id	sid	tid	token	lemma	pos
ejpam-977	1	1	9_xxx_jayram.dvi	9_xxx_jayram.dvi	NUM
ejpam-977	1	2	european	european	ADJ
ejpam-977	1	3	journal	journal	PROPN
ejpam-977	1	4	of	of	ADP
ejpam-977	1	5	pure	pure	ADJ
ejpam-977	1	6	and	and	CCONJ
ejpam-977	1	7	applied	apply	VERB
ejpam-977	1	8	mathematics	mathematic	NOUN
ejpam-977	1	9	vol	vol	NOUN
ejpam-977	1	10	.	.	PUNCT
ejpam-977	2	1	3	3	NUM
ejpam-977	2	2	,	,	PUNCT
ejpam-977	2	3	no	no	INTJ
ejpam-977	2	4	.	.	NOUN
ejpam-977	2	5	5	5	NUM
ejpam-977	2	6	,	,	PUNCT
ejpam-977	2	7	2010	2010	NUM
ejpam-977	2	8	,	,	PUNCT
ejpam-977	2	9	899	899	NUM
ejpam-977	2	10	-	-	SYM
ejpam-977	2	11	902	902	NUM
ejpam-977	2	12	issn	issn	PROPN
ejpam-977	2	13	1307	1307	NUM
ejpam-977	2	14	-	-	SYM
ejpam-977	2	15	5543	5543	NUM
ejpam-977	2	16	–	–	PUNCT
ejpam-977	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-977	2	18	a	a	DET
ejpam-977	2	19	note	note	NOUN
ejpam-977	2	20	on	on	ADP
ejpam-977	2	21	prüfer	prüfer	NOUN
ejpam-977	2	22	modules	module	NOUN
ejpam-977	2	23	c.	c.	PROPN
ejpam-977	2	24	jayaram1,∗and	jayaram1,∗and	PROPN
ejpam-977	2	25	v.c.c	v.c.c	NOUN
ejpam-977	2	26	.	.	PUNCT
ejpam-977	3	1	raju2	raju2	PROPN
ejpam-977	3	2	1	1	NUM
ejpam-977	3	3	the	the	DET
ejpam-977	3	4	university	university	NOUN
ejpam-977	3	5	of	of	ADP
ejpam-977	3	6	the	the	DET
ejpam-977	3	7	west	west	PROPN
ejpam-977	3	8	indies	indies	PROPN
ejpam-977	3	9	,	,	PUNCT
ejpam-977	3	10	department	department	NOUN
ejpam-977	3	11	of	of	ADP
ejpam-977	3	12	mathematics	mathematics	PROPN
ejpam-977	3	13	,	,	PUNCT
ejpam-977	3	14	p.o	p.o	PROPN
ejpam-977	3	15	.	.	PROPN
ejpam-977	3	16	box	box	PROPN
ejpam-977	3	17	64	64	NUM
ejpam-977	3	18	,	,	PUNCT
ejpam-977	3	19	bridgetown	bridgetown	PROPN
ejpam-977	3	20	,	,	PUNCT
ejpam-977	3	21	barbados	barbado	VERB
ejpam-977	3	22	2	2	NUM
ejpam-977	3	23	department	department	NOUN
ejpam-977	3	24	of	of	ADP
ejpam-977	3	25	mathematics	mathematic	NOUN
ejpam-977	3	26	,	,	PUNCT
ejpam-977	3	27	university	university	PROPN
ejpam-977	3	28	of	of	ADP
ejpam-977	3	29	botswana	botswana	PROPN
ejpam-977	3	30	,	,	PUNCT
ejpam-977	3	31	gaborone	gaborone	PROPN
ejpam-977	3	32	,	,	PUNCT
ejpam-977	3	33	botswana	botswana	PROPN
ejpam-977	3	34	abstract	abstract	NOUN
ejpam-977	3	35	.	.	PUNCT
ejpam-977	4	1	in	in	ADP
ejpam-977	4	2	this	this	DET
ejpam-977	4	3	paper	paper	NOUN
ejpam-977	4	4	we	we	PRON
ejpam-977	4	5	characterize	characterize	VERB
ejpam-977	4	6	prüfer	prüfer	NOUN
ejpam-977	4	7	modules	module	NOUN
ejpam-977	4	8	and	and	CCONJ
ejpam-977	4	9	dedekind	dedekind	NOUN
ejpam-977	4	10	modules	module	NOUN
ejpam-977	4	11	.	.	PUNCT
ejpam-977	5	1	2000	2000	NUM
ejpam-977	5	2	mathematics	mathematic	NOUN
ejpam-977	5	3	subject	subject	NOUN
ejpam-977	5	4	classifications	classification	NOUN
ejpam-977	5	5	:	:	PUNCT
ejpam-977	5	6	primary	primary	ADJ
ejpam-977	5	7	13c13	13c13	NUM
ejpam-977	5	8	;	;	PUNCT
ejpam-977	5	9	secondary	secondary	ADJ
ejpam-977	5	10	13c05	13c05	NUM
ejpam-977	5	11	,	,	PUNCT
ejpam-977	5	12	13a15	13a15	NUM
ejpam-977	5	13	key	key	ADJ
ejpam-977	5	14	words	word	NOUN
ejpam-977	5	15	and	and	CCONJ
ejpam-977	5	16	phrases	phrase	NOUN
ejpam-977	5	17	:	:	PUNCT
ejpam-977	5	18	multiplication	multiplication	NOUN
ejpam-977	5	19	module	module	NOUN
ejpam-977	5	20	,	,	PUNCT
ejpam-977	5	21	prüfer	prüfer	NOUN
ejpam-977	5	22	module	module	NOUN
ejpam-977	5	23	,	,	PUNCT
ejpam-977	5	24	dedekind	dedekind	NOUN
ejpam-977	5	25	module	module	NOUN
ejpam-977	5	26	,	,	PUNCT
ejpam-977	5	27	quasi	quasi	ADJ
ejpam-977	5	28	-	-	ADJ
ejpam-977	5	29	principal	principal	ADJ
ejpam-977	5	30	ideal	ideal	NOUN
ejpam-977	5	31	,	,	PUNCT
ejpam-977	5	32	quasi	quasi	ADJ
ejpam-977	5	33	-	-	ADJ
ejpam-977	5	34	cyclic	cyclic	ADJ
ejpam-977	5	35	module	module	NOUN
ejpam-977	5	36	.	.	PUNCT
ejpam-977	6	1	1	1	X
ejpam-977	6	2	.	.	X
ejpam-977	6	3	introduction	introduction	NOUN
ejpam-977	6	4	throughout	throughout	ADP
ejpam-977	6	5	this	this	DET
ejpam-977	6	6	paper	paper	NOUN
ejpam-977	6	7	r	r	NOUN
ejpam-977	6	8	denotes	denote	VERB
ejpam-977	6	9	a	a	DET
ejpam-977	6	10	commutative	commutative	ADJ
ejpam-977	6	11	ring	ring	NOUN
ejpam-977	6	12	with	with	ADP
ejpam-977	6	13	identity	identity	NOUN
ejpam-977	6	14	and	and	CCONJ
ejpam-977	6	15	m	m	NOUN
ejpam-977	6	16	denotes	denote	VERB
ejpam-977	6	17	a	a	DET
ejpam-977	6	18	unital	unital	ADJ
ejpam-977	6	19	r	r	NOUN
ejpam-977	6	20	-	-	PUNCT
ejpam-977	6	21	module	module	NOUN
ejpam-977	6	22	.	.	PUNCT
ejpam-977	7	1	l(r	l(r	PROPN
ejpam-977	7	2	)	)	PUNCT
ejpam-977	7	3	(	(	PUNCT
ejpam-977	7	4	l(m	l(m	PROPN
ejpam-977	7	5	)	)	PUNCT
ejpam-977	7	6	)	)	PUNCT
ejpam-977	7	7	denotes	denote	VERB
ejpam-977	7	8	the	the	DET
ejpam-977	7	9	lattice	lattice	NOUN
ejpam-977	7	10	of	of	ADP
ejpam-977	7	11	all	all	DET
ejpam-977	7	12	ideals	ideal	NOUN
ejpam-977	7	13	of	of	ADP
ejpam-977	7	14	r	r	NOUN
ejpam-977	7	15	(	(	PUNCT
ejpam-977	7	16	submodules	submodule	NOUN
ejpam-977	7	17	of	of	ADP
ejpam-977	7	18	m	m	PROPN
ejpam-977	7	19	)	)	PUNCT
ejpam-977	7	20	.	.	PUNCT
ejpam-977	8	1	for	for	ADP
ejpam-977	8	2	any	any	DET
ejpam-977	8	3	two	two	NUM
ejpam-977	8	4	submodules	submodule	NOUN
ejpam-977	8	5	n	n	PRON
ejpam-977	8	6	and	and	CCONJ
ejpam-977	8	7	k	k	PROPN
ejpam-977	8	8	of	of	ADP
ejpam-977	8	9	m	m	PROPN
ejpam-977	8	10	,	,	PUNCT
ejpam-977	8	11	the	the	DET
ejpam-977	8	12	ideal	ideal	NOUN
ejpam-977	8	13	{	{	PUNCT
ejpam-977	8	14	a	a	DET
ejpam-977	8	15	∈	∈	PROPN
ejpam-977	8	16	r	r	NOUN
ejpam-977	8	17	|	|	NOUN
ejpam-977	8	18	ak	ak	PROPN
ejpam-977	8	19	⊆	⊆	NUM
ejpam-977	8	20	n	n	CCONJ
ejpam-977	8	21	}	}	PUNCT
ejpam-977	8	22	will	will	AUX
ejpam-977	8	23	be	be	AUX
ejpam-977	8	24	denoted	denote	VERB
ejpam-977	8	25	by	by	ADP
ejpam-977	8	26	(	(	PUNCT
ejpam-977	8	27	n	n	NUM
ejpam-977	8	28	:	:	PUNCT
ejpam-977	8	29	k	k	X
ejpam-977	8	30	)	)	PUNCT
ejpam-977	8	31	.	.	PUNCT
ejpam-977	9	1	thus	thus	ADV
ejpam-977	9	2	(	(	PUNCT
ejpam-977	9	3	o	o	NOUN
ejpam-977	9	4	:	:	PUNCT
ejpam-977	9	5	m	m	VERB
ejpam-977	9	6	)	)	PUNCT
ejpam-977	9	7	is	be	AUX
ejpam-977	9	8	the	the	DET
ejpam-977	9	9	annihilator	annihilator	NOUN
ejpam-977	9	10	of	of	ADP
ejpam-977	9	11	m	m	PROPN
ejpam-977	9	12	.	.	PUNCT
ejpam-977	10	1	m	m	PROPN
ejpam-977	10	2	is	be	AUX
ejpam-977	10	3	said	say	VERB
ejpam-977	10	4	to	to	PART
ejpam-977	10	5	be	be	AUX
ejpam-977	10	6	a	a	DET
ejpam-977	10	7	faithful	faithful	ADJ
ejpam-977	10	8	module	module	NOUN
ejpam-977	10	9	if	if	SCONJ
ejpam-977	10	10	(	(	PUNCT
ejpam-977	10	11	o	o	NOUN
ejpam-977	10	12	:	:	PUNCT
ejpam-977	10	13	m	m	VERB
ejpam-977	10	14	)	)	PUNCT
ejpam-977	10	15	is	be	AUX
ejpam-977	10	16	the	the	DET
ejpam-977	10	17	zero	zero	NUM
ejpam-977	10	18	ideal	ideal	NOUN
ejpam-977	10	19	of	of	ADP
ejpam-977	10	20	r.	r.	PROPN
ejpam-977	10	21	m	m	PROPN
ejpam-977	10	22	is	be	AUX
ejpam-977	10	23	said	say	VERB
ejpam-977	10	24	to	to	PART
ejpam-977	10	25	be	be	AUX
ejpam-977	10	26	a	a	DET
ejpam-977	10	27	multiplication	multiplication	NOUN
ejpam-977	10	28	module	module	NOUN
ejpam-977	10	29	[	[	X
ejpam-977	10	30	4	4	X
ejpam-977	10	31	]	]	X
ejpam-977	10	32	if	if	SCONJ
ejpam-977	10	33	every	every	DET
ejpam-977	10	34	submodule	submodule	NOUN
ejpam-977	10	35	of	of	ADP
ejpam-977	10	36	m	m	PROPN
ejpam-977	10	37	is	be	AUX
ejpam-977	10	38	of	of	ADP
ejpam-977	10	39	the	the	DET
ejpam-977	10	40	form	form	NOUN
ejpam-977	10	41	i	i	PRON
ejpam-977	10	42	m	m	VERB
ejpam-977	10	43	,	,	PUNCT
ejpam-977	10	44	for	for	ADP
ejpam-977	10	45	some	some	DET
ejpam-977	10	46	ideal	ideal	ADJ
ejpam-977	10	47	i	i	PRON
ejpam-977	10	48	of	of	ADP
ejpam-977	10	49	r.	r.	PROPN
ejpam-977	10	50	according	accord	VERB
ejpam-977	10	51	to	to	ADP
ejpam-977	10	52	[	[	X
ejpam-977	10	53	7	7	NUM
ejpam-977	10	54	]	]	PUNCT
ejpam-977	10	55	,	,	PUNCT
ejpam-977	10	56	a	a	DET
ejpam-977	10	57	submodule	submodule	NOUN
ejpam-977	10	58	n	n	PROPN
ejpam-977	10	59	of	of	ADP
ejpam-977	10	60	m	m	PROPN
ejpam-977	10	61	is	be	AUX
ejpam-977	10	62	called	call	VERB
ejpam-977	10	63	meet	meet	ADJ
ejpam-977	10	64	-	-	PUNCT
ejpam-977	10	65	quasi	quasi	NOUN
ejpam-977	10	66	-	-	NOUN
ejpam-977	10	67	cyclic	cyclic	ADJ
ejpam-977	10	68	(	(	PUNCT
ejpam-977	10	69	or	or	CCONJ
ejpam-977	10	70	meet	meet	VERB
ejpam-977	10	71	principal	principal	NOUN
ejpam-977	10	72	in	in	ADP
ejpam-977	10	73	the	the	DET
ejpam-977	10	74	sense	sense	NOUN
ejpam-977	10	75	of	of	ADP
ejpam-977	10	76	[	[	X
ejpam-977	10	77	1	1	NUM
ejpam-977	10	78	,	,	PUNCT
