id	sid	tid	token	lemma	pos
iajs-747	1	1	ibn	ibn	PROPN
iajs-747	1	2	alhaitham	alhaitham	NOUN
iajs-747	1	3	j.	j.	PROPN
iajs-747	1	4	for	for	ADP
iajs-747	1	5	pure	pure	ADJ
iajs-747	1	6	&	&	CCONJ
iajs-747	1	7	appl	appl	PROPN
iajs-747	1	8	.	.	PUNCT
iajs-747	2	1	sci	sci	PROPN
iajs-747	2	2	.	.	PUNCT
iajs-747	2	3	vol.24	vol.24	NOUN
iajs-747	2	4	(	(	PUNCT
iajs-747	2	5	3	3	NUM
iajs-747	2	6	)	)	PUNCT
iajs-747	2	7	2011	2011	NUM
iajs-747	2	8	essentially	essentially	ADV
iajs-747	2	9	quasi	quasi	ADJ
iajs-747	2	10	-	-	ADJ
iajs-747	2	11	invertible	invertible	ADJ
iajs-747	2	12	submodules	submodule	NOUN
iajs-747	2	13	and	and	CCONJ
iajs-747	2	14	essentially	essentially	ADV
iajs-747	2	15	quasi	quasi	ADJ
iajs-747	2	16	-	-	ADJ
iajs-747	2	17	dedekind	dedekind	ADJ
iajs-747	2	18	modules	module	NOUN
iajs-747	2	19	i.m	i.m	PROPN
iajs-747	2	20	-	-	PUNCT
iajs-747	2	21	a	a	DET
iajs-747	2	22	hadi	hadi	NOUN
iajs-747	2	23	,	,	PUNCT
iajs-747	2	24	th	th	X
iajs-747	2	25	.	.	PUNCT
iajs-747	3	1	y.	y.	PROPN
iajs-747	3	2	ghawi	ghawi	PROPN
iajs-747	3	3	department	department	PROPN
iajs-747	3	4	of	of	ADP
iajs-747	3	5	mathematics	mathematics	PROPN
iajs-747	3	6	,	,	PUNCT
iajs-747	3	7	college	college	NOUN
iajs-747	3	8	of	of	ADP
iajs-747	3	9	education	education	PROPN
iajs-747	3	10	ibn	ibn	PROPN
iajs-747	3	11	al	al	PROPN
iajs-747	3	12	-	-	PUNCT
iajs-747	3	13	haitham	haitham	PROPN
iajs-747	3	14	university	university	PROPN
iajs-747	3	15	of	of	ADP
iajs-747	3	16	baghdad	baghdad	PROPN
iajs-747	3	17	department	department	PROPN
iajs-747	3	18	of	of	ADP
iajs-747	3	19	mathematics	mathematics	PROPN
iajs-747	3	20	,	,	PUNCT
iajs-747	3	21	college	college	NOUN
iajs-747	3	22	of	of	ADP
iajs-747	3	23	education	education	NOUN
iajs-747	3	24	,	,	PUNCT
iajs-747	3	25	university	university	NOUN
iajs-747	3	26	of	of	ADP
iajs-747	3	27	alqadisiya	alqadisiya	PROPN
iajs-747	3	28	received	receive	VERB
iajs-747	3	29	in	in	ADP
iajs-747	3	30	:	:	PUNCT
iajs-747	3	31	6	6	NUM
iajs-747	3	32	june	june	PROPN
iajs-747	3	33	2011	2011	NUM
iajs-747	3	34	accepted	accept	VERB
iajs-747	3	35	in	in	ADP
iajs-747	3	36	:	:	PUNCT
iajs-747	3	37	8	8	NUM
iajs-747	3	38	february	february	NOUN
iajs-747	3	39	2011	2011	NUM
iajs-747	3	40	abstract	abstract	ADV
iajs-747	3	41	let	let	VERB
iajs-747	3	42	r	r	PRON
iajs-747	3	43	be	be	AUX
iajs-747	3	44	a	a	DET
iajs-747	3	45	commutative	commutative	ADJ
iajs-747	3	46	ring	ring	NOUN
iajs-747	3	47	with	with	ADP
iajs-747	3	48	identity	identity	NOUN
iajs-747	3	49	.	.	PUNCT
iajs-747	4	1	in	in	ADP
iajs-747	4	2	this	this	DET
iajs-747	4	3	paper	paper	NOUN
iajs-747	4	4	we	we	PRON
iajs-747	4	5	study	study	VERB
iajs-747	4	6	the	the	DET
iajs-747	4	7	concepts	concept	NOUN
iajs-747	4	8	of	of	ADP
iajs-747	4	9	essentially	essentially	ADV
iajs-747	4	10	quasi	quasi	ADJ
iajs-747	4	11	-	-	ADJ
iajs-747	4	12	invertible	invertible	ADJ
iajs-747	4	13	submodules	submodule	NOUN
iajs-747	4	14	and	and	CCONJ
iajs-747	4	15	essentially	essentially	ADV
iajs-747	4	16	quasi	quasi	ADJ
iajs-747	4	17	-	-	ADJ
iajs-747	4	18	dedekind	dedekind	ADJ
iajs-747	4	19	modules	module	NOUN
iajs-747	4	20	as	as	ADP
iajs-747	4	21	a	a	DET
iajs-747	4	22	generalization	generalization	NOUN
iajs-747	4	23	of	of	ADP
iajs-747	4	24	quasi	quasi	ADJ
iajs-747	4	25	-	-	ADJ
iajs-747	4	26	invertible	invertible	ADJ
iajs-747	4	27	submodules	submodule	NOUN
iajs-747	4	28	and	and	CCONJ
iajs-747	4	29	quasi	quasi	ADJ
iajs-747	4	30	-	-	ADJ
iajs-747	4	31	dedekind	dedekind	ADJ
iajs-747	4	32	modules	module	NOUN
iajs-747	4	33	.	.	PUNCT
iajs-747	5	1	among	among	ADP
iajs-747	5	2	the	the	DET
iajs-747	5	3	results	result	NOUN
iajs-747	5	4	that	that	PRON
iajs-747	5	5	we	we	PRON
iajs-747	5	6	obtain	obtain	VERB
iajs-747	5	7	is	be	AUX
iajs-747	5	8	the	the	DET
iajs-747	5	9	following	follow	VERB
iajs-747	5	10	:	:	PUNCT
iajs-747	5	11	m	m	VERB
iajs-747	5	12	is	be	AUX
iajs-747	5	13	an	an	DET
iajs-747	5	14	essentially	essentially	ADV
iajs-747	5	15	quasi	quasi	ADJ
iajs-747	5	16	-	-	ADJ
iajs-747	5	17	dedekind	dedekind	ADJ
iajs-747	5	18	module	module	NOUN
iajs-747	5	19	if	if	SCONJ
iajs-747	5	20	and	and	CCONJ
iajs-747	5	21	only	only	ADV
iajs-747	5	22	if	if	SCONJ
iajs-747	5	23	m	m	NOUN
iajs-747	5	24	is	be	AUX
iajs-747	5	25	ak	ak	ADJ
iajs-747	5	26	-	-	PUNCT
iajs-747	5	27	nonsingular	nonsingular	ADJ
iajs-747	5	28	module	module	NOUN
iajs-747	5	29	,	,	PUNCT
iajs-747	5	30	where	where	SCONJ
iajs-747	5	31	a	a	DET
iajs-747	5	32	module	module	NOUN
iajs-747	5	33	m	m	NOUN
iajs-747	5	34	is	be	AUX
iajs-747	5	35	k	k	ADJ
iajs-747	5	36	-	-	ADJ
iajs-747	5	37	nonsingular	nonsingular	ADJ
iajs-747	5	38	if	if	SCONJ
iajs-747	5	39	,	,	PUNCT
iajs-747	5	40	for	for	ADP
iajs-747	5	41	each	each	PRON
iajs-747	5	42	)	)	PUNCT
iajs-747	5	43	(	(	PUNCT
iajs-747	5	44	mendf	mendf	ADJ
iajs-747	5	45	r	r	ADJ
iajs-747	5	46	,	,	PUNCT
iajs-747	5	47	kerf	kerf	NOUN
iajs-747	5	48	≤e	≤e	NOUN
iajs-747	5	49	m	m	VERB
iajs-747	5	50	implies	imply	VERB
iajs-747	5	51	f	f	PROPN
iajs-747	5	52	=	=	SYM
iajs-747	5	53	0	0	PROPN
iajs-747	5	54	.	.	PUNCT
iajs-747	6	1	kew	kew	VERB
iajs-747	6	2	words	word	NOUN
iajs-747	6	3	:	:	PUNCT
iajs-747	6	4	essentially	essentially	ADV
iajs-747	6	5	quasi	quasi	ADJ
iajs-747	6	6	-	-	ADJ
iajs-747	6	7	invertible	invertible	ADJ
iajs-747	6	8	submodules	submodule	NOUN
iajs-747	6	9	,	,	PUNCT
iajs-747	6	10	essentially	essentially	ADV
iajs-747	6	11	quasi	quasi	ADJ
iajs-747	6	12	-	-	ADJ
iajs-747	6	13	dedekind	dedekind	ADJ
iajs-747	6	14	modules	module	NOUN
iajs-747	6	15	.	.	PUNCT
iajs-747	7	1	introduction	introduction	NOUN
iajs-747	7	2	the	the	DET
iajs-747	7	3	concepts	concept	NOUN
iajs-747	7	4	of	of	ADP
iajs-747	7	5	a	a	DET
iajs-747	7	6	quasi	quasi	ADJ
iajs-747	7	7	-	-	ADJ
iajs-747	7	8	invertible	invertible	ADJ
iajs-747	7	9	submodule	submodule	NOUN
iajs-747	7	10	of	of	ADP
iajs-747	7	11	an	an	DET
iajs-747	7	12	r	r	NOUN
iajs-747	7	13	-	-	PUNCT
iajs-747	7	14	module	module	NOUN
iajs-747	7	15	and	and	CCONJ
iajs-747	7	16	quasi	quasi	ADJ
iajs-747	7	17	-	-	ADJ
iajs-747	7	18	dedekind	dedekind	ADJ
iajs-747	7	19	module	module	NOUN
iajs-747	7	20	were	be	AUX
iajs-747	7	21	introduced	introduce	VERB
iajs-747	7	22	in	in	ADP
iajs-747	7	23	[	[	X
iajs-747	7	24	5	5	NUM
iajs-747	7	25	]	]	PUNCT
iajs-747	7	26	.where	.where	ADP
iajs-747	7	27	a	a	DET
iajs-747	7	28	submodule	submodule	NOUN
iajs-747	7	29	n	n	PROPN
iajs-747	7	30	of	of	ADP
iajs-747	7	31	an	an	DET
iajs-747	7	32	r	r	NOUN
iajs-747	7	33	-	-	PUNCT
iajs-747	7	34	module	module	NOUN
iajs-747	7	35	m	m	NOUN
iajs-747	7	36	is	be	AUX
iajs-747	7	37	called	call	VERB
iajs-747	7	38	quasiinvertible	quasiinvertible	ADJ
iajs-747	7	39	if	if	SCONJ
iajs-747	7	40	0	0	NUM
iajs-747	7	41	)	)	PUNCT
iajs-747	7	42	,	,	PUNCT
iajs-747	7	43	(	(	PUNCT
iajs-747	7	44	mnmhom	mnmhom	PUNCT
iajs-747	7	45	,	,	PUNCT
iajs-747	7	46	and	and	CCONJ
iajs-747	7	47	an	an	DET
iajs-747	7	48	r	r	NOUN
iajs-747	7	49	-	-	PUNCT
iajs-747	7	50	module	module	NOUN
iajs-747	7	51	m	m	NOUN
iajs-747	7	52	is	be	AUX
iajs-747	7	53	called	call	VERB
iajs-747	7	54	quasi	quasi	ADJ
iajs-747	7	55	-	-	NOUN
iajs-747	7	56	dedekind	dedekind	ADJ
iajs-747	7	57	if	if	SCONJ
iajs-747	7	58	each	each	DET
iajs-747	7	59	nonzero	nonzero	PROPN
iajs-747	7	60	submodule	submodule	PROPN
iajs-747	7	61	of	of	ADP
iajs-747	7	62	m	m	PROPN
iajs-747	7	63	is	be	AUX
iajs-747	7	64	quasi	quasi	ADJ
iajs-747	7	65	-	-	ADJ
iajs-747	7	66	invertible	invertible	ADJ
iajs-747	7	67	.	.	PUNCT
iajs-747	8	1	as	as	ADP
iajs-747	8	2	a	a	DET
iajs-747	8	3	generalizations	generalization	NOUN
iajs-747	8	4	to	to	ADP
iajs-747	8	5	these	these	DET
iajs-747	8	6	concepts	concept	NOUN
iajs-747	8	7	we	we	PRON
iajs-747	8	8	introduce	introduce	VERB
iajs-747	8	9	the	the	DET
iajs-747	8	10	following	follow	VERB
iajs-747	8	11	concepts	concept	NOUN
iajs-747	8	12	:	:	PUNCT
iajs-747	8	13	we	we	PRON
iajs-747	8	14	call	call	VERB
iajs-747	8	15	a	a	DET
iajs-747	8	16	submodule	submodule	NOUN
iajs-747	8	17	n	n	PROPN
iajs-747	8	18	of	of	ADP
iajs-747	8	19	m	m	PROPN
iajs-747	8	20	is	be	AUX
iajs-747	8	21	essentially	essentially	ADV
iajs-747	8	22	quasiinvertible	quasiinvertible	ADJ
iajs-747	8	23	if	if	SCONJ
iajs-747	8	24	,	,	PUNCT
iajs-747	8	25	n	n	CCONJ
iajs-747	8	26	≤e	≤e	VERB
iajs-747	8	27	m	m	PRON
iajs-747	8	28	and	and	CCONJ
iajs-747	8	29	n	n	ADV
iajs-747	8	30	is	be	AUX
iajs-747	8	31	quasi	quasi	ADJ
iajs-747	8	32	-	-	ADJ
iajs-747	8	33	invertible	invertible	ADJ
iajs-747	8	34	.and	.and	PUNCT
iajs-747	8	35	an	an	DET
iajs-747	8	36	r	r	NOUN
iajs-747	8	37	-	-	PUNCT
iajs-747	8	38	module	module	NOUN
iajs-747	8	39	m	m	NOUN
iajs-747	8	40	is	be	AUX
iajs-747	8	41	called	call	VERB
iajs-747	8	42	essentially	essentially	ADV
iajs-747	8	43	quasi	quasi	ADJ
iajs-747	8	44	-	-	NOUN
iajs-747	8	45	dedekind	dedekind	ADJ
iajs-747	8	46	if	if	SCONJ
iajs-747	8	47	every	every	DET
iajs-747	8	48	essential	essential	ADJ
iajs-747	8	49	submodule	submodule	NOUN
iajs-747	8	50	n	n	PROPN
iajs-747	8	51	of	of	ADP
iajs-747	8	52	m	m	PROPN
iajs-747	8	53	is	be	AUX
iajs-747	8	54	quasi	quasi	ADJ
iajs-747	8	55	-	-	ADJ
iajs-747	8	56	invertible	invertible	ADJ
iajs-747	8	57	;	;	PUNCT
iajs-747	8	58	(	(	PUNCT
iajs-747	8	59	i.e	i.e	X
iajs-747	8	60	0	0	NUM
iajs-747	8	61	)	)	PUNCT
iajs-747	8	62	,	,	PUNCT
iajs-747	8	63	(	(	PUNCT
iajs-747	8	64	mnmhom	mnmhom	PUNCT
iajs-747	8	65	)	)	PUNCT
iajs-747	8	66	.	.	PUNCT
iajs-747	9	1	this	this	DET
iajs-747	9	2	paper	paper	NOUN
iajs-747	9	3	consists	consist	VERB
iajs-747	9	4	of	of	ADP
iajs-747	9	5	two	two	NUM
iajs-747	9	6	sections	section	NOUN
iajs-747	9	7	,	,	PUNCT
iajs-747	9	8	§	§	PROPN
iajs-747	9	9	1	1	NUM
iajs-747	9	10	is	be	AUX
iajs-747	9	11	devoted	devote	VERB
iajs-747	9	12	to	to	PART
iajs-747	9	13	study	study	VERB
iajs-747	9	14	essentially	essentially	ADV
iajs-747	9	15	quasi	quasi	ADJ
iajs-747	9	16	-	-	ADJ
iajs-747	9	17	invertible	invertible	ADJ
iajs-747	9	18	submodules	submodule	NOUN
iajs-747	9	19	,	,	PUNCT
iajs-747	9	20	in	in	ADP
iajs-747	9	21	§	§	NOUN
iajs-747	9	22	2	2	NUM
iajs-747	9	23	we	we	PRON
iajs-747	9	24	study	study	VERB
iajs-747	9	25	and	and	CCONJ
iajs-747	9	26	give	give	VERB
iajs-747	9	27	the	the	DET
iajs-747	9	28	basic	basic	ADJ
iajs-747	9	29	properties	property	NOUN
iajs-747	9	30	of	of	ADP
iajs-747	9	31	essentially	essentially	ADV
iajs-747	9	32	quasi	quasi	ADJ
iajs-747	9	33	-	-	ADJ
iajs-747	9	34	dedekind	dedekind	ADJ
iajs-747	9	35	modules	module	NOUN
iajs-747	9	36	.	.	PUNCT
iajs-747	10	1	this	this	DET
iajs-747	10	2	paper	paper	NOUN
iajs-747	10	3	represents	represent	VERB
iajs-747	10	4	a	a	DET
iajs-747	10	5	part	part	NOUN
iajs-747	10	6	of	of	ADP
iajs-747	10	7	the	the	DET
