpam-2026	77	2	c+	c+	NOUN
ejpam-2026	77	3	xd(y)]r	xd(y)]r	X
ejpam-2026	77	4	.	.	PUNCT
ejpam-2026	78	1	(	(	PUNCT
ejpam-2026	78	2	3	3	NUM
ejpam-2026	78	3	)	)	PUNCT
ejpam-2026	78	4	a+	a+	PUNCT
ejpam-2026	78	5	xb(y)(r	xb(y)(r	PUNCT
ejpam-2026	79	1	+	+	NUM
ejpam-2026	79	2	s	s	X
ejpam-2026	79	3	)	)	PUNCT
ejpam-2026	79	4	=	=	PUNCT
ejpam-2026	80	1	[	[	X
ejpam-2026	80	2	a+	a+	X
ejpam-2026	80	3	xb(y)]r	xb(y)]r	PUNCT
ejpam-2026	81	1	+	+	PUNCT
ejpam-2026	82	1	[	[	X
ejpam-2026	82	2	a+	a+	X
ejpam-2026	82	3	xb(y)]s	xb(y)]s	X
ejpam-2026	82	4	.	.	PUNCT
ejpam-2026	83	1	definition	definition	NOUN
ejpam-2026	83	2	2	2	NUM
ejpam-2026	83	3	.	.	PUNCT
ejpam-2026	84	1	let	let	VERB
ejpam-2026	84	2	r	r	PRON
ejpam-2026	84	3	be	be	AUX
ejpam-2026	84	4	a	a	DET
ejpam-2026	84	5	pseudo	pseudo	NOUN
ejpam-2026	84	6	-	-	NOUN
ejpam-2026	84	7	valuation	valuation	NOUN
ejpam-2026	84	8	ring	ring	NOUN
ejpam-2026	84	9	and	and	CCONJ
ejpam-2026	84	10	m	m	AUX
ejpam-2026	84	11	be	be	AUX
ejpam-2026	84	12	a	a	DET
ejpam-2026	84	13	module	module	NOUN
ejpam-2026	84	14	over	over	ADP
ejpam-2026	84	15	pv	pv	PROPN
ejpam-2026	84	16	r.	r.	PROPN
ejpam-2026	84	17	a	a	DET
ejpam-2026	84	18	subset	subset	NOUN
ejpam-2026	84	19	n	n	PROPN
ejpam-2026	84	20	of	of	ADP
ejpam-2026	84	21	m	m	PROPN
ejpam-2026	84	22	is	be	AUX
ejpam-2026	84	23	a	a	DET
ejpam-2026	84	24	submodule	submodule	NOUN
ejpam-2026	84	25	over	over	ADP
ejpam-2026	84	26	pseudo	pseudo	NOUN
ejpam-2026	84	27	-	-	NOUN
ejpam-2026	84	28	valuation	valuation	NOUN
ejpam-2026	84	29	ring	ring	NOUN
ejpam-2026	84	30	if	if	SCONJ
ejpam-2026	84	31	:	:	PUNCT
ejpam-2026	84	32	(	(	PUNCT
ejpam-2026	84	33	1	1	X
ejpam-2026	84	34	)	)	PUNCT
ejpam-2026	84	35	n−	n−	NOUN
ejpam-2026	84	36	ń	ń	NOUN
ejpam-2026	84	37	∈	∈	PROPN
ejpam-2026	84	38	n	n	PRON
ejpam-2026	84	39	∀	∀	X
ejpam-2026	84	40	n	n	CCONJ
ejpam-2026	84	41	,	,	PUNCT
ejpam-2026	84	42	ń	ń	PROPN
ejpam-2026	84	43	∈	∈	PROPN
ejpam-2026	84	44	n.	n.	NOUN
ejpam-2026	84	45	(	(	PUNCT
ejpam-2026	84	46	2	2	NUM
ejpam-2026	84	47	)	)	PUNCT
ejpam-2026	84	48	r	r	NOUN
ejpam-2026	84	49	∈	∈	NOUN
ejpam-2026	84	50	r	r	NOUN
ejpam-2026	84	51	,	,	PUNCT
ejpam-2026	84	52	n	n	PRON
ejpam-2026	84	53	∈	∈	NOUN
ejpam-2026	84	54	n	n	CCONJ
ejpam-2026	84	55	either	either	CCONJ
ejpam-2026	84	56	r	r	X
ejpam-2026	84	57	/	/	SYM
ejpam-2026	84	58	s	s	PART
ejpam-2026	84	59	n	n	PRON
ejpam-2026	84	60	∈	∈	PROPN
ejpam-2026	84	61	n	n	CCONJ
ejpam-2026	84	62	or	or	CCONJ
ejpam-2026	84	63	s	s	PROPN
ejpam-2026	84	64	/	/	SYM
ejpam-2026	84	65	rt	rt	PROPN
ejpam-2026	84	66	n	n	CCONJ
ejpam-2026	84	67	∈	∈	PROPN
ejpam-2026	84	68	n.	n.	NOUN
ejpam-2026	84	69	theorem	theorem	VERB
ejpam-2026	84	70	2	2	X
ejpam-2026	84	71	.	.	PUNCT
ejpam-2026	85	1	let	let	VERB
ejpam-2026	85	2	r	r	NOUN
ejpam-2026	85	3	is	be	AUX
ejpam-2026	85	4	a	a	DET
ejpam-2026	85	5	pv	pv	NOUN
ejpam-2026	85	6	r	r	NOUN
ejpam-2026	85	7	and	and	CCONJ
ejpam-2026	85	8	m	m	PROPN
ejpam-2026	85	9	is	be	AUX
ejpam-2026	85	10	a	a	DET
ejpam-2026	85	11	module	module	NOUN
ejpam-2026	85	12	.	.	PUNCT
ejpam-2026	86	1	a	a	DET
ejpam-2026	86	2	subset	subset	NOUN
ejpam-2026	86	3	n	n	NOUN
ejpam-2026	86	4	of	of	ADP
ejpam-2026	86	5	m	m	PROPN
ejpam-2026	86	6	is	be	AUX
ejpam-2026	86	7	a	a	DET
ejpam-2026	86	8	submodule	submodule	NOUN
ejpam-2026	86	9	over	over	ADP
ejpam-2026	86	10	pv	pv	PROPN
ejpam-2026	86	11	r⇔	r⇔	NOUN
ejpam-2026	86	12	∀	∀	X
ejpam-2026	86	13	r	r	NOUN
ejpam-2026	86	14	,	,	PUNCT
ejpam-2026	86	15	s	s	PROPN
ejpam-2026	86	16	,	,	PUNCT
ejpam-2026	86	17	t	t	PROPN
ejpam-2026	86	18	∈	∈	PROPN
ejpam-2026	86	19	r	r	X
ejpam-2026	86	20	,	,	PUNCT
ejpam-2026	86	21	n	n	PRON
ejpam-2026	86	22	∈	∈	NOUN
ejpam-2026	86	23	n	n	CCONJ
ejpam-2026	86	24	either	either	CCONJ
ejpam-2026	86	25	r	r	X
ejpam-2026	86	26	/	/	SYM
ejpam-2026	86	27	s	s	PART
ejpam-2026	86	28	n	n	PRON
ejpam-2026	86	29	∈	∈	PROPN
ejpam-2026	86	30	n	n	CCONJ
ejpam-2026	86	31	or	or	CCONJ
ejpam-2026	86	32	s	s	PROPN
ejpam-2026	86	33	/	/	SYM
ejpam-2026	86	34	rt	rt	PROPN
ejpam-2026	86	35	n	n	CCONJ
ejpam-2026	86	36	∈	∈	PROPN
ejpam-2026	86	37	n.	n.	NOUN
ejpam-2026	86	38	proof	proof	NOUN
ejpam-2026	86	39	.	.	PUNCT
ejpam-2026	87	1	if	if	SCONJ
ejpam-2026	87	2	r	r	NOUN
ejpam-2026	87	3	is	be	AUX
ejpam-2026	87	4	a	a	DET
ejpam-2026	87	5	pv	pv	NOUN
ejpam-2026	87	6	r	r	NOUN
ejpam-2026	87	7	and	and	CCONJ
ejpam-2026	87	8	n	n	NOUN
ejpam-2026	87	9	is	be	AUX
ejpam-2026	87	10	a	a	DET
ejpam-2026	87	11	subset	subset	NOUN
ejpam-2026	87	12	of	of	ADP
ejpam-2026	87	13	m	m	PROPN
ejpam-2026	87	14	,	,	PUNCT
ejpam-2026	87	15	where	where	SCONJ
ejpam-2026	87	16	m	m	NOUN
ejpam-2026	87	17	is	be	AUX
ejpam-2026	87	18	a	a	DET
ejpam-2026	87	19	module	module	NOUN
ejpam-2026	87	20	then	then	ADV
ejpam-2026	87	21	by	by	ADP
ejpam-2026	87	22	definition	definition	NOUN
ejpam-2026	87	23	,	,	PUNCT
ejpam-2026	87	24	we	we	PRON
ejpam-2026	87	25	can	can	AUX
ejpam-2026	87	26	write	write	VERB
ejpam-2026	87	27	that	that	PRON
ejpam-2026	87	28	∀	∀	PUNCT
ejpam-2026	88	1	r	r	NOUN
ejpam-2026	88	2	∈	∈	NOUN
ejpam-2026	88	3	r	r	NOUN
ejpam-2026	88	4	and	and	CCONJ
ejpam-2026	88	5	n	n	CCONJ
ejpam-2026	88	6	,	,	PUNCT
ejpam-2026	88	7	ń	ń	PROPN
ejpam-2026	88	8	∈	∈	PROPN
ejpam-2026	88	9	n	n	CCONJ
ejpam-2026	88	10	,	,	PUNCT
ejpam-2026	88	11	either	either	CCONJ
ejpam-2026	88	12	r	r	X
ejpam-2026	88	13	/	/	SYM
ejpam-2026	88	14	s	s	PART
ejpam-2026	88	15	n	n	PRON
ejpam-2026	88	16	∈	∈	PROPN
ejpam-2026	88	17	n	n	CCONJ
ejpam-2026	88	18	or	or	CCONJ
ejpam-2026	88	19	s	s	PROPN
ejpam-2026	88	20	/	/	SYM
ejpam-2026	88	21	rt	rt	PROPN
ejpam-2026	88	22	n	n	CCONJ
ejpam-2026	88	23	∈	∈	PROPN
ejpam-2026	88	24	n	n	X
ejpam-2026	88	25	.	.	PUNCT
ejpam-2026	89	1	conversely	conversely	ADV
ejpam-2026	89	2	,	,	PUNCT
ejpam-2026	89	3	suppose	suppose	VERB
ejpam-2026	89	4	that	that	SCONJ
ejpam-2026	89	5	n	n	PRON
ejpam-2026	89	6	is	be	AUX
ejpam-2026	89	7	a	a	DET
ejpam-2026	89	8	submodule	submodule	NOUN
ejpam-2026	89	9	over	over	ADP
ejpam-2026	89	10	pv	pv	NOUN
ejpam-2026	89	11	r	r	NOUN
ejpam-2026	89	12	,	,	PUNCT
ejpam-2026	89	13	since	since	SCONJ
ejpam-2026	89	14	we	we	PRON