ejpam-977	10	79	3	3	NUM
ejpam-977	10	80	]	]	PUNCT
ejpam-977	10	81	)	)	PUNCT
ejpam-977	10	82	if	if	SCONJ
ejpam-977	10	83	(	(	PUNCT
ejpam-977	10	84	b	b	NOUN
ejpam-977	10	85	∩	∩	NOUN
ejpam-977	10	86	(	(	PUNCT
ejpam-977	10	87	k	k	NOUN
ejpam-977	10	88	:	:	PUNCT
ejpam-977	10	89	n))n	n))n	PROPN
ejpam-977	10	90	=	=	SYM
ejpam-977	10	91	bn	bn	PROPN
ejpam-977	10	92	∩	∩	X
ejpam-977	10	93	k	k	PROPN
ejpam-977	10	94	for	for	ADP
ejpam-977	10	95	all	all	DET
ejpam-977	10	96	ideals	ideal	NOUN
ejpam-977	10	97	b	b	PROPN
ejpam-977	10	98	of	of	ADP
ejpam-977	10	99	r	r	NOUN
ejpam-977	10	100	and	and	CCONJ
ejpam-977	10	101	for	for	ADP
ejpam-977	10	102	all	all	DET
ejpam-977	10	103	submodules	submodule	NOUN
ejpam-977	10	104	k	k	PROPN
ejpam-977	10	105	of	of	ADP
ejpam-977	10	106	m	m	PROPN
ejpam-977	10	107	;	;	PUNCT
ejpam-977	10	108	n	n	X
ejpam-977	10	109	is	be	AUX
ejpam-977	10	110	called	call	VERB
ejpam-977	10	111	weak	weak	ADJ
ejpam-977	10	112	-	-	PUNCT
ejpam-977	10	113	join	join	NOUN
ejpam-977	10	114	-	-	PUNCT
ejpam-977	10	115	quasi	quasi	NOUN
ejpam-977	10	116	-	-	NOUN
ejpam-977	10	117	cyclic	cyclic	ADJ
ejpam-977	10	118	if	if	SCONJ
ejpam-977	10	119	(	(	PUNCT
ejpam-977	10	120	bn	bn	NOUN
ejpam-977	10	121	)	)	PUNCT
ejpam-977	10	122	:	:	PUNCT
ejpam-977	11	1	n	n	X
ejpam-977	11	2	=	=	SYM
ejpam-977	11	3	(	(	PUNCT
ejpam-977	11	4	0	0	NUM
ejpam-977	11	5	:	:	PUNCT
ejpam-977	11	6	n	n	X
ejpam-977	11	7	)	)	PUNCT
ejpam-977	12	1	+	+	CCONJ
ejpam-977	12	2	b	b	X
ejpam-977	12	3	for	for	ADP
ejpam-977	12	4	all	all	DET
ejpam-977	12	5	ideals	ideal	NOUN
ejpam-977	12	6	b	b	NOUN
ejpam-977	12	7	of	of	ADP
ejpam-977	12	8	r	r	NOUN
ejpam-977	12	9	;	;	PUNCT
ejpam-977	12	10	n	n	X
ejpam-977	12	11	is	be	AUX
ejpam-977	12	12	called	call	VERB
ejpam-977	12	13	join	join	NOUN
ejpam-977	12	14	-	-	PUNCT
ejpam-977	12	15	quasi	quasi	NOUN
ejpam-977	12	16	-	-	NOUN
ejpam-977	12	17	cyclic	cyclic	ADJ
ejpam-977	12	18	(	(	PUNCT
ejpam-977	12	19	or	or	CCONJ
ejpam-977	12	20	join	join	VERB
ejpam-977	12	21	principal	principal	NOUN
ejpam-977	12	22	in	in	ADP
ejpam-977	12	23	the	the	DET
ejpam-977	12	24	sense	sense	NOUN
ejpam-977	12	25	of	of	ADP
ejpam-977	12	26	[	[	X
ejpam-977	12	27	1	1	NUM
ejpam-977	12	28	]	]	PUNCT
ejpam-977	12	29	and	and	CCONJ
ejpam-977	12	30	[	[	X
ejpam-977	12	31	3	3	NUM
ejpam-977	12	32	]	]	PUNCT
ejpam-977	12	33	)	)	PUNCT
ejpam-977	13	1	if	if	SCONJ
ejpam-977	13	2	(	(	PUNCT
ejpam-977	13	3	k	k	X
ejpam-977	13	4	+	+	NOUN
ejpam-977	13	5	bn	bn	NUM
ejpam-977	13	6	)	)	PUNCT
ejpam-977	13	7	:	:	PUNCT
ejpam-977	13	8	n	n	X
ejpam-977	13	9	=	=	SYM
ejpam-977	13	10	(	(	PUNCT
ejpam-977	13	11	k	k	NOUN
ejpam-977	13	12	:	:	PUNCT
ejpam-977	13	13	n	n	X
ejpam-977	13	14	)	)	PUNCT
ejpam-977	14	1	+	+	CCONJ
ejpam-977	14	2	b	b	X
ejpam-977	14	3	for	for	ADP
ejpam-977	14	4	all	all	DET
ejpam-977	14	5	ideals	ideal	NOUN
ejpam-977	14	6	b	b	PROPN
ejpam-977	14	7	of	of	ADP
ejpam-977	14	8	r	r	NOUN
ejpam-977	14	9	and	and	CCONJ
ejpam-977	14	10	for	for	ADP
ejpam-977	14	11	all	all	DET
ejpam-977	14	12	submodules	submodule	NOUN
ejpam-977	14	13	k	k	PROPN
ejpam-977	14	14	of	of	ADP
ejpam-977	14	15	m	m	PROPN
ejpam-977	14	16	.	.	PUNCT
ejpam-977	15	1	n	n	PROPN
ejpam-977	15	2	is	be	AUX
ejpam-977	15	3	called	call	VERB
ejpam-977	15	4	quasi	quasi	ADJ
ejpam-977	15	5	-	-	ADJ
ejpam-977	15	6	cyclic	cyclic	ADJ
ejpam-977	15	7	[	[	X
ejpam-977	15	8	7	7	NUM
ejpam-977	15	9	]	]	X
ejpam-977	15	10	(	(	PUNCT
ejpam-977	15	11	or	or	CCONJ
ejpam-977	15	12	principal	principal	NOUN
ejpam-977	15	13	in	in	ADP
ejpam-977	15	14	the	the	DET
ejpam-977	15	15	sense	sense	NOUN
ejpam-977	15	16	of	of	ADP
ejpam-977	15	17	[	[	X
ejpam-977	15	18	1	1	NUM
ejpam-977	15	19	]	]	PUNCT
ejpam-977	15	20	and	and	CCONJ
ejpam-977	15	21	[	[	X
ejpam-977	15	22	3])if	3])if	NUM
ejpam-977	15	23	n	n	NOUN
ejpam-977	15	24	is	be	AUX
ejpam-977	15	25	both	both	PRON
ejpam-977	15	26	meet	meet	ADJ
ejpam-977	15	27	-	-	PUNCT
ejpam-977	15	28	quasi	quasi	NOUN
ejpam-977	15	29	-	-	ADJ
ejpam-977	15	30	cyclic	cyclic	ADJ
ejpam-977	15	31	and	and	CCONJ
ejpam-977	15	32	join	join	NOUN
ejpam-977	15	33	-	-	PUNCT
ejpam-977	15	34	quasi	quasi	NOUN
ejpam-977	15	35	-	-	ADJ
ejpam-977	15	36	cyclic	cyclic	ADJ
ejpam-977	15	37	.	.	PUNCT
ejpam-977	16	1	note	note	VERB
ejpam-977	16	2	that	that	SCONJ
ejpam-977	16	3	quasi	quasi	ADJ
ejpam-977	16	4	-	-	ADJ
ejpam-977	16	5	cyclic	cyclic	ADJ
ejpam-977	16	6	submodules	submodule	NOUN
ejpam-977	16	7	have	have	AUX
ejpam-977	16	8	been	be	AUX
ejpam-977	16	9	studied	study	VERB
ejpam-977	16	10	in	in	ADP
ejpam-977	16	11	[	[	X
ejpam-977	16	12	1	1	NUM
ejpam-977	16	13	]	]	PUNCT
ejpam-977	16	14	,	,	PUNCT
ejpam-977	17	1	[	[	X
ejpam-977	17	2	3	3	NUM
ejpam-977	17	3	]	]	PUNCT
ejpam-977	17	4	and	and	CCONJ
ejpam-977	17	5	[	[	X
ejpam-977	17	6	7	7	NUM
ejpam-977	17	7	]	]	PUNCT
ejpam-977	17	8	.	.	PUNCT
ejpam-977	18	1	for	for	ADP
ejpam-977	18	2	any	any	DET
ejpam-977	18	3	a	a	DET
ejpam-977	18	4	∈	∈	PROPN
ejpam-977	18	5	r	r	NOUN
ejpam-977	18	6	,	,	PUNCT
ejpam-977	18	7	the	the	DET
ejpam-977	18	8	principal	principal	ADJ
ejpam-977	18	9	ideal	ideal	NOUN
ejpam-977	18	10	generated	generate	VERB
ejpam-977	18	11	by	by	ADP
ejpam-977	18	12	a	a	PRON
ejpam-977	18	13	is	be	AUX
ejpam-977	18	14	denoted	denote	VERB
ejpam-977	18	15	by	by	ADP
ejpam-977	18	16	(	(	PUNCT
ejpam-977	18	17	a	a	NOUN
ejpam-977	18	18	)	)	PUNCT
ejpam-977	18	19	.	.	PUNCT
ejpam-977	19	1	recall	recall	VERB
ejpam-977	19	2	that	that	SCONJ
ejpam-977	19	3	an	an	DET
ejpam-977	19	4	ideal	ideal	NOUN
ejpam-977	19	5	i	i	PRON
ejpam-977	19	6	of	of	ADP
ejpam-977	19	7	r	r	NOUN
ejpam-977	19	8	is	be	AUX
ejpam-977	19	9	called	call	VERB
ejpam-977	19	10	a	a	DET
ejpam-977	19	11	multiplication	multiplication	NOUN
ejpam-977	19	12	ideal	ideal	NOUN
ejpam-977	19	13	if	if	SCONJ
ejpam-977	19	14	for	for	ADP
ejpam-977	19	15	every	every	DET
ejpam-977	19	16	ideal	ideal	NOUN
ejpam-977	19	17	j	j	PROPN
ejpam-977	19	18	⊆	⊆	NUM
ejpam-977	19	19	i	i	PRON
ejpam-977	19	20	,	,	PUNCT
ejpam-977	19	21	there	there	PRON
ejpam-977	19	22	exists	exist	VERB
ejpam-977	19	23	an	an	DET
ejpam-977	19	24	ideal	ideal	NOUN
ejpam-977	19	25	k	k	PROPN
ejpam-977	19	26	with	with	ADP
ejpam-977	19	27	j	j	PROPN
ejpam-977	19	28	=	=	SYM
ejpam-977	19	29	ki	ki	PROPN
ejpam-977	19	30	.	.	PUNCT
ejpam-977	20	1	an	an	DET
ejpam-977	20	2	ideal	ideal	ADJ
ejpam-977	20	3	i	i	PRON
ejpam-977	20	4	of	of	ADP
ejpam-977	20	5	r	r	NOUN
ejpam-977	20	6	is	be	AUX
ejpam-977	20	7	called	call	VERB
ejpam-977	20	8	weak	weak	ADJ
ejpam-977	20	9	join	join	NOUN
ejpam-977	20	10	principal	principal	NOUN
ejpam-977	20	11	if	if	SCONJ
ejpam-977	20	12	(	(	PUNCT
ejpam-977	20	13	ai	ai	INTJ
ejpam-977	20	14	:	:	PUNCT
ejpam-977	20	15	i	i	NOUN
ejpam-977	20	16	)	)	PUNCT
ejpam-977	20	17	=	=	SYM
ejpam-977	20	18	a+	a+	PUNCT
ejpam-977	20	19	(	(	PUNCT
ejpam-977	20	20	0	0	NUM
ejpam-977	20	21	:	:	PUNCT
ejpam-977	20	22	i	i	NOUN
ejpam-977	20	23	)	)	PUNCT
ejpam-977	20	24	for	for	ADP
ejpam-977	20	25	all	all	DET
ejpam-977	20	26	a	a	DET
ejpam-977	20	27	∈	∈	NOUN
ejpam-977	20	28	l(r	l(r	PROPN
ejpam-977	20	29	)	)	PUNCT
ejpam-977	20	30	.	.	PUNCT
ejpam-977	21	1	i	i	PRON
ejpam-977	21	2	is	be	AUX
ejpam-977	21	3	called	call	VERB
ejpam-977	21	4	join	join	NOUN
ejpam-977	21	5	principal	principal	NOUN
ejpam-977	21	6	if	if	SCONJ
ejpam-977	21	7	(	(	PUNCT
ejpam-977	21	8	a+	a+	X
ejpam-977	21	9	bi	bi	NOUN
ejpam-977	21	10	)	)	PUNCT
ejpam-977	21	11	:	:	PUNCT
ejpam-977	22	1	i	i	PRON
ejpam-977	22	2	=	=	PUNCT
ejpam-977	22	3	(	(	PUNCT
ejpam-977	22	4	a	a	X
ejpam-977	22	5	:	:	PUNCT
ejpam-977	22	6	i	i	NOUN
ejpam-977	22	7	)	)	PUNCT
ejpam-977	23	1	+	+	CCONJ
ejpam-977	23	2	b	b	X
ejpam-977	23	3	,	,	PUNCT
ejpam-977	23	4	for	for	ADP
ejpam-977	23	5	all	all	DET
ejpam-977	23	6	a	a	DET
ejpam-977	23	7	,	,	PUNCT
ejpam-977	23	8	b	b	NOUN
ejpam-977	23	9	∈	∈	PROPN
ejpam-977	23	10	l(r	l(r	PROPN
ejpam-977	23	11	)	)	PUNCT
ejpam-977	23	12	.	.	PUNCT
ejpam-977	24	1	an	an	DET
ejpam-977	24	2	ideal	ideal	ADJ
ejpam-977	24	3	i	i	PRON
ejpam-977	24	4	of	of	ADP
ejpam-977	24	5	r	r	NOUN
ejpam-977	24	6	is	be	AUX
ejpam-977	24	7	called	call	VERB
ejpam-977	24	8	a	a	DET
ejpam-977	24	9	quasi	quasi	ADJ
ejpam-977	24	10	-	-	ADJ
ejpam-977	24	11	principal	principal	ADJ
ejpam-977	24	12	ideal	ideal	NOUN
ejpam-977	25	1	[	[	X
ejpam-977	25	2	8	8	NUM
ejpam-977	25	3	,	,	PUNCT
ejpam-977	25	4	exercise	exercise	VERB
ejpam-977	25	5	10	10	NUM
ejpam-977	25	6	,	,	PUNCT
ejpam-977	25	7	page	page	NOUN