iajs-747	10	8	m	m	NOUN
iajs-747	10	9	.	.	PUNCT
iajs-747	11	1	sc	sc	PROPN
iajs-747	11	2	.	.	PUNCT
iajs-747	12	1	thesis	thesis	NOUN
iajs-747	12	2	written	write	VERB
iajs-747	12	3	by	by	ADP
iajs-747	12	4	the	the	DET
iajs-747	12	5	second	second	ADJ
iajs-747	12	6	author	author	NOUN
iajs-747	12	7	under	under	ADP
iajs-747	12	8	the	the	DET
iajs-747	12	9	supervision	supervision	NOUN
iajs-747	12	10	of	of	ADP
iajs-747	12	11	the	the	DET
iajs-747	12	12	first	first	ADJ
iajs-747	12	13	author	author	NOUN
iajs-747	12	14	and	and	CCONJ
iajs-747	12	15	was	be	AUX
iajs-747	12	16	submitted	submit	VERB
iajs-747	12	17	to	to	ADP
iajs-747	12	18	the	the	DET
iajs-747	12	19	college	college	NOUN
iajs-747	12	20	of	of	ADP
iajs-747	12	21	education	education	NOUN
iajs-747	12	22	–	–	PUNCT
iajs-747	12	23	ibn	ibn	NOUN
iajs-747	12	24	alhaitham	alhaitham	NOUN
iajs-747	12	25	,	,	PUNCT
iajs-747	12	26	university	university	NOUN
iajs-747	12	27	of	of	ADP
iajs-747	12	28	baghdad	baghdad	PROPN
iajs-747	12	29	,	,	PUNCT
iajs-747	12	30	2010	2010	NUM
iajs-747	12	31	.	.	PUNCT
iajs-747	13	1	ibn	ibn	PROPN
iajs-747	13	2	alhaitham	alhaitham	PROPN
iajs-747	13	3	j.	j.	PROPN
iajs-747	13	4	for	for	ADP
iajs-747	13	5	pure	pure	ADJ
iajs-747	13	6	&	&	CCONJ
iajs-747	13	7	appl	appl	PROPN
iajs-747	13	8	.	.	PUNCT
iajs-747	14	1	sci	sci	PROPN
iajs-747	14	2	.	.	PUNCT
iajs-747	14	3	vol.24	vol.24	NOUN
iajs-747	14	4	(	(	PUNCT
iajs-747	14	5	3	3	NUM
iajs-747	14	6	)	)	PUNCT
iajs-747	14	7	2011	2011	NUM
iajs-747	14	8	1	1	NUM
iajs-747	14	9	.	.	PUNCT
iajs-747	15	1	essentially	essentially	ADV
iajs-747	15	2	quasi	quasi	ADJ
iajs-747	15	3	-	-	ADJ
iajs-747	15	4	invertible	invertible	ADJ
iajs-747	15	5	submodules	submodule	NOUN
iajs-747	15	6	in	in	ADP
iajs-747	15	7	this	this	DET
iajs-747	15	8	section	section	NOUN
iajs-747	15	9	we	we	PRON
iajs-747	15	10	introduce	introduce	VERB
iajs-747	15	11	the	the	DET
iajs-747	15	12	concept	concept	NOUN
iajs-747	15	13	of	of	ADP
iajs-747	15	14	essentially	essentially	ADV
iajs-747	15	15	quasi	quasi	ADJ
iajs-747	15	16	-	-	ADJ
iajs-747	15	17	invertible	invertible	ADJ
iajs-747	15	18	submodules	submodule	NOUN
iajs-747	15	19	.	.	PUNCT
iajs-747	16	1	we	we	PRON
iajs-747	16	2	develop	develop	VERB
iajs-747	16	3	basic	basic	ADJ
iajs-747	16	4	properties	property	NOUN
iajs-747	16	5	of	of	ADP
iajs-747	16	6	essentially	essentially	ADV
iajs-747	16	7	quasi	quasi	ADJ
iajs-747	16	8	-	-	ADJ
iajs-747	16	9	invertible	invertible	ADJ
iajs-747	16	10	submodule	submodule	NOUN
iajs-747	16	11	.	.	PUNCT
iajs-747	17	1	we	we	PRON
iajs-747	17	2	start	start	VERB
iajs-747	17	3	with	with	ADP
iajs-747	17	4	the	the	DET
iajs-747	17	5	following	follow	VERB
iajs-747	17	6	definition	definition	NOUN
iajs-747	17	7	:	:	PUNCT
iajs-747	17	8	definition	definition	NOUN
iajs-747	17	9	(	(	PUNCT
iajs-747	17	10	1.1	1.1	NUM
iajs-747	17	11	)	)	PUNCT
iajs-747	17	12	let	let	VERB
iajs-747	17	13	m	m	PRON
iajs-747	17	14	be	be	AUX
iajs-747	17	15	an	an	DET
iajs-747	17	16	r	r	NOUN
iajs-747	17	17	-	-	PUNCT
iajs-747	17	18	module	module	NOUN
iajs-747	17	19	and	and	CCONJ
iajs-747	17	20	n	n	CCONJ
iajs-747	17	21	≤e	≤e	VERB
iajs-747	17	22	m	m	VERB
iajs-747	17	23	,	,	PUNCT
iajs-747	17	24	then	then	ADV
iajs-747	17	25	n	n	CCONJ
iajs-747	17	26	is	be	AUX
iajs-747	17	27	called	call	VERB
iajs-747	17	28	an	an	DET
iajs-747	17	29	essentially	essentially	ADV
iajs-747	17	30	quasi	quasi	ADJ
iajs-747	17	31	-	-	ADJ
iajs-747	17	32	invertible	invertible	ADJ
iajs-747	17	33	submodule	submodule	NOUN
iajs-747	17	34	of	of	ADP
iajs-747	17	35	m	m	PROPN
iajs-747	17	36	if	if	SCONJ
iajs-747	17	37	,	,	PUNCT
iajs-747	17	38	0	0	NUM
iajs-747	17	39	)	)	PUNCT
iajs-747	17	40	,	,	PUNCT
iajs-747	17	41	(	(	PUNCT
iajs-747	17	42	mnmhom	mnmhom	PUNCT
iajs-747	17	43	;	;	PUNCT
iajs-747	17	44	that	that	PRON
iajs-747	17	45	is	be	AUX
iajs-747	17	46	n	n	PRON
iajs-747	17	47	is	be	AUX
iajs-747	17	48	essentially	essentially	ADV
iajs-747	17	49	quasi	quasi	ADJ
iajs-747	17	50	-	-	ADJ
iajs-747	17	51	invertible	invertible	ADJ
iajs-747	17	52	if	if	SCONJ
iajs-747	17	53	,	,	PUNCT
iajs-747	17	54	n	n	CCONJ
iajs-747	17	55	≤e	≤e	VERB
iajs-747	17	56	m	m	PRON
iajs-747	17	57	and	and	CCONJ
iajs-747	17	58	n	n	ADV
iajs-747	17	59	is	be	AUX
iajs-747	17	60	quasi	quasi	ADJ
iajs-747	17	61	-	-	ADJ
iajs-747	17	62	invertible	invertible	ADJ
iajs-747	17	63	.	.	PUNCT
iajs-747	18	1	an	an	DET
iajs-747	18	2	ideal	ideal	ADJ
iajs-747	18	3	j	j	PROPN
iajs-747	18	4	in	in	ADP
iajs-747	18	5	a	a	DET
iajs-747	18	6	ring	ring	NOUN
iajs-747	18	7	r	r	NOUN
iajs-747	18	8	is	be	AUX
iajs-747	18	9	called	call	VERB
iajs-747	18	10	an	an	DET
iajs-747	18	11	essentially	essentially	ADV
iajs-747	18	12	quasi	quasi	ADJ
iajs-747	18	13	-	-	ADJ
iajs-747	18	14	invertible	invertible	ADJ
iajs-747	18	15	ideal	ideal	NOUN
iajs-747	18	16	of	of	ADP
iajs-747	18	17	r	r	NOUN
iajs-747	18	18	if	if	SCONJ
iajs-747	18	19	,	,	PUNCT
iajs-747	18	20	j	j	PROPN
iajs-747	18	21	is	be	AUX
iajs-747	18	22	an	an	DET
iajs-747	18	23	essentially	essentially	ADV
iajs-747	18	24	quasi	quasi	ADJ
iajs-747	18	25	-	-	ADJ
iajs-747	18	26	invertible	invertible	ADJ
iajs-747	18	27	r	r	NOUN
iajs-747	18	28	-	-	PUNCT
iajs-747	18	29	submodule	submodule	NOUN
iajs-747	18	30	of	of	ADP
iajs-747	18	31	r	r	NOUN
iajs-747	18	32	.	.	PUNCT
iajs-747	19	1	remarks	remark	NOUN
iajs-747	19	2	and	and	CCONJ
iajs-747	19	3	examples	example	NOUN
iajs-747	19	4	(	(	PUNCT
iajs-747	19	5	1.2	1.2	NUM
iajs-747	19	6	)	)	PUNCT
iajs-747	19	7	1	1	NUM
iajs-747	19	8	)	)	PUNCT
iajs-747	19	9	it	it	PRON
iajs-747	19	10	is	be	AUX
iajs-747	19	11	clear	clear	ADJ
iajs-747	19	12	that	that	SCONJ
iajs-747	19	13	every	every	DET
iajs-747	19	14	essentially	essentially	ADV
iajs-747	19	15	quasi	quasi	ADJ
iajs-747	19	16	-	-	ADJ
iajs-747	19	17	invertible	invertible	ADJ
iajs-747	19	18	submodule	submodule	NOUN
iajs-747	19	19	is	be	AUX
iajs-747	19	20	quasi	quasi	ADJ
iajs-747	19	21	-	-	ADJ
iajs-747	19	22	invertible	invertible	ADJ
iajs-747	19	23	submodule	submodule	NOUN
iajs-747	19	24	.	.	PUNCT
iajs-747	20	1	recall	recall	VERB
iajs-747	20	2	that	that	SCONJ
iajs-747	20	3	an	an	DET
iajs-747	20	4	r	r	NOUN
iajs-747	20	5	-	-	PUNCT
iajs-747	20	6	module	module	NOUN
iajs-747	20	7	m	m	NOUN
iajs-747	20	8	is	be	AUX
iajs-747	20	9	called	call	VERB
iajs-747	20	10	a	a	DET
iajs-747	20	11	semisimple	semisimple	NOUN
iajs-747	20	12	if	if	SCONJ
iajs-747	20	13	every	every	DET
iajs-747	20	14	submodule	submodule	NOUN
iajs-747	20	15	of	of	ADP
iajs-747	20	16	m	m	PROPN
iajs-747	20	17	is	be	AUX
iajs-747	20	18	a	a	DET
iajs-747	20	19	direct	direct	ADJ
iajs-747	20	20	summand	summand	NOUN
iajs-747	20	21	of	of	ADP
iajs-747	20	22	m	m	PRON
iajs-747	20	23	,	,	PUNCT
iajs-747	21	1	[	[	X
iajs-747	21	2	3	3	NUM
iajs-747	21	3	,	,	PUNCT
iajs-747	21	4	p.189	p.189	NOUN
iajs-747	21	5	]	]	PUNCT
iajs-747	21	6	.	.	PUNCT
iajs-747	22	1	2	2	X
iajs-747	22	2	)	)	PUNCT
iajs-747	22	3	if	if	SCONJ
iajs-747	22	4	m	m	NOUN
iajs-747	22	5	is	be	AUX
iajs-747	22	6	a	a	DET
iajs-747	22	7	semisimple	semisimple	ADJ
iajs-747	22	8	r	r	NOUN
iajs-747	22	9	-	-	PUNCT
iajs-747	22	10	module	module	NOUN
iajs-747	22	11	,	,	PUNCT
iajs-747	22	12	then	then	ADV
iajs-747	22	13	m	m	VERB
iajs-747	22	14	is	be	AUX
iajs-747	22	15	the	the	DET
iajs-747	22	16	only	only	ADJ
iajs-747	22	17	essentially	essentially	ADV
iajs-747	22	18	quasi	quasi	ADJ
iajs-747	22	19	-	-	ADJ
iajs-747	22	20	invertible	invertible	ADJ
iajs-747	22	21	submodule	submodule	NOUN
iajs-747	22	22	of	of	ADP
iajs-747	22	23	m	m	PROPN
iajs-747	22	24	.	.	PUNCT
iajs-747	23	1	3	3	X
iajs-747	23	2	)	)	PUNCT
iajs-747	23	3	consider	consider	VERB
iajs-747	23	4	z4	z4	NOUN
iajs-747	23	5	as	as	ADP
iajs-747	23	6	a	a	DET
iajs-747	23	7	z	z	NOUN
iajs-747	23	8	-	-	PUNCT
iajs-747	23	9	module	module	NOUN
iajs-747	23	10	,	,	PUNCT
iajs-747	23	11	)	)	PUNCT
iajs-747	23	12	2(n	2(n	X
iajs-747	23	13	≤e	≤e	VERB
iajs-747	23	14	z4	z4	PROPN
iajs-747	23	15	,	,	PUNCT
iajs-747	23	16	but	but	CCONJ
iajs-747	23	17	0)),2	0)),2	NUM
iajs-747	23	18	(	(	PUNCT
iajs-747	23	19	(	(	PUNCT
iajs-747	23	20	244	244	NUM
iajs-747	23	21			NUM
iajs-747	23	22	zzzhom	zzzhom	NOUN
iajs-747	23	23	,	,	PUNCT
iajs-747	23	24	so	so	CCONJ
iajs-747	23	25	)	)	PUNCT
iajs-747	23	26	2(n	2(n	NUM
iajs-747	23	27	is	be	AUX
iajs-747	23	28	not	not	PART
iajs-747	23	29	essentially	essentially	ADV
iajs-747	23	30	quasi	quasi	ADJ
iajs-747	23	31	-	-	ADJ
iajs-747	23	32	invertible	invertible	ADJ
iajs-747	23	33	submodule	submodule	NOUN
iajs-747	23	34	of	of	ADP
iajs-747	23	35	z4	z4	PROPN
iajs-747	23	36	,	,	PUNCT
iajs-747	23	37	similarly	similarly	ADV
iajs-747	23	38	in	in	ADP
iajs-747	23	39	the	the	DET
iajs-747	23	40	z	z	NOUN
iajs-747	23	41	-	-	PUNCT
iajs-747	23	42	module	module	NOUN
iajs-747	23	43	z20	z20	NOUN
iajs-747	23	44	,	,	PUNCT
iajs-747	23	45	)	)	PUNCT
iajs-747	23	46	2(n	2(n	X
iajs-747	23	47	≤e	≤e	VERB
iajs-747	23	48	z20	z20	NOUN
iajs-747	23	49	,	,	PUNCT
iajs-747	23	50	but	but	CCONJ
iajs-747	23	51	it	it	PRON
iajs-747	23	52	is	be	AUX
iajs-747	23	53	not	not	PART
iajs-747	23	54	quasi	quasi	ADJ
iajs-747	23	55	-	-	ADJ
iajs-747	23	56	invertible	invertible	ADJ
iajs-747	23	57	.	.	PUNCT
iajs-747	24	1	4	4	X
iajs-747	24	2	)	)	PUNCT
iajs-747	24	3	if	if	SCONJ
iajs-747	24	4	n	n	PRON
iajs-747	24	5	is	be	AUX
iajs-747	24	6	an	an	DET
iajs-747	24	7	essentially	essentially	ADV
iajs-747	24	8	quasi	quasi	ADJ
iajs-747	24	9	-	-	ADJ
iajs-747	24	10	invertible	invertible	ADJ
iajs-747	24	11	r	r	NOUN
iajs-747	24	12	-	-	PUNCT
iajs-747	24	13	submodule	submodule	NOUN
iajs-747	24	14	of	of	ADP
iajs-747	24	15	an	an	DET
iajs-747	24	16	r	r	NOUN
iajs-747	24	17	-	-	PUNCT
iajs-747	24	18	module	module	NOUN
iajs-747	24	19	m	m	NOUN
iajs-747	24	20	,	,	PUNCT
iajs-747	24	21	then	then	ADV
iajs-747	24	22	nannmann	nannmann	PROPN
iajs-747	24	23	rr	rr	PROPN
iajs-747	25	1			NUM
iajs-747	25	2	.	.	PUNCT
iajs-747	26	1	proof	proof	NOUN
iajs-747	26	2	:	:	PUNCT
iajs-747	26	3	it	it	PRON
iajs-747	26	4	is	be	AUX
iajs-747	26	5	clear	clear	ADJ
iajs-747	26	6	.	.	PUNCT
iajs-747	27	1	the	the	DET
iajs-747	27	2	converse	converse	NOUN
iajs-747	27	3	of	of	ADP
iajs-747	27	4	(	(	PUNCT
iajs-747	27	5	rem.and.ex	rem.and.ex	X
iajs-747	27	6	.	.	PUNCT
iajs-747	27	7	1.2(4	1.2(4	NUM
iajs-747	27	8	)	)	PUNCT
iajs-747	27	9	)	)	PUNCT
iajs-747	27	10	is	be	AUX
iajs-747	27	11	not	not	PART
iajs-747	27	12	true	true	ADJ
iajs-747	27	13	in	in	ADP
iajs-747	27	14	general	general	ADJ
iajs-747	27	15	,	,	PUNCT
iajs-747	27	16	for	for	ADP
iajs-747	27	17	example	example	NOUN
iajs-747	27	18	:	:	PUNCT
iajs-747	27	19	let	let	VERB
iajs-747	27	20	zzm	zzm	NOUN
iajs-747	27	21			NOUN
iajs-747	27	22	,	,	PUNCT