ejpam-2026	89	15	have	have	AUX
ejpam-2026	89	16	already	already	ADV
ejpam-2026	89	17	define	define	VERB
ejpam-2026	89	18	that	that	DET
ejpam-2026	89	19	∀	∀	NOUN
ejpam-2026	89	20	r	r	NOUN
ejpam-2026	89	21	,	,	PUNCT
ejpam-2026	89	22	s	s	PROPN
ejpam-2026	89	23	,	,	PUNCT
ejpam-2026	89	24	t	t	PROPN
ejpam-2026	89	25	∈	∈	PROPN
ejpam-2026	89	26	r	r	NOUN
ejpam-2026	89	27	where	where	SCONJ
ejpam-2026	89	28	t	t	PROPN
ejpam-2026	89	29	is	be	AUX
ejpam-2026	89	30	a	a	DET
ejpam-2026	89	31	non	non	ADJ
ejpam-2026	89	32	-	-	NOUN
ejpam-2026	89	33	unit	unit	NOUN
ejpam-2026	89	34	,	,	PUNCT
ejpam-2026	89	35	n	n	CCONJ
ejpam-2026	89	36	,	,	PUNCT
ejpam-2026	89	37	ń	ń	PROPN
ejpam-2026	89	38	∈	∈	PROPN
ejpam-2026	89	39	n	n	CCONJ
ejpam-2026	89	40	either	either	CCONJ
ejpam-2026	89	41	r	r	X
ejpam-2026	89	42	/	/	SYM
ejpam-2026	89	43	s	s	PART
ejpam-2026	89	44	n	n	PRON
ejpam-2026	89	45	∈	∈	PROPN
ejpam-2026	89	46	n	n	CCONJ
ejpam-2026	89	47	or	or	CCONJ
ejpam-2026	89	48	s	s	PROPN
ejpam-2026	89	49	/	/	SYM
ejpam-2026	89	50	rt	rt	PROPN
ejpam-2026	89	51	n	n	CCONJ
ejpam-2026	89	52	∈	∈	PROPN
ejpam-2026	89	53	n.now	n.now	NOUN
ejpam-2026	89	54	we	we	PRON
ejpam-2026	89	55	can	can	AUX
ejpam-2026	89	56	easily	easily	ADV
ejpam-2026	89	57	conclude	conclude	VERB
ejpam-2026	89	58	the	the	DET
ejpam-2026	89	59	properties	property	NOUN
ejpam-2026	89	60	of	of	ADP
ejpam-2026	89	61	submodule	submodule	NOUN
ejpam-2026	89	62	as	as	ADP
ejpam-2026	89	63	;	;	PUNCT
ejpam-2026	89	64	(	(	PUNCT
ejpam-2026	89	65	1	1	X
ejpam-2026	89	66	)	)	PUNCT
ejpam-2026	89	67	n−	n−	NOUN
ejpam-2026	89	68	ń	ń	NOUN
ejpam-2026	89	69	∈	∈	PROPN
ejpam-2026	89	70	n	n	PRON
ejpam-2026	89	71	∀	∀	X
ejpam-2026	89	72	n	n	CCONJ
ejpam-2026	89	73	,	,	PUNCT
ejpam-2026	89	74	ń	ń	PROPN
ejpam-2026	89	75	∈	∈	PROPN
ejpam-2026	89	76	n	n	CCONJ
ejpam-2026	89	77	(	(	PUNCT
ejpam-2026	89	78	closure	closure	NOUN
ejpam-2026	89	79	property	property	NOUN
ejpam-2026	89	80	holds	hold	NOUN
ejpam-2026	89	81	)	)	PUNCT
ejpam-2026	89	82	.	.	PUNCT
ejpam-2026	90	1	(	(	PUNCT
ejpam-2026	90	2	2	2	X
ejpam-2026	90	3	)	)	PUNCT
ejpam-2026	90	4	r	r	NOUN
ejpam-2026	90	5	∈	∈	NOUN
ejpam-2026	90	6	r	r	NOUN
ejpam-2026	90	7	,	,	PUNCT
ejpam-2026	90	8	n	n	PRON
ejpam-2026	90	9	∈	∈	NOUN
ejpam-2026	90	10	n	n	CCONJ
ejpam-2026	90	11	either	either	CCONJ
ejpam-2026	90	12	r	r	X
ejpam-2026	90	13	/	/	SYM
ejpam-2026	90	14	s	s	PART
ejpam-2026	90	15	n	n	PRON
ejpam-2026	90	16	∈	∈	PROPN
ejpam-2026	90	17	n	n	CCONJ
ejpam-2026	90	18	or	or	CCONJ
ejpam-2026	90	19	s	s	PROPN
ejpam-2026	90	20	/	/	SYM
ejpam-2026	90	21	rt	rt	PROPN
ejpam-2026	90	22	n	n	CCONJ
ejpam-2026	90	23	∈	∈	PROPN
ejpam-2026	90	24	n	n	CCONJ
ejpam-2026	90	25	(	(	PUNCT
ejpam-2026	90	26	by	by	ADP
ejpam-2026	90	27	definition	definition	NOUN
ejpam-2026	90	28	)	)	PUNCT
ejpam-2026	90	29	.	.	PUNCT
ejpam-2026	91	1	which	which	PRON
ejpam-2026	91	2	proves	prove	VERB
ejpam-2026	91	3	that	that	SCONJ
ejpam-2026	91	4	n	n	X
ejpam-2026	91	5	is	be	AUX
ejpam-2026	91	6	a	a	DET
ejpam-2026	91	7	submodule	submodule	NOUN
ejpam-2026	91	8	.	.	PUNCT
ejpam-2026	91	9	example	example	NOUN
ejpam-2026	92	1	2	2	NUM
ejpam-2026	92	2	.	.	PUNCT
ejpam-2026	92	3	let	let	VERB
ejpam-2026	92	4	r	r	NOUN
ejpam-2026	92	5	=	=	SYM
ejpam-2026	92	6	q+xq(y)[[x	q+xq(y)[[x	PROPN
ejpam-2026	92	7	]	]	X
ejpam-2026	92	8	]	]	X
ejpam-2026	92	9	is	be	AUX
ejpam-2026	92	10	a	a	DET
ejpam-2026	92	11	pv	pv	PROPN
ejpam-2026	92	12	r.	r.	NOUN
ejpam-2026	92	13	let	let	VERB
ejpam-2026	92	14	r[x	r[x	PROPN
ejpam-2026	92	15	,	,	PUNCT
ejpam-2026	92	16	y	y	PROPN
ejpam-2026	92	17	]	]	PUNCT
ejpam-2026	92	18	is	be	AUX
ejpam-2026	92	19	a	a	DET
ejpam-2026	92	20	submodule	submodule	NOUN
ejpam-2026	92	21	over	over	ADP
ejpam-2026	92	22	pseudo	pseudo	NOUN
ejpam-2026	92	23	valuation	valuation	NOUN
ejpam-2026	92	24	ring	ring	NOUN
ejpam-2026	92	25	if	if	SCONJ
ejpam-2026	92	26	for	for	ADP
ejpam-2026	92	27	all	all	DET
ejpam-2026	92	28	a	a	DET
ejpam-2026	92	29	+	+	NOUN
ejpam-2026	92	30	xb(y	xb(y	NUM
ejpam-2026	92	31	)	)	PUNCT
ejpam-2026	92	32	,	,	PUNCT
ejpam-2026	92	33	c	c	PROPN
ejpam-2026	92	34	+	+	PUNCT
ejpam-2026	92	35	xd(y	xd(y	NOUN
ejpam-2026	92	36	)	)	PUNCT
ejpam-2026	92	37	in	in	ADP
ejpam-2026	92	38	r	r	NOUN
ejpam-2026	92	39	and	and	CCONJ
ejpam-2026	92	40	p	p	NOUN
ejpam-2026	92	41	,	,	PUNCT
ejpam-2026	92	42	q	q	NOUN
ejpam-2026	92	43	in	in	ADP
ejpam-2026	92	44	r[x	r[x	PROPN
ejpam-2026	92	45	,	,	PUNCT
ejpam-2026	92	46	y	y	PROPN
ejpam-2026	92	47	]	]	PUNCT
ejpam-2026	92	48	where	where	SCONJ
ejpam-2026	92	49	a	a	DET
ejpam-2026	92	50	∈	∈	PROPN
ejpam-2026	92	51	q	q	NOUN
ejpam-2026	92	52	,	,	PUNCT
ejpam-2026	92	53	b(y	b(y	PROPN
ejpam-2026	92	54	)	)	PUNCT
ejpam-2026	92	55	∈	∈	PROPN
ejpam-2026	93	1	q(y)[[x	q(y)[[x	NUM
ejpam-2026	93	2	]	]	X
ejpam-2026	93	3	]	]	PUNCT
ejpam-2026	93	4	and	and	CCONJ
ejpam-2026	93	5	p	p	X
ejpam-2026	93	6	=	=	PROPN
ejpam-2026	93	7	f(x	f(x	PROPN
ejpam-2026	93	8	,	,	PUNCT
ejpam-2026	93	9	y	y	PROPN
ejpam-2026	93	10	)	)	PUNCT
ejpam-2026	93	11	,	,	PUNCT
ejpam-2026	93	12	q	q	NOUN
ejpam-2026	93	13	=	=	SYM
ejpam-2026	93	14	g(x	g(x	PROPN
ejpam-2026	93	15	,	,	PUNCT
ejpam-2026	93	16	y	y	NOUN
ejpam-2026	93	17	)	)	PUNCT
ejpam-2026	93	18	∈	∈	PROPN
ejpam-2026	93	19	r[x	r[x	NOUN
ejpam-2026	93	20	,	,	PUNCT
ejpam-2026	93	21	y	y	PROPN
ejpam-2026	93	22	]	]	PUNCT
ejpam-2026	93	23	,	,	PUNCT
ejpam-2026	93	24	we	we	PRON
ejpam-2026	93	25	have	have	VERB
ejpam-2026	93	26	(	(	PUNCT
ejpam-2026	93	27	1	1	X
ejpam-2026	93	28	)	)	PUNCT
ejpam-2026	93	29	p−	p−	NOUN
ejpam-2026	93	30	q	q	PROPN
ejpam-2026	93	31	∈	∈	PROPN
ejpam-2026	93	32	r[x	r[x	NOUN
ejpam-2026	93	33	,	,	PUNCT
ejpam-2026	93	34	y	y	PROPN
ejpam-2026	93	35	]	]	PUNCT
ejpam-2026	93	36	∀	∀	X
ejpam-2026	94	1	p	p	NOUN
ejpam-2026	94	2	,	,	PUNCT
ejpam-2026	94	3	q	q	PROPN
ejpam-2026	94	4	∈	∈	PROPN
ejpam-2026	94	5	r[x	r[x	NOUN
ejpam-2026	94	6	,	,	PUNCT
ejpam-2026	94	7	y	y	PROPN
ejpam-2026	94	8	]	]	PUNCT
ejpam-2026	94	9	.	.	PUNCT
ejpam-2026	95	1	(	(	PUNCT
ejpam-2026	95	2	2	2	NUM
ejpam-2026	95	3	)	)	PUNCT
ejpam-2026	95	4	a+xb(y	a+xb(y	VERB