ejpam-977	25	8	147	147	NUM
ejpam-977	25	9	]	]	PUNCT
ejpam-977	25	10	(	(	PUNCT
ejpam-977	25	11	or	or	CCONJ
ejpam-977	25	12	a	a	DET
ejpam-977	25	13	principal	principal	ADJ
ejpam-977	25	14	element	element	NOUN
ejpam-977	25	15	of	of	ADP
ejpam-977	25	16	l(r	l(r	PROPN
ejpam-977	25	17	)	)	PUNCT
ejpam-977	26	1	[	[	X
ejpam-977	26	2	9	9	NUM
ejpam-977	26	3	]	]	SYM
ejpam-977	26	4	)	)	PUNCT
ejpam-977	26	5	if	if	SCONJ
ejpam-977	26	6	it	it	PRON
ejpam-977	26	7	satisfies	satisfy	VERB
ejpam-977	26	8	the	the	DET
ejpam-977	26	9	identities	identity	NOUN
ejpam-977	26	10	∗corresponding	∗corresponde	VERB
ejpam-977	26	11	author	author	NOUN
ejpam-977	26	12	.	.	PUNCT
ejpam-977	27	1	email	email	NOUN
ejpam-977	27	2	addresses	address	NOUN
ejpam-977	27	3	:	:	PUNCT
ejpam-977	28	1	jayaram	jayaram	PROPN
ejpam-977	28	2	.	.	PROPN
ejpam-977	28	3	hillumu	hillumu	PROPN
ejpam-977	28	4	�	�	PROPN
ejpam-977	28	5	avehill.uwi.edu	avehill.uwi.edu	PROPN
ejpam-977	28	6	(	(	PUNCT
ejpam-977	28	7	c.	c.	PROPN
ejpam-977	28	8	jayaram	jayaram	PROPN
ejpam-977	28	9	)	)	PUNCT
ejpam-977	28	10	,	,	PUNCT
ejpam-977	28	11	varanasi�mopipi.ub.bw	varanasi�mopipi.ub.bw	NOUN
ejpam-977	28	12	(	(	PUNCT
ejpam-977	28	13	v.	v.	X
ejpam-977	28	14	raju	raju	X
ejpam-977	28	15	)	)	PUNCT
ejpam-977	28	16	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-977	29	1	899	899	NUM
ejpam-977	30	1	c	c	X
ejpam-977	30	2	©	©	VERB
ejpam-977	30	3	2010	2010	NUM
ejpam-977	30	4	ejpam	ejpam	NOUN
ejpam-977	30	5	all	all	DET
ejpam-977	30	6	rights	right	NOUN
ejpam-977	30	7	reserved	reserve	VERB
ejpam-977	30	8	.	.	PUNCT
ejpam-977	31	1	c.	c.	PROPN
ejpam-977	31	2	jayaram	jayaram	PROPN
ejpam-977	31	3	and	and	CCONJ
ejpam-977	31	4	v.	v.	ADP
ejpam-977	31	5	raju	raju	PROPN
ejpam-977	31	6	/	/	SYM
ejpam-977	31	7	eur	eur	PROPN
ejpam-977	31	8	.	.	PUNCT
ejpam-977	32	1	j.	j.	PROPN
ejpam-977	32	2	pure	pure	PROPN
ejpam-977	32	3	appl	appl	PROPN
ejpam-977	32	4	.	.	PROPN
ejpam-977	32	5	math	math	PROPN
ejpam-977	32	6	,	,	PUNCT
ejpam-977	32	7	3	3	NUM
ejpam-977	32	8	(	(	PUNCT
ejpam-977	32	9	2010	2010	NUM
ejpam-977	32	10	)	)	PUNCT
ejpam-977	32	11	,	,	PUNCT
ejpam-977	32	12	899	899	NUM
ejpam-977	32	13	-	-	SYM
ejpam-977	32	14	902	902	NUM
ejpam-977	32	15	900	900	NUM
ejpam-977	32	16	(	(	PUNCT
ejpam-977	32	17	i	i	NOUN
ejpam-977	32	18	)	)	PUNCT
ejpam-977	32	19	(	(	PUNCT
ejpam-977	32	20	a∩	a∩	PROPN
ejpam-977	32	21	(	(	PUNCT
ejpam-977	32	22	b	b	X
ejpam-977	32	23	:	:	PUNCT
ejpam-977	32	24	i))i	i))i	PROPN
ejpam-977	32	25	=	=	SYM
ejpam-977	32	26	ai	ai	VERB
ejpam-977	32	27	∩	∩	ADJ
ejpam-977	32	28	b	b	PROPN
ejpam-977	32	29	and	and	CCONJ
ejpam-977	32	30	(	(	PUNCT
ejpam-977	32	31	ii	ii	NOUN
ejpam-977	32	32	)	)	PUNCT
ejpam-977	32	33	(	(	PUNCT
ejpam-977	32	34	a+	a+	X
ejpam-977	32	35	bi	bi	NOUN
ejpam-977	32	36	)	)	PUNCT
ejpam-977	32	37	:	:	PUNCT
ejpam-977	33	1	i	i	PRON
ejpam-977	33	2	=	=	PUNCT
ejpam-977	33	3	(	(	PUNCT
ejpam-977	33	4	a	a	X
ejpam-977	33	5	:	:	PUNCT
ejpam-977	33	6	i	i	NOUN
ejpam-977	33	7	)	)	PUNCT
ejpam-977	34	1	+	+	CCONJ
ejpam-977	34	2	b	b	X
ejpam-977	34	3	,	,	PUNCT
ejpam-977	34	4	for	for	ADP
ejpam-977	34	5	all	all	DET
ejpam-977	34	6	a	a	DET
ejpam-977	34	7	,	,	PUNCT
ejpam-977	34	8	b	b	NOUN
ejpam-977	34	9	∈	∈	PROPN
ejpam-977	34	10	l(r	l(r	PROPN
ejpam-977	34	11	)	)	PUNCT
ejpam-977	34	12	.	.	PUNCT
ejpam-977	35	1	obviously	obviously	ADV
ejpam-977	35	2	,	,	PUNCT
ejpam-977	35	3	every	every	DET
ejpam-977	35	4	quasi	quasi	ADJ
ejpam-977	35	5	-	-	ADJ
ejpam-977	35	6	principal	principal	ADJ
ejpam-977	35	7	ideal	ideal	NOUN
ejpam-977	35	8	is	be	AUX
ejpam-977	35	9	a	a	DET
ejpam-977	35	10	multiplication	multiplication	NOUN
ejpam-977	35	11	ideal	ideal	ADJ
ejpam-977	35	12	.	.	PUNCT
ejpam-977	36	1	quasi	quasi	ADJ
ejpam-977	36	2	-	-	ADJ
ejpam-977	36	3	principal	principal	ADJ
ejpam-977	36	4	ideals	ideal	NOUN
ejpam-977	36	5	have	have	AUX
ejpam-977	36	6	been	be	AUX
ejpam-977	36	7	studied	study	VERB
ejpam-977	36	8	in	in	ADP
ejpam-977	36	9	[	[	X
ejpam-977	36	10	2	2	NUM
ejpam-977	36	11	,	,	PUNCT
ejpam-977	36	12	5	5	NUM
ejpam-977	36	13	,	,	PUNCT
ejpam-977	36	14	9	9	NUM
ejpam-977	36	15	]	]	PUNCT
ejpam-977	36	16	.	.	PUNCT
ejpam-977	37	1	let	let	VERB
ejpam-977	37	2	s	s	PRON
ejpam-977	37	3	be	be	AUX
ejpam-977	37	4	the	the	DET
ejpam-977	37	5	set	set	NOUN
ejpam-977	37	6	of	of	ADP
ejpam-977	37	7	all	all	DET
ejpam-977	37	8	non	non	ADJ
ejpam-977	37	9	-	-	ADJ
ejpam-977	37	10	zero	zero	ADJ
ejpam-977	37	11	divisors	divisor	NOUN
ejpam-977	37	12	of	of	ADP
ejpam-977	37	13	r	r	NOUN
ejpam-977	37	14	and	and	CCONJ
ejpam-977	37	15	let	let	VERB
ejpam-977	37	16	t	t	NOUN
ejpam-977	37	17	=	=	SYM
ejpam-977	37	18	{	{	PUNCT
ejpam-977	37	19	t	t	PROPN
ejpam-977	37	20	∈	∈	PROPN
ejpam-977	37	21	s	s	PART
ejpam-977	37	22	:	:	PUNCT
ejpam-977	37	23	tm	tm	PROPN
ejpam-977	37	24	=	=	PROPN
ejpam-977	37	25	0	0	NUM
ejpam-977	37	26	for	for	ADP
ejpam-977	37	27	some	some	DET
ejpam-977	37	28	m	m	NOUN
ejpam-977	37	29	∈	∈	NOUN
ejpam-977	37	30	m	m	NOUN
ejpam-977	37	31	implies	imply	VERB
ejpam-977	37	32	m	m	VERB
ejpam-977	37	33	=	=	NOUN
ejpam-977	37	34	0	0	NUM
ejpam-977	37	35	}	}	PUNCT
ejpam-977	37	36	.	.	PUNCT
ejpam-977	38	1	let	let	VERB
ejpam-977	38	2	rt	rt	PRON
ejpam-977	38	3	be	be	AUX
ejpam-977	38	4	the	the	DET
ejpam-977	38	5	localization	localization	NOUN
ejpam-977	38	6	of	of	ADP
ejpam-977	38	7	r	r	NOUN
ejpam-977	38	8	at	at	ADP
ejpam-977	38	9	t	t	PROPN
ejpam-977	38	10	.	.	PUNCT
ejpam-977	39	1	for	for	ADP
ejpam-977	39	2	any	any	DET
ejpam-977	39	3	non	non	ADJ
ejpam-977	39	4	-	-	ADJ
ejpam-977	39	5	zero	zero	NUM
ejpam-977	39	6	submodule	submodule	NOUN
ejpam-977	39	7	n	n	PROPN
ejpam-977	39	8	of	of	ADP
ejpam-977	39	9	m	m	PRON
ejpam-977	39	10	,	,	PUNCT
ejpam-977	39	11	let	let	VERB
ejpam-977	39	12	n−1	n−1	PROPN
ejpam-977	39	13	=	=	SYM
ejpam-977	39	14	{	{	PUNCT
ejpam-977	39	15	x	x	PUNCT
ejpam-977	39	16	∈	∈	PROPN
ejpam-977	39	17	rt	rt	NOUN
ejpam-977	39	18	:	:	PUNCT
ejpam-977	39	19	xn	xn	PROPN
ejpam-977	40	1	⊆	⊆	NUM
ejpam-977	40	2	m	m	NUM
ejpam-977	40	3	}	}	PUNCT
ejpam-977	40	4	.	.	PUNCT
ejpam-977	41	1	it	it	PRON
ejpam-977	41	2	is	be	AUX
ejpam-977	41	3	easily	easily	ADV
ejpam-977	41	4	seen	see	VERB
ejpam-977	41	5	that	that	SCONJ
ejpam-977	41	6	n−1	n−1	PROPN
ejpam-977	41	7	is	be	AUX
ejpam-977	41	8	an	an	DET
ejpam-977	41	9	r	r	NOUN
ejpam-977	41	10	-	-	PUNCT
ejpam-977	41	11	submodule	submodule	NOUN
ejpam-977	41	12	of	of	ADP
ejpam-977	41	13	rt	rt	PROPN
ejpam-977	41	14	,	,	PUNCT
ejpam-977	41	15	r	r	NOUN
ejpam-977	41	16	⊆	⊆	NUM
ejpam-977	41	17	n−1	n−1	PROPN
ejpam-977	41	18	and	and	CCONJ
ejpam-977	41	19	n−1n	n−1n	PROPN
ejpam-977	41	20	⊆	⊆	NUM
ejpam-977	41	21	m	m	NOUN
ejpam-977	41	22	.	.	PUNCT
ejpam-977	42	1	following	follow	VERB
ejpam-977	42	2	[	[	X
ejpam-977	42	3	10	10	NUM
ejpam-977	42	4	]	]	PUNCT
ejpam-977	42	5	,	,	PUNCT
ejpam-977	42	6	n	n	X
ejpam-977	42	7	is	be	AUX
ejpam-977	42	8	an	an	DET
ejpam-977	42	9	invertible	invertible	ADJ
ejpam-977	42	10	submodule	submodule	NOUN
ejpam-977	42	11	of	of	ADP
ejpam-977	42	12	m	m	PROPN
ejpam-977	42	13	if	if	SCONJ
ejpam-977	42	14	n−1n	n−1n	NOUN
ejpam-977	42	15	=	=	NOUN
ejpam-977	42	16	m	m	VERB
ejpam-977	42	17	.	.	PUNCT
ejpam-977	43	1	following	follow	VERB
ejpam-977	43	2	[	[	X
ejpam-977	43	3	10	10	NUM
ejpam-977	43	4	]	]	PUNCT
ejpam-977	43	5	,	,	PUNCT
ejpam-977	43	6	an	an	DET
ejpam-977	43	7	r	r	NOUN
ejpam-977	43	8	module	module	NOUN
ejpam-977	43	9	m	m	NOUN
ejpam-977	43	10	is	be	AUX
ejpam-977	43	11	called	call	VERB
ejpam-977	43	12	a	a	DET
ejpam-977	43	13	dedekind	dedekind	NOUN
ejpam-977	43	14	module	module	NOUN
ejpam-977	43	15	(	(	PUNCT
ejpam-977	43	16	prüfer	prüfer	NOUN
ejpam-977	43	17	module	module	NOUN
ejpam-977	43	18	)	)	PUNCT
ejpam-977	43	19	if	if	SCONJ
ejpam-977	43	20	every	every	DET
ejpam-977	43	21	non	non	ADJ
ejpam-977	43	22	-	-	ADJ
ejpam-977	43	23	zero	zero	NUM
ejpam-977	43	24	(	(	PUNCT
ejpam-977	43	25	finitely	finitely	ADV
ejpam-977	43	26	generated	generate	VERB
ejpam-977	43	27	)	)	PUNCT
ejpam-977	43	28	submodule	submodule	NOUN
ejpam-977	43	29	of	of	ADP
ejpam-977	43	30	m	m	PROPN
ejpam-977	43	31	is	be	AUX
ejpam-977	43	32	invertible	invertible	ADJ
ejpam-977	43	33	.	.	PUNCT
ejpam-977	44	1	dedekind	dedekind	ADJ
ejpam-977	44	2	modules	module	NOUN
ejpam-977	44	3	and	and	CCONJ
ejpam-977	44	4	prüfer	prüfer	NOUN
ejpam-977	44	5	modules	module	NOUN
ejpam-977	44	6	have	have	AUX
ejpam-977	44	7	been	be	AUX
