id	sid	tid	token	lemma	pos
ejpam-2589	1	1	compile	compile	NOUN
ejpam-2589	1	2	/	/	SYM
ejpam-2589	1	3	output.dvi	output.dvi	NOUN
ejpam-2589	1	4	european	european	ADJ
ejpam-2589	1	5	journal	journal	NOUN
ejpam-2589	1	6	of	of	ADP
ejpam-2589	1	7	pure	pure	ADJ
ejpam-2589	1	8	and	and	CCONJ
ejpam-2589	1	9	applied	apply	VERB
ejpam-2589	1	10	mathematics	mathematic	NOUN
ejpam-2589	1	11	vol	vol	NOUN
ejpam-2589	1	12	.	.	PROPN
ejpam-2589	2	1	9	9	NUM
ejpam-2589	2	2	,	,	PUNCT
ejpam-2589	2	3	no	no	INTJ
ejpam-2589	2	4	.	.	NOUN
ejpam-2589	2	5	1	1	NUM
ejpam-2589	2	6	,	,	PUNCT
ejpam-2589	2	7	2016	2016	NUM
ejpam-2589	2	8	,	,	PUNCT
ejpam-2589	2	9	1	1	NUM
ejpam-2589	2	10	-	-	SYM
ejpam-2589	2	11	2	2	NUM
ejpam-2589	2	12	issn	issn	PROPN
ejpam-2589	2	13	1307	1307	NUM
ejpam-2589	2	14	-	-	SYM
ejpam-2589	2	15	5543	5543	NUM
ejpam-2589	2	16	–	–	PUNCT
ejpam-2589	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-2589	2	18	a	a	DET
ejpam-2589	2	19	note	note	NOUN
ejpam-2589	2	20	on	on	ADP
ejpam-2589	2	21	prüfer	prüfer	NOUN
ejpam-2589	2	22	⋆-multiplication	⋆-multiplication	NOUN
ejpam-2589	2	23	domains	domain	NOUN
ejpam-2589	2	24	ii	ii	PROPN
ejpam-2589	2	25	olivier	olivier	PROPN
ejpam-2589	2	26	a.	a.	PROPN
ejpam-2589	2	27	heubo	heubo	PROPN
ejpam-2589	2	28	-	-	PUNCT
ejpam-2589	2	29	kwegna	kwegna	PROPN
ejpam-2589	2	30	department	department	PROPN
ejpam-2589	2	31	of	of	ADP
ejpam-2589	2	32	mathematical	mathematical	ADJ
ejpam-2589	2	33	sciences	sciences	PROPN
ejpam-2589	2	34	,	,	PUNCT
ejpam-2589	2	35	saginaw	saginaw	NOUN
ejpam-2589	2	36	valley	valley	PROPN
ejpam-2589	2	37	state	state	PROPN
ejpam-2589	2	38	university	university	PROPN
ejpam-2589	2	39	,	,	PUNCT
ejpam-2589	2	40	university	university	NOUN
ejpam-2589	2	41	center	center	NOUN
ejpam-2589	2	42	mi	mi	PROPN
ejpam-2589	2	43	48710	48710	NUM
ejpam-2589	2	44	,	,	PUNCT
ejpam-2589	2	45	usa	usa	PROPN
ejpam-2589	2	46	abstract	abstract	NOUN
ejpam-2589	2	47	.	.	PUNCT
ejpam-2589	3	1	we	we	PRON
ejpam-2589	3	2	bring	bring	VERB
ejpam-2589	3	3	some	some	DET
ejpam-2589	3	4	corrections	correction	NOUN
ejpam-2589	3	5	to	to	PART
ejpam-2589	3	6	corollary	corollary	VERB
ejpam-2589	3	7	1	1	NUM
ejpam-2589	3	8	of	of	ADP
ejpam-2589	3	9	[	[	X
ejpam-2589	3	10	3	3	NUM
ejpam-2589	3	11	]	]	PUNCT
ejpam-2589	3	12	.	.	PUNCT
ejpam-2589	4	1	in	in	ADP
ejpam-2589	4	2	[	[	X
ejpam-2589	4	3	3	3	NUM
ejpam-2589	4	4	]	]	PUNCT
ejpam-2589	4	5	,	,	PUNCT
ejpam-2589	4	6	we	we	PRON
ejpam-2589	4	7	attempted	attempt	VERB
ejpam-2589	4	8	to	to	PART
ejpam-2589	4	9	show	show	VERB
ejpam-2589	4	10	that	that	SCONJ
ejpam-2589	4	11	for	for	ADP
ejpam-2589	4	12	an	an	DET
ejpam-2589	4	13	arbitrary	arbitrary	ADJ
ejpam-2589	4	14	star	star	NOUN
ejpam-2589	4	15	operation	operation	NOUN
ejpam-2589	4	16	⋆	⋆	VERB
ejpam-2589	4	17	on	on	ADP
ejpam-2589	4	18	a	a	DET
ejpam-2589	4	19	domain	domain	NOUN
ejpam-2589	4	20	r	r	NOUN
ejpam-2589	4	21	,	,	PUNCT
ejpam-2589	4	22	the	the	DET
ejpam-2589	4	23	domain	domain	NOUN
ejpam-2589	4	24	r	r	NOUN
ejpam-2589	4	25	is	be	AUX
ejpam-2589	4	26	a	a	DET
ejpam-2589	4	27	prüfer	prüfer	NOUN
ejpam-2589	4	28	⋆-multiplication	⋆-multiplication	NOUN
ejpam-2589	4	29	domain	domain	NOUN
ejpam-2589	4	30	if	if	SCONJ
ejpam-2589	4	31	and	and	CCONJ
ejpam-2589	4	32	only	only	ADV
ejpam-2589	4	33	if	if	SCONJ
ejpam-2589	4	34	(	(	PUNCT
ejpam-2589	4	35	a	a	NOUN
ejpam-2589	4	36	)	)	PUNCT
ejpam-2589	4	37	∩	∩	NOUN
ejpam-2589	4	38	(	(	PUNCT
ejpam-2589	4	39	b	b	X
ejpam-2589	4	40	)	)	PUNCT
ejpam-2589	4	41	is	be	AUX
ejpam-2589	4	42	⋆	⋆	VERB
ejpam-2589	4	43	f	f	NOUN
ejpam-2589	4	44	-invertible	-invertible	ADJ
ejpam-2589	4	45	for	for	ADP
ejpam-2589	4	46	all	all	DET
ejpam-2589	4	47	a	a	PRON
ejpam-2589	4	48	,	,	PUNCT
ejpam-2589	4	49	b	b	X
ejpam-2589	4	50	∈	∈	PROPN
ejpam-2589	4	51	r	r	NOUN
ejpam-2589	4	52	\	\	PUNCT
ejpam-2589	4	53	{	{	PUNCT
ejpam-2589	4	54	0	0	NUM
ejpam-2589	4	55	}	}	PUNCT
ejpam-2589	4	56	.	.	PUNCT
ejpam-2589	5	1	we	we	PRON
ejpam-2589	5	2	show	show	VERB
ejpam-2589	5	3	in	in	ADP
ejpam-2589	5	4	this	this	DET
ejpam-2589	5	5	paper	paper	NOUN
ejpam-2589	5	6	that	that	SCONJ
ejpam-2589	5	7	the	the	DET
ejpam-2589	5	8	characterization	characterization	NOUN
ejpam-2589	5	9	does	do	AUX
ejpam-2589	5	10	not	not	PART
ejpam-2589	5	11	hold	hold	VERB
ejpam-2589	5	12	in	in	ADP
ejpam-2589	5	13	general	general	ADJ
ejpam-2589	5	14	and	and	CCONJ
ejpam-2589	5	15	we	we	PRON
ejpam-2589	5	16	restate	restate	VERB
ejpam-2589	5	17	[	[	X
ejpam-2589	5	18	3	3	NUM
ejpam-2589	5	19	,	,	PUNCT
ejpam-2589	5	20	corollary	corollary	NOUN
ejpam-2589	5	21	1	1	NUM
ejpam-2589	5	22	]	]	PUNCT
ejpam-2589	5	23	with	with	ADP
ejpam-2589	5	24	justification	justification	NOUN
ejpam-2589	5	25	and	and	CCONJ
ejpam-2589	5	26	proof	proof	NOUN
ejpam-2589	5	27	as	as	SCONJ
ejpam-2589	5	28	follows	follow	VERB
ejpam-2589	5	29	:	:	PUNCT
ejpam-2589	5	30	if	if	SCONJ
ejpam-2589	5	31	a	a	DET
ejpam-2589	5	32	domain	domain	NOUN
ejpam-2589	5	33	r	r	NOUN
ejpam-2589	5	34	is	be	AUX
ejpam-2589	5	35	a	a	DET
ejpam-2589	5	36	prüfer	prüfer	NOUN
ejpam-2589	5	37	⋆-multiplication	⋆-multiplication	NOUN
ejpam-2589	5	38	domain	domain	NOUN
ejpam-2589	5	39	,	,	PUNCT
ejpam-2589	5	40	then	then	ADV
ejpam-2589	5	41	(	(	PUNCT
ejpam-2589	5	42	a)∩	a)∩	X