ejpam-2026	95	5	)	)	PUNCT
ejpam-2026	95	6	∈	∈	PROPN
ejpam-2026	95	7	r	r	NOUN
ejpam-2026	95	8	,	,	PUNCT
ejpam-2026	95	9	p	p	PROPN
ejpam-2026	95	10	∈	∈	PROPN
ejpam-2026	95	11	r[x	r[x	NOUN
ejpam-2026	95	12	,	,	PUNCT
ejpam-2026	95	13	y	y	PROPN
ejpam-2026	95	14	]	]	PUNCT
ejpam-2026	95	15	,	,	PUNCT
ejpam-2026	95	16	a+xb(y)p	a+xb(y)p	PROPN
ejpam-2026	95	17	∈	∈	PROPN
ejpam-2026	95	18	r[x	r[x	NOUN
ejpam-2026	95	19	,	,	PUNCT
ejpam-2026	95	20	y	y	PROPN
ejpam-2026	95	21	]	]	PUNCT
ejpam-2026	95	22	and	and	CCONJ
ejpam-2026	95	23	either	either	CCONJ
ejpam-2026	95	24	[	[	X
ejpam-2026	95	25	a+xb(y)/c+xd(y)]p	a+xb(y)/c+xd(y)]p	PROPN
ejpam-2026	95	26	∈	∈	PROPN
ejpam-2026	95	27	r[x	r[x	NOUN
ejpam-2026	95	28	,	,	PUNCT
ejpam-2026	95	29	y	y	PROPN
ejpam-2026	95	30	]	]	PUNCT
ejpam-2026	95	31	or	or	CCONJ
ejpam-2026	95	32	[	[	X
ejpam-2026	95	33	a+	a+	X
ejpam-2026	95	34	xb(y)/(c+	xb(y)/(c+	PROPN
ejpam-2026	95	35	xd(y).m+	xd(y).m+	PROPN
ejpam-2026	95	36	n(y))]p	n(y))]p	PROPN
ejpam-2026	95	37	∈	∈	PROPN
ejpam-2026	95	38	r[x	r[x	PROPN
ejpam-2026	95	39	,	,	PUNCT
ejpam-2026	95	40	y	y	PROPN
ejpam-2026	95	41	]	]	PUNCT
ejpam-2026	95	42	(	(	PUNCT
ejpam-2026	95	43	by	by	ADP
ejpam-2026	95	44	definition	definition	NOUN
ejpam-2026	95	45	)	)	PUNCT
ejpam-2026	95	46	.	.	PUNCT
ejpam-2026	96	1	w.	w.	PROPN
ejpam-2026	96	2	a.	a.	PROPN
ejpam-2026	96	3	khan	khan	PROPN
ejpam-2026	96	4	/	/	SYM
ejpam-2026	96	5	eur	eur	PROPN
ejpam-2026	96	6	.	.	PUNCT
ejpam-2026	97	1	j.	j.	PROPN
ejpam-2026	97	2	pure	pure	PROPN
ejpam-2026	97	3	appl	appl	PROPN
ejpam-2026	97	4	.	.	PROPN
ejpam-2026	97	5	math	math	PROPN
ejpam-2026	97	6	,	,	PUNCT
ejpam-2026	97	7	10	10	NUM
ejpam-2026	97	8	(	(	PUNCT
ejpam-2026	97	9	3	3	NUM
ejpam-2026	97	10	)	)	PUNCT
ejpam-2026	97	11	(	(	PUNCT
ejpam-2026	97	12	2017	2017	NUM
ejpam-2026	97	13	)	)	PUNCT
ejpam-2026	97	14	,	,	PUNCT
ejpam-2026	97	15	516	516	NUM
ejpam-2026	97	16	-	-	SYM
ejpam-2026	97	17	520	520	NUM
ejpam-2026	97	18	519	519	NUM
ejpam-2026	97	19	3	3	NUM
ejpam-2026	97	20	.	.	PUNCT
ejpam-2026	97	21	module	module	NOUN
ejpam-2026	97	22	over	over	ADP
ejpam-2026	97	23	pseudo	pseudo	NOUN
ejpam-2026	97	24	-	-	NOUN
ejpam-2026	97	25	valuation	valuation	NOUN
ejpam-2026	97	26	domain	domain	NOUN
ejpam-2026	97	27	in	in	ADP
ejpam-2026	97	28	this	this	DET
ejpam-2026	97	29	section	section	NOUN
ejpam-2026	97	30	we	we	PRON
ejpam-2026	97	31	defined	define	VERB
ejpam-2026	97	32	module	module	NOUN
ejpam-2026	97	33	over	over	ADP
ejpam-2026	97	34	pseudo	pseudo	NOUN
ejpam-2026	97	35	-	-	NOUN
ejpam-2026	97	36	valuation	valuation	NOUN
ejpam-2026	97	37	domain	domain	NOUN
ejpam-2026	97	38	and	and	CCONJ
ejpam-2026	97	39	its	its	PRON
ejpam-2026	97	40	submodule	submodule	NOUN
ejpam-2026	97	41	.	.	PUNCT
ejpam-2026	98	1	by	by	ADP
ejpam-2026	98	2	[	[	X
ejpam-2026	98	3	5	5	NUM
ejpam-2026	98	4	]	]	PUNCT
ejpam-2026	98	5	let	let	VERB
ejpam-2026	98	6	r	r	PRON
ejpam-2026	98	7	be	be	AUX
ejpam-2026	98	8	a	a	DET
ejpam-2026	98	9	domain	domain	NOUN
ejpam-2026	98	10	with	with	ADP
ejpam-2026	98	11	quotient	quotient	NOUN
ejpam-2026	98	12	field	field	NOUN
ejpam-2026	98	13	k.	k.	PROPN
ejpam-2026	99	1	the	the	DET
ejpam-2026	99	2	following	follow	VERB
ejpam-2026	99	3	statements	statement	NOUN
ejpam-2026	99	4	are	be	AUX
ejpam-2026	99	5	equivalent	equivalent	ADJ
ejpam-2026	99	6	.	.	PUNCT
ejpam-2026	100	1	(	(	PUNCT
ejpam-2026	100	2	1	1	X
ejpam-2026	100	3	)	)	PUNCT
ejpam-2026	100	4	r	r	NOUN
ejpam-2026	100	5	is	be	AUX
ejpam-2026	100	6	a	a	DET
ejpam-2026	100	7	pseudo	pseudo	NOUN
ejpam-2026	100	8	-	-	ADJ
ejpam-2026	100	9	valuation	valuation	NOUN
ejpam-2026	100	10	domain	domain	NOUN
ejpam-2026	100	11	.	.	PUNCT
ejpam-2026	101	1	(	(	PUNCT
ejpam-2026	101	2	2	2	X
ejpam-2026	101	3	)	)	PUNCT
ejpam-2026	101	4	for	for	ADP
ejpam-2026	101	5	each	each	DET
ejpam-2026	101	6	x	x	SYM
ejpam-2026	101	7	∈	∈	PROPN
ejpam-2026	101	8	k	k	PROPN
ejpam-2026	101	9	−r	−r	PROPN
ejpam-2026	101	10	and	and	CCONJ
ejpam-2026	101	11	for	for	ADP
ejpam-2026	101	12	each	each	DET
ejpam-2026	101	13	nonunit	nonunit	NOUN
ejpam-2026	101	14	a	a	PRON
ejpam-2026	101	15	of	of	ADP
ejpam-2026	101	16	r	r	NOUN
ejpam-2026	101	17	,	,	PUNCT
ejpam-2026	101	18	we	we	PRON
ejpam-2026	101	19	have	have	VERB
ejpam-2026	101	20	(	(	PUNCT
ejpam-2026	101	21	x+	x+	X
ejpam-2026	101	22	a)r	a)r	NOUN
ejpam-2026	101	23	=	=	SYM
ejpam-2026	101	24	xr	xr	PROPN
ejpam-2026	101	25	.	.	PUNCT
ejpam-2026	102	1	(	(	PUNCT
ejpam-2026	102	2	3	3	X
ejpam-2026	102	3	)	)	PUNCT
ejpam-2026	102	4	if	if	SCONJ
ejpam-2026	102	5	module	module	NOUN
ejpam-2026	102	6	over	over	ADP
ejpam-2026	102	7	pseudo	pseudo	NOUN
ejpam-2026	102	8	-	-	NOUN
ejpam-2026	102	9	valuation	valuation	NOUN
ejpam-2026	102	10	domain	domain	NOUN
ejpam-2026	102	11	r	r	NOUN
ejpam-2026	102	12	consists	consist	VERB
ejpam-2026	102	13	of	of	ADP
ejpam-2026	102	14	an	an	DET
ejpam-2026	102	15	abelian	abelian	ADJ
ejpam-2026	102	16	group	group	NOUN
ejpam-2026	102	17	(	(	PUNCT
ejpam-2026	102	18	m,+	m,+	PROPN
ejpam-2026	102	19	)	)	PUNCT
ejpam-2026	102	20	and	and	CCONJ
ejpam-2026	102	21	an	an	DET
ejpam-2026	102	22	operation	operation	NOUN
ejpam-2026	102	23	r×m	r×m	NOUN
ejpam-2026	102	24	→m	→m	VERB
ejpam-2026	102	25	scalar	scalar	ADJ
ejpam-2026	102	26	multiplication	multiplication	NOUN
ejpam-2026	102	27	,	,	PUNCT
ejpam-2026	102	28	usually	usually	ADV
ejpam-2026	102	29	just	just	ADV
ejpam-2026	102	30	written	write	VERB
ejpam-2026	102	31	by	by	ADP
ejpam-2026	102	32	juxtaposition	juxtaposition	NOUN
ejpam-2026	102	33	.	.	PUNCT
ejpam-2026	103	1	definition	definition	NOUN
ejpam-2026	103	2	3	3	X
ejpam-2026	103	3	.	.	PUNCT
ejpam-2026	104	1	let	let	VERB
ejpam-2026	104	2	k	k	PRON
ejpam-2026	104	3	be	be	AUX
ejpam-2026	104	4	the	the	DET
ejpam-2026	104	5	quotient	quotient	NOUN
ejpam-2026	104	6	field	field	NOUN
ejpam-2026	104	7	of	of	ADP
ejpam-2026	104	8	r	r	NOUN
ejpam-2026	104	9	,	,	PUNCT
ejpam-2026	104	10	x	x	SYM
ejpam-2026	104	11	∈	∈	PROPN
ejpam-2026	104	12	k	k	PROPN
ejpam-2026	104	13	−r	−r	PROPN
ejpam-2026	104	14	,	,	PUNCT
ejpam-2026	104	15	a	a	DET