iajs-747	27	23	considered	consider	VERB
iajs-747	27	24	as	as	ADP
iajs-747	27	25	a	a	DET
iajs-747	27	26	z	z	NOUN
iajs-747	27	27	-	-	PUNCT
iajs-747	27	28	module	module	NOUN
iajs-747	27	29	and	and	CCONJ
iajs-747	27	30	let	let	VERB
iajs-747	27	31	mzn	mzn	PROPN
iajs-747	27	32			PROPN
iajs-747	27	33	)	)	PUNCT
iajs-747	27	34	0	0	NUM
iajs-747	28	1	(	(	PUNCT
iajs-747	28	2	,	,	PUNCT
iajs-747	28	3	then	then	ADV
iajs-747	28	4	it	it	PRON
iajs-747	28	5	is	be	AUX
iajs-747	28	6	clear	clear	ADJ
iajs-747	28	7	that	that	SCONJ
iajs-747	28	8	nannmann	nannmann	NOUN
iajs-747	28	9	rr	rr	NOUN
iajs-747	29	1			NOUN
iajs-747	30	1	=	=	SYM
iajs-747	31	1	(	(	PUNCT
iajs-747	31	2	0	0	NUM
iajs-747	31	3	)	)	PUNCT
iajs-747	31	4	,	,	PUNCT
iajs-747	31	5	but	but	CCONJ
iajs-747	31	6	n	n	PRON
iajs-747	31	7	is	be	AUX
iajs-747	31	8	not	not	PART
iajs-747	31	9	essentially	essentially	ADV
iajs-747	31	10	quasi	quasi	ADJ
iajs-747	31	11	-	-	ADJ
iajs-747	31	12	invertible	invertible	ADJ
iajs-747	31	13	submodule	submodule	NOUN
iajs-747	31	14	of	of	ADP
iajs-747	31	15	m	m	PROPN
iajs-747	31	16	,	,	PUNCT
iajs-747	31	17	since	since	SCONJ
iajs-747	31	18	n	n	PROPN
iajs-747	31	19	≰e	≰e	PROPN
iajs-747	31	20	m	m	NOUN
iajs-747	31	21	and	and	CCONJ
iajs-747	31	22	also	also	ADV
iajs-747	31	23	n	n	PRON
iajs-747	31	24	is	be	AUX
iajs-747	31	25	not	not	PART
iajs-747	31	26	quasi	quasi	ADJ
iajs-747	31	27	-	-	ADJ
iajs-747	31	28	invertible	invertible	ADJ
iajs-747	31	29	.	.	PUNCT
iajs-747	32	1	5	5	X
iajs-747	32	2	)	)	PUNCT
iajs-747	32	3	let	let	VERB
iajs-747	32	4	j	j	PROPN
iajs-747	32	5	be	be	AUX
iajs-747	32	6	an	an	DET
iajs-747	32	7	ideal	ideal	NOUN
iajs-747	32	8	of	of	ADP
iajs-747	32	9	a	a	DET
iajs-747	32	10	ring	ring	NOUN
iajs-747	32	11	r	r	NOUN
iajs-747	32	12	.	.	PUNCT
iajs-747	33	1	then	then	ADV
iajs-747	33	2	j	j	PROPN
iajs-747	33	3	is	be	AUX
iajs-747	33	4	an	an	DET
iajs-747	33	5	essentially	essentially	ADV
iajs-747	33	6	quasiinvertible	quasiinvertible	ADJ
iajs-747	33	7	if	if	SCONJ
iajs-747	33	8	and	and	CCONJ
iajs-747	33	9	only	only	ADV
iajs-747	33	10	if	if	SCONJ
iajs-747	33	11	0	0	NUM
iajs-747	33	12	)	)	PUNCT
iajs-747	33	13	(	(	PUNCT
iajs-747	33	14	jannr	jannr	X
iajs-747	33	15	.	.	PUNCT
iajs-747	34	1	proof	proof	NOUN
iajs-747	34	2	:	:	PUNCT
iajs-747	34	3	it	it	PRON
iajs-747	34	4	is	be	AUX
iajs-747	34	5	easy	easy	ADJ
iajs-747	34	6	.	.	PUNCT
iajs-747	35	1	6	6	X
iajs-747	35	2	)	)	PUNCT
iajs-747	35	3	let	let	VERB
iajs-747	35	4	j	j	PROPN
iajs-747	35	5	be	be	AUX
iajs-747	35	6	an	an	DET
iajs-747	35	7	ideal	ideal	NOUN
iajs-747	35	8	of	of	ADP
iajs-747	35	9	a	a	DET
iajs-747	35	10	ring	ring	NOUN
iajs-747	35	11	r	r	NOUN
iajs-747	35	12	.	.	PUNCT
iajs-747	36	1	the	the	DET
iajs-747	36	2	following	follow	VERB
iajs-747	36	3	statements	statement	NOUN
iajs-747	36	4	are	be	AUX
iajs-747	36	5	equivalent	equivalent	ADJ
iajs-747	36	6	:	:	PUNCT
iajs-747	36	7	a	a	X
iajs-747	36	8	)	)	PUNCT
iajs-747	36	9	j	j	PROPN
iajs-747	36	10	is	be	AUX
iajs-747	36	11	an	an	DET
iajs-747	36	12	essentially	essentially	ADV
iajs-747	36	13	quasiinvertible	quasiinvertible	ADJ
iajs-747	36	14	ideal	ideal	NOUN
iajs-747	36	15	of	of	ADP
iajs-747	36	16	r	r	NOUN
iajs-747	36	17	.	.	PUNCT
iajs-747	37	1	b	b	X
iajs-747	37	2	)	)	PUNCT
iajs-747	37	3	j	j	PROPN
iajs-747	37	4	is	be	AUX
iajs-747	37	5	a	a	DET
iajs-747	37	6	quasi	quasi	ADJ
iajs-747	37	7	-	-	ADJ
iajs-747	37	8	invertible	invertible	ADJ
iajs-747	37	9	ideal	ideal	NOUN
iajs-747	37	10	of	of	ADP
iajs-747	37	11	r	r	NOUN
iajs-747	37	12	.	.	PUNCT
iajs-747	38	1	c	c	X
iajs-747	38	2	)	)	PUNCT
iajs-747	38	3	0	0	NUM
iajs-747	38	4	)	)	PUNCT
iajs-747	38	5	(	(	PUNCT
iajs-747	38	6	jannr	jannr	X
iajs-747	38	7	.	.	PUNCT
iajs-747	39	1	proof	proof	NOUN
iajs-747	39	2	:	:	PUNCT
iajs-747	39	3	)	)	PUNCT
iajs-747	39	4	(	(	PUNCT
iajs-747	39	5	)	)	PUNCT
iajs-747	39	6	(	(	PUNCT
iajs-747	39	7	ca	can	AUX
iajs-747	39	8			VERB
iajs-747	39	9	:	:	PUNCT
iajs-747	39	10	it	it	PRON
iajs-747	39	11	follows	follow	VERB
iajs-747	39	12	by	by	ADP
iajs-747	39	13	(	(	PUNCT
iajs-747	39	14	rem.and.ex	rem.and.ex	X
iajs-747	39	15	.	.	PUNCT
iajs-747	40	1	1.2(5	1.2(5	NUM
iajs-747	40	2	)	)	PUNCT
iajs-747	40	3	)	)	PUNCT
iajs-747	40	4	.	.	PUNCT
iajs-747	41	1	)	)	PUNCT
iajs-747	41	2	(	(	PUNCT
iajs-747	41	3	)	)	PUNCT
iajs-747	41	4	(	(	PUNCT
iajs-747	41	5	cb	cb	PROPN
iajs-747	41	6			PROPN
iajs-747	41	7	:	:	PUNCT
iajs-747	41	8	it	it	PRON
iajs-747	41	9	follows	follow	VERB
iajs-747	41	10	by	by	ADP
iajs-747	41	11	[	[	X
iajs-747	41	12	5	5	NUM
iajs-747	41	13	,	,	PUNCT
iajs-747	41	14	prop	prop	NOUN
iajs-747	41	15	.	.	PUNCT
iajs-747	42	1	2.2	2.2	NUM
iajs-747	42	2	]	]	PUNCT
iajs-747	42	3	.	.	PUNCT
iajs-747	43	1	7	7	X
iajs-747	43	2	)	)	PUNCT
iajs-747	43	3	let	let	VERB
iajs-747	43	4	r	r	PRON
iajs-747	43	5	be	be	AUX
iajs-747	43	6	a	a	DET
iajs-747	43	7	ring	ring	NOUN
iajs-747	43	8	.	.	PUNCT
iajs-747	44	1	the	the	DET
iajs-747	44	2	following	follow	VERB
iajs-747	44	3	statements	statement	NOUN
iajs-747	44	4	are	be	AUX
iajs-747	44	5	equivalent	equivalent	ADJ
iajs-747	44	6	:	:	PUNCT
iajs-747	44	7	ibn	ibn	PROPN
iajs-747	44	8	alhaitham	alhaitham	NOUN
iajs-747	44	9	j.	j.	PROPN
iajs-747	44	10	for	for	ADP
iajs-747	44	11	pure	pure	ADJ
iajs-747	44	12	&	&	CCONJ
iajs-747	44	13	appl	appl	PROPN
iajs-747	44	14	.	.	PUNCT
iajs-747	45	1	sci	sci	PROPN
iajs-747	45	2	.	.	PUNCT
iajs-747	45	3	vol.24	vol.24	NOUN
iajs-747	45	4	(	(	PUNCT
iajs-747	45	5	3	3	NUM
iajs-747	45	6	)	)	PUNCT
iajs-747	45	7	2011	2011	NUM
iajs-747	45	8	a	a	X
iajs-747	45	9	)	)	PUNCT
iajs-747	45	10	r	r	NOUN
iajs-747	45	11	is	be	AUX
iajs-747	45	12	an	an	DET
iajs-747	45	13	integral	integral	ADJ
iajs-747	45	14	domain	domain	NOUN
iajs-747	45	15	.	.	PUNCT
iajs-747	46	1	b	b	X
iajs-747	46	2	)	)	PUNCT
iajs-747	46	3	r	r	NOUN
iajs-747	46	4	is	be	AUX
iajs-747	46	5	quasi	quasi	ADJ
iajs-747	46	6	-	-	ADJ
iajs-747	46	7	dedekind	dedekind	ADJ
iajs-747	46	8	.	.	PUNCT
iajs-747	47	1	proof	proof	NOUN
iajs-747	47	2	:	:	PUNCT
iajs-747	47	3	it	it	PRON
iajs-747	47	4	follows	follow	VERB
iajs-747	47	5	by	by	ADP
iajs-747	47	6	(	(	PUNCT
iajs-747	47	7	rem.and.ex	rem.and.ex	X
iajs-747	47	8	.	.	PUNCT
iajs-747	47	9	1.2(6	1.2(6	NUM
iajs-747	47	10	)	)	PUNCT
iajs-747	47	11	)	)	PUNCT
iajs-747	47	12	.	.	PUNCT
iajs-747	48	1	8)	8)	NUM
iajs-747	48	2	if	if	SCONJ
iajs-747	48	3	21	21	NUM
iajs-747	48	4	mmm	mmm	NOUN
iajs-747	48	5			NOUN
iajs-747	48	6	is	be	AUX
iajs-747	48	7	an	an	DET
iajs-747	48	8	r	r	NOUN
iajs-747	48	9	-	-	PUNCT
iajs-747	48	10	module	module	NOUN
iajs-747	48	11	,	,	PUNCT
iajs-747	48	12	and	and	CCONJ
iajs-747	48	13	k	k	PROPN
iajs-747	48	14	be	be	AUX
iajs-747	48	15	an	an	DET
iajs-747	48	16	essentially	essentially	ADV
iajs-747	48	17	quasiinvertible	quasiinvertible	ADJ
iajs-747	48	18	submodule	submodule	NOUN
iajs-747	48	19	in	in	ADP
iajs-747	48	20	mi	mi	PROPN
iajs-747	48	21	for	for	ADP
iajs-747	48	22	some	some	DET
iajs-747	48	23	i=	i=	ADJ
iajs-747	48	24	1,2	1,2	NUM
iajs-747	48	25	,	,	PUNCT
iajs-747	48	26	then	then	ADV
iajs-747	48	27	it	it	PRON
iajs-747	48	28	is	be	AUX
iajs-747	48	29	not	not	PART
iajs-747	48	30	necessarily	necessarily	ADV
iajs-747	48	31	that	that	SCONJ
iajs-747	48	32	k	k	PROPN
iajs-747	48	33	is	be	AUX
iajs-747	48	34	an	an	DET
iajs-747	48	35	essentially	essentially	ADV
iajs-747	48	36	quasi	quasi	ADJ
iajs-747	48	37	-	-	ADJ
iajs-747	48	38	invertible	invertible	ADJ
iajs-747	48	39	submodule	submodule	NOUN
iajs-747	48	40	of	of	ADP
iajs-747	48	41	m	m	PRON
iajs-747	48	42	,	,	PUNCT
iajs-747	48	43	for	for	ADP
iajs-747	48	44	example	example	NOUN
iajs-747	48	45	:	:	PUNCT
iajs-747	48	46	let	let	VERB
iajs-747	48	47	2	2	NUM
iajs-747	48	48	zzm	zzm	NOUN
iajs-747	48	49			NOUN
iajs-747	48	50	as	as	ADP
iajs-747	48	51	z	z	NOUN
iajs-747	48	52	-	-	PUNCT
iajs-747	48	53	module	module	NOUN
iajs-747	48	54	,	,	PUNCT
iajs-747	48	55	then	then	ADV
iajs-747	48	56	k	k	PROPN
iajs-747	48	57	=	=	SYM
iajs-747	48	58	z2	z2	PROPN
iajs-747	48	59	is	be	AUX
iajs-747	48	60	an	an	DET
iajs-747	48	61	essentially	essentially	ADV
iajs-747	48	62	quasiinvertible	quasiinvertible	ADJ
iajs-747	48	63	submodule	submodule	NOUN
iajs-747	48	64	of	of	ADP
iajs-747	48	65	z2	z2	PROPN
iajs-747	48	66	as	as	ADP
iajs-747	48	67	z	z	NOUN
iajs-747	48	68	-	-	PUNCT
iajs-747	48	69	module	module	NOUN
iajs-747	48	70	,	,	PUNCT
iajs-747	48	71	but	but	CCONJ
iajs-747	48	72	22	22	NUM
iajs-747	48	73	)	)	PUNCT
iajs-747	48	74	0	0	NUM
iajs-747	48	75	(	(	PUNCT
iajs-747	48	76	zz	zz	PROPN
iajs-747	48	77			PROPN
iajs-747	48	78	which	which	PRON
iajs-747	48	79	is	be	AUX
iajs-747	48	80	not	not	PART
iajs-747	48	81	essentially	essentially	ADV
iajs-747	48	82	quasi	quasi	ADJ
iajs-747	48	83	-	-	ADJ
iajs-747	48	84	invertible	invertible	ADJ
iajs-747	48	85	of	of	ADP
iajs-747	48	86	2	2	NUM
iajs-747	48	87	zzm	zzm	NOUN
iajs-747	48	88			NOUN
iajs-747	48	89	,	,	PUNCT
iajs-747	48	90	since	since	SCONJ
iajs-747	48	91	2)0	2)0	NUM
iajs-747	48	92	(	(	PUNCT
iajs-747	48	93	z	z	NOUN
iajs-747	48	94	≰e	≰e	PROPN
iajs-747	48	95	2	2	NUM
iajs-747	48	96	zz	zz	NOUN
iajs-747	48	97			PROPN
iajs-747	48	98	.	.	PUNCT
iajs-747	49	1	proposition	proposition	NOUN
iajs-747	49	2	(	(	PUNCT
iajs-747	49	3	1.3	1.3	NUM
iajs-747	49	4	)	)	PUNCT
iajs-747	49	5	let	let	VERB
iajs-747	49	6	m	m	PRON
iajs-747	49	7	be	be	AUX
iajs-747	49	8	an	an	DET
iajs-747	49	9	r	r	NOUN
iajs-747	49	10	-	-	PUNCT
iajs-747	49	11	module	module	NOUN
iajs-747	49	12	,	,	PUNCT
iajs-747	49	13	and	and	CCONJ
iajs-747	49	14	let	let	VERB
iajs-747	49	15	n1	n1	NOUN
iajs-747	49	16	,	,	PUNCT
iajs-747	49	17	n2	n2	ADJ
iajs-747	49	18	be	be	AUX
iajs-747	49	19	an	an	DET
iajs-747	49	20	essentially	essentially	ADV
iajs-747	49	21	quasiinvertible	quasiinvertible	ADJ
iajs-747	49	22	rsubmodules	rsubmodule	NOUN
iajs-747	49	23	of	of	ADP
iajs-747	49	24	m	m	PRON
iajs-747	49	25	,	,	PUNCT
iajs-747	49	26	then	then	ADV
iajs-747	49	27	21	21	NUM
iajs-747	49	28	nn	nn	NOUN
iajs-747	49	29			PROPN
iajs-747	49	30	is	be	AUX
iajs-747	49	31	an	an	DET
iajs-747	49	32	essentially	essentially	ADV
iajs-747	49	33	quasi	quasi	ADJ
iajs-747	49	34	-	-	ADJ
iajs-747	49	35	invertible	invertible	ADJ
iajs-747	49	36	r	r	NOUN
iajs-747	49	37	-	-	PUNCT
iajs-747	49	38	submodule	submodule	NOUN
iajs-747	49	39	of	of	ADP
iajs-747	49	40	m.	m.	NOUN
iajs-747	49	41	proof	proof	NOUN
iajs-747	49	42	:	:	PUNCT
iajs-747	49	43	since	since	SCONJ
iajs-747	49	44	n1	n1	PROPN
iajs-747	49	45	≤e	≤e	VERB
iajs-747	49	46	m	m	PROPN
iajs-747	49	47	,	,	PUNCT
iajs-747	49	48	n2	n2	PROPN
iajs-747	49	49	≤e	≤e	NOUN
iajs-747	49	50	m	m	VERB
iajs-747	49	51	then	then	ADV
iajs-747	49	52	0	0	NUM
iajs-747	49	53	)	)	PUNCT
iajs-747	49	54	,	,	PUNCT
