id	sid	tid	token	lemma	pos
ejpam-2429	1	1	compile	compile	NOUN
ejpam-2429	1	2	/	/	SYM
ejpam-2429	1	3	output.dvi	output.dvi	NOUN
ejpam-2429	1	4	european	european	ADJ
ejpam-2429	1	5	journal	journal	NOUN
ejpam-2429	1	6	of	of	ADP
ejpam-2429	1	7	pure	pure	ADJ
ejpam-2429	1	8	and	and	CCONJ
ejpam-2429	1	9	applied	apply	VERB
ejpam-2429	1	10	mathematics	mathematic	NOUN
ejpam-2429	1	11	vol	vol	NOUN
ejpam-2429	1	12	.	.	PROPN
ejpam-2429	1	13	8	8	NUM
ejpam-2429	1	14	,	,	PUNCT
ejpam-2429	1	15	no	no	INTJ
ejpam-2429	1	16	.	.	NOUN
ejpam-2429	1	17	4	4	NUM
ejpam-2429	1	18	,	,	PUNCT
ejpam-2429	1	19	2015	2015	NUM
ejpam-2429	1	20	,	,	PUNCT
ejpam-2429	1	21	458	458	NUM
ejpam-2429	1	22	-	-	SYM
ejpam-2429	1	23	461	461	NUM
ejpam-2429	1	24	issn	issn	PROPN
ejpam-2429	1	25	1307	1307	NUM
ejpam-2429	1	26	-	-	SYM
ejpam-2429	1	27	5543	5543	NUM
ejpam-2429	1	28	–	–	PUNCT
ejpam-2429	1	29	www.ejpam.com	www.ejpam.com	X
ejpam-2429	1	30	a	a	DET
ejpam-2429	1	31	note	note	NOUN
ejpam-2429	1	32	on	on	ADP
ejpam-2429	1	33	prüfer	prüfer	NOUN
ejpam-2429	1	34	⋆-multiplication	⋆-multiplication	NOUN
ejpam-2429	1	35	domains	domain	NOUN
ejpam-2429	1	36	olivier	olivier	PROPN
ejpam-2429	1	37	a.	a.	NOUN
ejpam-2429	1	38	heubo	heubo	PROPN
ejpam-2429	1	39	-	-	PUNCT
ejpam-2429	1	40	kwegna	kwegna	PROPN
ejpam-2429	1	41	department	department	PROPN
ejpam-2429	1	42	of	of	ADP
ejpam-2429	1	43	mathematical	mathematical	ADJ
ejpam-2429	1	44	sciences	sciences	PROPN
ejpam-2429	1	45	,	,	PUNCT
ejpam-2429	1	46	saginaw	saginaw	NOUN
ejpam-2429	1	47	valley	valley	PROPN
ejpam-2429	1	48	state	state	PROPN
ejpam-2429	1	49	university	university	PROPN
ejpam-2429	1	50	,	,	PUNCT
ejpam-2429	1	51	university	university	NOUN
ejpam-2429	1	52	center	center	NOUN
ejpam-2429	1	53	mi	mi	PROPN
ejpam-2429	1	54	48710	48710	NUM
ejpam-2429	1	55	,	,	PUNCT
ejpam-2429	1	56	usa	usa	PROPN
ejpam-2429	1	57	abstract	abstract	NOUN
ejpam-2429	1	58	.	.	PUNCT
ejpam-2429	2	1	in	in	ADP
ejpam-2429	2	2	this	this	DET
ejpam-2429	2	3	note	note	NOUN
ejpam-2429	2	4	,	,	PUNCT
ejpam-2429	2	5	we	we	PRON
ejpam-2429	2	6	prove	prove	VERB
ejpam-2429	2	7	that	that	SCONJ
ejpam-2429	2	8	for	for	ADP
ejpam-2429	2	9	an	an	DET
ejpam-2429	2	10	arbitrary	arbitrary	ADJ
ejpam-2429	2	11	star	star	NOUN
ejpam-2429	2	12	operation	operation	NOUN
ejpam-2429	2	13	⋆	⋆	VERB
ejpam-2429	2	14	on	on	ADP
ejpam-2429	2	15	a	a	DET
ejpam-2429	2	16	domain	domain	NOUN
ejpam-2429	2	17	r	r	NOUN
ejpam-2429	2	18	,	,	PUNCT
ejpam-2429	2	19	the	the	DET
ejpam-2429	2	20	domain	domain	NOUN
ejpam-2429	2	21	r	r	NOUN
ejpam-2429	2	22	is	be	AUX
ejpam-2429	2	23	a	a	DET
ejpam-2429	2	24	prüfer	prüfer	NOUN
ejpam-2429	2	25	⋆-multiplication	⋆-multiplication	NOUN
ejpam-2429	2	26	domain	domain	NOUN
ejpam-2429	2	27	if	if	SCONJ
ejpam-2429	2	28	every	every	DET
ejpam-2429	2	29	2	2	NUM
ejpam-2429	2	30	-	-	PUNCT
ejpam-2429	2	31	generated	generate	VERB
ejpam-2429	2	32	ideal	ideal	NOUN
ejpam-2429	2	33	of	of	ADP
ejpam-2429	2	34	r	r	NOUN
ejpam-2429	2	35	is	be	AUX
ejpam-2429	2	36	⋆	⋆	VERB
ejpam-2429	2	37	f	f	PROPN
ejpam-2429	2	38	-invertible	-invertible	PROPN
ejpam-2429	2	39	.	.	PUNCT
ejpam-2429	3	1	some	some	DET
ejpam-2429	3	2	characterizations	characterization	NOUN
ejpam-2429	3	3	of	of	ADP
ejpam-2429	3	4	prüfer-⋆	prüfer-⋆	NOUN
ejpam-2429	3	5	multiplication	multiplication	NOUN
ejpam-2429	3	6	domains	domain	NOUN
ejpam-2429	3	7	are	be	AUX
ejpam-2429	3	8	therefore	therefore	ADV
ejpam-2429	3	9	obtained	obtain	VERB
ejpam-2429	3	10	.	.	PUNCT
ejpam-2429	4	1	2010	2010	NUM
ejpam-2429	4	2	mathematics	mathematic	NOUN
ejpam-2429	4	3	subject	subject	NOUN
ejpam-2429	4	4	classifications	classification	NOUN
ejpam-2429	4	5	:	:	PUNCT
ejpam-2429	4	6	13a15	13a15	NUM
ejpam-2429	4	7	,	,	PUNCT
ejpam-2429	4	8	13a18	13a18	NUM
ejpam-2429	4	9	,	,	PUNCT
ejpam-2429	4	10	16w50	16w50	NUM
ejpam-2429	4	11	key	key	ADJ
ejpam-2429	4	12	words	word	NOUN
ejpam-2429	4	13	and	and	CCONJ
ejpam-2429	4	14	phrases	phrase	NOUN
ejpam-2429	4	15	:	:	PUNCT
ejpam-2429	4	16	star	star	NOUN
ejpam-2429	4	17	operation	operation	NOUN
ejpam-2429	4	18	;	;	PUNCT
ejpam-2429	4	19	⋆-ideal	⋆-ideal	NOUN
ejpam-2429	4	20	;	;	PUNCT
ejpam-2429	4	21	prüfer	prüfer	NOUN
ejpam-2429	4	22	⋆-multiplication	⋆-multiplication	NOUN
ejpam-2429	4	23	domain	domain	NOUN
ejpam-2429	4	24	1	1	NUM
ejpam-2429	4	25	.	.	PUNCT
ejpam-2429	4	26	introduction	introduction	NOUN
ejpam-2429	4	27	throughout	throughout	ADP
ejpam-2429	4	28	this	this	DET
ejpam-2429	4	29	note	note	NOUN
ejpam-2429	4	30	r	r	NOUN
ejpam-2429	4	31	denotes	denote	VERB
ejpam-2429	4	32	an	an	DET
ejpam-2429	4	33	integral	integral	ADJ
ejpam-2429	4	34	domain	domain	NOUN
ejpam-2429	4	35	with	with	ADP
ejpam-2429	4	36	quotient	quotient	NOUN
ejpam-2429	4	37	field	field	NOUN
ejpam-2429	4	38	k	k	PROPN
ejpam-2429	4	39	.	.	PUNCT
ejpam-2429	5	1	letf	letf	ADJ
ejpam-2429	5	2	(	(	PUNCT
ejpam-2429	5	3	r	r	NOUN
ejpam-2429	5	4	)	)	PUNCT
ejpam-2429	5	5	be	be	AUX
ejpam-2429	5	6	the	the	DET
ejpam-2429	5	7	set	set	NOUN
ejpam-2429	5	8	of	of	ADP
ejpam-2429	5	9	all	all	DET
ejpam-2429	5	10	nonzero	nonzero	ADJ
ejpam-2429	5	11	fractional	fractional	ADJ
ejpam-2429	5	12	ideals	ideal	NOUN
ejpam-2429	5	13	of	of	ADP
ejpam-2429	5	14	r	r	NOUN
ejpam-2429	5	15	and	and	CCONJ
ejpam-2429	5	16	f	f	PROPN
ejpam-2429	5	17	(	(	PUNCT
ejpam-2429	5	18	r	r	AUX
ejpam-2429	5	19	)	)	PUNCT
ejpam-2429	5	20	be	be	AUX
ejpam-2429	5	21	the	the	DET
ejpam-2429	5	22	set	set	NOUN
ejpam-2429	5	23	of	of	ADP
ejpam-2429	5	24	all	all	DET
ejpam-2429	5	25	nonzero	nonzero	NOUN
ejpam-2429	5	26	finitely	finitely	ADV
ejpam-2429	5	27	generated	generate	VERB
ejpam-2429	5	28	fractional	fractional	ADJ
ejpam-2429	5	29	ideals	ideal	NOUN
ejpam-2429	5	30	of	of	ADP
ejpam-2429	5	31	r.	r.	PROPN
ejpam-2429	5	32	a	a	DET
ejpam-2429	5	33	star	star	NOUN
ejpam-2429	5	34	operation	operation	NOUN
ejpam-2429	5	35	on	on	ADP
ejpam-2429	5	36	r	r	NOUN
ejpam-2429	5	37	is	be	AUX
ejpam-2429	5	38	a	a	DET
ejpam-2429	5	39	mapping	mapping	NOUN
ejpam-2429	5	40	a→	a→	X
ejpam-2429	5	41	a⋆	a⋆	ADV
ejpam-2429	5	42	off	off	ADV
ejpam-2429	5	43	(	(	PUNCT
ejpam-2429	5	44	r	r	NOUN
ejpam-2429	5	45	)	)	PUNCT
ejpam-2429	5	46	intof	intof	NOUN
ejpam-2429	5	47	(	(	PUNCT
ejpam-2429	5	48	r	r	NOUN
ejpam-2429	5	49	)	)	PUNCT
ejpam-2429	5	50	such	such	ADJ
ejpam-2429	5	51	that	that	PRON
ejpam-2429	5	52	for	for	ADP
ejpam-2429	5	53	all	all	DET
ejpam-2429	5	54	a	a	PRON
ejpam-2429	5	55	,	,	PUNCT
ejpam-2429	5	56	b	b	X
ejpam-2429	5	57	∈	∈	X
ejpam-2429	5	58	f	f	X
ejpam-2429	5	59	(	(	PUNCT
ejpam-2429	5	60	r	r	NOUN
ejpam-2429	5	61	)	)	PUNCT
ejpam-2429	5	62	and	and	CCONJ
ejpam-2429	5	63	for	for	ADP
ejpam-2429	5	64	all	all	DET
ejpam-2429	5	65	a	a	DET
ejpam-2429	5	66	∈	∈	NOUN
ejpam-2429	5	67	k	k	X
ejpam-2429	5	68	\	\	PROPN
ejpam-2429	5	69	{	{	PUNCT
ejpam-2429	5	70	0	0	NUM
ejpam-2429	5	71	}	}	PUNCT
ejpam-2429	5	72	,	,	PUNCT
ejpam-2429	5	73	(	(	PUNCT
ejpam-2429	5	74	i	i	NOUN
ejpam-2429	5	75	)	)	PUNCT
ejpam-2429	5	76	(	(	PUNCT
ejpam-2429	5	77	a)⋆	a)⋆	NOUN
ejpam-2429	5	78	=	=	SYM
ejpam-2429	5	79	(	(	PUNCT
ejpam-2429	5	80	a	a	NOUN
ejpam-2429	5	81	)	)	PUNCT
ejpam-2429	5	82	and	and	CCONJ
ejpam-2429	5	83	(	(	PUNCT
ejpam-2429	5	84	aa)⋆	aa)⋆	NOUN
ejpam-2429	5	85	=	=	PUNCT
ejpam-2429	5	86	aa⋆	aa⋆	NOUN
ejpam-2429	5	87	;	;	PUNCT
ejpam-2429	5	88	(	(	PUNCT
ejpam-2429	5	89	ii	ii	NOUN
ejpam-2429	5	90	)	)	PUNCT
ejpam-2429	5	91	a⊆	a⊆	NOUN
ejpam-2429	5	92	a⋆	a⋆	ADV
ejpam-2429	5	93	and	and	CCONJ
ejpam-2429	5	94	a⊆	a⊆	VERB
ejpam-2429	5	95	b⇒	b⇒	X
ejpam-2429	5	96	a⋆	a⋆	ADV
ejpam-2429	5	97	⊆	⊆	NUM
ejpam-2429	5	98	b⋆	b⋆	NOUN
ejpam-2429	5	99	,	,	PUNCT
ejpam-2429	5	100	and	and	CCONJ
ejpam-2429	5	101	(	(	PUNCT
ejpam-2429	5	102	iii	iii	X
ejpam-2429	5	103	)	)	PUNCT
ejpam-2429	5	104	a⋆⋆	a⋆⋆	NOUN
ejpam-2429	5	105	:	:	PUNCT
ejpam-2429	6	1	=	=	SYM
ejpam-2429	6	2	(	(	PUNCT
ejpam-2429	6	3	a⋆)⋆	a⋆)⋆	PROPN
ejpam-2429	6	4	=	=	PUNCT
ejpam-2429	6	5	a⋆.	a⋆.	NOUN
ejpam-2429	6	6	for	for	ADP
ejpam-2429	6	7	an	an	DET
ejpam-2429	6	8	overview	overview	NOUN