ejpam-2026	104	16	,	,	PUNCT
ejpam-2026	104	17	b	b	NOUN
ejpam-2026	104	18	,	,	PUNCT
ejpam-2026	104	19	r	r	NOUN
ejpam-2026	104	20	,	,	PUNCT
ejpam-2026	104	21	s	s	NOUN
ejpam-2026	104	22	∈	∈	PROPN
ejpam-2026	104	23	r	r	NOUN
ejpam-2026	104	24	and	and	CCONJ
ejpam-2026	104	25	m	m	PROPN
ejpam-2026	104	26	∈	∈	PROPN
ejpam-2026	104	27	m(an	m(an	ADJ
ejpam-2026	104	28	additive	additive	ADJ
ejpam-2026	104	29	abelian	abelian	ADJ
ejpam-2026	104	30	group	group	NOUN
ejpam-2026	104	31	)	)	PUNCT
ejpam-2026	104	32	be	be	AUX
ejpam-2026	104	33	a	a	DET
ejpam-2026	104	34	module	module	NOUN
ejpam-2026	104	35	over	over	ADP
ejpam-2026	104	36	r	r	NOUN
ejpam-2026	104	37	if	if	SCONJ
ejpam-2026	104	38	;	;	PUNCT
ejpam-2026	104	39	(	(	PUNCT
ejpam-2026	104	40	1	1	X
ejpam-2026	104	41	)	)	PUNCT
ejpam-2026	104	42	for	for	ADP
ejpam-2026	104	43	r	r	NOUN
ejpam-2026	104	44	∈	∈	PROPN
ejpam-2026	104	45	r	r	NOUN
ejpam-2026	104	46	,	,	PUNCT
ejpam-2026	104	47	m	m	VERB
ejpam-2026	104	48	∈	∈	NOUN
ejpam-2026	104	49	m	m	NOUN
ejpam-2026	104	50	and	and	CCONJ
ejpam-2026	104	51	for	for	ADP
ejpam-2026	104	52	each	each	DET
ejpam-2026	104	53	nonunit	nonunit	NOUN
ejpam-2026	104	54	a	a	PRON
ejpam-2026	104	55	of	of	ADP
ejpam-2026	104	56	r	r	NOUN
ejpam-2026	104	57	,	,	PUNCT
ejpam-2026	104	58	x−1am	x−1am	PROPN
ejpam-2026	104	59	∈m	∈m	NOUN
ejpam-2026	104	60	.	.	PUNCT
ejpam-2026	105	1	(	(	PUNCT
ejpam-2026	105	2	2	2	NUM
ejpam-2026	105	3	)	)	PUNCT
ejpam-2026	105	4	(	(	PUNCT
ejpam-2026	105	5	rs)m	rs)m	PROPN
ejpam-2026	105	6	=	=	SYM
ejpam-2026	105	7	r(sm	r(sm	PROPN
ejpam-2026	105	8	)	)	PUNCT
ejpam-2026	105	9	=	=	PUNCT
ejpam-2026	105	10	(	(	PUNCT
ejpam-2026	105	11	x−1ax−1b)m	x−1ax−1b)m	X
ejpam-2026	105	12	=	=	SYM
ejpam-2026	105	13	x−1a(x−1bm	x−1a(x−1bm	PROPN
ejpam-2026	105	14	)	)	PUNCT
ejpam-2026	105	15	(	(	PUNCT
ejpam-2026	106	1	3	3	X
ejpam-2026	106	2	)	)	PUNCT
ejpam-2026	106	3	(	(	PUNCT
ejpam-2026	106	4	r	r	NOUN
ejpam-2026	106	5	+	+	NOUN
ejpam-2026	106	6	s)m	s)m	ADJ
ejpam-2026	106	7	=	=	PUNCT
ejpam-2026	107	1	rm+	rm+	NOUN
ejpam-2026	107	2	sm	sm	NOUN
ejpam-2026	107	3	(	(	PUNCT
ejpam-2026	107	4	4	4	NUM
ejpam-2026	107	5	)	)	PUNCT
ejpam-2026	107	6	r(m+	r(m+	NOUN
ejpam-2026	107	7	n	n	CCONJ
ejpam-2026	107	8	)	)	PUNCT
ejpam-2026	107	9	=	=	SYM
ejpam-2026	108	1	rm+	rm+	PROPN
ejpam-2026	108	2	rn	rn	PROPN
ejpam-2026	108	3	example	example	NOUN
ejpam-2026	109	1	3	3	X
ejpam-2026	109	2	.	.	PUNCT
ejpam-2026	110	1	let	let	VERB
ejpam-2026	110	2	r	r	NOUN
ejpam-2026	110	3	=	=	PUNCT
ejpam-2026	110	4	z+xq[[x	z+xq[[x	NOUN
ejpam-2026	110	5	]	]	X
ejpam-2026	110	6	]	]	PUNCT
ejpam-2026	110	7	be	be	AUX
ejpam-2026	110	8	a	a	DET
ejpam-2026	110	9	pseudo	pseudo	NOUN
ejpam-2026	110	10	-	-	ADJ
ejpam-2026	110	11	valuation	valuation	NOUN
ejpam-2026	110	12	domain	domain	NOUN
ejpam-2026	111	1	where	where	SCONJ
ejpam-2026	111	2	q[[x	q[[x	NOUN
ejpam-2026	111	3	]	]	X
ejpam-2026	111	4	]	]	X
ejpam-2026	111	5	is	be	AUX
ejpam-2026	111	6	a	a	DET
ejpam-2026	111	7	maximal	maximal	ADJ
ejpam-2026	111	8	ideal	ideal	NOUN
ejpam-2026	111	9	and	and	CCONJ
ejpam-2026	111	10	take	take	VERB
ejpam-2026	111	11	q[x	q[x	PROPN
ejpam-2026	111	12	]	]	PUNCT
ejpam-2026	111	13	be	be	VERB
ejpam-2026	111	14	an	an	DET
ejpam-2026	111	15	additive	additive	ADJ
ejpam-2026	111	16	abelian	abelian	ADJ
ejpam-2026	111	17	group	group	NOUN
ejpam-2026	111	18	.	.	PUNCT
ejpam-2026	112	1	q[x	q[x	CCONJ
ejpam-2026	112	2	]	]	PUNCT
ejpam-2026	112	3	is	be	AUX
ejpam-2026	112	4	a	a	DET
ejpam-2026	112	5	module	module	NOUN
ejpam-2026	112	6	over	over	ADP
ejpam-2026	112	7	pseudo	pseudo	NOUN
ejpam-2026	112	8	-	-	NOUN
ejpam-2026	112	9	valuation	valuation	NOUN
ejpam-2026	112	10	domain	domain	NOUN
ejpam-2026	112	11	,	,	PUNCT
ejpam-2026	112	12	clearly	clearly	ADV
ejpam-2026	112	13	(	(	PUNCT
ejpam-2026	112	14	a+	a+	PUNCT
ejpam-2026	112	15	bx)r	bx)r	PROPN
ejpam-2026	112	16	∈	∈	PROPN
ejpam-2026	112	17	q[x	q[x	PROPN
ejpam-2026	112	18	]	]	PUNCT
ejpam-2026	112	19	where	where	SCONJ
ejpam-2026	112	20	a	a	DET
ejpam-2026	112	21	∈	∈	PROPN
ejpam-2026	112	22	z	z	PROPN
ejpam-2026	112	23	,	,	PUNCT
ejpam-2026	112	24	b	b	PROPN
ejpam-2026	112	25	∈	∈	PROPN
ejpam-2026	112	26	q[[x	q[[x	PROPN
ejpam-2026	112	27	]	]	X
ejpam-2026	112	28	]	]	PUNCT
ejpam-2026	112	29	and	and	CCONJ
ejpam-2026	112	30	r	r	NOUN
ejpam-2026	112	31	=	=	SYM
ejpam-2026	112	32	f(x	f(x	PROPN
ejpam-2026	112	33	)	)	PUNCT
ejpam-2026	112	34	∈	∈	PROPN
ejpam-2026	112	35	q[x	q[x	PROPN
ejpam-2026	112	36	]	]	PUNCT
ejpam-2026	112	37	,	,	PUNCT
ejpam-2026	112	38	∀	∀	X
ejpam-2026	112	39	a+	a+	X
ejpam-2026	112	40	xb	xb	PROPN
ejpam-2026	112	41	,	,	PUNCT
ejpam-2026	112	42	c+	c+	VERB
ejpam-2026	112	43	xd	xd	INTJ
ejpam-2026	112	44	is	be	AUX
ejpam-2026	112	45	in	in	ADP
ejpam-2026	112	46	r	r	NOUN
ejpam-2026	112	47	and	and	CCONJ
ejpam-2026	112	48	r	r	NOUN
ejpam-2026	112	49	,	,	PUNCT
ejpam-2026	112	50	s	s	PROPN
ejpam-2026	112	51	in	in	ADP
ejpam-2026	112	52	q[x	q[x	PROPN
ejpam-2026	112	53	]	]	PUNCT
ejpam-2026	112	54	we	we	PRON
ejpam-2026	112	55	have	have	VERB
ejpam-2026	112	56	;	;	PUNCT
ejpam-2026	112	57	(	(	PUNCT
ejpam-2026	112	58	1	1	X
ejpam-2026	112	59	)	)	PUNCT
ejpam-2026	113	1	[	[	X
ejpam-2026	113	2	(	(	PUNCT
ejpam-2026	113	3	a+	a+	PUNCT
ejpam-2026	113	4	xb)(c+	xb)(c+	PROPN
ejpam-2026	113	5	xd)]r	xd)]r	PROPN
ejpam-2026	113	6	=	=	PRON
ejpam-2026	113	7	a+	a+	PUNCT
ejpam-2026	113	8	xb((c+	xb((c+	PROPN
ejpam-2026	113	9	xd)r	xd)r	PROPN
ejpam-2026	113	10	)	)	PUNCT
ejpam-2026	113	11	(	(	PUNCT
ejpam-2026	113	12	2	2	X
ejpam-2026	113	13	)	)	PUNCT
ejpam-2026	113	14	(	(	PUNCT
ejpam-2026	113	15	a+	a+	X
ejpam-2026	113	16	xb+	xb+	NOUN
ejpam-2026	113	17	c+	c+	VERB
ejpam-2026	113	18	xd)r	xd)r	NOUN
ejpam-2026	113	19	=	=	SYM
ejpam-2026	113	20	a+	a+	PUNCT
ejpam-2026	113	21	xbr	xbr	PROPN
ejpam-2026	113	22	+	+	CCONJ
ejpam-2026	113	23	c+	c+	VERB
ejpam-2026	113	24	xdr	xdr	PROPN
ejpam-2026	113	25	(	(	PUNCT
ejpam-2026	113	26	3	3	NUM
ejpam-2026	113	27	)	)	PUNCT
ejpam-2026	113	28	a+	a+	PUNCT
ejpam-2026	113	29	xb(r	xb(r	NUM
ejpam-2026	113	30	+	+	SYM
ejpam-2026	113	31	s	s	X