ejpam-2589	5	43	(	(	PUNCT
ejpam-2589	5	44	b	b	X
ejpam-2589	5	45	)	)	PUNCT
ejpam-2589	5	46	is	be	AUX
ejpam-2589	5	47	⋆	⋆	VERB
ejpam-2589	5	48	f	f	NOUN
ejpam-2589	5	49	-invertible	-invertible	ADJ
ejpam-2589	5	50	for	for	ADP
ejpam-2589	5	51	all	all	DET
ejpam-2589	5	52	a	a	PRON
ejpam-2589	5	53	,	,	PUNCT
ejpam-2589	5	54	b	b	X
ejpam-2589	5	55	∈	∈	PROPN
ejpam-2589	5	56	r	r	NOUN
ejpam-2589	5	57	\	\	PUNCT
ejpam-2589	5	58	{	{	PUNCT
ejpam-2589	5	59	0	0	NUM
ejpam-2589	5	60	}	}	PUNCT
ejpam-2589	5	61	.	.	PUNCT
ejpam-2589	6	1	the	the	DET
ejpam-2589	6	2	converse	converse	NOUN
ejpam-2589	6	3	holds	hold	VERB
ejpam-2589	6	4	only	only	ADV
ejpam-2589	6	5	if	if	SCONJ
ejpam-2589	6	6	⋆	⋆	X
ejpam-2589	6	7	f	f	NOUN
ejpam-2589	6	8	=	=	SYM
ejpam-2589	6	9	t.	t.	NOUN
ejpam-2589	6	10	2010	2010	NUM
ejpam-2589	6	11	mathematics	mathematic	NOUN
ejpam-2589	6	12	subject	subject	NOUN
ejpam-2589	6	13	classifications	classification	NOUN
ejpam-2589	6	14	:	:	PUNCT
ejpam-2589	6	15	13a15	13a15	NUM
ejpam-2589	6	16	,	,	PUNCT
ejpam-2589	6	17	13a18	13a18	NUM
ejpam-2589	6	18	,	,	PUNCT
ejpam-2589	6	19	16w50	16w50	NUM
ejpam-2589	6	20	key	key	ADJ
ejpam-2589	6	21	words	word	NOUN
ejpam-2589	6	22	and	and	CCONJ
ejpam-2589	6	23	phrases	phrase	NOUN
ejpam-2589	6	24	:	:	PUNCT
ejpam-2589	6	25	star	star	NOUN
ejpam-2589	6	26	operation	operation	NOUN
ejpam-2589	6	27	;	;	PUNCT
ejpam-2589	6	28	⋆-ideal	⋆-ideal	NOUN
ejpam-2589	6	29	;	;	PUNCT
ejpam-2589	6	30	prüfer	prüfer	NOUN
ejpam-2589	6	31	⋆-multiplication	⋆-multiplication	NOUN
ejpam-2589	6	32	domain	domain	NOUN
ejpam-2589	6	33	in	in	ADP
ejpam-2589	6	34	[	[	X
ejpam-2589	6	35	3	3	NUM
ejpam-2589	6	36	,	,	PUNCT
ejpam-2589	6	37	corollary	corollary	ADJ
ejpam-2589	6	38	1	1	NUM
ejpam-2589	6	39	]	]	PUNCT
ejpam-2589	6	40	,	,	PUNCT
ejpam-2589	6	41	we	we	PRON
ejpam-2589	6	42	tried	try	VERB
ejpam-2589	6	43	to	to	PART
ejpam-2589	6	44	show	show	VERB
ejpam-2589	6	45	that	that	SCONJ
ejpam-2589	6	46	a	a	DET
ejpam-2589	6	47	prüfer	prüfer	NOUN
ejpam-2589	6	48	⋆-multiplication	⋆-multiplication	NOUN
ejpam-2589	6	49	domain	domain	NOUN
ejpam-2589	6	50	(	(	PUNCT
ejpam-2589	6	51	for	for	ADP
ejpam-2589	6	52	short	short	ADJ
ejpam-2589	6	53	p⋆md	p⋆md	NOUN
ejpam-2589	6	54	)	)	PUNCT
ejpam-2589	6	55	r	r	NOUN
ejpam-2589	6	56	is	be	AUX
ejpam-2589	6	57	characterized	characterize	VERB
ejpam-2589	6	58	by	by	ADP
ejpam-2589	6	59	(	(	PUNCT
ejpam-2589	6	60	a	a	PRON
ejpam-2589	6	61	)	)	PUNCT
ejpam-2589	6	62	∩	∩	NOUN
ejpam-2589	6	63	(	(	PUNCT
ejpam-2589	6	64	b	b	NOUN
ejpam-2589	6	65	)	)	PUNCT
ejpam-2589	6	66	being	be	AUX
ejpam-2589	6	67	⋆	⋆	VERB
ejpam-2589	6	68	f	f	NOUN
ejpam-2589	6	69	-invertible	-invertible	ADJ
ejpam-2589	6	70	for	for	ADP
ejpam-2589	6	71	all	all	DET
ejpam-2589	6	72	nonzero	nonzero	NOUN
ejpam-2589	6	73	a	a	PRON
ejpam-2589	6	74	,	,	PUNCT
ejpam-2589	6	75	b	b	PROPN
ejpam-2589	6	76	∈	∈	PROPN
ejpam-2589	6	77	r.	r.	PROPN
ejpam-2589	6	78	however	however	ADV
ejpam-2589	6	79	,	,	PUNCT
ejpam-2589	6	80	it	it	PRON
ejpam-2589	6	81	turns	turn	VERB
ejpam-2589	6	82	out	out	ADP
ejpam-2589	6	83	that	that	SCONJ
ejpam-2589	6	84	[	[	X
ejpam-2589	6	85	3	3	NUM
ejpam-2589	6	86	,	,	PUNCT
ejpam-2589	6	87	corollary	corollary	ADJ
ejpam-2589	6	88	1	1	NUM
ejpam-2589	6	89	]	]	PUNCT
ejpam-2589	6	90	is	be	AUX
ejpam-2589	6	91	not	not	PART
ejpam-2589	6	92	completely	completely	ADV
ejpam-2589	6	93	true	true	ADJ
ejpam-2589	6	94	and	and	CCONJ
ejpam-2589	6	95	needs	need	VERB
ejpam-2589	6	96	to	to	PART
ejpam-2589	6	97	be	be	AUX
ejpam-2589	6	98	adjusted	adjust	VERB
ejpam-2589	6	99	.	.	PUNCT
ejpam-2589	7	1	we	we	PRON
ejpam-2589	7	2	hereby	hereby	ADV
ejpam-2589	7	3	provide	provide	VERB
ejpam-2589	7	4	an	an	DET
ejpam-2589	7	5	adjustment	adjustment	NOUN
ejpam-2589	7	6	with	with	ADP
ejpam-2589	7	7	proof	proof	NOUN
ejpam-2589	7	8	of	of	ADP
ejpam-2589	7	9	[	[	X
ejpam-2589	7	10	3	3	NUM
ejpam-2589	7	11	,	,	PUNCT
ejpam-2589	7	12	corollary	corollary	ADJ
ejpam-2589	7	13	1	1	NUM
ejpam-2589	7	14	]	]	PUNCT
ejpam-2589	7	15	.	.	PUNCT
ejpam-2589	8	1	theorem	theorem	NOUN
ejpam-2589	8	2	1	1	NUM
ejpam-2589	8	3	.	.	PUNCT
ejpam-2589	9	1	if	if	SCONJ
ejpam-2589	9	2	r	r	NOUN
ejpam-2589	9	3	is	be	AUX
ejpam-2589	9	4	a	a	DET
ejpam-2589	9	5	p⋆md	p⋆md	ADJ
ejpam-2589	9	6	,	,	PUNCT
ejpam-2589	9	7	then	then	ADV
ejpam-2589	9	8	ar	ar	PROPN
ejpam-2589	9	9	∩	∩	NOUN
ejpam-2589	9	10	br	br	PROPN
ejpam-2589	9	11	is	be	AUX
ejpam-2589	9	12	⋆	⋆	PUNCT
ejpam-2589	9	13	f	f	NOUN
ejpam-2589	9	14	-invertible	-invertible	ADJ
ejpam-2589	9	15	for	for	ADP
ejpam-2589	9	16	every	every	DET
ejpam-2589	9	17	pair	pair	NOUN
ejpam-2589	9	18	of	of	ADP
ejpam-2589	9	19	nonzero	nonzero	PROPN
ejpam-2589	9	20	elements	element	NOUN
ejpam-2589	9	21	a	a	PRON
ejpam-2589	9	22	,	,	PUNCT
ejpam-2589	9	23	b	b	PROPN
ejpam-2589	9	24	∈	∈	PROPN
ejpam-2589	9	25	r.	r.	NOUN
ejpam-2589	9	26	the	the	DET
ejpam-2589	9	27	converse	converse	NOUN
ejpam-2589	9	28	holds	hold	VERB
ejpam-2589	9	29	only	only	ADV
ejpam-2589	9	30	if	if	SCONJ
ejpam-2589	9	31	⋆	⋆	X
ejpam-2589	9	32	f	f	NOUN
ejpam-2589	9	33	=	=	PUNCT
ejpam-2589	9	34	t.	t.	NOUN
ejpam-2589	9	35	proof	proof	NOUN
ejpam-2589	9	36	.	.	PUNCT
ejpam-2589	10	1	suppose	suppose	VERB
ejpam-2589	10	2	r	r	NOUN