ejpam-2429	6	9	of	of	ADP
ejpam-2429	6	10	star	star	NOUN
ejpam-2429	6	11	operations	operation	NOUN
ejpam-2429	6	12	,	,	PUNCT
ejpam-2429	6	13	the	the	DET
ejpam-2429	6	14	reader	reader	NOUN
ejpam-2429	6	15	may	may	AUX
ejpam-2429	6	16	refer	refer	VERB
ejpam-2429	6	17	to	to	ADP
ejpam-2429	6	18	[	[	X
ejpam-2429	6	19	5	5	NUM
ejpam-2429	6	20	,	,	PUNCT
ejpam-2429	6	21	sections	section	NOUN
ejpam-2429	6	22	32	32	NUM
ejpam-2429	6	23	and	and	CCONJ
ejpam-2429	6	24	34	34	NUM
ejpam-2429	6	25	]	]	PUNCT
ejpam-2429	6	26	.	.	PUNCT
ejpam-2429	7	1	given	give	VERB
ejpam-2429	7	2	a	a	DET
ejpam-2429	7	3	star	star	NOUN
ejpam-2429	7	4	operation	operation	NOUN
ejpam-2429	7	5	⋆	⋆	VERB
ejpam-2429	7	6	on	on	ADP
ejpam-2429	7	7	r	r	NOUN
ejpam-2429	7	8	,	,	PUNCT
ejpam-2429	7	9	one	one	PRON
ejpam-2429	7	10	can	can	AUX
ejpam-2429	7	11	construct	construct	VERB
ejpam-2429	7	12	a	a	DET
ejpam-2429	7	13	new	new	ADJ
ejpam-2429	7	14	star	star	NOUN
ejpam-2429	7	15	operation	operation	NOUN
ejpam-2429	7	16	⋆	⋆	PUNCT
ejpam-2429	7	17	f	f	PROPN
ejpam-2429	7	18	as	as	SCONJ
ejpam-2429	7	19	follows	follow	VERB
ejpam-2429	7	20	:	:	PUNCT
ejpam-2429	7	21	for	for	ADP
ejpam-2429	7	22	each	each	DET
ejpam-2429	7	23	a∈	a∈	PROPN
ejpam-2429	7	24	f(r	f(r	PROPN
ejpam-2429	7	25	)	)	PUNCT
ejpam-2429	7	26	,	,	PUNCT
ejpam-2429	7	27	a⋆	a⋆	NOUN
ejpam-2429	8	1	f	f	PROPN
ejpam-2429	8	2	=	=	PUNCT
ejpam-2429	8	3	∪{b⋆|b	∪{b⋆|b	NUM
ejpam-2429	8	4	⊆	⊆	NUM
ejpam-2429	8	5	a	a	PRON
ejpam-2429	8	6	and	and	CCONJ
ejpam-2429	8	7	b	b	NOUN
ejpam-2429	8	8	∈	∈	ADJ
ejpam-2429	8	9	f	f	X
ejpam-2429	8	10	(	(	PUNCT
ejpam-2429	8	11	r	r	NOUN
ejpam-2429	8	12	)	)	PUNCT
ejpam-2429	8	13	}	}	PUNCT
ejpam-2429	8	14	.	.	PUNCT
ejpam-2429	9	1	a	a	DET
ejpam-2429	9	2	star	star	NOUN
ejpam-2429	9	3	operation	operation	NOUN
ejpam-2429	9	4	is	be	AUX
ejpam-2429	9	5	said	say	VERB
ejpam-2429	9	6	to	to	PART
ejpam-2429	9	7	be	be	AUX
ejpam-2429	9	8	of	of	ADP
ejpam-2429	9	9	finite	finite	ADJ
ejpam-2429	9	10	type	type	NOUN
ejpam-2429	9	11	if	if	SCONJ
ejpam-2429	9	12	⋆	⋆	NOUN
ejpam-2429	9	13	f	f	NOUN
ejpam-2429	9	14	=	=	PUNCT
ejpam-2429	9	15	⋆.	⋆.	AUX
ejpam-2429	9	16	since	since	SCONJ
ejpam-2429	9	17	(	(	PUNCT
ejpam-2429	9	18	⋆	⋆	X
ejpam-2429	9	19	f	f	PROPN
ejpam-2429	9	20	)	)	PUNCT
ejpam-2429	9	21	f	f	PROPN
ejpam-2429	9	22	=	=	PUNCT
ejpam-2429	9	23	⋆	⋆	PUNCT
ejpam-2429	9	24	f	f	PROPN
ejpam-2429	9	25	,	,	PUNCT
ejpam-2429	9	26	⋆	⋆	X
ejpam-2429	9	27	f	f	PROPN
ejpam-2429	9	28	is	be	AUX
ejpam-2429	9	29	a	a	DET
ejpam-2429	9	30	finite	finite	ADJ
ejpam-2429	9	31	type	type	NOUN
ejpam-2429	9	32	star	star	NOUN
ejpam-2429	9	33	operation	operation	NOUN
ejpam-2429	9	34	for	for	ADP
ejpam-2429	9	35	any	any	DET
ejpam-2429	9	36	given	give	VERB
ejpam-2429	9	37	star	star	NOUN
ejpam-2429	9	38	operation	operation	NOUN
ejpam-2429	9	39	⋆	⋆	VERB
ejpam-2429	9	40	on	on	ADP
ejpam-2429	9	41	r.	r.	PROPN
ejpam-2429	9	42	note	note	VERB
ejpam-2429	9	43	that	that	SCONJ
ejpam-2429	9	44	d	d	X
ejpam-2429	9	45	f	f	X
ejpam-2429	9	46	=	=	SYM
ejpam-2429	9	47	d	d	PROPN
ejpam-2429	9	48	,	,	PUNCT
ejpam-2429	9	49	where	where	SCONJ
ejpam-2429	9	50	d	d	NOUN
ejpam-2429	9	51	is	be	AUX
ejpam-2429	9	52	the	the	DET
ejpam-2429	9	53	identity	identity	NOUN
ejpam-2429	9	54	star	star	NOUN
ejpam-2429	9	55	operation	operation	NOUN
ejpam-2429	9	56	and	and	CCONJ
ejpam-2429	9	57	if	if	SCONJ
ejpam-2429	9	58	⋆	⋆	ADJ
ejpam-2429	9	59	is	be	AUX
ejpam-2429	9	60	the	the	DET
ejpam-2429	9	61	v	v	NOUN
ejpam-2429	9	62	-	-	PUNCT
ejpam-2429	9	63	operation	operation	NOUN
ejpam-2429	9	64	we	we	PRON
ejpam-2429	9	65	denote	denote	VERB
ejpam-2429	9	66	v	v	ADP
ejpam-2429	9	67	f	f	NOUN
ejpam-2429	9	68	:	:	PUNCT
ejpam-2429	9	69	=	=	SYM
ejpam-2429	9	70	t	t	NOUN
ejpam-2429	9	71	and	and	CCONJ
ejpam-2429	9	72	call	call	VERB
ejpam-2429	9	73	it	it	PRON
ejpam-2429	9	74	the	the	DET
ejpam-2429	9	75	t	t	NOUN
ejpam-2429	9	76	-	-	PUNCT
ejpam-2429	9	77	operation	operation	NOUN
ejpam-2429	9	78	.	.	PUNCT
ejpam-2429	10	1	a	a	DET
ejpam-2429	10	2	nonzero	nonzero	PROPN
ejpam-2429	10	3	ideal	ideal	NOUN
ejpam-2429	10	4	a	a	PRON
ejpam-2429	10	5	of	of	ADP
ejpam-2429	10	6	r	r	NOUN
ejpam-2429	10	7	is	be	AUX
ejpam-2429	10	8	a	a	DET
ejpam-2429	10	9	⋆-ideal	⋆-ideal	NOUN
ejpam-2429	10	10	if	if	SCONJ
ejpam-2429	10	11	a⋆	a⋆	NOUN
ejpam-2429	10	12	=	=	VERB
ejpam-2429	10	13	a.	a.	NOUN
ejpam-2429	10	14	similarly	similarly	ADV
ejpam-2429	10	15	,	,	PUNCT
ejpam-2429	10	16	we	we	PRON
ejpam-2429	10	17	call	call	VERB
ejpam-2429	10	18	a	a	DET
ejpam-2429	10	19	⋆-ideal	⋆-ideal	NOUN
ejpam-2429	10	20	of	of	ADP
ejpam-2429	10	21	r	r	NOUN
ejpam-2429	10	22	a	a	DET
ejpam-2429	10	23	⋆-prime	⋆-prime	NOUN
ejpam-2429	10	24	ideal	ideal	NOUN
ejpam-2429	10	25	of	of	ADP
ejpam-2429	10	26	r	r	NOUN
ejpam-2429	10	27	if	if	SCONJ
ejpam-2429	10	28	it	it	PRON
ejpam-2429	10	29	is	be	AUX
ejpam-2429	10	30	also	also	ADV
ejpam-2429	10	31	a	a	DET
ejpam-2429	10	32	prime	prime	ADJ
ejpam-2429	10	33	ideal	ideal	NOUN
ejpam-2429	10	34	.	.	PUNCT
ejpam-2429	11	1	we	we	PRON
ejpam-2429	11	2	call	call	VERB
ejpam-2429	11	3	a	a	DET
ejpam-2429	11	4	maximal	maximal	ADJ
ejpam-2429	11	5	element	element	NOUN
ejpam-2429	11	6	in	in	ADP
ejpam-2429	11	7	the	the	DET
ejpam-2429	11	8	set	set	NOUN
ejpam-2429	11	9	of	of	ADP
ejpam-2429	11	10	all	all	DET
ejpam-2429	11	11	proper	proper	ADJ
ejpam-2429	11	12	⋆-ideals	⋆-ideal	NOUN
ejpam-2429	11	13	of	of	ADP
ejpam-2429	11	14	r	r	NOUN
ejpam-2429	11	15	a	a	DET
ejpam-2429	11	16	⋆-maximal	⋆-maximal	ADJ
ejpam-2429	11	17	ideal	ideal	NOUN
ejpam-2429	11	18	of	of	ADP
ejpam-2429	11	19	r.	r.	PROPN
ejpam-2429	11	20	we	we	PRON
ejpam-2429	11	21	denote	denote	VERB
ejpam-2429	11	22	spec⋆(r	spec⋆(r	PROPN
ejpam-2429	11	23	)	)	PUNCT
ejpam-2429	11	24	the	the	DET
ejpam-2429	11	25	set	set	NOUN
ejpam-2429	11	26	of	of	ADP
ejpam-2429	11	27	all	all	DET
ejpam-2429	11	28	⋆-prime	⋆-prime	NOUN
ejpam-2429	11	29	ideals	ideal	VERB
ejpam-2429	11	30	email	email	NOUN
ejpam-2429	11	31	address	address	NOUN
ejpam-2429	11	32	:	:	PUNCT
ejpam-2429	12	1	oheubokw@svsu.edu	oheubokw@svsu.edu	X
ejpam-2429	12	2	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-2429	13	1	458	458	NUM
ejpam-2429	14	1	c	c	X
ejpam-2429	14	2	©	©	PROPN
ejpam-2429	14	3	2015	2015	NUM
ejpam-2429	14	4	ejpam	ejpam	NOUN
ejpam-2429	14	5	all	all	DET
ejpam-2429	14	6	rights	right	NOUN
ejpam-2429	14	7	reserved	reserve	VERB
ejpam-2429	14	8	.	.	PUNCT
ejpam-2429	15	1	o.	o.	PROPN
ejpam-2429	15	2	heubo	heubo	PROPN
ejpam-2429	15	3	-	-	PUNCT
ejpam-2429	15	4	kwegna	kwegna	PROPN
ejpam-2429	15	5	/	/	PUNCT
ejpam-2429	15	6	eur	eur	PROPN
ejpam-2429	15	7	.	.	PUNCT
ejpam-2429	16	1	j.	j.	PROPN
ejpam-2429	16	2	pure	pure	PROPN
ejpam-2429	16	3	appl	appl	PROPN
ejpam-2429	16	4	.	.	PROPN
ejpam-2429	16	5	math	math	PROPN
ejpam-2429	16	6	,	,	PUNCT
ejpam-2429	16	7	8	8	NUM
ejpam-2429	16	8	(	(	PUNCT
ejpam-2429	16	9	2015	2015	NUM
ejpam-2429	16	10	)	)	PUNCT
ejpam-2429	16	11	,	,	PUNCT
ejpam-2429	16	12	458	458	NUM
ejpam-2429	16	13	-	-	SYM
ejpam-2429	16	14	461	461	NUM
ejpam-2429	16	15	459	459	NUM
ejpam-2429	16	16	of	of	ADP
ejpam-2429	16	17	r	r	NOUN
ejpam-2429	16	18	and	and	CCONJ
ejpam-2429	16	19	max⋆(r	max⋆(r	NUM
ejpam-2429	16	20	)	)	PUNCT
ejpam-2429	16	21	the	the	DET
ejpam-2429	16	22	set	set	NOUN
ejpam-2429	16	23	of	of	ADP
ejpam-2429	16	24	all	all	DET
ejpam-2429	16	25	⋆-maximal	⋆-maximal	ADJ
ejpam-2429	16	26	ideals	ideal	NOUN
ejpam-2429	16	27	of	of	ADP
ejpam-2429	16	28	r.	r.	PROPN
ejpam-2429	16	29	an	an	DET
ejpam-2429	16	30	a	a	DET
ejpam-2429	16	31	∈	∈	PROPN
ejpam-2429	16	32	f	f	X
ejpam-2429	16	33	(	(	PUNCT
ejpam-2429	16	34	r	r	NOUN
ejpam-2429	16	35	)	)	PUNCT
ejpam-2429	16	36	is	be	AUX
ejpam-2429	16	37	said	say	VERB
ejpam-2429	16	38	to	to	PART
ejpam-2429	16	39	be	be	AUX
ejpam-2429	16	40	⋆-invertible	⋆-invertible	ADJ
ejpam-2429	16	41	if	if	SCONJ
ejpam-2429	16	42	(	(	PUNCT
ejpam-2429	16	43	aa−1)⋆	aa−1)⋆	PROPN
ejpam-2429	16	44	=	=	SYM
ejpam-2429	16	45	r	r	NOUN
ejpam-2429	16	46	,	,	PUNCT
ejpam-2429	16	47	whereas	whereas	SCONJ