ejpam-2026	113	32	)	)	PUNCT
ejpam-2026	113	33	=	=	SYM
ejpam-2026	113	34	a+	a+	PUNCT
ejpam-2026	113	35	xbr	xbr	PROPN
ejpam-2026	113	36	+	+	CCONJ
ejpam-2026	113	37	a+	a+	PUNCT
ejpam-2026	113	38	xbs	xbs	PROPN
ejpam-2026	113	39	theorem	theorem	VERB
ejpam-2026	113	40	3	3	X
ejpam-2026	113	41	.	.	PUNCT
ejpam-2026	114	1	if	if	SCONJ
ejpam-2026	114	2	r	r	NOUN
ejpam-2026	114	3	is	be	AUX
ejpam-2026	114	4	pseudo	pseudo	NOUN
ejpam-2026	114	5	-	-	NOUN
ejpam-2026	114	6	valuation	valuation	NOUN
ejpam-2026	114	7	domain	domain	NOUN
ejpam-2026	114	8	and	and	CCONJ
ejpam-2026	114	9	m	m	NOUN
ejpam-2026	114	10	is	be	AUX
ejpam-2026	114	11	an	an	DET
ejpam-2026	114	12	abelian	abelian	ADJ
ejpam-2026	114	13	group	group	NOUN
ejpam-2026	114	14	then	then	ADV
ejpam-2026	114	15	m	m	VERB
ejpam-2026	114	16	is	be	AUX
ejpam-2026	114	17	module	module	NOUN
ejpam-2026	114	18	over	over	ADP
ejpam-2026	114	19	r⇔	r⇔	NOUN
ejpam-2026	114	20	(	(	PUNCT
ejpam-2026	114	21	a+	a+	PUNCT
ejpam-2026	114	22	xb)m	xb)m	PROPN
ejpam-2026	114	23	∈m	∈m	NOUN
ejpam-2026	114	24	.	.	PUNCT
ejpam-2026	115	1	proof	proof	NOUN
ejpam-2026	115	2	.	.	PUNCT
ejpam-2026	116	1	assume	assume	VERB
ejpam-2026	116	2	if	if	SCONJ
ejpam-2026	116	3	m	m	PROPN
ejpam-2026	116	4	is	be	AUX
ejpam-2026	116	5	a	a	DET
ejpam-2026	116	6	module	module	NOUN
ejpam-2026	116	7	over	over	ADP
ejpam-2026	116	8	pseudo	pseudo	NOUN
ejpam-2026	116	9	-	-	NOUN
ejpam-2026	116	10	valuation	valuation	NOUN
ejpam-2026	116	11	domain	domain	NOUN
ejpam-2026	116	12	r	r	NOUN
ejpam-2026	116	13	then	then	ADV
ejpam-2026	116	14	for	for	ADP
ejpam-2026	116	15	each	each	DET
ejpam-2026	116	16	non	non	ADJ
ejpam-2026	116	17	-	-	ADJ
ejpam-2026	116	18	unit	unit	NOUN
ejpam-2026	116	19	element	element	NOUN
ejpam-2026	116	20	a	a	PRON
ejpam-2026	116	21	in	in	ADP
ejpam-2026	116	22	r	r	NOUN
ejpam-2026	116	23	,	,	PUNCT
ejpam-2026	116	24	x−1am	x−1am	PROPN
ejpam-2026	116	25	∈m	∈m	NOUN
ejpam-2026	116	26	where	where	SCONJ
ejpam-2026	116	27	x	x	SYM
ejpam-2026	116	28	∈	∈	PROPN
ejpam-2026	116	29	k	k	PROPN
ejpam-2026	116	30	−r	−r	PROPN
ejpam-2026	116	31	.	.	PUNCT
ejpam-2026	117	1	conversely	conversely	ADV
ejpam-2026	117	2	,	,	PUNCT
ejpam-2026	117	3	let	let	VERB
ejpam-2026	117	4	x−1am	x−1am	PROPN
ejpam-2026	117	5	∈	∈	PROPN
ejpam-2026	117	6	m	m	PROPN
ejpam-2026	117	7	,	,	PUNCT
ejpam-2026	117	8	where	where	SCONJ
ejpam-2026	117	9	x	x	PUNCT
ejpam-2026	117	10	∈	∈	PROPN
ejpam-2026	117	11	k	k	NOUN
ejpam-2026	117	12	−	−	NOUN
ejpam-2026	117	13	r	r	NOUN
ejpam-2026	117	14	,	,	PUNCT
ejpam-2026	117	15	we	we	PRON
ejpam-2026	117	16	have	have	VERB
ejpam-2026	117	17	to	to	PART
ejpam-2026	117	18	prove	prove	VERB
ejpam-2026	117	19	that	that	SCONJ
ejpam-2026	117	20	m	m	PROPN
ejpam-2026	117	21	is	be	AUX
ejpam-2026	117	22	a	a	DET
ejpam-2026	117	23	module	module	NOUN
ejpam-2026	117	24	over	over	ADP
ejpam-2026	117	25	pseudo	pseudo	NOUN
ejpam-2026	117	26	-	-	NOUN
ejpam-2026	117	27	valuation	valuation	NOUN
ejpam-2026	117	28	domain	domain	NOUN
ejpam-2026	117	29	r.	r.	PROPN
ejpam-2026	117	30	as	as	ADP
ejpam-2026	117	31	x−1am	x−1am	PROPN
ejpam-2026	117	32	∈	∈	PROPN
ejpam-2026	117	33	m	m	NOUN
ejpam-2026	117	34	=	=	NOUN
ejpam-2026	117	35	⇒	⇒	NOUN
ejpam-2026	117	36	(	(	PUNCT
ejpam-2026	117	37	x−1ax−1b)m	x−1ax−1b)m	PROPN
ejpam-2026	117	38	∈	∈	PROPN
ejpam-2026	117	39	m	m	PRON
ejpam-2026	117	40	,	,	PUNCT
ejpam-2026	117	41	so	so	ADV
ejpam-2026	117	42	we	we	PRON
ejpam-2026	117	43	may	may	AUX
ejpam-2026	117	44	write	write	VERB
ejpam-2026	117	45	(	(	PUNCT
ejpam-2026	117	46	x−1ax−1b)m	x−1ax−1b)m	X
ejpam-2026	117	47	=	=	SYM
ejpam-2026	117	48	x−1a(x−1bm).also	x−1a(x−1bm).also	PROPN
ejpam-2026	117	49	,	,	PUNCT
ejpam-2026	118	1	if	if	SCONJ
ejpam-2026	118	2	x−1am	x−1am	PROPN
ejpam-2026	118	3	∈	∈	PROPN
ejpam-2026	118	4	m	m	VERB
ejpam-2026	118	5	then	then	ADV
ejpam-2026	118	6	(	(	PUNCT
ejpam-2026	118	7	x−1a	x−1a	X
ejpam-2026	118	8	+	+	NUM
ejpam-2026	118	9	x−1b)m	x−1b)m	NOUN
ejpam-2026	118	10	∈	∈	PROPN
ejpam-2026	118	11	m	m	NOUN
ejpam-2026	118	12	=	=	NOUN
ejpam-2026	118	13	⇒	⇒	NOUN
ejpam-2026	118	14	(	(	PUNCT
ejpam-2026	118	15	x−1a+	x−1a+	PROPN
ejpam-2026	118	16	x−1b)m	x−1b)m	PROPN
ejpam-2026	118	17	=	=	SYM
ejpam-2026	118	18	x−1am+	x−1am+	X
ejpam-2026	119	1	x−1bm.finally	x−1bm.finally	PROPN
ejpam-2026	119	2	x−1a	x−1a	PUNCT
ejpam-2026	119	3	(	(	PUNCT
ejpam-2026	119	4	m+	m+	NOUN
ejpam-2026	119	5	n	n	CCONJ
ejpam-2026	119	6	)	)	PUNCT
ejpam-2026	119	7	=	=	PUNCT
ejpam-2026	119	8	x−1am+	x−1am+	PROPN
ejpam-2026	119	9	x−1an	x−1an	PROPN
ejpam-2026	119	10	for	for	ADP
ejpam-2026	119	11	all	all	DET
ejpam-2026	119	12	x−1a	x−1a	PROPN
ejpam-2026	119	13	,	,	PUNCT
ejpam-2026	119	14	x−1b	x−1b	PROPN
ejpam-2026	119	15	∈	∈	PROPN
ejpam-2026	119	16	r	r	NOUN
ejpam-2026	119	17	and	and	CCONJ
ejpam-2026	119	18	m	m	PROPN
ejpam-2026	119	19	,	,	PUNCT
ejpam-2026	119	20	n	n	PROPN
ejpam-2026	119	21	∈	∈	PROPN
ejpam-2026	119	22	m	m	VERB
ejpam-2026	119	23	where	where	SCONJ
ejpam-2026	119	24	a	a	PRON
ejpam-2026	119	25	and	and	CCONJ
ejpam-2026	119	26	b	b	NOUN
ejpam-2026	119	27	are	be	AUX
ejpam-2026	119	28	non	non	ADJ
ejpam-2026	119	29	-	-	ADJ
ejpam-2026	119	30	unit	unit	ADJ
ejpam-2026	119	31	elements	element	NOUN
ejpam-2026	119	32	of	of	ADP
ejpam-2026	119	33	r.	r.	PROPN
ejpam-2026	119	34	as	as	SCONJ
ejpam-2026	119	35	the	the	DET
ejpam-2026	119	36	scaler	scaler	ADJ
ejpam-2026	119	37	multiplication	multiplication	NOUN
ejpam-2026	119	38	of	of	ADP
ejpam-2026	119	39	module	module	NOUN
ejpam-2026	119	40	is	be	AUX
ejpam-2026	119	41	given	give	VERB
ejpam-2026	119	42	so	so	SCONJ
ejpam-2026	119	43	we	we	PRON
ejpam-2026	119	44	can	can	AUX
ejpam-2026	119	45	write	write	VERB
ejpam-2026	119	46	the	the	DET
ejpam-2026	119	47	other	other	ADJ
ejpam-2026	119	48	properties	property	NOUN
ejpam-2026	119	49	of	of	ADP
ejpam-2026	119	50	a	a	DET
ejpam-2026	119	51	module	module	NOUN
ejpam-2026	119	52	.	.	PUNCT
ejpam-2026	120	1	suppose	suppose	VERB
ejpam-2026	120	2	m	m	PRON
ejpam-2026	120	3	is	be	AUX
ejpam-2026	120	4	r	r	NOUN
ejpam-2026	120	5	-	-	PUNCT
ejpam-2026	120	6	module	module	NOUN