ejpam-2589	10	3	is	be	AUX
ejpam-2589	10	4	a	a	DET
ejpam-2589	10	5	p⋆md	p⋆md	ADJ
ejpam-2589	10	6	.	.	PUNCT
ejpam-2589	11	1	note	note	VERB
ejpam-2589	11	2	that	that	SCONJ
ejpam-2589	11	3	we	we	PRON
ejpam-2589	11	4	have	have	VERB
ejpam-2589	11	5	(	(	PUNCT
ejpam-2589	11	6	ab)−1[(a)∩	ab)−1[(a)∩	NOUN
ejpam-2589	11	7	(	(	PUNCT
ejpam-2589	11	8	b	b	NOUN
ejpam-2589	11	9	)	)	PUNCT
ejpam-2589	11	10	]	]	PUNCT
ejpam-2589	12	1	=	=	PUNCT
ejpam-2589	12	2	(	(	PUNCT
ejpam-2589	12	3	a	a	PRON
ejpam-2589	12	4	,	,	PUNCT
ejpam-2589	12	5	b)−1	b)−1	NOUN
ejpam-2589	12	6	.	.	PUNCT
ejpam-2589	13	1	so	so	ADV
ejpam-2589	13	2	(	(	PUNCT
ejpam-2589	13	3	ab)−1[(a)∩(b)](a	ab)−1[(a)∩(b)](a	PROPN
ejpam-2589	13	4	,	,	PUNCT
ejpam-2589	13	5	b	b	NOUN
ejpam-2589	13	6	)	)	PUNCT
ejpam-2589	13	7	=	=	SYM
ejpam-2589	13	8	(	(	PUNCT
ejpam-2589	13	9	a	a	PRON
ejpam-2589	13	10	,	,	PUNCT
ejpam-2589	13	11	b)−1(a	b)−1(a	PROPN
ejpam-2589	13	12	,	,	PUNCT
ejpam-2589	13	13	b	b	NOUN
ejpam-2589	13	14	)	)	PUNCT
ejpam-2589	13	15	and	and	CCONJ
ejpam-2589	13	16	�	�	PROPN
ejpam-2589	13	17	(	(	PUNCT
ejpam-2589	13	18	ab)−1[(a)∩	ab)−1[(a)∩	PROPN
ejpam-2589	13	19	(	(	PUNCT
ejpam-2589	13	20	b)](a	b)](a	PROPN
ejpam-2589	13	21	,	,	PUNCT
ejpam-2589	13	22	b	b	X
ejpam-2589	13	23	)	)	PUNCT
ejpam-2589	13	24	�	�	PROPN
ejpam-2589	13	25	⋆	⋆	PUNCT
ejpam-2589	13	26	f	f	PROPN
ejpam-2589	13	27	=	=	SYM
ejpam-2589	13	28	�	�	PROPN
ejpam-2589	13	29	(	(	PUNCT
ejpam-2589	13	30	a	a	PRON
ejpam-2589	13	31	,	,	PUNCT
ejpam-2589	13	32	b)−1(a	b)−1(a	PROPN
ejpam-2589	13	33	,	,	PUNCT
ejpam-2589	13	34	b	b	NOUN
ejpam-2589	13	35	)	)	PUNCT
ejpam-2589	13	36	�	�	PROPN
ejpam-2589	13	37	⋆	⋆	PUNCT
ejpam-2589	13	38	f	f	PROPN
ejpam-2589	13	39	.	.	PUNCT
ejpam-2589	14	1	since	since	SCONJ
ejpam-2589	14	2	r	r	NOUN
ejpam-2589	14	3	is	be	AUX
ejpam-2589	14	4	a	a	DET
ejpam-2589	14	5	p⋆md	p⋆md	ADJ
ejpam-2589	14	6	,	,	PUNCT
ejpam-2589	14	7	(	(	PUNCT
ejpam-2589	14	8	a	a	DET
ejpam-2589	14	9	,	,	PUNCT
ejpam-2589	14	10	b	b	NOUN
ejpam-2589	14	11	)	)	PUNCT
ejpam-2589	14	12	is	be	AUX
ejpam-2589	14	13	⋆	⋆	ADJ
ejpam-2589	14	14	f	f	NOUN
ejpam-2589	14	15	-invertible	-invertible	ADJ
ejpam-2589	14	16	and	and	CCONJ
ejpam-2589	14	17	thus	thus	ADV
ejpam-2589	14	18	if	if	SCONJ
ejpam-2589	14	19	a	a	PRON
ejpam-2589	14	20	,	,	PUNCT
ejpam-2589	14	21	b	b	X
ejpam-2589	14	22	∈	∈	PROPN
ejpam-2589	14	23	r	r	NOUN
ejpam-2589	14	24	\	\	PUNCT
ejpam-2589	14	25	{	{	PUNCT
ejpam-2589	14	26	0	0	NUM
ejpam-2589	14	27	}	}	PUNCT
ejpam-2589	14	28	,	,	PUNCT
ejpam-2589	14	29	(	(	PUNCT
ejpam-2589	14	30	a)∩	a)∩	X
ejpam-2589	14	31	(	(	PUNCT
ejpam-2589	14	32	b	b	X
ejpam-2589	14	33	)	)	PUNCT
ejpam-2589	14	34	is	be	AUX
ejpam-2589	14	35	⋆	⋆	VERB
ejpam-2589	14	36	f	f	PROPN
ejpam-2589	14	37	-invertible	-invertible	PROPN
ejpam-2589	14	38	.	.	PUNCT
ejpam-2589	15	1	now	now	ADV
ejpam-2589	15	2	suppose	suppose	VERB
ejpam-2589	15	3	that	that	SCONJ
ejpam-2589	15	4	(	(	PUNCT
ejpam-2589	15	5	a)∩	a)∩	X
ejpam-2589	15	6	(	(	PUNCT
ejpam-2589	15	7	b	b	X
ejpam-2589	15	8	)	)	PUNCT
ejpam-2589	15	9	is	be	AUX
ejpam-2589	15	10	⋆	⋆	VERB
ejpam-2589	15	11	f	f	NOUN
ejpam-2589	15	12	-invertible	-invertible	ADJ
ejpam-2589	15	13	for	for	ADP
ejpam-2589	15	14	every	every	DET
ejpam-2589	15	15	pair	pair	NOUN
ejpam-2589	15	16	of	of	ADP
ejpam-2589	15	17	nonzero	nonzero	PROPN
ejpam-2589	15	18	elements	element	NOUN
ejpam-2589	15	19	a	a	PRON
ejpam-2589	15	20	,	,	PUNCT
ejpam-2589	15	21	b	b	PROPN
ejpam-2589	15	22	∈	∈	PROPN
ejpam-2589	15	23	r.	r.	PROPN
ejpam-2589	15	24	then	then	ADV
ejpam-2589	15	25	there	there	PRON
ejpam-2589	15	26	is	be	VERB
ejpam-2589	15	27	a	a	DET
ejpam-2589	15	28	fractional	fractional	ADJ
ejpam-2589	15	29	ideal	ideal	NOUN
ejpam-2589	15	30	a	a	DET
ejpam-2589	15	31	such	such	ADJ
ejpam-2589	15	32	that	that	SCONJ
ejpam-2589	15	33	(	(	PUNCT
ejpam-2589	15	34	a(ar	a(ar	X
ejpam-2589	15	35	∩	∩	ADJ
ejpam-2589	15	36	br))⋆	br))⋆	NOUN
ejpam-2589	15	37	f	f	PROPN
ejpam-2589	15	38	=	=	PUNCT
ejpam-2589	15	39	r.	r.	PROPN
ejpam-2589	15	40	that	that	PRON
ejpam-2589	15	41	is	be	AUX
ejpam-2589	15	42	,	,	PUNCT
ejpam-2589	15	43	a⋆	a⋆	ADJ
ejpam-2589	15	44	f	f	PROPN
ejpam-2589	15	45	=	=	SYM
ejpam-2589	15	46	(	(	PUNCT
ejpam-2589	15	47	ar	ar	PROPN
ejpam-2589	15	48	∩	∩	ADJ
ejpam-2589	15	49	br)−1	br)−1	NOUN
ejpam-2589	15	50	is	be	AUX
ejpam-2589	15	51	a	a	DET
ejpam-2589	15	52	divisorial	divisorial	ADJ
ejpam-2589	15	53	ideal	ideal	NOUN
ejpam-2589	15	54	and	and	CCONJ
ejpam-2589	15	55	because	because	SCONJ
ejpam-2589	15	56	a	a	PRON
ejpam-2589	15	57	is	be	AUX
ejpam-2589	15	58	of	of	ADP
ejpam-2589	15	59	finite	finite	ADJ
ejpam-2589	15	60	type	type	NOUN
ejpam-2589	15	61	,	,	PUNCT
ejpam-2589	15	62	we	we	PRON
ejpam-2589	15	63	deduce	deduce	VERB
ejpam-2589	15	64	from	from	ADP
ejpam-2589	15	65	discussion	discussion	NOUN
ejpam-2589	15	66	in	in	ADP
ejpam-2589	15	67	[	[	X
ejpam-2589	15	68	4	4	NUM
ejpam-2589	15	69	,	,	PUNCT
ejpam-2589	15	70	pp	pp	ADJ
ejpam-2589	15	71	.	.	PUNCT
ejpam-2589	16	1	433	433	NUM
ejpam-2589	16	2	-	-	SYM
ejpam-2589	16	3	434	434	NUM
ejpam-2589	16	4	]	]	PUNCT
ejpam-2589	17	1	that	that	SCONJ
ejpam-2589	17	2	a⋆	a⋆	ADJ
ejpam-2589	17	3	f	f	PROPN
ejpam-2589	17	4	=	=	SYM
ejpam-2589	17	5	av	av	PROPN
ejpam-2589	17	6	=	=	PUNCT