ejpam-2429	16	48	a	a	DET
ejpam-2429	16	49	domain	domain	NOUN
ejpam-2429	16	50	r	r	NOUN
ejpam-2429	16	51	is	be	AUX
ejpam-2429	16	52	a	a	DET
ejpam-2429	16	53	prüfer	prüfer	NOUN
ejpam-2429	16	54	⋆-multiplication	⋆-multiplication	NOUN
ejpam-2429	16	55	domain	domain	NOUN
ejpam-2429	16	56	(	(	PUNCT
ejpam-2429	16	57	in	in	ADP
ejpam-2429	16	58	short	short	ADJ
ejpam-2429	16	59	,	,	PUNCT
ejpam-2429	16	60	p⋆md	p⋆md	ADJ
ejpam-2429	16	61	)	)	PUNCT
ejpam-2429	16	62	if	if	SCONJ
ejpam-2429	16	63	every	every	DET
ejpam-2429	16	64	finitely	finitely	ADV
ejpam-2429	16	65	generated	generate	VERB
ejpam-2429	16	66	ideal	ideal	NOUN
ejpam-2429	16	67	a	a	PRON
ejpam-2429	16	68	of	of	ADP
ejpam-2429	16	69	r	r	NOUN
ejpam-2429	16	70	is	be	AUX
ejpam-2429	16	71	⋆	⋆	VERB
ejpam-2429	16	72	f	f	PROPN
ejpam-2429	16	73	-invertible	-invertible	ADJ
ejpam-2429	16	74	,	,	PUNCT
ejpam-2429	16	75	i.e.	i.e.	X
ejpam-2429	16	76	,	,	PUNCT
ejpam-2429	16	77	(	(	PUNCT
ejpam-2429	16	78	aa−1)⋆	aa−1)⋆	PROPN
ejpam-2429	16	79	f	f	PROPN
ejpam-2429	16	80	=	=	SYM
ejpam-2429	16	81	r	r	NOUN
ejpam-2429	16	82	for	for	ADP
ejpam-2429	16	83	any	any	DET
ejpam-2429	16	84	a∈	a∈	PROPN
ejpam-2429	16	85	f	f	PROPN
ejpam-2429	16	86	(	(	PUNCT
ejpam-2429	16	87	r	r	NOUN
ejpam-2429	16	88	)	)	PUNCT
ejpam-2429	16	89	.	.	PUNCT
ejpam-2429	17	1	thus	thus	ADV
ejpam-2429	17	2	a	a	DET
ejpam-2429	17	3	prüfer	prüfer	NOUN
ejpam-2429	17	4	domain	domain	NOUN
ejpam-2429	17	5	is	be	AUX
ejpam-2429	17	6	a	a	DET
ejpam-2429	17	7	pdmd	pdmd	NOUN
ejpam-2429	17	8	and	and	CCONJ
ejpam-2429	17	9	pvmd	pvmd	NOUN
ejpam-2429	17	10	is	be	AUX
ejpam-2429	17	11	often	often	ADV
ejpam-2429	17	12	called	call	VERB
ejpam-2429	17	13	a	a	DET
ejpam-2429	17	14	prüfer	prüfer	NOUN
ejpam-2429	17	15	multiplication	multiplication	NOUN
ejpam-2429	17	16	domain	domain	NOUN
ejpam-2429	17	17	.	.	PUNCT
ejpam-2429	18	1	many	many	ADJ
ejpam-2429	18	2	authors	author	NOUN
ejpam-2429	18	3	have	have	AUX
ejpam-2429	18	4	previously	previously	ADV
ejpam-2429	18	5	produced	produce	VERB
ejpam-2429	18	6	several	several	ADJ
ejpam-2429	18	7	characterizations	characterization	NOUN
ejpam-2429	18	8	of	of	ADP
ejpam-2429	18	9	prüfer-⋆	prüfer-⋆	NOUN
ejpam-2429	18	10	multiplication	multiplication	NOUN
ejpam-2429	18	11	domains	domain	NOUN
ejpam-2429	18	12	(	(	PUNCT
ejpam-2429	18	13	for	for	ADP
ejpam-2429	18	14	instance	instance	NOUN
ejpam-2429	18	15	see	see	VERB
ejpam-2429	18	16	[	[	X
ejpam-2429	18	17	1–3	1–3	NOUN
ejpam-2429	18	18	,	,	PUNCT
ejpam-2429	18	19	6	6	NUM
ejpam-2429	18	20	]	]	NUM
ejpam-2429	18	21	)	)	PUNCT
ejpam-2429	18	22	.	.	PUNCT
ejpam-2429	19	1	the	the	DET
ejpam-2429	19	2	aim	aim	NOUN
ejpam-2429	19	3	of	of	ADP
ejpam-2429	19	4	this	this	DET
ejpam-2429	19	5	note	note	NOUN
ejpam-2429	19	6	is	be	AUX
ejpam-2429	19	7	to	to	PART
ejpam-2429	19	8	provide	provide	VERB
ejpam-2429	19	9	some	some	DET
ejpam-2429	19	10	new	new	ADJ
ejpam-2429	19	11	characterizations	characterization	NOUN
ejpam-2429	19	12	of	of	ADP
ejpam-2429	19	13	prüfer-⋆multiplication	prüfer-⋆multiplication	NOUN
ejpam-2429	19	14	domains	domain	NOUN
ejpam-2429	19	15	.	.	PUNCT
ejpam-2429	20	1	we	we	PRON
ejpam-2429	20	2	precisely	precisely	ADV
ejpam-2429	20	3	show	show	VERB
ejpam-2429	20	4	that	that	SCONJ
ejpam-2429	20	5	a	a	DET
ejpam-2429	20	6	domain	domain	NOUN
ejpam-2429	20	7	r	r	NOUN
ejpam-2429	20	8	is	be	AUX
ejpam-2429	20	9	a	a	DET
ejpam-2429	20	10	p⋆md	p⋆md	ADJ
ejpam-2429	20	11	if	if	SCONJ
ejpam-2429	20	12	and	and	CCONJ
ejpam-2429	20	13	only	only	ADV
ejpam-2429	20	14	if	if	SCONJ
ejpam-2429	20	15	each	each	DET
ejpam-2429	20	16	2	2	NUM
ejpam-2429	20	17	-	-	PUNCT
ejpam-2429	20	18	generated	generate	VERB
ejpam-2429	20	19	ideal	ideal	NOUN
ejpam-2429	20	20	of	of	ADP
ejpam-2429	20	21	r	r	NOUN
ejpam-2429	20	22	is	be	AUX
ejpam-2429	20	23	⋆	⋆	PUNCT
ejpam-2429	20	24	f	f	PROPN
ejpam-2429	20	25	-invertible	-invertible	PROPN
ejpam-2429	20	26	.	.	PUNCT
ejpam-2429	21	1	note	note	VERB
ejpam-2429	21	2	that	that	SCONJ
ejpam-2429	21	3	this	this	DET
ejpam-2429	21	4	result	result	NOUN
ejpam-2429	21	5	is	be	AUX
ejpam-2429	21	6	a	a	DET
ejpam-2429	21	7	generalization	generalization	NOUN
ejpam-2429	21	8	of	of	ADP
ejpam-2429	21	9	the	the	DET
ejpam-2429	21	10	fact	fact	NOUN
ejpam-2429	21	11	that	that	SCONJ
ejpam-2429	21	12	a	a	DET
ejpam-2429	21	13	domain	domain	NOUN
ejpam-2429	21	14	is	be	AUX
ejpam-2429	21	15	prüfer	prüfer	NOUN
ejpam-2429	21	16	if	if	SCONJ
ejpam-2429	21	17	and	and	CCONJ
ejpam-2429	21	18	only	only	ADV
ejpam-2429	21	19	if	if	SCONJ
ejpam-2429	21	20	each	each	DET
ejpam-2429	21	21	2	2	NUM
ejpam-2429	21	22	-	-	PUNCT
ejpam-2429	21	23	generated	generate	VERB
ejpam-2429	21	24	ideal	ideal	NOUN
ejpam-2429	21	25	is	be	AUX
ejpam-2429	21	26	invertible	invertible	ADJ
ejpam-2429	21	27	[	[	X
ejpam-2429	21	28	9	9	NUM
ejpam-2429	21	29	,	,	PUNCT
ejpam-2429	21	30	page	page	NOUN
ejpam-2429	21	31	7	7	NUM
ejpam-2429	21	32	]	]	PUNCT
ejpam-2429	21	33	.	.	PUNCT
ejpam-2429	22	1	we	we	PRON
ejpam-2429	22	2	also	also	ADV
ejpam-2429	22	3	show	show	VERB
ejpam-2429	22	4	that	that	SCONJ
ejpam-2429	22	5	a	a	DET
ejpam-2429	22	6	domain	domain	NOUN
ejpam-2429	22	7	r	r	NOUN
ejpam-2429	22	8	is	be	AUX
ejpam-2429	22	9	a	a	DET
ejpam-2429	22	10	p⋆md	p⋆md	ADJ
ejpam-2429	22	11	if	if	SCONJ
ejpam-2429	22	12	and	and	CCONJ
ejpam-2429	22	13	only	only	ADV
ejpam-2429	22	14	if	if	SCONJ
ejpam-2429	22	15	(	(	PUNCT
ejpam-2429	22	16	a	a	NOUN
ejpam-2429	22	17	)	)	PUNCT
ejpam-2429	22	18	∩	∩	NOUN
ejpam-2429	22	19	(	(	PUNCT
ejpam-2429	22	20	b	b	X
ejpam-2429	22	21	)	)	PUNCT
ejpam-2429	22	22	is	be	AUX
ejpam-2429	22	23	⋆	⋆	VERB
ejpam-2429	22	24	f	f	NOUN
ejpam-2429	22	25	-invertible	-invertible	ADJ
ejpam-2429	22	26	for	for	ADP
ejpam-2429	22	27	all	all	DET
ejpam-2429	22	28	a	a	PRON
ejpam-2429	22	29	,	,	PUNCT
ejpam-2429	22	30	b	b	X
ejpam-2429	22	31	∈	∈	PROPN
ejpam-2429	22	32	r	r	NOUN
ejpam-2429	22	33	\	\	PUNCT
ejpam-2429	22	34	{	{	PUNCT
ejpam-2429	22	35	0	0	NUM
ejpam-2429	22	36	}	}	PUNCT
ejpam-2429	22	37	.	.	PUNCT
ejpam-2429	23	1	the	the	DET
ejpam-2429	23	2	latest	late	ADJ
ejpam-2429	23	3	result	result	NOUN
ejpam-2429	23	4	has	have	AUX
ejpam-2429	23	5	also	also	ADV
ejpam-2429	23	6	been	be	AUX
ejpam-2429	23	7	shown	show	VERB
ejpam-2429	23	8	in	in	ADP
ejpam-2429	23	9	the	the	DET
ejpam-2429	23	10	v	v	NOUN
ejpam-2429	23	11	-	-	PUNCT
ejpam-2429	23	12	domain	domain	NOUN
ejpam-2429	23	13	context	context	NOUN
ejpam-2429	23	14	[	[	X
ejpam-2429	23	15	8	8	NUM
ejpam-2429	23	16	]	]	PUNCT
ejpam-2429	23	17	and	and	CCONJ
ejpam-2429	23	18	in	in	ADP
ejpam-2429	23	19	the	the	DET
ejpam-2429	23	20	pvmd	pvmd	NOUN
ejpam-2429	23	21	context	context	NOUN
ejpam-2429	24	1	[	[	X
ejpam-2429	24	2	7	7	NUM
ejpam-2429	24	3	]	]	SYM
ejpam-2429	24	4	.	.	PUNCT
ejpam-2429	25	1	2	2	X
ejpam-2429	25	2	.	.	X
ejpam-2429	25	3	main	main	ADJ
ejpam-2429	25	4	results	result	NOUN
ejpam-2429	25	5	we	we	PRON
ejpam-2429	25	6	start	start	VERB
ejpam-2429	25	7	this	this	DET
ejpam-2429	25	8	section	section	NOUN
ejpam-2429	25	9	with	with	ADP
ejpam-2429	25	10	the	the	DET
ejpam-2429	25	11	recollection	recollection	NOUN
ejpam-2429	25	12	of	of	ADP
ejpam-2429	25	13	some	some	DET
ejpam-2429	25	14	facts	fact	NOUN
ejpam-2429	25	15	about	about	ADP
ejpam-2429	25	16	star	star	NOUN
ejpam-2429	25	17	operations	operation	NOUN
ejpam-2429	25	18	.	.	PUNCT
ejpam-2429	26	1	let	let	VERB
ejpam-2429	26	2	⋆	⋆	PRON
ejpam-2429	26	3	be	be	AUX
ejpam-2429	26	4	a	a	DET
ejpam-2429	26	5	star	star	NOUN
ejpam-2429	26	6	operation	operation	NOUN
ejpam-2429	26	7	on	on	ADP
ejpam-2429	26	8	r.	r.	PROPN
ejpam-2429	26	9	recall	recall	PROPN
ejpam-2429	26	10	that	that	PRON
ejpam-2429	26	11	⋆	⋆	VERB
ejpam-2429	26	12	is	be	AUX
ejpam-2429	26	13	stable	stable	ADJ
ejpam-2429	26	14	if	if	SCONJ
ejpam-2429	26	15	(	(	PUNCT
ejpam-2429	26	16	a∩	a∩	PROPN
ejpam-2429	26	17	b)⋆	b)⋆	PROPN
ejpam-2429	26	18	=	=	SYM
ejpam-2429	26	19	a⋆∩	a⋆∩	PROPN
ejpam-2429	26	20	b⋆	b⋆	X
ejpam-2429	26	21	for	for	ADP
ejpam-2429	26	22	all	all	DET