ejpam-977	44	8	extensively	extensively	ADV
ejpam-977	44	9	studied	study	VERB
ejpam-977	44	10	in	in	ADP
ejpam-977	44	11	[	[	X
ejpam-977	44	12	1	1	NUM
ejpam-977	44	13	]	]	PUNCT
ejpam-977	44	14	and	and	CCONJ
ejpam-977	44	15	[	[	X
ejpam-977	44	16	10	10	NUM
ejpam-977	44	17	]	]	PUNCT
ejpam-977	44	18	.	.	PUNCT
ejpam-977	45	1	in	in	ADP
ejpam-977	45	2	[	[	X
ejpam-977	45	3	1	1	NUM
ejpam-977	45	4	,	,	PUNCT
ejpam-977	45	5	theorem	theorem	VERB
ejpam-977	45	6	2.3	2.3	NUM
ejpam-977	45	7	]	]	PUNCT
ejpam-977	45	8	,	,	PUNCT
ejpam-977	45	9	it	it	PRON
ejpam-977	45	10	is	be	AUX
ejpam-977	45	11	proved	prove	VERB
ejpam-977	45	12	that	that	SCONJ
ejpam-977	45	13	if	if	SCONJ
ejpam-977	45	14	r	r	NOUN
ejpam-977	45	15	is	be	AUX
ejpam-977	45	16	an	an	DET
ejpam-977	45	17	integral	integral	ADJ
ejpam-977	45	18	domain	domain	NOUN
ejpam-977	45	19	and	and	CCONJ
ejpam-977	45	20	m	m	NOUN
ejpam-977	45	21	is	be	AUX
ejpam-977	45	22	a	a	DET
ejpam-977	45	23	faithful	faithful	ADJ
ejpam-977	45	24	multiplication	multiplication	NOUN
ejpam-977	45	25	r	r	NOUN
ejpam-977	45	26	-	-	PUNCT
ejpam-977	45	27	module	module	NOUN
ejpam-977	45	28	,	,	PUNCT
ejpam-977	45	29	then	then	ADV
ejpam-977	45	30	m	m	NOUN
ejpam-977	45	31	is	be	AUX
ejpam-977	45	32	a	a	DET
ejpam-977	45	33	prüfer	prüfer	NOUN
ejpam-977	45	34	module	module	NOUN
ejpam-977	45	35	if	if	SCONJ
ejpam-977	45	36	and	and	CCONJ
ejpam-977	45	37	only	only	ADV
ejpam-977	45	38	if	if	SCONJ
ejpam-977	45	39	every	every	DET
ejpam-977	45	40	finitely	finitely	ADV
ejpam-977	45	41	generated	generate	VERB
ejpam-977	45	42	submodule	submodule	NOUN
ejpam-977	45	43	of	of	ADP
ejpam-977	45	44	m	m	PROPN
ejpam-977	45	45	is	be	AUX
ejpam-977	45	46	principal	principal	ADJ
ejpam-977	45	47	.	.	PUNCT
ejpam-977	46	1	in	in	ADP
ejpam-977	46	2	this	this	DET
ejpam-977	46	3	paper	paper	NOUN
ejpam-977	46	4	we	we	PRON
ejpam-977	46	5	prove	prove	VERB
ejpam-977	46	6	that	that	SCONJ
ejpam-977	46	7	if	if	SCONJ
ejpam-977	46	8	m	m	NOUN
ejpam-977	46	9	is	be	AUX
ejpam-977	46	10	a	a	DET
ejpam-977	46	11	non	non	ADJ
ejpam-977	46	12	zero	zero	NUM
ejpam-977	46	13	faithful	faithful	ADJ
ejpam-977	46	14	multiplication	multiplication	NOUN
ejpam-977	46	15	rmodule	rmodule	NOUN
ejpam-977	46	16	,	,	PUNCT
ejpam-977	46	17	then	then	ADV
ejpam-977	46	18	r	r	NOUN
ejpam-977	46	19	is	be	AUX
ejpam-977	46	20	a	a	DET
ejpam-977	46	21	prüfer	prüfer	NOUN
ejpam-977	46	22	module	module	NOUN
ejpam-977	46	23	if	if	SCONJ
ejpam-977	46	24	and	and	CCONJ
ejpam-977	46	25	only	only	ADV
ejpam-977	46	26	if	if	SCONJ
ejpam-977	46	27	r	r	NOUN
ejpam-977	46	28	is	be	AUX
ejpam-977	46	29	an	an	DET
ejpam-977	46	30	integral	integral	ADJ
ejpam-977	46	31	domain	domain	NOUN
ejpam-977	46	32	and	and	CCONJ
ejpam-977	46	33	every	every	DET
ejpam-977	46	34	finitely	finitely	ADV
ejpam-977	46	35	generated	generate	VERB
ejpam-977	46	36	submodule	submodule	NOUN
ejpam-977	46	37	of	of	ADP
ejpam-977	46	38	m	m	PROPN
ejpam-977	46	39	is	be	AUX
ejpam-977	46	40	join	join	NOUN
ejpam-977	46	41	-	-	PUNCT
ejpam-977	46	42	quasi	quasi	NOUN
ejpam-977	46	43	-	-	NOUN
ejpam-977	46	44	cyclic	cyclic	ADJ
ejpam-977	46	45	(	(	PUNCT
ejpam-977	46	46	i.e.	i.e.	X
ejpam-977	46	47	,	,	PUNCT
ejpam-977	46	48	join	join	VERB
ejpam-977	46	49	principal	principal	NOUN
ejpam-977	46	50	)	)	PUNCT
ejpam-977	46	51	.	.	PUNCT
ejpam-977	47	1	next	next	ADV
ejpam-977	47	2	we	we	PRON
ejpam-977	47	3	show	show	VERB
ejpam-977	47	4	that	that	SCONJ
ejpam-977	47	5	if	if	SCONJ
ejpam-977	47	6	m	m	NOUN
ejpam-977	47	7	is	be	AUX
ejpam-977	47	8	a	a	DET
ejpam-977	47	9	non	non	ADJ
ejpam-977	47	10	zero	zero	NUM
ejpam-977	47	11	faithful	faithful	ADJ
ejpam-977	47	12	multiplication	multiplication	NOUN
ejpam-977	47	13	r	r	NOUN
ejpam-977	47	14	-	-	PUNCT
ejpam-977	47	15	module	module	NOUN
ejpam-977	47	16	,	,	PUNCT
ejpam-977	47	17	then	then	ADV
ejpam-977	47	18	r	r	NOUN
ejpam-977	47	19	is	be	AUX
ejpam-977	47	20	a	a	DET
ejpam-977	47	21	dedekind	dedekind	NOUN
ejpam-977	47	22	module	module	NOUN
ejpam-977	47	23	if	if	SCONJ
ejpam-977	47	24	and	and	CCONJ
ejpam-977	47	25	only	only	ADV
ejpam-977	47	26	if	if	SCONJ
ejpam-977	47	27	r	r	NOUN
ejpam-977	47	28	is	be	AUX
ejpam-977	47	29	an	an	DET
ejpam-977	47	30	integral	integral	ADJ
ejpam-977	47	31	domain	domain	NOUN
ejpam-977	47	32	and	and	CCONJ
ejpam-977	47	33	every	every	DET
ejpam-977	47	34	submodule	submodule	NOUN
ejpam-977	47	35	of	of	ADP
ejpam-977	47	36	m	m	PROPN
ejpam-977	47	37	is	be	AUX
ejpam-977	47	38	a	a	DET
ejpam-977	47	39	finitely	finitely	ADV
ejpam-977	47	40	generated	generate	VERB
ejpam-977	47	41	join	join	NOUN
ejpam-977	47	42	-	-	PUNCT
ejpam-977	47	43	quasi	quasi	NOUN
ejpam-977	47	44	-	-	ADJ
ejpam-977	47	45	cyclic	cyclic	ADJ
ejpam-977	47	46	submodule	submodule	NOUN
ejpam-977	47	47	of	of	ADP
ejpam-977	47	48	m	m	PROPN
ejpam-977	47	49	.	.	PUNCT
ejpam-977	48	1	for	for	ADP
ejpam-977	48	2	general	general	ADJ
ejpam-977	48	3	background	background	NOUN
ejpam-977	48	4	and	and	CCONJ
ejpam-977	48	5	terminology	terminology	NOUN
ejpam-977	48	6	,	,	PUNCT
ejpam-977	48	7	the	the	DET
ejpam-977	48	8	reader	reader	NOUN
ejpam-977	48	9	is	be	AUX
ejpam-977	48	10	referred	refer	VERB
ejpam-977	48	11	to	to	ADP
ejpam-977	48	12	[	[	X
ejpam-977	48	13	8	8	NUM
ejpam-977	48	14	]	]	SYM
ejpam-977	48	15	.	.	PUNCT
ejpam-977	49	1	2	2	X
ejpam-977	49	2	.	.	X
ejpam-977	49	3	prüfer	prüfer	NOUN
ejpam-977	49	4	modules	module	NOUN
ejpam-977	49	5	and	and	CCONJ
ejpam-977	49	6	dedekind	dedekind	NOUN
ejpam-977	49	7	modules	module	NOUN
ejpam-977	49	8	.	.	PUNCT
ejpam-977	50	1	in	in	ADP
ejpam-977	50	2	this	this	DET
ejpam-977	50	3	paper	paper	NOUN
ejpam-977	50	4	we	we	PRON
ejpam-977	50	5	establish	establish	VERB
ejpam-977	50	6	some	some	DET
ejpam-977	50	7	new	new	ADJ
ejpam-977	50	8	characterizations	characterization	NOUN
ejpam-977	50	9	for	for	ADP
ejpam-977	50	10	prüfer	prüfer	NOUN
ejpam-977	50	11	modules	module	NOUN
ejpam-977	50	12	and	and	CCONJ
ejpam-977	50	13	dedekind	dedekind	NOUN
ejpam-977	50	14	modules	module	NOUN
ejpam-977	50	15	.	.	PUNCT
ejpam-977	51	1	we	we	PRON
ejpam-977	51	2	shall	shall	AUX
ejpam-977	51	3	begin	begin	VERB
ejpam-977	51	4	with	with	ADP
ejpam-977	51	5	the	the	DET
ejpam-977	51	6	following	follow	VERB
ejpam-977	51	7	lemmas	lemmas	NOUN
ejpam-977	51	8	.	.	PUNCT
ejpam-977	52	1	lemma	lemma	PROPN
ejpam-977	52	2	1	1	X
ejpam-977	52	3	.	.	PUNCT
ejpam-977	52	4	suppose	suppose	VERB
ejpam-977	52	5	m	m	NOUN
ejpam-977	52	6	is	be	AUX
ejpam-977	52	7	a	a	DET
ejpam-977	52	8	non	non	ADJ
ejpam-977	52	9	zero	zero	NUM
ejpam-977	52	10	faithful	faithful	ADJ
ejpam-977	52	11	finitely	finitely	ADV
ejpam-977	52	12	generated	generate	VERB
ejpam-977	52	13	weak	weak	ADJ
ejpam-977	52	14	-	-	PUNCT
ejpam-977	52	15	join	join	NOUN
ejpam-977	52	16	-	-	PUNCT
ejpam-977	52	17	quasi	quasi	NOUN
ejpam-977	52	18	-	-	ADJ
ejpam-977	52	19	cyclic	cyclic	ADJ
ejpam-977	52	20	r	r	NOUN
ejpam-977	52	21	-	-	PUNCT
ejpam-977	52	22	module	module	NOUN
ejpam-977	52	23	and	and	CCONJ
ejpam-977	52	24	b	b	NOUN
ejpam-977	52	25	is	be	AUX
ejpam-977	52	26	an	an	DET
ejpam-977	52	27	ideal	ideal	NOUN
ejpam-977	52	28	of	of	ADP
ejpam-977	52	29	r.	r.	PROPN
ejpam-977	52	30	if	if	SCONJ
ejpam-977	52	31	bm	bm	PROPN
ejpam-977	52	32	is	be	AUX
ejpam-977	52	33	weak	weak	ADJ
ejpam-977	52	34	-	-	PUNCT
ejpam-977	52	35	join	join	NOUN
ejpam-977	52	36	-	-	PUNCT
ejpam-977	52	37	quasi	quasi	NOUN
ejpam-977	52	38	-	-	NOUN
ejpam-977	52	39	cyclic	cyclic	ADJ
ejpam-977	52	40	(	(	PUNCT
ejpam-977	52	41	join	join	NOUN
ejpam-977	52	42	-	-	PUNCT
ejpam-977	52	43	quasi	quasi	NOUN
ejpam-977	52	44	-	-	NOUN
ejpam-977	52	45	cyclic	cyclic	ADJ
ejpam-977	52	46	)	)	PUNCT
ejpam-977	52	47	,	,	PUNCT
ejpam-977	52	48	then	then	ADV
ejpam-977	52	49	b	b	PROPN
ejpam-977	52	50	is	be	AUX
ejpam-977	52	51	weak	weak	ADJ
ejpam-977	52	52	join	join	NOUN
ejpam-977	52	53	principal	principal	NOUN
ejpam-977	52	54	(	(	PUNCT
ejpam-977	52	55	join	join	VERB
ejpam-977	52	56	principal	principal	NOUN
ejpam-977	52	57	)	)	PUNCT
ejpam-977	52	58	.	.	PUNCT
ejpam-977	53	1	proof	proof	NOUN
ejpam-977	53	2	.	.	PUNCT
ejpam-977	54	1	let	let	VERB
ejpam-977	54	2	a	a	DET
ejpam-977	54	3	∈	∈	NOUN
ejpam-977	54	4	l(r	l(r	PROPN
ejpam-977	54	5	)	)	PUNCT
ejpam-977	54	6	.	.	PUNCT
ejpam-977	55	1	since	since	SCONJ
ejpam-977	55	2	m	m	PROPN
ejpam-977	55	3	is	be	AUX
ejpam-977	55	4	faithful	faithful	ADJ
ejpam-977	55	5	and	and	CCONJ
ejpam-977	55	6	weak	weak	ADJ