ejpam-2589	17	7	at	at	ADP
ejpam-2589	17	8	.	.	PUNCT
ejpam-2589	18	1	so	so	ADV
ejpam-2589	18	2	r	r	NOUN
ejpam-2589	18	3	is	be	AUX
ejpam-2589	18	4	a	a	DET
ejpam-2589	18	5	p⋆md	p⋆md	ADJ
ejpam-2589	18	6	only	only	ADV
ejpam-2589	18	7	if	if	SCONJ
ejpam-2589	18	8	⋆	⋆	X
ejpam-2589	18	9	f	f	X
ejpam-2589	18	10	=	=	PUNCT
ejpam-2589	19	1	t.	t.	NOUN
ejpam-2589	19	2	now	now	ADV
ejpam-2589	19	3	let	let	VERB
ejpam-2589	19	4	us	we	PRON
ejpam-2589	19	5	proceed	proceed	VERB
ejpam-2589	19	6	to	to	PART
ejpam-2589	19	7	show	show	VERB
ejpam-2589	19	8	that	that	SCONJ
ejpam-2589	19	9	there	there	PRON
ejpam-2589	19	10	is	be	VERB
ejpam-2589	19	11	a	a	DET
ejpam-2589	19	12	pathology	pathology	NOUN
ejpam-2589	19	13	in	in	ADP
ejpam-2589	19	14	[	[	X
ejpam-2589	19	15	3	3	NUM
ejpam-2589	19	16	,	,	PUNCT
ejpam-2589	19	17	corollary	corollary	ADJ
ejpam-2589	19	18	1	1	NUM
ejpam-2589	19	19	]	]	PUNCT
ejpam-2589	19	20	.	.	PUNCT
ejpam-2589	20	1	first	first	ADV
ejpam-2589	20	2	recall	recall	VERB
ejpam-2589	20	3	that	that	SCONJ
ejpam-2589	20	4	in	in	ADP
ejpam-2589	20	5	[	[	X
ejpam-2589	20	6	1	1	NUM
ejpam-2589	20	7	]	]	PUNCT
ejpam-2589	20	8	a	a	DET
ejpam-2589	20	9	generalized	generalized	ADJ
ejpam-2589	20	10	gcd	gcd	NOUN
ejpam-2589	20	11	domain	domain	NOUN
ejpam-2589	20	12	(	(	PUNCT
ejpam-2589	20	13	for	for	ADP
ejpam-2589	20	14	short	short	ADJ
ejpam-2589	20	15	ggcd	ggcd	VERB
ejpam-2589	20	16	domain	domain	NOUN
ejpam-2589	20	17	)	)	PUNCT
ejpam-2589	20	18	is	be	AUX
ejpam-2589	20	19	defined	define	VERB
ejpam-2589	20	20	as	as	ADP
ejpam-2589	20	21	a	a	DET
ejpam-2589	20	22	domain	domain	NOUN
ejpam-2589	20	23	for	for	ADP
ejpam-2589	20	24	which	which	PRON
ejpam-2589	20	25	the	the	DET
ejpam-2589	20	26	v	v	NOUN
ejpam-2589	20	27	-	-	PUNCT
ejpam-2589	20	28	image	image	NOUN
ejpam-2589	20	29	(	(	PUNCT
ejpam-2589	20	30	a	a	PRON
ejpam-2589	20	31	,	,	PUNCT
ejpam-2589	20	32	b)v	b)v	ADJ
ejpam-2589	20	33	of	of	ADP
ejpam-2589	20	34	the	the	DET
ejpam-2589	20	35	ideal	ideal	NOUN
ejpam-2589	20	36	generated	generate	VERB
ejpam-2589	20	37	by	by	ADP
ejpam-2589	20	38	each	each	DET
ejpam-2589	20	39	pair	pair	NOUN
ejpam-2589	20	40	of	of	ADP
ejpam-2589	20	41	nonzero	nonzero	PROPN
ejpam-2589	20	42	elements	element	NOUN
ejpam-2589	20	43	is	be	AUX
ejpam-2589	20	44	invertible	invertible	ADJ
ejpam-2589	20	45	.	.	PUNCT
ejpam-2589	21	1	note	note	VERB
ejpam-2589	21	2	that	that	SCONJ
ejpam-2589	21	3	(	(	PUNCT
ejpam-2589	21	4	1	1	NUM
ejpam-2589	21	5	ab	ab	PROPN
ejpam-2589	21	6	(	(	PUNCT
ejpam-2589	21	7	a	a	PRON
ejpam-2589	21	8	,	,	PUNCT
ejpam-2589	21	9	b))−1	b))−1	PROPN
ejpam-2589	21	10	=	=	X
ejpam-2589	21	11	ar∩	ar∩	NOUN
ejpam-2589	21	12	br	br	PROPN
ejpam-2589	21	13	.	.	PUNCT
ejpam-2589	22	1	but	but	CCONJ
ejpam-2589	22	2	then	then	ADV
ejpam-2589	22	3	we	we	PRON
ejpam-2589	22	4	also	also	ADV
ejpam-2589	22	5	have	have	VERB
ejpam-2589	22	6	(	(	PUNCT
ejpam-2589	22	7	1	1	NUM
ejpam-2589	22	8	ab	ab	PROPN
ejpam-2589	22	9	(	(	PUNCT
ejpam-2589	22	10	a	a	PRON
ejpam-2589	22	11	,	,	PUNCT
ejpam-2589	22	12	b))−1	b))−1	NOUN
ejpam-2589	22	13	=	=	SYM
ejpam-2589	22	14	(	(	PUNCT
ejpam-2589	22	15	1	1	NUM
ejpam-2589	22	16	ab	ab	PROPN
ejpam-2589	22	17	(	(	PUNCT
ejpam-2589	22	18	a	a	PRON
ejpam-2589	22	19	,	,	PUNCT
ejpam-2589	22	20	b)v	b)v	ADJ
ejpam-2589	22	21	)	)	PUNCT
ejpam-2589	22	22	−1	−1	NOUN
ejpam-2589	23	1	=	=	PUNCT
ejpam-2589	23	2	ar∩	ar∩	NOUN
ejpam-2589	23	3	br	br	PROPN
ejpam-2589	23	4	.	.	PUNCT
ejpam-2589	23	5	email	email	NOUN
ejpam-2589	23	6	address	address	NOUN
ejpam-2589	23	7	:	:	PUNCT
ejpam-2589	23	8	oheubokw@svsu.edu	oheubokw@svsu.edu	X
ejpam-2589	23	9	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-2589	24	1	1	1	NUM
ejpam-2589	24	2	c	c	X
ejpam-2589	24	3	©	©	PROPN
ejpam-2589	24	4	2016	2016	NUM
ejpam-2589	24	5	ejpam	ejpam	VERB
ejpam-2589	24	6	all	all	DET
ejpam-2589	24	7	rights	right	NOUN
ejpam-2589	24	8	reserved	reserve	VERB
ejpam-2589	24	9	.	.	PUNCT
ejpam-2589	25	1	references	reference	NOUN
ejpam-2589	25	2	2	2	NUM
ejpam-2589	25	3	now	now	ADV
ejpam-2589	25	4	the	the	DET
ejpam-2589	25	5	above	above	ADJ
ejpam-2589	25	6	two	two	NUM
ejpam-2589	25	7	equations	equation	NOUN
ejpam-2589	25	8	work	work	VERB
ejpam-2589	25	9	in	in	ADP
ejpam-2589	25	10	both	both	DET
ejpam-2589	25	11	prüfer	prüfer	NOUN
ejpam-2589	25	12	domains	domain	NOUN
ejpam-2589	25	13	(	(	PUNCT
ejpam-2589	25	14	domains	domain	NOUN
ejpam-2589	25	15	for	for	ADP
ejpam-2589	25	16	which	which	PRON
ejpam-2589	25	17	every	every	DET
ejpam-2589	25	18	two	two	NUM
ejpam-2589	25	19	generated	generate	VERB
ejpam-2589	25	20	nonzero	nonzero	PROPN
ejpam-2589	25	21	ideal	ideal	PROPN
ejpam-2589	25	22	is	be	AUX
ejpam-2589	25	23	invertible	invertible	ADJ
ejpam-2589	25	24	)	)	PUNCT
ejpam-2589	25	25	and	and	CCONJ
ejpam-2589	25	26	ggcd	ggcd	VERB
ejpam-2589	25	27	domains	domain	NOUN
ejpam-2589	25	28	.	.	PUNCT
ejpam-2589	26	1	in	in	ADP
ejpam-2589	26	2	fact	fact	NOUN
ejpam-2589	26	3	,	,	PUNCT
ejpam-2589	26	4	if	if	SCONJ
ejpam-2589	26	5	(	(	PUNCT
ejpam-2589	26	6	a	a	DET
ejpam-2589	26	7	,	,	PUNCT
ejpam-2589	26	8	b	b	NOUN
ejpam-2589	26	9	)	)	PUNCT
ejpam-2589	26	10	is	be	AUX
ejpam-2589	26	11	invertible	invertible	ADJ
ejpam-2589	26	12	then	then	ADV
ejpam-2589	26	13	(	(	PUNCT