ejpam-2429	26	23	a	a	PRON
ejpam-2429	26	24	,	,	PUNCT
ejpam-2429	26	25	b	b	X
ejpam-2429	26	26	∈	∈	X
ejpam-2429	26	27	f	f	X
ejpam-2429	26	28	(	(	PUNCT
ejpam-2429	26	29	r	r	NOUN
ejpam-2429	26	30	)	)	PUNCT
ejpam-2429	26	31	.	.	PUNCT
ejpam-2429	27	1	now	now	ADV
ejpam-2429	27	2	define	define	VERB
ejpam-2429	27	3	e⋆	e⋆	NUM
ejpam-2429	27	4	by	by	ADP
ejpam-2429	27	5	ae⋆	ae⋆	NOUN
ejpam-2429	27	6	:	:	PUNCT
ejpam-2429	27	7	=	=	SYM
ejpam-2429	27	8	∩{arm	∩{arm	PUNCT
ejpam-2429	28	1	|m	|m	NOUN
ejpam-2429	28	2	∈	∈	PROPN
ejpam-2429	28	3	max⋆	max⋆	PROPN
ejpam-2429	28	4	f	f	X
ejpam-2429	28	5	(	(	PUNCT
ejpam-2429	28	6	r	r	NOUN
ejpam-2429	28	7	)	)	PUNCT
ejpam-2429	28	8	}	}	PUNCT
ejpam-2429	28	9	,	,	PUNCT
ejpam-2429	28	10	for	for	ADP
ejpam-2429	28	11	all	all	DET
ejpam-2429	28	12	a	a	DET
ejpam-2429	28	13	∈	∈	NOUN
ejpam-2429	28	14	f	f	X
ejpam-2429	28	15	(	(	PUNCT
ejpam-2429	28	16	r	r	NOUN
ejpam-2429	28	17	)	)	PUNCT
ejpam-2429	28	18	.	.	PUNCT
ejpam-2429	29	1	then	then	ADV
ejpam-2429	29	2	it	it	PRON
ejpam-2429	29	3	is	be	AUX
ejpam-2429	29	4	well	well	ADV
ejpam-2429	29	5	known	know	VERB
ejpam-2429	29	6	that	that	SCONJ
ejpam-2429	29	7	e⋆	e⋆	PUNCT
ejpam-2429	29	8	is	be	AUX
ejpam-2429	29	9	a	a	DET
ejpam-2429	29	10	stable	stable	ADJ
ejpam-2429	29	11	star	star	NOUN
ejpam-2429	29	12	operation	operation	NOUN
ejpam-2429	29	13	on	on	ADP
ejpam-2429	29	14	r	r	NOUN
ejpam-2429	29	15	of	of	ADP
ejpam-2429	29	16	finite	finite	ADJ
ejpam-2429	29	17	type	type	NOUN
ejpam-2429	29	18	called	call	VERB
ejpam-2429	29	19	the	the	DET
ejpam-2429	29	20	stable	stable	ADJ
ejpam-2429	29	21	star	star	NOUN
ejpam-2429	29	22	operation	operation	NOUN
ejpam-2429	29	23	of	of	ADP
ejpam-2429	29	24	finite	finite	ADJ
ejpam-2429	29	25	type	type	NOUN
ejpam-2429	29	26	associated	associate	VERB
ejpam-2429	29	27	to	to	ADP
ejpam-2429	29	28	⋆.	⋆.	NUM
ejpam-2429	29	29	it	it	PRON
ejpam-2429	29	30	is	be	AUX
ejpam-2429	29	31	not	not	PART
ejpam-2429	29	32	hard	hard	ADJ
ejpam-2429	29	33	to	to	PART
ejpam-2429	29	34	see	see	VERB
ejpam-2429	29	35	that	that	DET
ejpam-2429	29	36	maxe⋆(r)=max⋆	maxe⋆(r)=max⋆	NOUN
ejpam-2429	29	37	f	f	X
ejpam-2429	29	38	(	(	PUNCT
ejpam-2429	29	39	r)[4	r)[4	NOUN
ejpam-2429	29	40	,	,	PUNCT
ejpam-2429	29	41	corollary	corollary	ADJ
ejpam-2429	29	42	3.5(2	3.5(2	NUM
ejpam-2429	29	43	)	)	PUNCT
ejpam-2429	29	44	]	]	PUNCT
ejpam-2429	29	45	.	.	PUNCT
ejpam-2429	30	1	from	from	ADP
ejpam-2429	30	2	the	the	DET
ejpam-2429	30	3	latest	late	ADJ
ejpam-2429	30	4	fact	fact	NOUN
ejpam-2429	30	5	,	,	PUNCT
ejpam-2429	30	6	it	it	PRON
ejpam-2429	30	7	then	then	ADV
ejpam-2429	30	8	follows	follow	VERB
ejpam-2429	30	9	that	that	SCONJ
ejpam-2429	30	10	an	an	DET
ejpam-2429	30	11	ideal	ideal	NOUN
ejpam-2429	30	12	a	a	PRON
ejpam-2429	30	13	is	be	AUX
ejpam-2429	30	14	e⋆-invertible	e⋆-invertible	ADJ
ejpam-2429	30	15	if	if	SCONJ
ejpam-2429	30	16	and	and	CCONJ
ejpam-2429	30	17	only	only	ADV
ejpam-2429	30	18	if	if	SCONJ
ejpam-2429	30	19	it	it	PRON
ejpam-2429	30	20	is	be	AUX
ejpam-2429	30	21	⋆	⋆	PUNCT
ejpam-2429	31	1	f	f	X
ejpam-2429	31	2	-invertible	-invertible	ADJ
ejpam-2429	31	3	(	(	PUNCT
ejpam-2429	31	4	in	in	ADP
ejpam-2429	31	5	fact	fact	NOUN
ejpam-2429	31	6	,	,	PUNCT
ejpam-2429	31	7	if	if	SCONJ
ejpam-2429	31	8	a	a	DET
ejpam-2429	31	9	star	star	NOUN
ejpam-2429	31	10	operation	operation	NOUN
ejpam-2429	31	11	⋆	⋆	NOUN
ejpam-2429	31	12	is	be	AUX
ejpam-2429	31	13	of	of	ADP
ejpam-2429	31	14	finite	finite	ADJ
ejpam-2429	31	15	type	type	NOUN
ejpam-2429	31	16	,	,	PUNCT
ejpam-2429	31	17	then	then	ADV
ejpam-2429	31	18	(	(	PUNCT
ejpam-2429	31	19	aa−1)⋆	aa−1)⋆	PROPN
ejpam-2429	31	20	=	=	PUNCT
ejpam-2429	31	21	r	r	NOUN
ejpam-2429	31	22	if	if	SCONJ
ejpam-2429	31	23	and	and	CCONJ
ejpam-2429	31	24	only	only	ADV
ejpam-2429	31	25	if	if	SCONJ
ejpam-2429	31	26	aa−1	aa−1	PROPN
ejpam-2429	31	27	6⊆	6⊆	NUM
ejpam-2429	31	28	m	m	VERB
ejpam-2429	31	29	for	for	ADP
ejpam-2429	31	30	all	all	DET
ejpam-2429	31	31	m	m	PROPN
ejpam-2429	31	32	∈	∈	PROPN
ejpam-2429	31	33	max⋆(r	max⋆(r	NOUN
ejpam-2429	31	34	)	)	PUNCT
ejpam-2429	31	35	)	)	PUNCT
ejpam-2429	31	36	.	.	PUNCT
ejpam-2429	32	1	from	from	ADP
ejpam-2429	32	2	this	this	DET
ejpam-2429	32	3	observation	observation	NOUN
ejpam-2429	32	4	it	it	PRON
ejpam-2429	32	5	then	then	ADV
ejpam-2429	32	6	follows	follow	VERB
ejpam-2429	32	7	that	that	SCONJ
ejpam-2429	32	8	p⋆md	p⋆md	ADJ
ejpam-2429	32	9	,	,	PUNCT
ejpam-2429	32	10	p⋆	p⋆	PROPN
ejpam-2429	32	11	f	f	PROPN
ejpam-2429	32	12	md	md	PROPN
ejpam-2429	32	13	,	,	PUNCT
ejpam-2429	32	14	and	and	CCONJ
ejpam-2429	32	15	pe⋆md	pe⋆md	DET
ejpam-2429	32	16	coincide	coincide	NOUN
ejpam-2429	32	17	.	.	PUNCT
ejpam-2429	33	1	lemma	lemma	PROPN
ejpam-2429	33	2	1	1	X
ejpam-2429	33	3	.	.	PUNCT
ejpam-2429	34	1	let	let	VERB
ejpam-2429	34	2	a	a	DET
ejpam-2429	34	3	be	be	AUX
ejpam-2429	34	4	a	a	DET
ejpam-2429	34	5	finitely	finitely	ADV
ejpam-2429	34	6	generated	generate	VERB
ejpam-2429	34	7	ideal	ideal	NOUN
ejpam-2429	34	8	of	of	ADP
ejpam-2429	34	9	r	r	NOUN
ejpam-2429	34	10	and	and	CCONJ
ejpam-2429	34	11	⋆	⋆	VERB
ejpam-2429	34	12	a	a	DET
ejpam-2429	34	13	star	star	NOUN
ejpam-2429	34	14	operation	operation	NOUN
ejpam-2429	34	15	on	on	ADP
ejpam-2429	34	16	r.	r.	PROPN
ejpam-2429	34	17	if	if	SCONJ
ejpam-2429	34	18	a	a	PRON
ejpam-2429	34	19	is	be	AUX
ejpam-2429	34	20	⋆	⋆	ADJ
ejpam-2429	35	1	f	f	PROPN
ejpam-2429	35	2	-invertible	-invertible	ADJ
ejpam-2429	35	3	,	,	PUNCT
ejpam-2429	35	4	then	then	ADV
ejpam-2429	35	5	arm	arm	NOUN
ejpam-2429	35	6	is	be	AUX
ejpam-2429	35	7	principal	principal	ADJ
ejpam-2429	35	8	for	for	ADP
ejpam-2429	35	9	every	every	DET
ejpam-2429	35	10	m	m	PROPN
ejpam-2429	35	11	∈	∈	PROPN
ejpam-2429	35	12	max⋆	max⋆	PROPN
ejpam-2429	35	13	f	f	X
ejpam-2429	35	14	(	(	PUNCT
ejpam-2429	35	15	r	r	NOUN
ejpam-2429	35	16	)	)	PUNCT
ejpam-2429	35	17	.	.	PUNCT
ejpam-2429	36	1	proof	proof	NOUN
ejpam-2429	36	2	.	.	PUNCT
ejpam-2429	37	1	suppose	suppose	VERB
ejpam-2429	37	2	that	that	SCONJ
ejpam-2429	37	3	a	a	PRON
ejpam-2429	37	4	is	be	AUX
ejpam-2429	37	5	⋆	⋆	ADJ
ejpam-2429	37	6	f	f	NOUN
ejpam-2429	37	7	-invertible	-invertible	PROPN
ejpam-2429	37	8	.	.	PUNCT
ejpam-2429	38	1	from	from	ADP
ejpam-2429	38	2	the	the	DET
ejpam-2429	38	3	above	above	ADJ
ejpam-2429	38	4	observation	observation	NOUN
ejpam-2429	38	5	,	,	PUNCT
ejpam-2429	38	6	it	it	PRON
ejpam-2429	38	7	follows	follow	VERB
ejpam-2429	38	8	that	that	SCONJ
ejpam-2429	38	9	a	a	PRON
ejpam-2429	38	10	is	be	AUX
ejpam-2429	38	11	e⋆-invertible	e⋆-invertible	ADJ
ejpam-2429	38	12	,	,	PUNCT
ejpam-2429	38	13	i.e.	i.e.	X
ejpam-2429	38	14	,	,	PUNCT
ejpam-2429	38	15	(	(	PUNCT
ejpam-2429	38	16	aa−1)e⋆	aa−1)e⋆	PROPN
ejpam-2429	39	1	=	=	SYM
ejpam-2429	39	2	r.	r.	NOUN
ejpam-2429	39	3	we	we	PRON
ejpam-2429	39	4	have	have	VERB
ejpam-2429	39	5	,	,	PUNCT
ejpam-2429	39	6	for	for	ADP
ejpam-2429	39	7	each	each	DET
ejpam-2429	39	8	maximal	maximal	ADJ
ejpam-2429	39	9	⋆	⋆	PUNCT
ejpam-2429	39	10	f	f	PROPN
ejpam-2429	39	11	-ideal	-ideal	PROPN
ejpam-2429	39	12	m	m	PROPN
ejpam-2429	39	13	,	,	PUNCT
ejpam-2429	39	14	rm	rm	PROPN
ejpam-2429	39	15	=	=	PUNCT
ejpam-2429	39	16	(	(	PUNCT
ejpam-2429	39	17	aa−1)e⋆rm	aa−1)e⋆rm	VERB
ejpam-2429	39	18	=	=	SYM
ejpam-2429	39	19	⋂	⋂	PROPN
ejpam-2429	39	20	{	{	PUNCT
ejpam-2429	39	21	(	(	PUNCT
ejpam-2429	39	22	aa−1)rn	aa−1)rn	NOUN
ejpam-2429	39	23	|n	|n	NOUN
ejpam-2429	39	24	∈max⋆	∈max⋆	VERB
ejpam-2429	39	25	f	f	PROPN
ejpam-2429	39	26	(	(	PUNCT
ejpam-2429	39	27	r)}rm	r)}rm	NOUN
ejpam-2429	39	28	=	=	SYM
ejpam-2429	39	29	(	(	PUNCT
ejpam-2429	39	30	aa−1)rm	aa−1)rm	VERB
ejpam-2429	39	31	[	[	X
ejpam-2429	39	32	4	4	NUM
ejpam-2429	39	33	,	,	PUNCT
ejpam-2429	39	34	lemma	lemma	PROPN
ejpam-2429	39	35	2.4.(1	2.4.(1	NUM
ejpam-2429	39	36	)	)	PUNCT
ejpam-2429	39	37	]	]	PUNCT
ejpam-2429	39	38	.	.	PUNCT
ejpam-2429	40	1	thus	thus	ADV
ejpam-2429	40	2	arm	arm	NOUN
ejpam-2429	40	3	is	be	AUX
ejpam-2429	40	4	invertible	invertible	ADJ
ejpam-2429	40	5	and	and	CCONJ
ejpam-2429	40	6	therefore	therefore	ADV