iajs-747	49	55	(	(	PUNCT
iajs-747	49	56	1	1	NUM
iajs-747	49	57	mnmhom	mnmhom	PUNCT
iajs-747	49	58	and	and	CCONJ
iajs-747	49	59	0	0	NUM
iajs-747	49	60	)	)	PUNCT
iajs-747	49	61	,	,	PUNCT
iajs-747	49	62	(	(	PUNCT
iajs-747	49	63	2	2	NUM
iajs-747	49	64	mnmhom	mnmhom	PUNCT
iajs-747	49	65	.	.	PUNCT
iajs-747	50	1	also	also	ADV
iajs-747	50	2	n1	n1	VERB
iajs-747	50	3	≤e	≤e	NOUN
iajs-747	50	4	m	m	VERB
iajs-747	50	5	,	,	PUNCT
iajs-747	50	6	n2≤e	n2≤e	PRON
iajs-747	50	7	m	m	NOUN
iajs-747	50	8	imply	imply	VERB
iajs-747	50	9	21	21	NUM
iajs-747	50	10	nn	nn	X
iajs-747	50	11			PROPN
iajs-747	50	12	≤e	≤e	NOUN
iajs-747	50	13	m	m	NOUN
iajs-747	50	14	.	.	PUNCT
iajs-747	51	1	but	but	CCONJ
iajs-747	51	2	)	)	PUNCT
iajs-747	51	3	,	,	PUNCT
iajs-747	51	4	(	(	PUNCT
iajs-747	51	5	)	)	PUNCT
iajs-747	51	6	,	,	PUNCT
iajs-747	51	7	(	(	PUNCT
iajs-747	51	8	)	)	PUNCT
iajs-747	51	9	,	,	PUNCT
iajs-747	51	10	(	(	PUNCT
iajs-747	51	11	2121	2121	NUM
iajs-747	51	12	mnmhommnmhommnnmhom	mnmhommnmhommnnmhom	NOUN
iajs-747	51	13			NOUN
iajs-747	51	14	.hence	.hence	ADP
iajs-747	51	15	0	0	NUM
iajs-747	51	16	)	)	PUNCT
iajs-747	52	1	,	,	PUNCT
iajs-747	52	2	(	(	PUNCT
iajs-747	52	3	21	21	NUM
iajs-747	52	4			PROPN
iajs-747	52	5	mnnmhom	mnnmhom	ADV
iajs-747	52	6	and	and	CCONJ
iajs-747	52	7	so	so	ADV
iajs-747	52	8	that	that	SCONJ
iajs-747	52	9	21	21	NUM
iajs-747	52	10	nn	nn	X
iajs-747	52	11			PROPN
iajs-747	52	12	is	be	AUX
iajs-747	52	13	an	an	DET
iajs-747	52	14	essentially	essentially	ADV
iajs-747	52	15	quasiinvertible	quasiinvertible	ADJ
iajs-747	52	16	rsubmodule	rsubmodule	NOUN
iajs-747	52	17	of	of	ADP
iajs-747	52	18	m	m	PROPN
iajs-747	52	19	.	.	PUNCT
iajs-747	53	1	the	the	DET
iajs-747	53	2	following	follow	VERB
iajs-747	53	3	lemma	lemma	PROPN
iajs-747	53	4	is	be	AUX
iajs-747	53	5	needed	need	VERB
iajs-747	53	6	for	for	ADP
iajs-747	53	7	the	the	DET
iajs-747	53	8	next	next	ADJ
iajs-747	53	9	proposition	proposition	NOUN
iajs-747	53	10	.	.	PUNCT
iajs-747	54	1	lemma	lemma	PROPN
iajs-747	54	2	(	(	PUNCT
iajs-747	54	3	1.4	1.4	NUM
iajs-747	54	4	)	)	PUNCT
iajs-747	54	5	let	let	VERB
iajs-747	54	6	m	m	PRON
iajs-747	54	7	be	be	AUX
iajs-747	54	8	an	an	DET
iajs-747	54	9	r	r	NOUN
iajs-747	54	10	-	-	PUNCT
iajs-747	54	11	module	module	NOUN
iajs-747	54	12	such	such	ADJ
iajs-747	54	13	that	that	PRON
iajs-747	54	14	for	for	ADP
iajs-747	54	15	each	each	DET
iajs-747	54	16	nonzero	nonzero	PROPN
iajs-747	54	17	submodule	submodule	PROPN
iajs-747	54	18	k	k	PROPN
iajs-747	54	19	of	of	ADP
iajs-747	54	20	m	m	PROPN
iajs-747	54	21	,	,	PUNCT
iajs-747	54	22	ppp	ppp	PROPN
iajs-747	54	23	mk	mk	PROPN
iajs-747	54	24	0	0	PROPN
iajs-747	54	25	for	for	ADP
iajs-747	54	26	each	each	DET
iajs-747	54	27	maximal	maximal	ADJ
iajs-747	54	28	ideal	ideal	NOUN
iajs-747	54	29	p	p	NOUN
iajs-747	54	30	of	of	ADP
iajs-747	54	31	r	r	NOUN
iajs-747	54	32	.	.	PUNCT
iajs-747	55	1	if	if	SCONJ
iajs-747	55	2	np	np	INTJ
iajs-747	55	3	≤e	≤e	VERB
iajs-747	55	4	m	m	VERB
iajs-747	55	5	p	p	NOUN
iajs-747	55	6	implies	imply	VERB
iajs-747	55	7	n	n	AUX
iajs-747	55	8	≤e	≤e	VERB
iajs-747	55	9	m	m	NOUN
iajs-747	55	10	.	.	PUNCT
iajs-747	56	1	proof	proof	NOUN
iajs-747	56	2	:	:	PUNCT
iajs-747	56	3	suppose	suppose	VERB
iajs-747	56	4	that	that	SCONJ
iajs-747	56	5	there	there	PRON
iajs-747	56	6	exists	exist	VERB
iajs-747	56	7	mu	mu	PROPN
iajs-747	56	8	0	0	PROPN
iajs-747	56	9	such	such	ADJ
iajs-747	56	10	that	that	PRON
iajs-747	56	11	0nu	0nu	NOUN
iajs-747	56	12	.hence	.hence	ADP
iajs-747	56	13	ppnu	ppnu	NOUN
iajs-747	56	14	0	0	NUM
iajs-747	56	15	)	)	PUNCT
iajs-747	56	16	(	(	PUNCT
iajs-747	56	17			PROPN
iajs-747	56	18	which	which	PRON
iajs-747	56	19	implies	imply	VERB
iajs-747	56	20	that	that	SCONJ
iajs-747	56	21	ppp	ppp	PROPN
iajs-747	56	22	nu	nu	PROPN
iajs-747	56	23	0	0	PROPN
iajs-747	56	24	,	,	PUNCT
iajs-747	56	25	but	but	CCONJ
iajs-747	56	26	ppp	ppp	PROPN
iajs-747	56	27	mu	mu	PROPN
iajs-747	56	28	0	0	PROPN
iajs-747	56	29	by	by	ADP
iajs-747	56	30	hypothesis	hypothesis	NOUN
iajs-747	56	31	,	,	PUNCT
iajs-747	56	32	so	so	SCONJ
iajs-747	56	33	that	that	SCONJ
iajs-747	56	34	np	np	INTJ
iajs-747	56	35	≰e	≰e	PROPN
iajs-747	56	36	m	m	NOUN
iajs-747	56	37	p	p	NOUN
iajs-747	56	38	which	which	PRON
iajs-747	56	39	is	be	AUX
iajs-747	56	40	a	a	DET
iajs-747	56	41	contradiction	contradiction	NOUN
iajs-747	56	42	.	.	PUNCT
iajs-747	57	1	proposition	proposition	NOUN
iajs-747	57	2	(	(	PUNCT
iajs-747	57	3	1.5	1.5	NUM
iajs-747	57	4	)	)	PUNCT
iajs-747	57	5	let	let	VERB
iajs-747	57	6	m	m	PRON
iajs-747	57	7	be	be	AUX
iajs-747	57	8	an	an	DET
iajs-747	57	9	r	r	NOUN
iajs-747	57	10	-	-	PUNCT
iajs-747	57	11	module	module	NOUN
iajs-747	57	12	,	,	PUNCT
iajs-747	57	13	n	n	CCONJ
iajs-747	57	14	≤	≤	NOUN
iajs-747	57	15	m	m	NOUN
iajs-747	57	16	.	.	PUNCT
iajs-747	58	1	if	if	SCONJ
iajs-747	58	2	np	np	PRON
iajs-747	58	3	is	be	AUX
iajs-747	58	4	an	an	DET
iajs-747	58	5	essentially	essentially	ADV
iajs-747	58	6	quasi	quasi	ADJ
iajs-747	58	7	-	-	ADJ
iajs-747	58	8	invertible	invertible	ADJ
iajs-747	58	9	rp	rp	NOUN
iajs-747	58	10	submodule	submodule	NOUN
iajs-747	58	11	of	of	ADP
iajs-747	58	12	rp	rp	NOUN
iajs-747	58	13	-	-	PUNCT
iajs-747	58	14	module	module	NOUN
iajs-747	58	15	m	m	NOUN
iajs-747	58	16	p	p	NOUN
iajs-747	58	17	(	(	PUNCT
iajs-747	58	18	for	for	ADP
iajs-747	58	19	each	each	DET
iajs-747	58	20	maximal	maximal	ADJ
iajs-747	58	21	ideal	ideal	NOUN
iajs-747	58	22	p	p	NOUN
iajs-747	58	23	of	of	ADP
iajs-747	58	24	r	r	NOUN
iajs-747	58	25	)	)	PUNCT
iajs-747	58	26	,	,	PUNCT
iajs-747	58	27	then	then	ADV
iajs-747	58	28	n	n	PRON
iajs-747	58	29	is	be	AUX
iajs-747	58	30	an	an	DET
iajs-747	58	31	essentially	essentially	ADV
iajs-747	58	32	quasi	quasi	ADJ
iajs-747	58	33	-	-	ADJ
iajs-747	58	34	invertible	invertible	ADJ
iajs-747	58	35	submodule	submodule	NOUN
iajs-747	58	36	of	of	ADP
iajs-747	58	37	an	an	DET
iajs-747	58	38	r	r	NOUN
iajs-747	58	39	-	-	PUNCT
iajs-747	58	40	module	module	NOUN
iajs-747	58	41	m.	m.	NOUN
iajs-747	58	42	proof	proof	NOUN
iajs-747	58	43	:	:	PUNCT
iajs-747	58	44	since	since	SCONJ
iajs-747	58	45	np	np	INTJ
iajs-747	58	46	is	be	VERB
iajs-747	58	47	an	an	DET
iajs-747	58	48	essentially	essentially	ADV
iajs-747	58	49	quasi	quasi	ADJ
iajs-747	58	50	-	-	ADJ
iajs-747	58	51	invertible	invertible	ADJ
iajs-747	58	52	rp	rp	NOUN
iajs-747	58	53	-	-	PUNCT
iajs-747	58	54	submodule	submodule	NOUN
iajs-747	58	55	of	of	ADP
iajs-747	58	56	m	m	PROPN
iajs-747	58	57	p	p	NOUN
iajs-747	58	58	,	,	PUNCT
iajs-747	58	59	0	0	NUM
iajs-747	58	60	)	)	PUNCT
iajs-747	58	61	,	,	PUNCT
iajs-747	58	62	(	(	PUNCT
iajs-747	58	63	ppp	ppp	ADP
iajs-747	58	64	mnmhom	mnmhom	VERB
iajs-747	58	65	.	.	PUNCT
iajs-747	59	1	but	but	CCONJ
iajs-747	59	2	by	by	ADP
iajs-747	59	3	[	[	X
iajs-747	59	4	4	4	NUM
iajs-747	59	5	,	,	PUNCT
iajs-747	59	6	ex.3	ex.3	PRON
iajs-747	59	7	,	,	PUNCT
iajs-747	59	8	p.75	p.75	PROPN
iajs-747	59	9	]	]	X
iajs-747	59	10	,	,	PUNCT
iajs-747	59	11	0	0	NUM
iajs-747	59	12	)	)	PUNCT
iajs-747	59	13	,	,	PUNCT
iajs-747	59	14	(	(	PUNCT
iajs-747	59	15	)	)	PUNCT
iajs-747	59	16	)	)	PUNCT
iajs-747	59	17	,	,	PUNCT
iajs-747	59	18	(	(	PUNCT
iajs-747	59	19	(	(	PUNCT
iajs-747	59	20			PROPN
iajs-747	59	21	pppp	pppp	PROPN
iajs-747	59	22	mnmhommnmhom	mnmhommnmhom	PROPN
iajs-747	59	23	,	,	PUNCT
iajs-747	59	24	thus	thus	ADV
iajs-747	59	25	0	0	NUM
iajs-747	59	26	)	)	PUNCT
iajs-747	59	27	)	)	PUNCT
iajs-747	59	28	,	,	PUNCT
iajs-747	59	29	(	(	PUNCT
iajs-747	59	30	(	(	PUNCT
iajs-747	59	31	pmnmhom	pmnmhom	PUNCT
iajs-747	59	32	and	and	CCONJ
iajs-747	59	33	by	by	ADP
iajs-747	59	34	[	[	X
iajs-747	59	35	4	4	NUM
iajs-747	59	36	,	,	PUNCT
iajs-747	59	37	prop.3.13	prop.3.13	NOUN
iajs-747	59	38	,	,	PUNCT
iajs-747	59	39	p.70	p.70	X
iajs-747	59	40	]	]	X
iajs-747	59	41	,	,	PUNCT
iajs-747	59	42	0	0	NUM
iajs-747	59	43	)	)	PUNCT
iajs-747	59	44	,	,	PUNCT
iajs-747	59	45	(	(	PUNCT
iajs-747	59	46	mnmhom	mnmhom	PUNCT
iajs-747	59	47	;	;	PUNCT
iajs-747	59	48	that	that	PRON
iajs-747	59	49	is	be	AUX
iajs-747	59	50	n	n	PRON
iajs-747	59	51	is	be	AUX
iajs-747	59	52	a	a	DET
iajs-747	59	53	quasi	quasi	ADJ
iajs-747	59	54	-	-	ADJ
iajs-747	59	55	invertible	invertible	ADJ
iajs-747	59	56	submodule	submodule	NOUN
iajs-747	59	57	of	of	ADP
iajs-747	59	58	m	m	PROPN
iajs-747	59	59	.	.	PUNCT
iajs-747	60	1	beside	beside	ADP
iajs-747	60	2	this	this	PRON
iajs-747	60	3	,	,	PUNCT
iajs-747	60	4	by	by	ADP
iajs-747	60	5	(	(	PUNCT
iajs-747	60	6	lemma	lemma	PROPN
iajs-747	60	7	(	(	PUNCT
iajs-747	60	8	1.4	1.4	NUM
iajs-747	60	9	)	)	PUNCT
iajs-747	60	10	)	)	PUNCT
iajs-747	60	11	,	,	PUNCT
iajs-747	60	12	n	n	CCONJ
iajs-747	60	13	≤e	≤e	VERB
iajs-747	60	14	m	m	NOUN
iajs-747	60	15	.	.	PUNCT
iajs-747	61	1	thus	thus	ADV
iajs-747	61	2	n	n	PRON
iajs-747	61	3	is	be	AUX
iajs-747	61	4	an	an	DET
iajs-747	61	5	essentially	essentially	ADV
iajs-747	61	6	quasi	quasi	ADJ
iajs-747	61	7	-	-	ADJ
iajs-747	61	8	invertible	invertible	ADJ
iajs-747	61	9	submodule	submodule	NOUN
iajs-747	61	10	of	of	ADP
iajs-747	61	11	m	m	PROPN
iajs-747	61	12	.	.	PUNCT
iajs-747	62	1	ibn	ibn	PROPN
iajs-747	62	2	alhaitham	alhaitham	PROPN
iajs-747	62	3	j.	j.	PROPN
iajs-747	62	4	for	for	ADP
iajs-747	62	5	pure	pure	ADJ
iajs-747	62	6	&	&	CCONJ
iajs-747	62	7	appl	appl	PROPN
iajs-747	62	8	.	.	PUNCT
iajs-747	63	1	sci	sci	PROPN
iajs-747	63	2	.	.	PUNCT
iajs-747	63	3	vol.24	vol.24	NOUN
iajs-747	63	4	(	(	PUNCT
iajs-747	63	5	3	3	NUM
iajs-747	63	6	)	)	PUNCT
iajs-747	63	7	2011	2011	NUM
iajs-747	63	8	recall	recall	VERB
iajs-747	63	9	that	that	SCONJ
iajs-747	63	10	an	an	DET
iajs-747	63	11	r	r	NOUN
iajs-747	63	12	-	-	PUNCT
iajs-747	63	13	submodule	submodule	NOUN
iajs-747	63	14	n	n	NOUN
iajs-747	63	15	of	of	ADP
iajs-747	63	16	an	an	DET
iajs-747	63	17	r	r	NOUN
iajs-747	63	18	-	-	PUNCT
iajs-747	63	19	module	module	NOUN
iajs-747	63	20	m	m	NOUN
iajs-747	63	21	is	be	AUX
iajs-747	63	22	called	call	VERB
iajs-747	63	23	a	a	DET
iajs-747	63	24	sqi	sqi	NOUN
iajs-747	63	25	-	-	PUNCT
iajs-747	63	26	submodule	submodule	NOUN
iajs-747	63	27	if	if	SCONJ
iajs-747	63	28	,	,	PUNCT
iajs-747	63	29	for	for	ADP
iajs-747	63	30	each	each	PRON
iajs-747	63	31	)	)	PUNCT
iajs-747	63	32	,	,	PUNCT
iajs-747	63	33	(	(	PUNCT
iajs-747	63	34	mnmhomf	mnmhomf	NOUN
iajs-747	63	35			PROPN
iajs-747	63	36	,	,	PUNCT
iajs-747	63	37	f(m	f(m	PROPN
iajs-747	63	38	/	/	SYM
iajs-747	63	39	n	n	CCONJ
iajs-747	63	40	)	)	PUNCT
iajs-747	63	41	is	be	AUX
iajs-747	63	42	a	a	DET
iajs-747	63	43	small	small	ADJ
iajs-747	63	44	submodule	submodule	NOUN
iajs-747	63	45	in	in	ADP
iajs-747	63	46	m	m	PROPN
iajs-747	63	47	,	,	PUNCT
iajs-747	64	1	[	[	X