ejpam-2589	26	14	a	a	DET
ejpam-2589	26	15	,	,	PUNCT
ejpam-2589	26	16	b	b	NOUN
ejpam-2589	26	17	)	)	PUNCT
ejpam-2589	26	18	is	be	AUX
ejpam-2589	26	19	divisorial	divisorial	ADJ
ejpam-2589	26	20	and	and	CCONJ
ejpam-2589	26	21	so	so	ADV
ejpam-2589	26	22	(	(	PUNCT
ejpam-2589	26	23	a	a	DET
ejpam-2589	26	24	,	,	PUNCT
ejpam-2589	26	25	b	b	NOUN
ejpam-2589	26	26	)	)	PUNCT
ejpam-2589	26	27	=	=	SYM
ejpam-2589	26	28	(	(	PUNCT
ejpam-2589	26	29	a	a	PRON
ejpam-2589	26	30	,	,	PUNCT
ejpam-2589	26	31	b)v	b)v	ADJ
ejpam-2589	26	32	in	in	ADP
ejpam-2589	26	33	the	the	DET
ejpam-2589	26	34	prüfer	prüfer	NOUN
ejpam-2589	26	35	domain	domain	NOUN
ejpam-2589	26	36	case	case	NOUN
ejpam-2589	26	37	.	.	PUNCT
ejpam-2589	27	1	on	on	ADP
ejpam-2589	27	2	the	the	DET
ejpam-2589	27	3	other	other	ADJ
ejpam-2589	27	4	hand	hand	NOUN
ejpam-2589	27	5	in	in	ADP
ejpam-2589	27	6	the	the	DET
ejpam-2589	27	7	ggcd	ggcd	VERB
ejpam-2589	27	8	domain	domain	NOUN
ejpam-2589	27	9	case	case	NOUN
ejpam-2589	27	10	ar∩	ar∩	NOUN
ejpam-2589	27	11	br	br	ADP
ejpam-2589	27	12	being	be	AUX
ejpam-2589	27	13	invertible	invertible	ADJ
ejpam-2589	27	14	works	work	NOUN
ejpam-2589	27	15	fine	fine	ADJ
ejpam-2589	27	16	because	because	SCONJ
ejpam-2589	27	17	1	1	NUM
ejpam-2589	27	18	ab	ab	PROPN
ejpam-2589	27	19	(	(	PUNCT
ejpam-2589	27	20	a	a	DET
ejpam-2589	27	21	,	,	PUNCT
ejpam-2589	27	22	b)v	b)v	ADJ
ejpam-2589	27	23	is	be	AUX
ejpam-2589	27	24	the	the	DET
ejpam-2589	27	25	inverse	inverse	NOUN
ejpam-2589	27	26	of	of	ADP
ejpam-2589	27	27	ar∩	ar∩	PROPN
ejpam-2589	27	28	br	br	PROPN
ejpam-2589	27	29	and	and	CCONJ
ejpam-2589	27	30	1	1	NUM
ejpam-2589	27	31	ab	ab	PROPN
ejpam-2589	27	32	(	(	PUNCT
ejpam-2589	27	33	a	a	DET
ejpam-2589	27	34	,	,	PUNCT
ejpam-2589	27	35	b)v	b)v	ADJ
ejpam-2589	27	36	is	be	AUX
ejpam-2589	27	37	invertible	invertible	ADJ
ejpam-2589	27	38	.	.	PUNCT
ejpam-2589	28	1	so	so	ADV
ejpam-2589	28	2	,	,	PUNCT
ejpam-2589	28	3	by	by	ADP
ejpam-2589	28	4	[	[	X
ejpam-2589	28	5	3	3	NUM
ejpam-2589	28	6	,	,	PUNCT
ejpam-2589	28	7	corollary	corollary	ADJ
ejpam-2589	28	8	1	1	NUM
ejpam-2589	28	9	]	]	PUNCT
ejpam-2589	28	10	,	,	PUNCT
ejpam-2589	28	11	ggcd	ggcd	VERB
ejpam-2589	28	12	domains	domain	NOUN
ejpam-2589	28	13	are	be	AUX
ejpam-2589	28	14	pdmds	pdmds	NOUN
ejpam-2589	28	15	.	.	PUNCT
ejpam-2589	29	1	but	but	CCONJ
ejpam-2589	29	2	then	then	ADV
ejpam-2589	29	3	we	we	PRON
ejpam-2589	29	4	have	have	VERB
ejpam-2589	29	5	the	the	DET
ejpam-2589	29	6	following	follow	VERB
ejpam-2589	29	7	observation	observation	NOUN
ejpam-2589	29	8	:	:	PUNCT
ejpam-2589	29	9	r	r	NOUN
ejpam-2589	29	10	is	be	AUX
ejpam-2589	29	11	a	a	DET
ejpam-2589	29	12	p⋆md	p⋆md	ADJ
ejpam-2589	29	13	if	if	SCONJ
ejpam-2589	30	1	and	and	CCONJ
ejpam-2589	30	2	only	only	ADV
ejpam-2589	30	3	if	if	SCONJ
ejpam-2589	30	4	every	every	DET
ejpam-2589	30	5	finitely	finitely	ADV
ejpam-2589	30	6	generated	generate	VERB
ejpam-2589	30	7	nonzero	nonzero	PROPN
ejpam-2589	30	8	ideal	ideal	NOUN
ejpam-2589	30	9	of	of	ADP
ejpam-2589	30	10	r	r	NOUN
ejpam-2589	30	11	is	be	AUX
ejpam-2589	30	12	⋆	⋆	ADJ
ejpam-2589	30	13	f	f	PROPN
ejpam-2589	30	14	-invertible	-invertible	PROPN
ejpam-2589	30	15	.	.	PUNCT
ejpam-2589	31	1	that	that	PRON
ejpam-2589	31	2	means	mean	VERB
ejpam-2589	31	3	for	for	ADP
ejpam-2589	31	4	every	every	DET
ejpam-2589	31	5	finitely	finitely	ADV
ejpam-2589	31	6	generated	generate	VERB
ejpam-2589	31	7	ideal	ideal	NOUN
ejpam-2589	31	8	a	a	PRON
ejpam-2589	31	9	we	we	PRON
ejpam-2589	31	10	have	have	VERB
ejpam-2589	31	11	a⋆	a⋆	ADJ
ejpam-2589	31	12	f	f	PROPN
ejpam-2589	31	13	=	=	SYM
ejpam-2589	31	14	av	av	PROPN
ejpam-2589	32	1	=	=	PUNCT
ejpam-2589	32	2	at	at	ADP
ejpam-2589	32	3	.	.	PUNCT
ejpam-2589	33	1	so	so	ADV
ejpam-2589	33	2	⋆	⋆	X
ejpam-2589	33	3	f	f	PROPN
ejpam-2589	33	4	=	=	SYM
ejpam-2589	33	5	t	t	PROPN
ejpam-2589	33	6	in	in	ADP
ejpam-2589	33	7	a	a	DET
ejpam-2589	33	8	p⋆md	p⋆md	ADJ
ejpam-2589	33	9	(	(	PUNCT
ejpam-2589	33	10	see	see	VERB
ejpam-2589	33	11	[	[	X
ejpam-2589	33	12	4	4	NUM
ejpam-2589	33	13	,	,	PUNCT
ejpam-2589	33	14	pp	pp	ADJ
ejpam-2589	33	15	.	.	PUNCT
ejpam-2589	34	1	433	433	NUM
ejpam-2589	34	2	-	-	SYM
ejpam-2589	34	3	434	434	NUM
ejpam-2589	34	4	]	]	PUNCT
ejpam-2589	34	5	and	and	CCONJ
ejpam-2589	34	6	[	[	X
ejpam-2589	34	7	5	5	NUM
ejpam-2589	34	8	]	]	NUM
ejpam-2589	34	9	)	)	PUNCT
ejpam-2589	34	10	.	.	PUNCT
ejpam-2589	35	1	so	so	ADV
ejpam-2589	35	2	this	this	PRON
ejpam-2589	35	3	means	mean	VERB
ejpam-2589	35	4	that	that	SCONJ
ejpam-2589	35	5	in	in	ADP
ejpam-2589	35	6	a	a	DET
ejpam-2589	35	7	pdmd	pdmd	NOUN
ejpam-2589	35	8	,	,	PUNCT
ejpam-2589	35	9	d	d	X
ejpam-2589	35	10	=	=	PUNCT
ejpam-2589	35	11	t.	t.	NOUN
ejpam-2589	35	12	that	that	PRON
ejpam-2589	35	13	is	be	AUX
ejpam-2589	35	14	a	a	DET
ejpam-2589	35	15	pdmd	pdmd	NOUN
ejpam-2589	35	16	is	be	AUX
ejpam-2589	35	17	a	a	DET
ejpam-2589	35	18	prüfer	prüfer	NOUN
ejpam-2589	35	19	domain	domain	NOUN
ejpam-2589	35	20	.	.	PUNCT
ejpam-2589	36	1	of	of	ADV
ejpam-2589	36	2	course	course	NOUN
ejpam-2589	36	3	d	d	PROPN
ejpam-2589	36	4	6=	6=	PROPN
ejpam-2589	36	5	t	t	NOUN
ejpam-2589	36	6	in	in	ADP
ejpam-2589	36	7	a	a	DET
ejpam-2589	36	8	ggcd	ggcd	ADJ
ejpam-2589	36	9	domain	domain	NOUN