ejpam-2429	40	7	principal	principal	ADJ
ejpam-2429	40	8	.	.	PUNCT
ejpam-2429	41	1	theorem	theorem	NOUN
ejpam-2429	41	2	1	1	NUM
ejpam-2429	41	3	.	.	PUNCT
ejpam-2429	42	1	let	let	VERB
ejpam-2429	42	2	r	r	NOUN
ejpam-2429	42	3	be	be	AUX
ejpam-2429	42	4	an	an	DET
ejpam-2429	42	5	integral	integral	ADJ
ejpam-2429	42	6	domain	domain	NOUN
ejpam-2429	42	7	and	and	CCONJ
ejpam-2429	42	8	let	let	VERB
ejpam-2429	42	9	⋆	⋆	PUNCT
ejpam-2429	42	10	be	be	AUX
ejpam-2429	42	11	a	a	DET
ejpam-2429	42	12	star	star	NOUN
ejpam-2429	42	13	operation	operation	NOUN
ejpam-2429	42	14	on	on	ADP
ejpam-2429	42	15	r.	r.	PROPN
ejpam-2429	42	16	then	then	ADV
ejpam-2429	42	17	the	the	DET
ejpam-2429	42	18	following	follow	VERB
ejpam-2429	42	19	statements	statement	NOUN
ejpam-2429	42	20	are	be	AUX
ejpam-2429	42	21	equivalent	equivalent	ADJ
ejpam-2429	42	22	for	for	ADP
ejpam-2429	42	23	an	an	DET
ejpam-2429	42	24	integral	integral	ADJ
ejpam-2429	42	25	domain	domain	NOUN
ejpam-2429	42	26	r.	r.	NOUN
ejpam-2429	42	27	(	(	PUNCT
ejpam-2429	42	28	i	i	NOUN
ejpam-2429	42	29	)	)	PUNCT
ejpam-2429	42	30	rm	rm	PROPN
ejpam-2429	42	31	is	be	AUX
ejpam-2429	42	32	a	a	DET
ejpam-2429	42	33	valuation	valuation	NOUN
ejpam-2429	42	34	domain	domain	NOUN
ejpam-2429	42	35	for	for	ADP
ejpam-2429	42	36	all	all	DET
ejpam-2429	42	37	m	m	PROPN
ejpam-2429	42	38	∈	∈	PROPN
ejpam-2429	42	39	max⋆	max⋆	PROPN
ejpam-2429	42	40	f	f	X
ejpam-2429	42	41	(	(	PUNCT
ejpam-2429	42	42	r	r	NOUN
ejpam-2429	42	43	)	)	PUNCT
ejpam-2429	42	44	.	.	PUNCT
ejpam-2429	43	1	(	(	PUNCT
ejpam-2429	43	2	ii	ii	X
ejpam-2429	43	3	)	)	PUNCT
ejpam-2429	43	4	r	r	NOUN
ejpam-2429	43	5	is	be	AUX
ejpam-2429	43	6	a	a	DET
ejpam-2429	43	7	p⋆md	p⋆md	NOUN
ejpam-2429	43	8	.	.	PUNCT
ejpam-2429	44	1	(	(	PUNCT
ejpam-2429	44	2	iii	iii	X
ejpam-2429	44	3	)	)	PUNCT
ejpam-2429	44	4	every	every	DET
ejpam-2429	44	5	nonzero	nonzero	NOUN
ejpam-2429	44	6	fractional	fractional	ADJ
ejpam-2429	44	7	finitely	finitely	ADV
ejpam-2429	44	8	generated	generate	VERB
ejpam-2429	44	9	ideal	ideal	NOUN
ejpam-2429	44	10	of	of	ADP
ejpam-2429	44	11	r	r	NOUN
ejpam-2429	44	12	is	be	AUX
ejpam-2429	44	13	⋆	⋆	ADJ
ejpam-2429	44	14	f	f	PROPN
ejpam-2429	44	15	-invertible	-invertible	PROPN
ejpam-2429	44	16	.	.	PUNCT
ejpam-2429	45	1	(	(	PUNCT
ejpam-2429	45	2	iv	iv	X
ejpam-2429	45	3	)	)	PUNCT
ejpam-2429	45	4	every	every	DET
ejpam-2429	45	5	nonzero	nonzero	NOUN
ejpam-2429	45	6	fractional	fractional	ADJ
ejpam-2429	45	7	2	2	NUM
ejpam-2429	45	8	-	-	PUNCT
ejpam-2429	45	9	generated	generate	VERB
ejpam-2429	45	10	ideal	ideal	NOUN
ejpam-2429	45	11	is	be	AUX
ejpam-2429	45	12	⋆	⋆	VERB
ejpam-2429	45	13	f	f	PROPN
ejpam-2429	45	14	-invertible	-invertible	PROPN
ejpam-2429	45	15	.	.	PUNCT
ejpam-2429	46	1	references	reference	NOUN
ejpam-2429	46	2	460	460	NUM
ejpam-2429	46	3	proof	proof	NOUN
ejpam-2429	46	4	.	.	PUNCT
ejpam-2429	47	1	for	for	ADP
ejpam-2429	47	2	(	(	PUNCT
ejpam-2429	47	3	i)⇔	i)⇔	PROPN
ejpam-2429	47	4	(	(	PUNCT
ejpam-2429	47	5	ii	ii	NOUN
ejpam-2429	47	6	)	)	PUNCT
ejpam-2429	47	7	(	(	PUNCT
ejpam-2429	47	8	see	see	VERB
ejpam-2429	47	9	[	[	X
ejpam-2429	47	10	1	1	NUM
ejpam-2429	47	11	,	,	PUNCT
ejpam-2429	47	12	corollary	corollary	ADJ
ejpam-2429	47	13	1.2	1.2	NUM
ejpam-2429	47	14	]	]	PUNCT
ejpam-2429	47	15	)	)	PUNCT
ejpam-2429	47	16	.	.	PUNCT
ejpam-2429	48	1	(	(	PUNCT
ejpam-2429	48	2	ii	ii	NOUN
ejpam-2429	48	3	)	)	PUNCT
ejpam-2429	48	4	⇒	⇒	NOUN
ejpam-2429	48	5	(	(	PUNCT
ejpam-2429	48	6	iii	iii	NOUN
ejpam-2429	48	7	)	)	PUNCT
ejpam-2429	48	8	and	and	CCONJ
ejpam-2429	48	9	(	(	PUNCT
ejpam-2429	48	10	iii	iii	X
ejpam-2429	48	11	)	)	PUNCT
ejpam-2429	48	12	⇒	⇒	NOUN
ejpam-2429	48	13	(	(	PUNCT
ejpam-2429	48	14	iv	iv	X
ejpam-2429	48	15	)	)	PUNCT
ejpam-2429	48	16	are	be	AUX
ejpam-2429	48	17	clear	clear	ADJ
ejpam-2429	48	18	.	.	PUNCT
ejpam-2429	49	1	so	so	ADV
ejpam-2429	49	2	it	it	PRON
ejpam-2429	49	3	remains	remain	VERB
ejpam-2429	49	4	to	to	PART
ejpam-2429	49	5	prove	prove	VERB
ejpam-2429	49	6	that	that	SCONJ
ejpam-2429	49	7	(	(	PUNCT
ejpam-2429	49	8	iv)⇒	iv)⇒	X
ejpam-2429	49	9	(	(	PUNCT
ejpam-2429	49	10	i	i	NOUN
ejpam-2429	49	11	)	)	PUNCT
ejpam-2429	49	12	.	.	PUNCT
ejpam-2429	50	1	let	let	VERB
ejpam-2429	50	2	x	x	PRON
ejpam-2429	50	3	,	,	PUNCT
ejpam-2429	50	4	y	y	PROPN
ejpam-2429	50	5	∈	∈	PROPN
ejpam-2429	50	6	r	r	NOUN
ejpam-2429	50	7	,	,	PUNCT
ejpam-2429	50	8	note	note	VERB
ejpam-2429	50	9	that	that	SCONJ
ejpam-2429	50	10	if	if	SCONJ
ejpam-2429	50	11	p	p	NOUN
ejpam-2429	50	12	is	be	AUX
ejpam-2429	50	13	a	a	DET
ejpam-2429	50	14	prime	prime	ADJ
ejpam-2429	50	15	ideal	ideal	NOUN
ejpam-2429	50	16	of	of	ADP
ejpam-2429	50	17	r	r	NOUN
ejpam-2429	50	18	,	,	PUNCT
ejpam-2429	50	19	we	we	PRON
ejpam-2429	50	20	have	have	VERB
ejpam-2429	50	21	xrp	xrp	PROPN
ejpam-2429	50	22	+	+	CCONJ
ejpam-2429	50	23	yrp	yrp	NOUN
ejpam-2429	51	1	=	=	SYM
ejpam-2429	51	2	(	(	PUNCT
ejpam-2429	51	3	a	a	PRON
ejpam-2429	51	4	,	,	PUNCT
ejpam-2429	51	5	b)rp	b)rp	PROPN
ejpam-2429	51	6	for	for	ADP
ejpam-2429	51	7	some	some	DET
ejpam-2429	51	8	a	a	PRON
ejpam-2429	51	9	,	,	PUNCT
ejpam-2429	51	10	b	b	PROPN
ejpam-2429	51	11	∈	∈	PROPN
ejpam-2429	51	12	r.	r.	NOUN
ejpam-2429	52	1	but	but	CCONJ
ejpam-2429	52	2	if	if	SCONJ
ejpam-2429	52	3	p	p	NOUN
ejpam-2429	52	4	is	be	AUX
ejpam-2429	52	5	a	a	DET
ejpam-2429	52	6	⋆	⋆	NOUN
ejpam-2429	52	7	f	f	NOUN
ejpam-2429	52	8	-maximal	-maximal	ADJ
ejpam-2429	52	9	ideal	ideal	NOUN
ejpam-2429	52	10	of	of	ADP
ejpam-2429	52	11	r	r	NOUN
ejpam-2429	52	12	then	then	ADV
ejpam-2429	52	13	,	,	PUNCT
ejpam-2429	52	14	by	by	ADP
ejpam-2429	52	15	lemma	lemma	PROPN
ejpam-2429	52	16	1	1	NUM
ejpam-2429	52	17	,	,	PUNCT
ejpam-2429	52	18	(	(	PUNCT
ejpam-2429	52	19	a	a	PRON
ejpam-2429	52	20	,	,	PUNCT
ejpam-2429	52	21	b)rp	b)rp	PROPN
ejpam-2429	52	22	is	be	AUX
ejpam-2429	52	23	principal	principal	ADJ
ejpam-2429	52	24	,	,	PUNCT
ejpam-2429	52	25	that	that	ADV
ejpam-2429	52	26	is	is	ADV
ejpam-2429	52	27	,	,	PUNCT
ejpam-2429	52	28	rp	rp	NOUN
ejpam-2429	52	29	is	be	AUX
ejpam-2429	52	30	a	a	DET
ejpam-2429	52	31	valuation	valuation	NOUN
ejpam-2429	52	32	domain	domain	NOUN
ejpam-2429	52	33	.	.	PUNCT
ejpam-2429	53	1	corollary	corollary	ADJ
ejpam-2429	53	2	1	1	NUM
ejpam-2429	53	3	.	.	PUNCT
ejpam-2429	54	1	a	a	DET
ejpam-2429	54	2	domain	domain	NOUN
ejpam-2429	54	3	r	r	NOUN
ejpam-2429	54	4	is	be	AUX
ejpam-2429	54	5	a	a	DET
ejpam-2429	54	6	p⋆md	p⋆md	ADJ
ejpam-2429	54	7	if	if	SCONJ
ejpam-2429	54	8	and	and	CCONJ
ejpam-2429	54	9	only	only	ADV
ejpam-2429	54	10	if	if	SCONJ
ejpam-2429	54	11	(	(	PUNCT
ejpam-2429	54	12	a)∩	a)∩	X
ejpam-2429	54	13	(	(	PUNCT
ejpam-2429	54	14	b	b	X
ejpam-2429	54	15	)	)	PUNCT
ejpam-2429	54	16	is	be	AUX
ejpam-2429	54	17	⋆	⋆	VERB
ejpam-2429	54	18	f	f	NOUN
ejpam-2429	54	19	-invertible	-invertible	ADJ
ejpam-2429	54	20	for	for	ADP
ejpam-2429	54	21	all	all	DET
ejpam-2429	54	22	a	a	PRON
ejpam-2429	54	23	,	,	PUNCT
ejpam-2429	54	24	b	b	X
ejpam-2429	54	25	∈	∈	PROPN
ejpam-2429	54	26	r	r	NOUN
ejpam-2429	54	27	\	\	PUNCT
ejpam-2429	54	28	{	{	PUNCT
ejpam-2429	54	29	0	0	NUM
ejpam-2429	54	30	}	}	PUNCT
ejpam-2429	54	31	.	.	PUNCT
ejpam-2429	55	1	proof	proof	NOUN
ejpam-2429	55	2	.	.	PUNCT
ejpam-2429	56	1	note	note	VERB
ejpam-2429	56	2	that	that	SCONJ
ejpam-2429	56	3	we	we	PRON
ejpam-2429	56	4	have	have	VERB
ejpam-2429	56	5	(	(	PUNCT
ejpam-2429	56	6	ab)−1[(a)∩	ab)−1[(a)∩	NOUN
ejpam-2429	56	7	(	(	PUNCT
ejpam-2429	56	8	b	b	NOUN
ejpam-2429	56	9	)	)	PUNCT
ejpam-2429	56	10	]	]	PUNCT
ejpam-2429	57	1	=	=	PUNCT
ejpam-2429	57	2	(	(	PUNCT
ejpam-2429	57	3	a	a	PRON
ejpam-2429	57	4	,	,	PUNCT
ejpam-2429	57	5	b)−1	b)−1	NOUN
ejpam-2429	57	6	.	.	PUNCT
ejpam-2429	58	1	so	so	ADV
ejpam-2429	58	2	(	(	PUNCT
ejpam-2429	58	3	ab)−1[(a	ab)−1[(a	ADJ
ejpam-2429	58	4	)	)	PUNCT
ejpam-2429	58	5	∩	∩	NOUN
ejpam-2429	58	6	(	(	PUNCT
ejpam-2429	58	7	b)](a	b)](a	PROPN
ejpam-2429	58	8	,	,	PUNCT
ejpam-2429	58	9	b	b	NOUN
ejpam-2429	58	10	)	)	PUNCT
ejpam-2429	58	11	=	=	SYM
ejpam-2429	58	12	(	(	PUNCT