ejpam-977	55	7	-	-	PUNCT
ejpam-977	55	8	join	join	NOUN
ejpam-977	55	9	-	-	PUNCT
ejpam-977	55	10	quasi	quasi	NOUN
ejpam-977	55	11	-	-	NOUN
ejpam-977	55	12	cyclic	cyclic	ADJ
ejpam-977	55	13	,	,	PUNCT
ejpam-977	55	14	we	we	PRON
ejpam-977	55	15	have	have	VERB
ejpam-977	55	16	(	(	PUNCT
ejpam-977	55	17	ab	ab	NOUN
ejpam-977	55	18	:	:	PUNCT
ejpam-977	55	19	b	b	X
ejpam-977	55	20	)	)	PUNCT
ejpam-977	55	21	=	=	SYM
ejpam-977	55	22	(	(	PUNCT
ejpam-977	55	23	abm	abm	PROPN
ejpam-977	55	24	:	:	PUNCT
ejpam-977	55	25	bm	bm	PROPN
ejpam-977	55	26	)	)	PUNCT
ejpam-977	55	27	.	.	PUNCT
ejpam-977	56	1	as	as	SCONJ
ejpam-977	56	2	bm	bm	PROPN
ejpam-977	56	3	is	be	AUX
ejpam-977	56	4	weak	weak	ADJ
ejpam-977	56	5	-	-	PUNCT
ejpam-977	56	6	join	join	NOUN
ejpam-977	56	7	-	-	PUNCT
ejpam-977	56	8	quasi	quasi	NOUN
ejpam-977	56	9	-	-	NOUN
ejpam-977	56	10	cyclic	cyclic	ADJ
ejpam-977	56	11	,	,	PUNCT
ejpam-977	56	12	we	we	PRON
ejpam-977	56	13	have	have	VERB
ejpam-977	56	14	(	(	PUNCT
ejpam-977	56	15	abm	abm	PROPN
ejpam-977	56	16	:	:	PUNCT
ejpam-977	56	17	bm	bm	PROPN
ejpam-977	56	18	)	)	PUNCT
ejpam-977	56	19	=	=	SYM
ejpam-977	56	20	a+(0	a+(0	NOUN
ejpam-977	56	21	:	:	PUNCT
ejpam-977	56	22	bm	bm	PROPN
ejpam-977	56	23	)	)	PUNCT
ejpam-977	56	24	=	=	PUNCT
ejpam-977	57	1	a+((0	a+((0	X
ejpam-977	57	2	:	:	PUNCT
ejpam-977	57	3	m	m	X
ejpam-977	57	4	)	)	PUNCT
ejpam-977	57	5	:	:	PUNCT
ejpam-977	57	6	b	b	X
ejpam-977	57	7	)	)	PUNCT
ejpam-977	57	8	=	=	SYM
ejpam-977	57	9	a+	a+	PUNCT
ejpam-977	57	10	(	(	PUNCT
ejpam-977	57	11	0	0	NUM
ejpam-977	57	12	:	:	PUNCT
ejpam-977	57	13	b	b	X
ejpam-977	57	14	)	)	PUNCT
ejpam-977	57	15	.	.	PUNCT
ejpam-977	58	1	therefore	therefore	ADV
ejpam-977	58	2	b	b	PROPN
ejpam-977	58	3	is	be	AUX
ejpam-977	58	4	weak	weak	ADJ
ejpam-977	58	5	join	join	NOUN
ejpam-977	58	6	principal	principal	NOUN
ejpam-977	58	7	.	.	PUNCT
ejpam-977	59	1	let	let	VERB
ejpam-977	59	2	a	a	PRON
ejpam-977	59	3	,	,	PUNCT
ejpam-977	59	4	c	c	PROPN
ejpam-977	59	5	∈	∈	PROPN
ejpam-977	59	6	l(r	l(r	PROPN
ejpam-977	59	7	)	)	PUNCT
ejpam-977	59	8	.	.	PUNCT
ejpam-977	60	1	since	since	SCONJ
ejpam-977	60	2	m	m	PROPN
ejpam-977	60	3	is	be	AUX
ejpam-977	60	4	faithful	faithful	ADJ
ejpam-977	60	5	and	and	CCONJ
ejpam-977	60	6	weak	weak	ADJ
ejpam-977	60	7	-	-	PUNCT
ejpam-977	60	8	join	join	NOUN
ejpam-977	60	9	-	-	PUNCT
ejpam-977	60	10	quasi	quasi	NOUN
ejpam-977	60	11	-	-	NOUN
ejpam-977	60	12	cyclic	cyclic	ADJ
ejpam-977	60	13	,	,	PUNCT
ejpam-977	60	14	it	it	PRON
ejpam-977	60	15	follows	follow	VERB
ejpam-977	60	16	that	that	SCONJ
ejpam-977	60	17	(	(	PUNCT
ejpam-977	60	18	(	(	PUNCT
ejpam-977	60	19	ab+	ab+	PROPN
ejpam-977	60	20	c	c	PROPN
ejpam-977	60	21	)	)	PUNCT
ejpam-977	60	22	:	:	PUNCT
ejpam-977	61	1	b	b	X
ejpam-977	61	2	)	)	PUNCT
ejpam-977	61	3	=	=	SYM
ejpam-977	61	4	(	(	PUNCT
ejpam-977	61	5	(	(	PUNCT
ejpam-977	61	6	abm	abm	PROPN
ejpam-977	61	7	+	+	PROPN
ejpam-977	61	8	c	c	NOUN
ejpam-977	61	9	m	m	PROPN
ejpam-977	61	10	)	)	PUNCT
ejpam-977	61	11	:	:	PUNCT
ejpam-977	61	12	bm	bm	PROPN
ejpam-977	61	13	)	)	PUNCT
ejpam-977	61	14	.	.	PUNCT
ejpam-977	62	1	as	as	SCONJ
ejpam-977	62	2	bm	bm	PROPN
ejpam-977	62	3	is	be	AUX
ejpam-977	62	4	join	join	NOUN
ejpam-977	62	5	-	-	PUNCT
ejpam-977	62	6	quasi	quasi	NOUN
ejpam-977	62	7	-	-	NOUN
ejpam-977	62	8	cyclic	cyclic	ADJ
ejpam-977	62	9	,	,	PUNCT
ejpam-977	62	10	we	we	PRON
ejpam-977	62	11	have	have	VERB
ejpam-977	62	12	(	(	PUNCT
ejpam-977	62	13	(	(	PUNCT
ejpam-977	62	14	abm	abm	PROPN
ejpam-977	62	15	+	+	PROPN
ejpam-977	62	16	c	c	NOUN
ejpam-977	62	17	m	m	PROPN
ejpam-977	62	18	)	)	PUNCT
ejpam-977	62	19	:	:	PUNCT
ejpam-977	62	20	bm	bm	PROPN
ejpam-977	62	21	)	)	PUNCT
ejpam-977	62	22	=	=	PRON
ejpam-977	62	23	a+	a+	PUNCT
ejpam-977	62	24	(	(	PUNCT
ejpam-977	62	25	c	c	NOUN
ejpam-977	62	26	m	m	VERB
ejpam-977	62	27	:	:	PUNCT
ejpam-977	62	28	bm	bm	X
ejpam-977	62	29	)	)	PUNCT
ejpam-977	62	30	=	=	PRON
ejpam-977	62	31	a+	a+	PUNCT
ejpam-977	62	32	(	(	PUNCT
ejpam-977	62	33	c	c	NOUN
ejpam-977	62	34	:	:	PUNCT
ejpam-977	62	35	b	b	X
ejpam-977	62	36	)	)	PUNCT
ejpam-977	62	37	since	since	SCONJ
ejpam-977	62	38	m	m	PROPN
ejpam-977	62	39	is	be	AUX
ejpam-977	62	40	faithful	faithful	ADJ
ejpam-977	62	41	and	and	CCONJ
ejpam-977	62	42	weak	weak	ADJ
ejpam-977	62	43	-	-	PUNCT
ejpam-977	62	44	join	join	NOUN
ejpam-977	62	45	-	-	PUNCT
ejpam-977	62	46	quasi	quasi	NOUN
ejpam-977	62	47	-	-	NOUN
ejpam-977	62	48	cyclic	cyclic	ADJ
ejpam-977	62	49	.	.	PUNCT
ejpam-977	63	1	therefore	therefore	ADV
ejpam-977	63	2	b	b	PROPN
ejpam-977	63	3	is	be	AUX
ejpam-977	63	4	join	join	VERB
ejpam-977	63	5	principal	principal	NOUN
ejpam-977	63	6	.	.	PUNCT
ejpam-977	64	1	lemma	lemma	PROPN
ejpam-977	64	2	2	2	X
ejpam-977	64	3	.	.	PUNCT
ejpam-977	64	4	suppose	suppose	VERB
ejpam-977	64	5	m	m	NOUN
ejpam-977	64	6	is	be	AUX
ejpam-977	64	7	a	a	DET
ejpam-977	64	8	non	non	ADJ
ejpam-977	64	9	zero	zero	NUM
ejpam-977	64	10	faithful	faithful	ADJ
ejpam-977	64	11	finitely	finitely	ADV
ejpam-977	64	12	generated	generate	VERB
ejpam-977	64	13	weak	weak	ADJ
ejpam-977	64	14	-	-	PUNCT
ejpam-977	64	15	join	join	NOUN
ejpam-977	64	16	-	-	PUNCT
ejpam-977	64	17	quasi	quasi	NOUN
ejpam-977	64	18	-	-	ADJ
ejpam-977	64	19	cyclic	cyclic	ADJ
ejpam-977	64	20	r	r	NOUN
ejpam-977	64	21	-	-	PUNCT
ejpam-977	64	22	module	module	NOUN
ejpam-977	64	23	.	.	PUNCT
ejpam-977	65	1	suppose	suppose	VERB
ejpam-977	65	2	r	r	NOUN
ejpam-977	65	3	is	be	AUX
ejpam-977	65	4	an	an	DET
ejpam-977	65	5	integral	integral	ADJ
ejpam-977	65	6	domain	domain	NOUN
ejpam-977	65	7	and	and	CCONJ
ejpam-977	65	8	b	b	NOUN
ejpam-977	65	9	is	be	AUX
ejpam-977	65	10	a	a	DET
ejpam-977	65	11	finitely	finitely	ADV
ejpam-977	65	12	generated	generate	VERB
ejpam-977	65	13	ideal	ideal	NOUN
ejpam-977	65	14	of	of	ADP
ejpam-977	65	15	r.	r.	PROPN
ejpam-977	65	16	if	if	SCONJ
ejpam-977	65	17	bm	bm	PROPN
ejpam-977	65	18	is	be	AUX
ejpam-977	65	19	weak	weak	ADJ
ejpam-977	65	20	-	-	PUNCT
ejpam-977	65	21	joinquasi	joinquasi	NOUN
ejpam-977	65	22	-	-	PUNCT
ejpam-977	65	23	cyclic	cyclic	ADJ
ejpam-977	65	24	,	,	PUNCT
ejpam-977	65	25	then	then	ADV
ejpam-977	65	26	b	b	NOUN
ejpam-977	65	27	is	be	AUX
ejpam-977	65	28	quasi	quasi	ADJ
ejpam-977	65	29	-	-	NOUN
ejpam-977	65	30	principal	principal	ADJ
ejpam-977	65	31	.	.	PUNCT
ejpam-977	66	1	proof	proof	NOUN
ejpam-977	66	2	.	.	PUNCT
ejpam-977	67	1	by	by	ADP
ejpam-977	67	2	lemma	lemma	PROPN
ejpam-977	67	3	1	1	NUM
ejpam-977	67	4	,	,	PUNCT
ejpam-977	67	5	b	b	NOUN
ejpam-977	67	6	is	be	AUX
ejpam-977	67	7	weak	weak	ADJ
ejpam-977	67	8	join	join	NOUN
ejpam-977	67	9	principal	principal	NOUN
ejpam-977	67	10	,	,	PUNCT
ejpam-977	67	11	so	so	ADV
ejpam-977	67	12	by	by	ADP
ejpam-977	67	13	[	[	X
ejpam-977	67	14	2	2	NUM
ejpam-977	67	15	,	,	PUNCT
ejpam-977	67	16	theorem	theorem	VERB
ejpam-977	67	17	4	4	NUM
ejpam-977	67	18	]	]	PUNCT
ejpam-977	67	19	,	,	PUNCT
ejpam-977	67	20	b	b	NOUN
ejpam-977	67	21	is	be	AUX
ejpam-977	67	22	quasi	quasi	ADJ
ejpam-977	67	23	-	-	NOUN
ejpam-977	67	24	principal	principal	ADJ
ejpam-977	67	25	.	.	PUNCT
ejpam-977	68	1	c.	c.	PROPN
ejpam-977	68	2	jayaram	jayaram	PROPN
ejpam-977	68	3	and	and	CCONJ
ejpam-977	68	4	v.	v.	ADP
ejpam-977	68	5	raju	raju	PROPN
ejpam-977	68	6	/	/	SYM
ejpam-977	68	7	eur	eur	PROPN
ejpam-977	68	8	.	.	PUNCT
ejpam-977	69	1	j.	j.	PROPN
ejpam-977	69	2	pure	pure	PROPN
ejpam-977	69	3	appl	appl	PROPN
ejpam-977	69	4	.	.	PROPN
ejpam-977	69	5	math	math	PROPN
ejpam-977	69	6	,	,	PUNCT
ejpam-977	69	7	3	3	NUM
ejpam-977	69	8	(	(	PUNCT
ejpam-977	69	9	2010	2010	NUM
ejpam-977	69	10	)	)	PUNCT
ejpam-977	69	11	,	,	PUNCT
ejpam-977	69	12	899	899	NUM
ejpam-977	69	13	-	-	SYM
ejpam-977	69	14	902	902	NUM
ejpam-977	69	15	901	901	NUM
ejpam-977	69	16	lemma	lemma	PROPN
ejpam-977	69	17	3	3	X
ejpam-977	69	18	.	.	PUNCT
ejpam-977	69	19	suppose	suppose	VERB
ejpam-977	69	20	r	r	NOUN
ejpam-977	69	21	is	be	AUX
ejpam-977	69	22	an	an	DET
ejpam-977	69	23	integral	integral	ADJ
ejpam-977	69	24	domain	domain	NOUN
ejpam-977	69	25	and	and	CCONJ
ejpam-977	69	26	m	m	NOUN
ejpam-977	69	27	is	be	AUX
ejpam-977	69	28	a	a	DET
ejpam-977	69	29	non	non	ADJ
ejpam-977	69	30	zero	zero	NUM
ejpam-977	69	31	faithful	faithful	ADJ
ejpam-977	69	32	finitely	finitely	ADV
ejpam-977	69	33	generated	generate	VERB
ejpam-977	69	34	rmodule	rmodule	NOUN
ejpam-977	69	35	.	.	PUNCT
ejpam-977	70	1	if	if	SCONJ
ejpam-977	70	2	every	every	DET