ejpam-2589	36	10	,	,	PUNCT
ejpam-2589	36	11	generally	generally	ADV
ejpam-2589	36	12	,	,	PUNCT
ejpam-2589	36	13	as	as	ADP
ejpam-2589	36	14	the	the	DET
ejpam-2589	36	15	example	example	NOUN
ejpam-2589	36	16	below	below	ADP
ejpam-2589	36	17	shows	show	NOUN
ejpam-2589	36	18	.	.	PUNCT
ejpam-2589	37	1	example	example	NOUN
ejpam-2589	38	1	1	1	NUM
ejpam-2589	38	2	.	.	PUNCT
ejpam-2589	39	1	let	let	VERB
ejpam-2589	39	2	r	r	PRON
ejpam-2589	39	3	be	be	AUX
ejpam-2589	39	4	a	a	DET
ejpam-2589	39	5	dedekind	dedekind	ADJ
ejpam-2589	39	6	domain	domain	NOUN
ejpam-2589	39	7	(	(	PUNCT
ejpam-2589	39	8	note	note	VERB
ejpam-2589	39	9	that	that	SCONJ
ejpam-2589	39	10	a	a	DET
ejpam-2589	39	11	dedekind	dedekind	NOUN
ejpam-2589	39	12	domain	domain	NOUN
ejpam-2589	39	13	is	be	AUX
ejpam-2589	39	14	a	a	DET
ejpam-2589	39	15	ggcd	ggcd	ADJ
ejpam-2589	39	16	domain	domain	NOUN
ejpam-2589	39	17	)	)	PUNCT
ejpam-2589	39	18	that	that	PRON
ejpam-2589	39	19	is	be	AUX
ejpam-2589	39	20	not	not	PART
ejpam-2589	39	21	a	a	DET
ejpam-2589	39	22	field	field	NOUN
ejpam-2589	39	23	.	.	PUNCT
ejpam-2589	40	1	according	accord	VERB
ejpam-2589	40	2	to	to	ADP
ejpam-2589	40	3	[	[	X
ejpam-2589	40	4	1	1	NUM
ejpam-2589	40	5	]	]	PUNCT
ejpam-2589	40	6	,	,	PUNCT
ejpam-2589	40	7	the	the	DET
ejpam-2589	40	8	polynomial	polynomial	ADJ
ejpam-2589	40	9	ring	ring	NOUN
ejpam-2589	40	10	r[x	r[x	PROPN
ejpam-2589	40	11	]	]	PUNCT
ejpam-2589	40	12	is	be	AUX
ejpam-2589	40	13	a	a	DET
ejpam-2589	40	14	ggcd	ggcd	ADJ
ejpam-2589	40	15	domain	domain	NOUN
ejpam-2589	40	16	.	.	PUNCT
ejpam-2589	41	1	so	so	ADV
ejpam-2589	41	2	in	in	ADP
ejpam-2589	41	3	d	d	PROPN
ejpam-2589	41	4	=	=	SYM
ejpam-2589	41	5	r[x	r[x	NOUN
ejpam-2589	41	6	]	]	PUNCT
ejpam-2589	41	7	for	for	ADP
ejpam-2589	41	8	every	every	DET
ejpam-2589	41	9	pair	pair	NOUN
ejpam-2589	41	10	f	f	NOUN
ejpam-2589	41	11	,	,	PUNCT
ejpam-2589	41	12	g	g	PROPN
ejpam-2589	41	13	∈	∈	PROPN
ejpam-2589	41	14	d	d	X
ejpam-2589	41	15	\	\	X
ejpam-2589	41	16	{	{	PUNCT
ejpam-2589	41	17	0	0	NUM
ejpam-2589	41	18	}	}	PUNCT
ejpam-2589	41	19	we	we	PRON
ejpam-2589	41	20	have	have	VERB
ejpam-2589	41	21	f	f	PROPN
ejpam-2589	41	22	d∩	d∩	PROPN
ejpam-2589	41	23	gd	gd	PROPN
ejpam-2589	41	24	invertible	invertible	VERB
ejpam-2589	41	25	and	and	CCONJ
ejpam-2589	41	26	hence	hence	ADV
ejpam-2589	41	27	d	d	X
ejpam-2589	41	28	-	-	PUNCT
ejpam-2589	41	29	invertible	invertible	ADJ
ejpam-2589	41	30	.	.	PUNCT
ejpam-2589	42	1	so	so	ADV
ejpam-2589	42	2	d	d	PRON
ejpam-2589	42	3	is	be	AUX
ejpam-2589	42	4	a	a	DET
ejpam-2589	42	5	pdmd	pdmd	NOUN
ejpam-2589	42	6	by	by	ADP
ejpam-2589	42	7	[	[	X
ejpam-2589	42	8	3	3	NUM
ejpam-2589	42	9	,	,	PUNCT
ejpam-2589	42	10	corollary	corollary	ADJ
ejpam-2589	42	11	1	1	NUM
ejpam-2589	42	12	]	]	PUNCT
ejpam-2589	42	13	.	.	PUNCT
ejpam-2589	43	1	but	but	CCONJ
ejpam-2589	43	2	there	there	PRON
ejpam-2589	43	3	are	be	VERB
ejpam-2589	43	4	maximal	maximal	ADJ
ejpam-2589	43	5	d	d	NOUN
ejpam-2589	43	6	-	-	NOUN
ejpam-2589	43	7	ideals	ideal	NOUN
ejpam-2589	43	8	such	such	ADJ
ejpam-2589	43	9	as	as	ADP
ejpam-2589	43	10	m	m	PROPN
ejpam-2589	43	11	=	=	NOUN
ejpam-2589	43	12	p	p	PROPN
ejpam-2589	44	1	+	+	NUM
ejpam-2589	44	2	xr[x	xr[x	PROPN
ejpam-2589	44	3	]	]	X
ejpam-2589	44	4	,	,	PUNCT
ejpam-2589	44	5	with	with	ADP
ejpam-2589	44	6	p	p	PROPN
ejpam-2589	44	7	a	a	DET
ejpam-2589	44	8	nonzero	nonzero	NOUN
ejpam-2589	44	9	prime	prime	NOUN
ejpam-2589	44	10	of	of	ADP
ejpam-2589	44	11	r	r	NOUN
ejpam-2589	44	12	for	for	ADP
ejpam-2589	44	13	which	which	PRON
ejpam-2589	44	14	dm	dm	NOUN
ejpam-2589	44	15	is	be	AUX
ejpam-2589	44	16	not	not	PART
ejpam-2589	44	17	a	a	DET
ejpam-2589	44	18	valuation	valuation	NOUN
ejpam-2589	44	19	domain	domain	NOUN
ejpam-2589	44	20	.	.	PUNCT
ejpam-2589	45	1	now	now	ADV
ejpam-2589	45	2	pvmds	pvmds	NOUN
ejpam-2589	45	3	do	do	AUX
ejpam-2589	45	4	not	not	PART
ejpam-2589	45	5	suffer	suffer	VERB
ejpam-2589	45	6	from	from	ADP
ejpam-2589	45	7	the	the	DET
ejpam-2589	45	8	malady	malady	ADJ
ejpam-2589	45	9	p⋆mds	p⋆mds	PROPN
ejpam-2589	45	10	suffer	suffer	VERB
ejpam-2589	45	11	from	from	ADP
ejpam-2589	45	12	because	because	SCONJ
ejpam-2589	45	13	in	in	ADP
ejpam-2589	45	14	the	the	DET
ejpam-2589	45	15	pvmds	pvmds	NOUN
ejpam-2589	45	16	case	case	NOUN
ejpam-2589	45	17	ar	ar	NOUN
ejpam-2589	45	18	∩	∩	NOUN
ejpam-2589	46	1	br	br	PROPN
ejpam-2589	46	2	being	be	AUX
ejpam-2589	46	3	t	t	NOUN
ejpam-2589	46	4	-	-	PUNCT
ejpam-2589	46	5	invertible	invertible	ADJ
ejpam-2589	46	6	gives	give	NOUN
ejpam-2589	46	7	(	(	PUNCT
ejpam-2589	46	8	a	a	DET
ejpam-2589	46	9	,	,	PUNCT
ejpam-2589	46	10	b)v	b)v	ADJ
ejpam-2589	46	11	being	be	AUX
ejpam-2589	46	12	t	t	NOUN
ejpam-2589	46	13	-	-	PUNCT
ejpam-2589	46	14	invertible	invertible	ADJ
ejpam-2589	46	15	which	which	PRON
ejpam-2589	46	16	is	be	AUX
ejpam-2589	46	17	equivalent	equivalent	ADJ
ejpam-2589	46	18	to	to	ADP
ejpam-2589	46	19	(	(	PUNCT
ejpam-2589	46	20	a	a	DET
ejpam-2589	46	21	,	,	PUNCT
ejpam-2589	46	22	b	b	NOUN
ejpam-2589	46	23	)	)	PUNCT
ejpam-2589	46	24	being	be	AUX
ejpam-2589	46	25	t	t	NOUN
ejpam-2589	46	26	-	-	PUNCT
ejpam-2589	46	27	invertible	invertible	ADJ
ejpam-2589	46	28	because	because	SCONJ
ejpam-2589	46	29	(	(	PUNCT
ejpam-2589	46	30	1	1	NUM