ejpam-2429	58	13	a	a	PRON
ejpam-2429	58	14	,	,	PUNCT
ejpam-2429	58	15	b)−1(a	b)−1(a	PROPN
ejpam-2429	58	16	,	,	PUNCT
ejpam-2429	58	17	b	b	NOUN
ejpam-2429	58	18	)	)	PUNCT
ejpam-2429	58	19	and	and	CCONJ
ejpam-2429	58	20	�	�	PROPN
ejpam-2429	58	21	(	(	PUNCT
ejpam-2429	58	22	ab)−1[(a)∩	ab)−1[(a)∩	PROPN
ejpam-2429	58	23	(	(	PUNCT
ejpam-2429	58	24	b)](a	b)](a	PROPN
ejpam-2429	58	25	,	,	PUNCT
ejpam-2429	58	26	b	b	X
ejpam-2429	58	27	)	)	PUNCT
ejpam-2429	58	28	�	�	PROPN
ejpam-2429	58	29	⋆	⋆	PUNCT
ejpam-2429	58	30	f	f	PROPN
ejpam-2429	58	31	=	=	SYM
ejpam-2429	58	32	�	�	PROPN
ejpam-2429	58	33	(	(	PUNCT
ejpam-2429	58	34	a	a	PRON
ejpam-2429	58	35	,	,	PUNCT
ejpam-2429	58	36	b)−1(a	b)−1(a	PROPN
ejpam-2429	58	37	,	,	PUNCT
ejpam-2429	58	38	b	b	NOUN
ejpam-2429	58	39	)	)	PUNCT
ejpam-2429	58	40	�	�	PROPN
ejpam-2429	58	41	⋆	⋆	PUNCT
ejpam-2429	58	42	f	f	PROPN
ejpam-2429	58	43	.	.	PUNCT
ejpam-2429	59	1	thus	thus	ADV
ejpam-2429	59	2	if	if	SCONJ
ejpam-2429	59	3	a	a	PRON
ejpam-2429	59	4	,	,	PUNCT
ejpam-2429	59	5	b	b	X
ejpam-2429	59	6	∈	∈	PROPN
ejpam-2429	59	7	r	r	NOUN
ejpam-2429	59	8	\	\	PUNCT
ejpam-2429	59	9	{	{	PUNCT
ejpam-2429	59	10	0	0	NUM
ejpam-2429	59	11	}	}	PUNCT
ejpam-2429	59	12	,	,	PUNCT
ejpam-2429	59	13	(	(	PUNCT
ejpam-2429	59	14	a)∩	a)∩	X
ejpam-2429	59	15	(	(	PUNCT
ejpam-2429	59	16	b	b	X
ejpam-2429	59	17	)	)	PUNCT
ejpam-2429	59	18	is	be	AUX
ejpam-2429	59	19	⋆	⋆	VERB
ejpam-2429	59	20	f	f	PROPN
ejpam-2429	59	21	invertible	invertible	ADJ
ejpam-2429	59	22	if	if	SCONJ
ejpam-2429	59	23	and	and	CCONJ
ejpam-2429	59	24	only	only	ADV
ejpam-2429	59	25	if	if	SCONJ
ejpam-2429	59	26	(	(	PUNCT
ejpam-2429	59	27	a	a	DET
ejpam-2429	59	28	,	,	PUNCT
ejpam-2429	59	29	b	b	NOUN
ejpam-2429	59	30	)	)	PUNCT
ejpam-2429	59	31	is	be	AUX
ejpam-2429	59	32	⋆	⋆	VERB
ejpam-2429	59	33	f	f	PROPN
ejpam-2429	59	34	-invertible	-invertible	ADJ
ejpam-2429	59	35	.	.	PUNCT
ejpam-2429	60	1	hence	hence	ADV
ejpam-2429	60	2	r	r	NOUN
ejpam-2429	60	3	is	be	AUX
ejpam-2429	60	4	a	a	DET
ejpam-2429	60	5	p⋆md	p⋆md	ADJ
ejpam-2429	60	6	if	if	SCONJ
ejpam-2429	60	7	and	and	CCONJ
ejpam-2429	60	8	only	only	ADV
ejpam-2429	60	9	if	if	SCONJ
ejpam-2429	60	10	(	(	PUNCT
ejpam-2429	60	11	a)∩	a)∩	X
ejpam-2429	60	12	(	(	PUNCT
ejpam-2429	60	13	b	b	X
ejpam-2429	60	14	)	)	PUNCT
ejpam-2429	60	15	is	be	AUX
ejpam-2429	60	16	⋆	⋆	VERB
ejpam-2429	60	17	f	f	NOUN
ejpam-2429	60	18	-invertible	-invertible	ADJ
ejpam-2429	60	19	for	for	ADP
ejpam-2429	60	20	all	all	DET
ejpam-2429	60	21	a	a	PRON
ejpam-2429	60	22	,	,	PUNCT
ejpam-2429	60	23	b	b	X
ejpam-2429	60	24	∈	∈	PROPN
ejpam-2429	61	1	r	r	NOUN
ejpam-2429	61	2	\	\	PUNCT
ejpam-2429	61	3	{	{	PUNCT
ejpam-2429	61	4	0	0	NUM
ejpam-2429	61	5	}	}	PUNCT
ejpam-2429	61	6	by	by	ADP
ejpam-2429	61	7	theorem	theorem	NOUN
ejpam-2429	61	8	1(iv	1(iv	NUM
ejpam-2429	61	9	)	)	PUNCT
ejpam-2429	61	10	.	.	PUNCT
ejpam-2429	62	1	recall	recall	VERB
ejpam-2429	62	2	that	that	SCONJ
ejpam-2429	62	3	a	a	DET
ejpam-2429	62	4	⋆-ideal	⋆-ideal	NOUN
ejpam-2429	62	5	a	a	PRON
ejpam-2429	62	6	of	of	ADP
ejpam-2429	62	7	r	r	NOUN
ejpam-2429	62	8	is	be	AUX
ejpam-2429	62	9	of	of	ADP
ejpam-2429	62	10	finite	finite	ADJ
ejpam-2429	62	11	type	type	NOUN
ejpam-2429	62	12	if	if	SCONJ
ejpam-2429	62	13	a=	a=	PROPN
ejpam-2429	62	14	(	(	PUNCT
ejpam-2429	62	15	a1	a1	NOUN
ejpam-2429	62	16	,	,	PUNCT
ejpam-2429	62	17	.	.	PUNCT
ejpam-2429	62	18	.	.	PUNCT
ejpam-2429	63	1	.	.	PUNCT
ejpam-2429	64	1	,	,	PUNCT
ejpam-2429	64	2	an	an	X
ejpam-2429	64	3	)	)	PUNCT
ejpam-2429	64	4	⋆	⋆	NOUN
ejpam-2429	64	5	for	for	ADP
ejpam-2429	64	6	some	some	PRON
ejpam-2429	64	7	(	(	PUNCT
ejpam-2429	64	8	0	0	NUM
ejpam-2429	64	9	)	)	PUNCT
ejpam-2429	64	10	6=	6=	NUM
ejpam-2429	64	11	(	(	PUNCT
ejpam-2429	64	12	a1	a1	NOUN
ejpam-2429	64	13	,	,	PUNCT
ejpam-2429	64	14	.	.	PUNCT
ejpam-2429	64	15	.	.	PUNCT
ejpam-2429	65	1	.	.	PUNCT
ejpam-2429	66	1	,	,	PUNCT
ejpam-2429	66	2	an	an	X
ejpam-2429	66	3	)	)	PUNCT
ejpam-2429	66	4	⊆	⊆	NUM
ejpam-2429	66	5	a.	a.	NOUN
ejpam-2429	66	6	note	note	NOUN
ejpam-2429	66	7	that	that	SCONJ
ejpam-2429	66	8	if	if	SCONJ
ejpam-2429	66	9	⋆=	⋆=	PRON
ejpam-2429	66	10	⋆	⋆	AUX
ejpam-2429	66	11	f	f	PROPN
ejpam-2429	66	12	,	,	PUNCT
ejpam-2429	66	13	then	then	ADV
ejpam-2429	66	14	a⋆	a⋆	ADV
ejpam-2429	66	15	is	be	AUX
ejpam-2429	66	16	of	of	ADP
ejpam-2429	66	17	finite	finite	ADJ
ejpam-2429	66	18	type	type	NOUN
ejpam-2429	67	1	if	if	SCONJ
ejpam-2429	67	2	and	and	CCONJ
ejpam-2429	67	3	only	only	ADV
ejpam-2429	67	4	if	if	SCONJ
ejpam-2429	67	5	a⋆	a⋆	ADV
ejpam-2429	67	6	=	=	SYM
ejpam-2429	67	7	(	(	PUNCT
ejpam-2429	67	8	a1	a1	PROPN
ejpam-2429	67	9	,	,	PUNCT
ejpam-2429	67	10	.	.	PUNCT
ejpam-2429	67	11	.	.	PUNCT
ejpam-2429	68	1	.	.	PUNCT
ejpam-2429	69	1	,	,	PUNCT
ejpam-2429	69	2	an	an	X
ejpam-2429	69	3	)	)	PUNCT
ejpam-2429	69	4	⋆	⋆	NOUN
ejpam-2429	69	5	for	for	ADP
ejpam-2429	69	6	some	some	PRON
ejpam-2429	69	7	(	(	PUNCT
ejpam-2429	69	8	0	0	NUM
ejpam-2429	69	9	)	)	PUNCT
ejpam-2429	69	10	6=	6=	NUM
ejpam-2429	69	11	(	(	PUNCT
ejpam-2429	69	12	a1	a1	NOUN
ejpam-2429	69	13	,	,	PUNCT
ejpam-2429	69	14	.	.	PUNCT
ejpam-2429	69	15	.	.	PUNCT
ejpam-2429	70	1	.	.	PUNCT
ejpam-2429	71	1	,	,	PUNCT
ejpam-2429	71	2	an	an	X
ejpam-2429	71	3	)	)	PUNCT
ejpam-2429	71	4	⊆	⊆	NUM
ejpam-2429	71	5	a.	a.	NOUN
ejpam-2429	71	6	if	if	SCONJ
ejpam-2429	71	7	⋆	⋆	ADJ
ejpam-2429	71	8	is	be	AUX
ejpam-2429	71	9	a	a	DET
ejpam-2429	71	10	star	star	NOUN
ejpam-2429	71	11	operation	operation	NOUN
ejpam-2429	71	12	of	of	ADP
ejpam-2429	71	13	finite	finite	PROPN
ejpam-2429	71	14	type	type	NOUN
ejpam-2429	71	15	,	,	PUNCT
ejpam-2429	71	16	then	then	ADV
ejpam-2429	71	17	a	a	DET
ejpam-2429	71	18	⋆-invertible	⋆-invertible	ADJ
ejpam-2429	71	19	ideal	ideal	NOUN
ejpam-2429	71	20	is	be	AUX
ejpam-2429	71	21	of	of	ADP
ejpam-2429	71	22	finite	finite	ADJ
ejpam-2429	71	23	type	type	NOUN
ejpam-2429	71	24	.	.	PUNCT
ejpam-2429	72	1	also	also	ADV
ejpam-2429	72	2	note	note	VERB
ejpam-2429	72	3	that	that	SCONJ
ejpam-2429	72	4	from	from	ADP
ejpam-2429	72	5	[	[	X
ejpam-2429	72	6	5	5	NUM
ejpam-2429	72	7	,	,	PUNCT
ejpam-2429	72	8	proposition	proposition	NOUN
ejpam-2429	72	9	32.2(b	32.2(b	NUM
ejpam-2429	72	10	)	)	PUNCT
ejpam-2429	72	11	]	]	PUNCT
ejpam-2429	72	12	and	and	CCONJ
ejpam-2429	72	13	the	the	DET
ejpam-2429	72	14	fact	fact	NOUN
ejpam-2429	72	15	that	that	SCONJ
ejpam-2429	72	16	(	(	PUNCT
ejpam-2429	72	17	z)⋆	z)⋆	PROPN
ejpam-2429	72	18	=	=	SYM
ejpam-2429	72	19	(	(	PUNCT
ejpam-2429	72	20	z	z	NOUN
ejpam-2429	72	21	)	)	PUNCT
ejpam-2429	72	22	for	for	ADP
ejpam-2429	72	23	any	any	DET
ejpam-2429	72	24	z	z	PROPN
ejpam-2429	72	25	∈	∈	PROPN
ejpam-2429	72	26	k	k	NOUN
ejpam-2429	72	27	,	,	PUNCT
ejpam-2429	72	28	we	we	PRON
ejpam-2429	72	29	have	have	AUX
ejpam-2429	72	30	(	(	PUNCT
ejpam-2429	72	31	(	(	PUNCT
ejpam-2429	72	32	a)∩	a)∩	X
ejpam-2429	72	33	(	(	PUNCT
ejpam-2429	72	34	b))⋆	b))⋆	NOUN
ejpam-2429	72	35	=	=	X
ejpam-2429	72	36	(	(	PUNCT
ejpam-2429	72	37	a	a	NOUN
ejpam-2429	72	38	)	)	PUNCT
ejpam-2429	72	39	∩	∩	NOUN
ejpam-2429	72	40	(	(	PUNCT
ejpam-2429	72	41	b	b	NOUN
ejpam-2429	72	42	)	)	PUNCT
ejpam-2429	72	43	for	for	ADP
ejpam-2429	72	44	any	any	DET
ejpam-2429	72	45	star	star	NOUN
ejpam-2429	72	46	operation	operation	NOUN
ejpam-2429	72	47	⋆	⋆	VERB
ejpam-2429	72	48	on	on	ADP
ejpam-2429	72	49	r.	r.	PROPN
ejpam-2429	72	50	thus	thus	ADV
ejpam-2429	72	51	(	(	PUNCT
ejpam-2429	72	52	a)∩	a)∩	X
ejpam-2429	72	53	(	(	PUNCT
ejpam-2429	72	54	b	b	X
ejpam-2429	72	55	)	)	PUNCT
ejpam-2429	72	56	is	be	AUX
ejpam-2429	72	57	a	a	DET
ejpam-2429	72	58	⋆-ideal	⋆-ideal	NOUN
ejpam-2429	72	59	of	of	ADP
ejpam-2429	72	60	r	r	NOUN
ejpam-2429	72	61	for	for	ADP
ejpam-2429	72	62	all	all	DET
ejpam-2429	72	63	a	a	PRON
ejpam-2429	72	64	,	,	PUNCT
ejpam-2429	72	65	b	b	X
ejpam-2429	72	66	∈	∈	PROPN
ejpam-2429	72	67	r	r	NOUN
ejpam-2429	72	68	\	\	PUNCT
ejpam-2429	72	69	{	{	PUNCT
ejpam-2429	72	70	0	0	NUM
ejpam-2429	72	71	}	}	PUNCT
ejpam-2429	72	72	.	.	PUNCT
ejpam-2429	73	1	corollary	corollary	ADJ
ejpam-2429	73	2	2	2	NUM