iajs-747	64	2	6	6	NUM
iajs-747	64	3	,	,	PUNCT
iajs-747	64	4	p.44	p.44	NOUN
iajs-747	64	5	]	]	PUNCT
iajs-747	64	6	.	.	PUNCT
iajs-747	65	1	and	and	CCONJ
iajs-747	65	2	an	an	DET
iajs-747	65	3	r	r	NOUN
iajs-747	65	4	-	-	PUNCT
iajs-747	65	5	submodule	submodule	NOUN
iajs-747	65	6	n	n	NOUN
iajs-747	65	7	of	of	ADP
iajs-747	65	8	an	an	DET
iajs-747	65	9	r	r	NOUN
iajs-747	65	10	-	-	PUNCT
iajs-747	65	11	module	module	NOUN
iajs-747	65	12	m	m	NOUN
iajs-747	65	13	is	be	AUX
iajs-747	65	14	called	call	VERB
iajs-747	65	15	a	a	DET
iajs-747	65	16	small	small	ADJ
iajs-747	65	17	submodule	submodule	NOUN
iajs-747	65	18	of	of	ADP
iajs-747	65	19	m	m	PROPN
iajs-747	65	20	(	(	PUNCT
iajs-747	65	21	n	n	CCONJ
iajs-747	65	22	≪	≪	ADJ
iajs-747	65	23	m	m	PRON
iajs-747	65	24	,	,	PUNCT
iajs-747	65	25	for	for	ADP
iajs-747	65	26	short	short	ADJ
iajs-747	65	27	)	)	PUNCT
iajs-747	65	28	if	if	SCONJ
iajs-747	65	29	,	,	PUNCT
iajs-747	65	30	for	for	ADP
iajs-747	65	31	all	all	DET
iajs-747	65	32	k	k	PROPN
iajs-747	65	33	≤	≤	NUM
iajs-747	65	34	m	m	VERB
iajs-747	65	35	with	with	ADP
iajs-747	65	36	n+k	n+k	PROPN
iajs-747	65	37	=	=	PUNCT
iajs-747	65	38	m	m	NOUN
iajs-747	65	39	implies	imply	VERB
iajs-747	65	40	k	k	PROPN
iajs-747	65	41	=	=	PUNCT
iajs-747	65	42	m	m	PROPN
iajs-747	65	43	,	,	PUNCT
iajs-747	65	44	[	[	X
iajs-747	65	45	3	3	NUM
iajs-747	65	46	,	,	PUNCT
iajs-747	65	47	p.106	p.106	NOUN
iajs-747	65	48	]	]	PUNCT
iajs-747	65	49	.	.	PUNCT
iajs-747	66	1	remark	remark	NOUN
iajs-747	66	2	(	(	PUNCT
iajs-747	66	3	1.6	1.6	NUM
iajs-747	66	4	)	)	PUNCT
iajs-747	66	5	it	it	PRON
iajs-747	66	6	is	be	AUX
iajs-747	66	7	clear	clear	ADJ
iajs-747	66	8	that	that	SCONJ
iajs-747	66	9	every	every	DET
iajs-747	66	10	quasi	quasi	ADJ
iajs-747	66	11	-	-	ADJ
iajs-747	66	12	invertible	invertible	ADJ
iajs-747	66	13	submodule	submodule	NOUN
iajs-747	66	14	is	be	AUX
iajs-747	66	15	an	an	DET
iajs-747	66	16	sqi	sqi	NOUN
iajs-747	66	17	-	-	PUNCT
iajs-747	66	18	submodule	submodule	NOUN
iajs-747	66	19	and	and	CCONJ
iajs-747	66	20	hence	hence	ADV
iajs-747	66	21	every	every	DET
iajs-747	66	22	essentially	essentially	ADV
iajs-747	66	23	quasi	quasi	ADJ
iajs-747	66	24	-	-	ADJ
iajs-747	66	25	invertible	invertible	ADJ
iajs-747	66	26	submodule	submodule	NOUN
iajs-747	66	27	is	be	AUX
iajs-747	66	28	an	an	DET
iajs-747	66	29	sqi	sqi	NOUN
iajs-747	66	30	-	-	PUNCT
iajs-747	66	31	submodule	submodule	NOUN
iajs-747	66	32	.	.	PUNCT
iajs-747	67	1	the	the	DET
iajs-747	67	2	converse	converse	NOUN
iajs-747	67	3	of	of	ADP
iajs-747	67	4	(	(	PUNCT
iajs-747	67	5	remark	remark	NOUN
iajs-747	67	6	1.6	1.6	NUM
iajs-747	67	7	)	)	PUNCT
iajs-747	67	8	is	be	AUX
iajs-747	67	9	not	not	PART
iajs-747	67	10	true	true	ADJ
iajs-747	67	11	in	in	ADP
iajs-747	67	12	general	general	ADJ
iajs-747	67	13	,	,	PUNCT
iajs-747	67	14	consider	consider	VERB
iajs-747	67	15	the	the	DET
iajs-747	67	16	following	follow	VERB
iajs-747	67	17	example	example	NOUN
iajs-747	67	18	.	.	PUNCT
iajs-747	68	1	example	example	NOUN
iajs-747	68	2	(	(	PUNCT
iajs-747	68	3	1.7	1.7	NUM
iajs-747	68	4	)	)	PUNCT
iajs-747	68	5	consider	consider	VERB
iajs-747	68	6	the	the	DET
iajs-747	68	7	z	z	NOUN
iajs-747	68	8	-	-	PUNCT
iajs-747	68	9	module	module	NOUN
iajs-747	68	10	z4	z4	NOUN
iajs-747	68	11	,	,	PUNCT
iajs-747	68	12	)	)	PUNCT
iajs-747	68	13	2(n	2(n	NOUN
iajs-747	68	14	,	,	PUNCT
iajs-747	68	15	then	then	ADV
iajs-747	68	16	n	n	PRON
iajs-747	68	17	is	be	AUX
iajs-747	68	18	an	an	DET
iajs-747	68	19	sqi	sqi	NOUN
iajs-747	68	20	-	-	PUNCT
iajs-747	68	21	submodule	submodule	NOUN
iajs-747	68	22	of	of	ADP
iajs-747	68	23	z4	z4	PROPN
iajs-747	68	24	,	,	PUNCT
iajs-747	68	25	since	since	SCONJ
iajs-747	68	26	for	for	ADP
iajs-747	68	27	all	all	PRON
iajs-747	68	28	)	)	PUNCT
iajs-747	68	29	)	)	PUNCT
iajs-747	68	30	,	,	PUNCT
iajs-747	68	31	2	2	X
iajs-747	68	32	(	(	PUNCT
iajs-747	68	33	(	(	PUNCT
iajs-747	68	34	44	44	NUM
iajs-747	68	35	zzhomf	zzhomf	NOUN
iajs-747	68	36			PROPN
iajs-747	68	37	,	,	PUNCT
iajs-747	68	38	then	then	ADV
iajs-747	68	39	)	)	PUNCT
iajs-747	68	40	2	2	X
iajs-747	68	41	(	(	PUNCT
iajs-747	68	42	(	(	PUNCT
iajs-747	68	43	4zf	4zf	ADJ
iajs-747	68	44	≨	≨	NOUN
iajs-747	68	45	z4	z4	X
iajs-747	68	46	,	,	PUNCT
iajs-747	68	47	and	and	CCONJ
iajs-747	68	48	every	every	DET
iajs-747	68	49	proper	proper	ADJ
iajs-747	68	50	submodule	submodule	NOUN
iajs-747	68	51	of	of	ADP
iajs-747	68	52	z4	z4	PROPN
iajs-747	68	53	is	be	AUX
iajs-747	68	54	a	a	DET
iajs-747	68	55	small	small	ADJ
iajs-747	68	56	in	in	ADP
iajs-747	68	57	z4	z4	PROPN
iajs-747	68	58	,	,	PUNCT
iajs-747	68	59	so	so	ADV
iajs-747	68	60	)	)	PUNCT
iajs-747	68	61	2	2	NUM
iajs-747	68	62	(	(	PUNCT
iajs-747	68	63	(	(	PUNCT
iajs-747	68	64	4zf	4zf	ADJ
iajs-747	68	65	≪	≪	VERB
iajs-747	68	66	z4	z4	NOUN
iajs-747	68	67	,	,	PUNCT
iajs-747	68	68	but	but	CCONJ
iajs-747	68	69	it	it	PRON
iajs-747	68	70	is	be	AUX
iajs-747	68	71	known	know	VERB
iajs-747	68	72	that	that	SCONJ
iajs-747	68	73	)	)	PUNCT
iajs-747	69	1	2(n	2(n	NUM
iajs-747	69	2	is	be	AUX
iajs-747	69	3	not	not	PART
iajs-747	69	4	essentially	essentially	ADV
iajs-747	69	5	quasiinvertible	quasiinvertible	ADJ
iajs-747	69	6	in	in	ADP
iajs-747	69	7	z4	z4	PROPN
iajs-747	69	8	,	,	PUNCT
iajs-747	69	9	(	(	PUNCT
iajs-747	69	10	see	see	VERB
iajs-747	69	11	rem.and.ex	rem.and.ex	PROPN
iajs-747	69	12	.	.	PUNCT
iajs-747	70	1	1.2(3	1.2(3	NUM
iajs-747	70	2	)	)	PUNCT
iajs-747	70	3	)	)	PUNCT
iajs-747	70	4	.	.	PUNCT
iajs-747	71	1	2	2	X
iajs-747	71	2	.	.	PUNCT
iajs-747	72	1	essentially	essentially	ADV
iajs-747	72	2	quasi	quasi	ADJ
iajs-747	72	3	-	-	ADJ
iajs-747	72	4	dedekind	dedekind	ADJ
iajs-747	72	5	modules	module	NOUN
iajs-747	72	6	in	in	ADP
iajs-747	72	7	this	this	DET
iajs-747	72	8	section	section	NOUN
iajs-747	72	9	we	we	PRON
iajs-747	72	10	give	give	VERB
iajs-747	72	11	the	the	DET
iajs-747	72	12	definition	definition	NOUN
iajs-747	72	13	of	of	ADP
iajs-747	72	14	essentially	essentially	ADV
iajs-747	72	15	quasi	quasi	ADJ
iajs-747	72	16	-	-	ADJ
iajs-747	72	17	dedekind	dedekind	ADJ
iajs-747	72	18	module	module	NOUN
iajs-747	72	19	with	with	ADP
iajs-747	72	20	some	some	DET
iajs-747	72	21	examples	example	NOUN
iajs-747	72	22	.	.	PUNCT
iajs-747	73	1	we	we	PRON
iajs-747	73	2	prove	prove	VERB
iajs-747	73	3	that	that	SCONJ
iajs-747	73	4	essentially	essentially	ADV
iajs-747	73	5	quasi	quasi	ADJ
iajs-747	73	6	-	-	ADJ
iajs-747	73	7	dedekind	dedekind	ADJ
iajs-747	73	8	module	module	NOUN
iajs-747	73	9	and	and	CCONJ
iajs-747	73	10	k	k	ADJ
iajs-747	73	11	-	-	PUNCT
iajs-747	73	12	nonsingular	nonsingular	ADJ
iajs-747	73	13	module	module	NOUN
iajs-747	73	14	which	which	PRON
iajs-747	73	15	is	be	AUX
iajs-747	73	16	introduced	introduce	VERB
iajs-747	73	17	by	by	ADP
iajs-747	73	18	[	[	X
iajs-747	73	19	8	8	NUM
iajs-747	73	20	]	]	PUNCT
iajs-747	73	21	are	be	AUX
iajs-747	73	22	equivalent	equivalent	ADJ
iajs-747	73	23	.we	.we	PUNCT
iajs-747	73	24	give	give	VERB
iajs-747	73	25	conditions	condition	NOUN
iajs-747	73	26	under	under	ADP
iajs-747	73	27	which	which	PRON
iajs-747	73	28	submodule	submodule	NOUN
iajs-747	73	29	(	(	PUNCT
iajs-747	73	30	resp	resp	NOUN
iajs-747	73	31	.	.	PUNCT
iajs-747	74	1	quotient	quotient	NOUN
iajs-747	74	2	module	module	NOUN
iajs-747	74	3	)	)	PUNCT
iajs-747	74	4	of	of	ADP
iajs-747	74	5	essentially	essentially	ADV
iajs-747	74	6	quasi	quasi	ADJ
iajs-747	74	7	-	-	ADJ
iajs-747	74	8	dedekind	dedekind	ADJ
iajs-747	74	9	is	be	AUX
iajs-747	74	10	essentially	essentially	ADV
iajs-747	74	11	quasidedekind	quasidedekind	ADJ
iajs-747	74	12	.	.	PUNCT
iajs-747	75	1	definition	definition	NOUN
iajs-747	75	2	(	(	PUNCT
iajs-747	75	3	2.1	2.1	NUM
iajs-747	75	4	)	)	PUNCT
iajs-747	75	5	an	an	DET
iajs-747	75	6	r	r	NOUN
iajs-747	75	7	-	-	PUNCT
iajs-747	75	8	module	module	NOUN
iajs-747	75	9	m	m	NOUN
iajs-747	75	10	is	be	AUX
iajs-747	75	11	called	call	VERB
iajs-747	75	12	essentially	essentially	ADV
iajs-747	75	13	quasi	quasi	ADJ
iajs-747	75	14	-	-	NOUN
iajs-747	75	15	dedekind	dedekind	ADJ
iajs-747	75	16	if	if	SCONJ
iajs-747	75	17	,	,	PUNCT
iajs-747	75	18	0	0	NUM
iajs-747	75	19	)	)	PUNCT
iajs-747	75	20	,	,	PUNCT
iajs-747	75	21	(	(	PUNCT
iajs-747	75	22	mnmhom	mnmhom	PUNCT
iajs-747	75	23	for	for	SCONJ
iajs-747	75	24	all	all	DET
iajs-747	75	25	n	n	ADV
iajs-747	75	26	≤e	≤e	VERB
iajs-747	75	27	m	m	NOUN
iajs-747	75	28	.	.	PUNCT
iajs-747	76	1	a	a	DET
iajs-747	76	2	ring	ring	NOUN
iajs-747	76	3	r	r	NOUN
iajs-747	76	4	is	be	AUX
iajs-747	76	5	essentially	essentially	ADV
iajs-747	76	6	quasi	quasi	ADJ
iajs-747	76	7	-	-	NOUN
iajs-747	76	8	dedekind	dedekind	ADJ
iajs-747	76	9	if	if	SCONJ
iajs-747	76	10	r	r	NOUN
iajs-747	76	11	is	be	AUX
iajs-747	76	12	an	an	DET
iajs-747	76	13	essentially	essentially	ADV
iajs-747	76	14	quasidedekind	quasidedekind	ADJ
iajs-747	76	15	r	r	NOUN
iajs-747	76	16	-	-	PUNCT
iajs-747	76	17	module	module	NOUN
iajs-747	76	18	.	.	PUNCT
iajs-747	77	1	remarks	remark	NOUN
iajs-747	77	2	and	and	CCONJ
iajs-747	77	3	examples	example	NOUN
iajs-747	77	4	(	(	PUNCT
iajs-747	77	5	2.2	2.2	NUM
iajs-747	77	6	)	)	PUNCT
iajs-747	77	7	1	1	NUM
iajs-747	77	8	)	)	PUNCT
iajs-747	77	9	it	it	PRON
iajs-747	77	10	is	be	AUX
iajs-747	77	11	clear	clear	ADJ
iajs-747	77	12	that	that	SCONJ
iajs-747	77	13	every	every	DET
iajs-747	77	14	quasi	quasi	ADJ
iajs-747	77	15	-	-	ADJ
iajs-747	77	16	dedekind	dedekind	ADJ
iajs-747	77	17	module	module	NOUN
iajs-747	77	18	is	be	AUX
iajs-747	77	19	an	an	DET
iajs-747	77	20	essentially	essentially	ADV
iajs-747	77	21	quasidedekind	quasidedekind	NOUN
iajs-747	77	22	module	module	NOUN
iajs-747	77	23	,	,	PUNCT
iajs-747	77	24	but	but	CCONJ
iajs-747	77	25	the	the	DET
iajs-747	77	26	converse	converse	NOUN
iajs-747	77	27	is	be	AUX
iajs-747	77	28	not	not	PART
iajs-747	77	29	true	true	ADJ
iajs-747	77	30	in	in	ADP
iajs-747	77	31	general	general	ADJ
iajs-747	77	32	,	,	PUNCT
iajs-747	77	33	for	for	ADP
iajs-747	77	34	example	example	NOUN
iajs-747	77	35	:	:	PUNCT
iajs-747	77	36	each	each	PRON
iajs-747	77	37	of	of	ADP
iajs-747	77	38	z10	z10	NOUN
iajs-747	77	39	,	,	PUNCT
iajs-747	77	40	z15	z15	PROPN
iajs-747	77	41	are	be	AUX
iajs-747	77	42	essentially	essentially	ADV
iajs-747	77	43	quasi	quasi	ADJ
iajs-747	77	44	-	-	NOUN
iajs-747	77	45	dedekind	dedekind	ADJ
iajs-747	77	46	as	as	ADP
iajs-747	77	47	a	a	DET
iajs-747	77	48	z	z	NOUN
iajs-747	77	49	-	-	PUNCT
iajs-747	77	50	module	module	NOUN
iajs-747	77	51	,	,	PUNCT
iajs-747	77	52	but	but	CCONJ
iajs-747	77	53	it	it	PRON
iajs-747	77	54	is	be	AUX
iajs-747	77	55	not	not	PART
iajs-747	77	56	quasi	quasi	ADJ
iajs-747	77	57	-	-	NOUN
iajs-747	77	58	dedekind	dedekind	ADJ