ejpam-2026	120	7	and	and	CCONJ
ejpam-2026	120	8	n	n	NOUN
ejpam-2026	120	9	is	be	AUX
ejpam-2026	120	10	a	a	DET
ejpam-2026	120	11	subgroup	subgroup	NOUN
ejpam-2026	120	12	of	of	ADP
ejpam-2026	120	13	m	m	PROPN
ejpam-2026	120	14	.	.	PUNCT
ejpam-2026	121	1	n	n	PRON
ejpam-2026	121	2	is	be	AUX
ejpam-2026	121	3	said	say	VERB
ejpam-2026	121	4	to	to	PART
ejpam-2026	121	5	be	be	AUX
ejpam-2026	121	6	a	a	DET
ejpam-2026	121	7	submodule	submodule	NOUN
ejpam-2026	121	8	(	(	PUNCT
ejpam-2026	121	9	or	or	CCONJ
ejpam-2026	121	10	r	r	NOUN
ejpam-2026	121	11	-	-	PUNCT
ejpam-2026	121	12	submodule	submodule	NOUN
ejpam-2026	121	13	,	,	PUNCT
ejpam-2026	121	14	to	to	PART
ejpam-2026	121	15	be	be	AUX
ejpam-2026	121	16	more	more	ADV
ejpam-2026	121	17	explicit	explicit	ADJ
ejpam-2026	121	18	)	)	PUNCT
ejpam-2026	121	19	if	if	SCONJ
ejpam-2026	121	20	,	,	PUNCT
ejpam-2026	121	21	for	for	ADP
ejpam-2026	121	22	any	any	DET
ejpam-2026	121	23	n	n	PRON
ejpam-2026	121	24	∈	∈	NOUN
ejpam-2026	121	25	n	n	NOUN
ejpam-2026	121	26	,	,	PUNCT
ejpam-2026	121	27	r	r	NOUN
ejpam-2026	121	28	∈	∈	PROPN
ejpam-2026	121	29	r	r	NOUN
ejpam-2026	121	30	,	,	PUNCT
ejpam-2026	121	31	the	the	DET
ejpam-2026	121	32	product	product	NOUN
ejpam-2026	121	33	rn	rn	PROPN
ejpam-2026	121	34	is	be	AUX
ejpam-2026	121	35	in	in	ADP
ejpam-2026	121	36	n	n	PROPN
ejpam-2026	121	37	.	.	PUNCT
ejpam-2026	122	1	definition	definition	NOUN
ejpam-2026	122	2	4	4	NUM
ejpam-2026	122	3	.	.	PUNCT
ejpam-2026	123	1	let	let	VERB
ejpam-2026	123	2	r	r	PRON
ejpam-2026	123	3	be	be	AUX
ejpam-2026	123	4	a	a	DET
ejpam-2026	123	5	pseudo	pseudo	NOUN
ejpam-2026	123	6	-	-	ADJ
ejpam-2026	123	7	valuation	valuation	NOUN
ejpam-2026	123	8	domain	domain	NOUN
ejpam-2026	123	9	and	and	CCONJ
ejpam-2026	123	10	m	m	AUX
ejpam-2026	123	11	be	be	AUX
ejpam-2026	123	12	a	a	DET
ejpam-2026	123	13	r−module	r−module	NOUN
ejpam-2026	123	14	.	.	PUNCT
ejpam-2026	124	1	if	if	SCONJ
ejpam-2026	124	2	n	n	PRON
ejpam-2026	124	3	is	be	AUX
ejpam-2026	124	4	a	a	DET
ejpam-2026	124	5	subset	subset	NOUN
ejpam-2026	124	6	of	of	ADP
ejpam-2026	124	7	r−module	r−module	NOUN
ejpam-2026	124	8	m	m	VERB
ejpam-2026	124	9	then	then	ADV
ejpam-2026	124	10	n	n	PRON
ejpam-2026	124	11	is	be	AUX
ejpam-2026	124	12	a	a	DET
ejpam-2026	124	13	submodule	submodule	NOUN
ejpam-2026	124	14	of	of	ADP
ejpam-2026	124	15	m	m	PROPN
ejpam-2026	124	16	if	if	SCONJ
ejpam-2026	124	17	;	;	PUNCT
ejpam-2026	124	18	(	(	PUNCT
ejpam-2026	124	19	1	1	X
ejpam-2026	124	20	)	)	PUNCT
ejpam-2026	124	21	n−	n−	NOUN
ejpam-2026	124	22	ń	ń	NOUN
ejpam-2026	124	23	∈	∈	PROPN
ejpam-2026	124	24	n	n	ADV
ejpam-2026	124	25	for	for	ADP
ejpam-2026	124	26	all	all	DET
ejpam-2026	124	27	n	n	CCONJ
ejpam-2026	124	28	,	,	PUNCT
ejpam-2026	124	29	ń	ń	PROPN
ejpam-2026	124	30	∈	∈	PROPN
ejpam-2026	124	31	n.	n.	NOUN
ejpam-2026	124	32	(	(	PUNCT
ejpam-2026	124	33	2	2	NUM
ejpam-2026	124	34	)	)	PUNCT
ejpam-2026	124	35	for	for	ADP
ejpam-2026	124	36	each	each	DET
ejpam-2026	124	37	x	x	SYM
ejpam-2026	124	38	∈	∈	PROPN
ejpam-2026	125	1	k	k	NOUN
ejpam-2026	125	2	−	−	NOUN
ejpam-2026	125	3	r	r	NOUN
ejpam-2026	125	4	,	,	PUNCT
ejpam-2026	125	5	where	where	SCONJ
ejpam-2026	125	6	k	k	PROPN
ejpam-2026	125	7	be	be	AUX
ejpam-2026	125	8	a	a	DET
ejpam-2026	125	9	quotient	quotient	NOUN
ejpam-2026	125	10	field	field	NOUN
ejpam-2026	125	11	and	and	CCONJ
ejpam-2026	125	12	for	for	ADP
ejpam-2026	125	13	each	each	DET
ejpam-2026	125	14	nonunit	nonunit	NOUN
ejpam-2026	125	15	a	a	PRON
ejpam-2026	125	16	of	of	ADP
ejpam-2026	125	17	r	r	NOUN
ejpam-2026	125	18	,	,	PUNCT
ejpam-2026	125	19	x−1a	x−1a	X
ejpam-2026	125	20	∈	∈	PROPN
ejpam-2026	126	1	r	r	NOUN
ejpam-2026	126	2	,	,	PUNCT
ejpam-2026	126	3	n	n	NOUN
ejpam-2026	126	4	∈	∈	NOUN
ejpam-2026	126	5	n	n	NOUN
ejpam-2026	126	6	,	,	PUNCT
ejpam-2026	126	7	x−1an	x−1an	PROPN
ejpam-2026	126	8	∈	∈	PROPN
ejpam-2026	126	9	n.	n.	NOUN
ejpam-2026	126	10	references	reference	VERB
ejpam-2026	126	11	520	520	NUM
ejpam-2026	126	12	example	example	NOUN
ejpam-2026	126	13	4	4	NUM
ejpam-2026	126	14	.	.	PUNCT
ejpam-2026	126	15	suppose	suppose	VERB
ejpam-2026	126	16	m	m	VERB
ejpam-2026	126	17	=	=	SYM
ejpam-2026	126	18	c[x	c[x	NOUN
ejpam-2026	126	19	]	]	X
ejpam-2026	126	20	is	be	AUX
ejpam-2026	126	21	a	a	DET
ejpam-2026	126	22	r	r	NOUN
ejpam-2026	126	23	−	−	NOUN
ejpam-2026	126	24	module	module	NOUN
ejpam-2026	126	25	where	where	SCONJ
ejpam-2026	126	26	r	r	NOUN
ejpam-2026	126	27	=	=	SYM
ejpam-2026	126	28	z	z	NOUN
ejpam-2026	127	1	+	+	NOUN
ejpam-2026	127	2	xq[[x	xq[[x	PROPN
ejpam-2026	127	3	]	]	X
ejpam-2026	127	4	]	]	X
ejpam-2026	127	5	is	be	AUX
ejpam-2026	127	6	pseudovaluation	pseudovaluation	NOUN
ejpam-2026	127	7	domain	domain	NOUN
ejpam-2026	127	8	and	and	CCONJ
ejpam-2026	127	9	n	n	NOUN
ejpam-2026	127	10	=	=	PUNCT
ejpam-2026	127	11	r[x	r[x	PROPN
ejpam-2026	127	12	]	]	PUNCT
ejpam-2026	127	13	is	be	AUX
ejpam-2026	127	14	a	a	DET
ejpam-2026	127	15	subgroup	subgroup	NOUN
ejpam-2026	127	16	of	of	ADP
ejpam-2026	127	17	m	m	PROPN
ejpam-2026	127	18	.	.	PUNCT
ejpam-2026	128	1	n	n	PRON
ejpam-2026	128	2	is	be	AUX
ejpam-2026	128	3	a	a	DET
ejpam-2026	128	4	submodule	submodule	NOUN
ejpam-2026	128	5	(	(	PUNCT
ejpam-2026	128	6	or	or	CCONJ
ejpam-2026	128	7	r−submodule	r−submodule	PROPN
ejpam-2026	128	8	,	,	PUNCT
ejpam-2026	128	9	to	to	PART
ejpam-2026	128	10	be	be	AUX
ejpam-2026	128	11	more	more	ADV
ejpam-2026	128	12	explicit	explicit	ADJ
ejpam-2026	128	13	)	)	PUNCT
ejpam-2026	128	14	since	since	SCONJ
ejpam-2026	128	15	for	for	ADP
ejpam-2026	128	16	any	any	DET
ejpam-2026	128	17	n	n	NOUN
ejpam-2026	128	18	=	=	SYM
ejpam-2026	128	19	f(x	f(x	PROPN
ejpam-2026	128	20	)	)	PUNCT
ejpam-2026	128	21	∈	∈	PROPN
ejpam-2026	128	22	n	n	PRON
ejpam-2026	128	23	and	and	CCONJ
ejpam-2026	128	24	any	any	DET
ejpam-2026	128	25	a+xb	a+xb	ADJ
ejpam-2026	128	26	in	in	ADP
ejpam-2026	128	27	r	r	NOUN
ejpam-2026	128	28	,	,	PUNCT
ejpam-2026	128	29	the	the	DET
ejpam-2026	128	30	product	product	NOUN
ejpam-2026	128	31	n(a+xb	n(a+xb	NUM
ejpam-2026	128	32	)	)	PUNCT
ejpam-2026	128	33	is	be	AUX