ejpam-2429	73	3	.	.	PUNCT
ejpam-2429	74	1	let	let	VERB
ejpam-2429	74	2	r	r	PRON
ejpam-2429	74	3	be	be	AUX
ejpam-2429	74	4	an	an	DET
ejpam-2429	74	5	integral	integral	ADJ
ejpam-2429	74	6	domain	domain	NOUN
ejpam-2429	74	7	such	such	ADJ
ejpam-2429	74	8	that	that	DET
ejpam-2429	74	9	�	�	PROPN
ejpam-2429	74	10	(	(	PUNCT
ejpam-2429	74	11	ab)−1[(a)∩	ab)−1[(a)∩	PROPN
ejpam-2429	74	12	(	(	PUNCT
ejpam-2429	74	13	b)](a	b)](a	PROPN
ejpam-2429	74	14	,	,	PUNCT
ejpam-2429	74	15	b	b	X
ejpam-2429	74	16	)	)	PUNCT
ejpam-2429	74	17	�	�	PROPN
ejpam-2429	74	18	⋆	⋆	PUNCT
ejpam-2429	74	19	=	=	SYM
ejpam-2429	74	20	r.	r.	PROPN
ejpam-2429	74	21	then	then	ADV
ejpam-2429	74	22	r	r	NOUN
ejpam-2429	74	23	is	be	AUX
ejpam-2429	74	24	a	a	DET
ejpam-2429	74	25	p⋆md	p⋆md	ADJ
ejpam-2429	74	26	if	if	SCONJ
ejpam-2429	75	1	and	and	CCONJ
ejpam-2429	75	2	only	only	ADV
ejpam-2429	75	3	if	if	SCONJ
ejpam-2429	75	4	(	(	PUNCT
ejpam-2429	75	5	a)∩	a)∩	X
ejpam-2429	75	6	(	(	PUNCT
ejpam-2429	75	7	b	b	NOUN
ejpam-2429	75	8	)	)	PUNCT
ejpam-2429	75	9	is	be	AUX
ejpam-2429	75	10	of	of	ADP
ejpam-2429	75	11	finite	finite	ADJ
ejpam-2429	75	12	type	type	NOUN
ejpam-2429	75	13	.	.	PUNCT
ejpam-2429	76	1	proof	proof	NOUN
ejpam-2429	76	2	.	.	PUNCT
ejpam-2429	77	1	suppose	suppose	VERB
ejpam-2429	77	2	that	that	SCONJ
ejpam-2429	77	3	r	r	NOUN
ejpam-2429	77	4	is	be	AUX
ejpam-2429	77	5	a	a	DET
ejpam-2429	77	6	p⋆md	p⋆md	NOUN
ejpam-2429	77	7	.	.	PUNCT
ejpam-2429	78	1	then	then	ADV
ejpam-2429	78	2	(	(	PUNCT
ejpam-2429	78	3	a)∩(b	a)∩(b	X
ejpam-2429	78	4	)	)	PUNCT
ejpam-2429	78	5	is	be	AUX
ejpam-2429	78	6	⋆	⋆	PUNCT
ejpam-2429	78	7	f	f	NOUN
ejpam-2429	78	8	-invertible	-invertible	ADJ
ejpam-2429	78	9	by	by	ADP
ejpam-2429	78	10	corollary	corollary	ADJ
ejpam-2429	78	11	1	1	NUM
ejpam-2429	78	12	.	.	PUNCT
ejpam-2429	79	1	so	so	ADV
ejpam-2429	79	2	(	(	PUNCT
ejpam-2429	79	3	a)∩(b	a)∩(b	PROPN
ejpam-2429	79	4	)	)	PUNCT
ejpam-2429	79	5	is	be	AUX
ejpam-2429	79	6	of	of	ADP
ejpam-2429	79	7	finite	finite	ADJ
ejpam-2429	79	8	type	type	NOUN
ejpam-2429	79	9	following	follow	VERB
ejpam-2429	79	10	the	the	DET
ejpam-2429	79	11	above	above	ADJ
ejpam-2429	79	12	discussion	discussion	NOUN
ejpam-2429	79	13	.	.	PUNCT
ejpam-2429	80	1	conversely	conversely	ADV
ejpam-2429	80	2	if	if	SCONJ
ejpam-2429	80	3	we	we	PRON
ejpam-2429	80	4	assume	assume	VERB
ejpam-2429	80	5	that	that	SCONJ
ejpam-2429	80	6	(	(	PUNCT
ejpam-2429	80	7	a	a	NOUN
ejpam-2429	80	8	)	)	PUNCT
ejpam-2429	80	9	∩	∩	NOUN
ejpam-2429	80	10	(	(	PUNCT
ejpam-2429	80	11	b	b	NOUN
ejpam-2429	80	12	)	)	PUNCT
ejpam-2429	80	13	is	be	AUX
ejpam-2429	80	14	of	of	ADP
ejpam-2429	80	15	finite	finite	ADJ
ejpam-2429	80	16	type	type	NOUN
ejpam-2429	80	17	,	,	PUNCT
ejpam-2429	80	18	then	then	ADV
ejpam-2429	80	19	from	from	ADP
ejpam-2429	80	20	�	�	PROPN
ejpam-2429	80	21	(	(	PUNCT
ejpam-2429	80	22	ab)−1[(a)∩	ab)−1[(a)∩	PROPN
ejpam-2429	80	23	(	(	PUNCT
ejpam-2429	80	24	b)](a	b)](a	PROPN
ejpam-2429	80	25	,	,	PUNCT
ejpam-2429	80	26	b	b	X
ejpam-2429	80	27	)	)	PUNCT
ejpam-2429	80	28	�	�	PROPN
ejpam-2429	80	29	⋆	⋆	NOUN
ejpam-2429	80	30	=	=	SYM
ejpam-2429	80	31	r	r	NOUN
ejpam-2429	80	32	,	,	PUNCT
ejpam-2429	80	33	it	it	PRON
ejpam-2429	80	34	follows	follow	VERB
ejpam-2429	80	35	that	that	SCONJ
ejpam-2429	80	36	(	(	PUNCT
ejpam-2429	80	37	a	a	NOUN
ejpam-2429	80	38	)	)	PUNCT
ejpam-2429	80	39	∩	∩	NOUN
ejpam-2429	80	40	(	(	PUNCT
ejpam-2429	80	41	b	b	X
ejpam-2429	80	42	)	)	PUNCT
ejpam-2429	80	43	is	be	AUX
ejpam-2429	80	44	⋆	⋆	VERB
ejpam-2429	80	45	f	f	NOUN
ejpam-2429	80	46	-invertible	-invertible	ADJ
ejpam-2429	80	47	and	and	CCONJ
ejpam-2429	80	48	hence	hence	ADV
ejpam-2429	80	49	r	r	NOUN
ejpam-2429	80	50	is	be	AUX
ejpam-2429	80	51	a	a	DET
ejpam-2429	80	52	p⋆md	p⋆md	ADJ
ejpam-2429	80	53	by	by	ADP
ejpam-2429	80	54	corollary	corollary	ADJ
ejpam-2429	80	55	1	1	NUM
ejpam-2429	80	56	.	.	PUNCT
ejpam-2429	80	57	remark	remark	NOUN
ejpam-2429	80	58	1	1	NUM
ejpam-2429	80	59	.	.	PUNCT
ejpam-2429	80	60	note	note	VERB
ejpam-2429	80	61	that	that	SCONJ
ejpam-2429	80	62	the	the	DET
ejpam-2429	80	63	preceding	precede	VERB
ejpam-2429	80	64	theorem	theorem	NOUN
ejpam-2429	80	65	and	and	CCONJ
ejpam-2429	80	66	corollaries	corollary	NOUN
ejpam-2429	80	67	give	give	VERB
ejpam-2429	80	68	new	new	ADJ
ejpam-2429	80	69	characterizations	characterization	NOUN
ejpam-2429	80	70	of	of	ADP
ejpam-2429	80	71	prüfer	prüfer	NOUN
ejpam-2429	80	72	⋆-multiplication	⋆-multiplication	NOUN
ejpam-2429	80	73	domains	domain	NOUN
ejpam-2429	80	74	which	which	PRON
ejpam-2429	80	75	generalize	generalize	VERB
ejpam-2429	80	76	some	some	PRON
ejpam-2429	80	77	of	of	ADP
ejpam-2429	80	78	the	the	DET
ejpam-2429	80	79	classical	classical	ADJ
ejpam-2429	80	80	characterizations	characterization	NOUN
ejpam-2429	80	81	of	of	ADP
ejpam-2429	80	82	prüfer	prüfer	NOUN
ejpam-2429	80	83	vmultiplication	vmultiplication	NOUN
ejpam-2429	80	84	domains	domain	NOUN
ejpam-2429	80	85	(	(	PUNCT
ejpam-2429	80	86	see	see	VERB
ejpam-2429	80	87	[	[	X
ejpam-2429	80	88	7	7	NUM
ejpam-2429	80	89	,	,	PUNCT
ejpam-2429	80	90	lemma	lemma	PROPN
ejpam-2429	80	91	1.7	1.7	NUM
ejpam-2429	80	92	,	,	PUNCT
ejpam-2429	80	93	corollary	corollary	ADJ
ejpam-2429	80	94	1.8	1.8	NUM
ejpam-2429	80	95	,	,	PUNCT
ejpam-2429	80	96	and	and	CCONJ
ejpam-2429	80	97	corollary	corollary	ADJ
ejpam-2429	80	98	1.9	1.9	NUM
ejpam-2429	80	99	]	]	PUNCT
ejpam-2429	80	100	)	)	PUNCT
ejpam-2429	80	101	.	.	PUNCT
ejpam-2429	81	1	acknowledgements	acknowledgement	VERB
ejpam-2429	81	2	the	the	DET
ejpam-2429	81	3	author	author	NOUN
ejpam-2429	81	4	wishes	wish	VERB
ejpam-2429	81	5	to	to	PART
ejpam-2429	81	6	express	express	VERB
ejpam-2429	81	7	his	his	PRON
ejpam-2429	81	8	gratitude	gratitude	NOUN
ejpam-2429	81	9	to	to	ADP
ejpam-2429	81	10	bruce	bruce	PROPN
ejpam-2429	81	11	olberding	olberde	VERB
ejpam-2429	81	12	for	for	ADP
ejpam-2429	81	13	bringing	bring	VERB
ejpam-2429	81	14	up	up	ADP
ejpam-2429	81	15	the	the	DET
ejpam-2429	81	16	problem	problem	NOUN
ejpam-2429	81	17	treated	treat	VERB
ejpam-2429	81	18	in	in	ADP
ejpam-2429	81	19	this	this	DET
ejpam-2429	81	20	paper	paper	NOUN
ejpam-2429	81	21	following	follow	VERB
ejpam-2429	81	22	some	some	DET
ejpam-2429	81	23	discussions	discussion	NOUN
ejpam-2429	81	24	on	on	ADP
ejpam-2429	81	25	prüfer-⋆multiplication	prüfer-⋆multiplication	NOUN
ejpam-2429	81	26	domains	domain	NOUN
ejpam-2429	81	27	.	.	PUNCT
ejpam-2429	82	1	references	reference	NOUN
ejpam-2429	82	2	[	[	X
ejpam-2429	82	3	1	1	X
ejpam-2429	82	4	]	]	PUNCT
ejpam-2429	82	5	d.	d.	PROPN
ejpam-2429	82	6	d.	d.	PROPN
ejpam-2429	82	7	anderson	anderson	PROPN
ejpam-2429	82	8	,	,	PUNCT
ejpam-2429	82	9	d.	d.	PROPN
ejpam-2429	82	10	f.	f.	PROPN
ejpam-2429	82	11	anderson	anderson	PROPN
ejpam-2429	82	12	,	,	PUNCT
ejpam-2429	82	13	m.	m.	PROPN
ejpam-2429	82	14	fontana	fontana	PROPN
ejpam-2429	82	15	,	,	PUNCT
ejpam-2429	82	16	and	and	CCONJ
ejpam-2429	82	17	m.	m.	PROPN
ejpam-2429	82	18	zafrullah	zafrullah	PROPN
ejpam-2429	82	19	.	.	PUNCT
ejpam-2429	83	1	on	on	ADP
ejpam-2429	83	2	v	v	NOUN
ejpam-2429	83	3	-	-	PUNCT
ejpam-2429	83	4	domains	domain	NOUN
ejpam-2429	83	5	and	and	CCONJ
ejpam-2429	83	6	star	star	NOUN
ejpam-2429	83	7	operations	operation	NOUN
ejpam-2429	83	8	,	,	PUNCT
ejpam-2429	83	9	communications	communication	NOUN
ejpam-2429	83	10	in	in	ADP
ejpam-2429	83	11	algebra	algebra	NOUN
ejpam-2429	83	12	,	,	PUNCT
ejpam-2429	83	13	2	2	NUM
ejpam-2429	83	14	,	,	PUNCT
ejpam-2429	83	15	141	141	NUM
ejpam-2429	83	16	-	-	SYM
ejpam-2429	83	17	145	145	NUM
ejpam-2429	83	18	.	.	PUNCT
ejpam-2429	83	19	2008	2008	NUM
ejpam-2429	83	20	.	.	PUNCT
ejpam-2429	84	1	[	[	X
ejpam-2429	84	2	2	2	X
ejpam-2429	84	3	]	]	PUNCT
ejpam-2429	84	4	d.	d.	PROPN
ejpam-2429	84	5	f.	f.	PROPN
ejpam-2429	84	6	anderson	anderson	PROPN
ejpam-2429	84	7	,	,	PUNCT
ejpam-2429	84	8	m.	m.	PROPN
ejpam-2429	84	9	fontana	fontana	PROPN
ejpam-2429	84	10	,	,	PUNCT
ejpam-2429	84	11	and	and	CCONJ
ejpam-2429	84	12	m.	m.	PROPN