ejpam-2589	46	31	ab	ab	PROPN
ejpam-2589	46	32	(	(	PUNCT
ejpam-2589	46	33	a	a	PRON
ejpam-2589	46	34	,	,	PUNCT
ejpam-2589	46	35	b)(ar	b)(ar	NOUN
ejpam-2589	46	36	∩	∩	NOUN
ejpam-2589	46	37	br))t	br))t	PROPN
ejpam-2589	46	38	=	=	PUNCT
ejpam-2589	46	39	(	(	PUNCT
ejpam-2589	46	40	1	1	NUM
ejpam-2589	46	41	ab	ab	PROPN
ejpam-2589	46	42	(	(	PUNCT
ejpam-2589	46	43	a	a	PRON
ejpam-2589	46	44	,	,	PUNCT
ejpam-2589	46	45	b)t(ar	b)t(ar	NOUN
ejpam-2589	46	46	∩	∩	VERB
ejpam-2589	46	47	br))t	br))t	PROPN
ejpam-2589	46	48	=	=	PUNCT
ejpam-2589	46	49	(	(	PUNCT
ejpam-2589	46	50	1	1	NUM
ejpam-2589	46	51	ab	ab	PROPN
ejpam-2589	46	52	(	(	PUNCT
ejpam-2589	46	53	a	a	PRON
ejpam-2589	46	54	,	,	PUNCT
ejpam-2589	46	55	b)v(ar	b)v(ar	NOUN
ejpam-2589	46	56	∩	∩	ADJ
ejpam-2589	46	57	br))t	br))t	PROPN
ejpam-2589	46	58	,	,	PUNCT
ejpam-2589	46	59	because	because	SCONJ
ejpam-2589	46	60	(	(	PUNCT
ejpam-2589	46	61	a	a	X
ejpam-2589	46	62	,	,	PUNCT
ejpam-2589	46	63	b)t	b)t	NOUN
ejpam-2589	46	64	=	=	SYM
ejpam-2589	46	65	(	(	PUNCT
ejpam-2589	46	66	a	a	DET
ejpam-2589	46	67	,	,	PUNCT
ejpam-2589	46	68	b)v	b)v	ADJ
ejpam-2589	46	69	.	.	PUNCT
ejpam-2589	47	1	similarly	similarly	ADV
ejpam-2589	47	2	one	one	PRON
ejpam-2589	47	3	may	may	AUX
ejpam-2589	47	4	note	note	VERB
ejpam-2589	47	5	that	that	SCONJ
ejpam-2589	47	6	the	the	DET
ejpam-2589	47	7	v	v	NOUN
ejpam-2589	47	8	-	-	PUNCT
ejpam-2589	47	9	domains	domain	NOUN
ejpam-2589	47	10	do	do	AUX
ejpam-2589	47	11	not	not	PART
ejpam-2589	47	12	suffer	suffer	VERB
ejpam-2589	47	13	from	from	ADP
ejpam-2589	47	14	this	this	DET
ejpam-2589	47	15	problem	problem	NOUN
ejpam-2589	47	16	because	because	SCONJ
ejpam-2589	47	17	(	(	PUNCT
ejpam-2589	47	18	a	a	DET
ejpam-2589	47	19	,	,	PUNCT
ejpam-2589	47	20	b	b	NOUN
ejpam-2589	47	21	)	)	PUNCT
ejpam-2589	47	22	is	be	AUX
ejpam-2589	47	23	v	v	ADV
ejpam-2589	47	24	-	-	PUNCT
ejpam-2589	47	25	invertible	invertible	ADJ
ejpam-2589	47	26	if	if	SCONJ
ejpam-2589	47	27	and	and	CCONJ
ejpam-2589	47	28	only	only	ADV
ejpam-2589	47	29	if	if	SCONJ
ejpam-2589	47	30	(	(	PUNCT
ejpam-2589	47	31	a	a	DET
ejpam-2589	47	32	,	,	PUNCT
ejpam-2589	47	33	b)v	b)v	ADJ
ejpam-2589	47	34	is	be	AUX
ejpam-2589	47	35	v	v	NOUN
ejpam-2589	47	36	-	-	PUNCT
ejpam-2589	47	37	invertible	invertible	ADJ
ejpam-2589	47	38	.	.	PUNCT
ejpam-2589	48	1	finally	finally	ADV
ejpam-2589	48	2	the	the	DET
ejpam-2589	48	3	ggcd	ggcd	VERB
ejpam-2589	48	4	domains	domain	NOUN
ejpam-2589	48	5	fall	fall	VERB
ejpam-2589	48	6	under	under	ADP
ejpam-2589	48	7	mixed	mixed	ADJ
ejpam-2589	48	8	invertibility	invertibility	NOUN
ejpam-2589	48	9	as	as	ADP
ejpam-2589	48	10	(	(	PUNCT
ejpam-2589	48	11	d	d	NOUN
ejpam-2589	48	12	,	,	PUNCT
ejpam-2589	48	13	v)-prüfer	v)-prüfer	X
ejpam-2589	48	14	i.e.	i.e.	X
ejpam-2589	48	15	domains	domain	NOUN
ejpam-2589	48	16	in	in	ADP
ejpam-2589	48	17	which	which	PRON
ejpam-2589	48	18	av	av	PRON
ejpam-2589	48	19	is	be	AUX
ejpam-2589	48	20	invertible	invertible	ADJ
ejpam-2589	48	21	for	for	SCONJ
ejpam-2589	48	22	each	each	DET
ejpam-2589	48	23	nonzero	nonzero	NOUN
ejpam-2589	48	24	finitely	finitely	ADV
ejpam-2589	48	25	generated	generate	VERB
ejpam-2589	48	26	ideal	ideal	ADJ
ejpam-2589	48	27	a.	a.	NOUN
ejpam-2589	48	28	these	these	PRON
ejpam-2589	48	29	may	may	AUX
ejpam-2589	48	30	serve	serve	VERB
ejpam-2589	48	31	as	as	ADP
ejpam-2589	48	32	pvmds	pvmds	NOUN
ejpam-2589	48	33	that	that	PRON
ejpam-2589	48	34	are	be	AUX
ejpam-2589	48	35	not	not	PART
ejpam-2589	48	36	p⋆mds	p⋆mds	ADJ
ejpam-2589	48	37	for	for	ADP
ejpam-2589	48	38	any	any	DET
ejpam-2589	48	39	⋆	⋆	NOUN
ejpam-2589	48	40	6=	6=	PROPN
ejpam-2589	48	41	v	v	ADP
ejpam-2589	48	42	,	,	PUNCT
ejpam-2589	48	43	t	t	PROPN
ejpam-2589	48	44	,	,	PUNCT
ejpam-2589	48	45	w	w	PROPN
ejpam-2589	48	46	(	(	PUNCT
ejpam-2589	48	47	see	see	VERB
ejpam-2589	48	48	section	section	NOUN
ejpam-2589	48	49	on	on	ADP
ejpam-2589	48	50	⋆-prüfer	⋆-prüfer	NOUN
ejpam-2589	48	51	domains	domain	NOUN
ejpam-2589	48	52	in	in	ADP
ejpam-2589	48	53	[	[	X
ejpam-2589	48	54	2	2	NUM
ejpam-2589	48	55	]	]	PUNCT
ejpam-2589	48	56	)	)	PUNCT
ejpam-2589	48	57	.	.	PUNCT
ejpam-2589	49	1	acknowledgements	acknowledgement	VERB
ejpam-2589	49	2	the	the	DET
ejpam-2589	49	3	author	author	NOUN
ejpam-2589	49	4	is	be	AUX
ejpam-2589	49	5	deeply	deeply	ADV
ejpam-2589	49	6	indebted	indebted	ADJ
ejpam-2589	49	7	to	to	ADP
ejpam-2589	49	8	muhammad	muhammad	PROPN
ejpam-2589	49	9	zafrullah	zafrullah	PROPN
ejpam-2589	49	10	for	for	ADP
ejpam-2589	49	11	bringing	bring	VERB
ejpam-2589	49	12	up	up	ADP
ejpam-2589	49	13	my	my	PRON
ejpam-2589	49	14	attention	attention	NOUN
ejpam-2589	49	15	toward	toward	ADP
ejpam-2589	49	16	the	the	DET
ejpam-2589	49	17	insufficiency	insufficiency	NOUN
ejpam-2589	49	18	of	of	ADP
ejpam-2589	49	19	[	[	X
ejpam-2589	49	20	3	3	NUM
ejpam-2589	49	21	,	,	PUNCT
ejpam-2589	49	22	corollary	corollary	ADJ
ejpam-2589	49	23	1	1	NUM
ejpam-2589	49	24	]	]	PUNCT
ejpam-2589	49	25	treated	treat	VERB
ejpam-2589	49	26	in	in	ADP
ejpam-2589	49	27	this	this	DET
ejpam-2589	49	28	paper	paper	NOUN
ejpam-2589	49	29	.	.	PUNCT
ejpam-2589	50	1	references	reference	NOUN
ejpam-2589	50	2	[	[	X
ejpam-2589	50	3	1	1	NUM
ejpam-2589	50	4	]	]	X
ejpam-2589	50	5	d.d	d.d	PROPN