ejpam-977	70	3	finitely	finitely	ADV
ejpam-977	70	4	generated	generate	VERB
ejpam-977	70	5	submodule	submodule	NOUN
ejpam-977	70	6	of	of	ADP
ejpam-977	70	7	m	m	PROPN
ejpam-977	70	8	is	be	AUX
ejpam-977	70	9	weak	weak	ADJ
ejpam-977	70	10	-	-	PUNCT
ejpam-977	70	11	join	join	NOUN
ejpam-977	70	12	-	-	PUNCT
ejpam-977	70	13	quasi	quasi	NOUN
ejpam-977	70	14	-	-	NOUN
ejpam-977	70	15	cyclic	cyclic	ADJ
ejpam-977	70	16	,	,	PUNCT
ejpam-977	70	17	then	then	ADV
ejpam-977	70	18	r	r	NOUN
ejpam-977	70	19	is	be	AUX
ejpam-977	70	20	a	a	DET
ejpam-977	70	21	prüfer	prüfer	NOUN
ejpam-977	70	22	domain	domain	NOUN
ejpam-977	70	23	.	.	PUNCT
ejpam-977	71	1	proof	proof	NOUN
ejpam-977	71	2	.	.	PUNCT
ejpam-977	72	1	let	let	VERB
ejpam-977	72	2	i	i	PRON
ejpam-977	72	3	be	be	AUX
ejpam-977	72	4	a	a	DET
ejpam-977	72	5	finitely	finitely	ADV
ejpam-977	72	6	generated	generate	VERB
ejpam-977	72	7	ideal	ideal	NOUN
ejpam-977	72	8	of	of	ADP
ejpam-977	72	9	r.	r.	PROPN
ejpam-977	72	10	then	then	ADV
ejpam-977	72	11	i	i	PRON
ejpam-977	72	12	m	m	VERB
ejpam-977	72	13	is	be	AUX
ejpam-977	72	14	finitely	finitely	ADV
ejpam-977	72	15	generated	generate	VERB
ejpam-977	72	16	,	,	PUNCT
ejpam-977	72	17	so	so	CCONJ
ejpam-977	72	18	i	i	PRON
ejpam-977	72	19	m	m	VERB
ejpam-977	72	20	is	be	AUX
ejpam-977	72	21	weak	weak	ADJ
ejpam-977	72	22	-	-	PUNCT
ejpam-977	72	23	join	join	NOUN
ejpam-977	72	24	-	-	PUNCT
ejpam-977	72	25	quasi	quasi	NOUN
ejpam-977	72	26	-	-	NOUN
ejpam-977	72	27	cyclic	cyclic	ADJ
ejpam-977	72	28	.	.	PUNCT
ejpam-977	73	1	by	by	ADP
ejpam-977	73	2	lemma	lemma	PROPN
ejpam-977	73	3	2	2	NUM
ejpam-977	73	4	,	,	PUNCT
ejpam-977	73	5	i	i	PRON
ejpam-977	73	6	is	be	AUX
ejpam-977	73	7	quasi	quasi	ADJ
ejpam-977	73	8	-	-	NOUN
ejpam-977	73	9	principal	principal	ADJ
ejpam-977	73	10	and	and	CCONJ
ejpam-977	73	11	hence	hence	ADV
ejpam-977	73	12	r	r	NOUN
ejpam-977	73	13	is	be	AUX
ejpam-977	73	14	a	a	DET
ejpam-977	73	15	prüfer	prüfer	NOUN
ejpam-977	73	16	domain	domain	NOUN
ejpam-977	73	17	[	[	X
ejpam-977	73	18	8	8	NUM
ejpam-977	73	19	,	,	PUNCT
ejpam-977	73	20	page	page	NOUN
ejpam-977	73	21	147	147	NUM
ejpam-977	73	22	,	,	PUNCT
ejpam-977	73	23	ex	ex	NOUN
ejpam-977	73	24	.	.	NOUN
ejpam-977	73	25	10(e	10(e	NUM
ejpam-977	73	26	)	)	PUNCT
ejpam-977	73	27	]	]	PUNCT
ejpam-977	73	28	.	.	PUNCT
ejpam-977	74	1	lemma	lemma	PROPN
ejpam-977	74	2	4	4	X
ejpam-977	74	3	.	.	PUNCT
ejpam-977	74	4	suppose	suppose	VERB
ejpam-977	74	5	r	r	NOUN
ejpam-977	74	6	is	be	AUX
ejpam-977	74	7	an	an	DET
ejpam-977	74	8	arithmetical	arithmetical	ADJ
ejpam-977	74	9	ring	ring	NOUN
ejpam-977	74	10	and	and	CCONJ
ejpam-977	74	11	m	m	NOUN
ejpam-977	74	12	is	be	AUX
ejpam-977	74	13	a	a	DET
ejpam-977	74	14	non	non	ADJ
ejpam-977	74	15	zero	zero	NUM
ejpam-977	74	16	finitely	finitely	ADV
ejpam-977	74	17	generated	generate	VERB
ejpam-977	74	18	r	r	NOUN
ejpam-977	74	19	-	-	PUNCT
ejpam-977	74	20	module	module	NOUN
ejpam-977	74	21	.	.	PUNCT
ejpam-977	75	1	then	then	ADV
ejpam-977	75	2	every	every	DET
ejpam-977	75	3	finitely	finitely	ADV
ejpam-977	75	4	generated	generate	VERB
ejpam-977	75	5	submodule	submodule	NOUN
ejpam-977	75	6	of	of	ADP
ejpam-977	75	7	m	m	PROPN
ejpam-977	75	8	is	be	AUX
ejpam-977	75	9	join	join	NOUN
ejpam-977	75	10	-	-	PUNCT
ejpam-977	75	11	quasi	quasi	NOUN
ejpam-977	75	12	-	-	ADJ
ejpam-977	75	13	cyclic	cyclic	ADJ
ejpam-977	75	14	.	.	PUNCT
ejpam-977	76	1	proof	proof	NOUN
ejpam-977	76	2	.	.	PUNCT
ejpam-977	77	1	let	let	VERB
ejpam-977	77	2	n	n	PRON
ejpam-977	77	3	be	be	AUX
ejpam-977	77	4	a	a	DET
ejpam-977	77	5	finitely	finitely	ADV
ejpam-977	77	6	generated	generate	VERB
ejpam-977	77	7	submodule	submodule	NOUN
ejpam-977	77	8	of	of	ADP
ejpam-977	77	9	m	m	PROPN
ejpam-977	77	10	.	.	PUNCT
ejpam-977	78	1	it	it	PRON
ejpam-977	78	2	is	be	AUX
ejpam-977	78	3	enough	enough	ADJ
ejpam-977	78	4	to	to	PART
ejpam-977	78	5	show	show	VERB
ejpam-977	78	6	that	that	SCONJ
ejpam-977	78	7	n	n	NOUN
ejpam-977	78	8	is	be	AUX
ejpam-977	78	9	locally	locally	ADV
ejpam-977	78	10	join	join	NOUN
ejpam-977	78	11	-	-	PUNCT
ejpam-977	78	12	quasi	quasi	NOUN
ejpam-977	78	13	-	-	NOUN
ejpam-977	78	14	cyclic	cyclic	ADJ
ejpam-977	78	15	.	.	PUNCT
ejpam-977	79	1	assume	assume	VERB
ejpam-977	79	2	that	that	SCONJ
ejpam-977	79	3	r	r	NOUN
ejpam-977	79	4	is	be	AUX
ejpam-977	79	5	a	a	DET
ejpam-977	79	6	valuation	valuation	NOUN
ejpam-977	79	7	ring(i.e	ring(i.e	NOUN
ejpam-977	79	8	.	.	PUNCT
ejpam-977	79	9	,	,	PUNCT
ejpam-977	79	10	any	any	DET
ejpam-977	79	11	two	two	NUM
ejpam-977	79	12	ideals	ideal	NOUN
ejpam-977	79	13	are	be	AUX
ejpam-977	79	14	comparable	comparable	ADJ
ejpam-977	79	15	)	)	PUNCT
ejpam-977	79	16	.	.	PUNCT
ejpam-977	80	1	let	let	VERB
ejpam-977	80	2	a	a	DET
ejpam-977	80	3	∈	∈	NOUN
ejpam-977	80	4	l(r	l(r	PROPN
ejpam-977	80	5	)	)	PUNCT
ejpam-977	80	6	and	and	CCONJ
ejpam-977	80	7	b	b	X
ejpam-977	80	8	∈	∈	PROPN
ejpam-977	80	9	l(m	l(m	PROPN
ejpam-977	80	10	)	)	PUNCT
ejpam-977	80	11	.	.	PUNCT
ejpam-977	81	1	clearly	clearly	ADV
ejpam-977	81	2	,	,	PUNCT
ejpam-977	81	3	a+	a+	PUNCT
ejpam-977	81	4	(	(	PUNCT
ejpam-977	81	5	b	b	X
ejpam-977	81	6	:	:	PUNCT
ejpam-977	81	7	n	n	CCONJ
ejpam-977	81	8	)	)	PUNCT
ejpam-977	81	9	⊆	⊆	NUM
ejpam-977	81	10	(	(	PUNCT
ejpam-977	81	11	(	(	PUNCT
ejpam-977	81	12	an	an	DET
ejpam-977	81	13	+	+	NUM
ejpam-977	81	14	b	b	NOUN
ejpam-977	81	15	)	)	PUNCT
ejpam-977	81	16	:	:	PUNCT
ejpam-977	81	17	n	n	CCONJ
ejpam-977	81	18	)	)	PUNCT
ejpam-977	81	19	.	.	PUNCT
ejpam-977	82	1	let	let	VERB
ejpam-977	82	2	a	a	DET
ejpam-977	82	3	∈	∈	NOUN
ejpam-977	82	4	(	(	PUNCT
ejpam-977	82	5	(	(	PUNCT
ejpam-977	82	6	an	an	DET
ejpam-977	82	7	+	+	NUM
ejpam-977	82	8	b	b	NOUN
ejpam-977	82	9	)	)	PUNCT
ejpam-977	82	10	:	:	PUNCT
ejpam-977	82	11	n	n	CCONJ
ejpam-977	82	12	)	)	PUNCT
ejpam-977	82	13	.	.	PUNCT
ejpam-977	83	1	then	then	ADV
ejpam-977	83	2	an	an	DET
ejpam-977	83	3	⊆	⊆	NUM
ejpam-977	83	4	an	an	DET
ejpam-977	83	5	+	+	NOUN
ejpam-977	83	6	b.	b.	NOUN
ejpam-977	83	7	we	we	PRON
ejpam-977	83	8	have	have	VERB
ejpam-977	83	9	either	either	CCONJ
ejpam-977	83	10	(	(	PUNCT
ejpam-977	83	11	a	a	X
ejpam-977	83	12	)	)	PUNCT
ejpam-977	83	13	⊆	⊆	NUM
ejpam-977	83	14	a	a	PRON
ejpam-977	83	15	or	or	CCONJ
ejpam-977	83	16	a⊆	a⊆	PROPN
ejpam-977	83	17	(	(	PUNCT
ejpam-977	83	18	a	a	NOUN
ejpam-977	83	19	)	)	PUNCT
ejpam-977	83	20	.	.	PUNCT
ejpam-977	84	1	if	if	SCONJ
ejpam-977	84	2	(	(	PUNCT
ejpam-977	84	3	a	a	X
ejpam-977	84	4	)	)	PUNCT
ejpam-977	84	5	⊆	⊆	NUM
ejpam-977	84	6	a	a	PRON
ejpam-977	84	7	,	,	PUNCT
ejpam-977	84	8	then	then	ADV
ejpam-977	84	9	we	we	PRON
ejpam-977	84	10	are	be	AUX
ejpam-977	84	11	through	through	ADP
ejpam-977	84	12	.	.	PUNCT
ejpam-977	84	13	suppose	suppose	VERB
ejpam-977	84	14	a⊂	a⊂	NOUN
ejpam-977	84	15	(	(	PUNCT
ejpam-977	84	16	a	a	NOUN
ejpam-977	84	17	)	)	PUNCT
ejpam-977	84	18	.	.	PUNCT
ejpam-977	85	1	as	as	SCONJ
ejpam-977	85	2	(	(	PUNCT
ejpam-977	85	3	a	a	X
ejpam-977	85	4	)	)	PUNCT
ejpam-977	85	5	is	be	AUX
ejpam-977	85	6	a	a	DET
ejpam-977	85	7	multiplication	multiplication	NOUN
ejpam-977	85	8	ideal	ideal	ADJ
ejpam-977	85	9	,	,	PUNCT
ejpam-977	85	10	it	it	PRON
ejpam-977	85	11	follows	follow	VERB
ejpam-977	85	12	that	that	SCONJ
ejpam-977	85	13	a=	a=	VERB
ejpam-977	85	14	j(a	j(a	NOUN
ejpam-977	85	15	)	)	PUNCT
ejpam-977	85	16	for	for	ADP
ejpam-977	85	17	some	some	DET
ejpam-977	85	18	proper	proper	ADJ
ejpam-977	85	19	ideal	ideal	ADJ
ejpam-977	85	20	j	j	PROPN
ejpam-977	85	21	of	of	ADP
ejpam-977	85	22	r.	r.	PROPN
ejpam-977	85	23	so	so	ADV
ejpam-977	85	24	an	an	DET
ejpam-977	85	25	⊆	⊆	NUM
ejpam-977	85	26	j(a)n	j(a)n	PROPN
ejpam-977	85	27	+	+	PROPN
ejpam-977	85	28	b	b	NOUN
ejpam-977	85	29	,	,	PUNCT
ejpam-977	85	30	so	so	ADV
ejpam-977	85	31	by	by	ADP
ejpam-977	85	32	nakayama	nakayama	PROPN
ejpam-977	85	33	’s	’s	PART
ejpam-977	85	34	lemma	lemma	PROPN
ejpam-977	85	35	an	an	DET
ejpam-977	85	36	⊆	⊆	NUM
ejpam-977	85	37	b	b	NOUN
ejpam-977	85	38	and	and	CCONJ
ejpam-977	85	39	hence	hence	ADV
ejpam-977	85	40	a	a	DET
ejpam-977	85	41	∈	∈	NOUN
ejpam-977	85	42	(	(	PUNCT
ejpam-977	85	43	b	b	NOUN
ejpam-977	85	44	:	:	PUNCT
ejpam-977	85	45	n	n	CCONJ
ejpam-977	85	46	)	)	PUNCT
ejpam-977	85	47	.	.	PUNCT
ejpam-977	86	1	therefore	therefore	ADV
ejpam-977	86	2	n	n	PROPN
ejpam-977	86	3	is	be	AUX
ejpam-977	86	4	join	join	NOUN
ejpam-977	86	5	-	-	PUNCT
ejpam-977	86	6	quasi	quasi	NOUN
ejpam-977	86	7	-	-	NOUN
ejpam-977	86	8	cyclic	cyclic	ADJ