ejpam-2026	128	34	in	in	ADP
ejpam-2026	128	35	n	n	PRON
ejpam-2026	129	1	where	where	SCONJ
ejpam-2026	129	2	a	a	DET
ejpam-2026	129	3	∈	∈	PROPN
ejpam-2026	129	4	z	z	PROPN
ejpam-2026	129	5	,	,	PUNCT
ejpam-2026	129	6	b	b	PROPN
ejpam-2026	129	7	=	=	SYM
ejpam-2026	129	8	f(x	f(x	PROPN
ejpam-2026	129	9	)	)	PUNCT
ejpam-2026	129	10	∈	∈	PROPN
ejpam-2026	129	11	q[[x	q[[x	PROPN
ejpam-2026	129	12	]	]	X
ejpam-2026	129	13	]	]	PUNCT
ejpam-2026	129	14	.	.	PUNCT
ejpam-2026	130	1	theorem	theorem	ADJ
ejpam-2026	130	2	4	4	NUM
ejpam-2026	130	3	.	.	PUNCT
ejpam-2026	131	1	if	if	SCONJ
ejpam-2026	131	2	r	r	NOUN
ejpam-2026	131	3	is	be	AUX
ejpam-2026	131	4	pseudo	pseudo	NOUN
ejpam-2026	131	5	-	-	NOUN
ejpam-2026	131	6	valuation	valuation	NOUN
ejpam-2026	131	7	domain	domain	NOUN
ejpam-2026	131	8	,	,	PUNCT
ejpam-2026	131	9	m	m	VERB
ejpam-2026	131	10	is	be	AUX
ejpam-2026	131	11	an	an	DET
ejpam-2026	131	12	abelian	abelian	ADJ
ejpam-2026	131	13	group	group	NOUN
ejpam-2026	131	14	and	and	CCONJ
ejpam-2026	131	15	n	n	NOUN
ejpam-2026	131	16	is	be	AUX
ejpam-2026	131	17	a	a	DET
ejpam-2026	131	18	subgroup	subgroup	NOUN
ejpam-2026	131	19	of	of	ADP
ejpam-2026	131	20	m	m	PRON
ejpam-2026	131	21	then	then	ADV
ejpam-2026	131	22	n	n	ADV
ejpam-2026	131	23	is	be	AUX
ejpam-2026	131	24	r−	r−	PROPN
ejpam-2026	131	25	submodule	submodule	NOUN
ejpam-2026	131	26	of	of	ADP
ejpam-2026	131	27	m	m	PROPN
ejpam-2026	131	28	⇐	⇐	ADJ
ejpam-2026	131	29	⇒	⇒	NOUN
ejpam-2026	131	30	x−1an	x−1an	PROPN
ejpam-2026	131	31	∈	∈	PROPN
ejpam-2026	131	32	n	n	CCONJ
ejpam-2026	131	33	where	where	SCONJ
ejpam-2026	131	34	x	x	SYM
ejpam-2026	131	35	∈	∈	PROPN
ejpam-2026	131	36	k	k	PROPN
ejpam-2026	131	37	−r	−r	PROPN
ejpam-2026	131	38	.	.	PUNCT
ejpam-2026	132	1	proof	proof	NOUN
ejpam-2026	132	2	.	.	PUNCT
ejpam-2026	133	1	assume	assume	VERB
ejpam-2026	133	2	that	that	SCONJ
ejpam-2026	133	3	n	n	PRON
ejpam-2026	133	4	is	be	AUX
ejpam-2026	133	5	r−submodule	r−submodule	NOUN
ejpam-2026	133	6	of	of	ADP
ejpam-2026	133	7	m	m	VERB
ejpam-2026	133	8	therefore	therefore	ADV
ejpam-2026	133	9	n	n	PRON
ejpam-2026	133	10	is	be	AUX
ejpam-2026	133	11	closed	close	VERB
ejpam-2026	133	12	under	under	ADP
ejpam-2026	133	13	the	the	DET
ejpam-2026	133	14	action	action	NOUN
ejpam-2026	133	15	of	of	ADP
ejpam-2026	133	16	ring	ring	NOUN
ejpam-2026	133	17	elements	element	NOUN
ejpam-2026	133	18	,	,	PUNCT
ejpam-2026	133	19	so	so	SCONJ
ejpam-2026	133	20	we	we	PRON
ejpam-2026	133	21	have	have	VERB
ejpam-2026	133	22	x−1an	x−1an	PROPN
ejpam-2026	133	23	∈	∈	PROPN
ejpam-2026	133	24	n	n	CCONJ
ejpam-2026	133	25	where	where	SCONJ
ejpam-2026	133	26	x	x	PUNCT
ejpam-2026	133	27	∈	∈	PROPN
ejpam-2026	133	28	k	k	NOUN
ejpam-2026	133	29	−	−	NOUN
ejpam-2026	133	30	r	r	NOUN
ejpam-2026	133	31	and	and	CCONJ
ejpam-2026	133	32	a	a	DET
ejpam-2026	133	33	non	non	ADJ
ejpam-2026	133	34	-	-	ADJ
ejpam-2026	133	35	unit	unit	ADJ
ejpam-2026	133	36	element	element	NOUN
ejpam-2026	133	37	a	a	PRON
ejpam-2026	133	38	in	in	ADP
ejpam-2026	133	39	r.conversely	r.conversely	ADV
ejpam-2026	133	40	,	,	PUNCT
ejpam-2026	133	41	assume	assume	VERB
ejpam-2026	133	42	x−1an	x−1an	PROPN
ejpam-2026	133	43	∈	∈	PROPN
ejpam-2026	134	1	n	n	NOUN
ejpam-2026	134	2	,	,	PUNCT
ejpam-2026	134	3	where	where	SCONJ
ejpam-2026	134	4	x	x	PUNCT
ejpam-2026	134	5	∈	∈	PROPN
ejpam-2026	134	6	k	k	NOUN
ejpam-2026	135	1	−	−	NOUN
ejpam-2026	135	2	r	r	NOUN
ejpam-2026	135	3	we	we	PRON
ejpam-2026	135	4	have	have	VERB
ejpam-2026	135	5	to	to	PART
ejpam-2026	135	6	prove	prove	VERB
ejpam-2026	135	7	that	that	SCONJ
ejpam-2026	135	8	n	n	NOUN
ejpam-2026	135	9	is	be	AUX
ejpam-2026	135	10	r−submodule	r−submodule	NOUN
ejpam-2026	135	11	of	of	ADP
ejpam-2026	135	12	m.	m.	NOUN
ejpam-2026	135	13	as	as	ADP
ejpam-2026	135	14	x−1an	x−1an	PROPN
ejpam-2026	135	15	∈	∈	PROPN
ejpam-2026	135	16	n	n	CCONJ
ejpam-2026	135	17	we	we	PRON
ejpam-2026	135	18	have	have	VERB
ejpam-2026	135	19	;	;	PUNCT
ejpam-2026	135	20	(	(	PUNCT
ejpam-2026	135	21	1	1	X
ejpam-2026	135	22	)	)	PUNCT
ejpam-2026	135	23	x−1an	x−1an	NOUN
ejpam-2026	135	24	∈	∈	PROPN
ejpam-2026	135	25	n	n	PRON
ejpam-2026	135	26	∀	∀	X
ejpam-2026	135	27	x−1a	x−1a	X
ejpam-2026	136	1	∈	∈	PROPN
ejpam-2026	137	1	r	r	NOUN
ejpam-2026	137	2	,	,	PUNCT
ejpam-2026	137	3	n	n	PROPN
ejpam-2026	137	4	∈	∈	PROPN
ejpam-2026	137	5	n.	n.	NOUN
ejpam-2026	137	6	given	give	VERB
ejpam-2026	137	7	that	that	SCONJ
ejpam-2026	137	8	n	n	NOUN
ejpam-2026	137	9	is	be	AUX
ejpam-2026	137	10	a	a	DET
ejpam-2026	137	11	subgroup	subgroup	NOUN
ejpam-2026	137	12	of	of	ADP
ejpam-2026	137	13	m	m	PRON
ejpam-2026	137	14	so	so	ADV
ejpam-2026	137	15	it	it	PRON
ejpam-2026	137	16	is	be	AUX
ejpam-2026	137	17	closed	close	VERB
ejpam-2026	137	18	under	under	ADP
ejpam-2026	137	19	subtraction	subtraction	NOUN
ejpam-2026	137	20	then	then	ADV
ejpam-2026	137	21	2nd	2nd	ADJ
ejpam-2026	137	22	condition	condition	NOUN
ejpam-2026	137	23	of	of	ADP
ejpam-2026	137	24	r−	r−	PROPN
ejpam-2026	137	25	submodule	submodule	NOUN
ejpam-2026	137	26	is	be	AUX
ejpam-2026	137	27	satisfied	satisfied	ADJ
ejpam-2026	137	28	so	so	ADV
ejpam-2026	137	29	;	;	PUNCT
ejpam-2026	137	30	(	(	PUNCT
ejpam-2026	137	31	2	2	X
ejpam-2026	137	32	)	)	PUNCT
ejpam-2026	137	33	n−	n−	NOUN
ejpam-2026	137	34	n′	n′	PROPN
ejpam-2026	137	35	∈	∈	PROPN
ejpam-2026	137	36	n	n	CCONJ
ejpam-2026	137	37	∀	∀	X
ejpam-2026	137	38	n	n	CCONJ
ejpam-2026	137	39	,	,	PUNCT
ejpam-2026	137	40	n′	n′	PROPN
ejpam-2026	137	41	∈	∈	PROPN
ejpam-2026	137	42	n	n	CCONJ
ejpam-2026	137	43	thus	thus	ADV
ejpam-2026	137	44	n	n	PRON
ejpam-2026	137	45	is	be	AUX
ejpam-2026	137	46	a	a	DET
ejpam-2026	137	47	r−submodule	r−submodule	NOUN
ejpam-2026	137	48	of	of	ADP
ejpam-2026	137	49	m.	m.	NOUN
ejpam-2026	137	50	references	reference	NOUN
ejpam-2026	137	51	[	[	X
ejpam-2026	137	52	1	1	NUM
ejpam-2026	137	53	]	]	X
ejpam-2026	137	54	badawi	badawi	ADJ
ejpam-2026	137	55	,	,	PUNCT
ejpam-2026	137	56	a.	a.	NOUN
ejpam-2026	137	57	:	:	PUNCT
ejpam-2026	137	58	on	on	ADP
ejpam-2026	137	59	domains	domain	NOUN
ejpam-2026	137	60	which	which	PRON