iajs-747	77	59	.	.	PUNCT
iajs-747	78	1	2	2	X
iajs-747	78	2	)	)	PUNCT
iajs-747	78	3	every	every	DET
iajs-747	78	4	integral	integral	ADJ
iajs-747	78	5	domain	domain	NOUN
iajs-747	78	6	r	r	NOUN
iajs-747	78	7	is	be	AUX
iajs-747	78	8	an	an	DET
iajs-747	78	9	essentially	essentially	ADV
iajs-747	78	10	quasi	quasi	ADJ
iajs-747	78	11	-	-	ADJ
iajs-747	78	12	dedekind	dedekind	ADJ
iajs-747	78	13	r	r	NOUN
iajs-747	78	14	-	-	NOUN
iajs-747	78	15	module	module	NOUN
iajs-747	78	16	,	,	PUNCT
iajs-747	78	17	by	by	ADP
iajs-747	78	18	[	[	PUNCT
iajs-747	78	19	5	5	NUM
iajs-747	78	20	,	,	PUNCT
iajs-747	78	21	ex	ex	X
iajs-747	78	22	1.4	1.4	NUM
iajs-747	78	23	,	,	PUNCT
iajs-747	78	24	p.24	p.24	PROPN
iajs-747	78	25	]	]	PUNCT
iajs-747	78	26	and	and	CCONJ
iajs-747	78	27	(	(	PUNCT
iajs-747	78	28	rem.and.ex	rem.and.ex	PROPN
iajs-747	78	29	2.2(1	2.2(1	NUM
iajs-747	78	30	)	)	PUNCT
iajs-747	78	31	)	)	PUNCT
iajs-747	78	32	.	.	PUNCT
iajs-747	79	1	3	3	X
iajs-747	79	2	)	)	PUNCT
iajs-747	79	3	z4	z4	PROPN
iajs-747	79	4	as	as	ADP
iajs-747	79	5	a	a	DET
iajs-747	79	6	z	z	NOUN
iajs-747	79	7	-	-	PUNCT
iajs-747	79	8	module	module	NOUN
iajs-747	79	9	is	be	AUX
iajs-747	79	10	not	not	PART
iajs-747	79	11	essentially	essentially	ADV
iajs-747	79	12	quasi	quasi	ADJ
iajs-747	79	13	-	-	ADJ
iajs-747	79	14	dedekind	dedekind	ADJ
iajs-747	79	15	,	,	PUNCT
iajs-747	79	16	since	since	SCONJ
iajs-747	79	17	)	)	PUNCT
iajs-747	79	18	2	2	NUM
iajs-747	79	19	(	(	PUNCT
iajs-747	79	20	≤	≤	X
iajs-747	79	21	e	e	X
iajs-747	79	22	z4	z4	X
iajs-747	79	23	,	,	PUNCT
iajs-747	79	24	but	but	CCONJ
iajs-747	79	25	0)),2	0)),2	NUM
iajs-747	79	26	(	(	PUNCT
iajs-747	79	27	(	(	PUNCT
iajs-747	79	28	244	244	NUM
iajs-747	79	29			NUM
iajs-747	79	30	zzzhom	zzzhom	NOUN
iajs-747	79	31	.	.	PUNCT
iajs-747	80	1	4	4	X
iajs-747	80	2	)	)	PUNCT
iajs-747	80	3	let	let	VERB
iajs-747	80	4	m	m	VERB
iajs-747	80	5	=	=	ADJ
iajs-747	80	6	zp	zp	X
iajs-747	80	7	∞	∞	PROPN
iajs-747	80	8	as	as	ADP
iajs-747	80	9	a	a	DET
iajs-747	80	10	z	z	NOUN
iajs-747	80	11	-	-	PUNCT
iajs-747	80	12	module	module	NOUN
iajs-747	80	13	.	.	PUNCT
iajs-747	81	1	then	then	ADV
iajs-747	81	2	m	m	VERB
iajs-747	81	3	is	be	AUX
iajs-747	81	4	not	not	PART
iajs-747	81	5	essentially	essentially	ADV
iajs-747	81	6	quasidedekind	quasidedekind	ADJ
iajs-747	81	7	,	,	PUNCT
iajs-747	81	8	but	but	CCONJ
iajs-747	81	9	)	)	PUNCT
iajs-747	81	10	(	(	PUNCT
iajs-747	81	11	mendz	mendz	VERB
iajs-747	81	12	(	(	PUNCT
iajs-747	81	13	is	be	AUX
iajs-747	81	14	the	the	DET
iajs-747	81	15	ring	ring	NOUN
iajs-747	81	16	of	of	ADP
iajs-747	81	17	p	p	NOUN
iajs-747	81	18	-	-	PUNCT
iajs-747	81	19	adic	adic	ADJ
iajs-747	81	20	integers	integer	NOUN
iajs-747	81	21	)	)	PUNCT
iajs-747	81	22	is	be	AUX
iajs-747	81	23	a	a	DET
iajs-747	81	24	commutative	commutative	ADJ
iajs-747	81	25	domain	domain	NOUN
iajs-747	81	26	[	[	X
iajs-747	81	27	see	see	VERB
iajs-747	81	28	ex	ex	X
iajs-747	81	29	4.1.2	4.1.2	NUM
iajs-747	81	30	,	,	PUNCT
iajs-747	81	31	8	8	NUM
iajs-747	81	32	]	]	PUNCT
iajs-747	81	33	,	,	PUNCT
iajs-747	81	34	so	so	ADV
iajs-747	81	35	)	)	PUNCT
iajs-747	81	36	(	(	PUNCT
iajs-747	81	37	mendz	mendz	X
iajs-747	81	38	is	be	AUX
iajs-747	81	39	essentially	essentially	ADV
iajs-747	81	40	quasi	quasi	ADJ
iajs-747	81	41	-	-	ADJ
iajs-747	81	42	dedekind	dedekind	ADJ
iajs-747	81	43	,	,	PUNCT
iajs-747	81	44	by	by	ADP
iajs-747	81	45	(	(	PUNCT
iajs-747	81	46	rem.and.ex	rem.and.ex	NOUN
iajs-747	81	47	2.2(2	2.2(2	NUM
iajs-747	81	48	)	)	PUNCT
iajs-747	81	49	)	)	PUNCT
iajs-747	81	50	.	.	PUNCT
iajs-747	82	1	5	5	X
iajs-747	82	2	)	)	PUNCT
iajs-747	82	3	let	let	VERB
iajs-747	82	4	m	m	PRON
iajs-747	82	5	be	be	AUX
iajs-747	82	6	a	a	DET
iajs-747	82	7	uniform	uniform	ADJ
iajs-747	82	8	r	r	NOUN
iajs-747	82	9	-	-	PUNCT
iajs-747	82	10	module	module	NOUN
iajs-747	82	11	.	.	PUNCT
iajs-747	83	1	then	then	ADV
iajs-747	83	2	m	m	PROPN
iajs-747	83	3	is	be	AUX
iajs-747	83	4	a	a	DET
iajs-747	83	5	quasi	quasi	ADJ
iajs-747	83	6	-	-	ADJ
iajs-747	83	7	dedekind	dedekind	ADJ
iajs-747	83	8	r	r	NOUN
iajs-747	83	9	-	-	PUNCT
iajs-747	83	10	module	module	NOUN
iajs-747	83	11	if	if	SCONJ
iajs-747	83	12	and	and	CCONJ
iajs-747	83	13	only	only	ADV
iajs-747	83	14	if	if	SCONJ
iajs-747	83	15	m	m	NOUN
iajs-747	83	16	is	be	AUX
iajs-747	83	17	an	an	DET
iajs-747	83	18	essentially	essentially	ADV
iajs-747	83	19	quasi	quasi	ADJ
iajs-747	83	20	-	-	ADJ
iajs-747	83	21	dedekind	dedekind	ADJ
iajs-747	83	22	r	r	NOUN
iajs-747	83	23	-	-	NOUN
iajs-747	83	24	module	module	NOUN
iajs-747	83	25	.	.	PUNCT
iajs-747	84	1	ibn	ibn	PROPN
iajs-747	84	2	alhaitham	alhaitham	PROPN
iajs-747	84	3	j.	j.	PROPN
iajs-747	84	4	for	for	ADP
iajs-747	84	5	pure	pure	ADJ
iajs-747	84	6	&	&	CCONJ
iajs-747	84	7	appl	appl	PROPN
iajs-747	84	8	.	.	PUNCT
iajs-747	85	1	sci	sci	PROPN
iajs-747	85	2	.	.	PUNCT
iajs-747	85	3	vol.24	vol.24	NOUN
iajs-747	85	4	(	(	PUNCT
iajs-747	85	5	3	3	NUM
iajs-747	85	6	)	)	PUNCT
iajs-747	85	7	2011	2011	NUM
iajs-747	85	8	proof	proof	NOUN
iajs-747	85	9	:	:	PUNCT
iajs-747	85	10	it	it	PRON
iajs-747	85	11	is	be	AUX
iajs-747	85	12	clear	clear	ADJ
iajs-747	85	13	.	.	PUNCT
iajs-747	86	1	roman	roman	PROPN
iajs-747	86	2	c.s	c.s	VERB
iajs-747	86	3	in	in	ADP
iajs-747	86	4	[	[	X
iajs-747	86	5	8	8	NUM
iajs-747	86	6	]	]	PUNCT
iajs-747	86	7	,	,	PUNCT
iajs-747	86	8	introduce	introduce	VERB
iajs-747	86	9	the	the	DET
iajs-747	86	10	following	follow	VERB
iajs-747	86	11	:	:	PUNCT
iajs-747	86	12	"	"	PUNCT
iajs-747	86	13	an	an	DET
iajs-747	86	14	r	r	NOUN
iajs-747	86	15	-	-	PUNCT
iajs-747	86	16	module	module	NOUN
iajs-747	86	17	m	m	NOUN
iajs-747	86	18	is	be	AUX
iajs-747	86	19	called	call	VERB
iajs-747	86	20	k	k	ADJ
iajs-747	86	21	-	-	ADJ
iajs-747	86	22	nonsingular	nonsingular	ADJ
iajs-747	86	23	if	if	SCONJ
iajs-747	86	24	,	,	PUNCT
iajs-747	86	25	for	for	ADP
iajs-747	86	26	each	each	PRON
iajs-747	86	27	)	)	PUNCT
iajs-747	86	28	(	(	PUNCT
iajs-747	86	29	mendf	mendf	ADJ
iajs-747	86	30	r	r	ADJ
iajs-747	86	31	,	,	PUNCT
iajs-747	86	32	kerf	kerf	NOUN
iajs-747	86	33	≤	≤	NUM
iajs-747	86	34	e	e	X
iajs-747	86	35	m	m	NOUN
iajs-747	86	36	implies	imply	VERB
iajs-747	86	37	f	f	PROPN
iajs-747	86	38	=	=	SYM
iajs-747	86	39	0	0	NUM
iajs-747	86	40	"	"	PUNCT
iajs-747	86	41	.	.	PUNCT
iajs-747	87	1	however	however	ADV
iajs-747	87	2	we	we	PRON
iajs-747	87	3	prove	prove	VERB
iajs-747	87	4	the	the	DET
iajs-747	87	5	following	follow	VERB
iajs-747	87	6	:	:	PUNCT
iajs-747	87	7	theorem	theorem	NOUN
iajs-747	87	8	(	(	PUNCT
iajs-747	87	9	2.3	2.3	NUM
iajs-747	87	10	)	)	PUNCT
iajs-747	87	11	let	let	VERB
iajs-747	87	12	m	m	PRON
iajs-747	87	13	be	be	AUX
iajs-747	87	14	an	an	DET
iajs-747	87	15	r	r	NOUN
iajs-747	87	16	-	-	PUNCT
iajs-747	87	17	module	module	NOUN
iajs-747	87	18	.	.	PUNCT
iajs-747	88	1	then	then	ADV
iajs-747	88	2	m	m	VERB
iajs-747	88	3	is	be	AUX
iajs-747	88	4	an	an	DET
iajs-747	88	5	essentially	essentially	ADV
iajs-747	88	6	quasi	quasi	ADJ
iajs-747	88	7	-	-	ADJ
iajs-747	88	8	dedekind	dedekind	ADJ
iajs-747	88	9	r	r	NOUN
iajs-747	88	10	-	-	PUNCT
iajs-747	88	11	module	module	NOUN
iajs-747	88	12	if	if	SCONJ
iajs-747	88	13	and	and	CCONJ
iajs-747	88	14	only	only	ADV
iajs-747	88	15	if	if	SCONJ
iajs-747	88	16	m	m	NOUN
iajs-747	88	17	is	be	AUX
iajs-747	88	18	a	a	DET
iajs-747	88	19	k	k	ADJ
iajs-747	88	20	-	-	ADJ
iajs-747	88	21	nonsingular	nonsingular	ADJ
iajs-747	88	22	r	r	NOUN
iajs-747	88	23	-	-	PUNCT
iajs-747	88	24	module	module	NOUN
iajs-747	88	25	.	.	PUNCT
iajs-747	89	1	proof	proof	NOUN
iajs-747	89	2	:	:	PUNCT
iajs-747	89	3	)	)	PUNCT
iajs-747	89	4			NOUN
iajs-747	89	5	let	let	VERB
iajs-747	89	6	)	)	PUNCT
iajs-747	89	7	(	(	PUNCT
iajs-747	89	8	mendf	mendf	ADJ
iajs-747	89	9	r	r	NOUN
iajs-747	89	10			NOUN
iajs-747	89	11	,	,	PUNCT
iajs-747	89	12	0f	0f	NUM
iajs-747	89	13	.	.	PUNCT
iajs-747	90	1	suppose	suppose	VERB
iajs-747	90	2	that	that	SCONJ
iajs-747	90	3	kerf	kerf	NOUN
iajs-747	90	4	≤	≤	NUM
iajs-747	90	5	e	e	VERB
iajs-747	90	6	m	m	PROPN
iajs-747	90	7	,	,	PUNCT
iajs-747	90	8	defined	define	VERB
iajs-747	90	9	mkerfmg	mkerfmg	NOUN
iajs-747	90	10			NUM
iajs-747	90	11	:	:	PUNCT
iajs-747	90	12	by	by	ADP
iajs-747	90	13	g	g	PROPN
iajs-747	90	14	(	(	PUNCT
iajs-747	90	15	m+kerf	m+kerf	PROPN
iajs-747	90	16	)	)	PUNCT
iajs-747	91	1	=	=	SYM
iajs-747	91	2	f	f	PROPN
iajs-747	91	3	(	(	PUNCT
iajs-747	91	4	m	m	PROPN
iajs-747	91	5	)	)	PUNCT
iajs-747	91	6	for	for	ADP
iajs-747	91	7	all	all	DET
iajs-747	91	8	mm	mm	NOUN
iajs-747	91	9	.	.	PUNCT
iajs-747	92	1	it	it	PRON
iajs-747	92	2	is	be	AUX
iajs-747	92	3	easy	easy	ADJ
iajs-747	92	4	to	to	PART
iajs-747	92	5	see	see	VERB
iajs-747	92	6	that	that	SCONJ
iajs-747	92	7	g	g	PROPN
iajs-747	92	8	is	be	AUX
iajs-747	92	9	well	well	ADV
iajs-747	92	10	-	-	PUNCT
iajs-747	92	11	defined	define	VERB
iajs-747	92	12	and	and	CCONJ
iajs-747	92	13	g	g	NOUN
iajs-747	92	14	is	be	AUX
iajs-747	92	15	a	a	DET
iajs-747	92	16	nonzero	nonzero	ADJ
iajs-747	92	17	homomorphism	homomorphism	NOUN
iajs-747	92	18	.	.	PUNCT
iajs-747	93	1	thus	thus	ADV
iajs-747	93	2	0	0	NUM
iajs-747	93	3	)	)	PUNCT
iajs-747	93	4	,	,	PUNCT
iajs-747	93	5	(	(	PUNCT
iajs-747	93	6	mkerfmhom	mkerfmhom	X
iajs-747	93	7	which	which	PRON
iajs-747	93	8	is	be	AUX
iajs-747	93	9	a	a	DET
iajs-747	93	10	contradiction	contradiction	NOUN
iajs-747	93	11	,	,	PUNCT
iajs-747	93	12	since	since	SCONJ
iajs-747	93	13	m	m	PROPN
iajs-747	93	14	is	be	AUX
iajs-747	93	15	an	an	DET
iajs-747	93	16	essentially	essentially	ADV
iajs-747	93	17	quasi	quasi	ADJ
iajs-747	93	18	-	-	ADJ
iajs-747	93	19	dedekind	dedekind	ADJ
iajs-747	93	20	r	r	NOUN
iajs-747	93	21	-	-	NOUN
iajs-747	93	22	module	module	NOUN
iajs-747	93	23	.	.	PUNCT
iajs-747	93	24	)	)	PUNCT
iajs-747	94	1			PROPN
iajs-747	94	2	n	n	CCONJ
iajs-747	95	1	≤	≤	NUM
iajs-747	95	2	e	e	X
iajs-747	95	3	m	m	PROPN
iajs-747	95	4	.	.	PUNCT
iajs-747	96	1	suppose	suppose	VERB
iajs-747	96	2	that	that	SCONJ
iajs-747	96	3	there	there	PRON
iajs-747	96	4	exists	exist	VERB
iajs-747	96	5	mnmf	mnmf	NOUN
iajs-747	96	6			NOUN
iajs-747	96	7	:	:	PUNCT
iajs-747	96	8	and	and	CCONJ
iajs-747	96	9	0f	0f	NUM
iajs-747	96	10	.	.	PUNCT
iajs-747	97	1	we	we	PRON
iajs-747	97	2	have	have	AUX
iajs-747	97	3	mnmm	mnmm	VERB
iajs-747	97	4	f	f	PROPN
iajs-747	97	5	,	,	PUNCT
iajs-747	97	6	where	where	SCONJ
iajs-747	97	7	π	π	PROPN
iajs-747	97	8	is	be	AUX
iajs-747	97	9	the	the	DET
iajs-747	97	10	canonical	canonical	ADJ
iajs-747	97	11	projection	projection	NOUN
iajs-747	97	12	.let	.let	NOUN
iajs-747	97	13	)	)	PUNCT
iajs-747	98	1	(	(	PUNCT
iajs-747	98	2	mendfo	mendfo	NOUN
iajs-747	98	3	r	r	NOUN
iajs-747	98	4			X
iajs-747	98	5	.	.	PUNCT