ejpam-2589	50	6	.	.	PROPN
ejpam-2589	50	7	anderson	anderson	PROPN
ejpam-2589	50	8	and	and	CCONJ
ejpam-2589	50	9	d.f	d.f	PROPN
ejpam-2589	50	10	.	.	PROPN
ejpam-2589	50	11	anderson	anderson	PROPN
ejpam-2589	50	12	.	.	PUNCT
ejpam-2589	51	1	generalized	generalize	VERB
ejpam-2589	51	2	gcd	gcd	NOUN
ejpam-2589	51	3	domains	domain	NOUN
ejpam-2589	51	4	,	,	PUNCT
ejpam-2589	51	5	commentarii	commentarii	PROPN
ejpam-2589	51	6	mathematici	mathematici	PROPN
ejpam-2589	51	7	universitatis	universitatis	PROPN
ejpam-2589	51	8	sancti	sancti	PROPN
ejpam-2589	51	9	pauli	pauli	PROPN
ejpam-2589	51	10	,	,	PUNCT
ejpam-2589	51	11	28	28	NUM
ejpam-2589	51	12	,	,	PUNCT
ejpam-2589	51	13	215	215	NUM
ejpam-2589	51	14	-	-	SYM
ejpam-2589	51	15	221	221	NUM
ejpam-2589	51	16	,	,	PUNCT
ejpam-2589	51	17	1979	1979	NUM
ejpam-2589	51	18	.	.	PUNCT
ejpam-2589	52	1	[	[	X
ejpam-2589	52	2	2	2	X
ejpam-2589	52	3	]	]	X
ejpam-2589	52	4	d.	d.	PROPN
ejpam-2589	52	5	d.	d.	PROPN
ejpam-2589	52	6	anderson	anderson	PROPN
ejpam-2589	52	7	,	,	PUNCT
ejpam-2589	52	8	d.	d.	PROPN
ejpam-2589	52	9	f.	f.	PROPN
ejpam-2589	52	10	anderson	anderson	PROPN
ejpam-2589	52	11	,	,	PUNCT
ejpam-2589	52	12	m.	m.	PROPN
ejpam-2589	52	13	fontana	fontana	PROPN
ejpam-2589	52	14	,	,	PUNCT
ejpam-2589	52	15	and	and	CCONJ
ejpam-2589	52	16	m.	m.	PROPN
ejpam-2589	52	17	zafrullah	zafrullah	PROPN
ejpam-2589	52	18	.	.	PUNCT
ejpam-2589	53	1	on	on	ADP
ejpam-2589	53	2	v	v	NOUN
ejpam-2589	53	3	-	-	PUNCT
ejpam-2589	53	4	domains	domain	NOUN
ejpam-2589	53	5	and	and	CCONJ
ejpam-2589	53	6	star	star	NOUN
ejpam-2589	53	7	operations	operation	NOUN
ejpam-2589	53	8	.	.	PUNCT
ejpam-2589	54	1	communications	communication	NOUN
ejpam-2589	54	2	in	in	ADP
ejpam-2589	54	3	algebra	algebra	NOUN
ejpam-2589	54	4	,	,	PUNCT
ejpam-2589	54	5	2	2	NUM
ejpam-2589	54	6	:	:	SYM
ejpam-2589	54	7	141	141	NUM
ejpam-2589	54	8	-	-	SYM
ejpam-2589	54	9	145	145	NUM
ejpam-2589	54	10	,	,	PUNCT
ejpam-2589	54	11	2008	2008	NUM
ejpam-2589	54	12	.	.	PUNCT
ejpam-2589	55	1	[	[	X
ejpam-2589	55	2	3	3	NUM
ejpam-2589	55	3	]	]	X
ejpam-2589	55	4	o.a	o.a	PROPN
ejpam-2589	55	5	.	.	PROPN
ejpam-2589	55	6	heubo	heubo	PROPN
ejpam-2589	55	7	-	-	PUNCT
ejpam-2589	55	8	kwegna	kwegna	PROPN
ejpam-2589	55	9	.	.	PUNCT
ejpam-2589	56	1	a	a	DET
ejpam-2589	56	2	note	note	NOUN
ejpam-2589	56	3	on	on	ADP
ejpam-2589	56	4	prüfer	prüfer	NOUN
ejpam-2589	56	5	⋆-multiplication	⋆-multiplication	NOUN
ejpam-2589	56	6	domains	domain	NOUN
ejpam-2589	56	7	,	,	PUNCT
ejpam-2589	56	8	european	european	ADJ
ejpam-2589	56	9	journal	journal	NOUN
ejpam-2589	56	10	of	of	ADP
ejpam-2589	56	11	pure	pure	ADJ
ejpam-2589	56	12	and	and	CCONJ
ejpam-2589	56	13	applied	applied	ADJ
ejpam-2589	56	14	mathematics	mathematic	NOUN
ejpam-2589	56	15	,	,	PUNCT
ejpam-2589	56	16	8(4	8(4	NUM
ejpam-2589	56	17	):	):	PUNCT
ejpam-2589	56	18	458	458	NUM
ejpam-2589	56	19	-	-	SYM
ejpam-2589	56	20	461	461	NUM
ejpam-2589	56	21	,	,	PUNCT
ejpam-2589	56	22	2015	2015	NUM
ejpam-2589	56	23	.	.	PUNCT
ejpam-2589	57	1	[	[	X
ejpam-2589	57	2	4	4	NUM
ejpam-2589	57	3	]	]	X
ejpam-2589	57	4	m.	m.	NOUN
ejpam-2589	57	5	zafrullah	zafrullah	PROPN
ejpam-2589	57	6	.	.	PUNCT
ejpam-2589	58	1	putting	put	VERB
ejpam-2589	58	2	t	t	NOUN
ejpam-2589	58	3	-	-	PUNCT
ejpam-2589	58	4	invertibility	invertibility	NOUN
ejpam-2589	58	5	to	to	PART
ejpam-2589	58	6	use	use	VERB
ejpam-2589	58	7	,	,	PUNCT
ejpam-2589	58	8	non	non	ADJ
ejpam-2589	58	9	-	-	ADJ
ejpam-2589	58	10	noetherian	noetherian	ADJ
ejpam-2589	58	11	commutative	commutative	ADJ
ejpam-2589	58	12	ring	ring	NOUN
ejpam-2589	58	13	theory	theory	NOUN
ejpam-2589	58	14	,	,	PUNCT
ejpam-2589	58	15	429	429	NUM
ejpam-2589	58	16	–	–	SYM
ejpam-2589	58	17	457	457	NUM
ejpam-2589	58	18	,	,	PUNCT
ejpam-2589	58	19	mathematics	mathematic	NOUN
ejpam-2589	58	20	and	and	CCONJ
ejpam-2589	58	21	its	its	PRON
ejpam-2589	58	22	applications	application	NOUN
ejpam-2589	58	23	,	,	PUNCT
ejpam-2589	58	24	520	520	NUM
ejpam-2589	58	25	,	,	PUNCT
ejpam-2589	58	26	kluwer	kluwer	NOUN
ejpam-2589	58	27	acad	acad	PROPN
ejpam-2589	58	28	.	.	PUNCT
ejpam-2589	59	1	publ	publ	PROPN
ejpam-2589	59	2	.	.	PUNCT
ejpam-2589	59	3	,	,	PUNCT
ejpam-2589	59	4	dordrecht	dordrecht	PROPN
ejpam-2589	59	5	,	,	PUNCT
ejpam-2589	59	6	2000	2000	NUM
ejpam-2589	59	7	.	.	PUNCT
ejpam-2589	60	1	[	[	X
ejpam-2589	60	2	5	5	NUM
ejpam-2589	60	3	]	]	PUNCT
ejpam-2589	60	4	m.	m.	NOUN
ejpam-2589	60	5	zafrullah	zafrullah	PROPN
ejpam-2589	60	6	.	.	PUNCT
ejpam-2589	61	1	t	t	PROPN
ejpam-2589	61	2	-	-	PUNCT
ejpam-2589	61	3	invertibility	invertibility	NOUN
ejpam-2589	61	4	and	and	CCONJ
ejpam-2589	61	5	bazzoni	bazzoni	ADJ
ejpam-2589	61	6	-	-	PUNCT
ejpam-2589	61	7	like	like	ADJ
ejpam-2589	61	8	statements	statement	NOUN
ejpam-2589	61	9	,	,	PUNCT
ejpam-2589	61	10	journal	journal	NOUN
ejpam-2589	61	11	of	of	ADP
ejpam-2589	61	12	pure	pure	ADJ
ejpam-2589	61	13	and	and	CCONJ
ejpam-2589	61	14	applied	applied	ADJ
ejpam-2589	61	15	algebra	algebra	NOUN
ejpam-2589	61	16	,	,	PUNCT
ejpam-2589	61	17	214	214	NUM
ejpam-2589	61	18	:	:	PUNCT
ejpam-2589	61	19	654	654	NUM
ejpam-2589	61	20	-	-	SYM
ejpam-2589	61	21	657	657	NUM
ejpam-2589	61	22	,	,	PUNCT
ejpam-2589	61	23	2010	2010	NUM
ejpam-2589	61	24	.	.	PUNCT