ejpam-2026	137	61	have	have	VERB
ejpam-2026	137	62	prime	prime	ADJ
ejpam-2026	137	63	ideals	ideal	NOUN
ejpam-2026	137	64	that	that	PRON
ejpam-2026	137	65	are	be	AUX
ejpam-2026	137	66	linearly	linearly	ADV
ejpam-2026	137	67	ordered	order	VERB
ejpam-2026	137	68	.	.	PUNCT
ejpam-2026	138	1	communication	communication	NOUN
ejpam-2026	138	2	in	in	ADP
ejpam-2026	138	3	algebra	algebra	NOUN
ejpam-2026	138	4	.	.	PUNCT
ejpam-2026	139	1	vol	vol	NOUN
ejpam-2026	139	2	.	.	PROPN
ejpam-2026	140	1	23	23	NUM
ejpam-2026	140	2	,	,	PUNCT
ejpam-2026	140	3	no	no	INTJ
ejpam-2026	140	4	.	.	NOUN
ejpam-2026	140	5	12	12	NUM
ejpam-2026	140	6	,	,	PUNCT
ejpam-2026	140	7	pp	pp	ADJ
ejpam-2026	140	8	.	.	PUNCT
ejpam-2026	141	1	4365–4373	4365–4373	NUM
ejpam-2026	141	2	,	,	PUNCT
ejpam-2026	141	3	1995	1995	NUM
ejpam-2026	141	4	.	.	PUNCT
ejpam-2026	142	1	[	[	X
ejpam-2026	142	2	2	2	NUM
ejpam-2026	142	3	]	]	X
ejpam-2026	142	4	david	david	PROPN
ejpam-2026	142	5	.	.	PUNCT
ejpam-2026	142	6	s.	s.	PROPN
ejpam-2026	142	7	dummit	dummit	PROPN
ejpam-2026	142	8	.	.	PUNCT
ejpam-2026	142	9	:	:	PUNCT
ejpam-2026	143	1	richard	richard	PROPN
ejpam-2026	143	2	m.	m.	PROPN
ejpam-2026	143	3	foote	foote	PROPN
ejpam-2026	143	4	.	.	PUNCT
ejpam-2026	144	1	abstract	abstract	ADJ
ejpam-2026	144	2	algebra	algebra	PROPN
ejpam-2026	144	3	.	.	PUNCT
ejpam-2026	145	1	2nd	2nd	PROPN
ejpam-2026	145	2	edition	edition	PROPN
ejpam-2026	145	3	,	,	PUNCT
ejpam-2026	145	4	(	(	PUNCT
ejpam-2026	145	5	2005	2005	NUM
ejpam-2026	145	6	)	)	PUNCT
ejpam-2026	145	7	,	,	PUNCT
ejpam-2026	145	8	997151429x	997151429x	NUM
ejpam-2026	145	9	,	,	PUNCT
ejpam-2026	145	10	9789971514297	9789971514297	NUM
ejpam-2026	145	11	,	,	PUNCT
ejpam-2026	145	12	978	978	NUM
ejpam-2026	145	13	-	-	SYM
ejpam-2026	145	14	9971514297	9971514297	NUM
ejpam-2026	145	15	,	,	PUNCT
ejpam-2026	145	16	john	john	PROPN
ejpam-2026	145	17	wiley	wiley	PROPN
ejpam-2026	145	18	&	&	CCONJ
ejpam-2026	145	19	sons	son	NOUN
ejpam-2026	145	20	.	.	PUNCT
ejpam-2026	146	1	[	[	X
ejpam-2026	146	2	3	3	NUM
ejpam-2026	146	3	]	]	PUNCT
ejpam-2026	146	4	eisenbud	eisenbud	NOUN
ejpam-2026	146	5	,	,	PUNCT
ejpam-2026	146	6	d.	d.	PROPN
ejpam-2026	146	7	and	and	CCONJ
ejpam-2026	146	8	robin	robin	PROPN
ejpam-2026	146	9	,	,	PUNCT
ejpam-2026	146	10	j.	j.	PROPN
ejpam-2026	146	11	c.	c.	PROPN
ejpam-2026	146	12	:	:	PUNCT
ejpam-2026	146	13	modules	module	NOUN
ejpam-2026	146	14	over	over	ADP
ejpam-2026	146	15	dedekind	dedekind	ADJ
ejpam-2026	146	16	prime	prime	ADJ
ejpam-2026	146	17	rings	ring	NOUN
ejpam-2026	146	18	.	.	PUNCT
ejpam-2026	147	1	j.	j.	PROPN
ejpam-2026	147	2	algebra	algebra	PROPN
ejpam-2026	147	3	,	,	PUNCT
ejpam-2026	147	4	vol	vol	NOUN
ejpam-2026	147	5	.	.	PROPN
ejpam-2026	147	6	16	16	NUM
ejpam-2026	147	7	,	,	PUNCT
ejpam-2026	147	8	67	67	NUM
ejpam-2026	147	9	-	-	SYM
ejpam-2026	147	10	85	85	NUM
ejpam-2026	147	11	,	,	PUNCT
ejpam-2026	147	12	(	(	PUNCT
ejpam-2026	147	13	1970	1970	NUM
ejpam-2026	147	14	)	)	PUNCT
ejpam-2026	148	1	[	[	X
ejpam-2026	148	2	4	4	NUM
ejpam-2026	148	3	]	]	X
ejpam-2026	148	4	gilmer	gilmer	X
ejpam-2026	148	5	,	,	PUNCT
ejpam-2026	148	6	r.	r.	PROPN
ejpam-2026	148	7	multiplicative	multiplicative	PROPN
ejpam-2026	148	8	ideal	ideal	PROPN
ejpam-2026	148	9	theory	theory	NOUN
ejpam-2026	148	10	.	.	PUNCT
ejpam-2026	148	11	:	:	PUNCT
ejpam-2026	148	12	queens	queen	VERB
ejpam-2026	148	13	papers	paper	NOUN
ejpam-2026	148	14	on	on	ADP
ejpam-2026	148	15	pure	pure	ADJ
ejpam-2026	148	16	and	and	CCONJ
ejpam-2026	148	17	applied	applied	ADJ
ejpam-2026	148	18	mathematics	mathematic	NOUN
ejpam-2026	148	19	,	,	PUNCT
ejpam-2026	148	20	no.12	no.12	PROPN
ejpam-2026	148	21	,	,	PUNCT
ejpam-2026	148	22	queens	queen	VERB
ejpam-2026	148	23	university	university	NOUN
ejpam-2026	148	24	press	press	NOUN
ejpam-2026	148	25	,	,	PUNCT
ejpam-2026	148	26	kingston	kingston	PROPN
ejpam-2026	148	27	,	,	PUNCT
ejpam-2026	148	28	ontario	ontario	PROPN
ejpam-2026	148	29	,	,	PUNCT
ejpam-2026	148	30	1968	1968	NUM
ejpam-2026	148	31	.	.	PUNCT
ejpam-2026	149	1	[	[	X
ejpam-2026	149	2	5	5	NUM
ejpam-2026	149	3	]	]	PUNCT
ejpam-2026	149	4	hedstorm	hedstorm	NOUN
ejpam-2026	149	5	,	,	PUNCT
ejpam-2026	149	6	j.	j.	PROPN
ejpam-2026	149	7	r.	r.	PROPN
ejpam-2026	149	8	,	,	PUNCT
ejpam-2026	149	9	and	and	CCONJ
ejpam-2026	149	10	houston	houston	PROPN
ejpam-2026	149	11	,	,	PUNCT
ejpam-2026	149	12	e.	e.	PROPN
ejpam-2026	149	13	g.	g.	PROPN
ejpam-2026	149	14	:	:	PUNCT
ejpam-2026	149	15	pseudo	pseudo	NOUN
ejpam-2026	149	16	-	-	PUNCT
ejpam-2026	149	17	valuation	valuation	NOUN
ejpam-2026	149	18	domains	domain	NOUN
ejpam-2026	149	19	.	.	PUNCT
ejpam-2026	149	20	pacific	pacific	PROPN
ejpam-2026	149	21	j.	j.	PROPN
ejpam-2026	149	22	math	math	PROPN
ejpam-2026	149	23	.	.	PUNCT
ejpam-2026	150	1	75	75	NUM
ejpam-2026	150	2	(	(	PUNCT
ejpam-2026	150	3	1978	1978	NUM
ejpam-2026	150	4	)	)	PUNCT
ejpam-2026	150	5	,	,	PUNCT
ejpam-2026	150	6	1	1	NUM
ejpam-2026	150	7	,	,	PUNCT
ejpam-2026	150	8	137	137	NUM
ejpam-2026	150	9	-	-	SYM
ejpam-2026	150	10	147	147	NUM
ejpam-2026	150	11	.	.	PUNCT
ejpam-2026	151	1	[	[	X
ejpam-2026	151	2	6	6	NUM
ejpam-2026	151	3	]	]	X
ejpam-2026	151	4	kaplansky	kaplansky	PROPN
ejpam-2026	151	5	,	,	PUNCT
ejpam-2026	151	6	i.	i.	NOUN
ejpam-2026	151	7	:	:	PUNCT
ejpam-2026	151	8	modules	module	NOUN
ejpam-2026	151	9	over	over	ADP
ejpam-2026	151	10	dedekind	dedekind	NOUN
ejpam-2026	151	11	rings	ring	NOUN
ejpam-2026	151	12	and	and	CCONJ
ejpam-2026	151	13	valuation	valuation	NOUN
ejpam-2026	151	14	rings	ring	NOUN
ejpam-2026	151	15	.	.	PUNCT
ejpam-2026	152	1	trans	trans	PROPN
ejpam-2026	152	2	.	.	PUNCT
ejpam-2026	153	1	amer	amer	PROPN
ejpam-2026	153	2	.	.	PUNCT
ejpam-2026	153	3	math	math	PROPN
ejpam-2026	153	4	.	.	PUNCT
ejpam-2026	154	1	soc	soc	PROPN
ejpam-2026	154	2	.	.	PUNCT
ejpam-2026	155	1	72	72	NUM
ejpam-2026	155	2	(	(	PUNCT
ejpam-2026	155	3	1952	1952	NUM
ejpam-2026	155	4	)	)	PUNCT
ejpam-2026	155	5	,	,	PUNCT
ejpam-2026	156	1	pp	pp	ADP
ejpam-2026	156	2	.	.	PUNCT
ejpam-2026	157	1	327–340	327–340	NUM
ejpam-2026	157	2	.	.	PUNCT