iajs-747	99	1	kern	kern	ADJ
iajs-747	99	2			PROPN
iajs-747	99	3	and	and	CCONJ
iajs-747	99	4	n≤	n≤	PRON
iajs-747	99	5	em	em	PRON
iajs-747	99	6	implies	imply	VERB
iajs-747	99	7	ker	ker	X
iajs-747	99	8	≤em	≤em	ADJ
iajs-747	99	9	,	,	PUNCT
iajs-747	99	10	(	(	PUNCT
iajs-747	99	11	)	)	PUNCT
iajs-747	99	12	(	(	PUNCT
iajs-747	99	13	)	)	PUNCT
iajs-747	99	14	(	(	PUNCT
iajs-747	99	15	)	)	PUNCT
iajs-747	99	16	0	0	NUM
iajs-747	99	17	m	m	VERB
iajs-747	99	18	fo	fo	ADP
iajs-747	99	19	m	m	PROPN
iajs-747	99	20	f	f	PROPN
iajs-747	99	21	m	m	PROPN
iajs-747	99	22	n	n	PROPN
iajs-747	100	1			VERB
iajs-747	101	1			NUM
iajs-747	101	2			NOUN
iajs-747	101	3	which	which	PRON
iajs-747	101	4	is	be	AUX
iajs-747	101	5	a	a	DET
iajs-747	101	6	contradiction	contradiction	NOUN
iajs-747	101	7	with	with	ADP
iajs-747	101	8	m	m	PROPN
iajs-747	101	9	is	be	AUX
iajs-747	101	10	a	a	DET
iajs-747	101	11	k	k	ADJ
iajs-747	101	12	-	-	ADJ
iajs-747	101	13	nonsingular	nonsingular	ADJ
iajs-747	101	14	r	r	NOUN
iajs-747	101	15	-	-	PUNCT
iajs-747	101	16	module	module	NOUN
iajs-747	101	17	.	.	PUNCT
iajs-747	102	1	although	although	SCONJ
iajs-747	102	2	the	the	DET
iajs-747	102	3	concepts	concept	NOUN
iajs-747	102	4	of	of	ADP
iajs-747	102	5	essentially	essentially	ADV
iajs-747	102	6	quasi	quasi	ADJ
iajs-747	102	7	-	-	ADJ
iajs-747	102	8	dedekind	dedekind	ADJ
iajs-747	102	9	module	module	NOUN
iajs-747	102	10	and	and	CCONJ
iajs-747	102	11	k	k	ADJ
iajs-747	102	12	-	-	PUNCT
iajs-747	102	13	nonsingular	nonsingular	ADJ
iajs-747	102	14	module	module	NOUN
iajs-747	102	15	are	be	AUX
iajs-747	102	16	equivalent	equivalent	ADJ
iajs-747	102	17	,	,	PUNCT
iajs-747	102	18	but	but	CCONJ
iajs-747	102	19	we	we	PRON
iajs-747	102	20	see	see	VERB
iajs-747	102	21	that	that	SCONJ
iajs-747	102	22	it	it	PRON
iajs-747	102	23	is	be	AUX
iajs-747	102	24	convenient	convenient	ADJ
iajs-747	102	25	to	to	PART
iajs-747	102	26	use	use	VERB
iajs-747	102	27	the	the	DET
iajs-747	102	28	notion	notion	NOUN
iajs-747	102	29	essentially	essentially	ADV
iajs-747	102	30	quasidedekind	quasidedekind	VERB
iajs-747	102	31	in	in	ADP
iajs-747	102	32	this	this	DET
iajs-747	102	33	paper	paper	NOUN
iajs-747	102	34	.	.	PUNCT
iajs-747	103	1	proposition	proposition	NOUN
iajs-747	103	2	(	(	PUNCT
iajs-747	103	3	2.4	2.4	NUM
iajs-747	103	4	)	)	PUNCT
iajs-747	103	5	every	every	DET
iajs-747	103	6	semisimple	semisimple	NOUN
iajs-747	103	7	r	r	NOUN
iajs-747	103	8	-	-	PUNCT
iajs-747	103	9	module	module	NOUN
iajs-747	103	10	is	be	AUX
iajs-747	103	11	an	an	DET
iajs-747	103	12	essentially	essentially	ADV
iajs-747	103	13	quasi	quasi	ADJ
iajs-747	103	14	-	-	ADJ
iajs-747	103	15	dedekind	dedekind	ADJ
iajs-747	103	16	r	r	NOUN
iajs-747	103	17	-	-	PUNCT
iajs-747	103	18	module	module	NOUN
iajs-747	103	19	.	.	PUNCT
iajs-747	104	1	proof	proof	NOUN
iajs-747	104	2	:	:	PUNCT
iajs-747	104	3	it	it	PRON
iajs-747	104	4	is	be	AUX
iajs-747	104	5	easy	easy	ADJ
iajs-747	104	6	.	.	PUNCT
iajs-747	105	1	the	the	DET
iajs-747	105	2	converse	converse	NOUN
iajs-747	105	3	of	of	ADP
iajs-747	105	4	(	(	PUNCT
iajs-747	105	5	prop	prop	NOUN
iajs-747	105	6	2.4	2.4	NUM
iajs-747	105	7	)	)	PUNCT
iajs-747	105	8	is	be	AUX
iajs-747	105	9	not	not	PART
iajs-747	105	10	true	true	ADJ
iajs-747	105	11	in	in	ADP
iajs-747	105	12	general	general	ADJ
iajs-747	105	13	,	,	PUNCT
iajs-747	105	14	consider	consider	VERB
iajs-747	105	15	the	the	DET
iajs-747	105	16	following	follow	VERB
iajs-747	105	17	example	example	NOUN
iajs-747	105	18	.	.	PUNCT
iajs-747	106	1	example	example	NOUN
iajs-747	106	2	(	(	PUNCT
iajs-747	106	3	2.5	2.5	NUM
iajs-747	106	4	)	)	PUNCT
iajs-747	106	5	it	it	PRON
iajs-747	106	6	is	be	AUX
iajs-747	106	7	known	know	VERB
iajs-747	106	8	that	that	SCONJ
iajs-747	106	9	z	z	NOUN
iajs-747	106	10	as	as	ADP
iajs-747	106	11	a	a	DET
iajs-747	106	12	z	z	NOUN
iajs-747	106	13	-	-	PUNCT
iajs-747	106	14	module	module	NOUN
iajs-747	106	15	is	be	AUX
iajs-747	106	16	essentially	essentially	ADV
iajs-747	106	17	quasi	quasi	ADJ
iajs-747	106	18	-	-	ADJ
iajs-747	106	19	dedekind	dedekind	ADJ
iajs-747	106	20	,	,	PUNCT
iajs-747	106	21	but	but	CCONJ
iajs-747	106	22	it	it	PRON
iajs-747	106	23	is	be	AUX
iajs-747	106	24	not	not	PART
iajs-747	106	25	semisimple	semisimple	ADJ
iajs-747	106	26	.	.	PUNCT
iajs-747	107	1	recall	recall	VERB
iajs-747	107	2	that	that	SCONJ
iajs-747	107	3	an	an	DET
iajs-747	107	4	ideal	ideal	NOUN
iajs-747	107	5	i	i	PRON
iajs-747	107	6	of	of	ADP
iajs-747	107	7	a	a	DET
iajs-747	107	8	ring	ring	NOUN
iajs-747	107	9	r	r	NOUN
iajs-747	107	10	is	be	AUX
iajs-747	107	11	semiprime	semiprime	NOUN
iajs-747	107	12	if	if	SCONJ
iajs-747	107	13	,	,	PUNCT
iajs-747	107	14	for	for	ADP
iajs-747	107	15	all	all	PRON
iajs-747	107	16	rr	rr	PROPN
iajs-747	107	17			NOUN
iajs-747	107	18	with	with	ADP
iajs-747	107	19	ir	ir	PROPN
iajs-747	107	20	2	2	PUNCT
iajs-747	107	21	implies	imply	VERB
iajs-747	107	22	ir	ir	PROPN
iajs-747	107	23			NOUN
iajs-747	107	24	[	[	X
iajs-747	107	25	or	or	CCONJ
iajs-747	107	26	,	,	PUNCT
iajs-747	107	27	for	for	SCONJ
iajs-747	107	28	all	all	DET
iajs-747	107	29	ideal	ideal	ADJ
iajs-747	107	30	a	a	PRON
iajs-747	107	31	of	of	ADP
iajs-747	107	32	r	r	NOUN
iajs-747	107	33	with	with	ADP
iajs-747	107	34	ia	ia	PROPN
iajs-747	107	35	2	2	NOUN
iajs-747	107	36	implies	imply	VERB
iajs-747	107	37	ia	ia	PROPN
iajs-747	107	38			PROPN
iajs-747	107	39	]	]	PUNCT
iajs-747	107	40	.and	.and	PUNCT
iajs-747	108	1	a	a	DET
iajs-747	108	2	ring	ring	NOUN
iajs-747	108	3	r	r	NOUN
iajs-747	108	4	is	be	AUX
iajs-747	108	5	called	call	VERB
iajs-747	108	6	semiprime	semiprime	NOUN
iajs-747	108	7	if	if	SCONJ
iajs-747	108	8	(	(	PUNCT
iajs-747	108	9	0	0	X
iajs-747	108	10	)	)	PUNCT
iajs-747	108	11	is	be	AUX
iajs-747	108	12	a	a	DET
iajs-747	108	13	semiprime	semiprime	NOUN
iajs-747	108	14	ideal	ideal	NOUN
iajs-747	108	15	of	of	ADP
iajs-747	108	16	r	r	NOUN
iajs-747	108	17	;	;	PUNCT
iajs-747	108	18	i.e	i.e	CCONJ
iajs-747	108	19	r	r	NOUN
iajs-747	108	20	does	do	AUX
iajs-747	108	21	not	not	PART
iajs-747	108	22	contain	contain	VERB
iajs-747	108	23	nonzero	nonzero	ADJ
iajs-747	108	24	nilpotent	nilpotent	ADJ
iajs-747	108	25	ideals	ideal	NOUN
iajs-747	108	26	,	,	PUNCT
iajs-747	108	27	[	[	X
iajs-747	108	28	2	2	NUM
iajs-747	108	29	]	]	PUNCT
iajs-747	108	30	.	.	PUNCT
iajs-747	109	1	proposition	proposition	NOUN
iajs-747	109	2	(	(	PUNCT
iajs-747	109	3	2.6	2.6	NUM
iajs-747	109	4	)	)	PUNCT
iajs-747	109	5	let	let	VERB
iajs-747	109	6	r	r	PRON
iajs-747	109	7	be	be	AUX
iajs-747	109	8	a	a	DET
iajs-747	109	9	ring	ring	NOUN
iajs-747	109	10	.	.	PUNCT
iajs-747	110	1	the	the	DET
iajs-747	110	2	following	follow	VERB
iajs-747	110	3	statements	statement	NOUN
iajs-747	110	4	are	be	AUX
iajs-747	110	5	equivalent	equivalent	ADJ
iajs-747	110	6	:	:	PUNCT
iajs-747	110	7	1	1	X
iajs-747	110	8	)	)	PUNCT
iajs-747	110	9	r	r	NOUN
iajs-747	110	10	is	be	AUX
iajs-747	110	11	an	an	DET
iajs-747	110	12	essentially	essentially	ADV
iajs-747	110	13	quasi	quasi	ADJ
iajs-747	110	14	-	-	ADJ
iajs-747	110	15	dedekind	dedekind	ADJ
iajs-747	110	16	ring	ring	NOUN
iajs-747	110	17	.	.	PUNCT
iajs-747	111	1	2	2	X
iajs-747	111	2	)	)	PUNCT
iajs-747	111	3	r	r	NOUN
iajs-747	111	4	is	be	AUX
iajs-747	111	5	a	a	DET
iajs-747	111	6	semiprime	semiprime	NOUN
iajs-747	111	7	ring	ring	NOUN
iajs-747	111	8	.	.	PUNCT
iajs-747	112	1	3	3	X
iajs-747	112	2	)	)	PUNCT
iajs-747	112	3	z(r	z(r	NOUN
iajs-747	112	4	)	)	PUNCT
iajs-747	113	1	=	=	SYM
iajs-747	113	2	0	0	PUNCT
iajs-747	114	1	(	(	PUNCT
iajs-747	114	2	r	r	NOUN
iajs-747	114	3	is	be	AUX
iajs-747	114	4	a	a	DET
iajs-747	114	5	nonsingular	nonsingular	ADJ
iajs-747	114	6	ring	ring	NOUN
iajs-747	114	7	)	)	PUNCT
iajs-747	114	8	.	.	PUNCT
iajs-747	115	1	proof	proof	NOUN
iajs-747	115	2	:	:	PUNCT
iajs-747	115	3	)	)	PUNCT
iajs-747	116	1	3()2	3()2	NUM
iajs-747	116	2	(	(	PUNCT
iajs-747	116	3			NOUN
iajs-747	116	4	:	:	PUNCT
iajs-747	116	5	it	it	PRON
iajs-747	116	6	is	be	AUX
iajs-747	116	7	follows	follow	VERB
iajs-747	116	8	by	by	ADP
iajs-747	116	9	[	[	X
iajs-747	116	10	2	2	NUM
iajs-747	116	11	,	,	PUNCT
iajs-747	116	12	prop	prop	NOUN
iajs-747	116	13	1.27	1.27	NUM
iajs-747	116	14	,	,	PUNCT
iajs-747	116	15	p.35	p.35	X
iajs-747	116	16	]	]	PUNCT
iajs-747	116	17	)	)	PUNCT
iajs-747	117	1	1()2	1()2	NUM
iajs-747	117	2	(	(	PUNCT
iajs-747	117	3			NOUN
iajs-747	117	4	:	:	PUNCT
iajs-747	117	5	let	let	VERB
iajs-747	117	6	)	)	PUNCT
iajs-747	117	7	(	(	PUNCT
iajs-747	117	8	rendf	rendf	NOUN
iajs-747	117	9	r	r	VERB
iajs-747	117	10	such	such	ADJ
iajs-747	117	11	that	that	DET
iajs-747	117	12	kerf	kerf	NOUN
iajs-747	117	13	≤	≤	NUM
iajs-747	117	14	e	e	NOUN
iajs-747	117	15	r	r	NOUN
iajs-747	117	16	.	.	PUNCT
iajs-747	118	1	to	to	PART
iajs-747	118	2	prove	prove	VERB
iajs-747	118	3	f	f	PROPN
iajs-747	118	4	=	=	SYM
iajs-747	118	5	0	0	PROPN
iajs-747	118	6	.	.	PUNCT
iajs-747	119	1	suppose	suppose	VERB
iajs-747	119	2	that	that	SCONJ
iajs-747	119	3	0f	0f	NUM
iajs-747	119	4	,	,	PUNCT
iajs-747	119	5	there	there	PRON
iajs-747	119	6	exists	exist	VERB
iajs-747	119	7	rr0	rr0	PROPN
iajs-747	119	8	such	such	ADJ
iajs-747	119	9	that	that	DET
iajs-747	119	10	f(a	f(a	NOUN
iajs-747	119	11	)	)	PUNCT
iajs-747	120	1	=	=	SYM
iajs-747	120	2	ra	ra	PROPN
iajs-747	120	3	for	for	ADP
iajs-747	120	4	all	all	DET
iajs-747	120	5	ra	ra	NOUN
iajs-747	120	6	.	.	PUNCT
iajs-747	121	1	since	since	SCONJ
iajs-747	121	2	kerf	kerf	NOUN
iajs-747	121	3	≤e	≤e	NOUN
iajs-747	121	4	r	r	NOUN
iajs-747	121	5	and	and	CCONJ
iajs-747	121	6	rr0	rr0	PROPN
iajs-747	121	7	,	,	PUNCT
iajs-747	121	8	then	then	ADV
iajs-747	121	9	there	there	PRON
iajs-747	121	10	exists	exist	VERB
iajs-747	121	11	rt	rt	NOUN
iajs-747	121	12	0	0	ADP
iajs-747	121	13	such	such	ADJ
iajs-747	121	14	that	that	DET
iajs-747	121	15	kerfrt	kerfrt	NOUN
iajs-747	121	16	0	0	ADV
iajs-747	121	17	,	,	PUNCT
iajs-747	121	18	hence	hence	ADV
iajs-747	121	19	0	0	NUM
iajs-747	121	20	=	=	SYM
iajs-747	121	21	f(rt	f(rt	NOUN
iajs-747	121	22	)	)	PUNCT
iajs-747	121	23	=	=	SYM
iajs-747	121	24	rf(t	rf(t	X
iajs-747	121	25	)	)	PUNCT
iajs-747	122	1	=	=	SYM
iajs-747	122	2	r2	r2	PROPN
iajs-747	122	3	t	t	PROPN
iajs-747	122	4	.	.	PUNCT
iajs-747	123	1	this	this	PRON
iajs-747	123	2	implies	imply	VERB
iajs-747	123	3	(	(	PUNCT
iajs-747	123	4	rt)2	rt)2	NOUN
iajs-747	123	5	=	=	SYM
iajs-747	123	6	0	0	PUNCT
iajs-747	124	1	and	and	CCONJ
iajs-747	124	2	since	since	SCONJ
iajs-747	124	3	r	r	NOUN
iajs-747	124	4	is	be	AUX
iajs-747	124	5	semiprime	semiprime	NOUN
iajs-747	124	6	,	,	PUNCT
iajs-747	124	7	rt	rt	PROPN
iajs-747	124	8	=	=	PUNCT
iajs-747	124	9	0	0	NUM
iajs-747	124	10	which	which	PRON