ejpam-2429	84	13	zafrullah	zafrullah	PROPN
ejpam-2429	84	14	.	.	PUNCT
ejpam-2429	85	1	some	some	DET
ejpam-2429	85	2	remarks	remark	NOUN
ejpam-2429	85	3	on	on	ADP
ejpam-2429	85	4	prüfer	prüfer	NOUN
ejpam-2429	85	5	⋆-multiplication	⋆-multiplication	NOUN
ejpam-2429	85	6	domains	domain	NOUN
ejpam-2429	85	7	and	and	CCONJ
ejpam-2429	85	8	class	class	NOUN
ejpam-2429	85	9	groups	group	NOUN
ejpam-2429	85	10	,	,	PUNCT
ejpam-2429	85	11	journal	journal	NOUN
ejpam-2429	85	12	of	of	ADP
ejpam-2429	85	13	algebra	algebra	PROPN
ejpam-2429	85	14	,	,	PUNCT
ejpam-2429	85	15	319	319	NUM
ejpam-2429	85	16	,	,	PUNCT
ejpam-2429	85	17	272	272	NUM
ejpam-2429	85	18	-	-	SYM
ejpam-2429	85	19	295	295	NUM
ejpam-2429	85	20	.	.	PUNCT
ejpam-2429	85	21	2008	2008	NUM
ejpam-2429	85	22	.	.	PUNCT
ejpam-2429	86	1	[	[	X
ejpam-2429	86	2	3	3	X
ejpam-2429	86	3	]	]	X
ejpam-2429	86	4	g.	g.	PROPN
ejpam-2429	86	5	w.	w.	PROPN
ejpam-2429	86	6	chang	chang	PROPN
ejpam-2429	86	7	.	.	PUNCT
ejpam-2429	87	1	prüfer	prüfer	NOUN
ejpam-2429	87	2	∗-multiplication	∗-multiplication	NOUN
ejpam-2429	87	3	domains	domain	NOUN
ejpam-2429	87	4	,	,	PUNCT
ejpam-2429	87	5	nagata	nagata	NOUN
ejpam-2429	87	6	rings	ring	NOUN
ejpam-2429	87	7	,	,	PUNCT
ejpam-2429	87	8	and	and	CCONJ
ejpam-2429	87	9	kroneker	kroneker	PROPN
ejpam-2429	87	10	function	function	PROPN
ejpam-2429	87	11	rings	ring	NOUN
ejpam-2429	87	12	,	,	PUNCT
ejpam-2429	87	13	journal	journal	NOUN
ejpam-2429	87	14	of	of	ADP
ejpam-2429	87	15	algebra	algebra	PROPN
ejpam-2429	87	16	,	,	PUNCT
ejpam-2429	87	17	319(1	319(1	NUM
ejpam-2429	87	18	)	)	PUNCT
ejpam-2429	87	19	,	,	PUNCT
ejpam-2429	87	20	309	309	NUM
ejpam-2429	87	21	-	-	SYM
ejpam-2429	87	22	319	319	NUM
ejpam-2429	87	23	.	.	PUNCT
ejpam-2429	87	24	2008	2008	NUM
ejpam-2429	87	25	.	.	PUNCT
ejpam-2429	88	1	references	reference	NOUN
ejpam-2429	88	2	461	461	NUM
ejpam-2429	88	3	[	[	X
ejpam-2429	88	4	4	4	NUM
ejpam-2429	88	5	]	]	PUNCT
ejpam-2429	88	6	m.	m.	PROPN
ejpam-2429	88	7	fontana	fontana	PROPN
ejpam-2429	88	8	and	and	CCONJ
ejpam-2429	88	9	k.a	k.a	PROPN
ejpam-2429	88	10	.	.	PROPN
ejpam-2429	88	11	loper	loper	PROPN
ejpam-2429	88	12	.	.	PUNCT
ejpam-2429	89	1	nagata	nagata	PROPN
ejpam-2429	89	2	rings	rings	PROPN
ejpam-2429	89	3	.	.	PUNCT
ejpam-2429	90	1	kronecker	kronecker	NOUN
ejpam-2429	90	2	function	function	NOUN
ejpam-2429	90	3	rings	ring	NOUN
ejpam-2429	90	4	and	and	CCONJ
ejpam-2429	90	5	related	relate	VERB
ejpam-2429	90	6	smistar	smistar	NOUN
ejpam-2429	90	7	operations	operation	NOUN
ejpam-2429	90	8	,	,	PUNCT
ejpam-2429	90	9	communications	communication	NOUN
ejpam-2429	90	10	in	in	ADP
ejpam-2429	90	11	algebra	algebra	NOUN
ejpam-2429	90	12	,	,	PUNCT
ejpam-2429	90	13	31	31	NUM
ejpam-2429	90	14	,	,	PUNCT
ejpam-2429	90	15	4775	4775	NUM
ejpam-2429	90	16	-	-	SYM
ejpam-2429	90	17	4805	4805	NUM
ejpam-2429	90	18	.	.	PUNCT
ejpam-2429	91	1	2003	2003	NUM
ejpam-2429	91	2	.	.	PUNCT
ejpam-2429	92	1	[	[	X
ejpam-2429	92	2	5	5	NUM
ejpam-2429	92	3	]	]	PUNCT
ejpam-2429	92	4	r.	r.	PROPN
ejpam-2429	92	5	gilmer	gilmer	PROPN
ejpam-2429	92	6	.	.	PUNCT
ejpam-2429	93	1	multiplicative	multiplicative	PROPN
ejpam-2429	93	2	ideal	ideal	PROPN
ejpam-2429	93	3	theory	theory	NOUN
ejpam-2429	93	4	,	,	PUNCT
ejpam-2429	93	5	corrected	correct	VERB
ejpam-2429	93	6	reprint	reprint	NOUN
ejpam-2429	93	7	of	of	ADP
ejpam-2429	93	8	the	the	DET
ejpam-2429	93	9	1972	1972	NUM
ejpam-2429	93	10	edition	edition	NOUN
ejpam-2429	93	11	.	.	PUNCT
ejpam-2429	94	1	queen	queen	PROPN
ejpam-2429	94	2	’s	’s	PART
ejpam-2429	94	3	papers	paper	NOUN
ejpam-2429	94	4	in	in	ADP
ejpam-2429	94	5	pure	pure	ADJ
ejpam-2429	94	6	and	and	CCONJ
ejpam-2429	94	7	applied	applied	ADJ
ejpam-2429	94	8	mathematics	mathematic	NOUN
ejpam-2429	94	9	,	,	PUNCT
ejpam-2429	94	10	90	90	NUM
ejpam-2429	94	11	.	.	PUNCT
ejpam-2429	95	1	queen	queen	PROPN
ejpam-2429	95	2	’s	’s	PART
ejpam-2429	95	3	university	university	PROPN
ejpam-2429	95	4	,	,	PUNCT
ejpam-2429	95	5	kingston	kingston	PROPN
ejpam-2429	95	6	,	,	PUNCT
ejpam-2429	95	7	on	on	ADP
ejpam-2429	95	8	,	,	PUNCT
ejpam-2429	95	9	1992	1992	NUM
ejpam-2429	95	10	.	.	PUNCT
ejpam-2429	96	1	[	[	X
ejpam-2429	96	2	6	6	NUM
ejpam-2429	96	3	]	]	PUNCT
ejpam-2429	96	4	e.	e.	PROPN
ejpam-2429	96	5	g.	g.	PROPN
ejpam-2429	96	6	houston	houston	PROPN
ejpam-2429	96	7	,	,	PUNCT
ejpam-2429	96	8	s.	s.	PROPN
ejpam-2429	96	9	b.	b.	PROPN
ejpam-2429	96	10	malik	malik	PROPN
ejpam-2429	96	11	,	,	PUNCT
ejpam-2429	96	12	and	and	CCONJ
ejpam-2429	96	13	j.l	j.l	PROPN
ejpam-2429	96	14	.	.	PROPN
ejpam-2429	96	15	mott	mott	PROPN
ejpam-2429	96	16	.	.	PUNCT
ejpam-2429	97	1	characterization	characterization	NOUN
ejpam-2429	97	2	of	of	ADP
ejpam-2429	97	3	⋆-multiplication	⋆-multiplication	NOUN
ejpam-2429	97	4	domains	domain	NOUN
ejpam-2429	97	5	,	,	PUNCT
ejpam-2429	97	6	canadian	canadian	ADJ
ejpam-2429	97	7	mathematics	mathematic	NOUN
ejpam-2429	97	8	bulletin	bulletin	NOUN
ejpam-2429	97	9	,	,	PUNCT
ejpam-2429	97	10	27	27	NUM
ejpam-2429	97	11	,	,	PUNCT
ejpam-2429	97	12	48	48	NUM
ejpam-2429	97	13	-	-	SYM
ejpam-2429	97	14	52	52	NUM
ejpam-2429	97	15	.	.	NOUN
ejpam-2429	97	16	1984	1984	NUM
ejpam-2429	97	17	.	.	PUNCT
ejpam-2429	98	1	[	[	X
ejpam-2429	98	2	7	7	X
ejpam-2429	98	3	]	]	X
ejpam-2429	98	4	s.	s.	PROPN
ejpam-2429	98	5	malik	malik	PROPN
ejpam-2429	98	6	,	,	PUNCT
ejpam-2429	98	7	j.	j.	PROPN
ejpam-2429	98	8	mott	mott	PROPN
ejpam-2429	98	9	,	,	PUNCT
ejpam-2429	98	10	and	and	CCONJ
ejpam-2429	98	11	m.	m.	PROPN
ejpam-2429	98	12	zafrullah	zafrullah	PROPN
ejpam-2429	98	13	.	.	PUNCT
ejpam-2429	99	1	on	on	ADP
ejpam-2429	99	2	t	t	PROPN
ejpam-2429	99	3	-	-	PUNCT
ejpam-2429	99	4	invertibility	invertibility	NOUN
ejpam-2429	99	5	,	,	PUNCT
ejpam-2429	99	6	communications	communication	NOUN
ejpam-2429	99	7	in	in	ADP
ejpam-2429	99	8	algebra	algebra	NOUN
ejpam-2429	99	9	,	,	PUNCT
ejpam-2429	99	10	16	16	NUM
ejpam-2429	99	11	,	,	PUNCT
ejpam-2429	99	12	149	149	NUM
ejpam-2429	99	13	-	-	SYM
ejpam-2429	99	14	170	170	NUM
ejpam-2429	99	15	.	.	PUNCT
ejpam-2429	99	16	1988	1988	NUM
ejpam-2429	99	17	.	.	PUNCT
ejpam-2429	100	1	[	[	X
ejpam-2429	100	2	8	8	X
ejpam-2429	100	3	]	]	PUNCT
ejpam-2429	100	4	j.	j.	PROPN
ejpam-2429	100	5	mott	mott	PROPN
ejpam-2429	100	6	,	,	PUNCT
ejpam-2429	100	7	b.	b.	PROPN
ejpam-2429	100	8	nashier	nashier	PROPN
ejpam-2429	100	9	,	,	PUNCT
ejpam-2429	100	10	and	and	CCONJ
ejpam-2429	100	11	m.	m.	PROPN
ejpam-2429	100	12	zafrullah	zafrullah	PROPN
ejpam-2429	100	13	.	.	PUNCT
ejpam-2429	101	1	contents	content	NOUN
ejpam-2429	101	2	of	of	ADP
ejpam-2429	101	3	polynomials	polynomial	NOUN
ejpam-2429	101	4	and	and	CCONJ
ejpam-2429	101	5	invertibility	invertibility	NOUN
ejpam-2429	101	6	,	,	PUNCT
ejpam-2429	101	7	communications	communication	NOUN
ejpam-2429	101	8	in	in	ADP
ejpam-2429	101	9	algebra	algebra	NOUN
ejpam-2429	101	10	,	,	PUNCT
ejpam-2429	101	11	18(5	18(5	NUM
ejpam-2429	101	12	)	)	PUNCT
ejpam-2429	101	13	,	,	PUNCT
ejpam-2429	101	14	1569	1569	NUM
ejpam-2429	101	15	-	-	SYM
ejpam-2429	101	16	1583	1583	NUM
ejpam-2429	101	17	.	.	PUNCT
ejpam-2429	102	1	1990	1990	NUM
ejpam-2429	102	2	.	.	PUNCT
ejpam-2429	103	1	[	[	X
ejpam-2429	103	2	9	9	NUM
ejpam-2429	103	3	]	]	X
ejpam-2429	103	4	h.	h.	NOUN
ejpam-2429	103	5	prüfer	prüfer	NOUN
ejpam-2429	103	6	.	.	PUNCT
ejpam-2429	104	1	untersuchen	untersuchen	PROPN
ejpam-2429	104	2	über	über	PROPN
ejpam-2429	104	3	teilbarkeitseigenshafen	teilbarkeitseigenshafen	NOUN
ejpam-2429	104	4	in	in	ADP
ejpam-2429	104	5	körper	körper	NOUN
ejpam-2429	104	6	,	,	PUNCT
ejpam-2429	104	7	journal	journal	NOUN
ejpam-2429	104	8	of	of	ADP
ejpam-2429	104	9	reine	reine	PROPN
ejpam-2429	104	10	angew	angew	PROPN
ejpam-2429	104	11	of	of	ADP
ejpam-2429	104	12	mathematics	mathematic	NOUN
ejpam-2429	104	13	,	,	PUNCT
ejpam-2429	104	14	168	168	NUM
ejpam-2429	104	15	,	,	PUNCT
ejpam-2429	104	16	1	1	NUM
ejpam-2429	104	17	-	-	SYM
ejpam-2429	104	18	36	36	NUM
ejpam-2429	104	19	.	.	PUNCT
ejpam-2429	104	20	1932	1932	NUM
ejpam-2429	104	21	.	.	PUNCT