ejpam-977	86	9	.	.	PUNCT
ejpam-977	87	1	lemma	lemma	PROPN
ejpam-977	87	2	5	5	X
ejpam-977	87	3	.	.	PUNCT
ejpam-977	87	4	suppose	suppose	VERB
ejpam-977	87	5	r	r	NOUN
ejpam-977	87	6	is	be	AUX
ejpam-977	87	7	an	an	DET
ejpam-977	87	8	integral	integral	ADJ
ejpam-977	87	9	domain	domain	NOUN
ejpam-977	87	10	and	and	CCONJ
ejpam-977	87	11	m	m	NOUN
ejpam-977	87	12	is	be	AUX
ejpam-977	87	13	a	a	DET
ejpam-977	87	14	non	non	ADJ
ejpam-977	87	15	zero	zero	NUM
ejpam-977	87	16	faithful	faithful	ADJ
ejpam-977	87	17	finitely	finitely	ADV
ejpam-977	87	18	generated	generate	VERB
ejpam-977	87	19	rmodule	rmodule	NOUN
ejpam-977	87	20	.	.	PUNCT
ejpam-977	88	1	then	then	ADV
ejpam-977	88	2	r	r	NOUN
ejpam-977	88	3	is	be	AUX
ejpam-977	88	4	a	a	DET
ejpam-977	88	5	prüfer	prüfer	NOUN
ejpam-977	88	6	domain	domain	NOUN
ejpam-977	88	7	if	if	SCONJ
ejpam-977	88	8	and	and	CCONJ
ejpam-977	88	9	only	only	ADV
ejpam-977	88	10	if	if	SCONJ
ejpam-977	88	11	every	every	DET
ejpam-977	88	12	finitely	finitely	ADV
ejpam-977	88	13	generated	generate	VERB
ejpam-977	88	14	submodule	submodule	NOUN
ejpam-977	88	15	of	of	ADP
ejpam-977	88	16	m	m	PROPN
ejpam-977	88	17	is	be	AUX
ejpam-977	88	18	join	join	NOUN
ejpam-977	88	19	-	-	PUNCT
ejpam-977	88	20	quasi	quasi	NOUN
ejpam-977	88	21	-	-	ADJ
ejpam-977	88	22	cyclic	cyclic	ADJ
ejpam-977	88	23	.	.	PUNCT
ejpam-977	89	1	proof	proof	NOUN
ejpam-977	89	2	.	.	PUNCT
ejpam-977	90	1	the	the	DET
ejpam-977	90	2	proof	proof	NOUN
ejpam-977	90	3	of	of	ADP
ejpam-977	90	4	the	the	DET
ejpam-977	90	5	lemma	lemma	PROPN
ejpam-977	90	6	follows	follow	VERB
ejpam-977	90	7	from	from	ADP
ejpam-977	90	8	lemma	lemma	PROPN
ejpam-977	90	9	3	3	PROPN
ejpam-977	90	10	and	and	CCONJ
ejpam-977	90	11	lemma	lemma	PROPN
ejpam-977	90	12	4	4	NUM
ejpam-977	90	13	.	.	PUNCT
ejpam-977	90	14	theorem	theorem	NOUN
ejpam-977	90	15	1	1	NUM
ejpam-977	90	16	.	.	PUNCT
ejpam-977	91	1	suppose	suppose	VERB
ejpam-977	91	2	m	m	NOUN
ejpam-977	91	3	is	be	AUX
ejpam-977	91	4	a	a	DET
ejpam-977	91	5	non	non	ADJ
ejpam-977	91	6	zero	zero	NUM
ejpam-977	91	7	faithful	faithful	ADJ
ejpam-977	91	8	multiplication	multiplication	NOUN
ejpam-977	91	9	r	r	NOUN
ejpam-977	91	10	-	-	PUNCT
ejpam-977	91	11	module	module	NOUN
ejpam-977	91	12	.	.	PUNCT
ejpam-977	92	1	then	then	ADV
ejpam-977	92	2	m	m	PROPN
ejpam-977	92	3	is	be	AUX
ejpam-977	92	4	a	a	DET
ejpam-977	92	5	prüfer	prüfer	NOUN
ejpam-977	92	6	module	module	NOUN
ejpam-977	92	7	if	if	SCONJ
ejpam-977	92	8	and	and	CCONJ
ejpam-977	92	9	only	only	ADV
ejpam-977	92	10	if	if	SCONJ
ejpam-977	92	11	r	r	NOUN
ejpam-977	92	12	is	be	AUX
ejpam-977	92	13	an	an	DET
ejpam-977	92	14	integral	integral	ADJ
ejpam-977	92	15	domain	domain	NOUN
ejpam-977	92	16	and	and	CCONJ
ejpam-977	92	17	every	every	DET
ejpam-977	92	18	finitely	finitely	ADV
ejpam-977	92	19	generated	generate	VERB
ejpam-977	92	20	submodule	submodule	NOUN
ejpam-977	92	21	of	of	ADP
ejpam-977	92	22	m	m	PROPN
ejpam-977	92	23	is	be	AUX
ejpam-977	92	24	join	join	NOUN
ejpam-977	92	25	-	-	PUNCT
ejpam-977	92	26	quasi	quasi	NOUN
ejpam-977	92	27	-	-	ADJ
ejpam-977	92	28	cyclic	cyclic	ADJ
ejpam-977	92	29	.	.	PUNCT
ejpam-977	93	1	proof	proof	NOUN
ejpam-977	93	2	.	.	PUNCT
ejpam-977	94	1	suppose	suppose	VERB
ejpam-977	94	2	m	m	PRON
ejpam-977	94	3	is	be	AUX
ejpam-977	94	4	a	a	DET
ejpam-977	94	5	prüfer	prüfer	NOUN
ejpam-977	94	6	module	module	NOUN
ejpam-977	94	7	.	.	PUNCT
ejpam-977	95	1	then	then	ADV
ejpam-977	95	2	by	by	ADP
ejpam-977	95	3	[	[	X
ejpam-977	95	4	10	10	NUM
ejpam-977	95	5	,	,	PUNCT
ejpam-977	95	6	theorem	theorem	VERB
ejpam-977	95	7	3.6	3.6	NUM
ejpam-977	95	8	]	]	PUNCT
ejpam-977	95	9	,	,	PUNCT
ejpam-977	95	10	r	r	NOUN
ejpam-977	95	11	is	be	AUX
ejpam-977	95	12	a	a	DET
ejpam-977	95	13	prüfer	prüfer	NOUN
ejpam-977	95	14	domain	domain	NOUN
ejpam-977	95	15	.	.	PUNCT
ejpam-977	96	1	as	as	SCONJ
ejpam-977	96	2	r	r	NOUN
ejpam-977	96	3	is	be	AUX
ejpam-977	96	4	an	an	DET
ejpam-977	96	5	integral	integral	ADJ
ejpam-977	96	6	domain	domain	NOUN
ejpam-977	96	7	and	and	CCONJ
ejpam-977	96	8	m	m	NOUN
ejpam-977	96	9	is	be	AUX
ejpam-977	96	10	a	a	DET
ejpam-977	96	11	non	non	ADJ
ejpam-977	96	12	zero	zero	NUM
ejpam-977	96	13	faithful	faithful	ADJ
ejpam-977	96	14	multiplication	multiplication	NOUN
ejpam-977	96	15	r	r	NOUN
ejpam-977	96	16	-	-	PUNCT
ejpam-977	96	17	module	module	NOUN
ejpam-977	96	18	,	,	PUNCT
ejpam-977	96	19	by	by	ADP
ejpam-977	96	20	[	[	X
ejpam-977	96	21	6	6	NUM
ejpam-977	96	22	,	,	PUNCT
ejpam-977	96	23	proposition	proposition	NOUN
ejpam-977	96	24	3.4	3.4	NUM
ejpam-977	96	25	]	]	PUNCT
ejpam-977	96	26	,	,	PUNCT
ejpam-977	96	27	m	m	VERB
ejpam-977	96	28	is	be	AUX
ejpam-977	96	29	finitely	finitely	ADV
ejpam-977	96	30	generated	generate	VERB
ejpam-977	96	31	.	.	PUNCT
ejpam-977	97	1	again	again	ADV
ejpam-977	97	2	by	by	ADP
ejpam-977	97	3	lemma	lemma	PROPN
ejpam-977	97	4	5	5	NUM
ejpam-977	97	5	,	,	PUNCT
ejpam-977	97	6	every	every	DET
ejpam-977	97	7	finitely	finitely	ADV
ejpam-977	97	8	generated	generate	VERB
ejpam-977	97	9	submodule	submodule	NOUN
ejpam-977	97	10	of	of	ADP
ejpam-977	97	11	m	m	PROPN
ejpam-977	97	12	is	be	AUX
ejpam-977	97	13	join	join	NOUN
ejpam-977	97	14	-	-	PUNCT
ejpam-977	97	15	quasi	quasi	NOUN
ejpam-977	97	16	-	-	NOUN
ejpam-977	97	17	cyclic	cyclic	ADJ
ejpam-977	97	18	.	.	PUNCT
ejpam-977	98	1	the	the	DET
ejpam-977	98	2	converse	converse	NOUN
ejpam-977	98	3	part	part	NOUN
ejpam-977	98	4	follows	follow	VERB
ejpam-977	98	5	from	from	ADP
ejpam-977	98	6	[	[	X
ejpam-977	98	7	6	6	NUM
ejpam-977	98	8	,	,	PUNCT
ejpam-977	98	9	proposition	proposition	NOUN
ejpam-977	98	10	3.4	3.4	NUM
ejpam-977	98	11	]	]	PUNCT
ejpam-977	98	12	,	,	PUNCT
ejpam-977	98	13	lemma	lemma	PROPN
ejpam-977	98	14	5	5	NUM
ejpam-977	98	15	and	and	CCONJ
ejpam-977	98	16	[	[	X
ejpam-977	98	17	10	10	NUM
ejpam-977	98	18	,	,	PUNCT
ejpam-977	98	19	theorem	theorem	VERB
ejpam-977	98	20	3.6	3.6	NUM
ejpam-977	98	21	]	]	PUNCT
ejpam-977	98	22	.	.	PUNCT
ejpam-977	99	1	theorem	theorem	NOUN
ejpam-977	99	2	2	2	NUM
ejpam-977	99	3	.	.	PUNCT
ejpam-977	99	4	suppose	suppose	VERB
ejpam-977	99	5	m	m	NOUN
ejpam-977	99	6	is	be	AUX
ejpam-977	99	7	a	a	DET
ejpam-977	99	8	non	non	ADJ
ejpam-977	99	9	zero	zero	NUM
ejpam-977	99	10	faithful	faithful	ADJ
ejpam-977	99	11	multiplication	multiplication	NOUN
ejpam-977	99	12	r	r	NOUN
ejpam-977	99	13	-	-	PUNCT
ejpam-977	99	14	module	module	NOUN
ejpam-977	99	15	.	.	PUNCT
ejpam-977	100	1	then	then	ADV
ejpam-977	100	2	m	m	PROPN
ejpam-977	100	3	is	be	AUX
ejpam-977	100	4	a	a	DET
ejpam-977	100	5	dedekind	dedekind	NOUN
ejpam-977	100	6	module	module	NOUN
ejpam-977	100	7	if	if	SCONJ
ejpam-977	100	8	and	and	CCONJ
ejpam-977	100	9	only	only	ADV
ejpam-977	100	10	if	if	SCONJ
ejpam-977	100	11	r	r	NOUN
ejpam-977	100	12	is	be	AUX
ejpam-977	100	13	an	an	DET
ejpam-977	100	14	integral	integral	ADJ
ejpam-977	100	15	domain	domain	NOUN
ejpam-977	100	16	and	and	CCONJ
ejpam-977	100	17	every	every	DET
ejpam-977	100	18	submodule	submodule	NOUN
ejpam-977	100	19	of	of	ADP
ejpam-977	100	20	m	m	PROPN
ejpam-977	100	21	is	be	AUX
ejpam-977	100	22	a	a	DET
ejpam-977	100	23	finitely	finitely	ADV
ejpam-977	100	24	generated	generate	VERB
ejpam-977	100	25	join	join	NOUN
ejpam-977	100	26	-	-	PUNCT
ejpam-977	100	27	quasi	quasi	NOUN
ejpam-977	100	28	-	-	ADJ
ejpam-977	100	29	cyclic	cyclic	ADJ
ejpam-977	100	30	submodule	submodule	NOUN
ejpam-977	100	31	of	of	ADP
ejpam-977	100	32	m.	m.	NOUN
ejpam-977	100	33	proof	proof	NOUN
ejpam-977	100	34	.	.	PUNCT
ejpam-977	101	1	the	the	DET
ejpam-977	101	2	proof	proof	NOUN
ejpam-977	101	3	of	of	ADP
ejpam-977	101	4	the	the	DET
ejpam-977	101	5	theorem	theorem	NOUN
ejpam-977	101	6	follows	follow	VERB
ejpam-977	101	7	from	from	ADP
ejpam-977	101	8	theorem	theorem	ADJ
ejpam-977	101	9	1	1	NUM
ejpam-977	101	10	and	and	CCONJ
ejpam-977	101	11	[	[	X
ejpam-977	101	12	1	1	NUM
ejpam-977	101	13	,	,	PUNCT
ejpam-977	101	14	theorem	theorem	VERB
ejpam-977	101	15	3.4	3.4	NUM
ejpam-977	101	16	]	]	PUNCT
ejpam-977	101	17	.	.	PUNCT
ejpam-977	102	1	references	reference	NOUN
ejpam-977	102	2	902	902	NUM
ejpam-977	102	3	references	reference	NOUN
ejpam-977	102	4	[	[	X
ejpam-977	102	5	1	1	NUM
ejpam-977	102	6	]	]	X
ejpam-977	102	7	m.m	m.m	PROPN
ejpam-977	102	8	.	.	PROPN
ejpam-977	102	9	ali	ali	PROPN
ejpam-977	102	10	,	,	PUNCT
ejpam-977	102	11	invertibility	invertibility	NOUN
ejpam-977	102	12	of	of	ADP
ejpam-977	102	13	multiplication	multiplication	NOUN
ejpam-977	102	14	modules	module	NOUN
ejpam-977	102	15	,	,	PUNCT
ejpam-977	102	16	new	new	PROPN
ejpam-977	102	17	zealand	zealand	PROPN
ejpam-977	102	18	journal	journal	PROPN
ejpam-977	102	19	of	of	ADP
ejpam-977	102	20	mathematics	mathematic	NOUN
ejpam-977	102	21	,	,	PUNCT
ejpam-977	102	22	35	35	NUM
ejpam-977	102	23	,	,	PUNCT
ejpam-977	102	24	17–29	17–29	NUM
ejpam-977	102	25	.	.	NOUN
ejpam-977	102	26	2006	2006	NUM
ejpam-977	102	27	.	.	PUNCT
ejpam-977	103	1	[	[	X
ejpam-977	103	2	2	2	X
ejpam-977	103	3	]	]	X
ejpam-977	103	4	d.	d.	PROPN
ejpam-977	103	5	d.	d.	PROPN
ejpam-977	103	6	anderson	anderson	PROPN
ejpam-977	103	7	and	and	CCONJ
ejpam-977	103	8	e.w	e.w	PROPN
ejpam-977	103	9	johnson	johnson	PROPN
ejpam-977	103	10	,	,	PUNCT
ejpam-977	103	11	dilworth	dilworth	PROPN
ejpam-977	103	12	’s	’s	PART
ejpam-977	103	13	principal	principal	ADJ
ejpam-977	103	14	elements	element	NOUN
ejpam-977	103	15	,	,	PUNCT
ejpam-977	103	16	algebra	algebra	NOUN
ejpam-977	103	17	universalis	universali	VERB
ejpam-977	103	18	,	,	PUNCT
ejpam-977	103	19	36	36	NUM
ejpam-977	103	20	,	,	PUNCT
ejpam-977	103	21	392	392	NUM
ejpam-977	103	22	-	-	SYM
ejpam-977	103	23	404	404	NUM
ejpam-977	103	24	.	.	PUNCT
ejpam-977	103	25	1996	1996	NUM
ejpam-977	103	26	.	.	PUNCT
ejpam-977	104	1	[	[	X
ejpam-977	104	2	3	3	X
ejpam-977	104	3	]	]	X
ejpam-977	104	4	d.	d.	PROPN
ejpam-977	104	5	d.	d.	PROPN
ejpam-977	104	6	anderson	anderson	PROPN
ejpam-977	104	7	,	,	PUNCT
ejpam-977	104	8	cancellation	cancellation	NOUN
ejpam-977	104	9	modules	module	NOUN
ejpam-977	104	10	and	and	CCONJ
ejpam-977	104	11	related	related	ADJ
ejpam-977	104	12	modules	module	NOUN
ejpam-977	104	13	,	,	PUNCT
ejpam-977	104	14	lecture	lecture	NOUN
ejpam-977	104	15	notes	note	NOUN
ejpam-977	104	16	in	in	ADP
ejpam-977	104	17	pure	pure	ADJ
ejpam-977	104	18	and	and	CCONJ
ejpam-977	104	19	applied	applied	ADJ
ejpam-977	104	20	mathematics	mathematic	NOUN
ejpam-977	104	21	,	,	PUNCT
ejpam-977	104	22	220	220	NUM
ejpam-977	104	23	,	,	PUNCT
ejpam-977	104	24	dekker	dekker	NOUN
ejpam-977	104	25	,	,	PUNCT
ejpam-977	104	26	new	new	PROPN
ejpam-977	104	27	york	york	PROPN
ejpam-977	104	28	,	,	PUNCT
ejpam-977	104	29	pp	pp	PROPN
ejpam-977	104	30	.	.	PUNCT
ejpam-977	105	1	13	13	NUM
ejpam-977	105	2	-	-	SYM
ejpam-977	105	3	25	25	NUM
ejpam-977	105	4	.	.	PUNCT
ejpam-977	106	1	2001	2001	NUM
ejpam-977	106	2	.	.	PUNCT
ejpam-977	107	1	[	[	X
ejpam-977	107	2	4	4	NUM
ejpam-977	107	3	]	]	PUNCT
ejpam-977	107	4	a.	a.	NOUN
ejpam-977	107	5	barnard	barnard	PROPN
ejpam-977	107	6	,	,	PUNCT
ejpam-977	107	7	multiplication	multiplication	NOUN
ejpam-977	107	8	modules	module	NOUN
ejpam-977	107	9	,	,	PUNCT
ejpam-977	107	10	journal	journal	NOUN
ejpam-977	107	11	of	of	ADP
ejpam-977	107	12	algebra	algebra	PROPN
ejpam-977	107	13	,	,	PUNCT
ejpam-977	107	14	71	71	NUM
ejpam-977	107	15	,	,	PUNCT
ejpam-977	107	16	174	174	NUM
ejpam-977	107	17	-	-	SYM
ejpam-977	107	18	178	178	NUM
ejpam-977	107	19	.	.	PUNCT
ejpam-977	107	20	1981	1981	NUM
ejpam-977	107	21	.	.	PUNCT
ejpam-977	108	1	[	[	X
ejpam-977	108	2	5	5	X
ejpam-977	108	3	]	]	PUNCT
ejpam-977	108	4	l.	l.	PROPN
ejpam-977	108	5	becerra	becerra	PROPN
ejpam-977	108	6	and	and	CCONJ
ejpam-977	108	7	j.a	j.a	PROPN
ejpam-977	108	8	.	.	PROPN
ejpam-977	108	9	johnson	johnson	PROPN
ejpam-977	108	10	,	,	PUNCT
ejpam-977	108	11	a	a	DET
ejpam-977	108	12	note	note	NOUN
ejpam-977	108	13	on	on	ADP
ejpam-977	108	14	quasi	quasi	ADJ
ejpam-977	108	15	-	-	ADJ
ejpam-977	108	16	principal	principal	ADJ
ejpam-977	108	17	ideals	ideal	NOUN
ejpam-977	108	18	,	,	PUNCT
ejpam-977	108	19	tamkang	tamkang	PROPN
ejpam-977	108	20	journal	journal	PROPN
ejpam-977	108	21	of	of	ADP
ejpam-977	108	22	mathematics	mathematic	NOUN
ejpam-977	108	23	,	,	PUNCT
ejpam-977	108	24	15	15	NUM
ejpam-977	108	25	,	,	PUNCT
ejpam-977	108	26	77	77	NUM
ejpam-977	108	27	-	-	SYM
ejpam-977	108	28	82	82	NUM
ejpam-977	108	29	.	.	NOUN
ejpam-977	108	30	1984	1984	NUM
ejpam-977	108	31	.	.	PUNCT
ejpam-977	109	1	[	[	X
ejpam-977	109	2	6	6	NUM
ejpam-977	109	3	]	]	PUNCT
ejpam-977	109	4	z.	z.	PROPN
ejpam-977	109	5	el	el	PROPN
ejpam-977	109	6	-	-	PUNCT
ejpam-977	109	7	bast	bast	NOUN
ejpam-977	109	8	and	and	CCONJ
ejpam-977	109	9	p.f	p.f	PROPN
ejpam-977	109	10	.	.	PROPN
ejpam-977	109	11	smith	smith	PROPN
ejpam-977	109	12	,	,	PUNCT
ejpam-977	109	13	multiplication	multiplication	NOUN
ejpam-977	109	14	modules	module	NOUN
ejpam-977	109	15	,	,	PUNCT
ejpam-977	109	16	communications	communication	NOUN
ejpam-977	109	17	in	in	ADP
ejpam-977	109	18	algebra	algebra	NOUN
ejpam-977	109	19	,	,	PUNCT
ejpam-977	109	20	16(4	16(4	NUM
ejpam-977	109	21	)	)	PUNCT
ejpam-977	109	22	,	,	PUNCT
ejpam-977	109	23	755	755	NUM
ejpam-977	109	24	-	-	SYM
ejpam-977	109	25	779	779	NUM
ejpam-977	109	26	.	.	NUM
ejpam-977	109	27	1988	1988	NUM
ejpam-977	109	28	.	.	PUNCT
ejpam-977	110	1	[	[	X
ejpam-977	110	2	7	7	X
ejpam-977	110	3	]	]	PUNCT
ejpam-977	110	4	j.	j.	PROPN
ejpam-977	110	5	a.	a.	PROPN
ejpam-977	110	6	johnson	johnson	PROPN
ejpam-977	110	7	and	and	CCONJ
ejpam-977	110	8	m.	m.	PROPN
ejpam-977	110	9	b.	b.	PROPN
ejpam-977	110	10	taylor	taylor	PROPN
ejpam-977	110	11	,	,	PUNCT
ejpam-977	110	12	characterizations	characterization	NOUN
ejpam-977	110	13	of	of	ADP
ejpam-977	110	14	quasi	quasi	ADJ
ejpam-977	110	15	-	-	ADJ
ejpam-977	110	16	cyclic	cyclic	ADJ
ejpam-977	110	17	submodules	submodule	NOUN
ejpam-977	110	18	,	,	PUNCT
ejpam-977	110	19	mathematica	mathematica	PROPN
ejpam-977	110	20	japonica	japonica	PROPN
ejpam-977	110	21	,	,	PUNCT
ejpam-977	110	22	35(5	35(5	NUM
ejpam-977	110	23	)	)	PUNCT
ejpam-977	110	24	,	,	PUNCT
ejpam-977	110	25	761	761	NUM
ejpam-977	110	26	-	-	SYM
ejpam-977	110	27	766	766	NUM
ejpam-977	110	28	.	.	NOUN
ejpam-977	110	29	1989	1989	NUM
ejpam-977	110	30	.	.	PUNCT
ejpam-977	111	1	[	[	X
ejpam-977	111	2	8	8	NUM
ejpam-977	111	3	]	]	X
ejpam-977	111	4	m.d	m.d	PROPN
ejpam-977	111	5	.	.	PROPN
ejpam-977	111	6	larsen	larsen	PROPN
ejpam-977	111	7	and	and	CCONJ
ejpam-977	111	8	p.j	p.j	PROPN
ejpam-977	111	9	.	.	PROPN
ejpam-977	111	10	mccarthy	mccarthy	PROPN
ejpam-977	111	11	,	,	PUNCT
ejpam-977	111	12	multiplicative	multiplicative	ADJ
ejpam-977	111	13	theory	theory	NOUN
ejpam-977	111	14	of	of	ADP
ejpam-977	111	15	ideals	ideal	NOUN
ejpam-977	111	16	(	(	PUNCT
ejpam-977	111	17	academic	academic	ADJ
ejpam-977	111	18	press	press	NOUN
ejpam-977	111	19	,	,	PUNCT
ejpam-977	111	20	new	new	PROPN
ejpam-977	111	21	york	york	PROPN
ejpam-977	111	22	)	)	PUNCT
ejpam-977	111	23	,	,	PUNCT
ejpam-977	111	24	1971	1971	NUM
ejpam-977	111	25	.	.	PUNCT
ejpam-977	112	1	[	[	X
ejpam-977	112	2	9	9	NUM
ejpam-977	112	3	]	]	X
ejpam-977	112	4	p.j	p.j	PROPN
ejpam-977	112	5	.	.	PROPN
ejpam-977	112	6	mccarthy	mccarthy	PROPN
ejpam-977	112	7	,	,	PUNCT
ejpam-977	112	8	principal	principal	ADJ
ejpam-977	112	9	elements	element	NOUN
ejpam-977	112	10	of	of	ADP
ejpam-977	112	11	lattices	lattice	NOUN
ejpam-977	112	12	of	of	ADP
ejpam-977	112	13	ideals	ideal	NOUN
ejpam-977	112	14	,	,	PUNCT
ejpam-977	112	15	proc	proc	NOUN
ejpam-977	112	16	.	.	PUNCT
ejpam-977	113	1	amer	amer	PROPN
ejpam-977	113	2	.	.	PUNCT
ejpam-977	113	3	math	math	PROPN
ejpam-977	113	4	.	.	PUNCT
ejpam-977	114	1	soc	soc	PROPN
ejpam-977	114	2	.	.	PUNCT
ejpam-977	114	3	,	,	PUNCT
ejpam-977	114	4	30	30	NUM
ejpam-977	114	5	,	,	PUNCT
ejpam-977	114	6	43	43	NUM
ejpam-977	114	7	-	-	SYM
ejpam-977	114	8	45	45	NUM
ejpam-977	114	9	.	.	NOUN
ejpam-977	114	10	1971	1971	NUM
ejpam-977	114	11	.	.	PUNCT
ejpam-977	115	1	[	[	X
ejpam-977	115	2	10	10	NUM
ejpam-977	115	3	]	]	X
ejpam-977	115	4	a.g	a.g	PROPN
ejpam-977	115	5	.	.	PROPN
ejpam-977	115	6	naoum	naoum	PROPN
ejpam-977	115	7	and	and	CCONJ
ejpam-977	115	8	f.a	f.a	PROPN
ejpam-977	115	9	.	.	PROPN
ejpam-977	115	10	al	al	PROPN
ejpam-977	115	11	-	-	PUNCT
ejpam-977	115	12	alwan	alwan	PROPN
ejpam-977	115	13	,	,	PUNCT
ejpam-977	115	14	dedekind	dedekind	NOUN
ejpam-977	115	15	modules	module	NOUN
ejpam-977	115	16	,	,	PUNCT
ejpam-977	115	17	communications	communication	NOUN
ejpam-977	115	18	in	in	ADP
ejpam-977	115	19	algebra	algebra	NOUN
ejpam-977	115	20	,	,	PUNCT
ejpam-977	115	21	24	24	NUM
ejpam-977	115	22	,	,	PUNCT
ejpam-977	115	23	397	397	NUM
ejpam-977	115	24	-	-	SYM
ejpam-977	115	25	412	412	NUM
ejpam-977	115	26	.	.	PUNCT
ejpam-977	115	27	1996	1996	NUM
ejpam-977	115	28	.	.	PUNCT