iajs-747	124	11	is	be	AUX
iajs-747	124	12	a	a	DET
iajs-747	124	13	contradiction	contradiction	NOUN
iajs-747	124	14	.	.	PUNCT
iajs-747	125	1	thus	thus	ADV
iajs-747	125	2	f	f	X
iajs-747	125	3	=	=	SYM
iajs-747	125	4	0	0	NUM
iajs-747	125	5	and	and	CCONJ
iajs-747	125	6	r	r	NOUN
iajs-747	125	7	is	be	AUX
iajs-747	125	8	essentially	essentially	ADV
iajs-747	125	9	quasi	quasi	ADJ
iajs-747	125	10	-	-	NOUN
iajs-747	125	11	dedekind	dedekind	ADJ
iajs-747	125	12	.	.	PUNCT
iajs-747	126	1	ibn	ibn	PROPN
iajs-747	126	2	alhaitham	alhaitham	PROPN
iajs-747	126	3	j.	j.	PROPN
iajs-747	126	4	for	for	ADP
iajs-747	126	5	pure	pure	ADJ
iajs-747	126	6	&	&	CCONJ
iajs-747	126	7	appl	appl	PROPN
iajs-747	126	8	.	.	PUNCT
iajs-747	127	1	sci	sci	PROPN
iajs-747	127	2	.	.	PUNCT
iajs-747	127	3	vol.24	vol.24	NOUN
iajs-747	127	4	(	(	PUNCT
iajs-747	127	5	3	3	NUM
iajs-747	127	6	)	)	PUNCT
iajs-747	127	7	2011	2011	NUM
iajs-747	127	8	)	)	PUNCT
iajs-747	128	1	3()1	3()1	NUM
iajs-747	128	2	(	(	PUNCT
iajs-747	128	3			NOUN
iajs-747	128	4	:	:	PUNCT
iajs-747	128	5	suppose	suppose	VERB
iajs-747	128	6	that	that	SCONJ
iajs-747	128	7	0	0	NUM
iajs-747	128	8	)	)	PUNCT
iajs-747	128	9	(	(	PUNCT
iajs-747	128	10	rz	rz	NOUN
iajs-747	128	11	.	.	PUNCT
iajs-747	129	1	then	then	ADV
iajs-747	129	2	there	there	PRON
iajs-747	129	3	exists	exist	VERB
iajs-747	129	4	)	)	PUNCT
iajs-747	129	5	(	(	PUNCT
iajs-747	129	6	0	0	NUM
iajs-747	129	7	rza	rza	ADJ
iajs-747	129	8	and	and	CCONJ
iajs-747	129	9	hence	hence	ADV
iajs-747	129	10	)	)	PUNCT
iajs-747	129	11	(	(	PUNCT
iajs-747	129	12	aann	aann	PROPN
iajs-747	129	13	r	r	NOUN
iajs-747	129	14	≤	≤	NUM
iajs-747	129	15	e	e	NOUN
iajs-747	129	16	r	r	NOUN
iajs-747	129	17	,	,	PUNCT
iajs-747	129	18	this	this	PRON
iajs-747	129	19	implies	imply	VERB
iajs-747	129	20	)	)	PUNCT
iajs-747	129	21	(	(	PUNCT
iajs-747	129	22	aann	aann	PROPN
iajs-747	129	23	r	r	NOUN
iajs-747	129	24	is	be	AUX
iajs-747	129	25	a	a	DET
iajs-747	129	26	quasi	quasi	ADJ
iajs-747	129	27	-	-	ADJ
iajs-747	129	28	invertible	invertible	ADJ
iajs-747	129	29	ideal	ideal	NOUN
iajs-747	129	30	and	and	CCONJ
iajs-747	129	31	so	so	ADV
iajs-747	129	32	that	that	SCONJ
iajs-747	129	33	by	by	ADP
iajs-747	129	34	(	(	PUNCT
iajs-747	129	35	5	5	NUM
iajs-747	129	36	,	,	PUNCT
iajs-747	129	37	prop	prop	NOUN
iajs-747	129	38	2.2	2.2	NUM
iajs-747	129	39	)	)	PUNCT
iajs-747	129	40	,	,	PUNCT
iajs-747	129	41	0	0	NUM
iajs-747	129	42	)	)	PUNCT
iajs-747	129	43	)	)	PUNCT
iajs-747	129	44	(	(	PUNCT
iajs-747	129	45	(	(	PUNCT
iajs-747	129	46	aannann	aannann	PUNCT
iajs-747	129	47	rr	rr	INTJ
iajs-747	129	48	,	,	PUNCT
iajs-747	129	49	but	but	CCONJ
iajs-747	129	50	)	)	PUNCT
iajs-747	129	51	)	)	PUNCT
iajs-747	129	52	(	(	PUNCT
iajs-747	129	53	(	(	PUNCT
iajs-747	129	54	)	)	PUNCT
iajs-747	129	55	(	(	PUNCT
iajs-747	129	56	aannanna	aannanna	PROPN
iajs-747	129	57	rr	rr	PROPN
iajs-747	129	58			PROPN
iajs-747	129	59	,	,	PUNCT
iajs-747	129	60	hence	hence	ADV
iajs-747	129	61	a	a	DET
iajs-747	129	62	=	=	SYM
iajs-747	129	63	0	0	NUM
iajs-747	129	64	which	which	PRON
iajs-747	129	65	is	be	AUX
iajs-747	129	66	a	a	DET
iajs-747	129	67	contradiction	contradiction	NOUN
iajs-747	129	68	.	.	PUNCT
iajs-747	130	1	proposition	proposition	NOUN
iajs-747	130	2	(	(	PUNCT
iajs-747	130	3	2.7	2.7	NUM
iajs-747	130	4	)	)	PUNCT
iajs-747	130	5	let	let	VERB
iajs-747	130	6	r	r	PRON
iajs-747	130	7	be	be	AUX
iajs-747	130	8	a	a	DET
iajs-747	130	9	ring	ring	NOUN
iajs-747	130	10	.	.	PUNCT
iajs-747	131	1	then	then	ADV
iajs-747	131	2	r	r	NOUN
iajs-747	131	3	is	be	AUX
iajs-747	131	4	essentially	essentially	ADV
iajs-747	131	5	quasi	quasi	ADJ
iajs-747	131	6	-	-	ADJ
iajs-747	131	7	dedekind	dedekind	ADJ
iajs-747	131	8	if	if	SCONJ
iajs-747	131	9	and	and	CCONJ
iajs-747	131	10	only	only	ADV
iajs-747	131	11	if	if	SCONJ
iajs-747	131	12	r[x	r[x	PROPN
iajs-747	131	13	]	]	X
iajs-747	131	14	is	be	AUX
iajs-747	131	15	essentially	essentially	ADV
iajs-747	131	16	quasi	quasi	ADJ
iajs-747	131	17	-	-	ADJ
iajs-747	131	18	dedekind	dedekind	ADJ
iajs-747	131	19	,	,	PUNCT
iajs-747	131	20	where	where	SCONJ
iajs-747	131	21	r[x	r[x	NOUN
iajs-747	131	22	]	]	X
iajs-747	131	23	is	be	AUX
iajs-747	131	24	the	the	DET
iajs-747	131	25	ring	ring	NOUN
iajs-747	131	26	of	of	ADP
iajs-747	131	27	polynomials	polynomial	NOUN
iajs-747	131	28	with	with	ADP
iajs-747	131	29	one	one	NUM
iajs-747	131	30	indeterminate	indeterminate	NOUN
iajs-747	131	31	x	x	X
iajs-747	131	32	.	.	PUNCT
iajs-747	132	1	proof	proof	NOUN
iajs-747	132	2	:	:	PUNCT
iajs-747	132	3	)	)	PUNCT
iajs-747	132	4			NOUN
iajs-747	132	5	suppose	suppose	VERB
iajs-747	132	6	that	that	SCONJ
iajs-747	132	7	r	r	NOUN
iajs-747	132	8	is	be	AUX
iajs-747	132	9	essentially	essentially	ADV
iajs-747	132	10	quasi	quasi	ADJ
iajs-747	132	11	-	-	ADJ
iajs-747	132	12	dedekind	dedekind	ADJ
iajs-747	132	13	,	,	PUNCT
iajs-747	132	14	so	so	CCONJ
iajs-747	132	15	by	by	ADP
iajs-747	132	16	(	(	PUNCT
iajs-747	132	17	prop	prop	NOUN
iajs-747	132	18	2.6	2.6	NUM
iajs-747	132	19	)	)	PUNCT
iajs-747	132	20	r	r	NOUN
iajs-747	132	21	is	be	AUX
iajs-747	132	22	a	a	DET
iajs-747	132	23	nonsingular	nonsingular	ADJ
iajs-747	132	24	ring	ring	NOUN
iajs-747	132	25	,	,	PUNCT
iajs-747	132	26	and	and	CCONJ
iajs-747	132	27	hence	hence	ADV
iajs-747	132	28	by	by	ADP
iajs-747	132	29	[	[	X
iajs-747	132	30	2	2	NUM
iajs-747	132	31	,	,	PUNCT
iajs-747	132	32	ex	ex	NOUN
iajs-747	132	33	.	.	NOUN
iajs-747	132	34	13	13	NUM
iajs-747	132	35	,	,	PUNCT
iajs-747	132	36	p.37	p.37	X
iajs-747	132	37	]	]	PUNCT
iajs-747	132	38	,	,	PUNCT
iajs-747	132	39	r[x	r[x	PROPN
iajs-747	132	40	]	]	X
iajs-747	132	41	is	be	AUX
iajs-747	132	42	a	a	DET
iajs-747	132	43	nonsingular	nonsingular	ADJ
iajs-747	132	44	ring	ring	NOUN
iajs-747	132	45	.	.	PUNCT
iajs-747	133	1	thus	thus	ADV
iajs-747	133	2	r[x	r[x	NOUN
iajs-747	133	3	]	]	PUNCT
iajs-747	133	4	is	be	AUX
iajs-747	133	5	essentially	essentially	ADV
iajs-747	133	6	quasi	quasi	ADJ
iajs-747	133	7	-	-	ADJ
iajs-747	133	8	dedekind	dedekind	ADJ
iajs-747	133	9	,	,	PUNCT
iajs-747	133	10	by	by	ADP
iajs-747	133	11	(	(	PUNCT
iajs-747	133	12	prop	prop	NOUN
iajs-747	133	13	2.6	2.6	NUM
iajs-747	133	14	)	)	PUNCT
iajs-747	133	15	.	.	PUNCT
iajs-747	133	16	)	)	PUNCT
iajs-747	134	1			PROPN
iajs-747	134	2	suppose	suppose	VERB
iajs-747	134	3	that	that	SCONJ
iajs-747	134	4	r	r	NOUN
iajs-747	134	5	is	be	AUX
iajs-747	134	6	not	not	PART
iajs-747	134	7	essentially	essentially	ADV
iajs-747	134	8	quasi	quasi	ADJ
iajs-747	134	9	-	-	ADJ
iajs-747	134	10	dedekind	dedekind	ADJ
iajs-747	134	11	,	,	PUNCT
iajs-747	134	12	so	so	CCONJ
iajs-747	134	13	by	by	ADP
iajs-747	134	14	(	(	PUNCT
iajs-747	134	15	prop	prop	NOUN
iajs-747	134	16	2.6	2.6	NUM
iajs-747	134	17	)	)	PUNCT
iajs-747	134	18	,	,	PUNCT
iajs-747	134	19	r	r	NOUN
iajs-747	134	20	is	be	AUX
iajs-747	134	21	not	not	PART
iajs-747	134	22	a	a	DET
iajs-747	134	23	semiprime	semiprime	NOUN
iajs-747	134	24	ring	ring	NOUN
iajs-747	134	25	;	;	PUNCT
iajs-747	134	26	that	that	PRON
iajs-747	134	27	is	be	AUX
iajs-747	134	28	there	there	PRON
iajs-747	134	29	exists	exist	VERB
iajs-747	134	30	)	)	PUNCT
iajs-747	134	31	(	(	PUNCT
iajs-747	134	32	rla	rla	PROPN
iajs-747	134	33	and	and	CCONJ
iajs-747	134	34	oa	oa	PROPN
iajs-747	134	35			NOUN
iajs-747	134	36	,	,	PUNCT
iajs-747	134	37	where	where	SCONJ
iajs-747	134	38	0	0	NUM
iajs-747	134	39	:	:	PUNCT
iajs-747	134	40	{	{	PUNCT
iajs-747	134	41	)	)	PUNCT
iajs-747	134	42	(	(	PUNCT
iajs-747	134	43			NOUN
iajs-747	134	44	nxrxrl	nxrxrl	ADV
iajs-747	134	45	,	,	PUNCT
iajs-747	134	46	for	for	ADP
iajs-747	134	47	some	some	DET
iajs-747	134	48	}	}	PUNCT
iajs-747	134	49	nn	nn	X
iajs-747	134	50			PROPN
iajs-747	134	51	,	,	PUNCT
iajs-747	134	52	then	then	ADV
iajs-747	134	53	an	an	DET
iajs-747	134	54	=	=	NOUN
iajs-747	134	55	0	0	NUM
iajs-747	134	56	,	,	PUNCT
iajs-747	134	57	for	for	ADP
iajs-747	134	58	some	some	DET
iajs-747	134	59	n	n	NOUN
iajs-747	134	60	n	n	PROPN
iajs-747	134	61	.	.	PUNCT
iajs-747	135	1	define	define	VERB
iajs-747	135	2	0	0	NUM
iajs-747	135	3	)	)	PUNCT
iajs-747	135	4	(	(	PUNCT
iajs-747	135	5			X
iajs-747	135	6	axf	axf	ADV
iajs-747	135	7	,	,	PUNCT
iajs-747	136	1	so	so	SCONJ
iajs-747	136	2	]	]	X
iajs-747	136	3	[	[	X
iajs-747	136	4	)	)	PUNCT
iajs-747	136	5	(	(	PUNCT
iajs-747	136	6	xrxf	xrxf	PROPN
iajs-747	136	7			PROPN
iajs-747	136	8	,	,	PUNCT
iajs-747	136	9	and	and	CCONJ
iajs-747	136	10	r[x	r[x	NOUN
iajs-747	136	11	]	]	X
iajs-747	136	12	is	be	AUX
iajs-747	136	13	a	a	DET
iajs-747	136	14	semiprime	semiprime	NOUN
iajs-747	136	15	ring	ring	NOUN
iajs-747	136	16	,	,	PUNCT
iajs-747	136	17	by	by	ADP
iajs-747	136	18	(	(	PUNCT
iajs-747	136	19	prop	prop	NOUN
iajs-747	136	20	2.6	2.6	NUM
iajs-747	136	21	)	)	PUNCT
iajs-747	136	22	.	.	PUNCT
iajs-747	137	1	on	on	ADP
iajs-747	137	2	the	the	DET
iajs-747	137	3	other	other	ADJ
iajs-747	137	4	hand	hand	NOUN
iajs-747	137	5	[	[	PUNCT
iajs-747	137	6	f(x	f(x	PROPN
iajs-747	137	7	)	)	PUNCT
iajs-747	137	8	]	]	PUNCT
iajs-747	138	1	n	n	PROPN
iajs-747	138	2	=	=	PUNCT
iajs-747	138	3	a	a	DET
iajs-747	138	4	n	n	NOUN
iajs-747	138	5	=	=	SYM
iajs-747	138	6	0	0	NUM
iajs-747	138	7	,	,	PUNCT
iajs-747	138	8	implies	imply	VERB
iajs-747	138	9	0	0	NUM
iajs-747	138	10	]	]	PUNCT
iajs-747	138	11	)	)	PUNCT
iajs-747	139	1	[	[	X
iajs-747	139	2	(	(	PUNCT
iajs-747	139	3	)	)	PUNCT
iajs-747	139	4	(	(	PUNCT
iajs-747	139	5			NOUN
iajs-747	139	6	xrlxf	xrlxf	PROPN
iajs-747	139	7	.	.	PUNCT
iajs-747	140	1	it	it	PRON
iajs-747	140	2	follows	follow	VERB
iajs-747	140	3	that	that	SCONJ
iajs-747	140	4	f	f	PROPN
iajs-747	140	5	=	=	PUNCT
iajs-747	140	6	0	0	NUM
iajs-747	140	7	which	which	PRON
iajs-747	140	8	is	be	AUX
iajs-747	140	9	a	a	DET
iajs-747	140	10	contradiction	contradiction	NOUN
iajs-747	140	11	.	.	PUNCT
iajs-747	141	1	thus	thus	ADV
iajs-747	141	2	r	r	NOUN
iajs-747	141	3	is	be	AUX
iajs-747	141	4	essentially	essentially	ADV
iajs-747	141	5	quasi	quasi	ADJ
iajs-747	141	6	-	-	ADJ
iajs-747	141	7	dedekind	dedekind	ADJ
iajs-747	141	8	.	.	PUNCT
iajs-747	142	1	proposition	proposition	NOUN
iajs-747	142	2	(	(	PUNCT
iajs-747	142	3	2.8	2.8	NUM
iajs-747	142	4	)	)	PUNCT
iajs-747	142	5	let	let	VERB
iajs-747	142	6	m	m	PRON
iajs-747	142	7	be	be	AUX
iajs-747	142	8	a	a	DET
iajs-747	142	9	faithful	faithful	ADJ
iajs-747	142	10	r	r	NOUN
iajs-747	142	11	-	-	PUNCT
iajs-747	142	12	module	module	NOUN
iajs-747	142	13	.	.	PUNCT
iajs-747	143	1	then	then	ADV
iajs-747	143	2	r	r	NOUN
iajs-747	143	3	is	be	AUX
iajs-747	143	4	essentially	essentially	ADV
iajs-747	143	5	quasidedekind	quasidedekind	ADJ
iajs-747	143	6	if	if	SCONJ
iajs-747	144	1	and	and	CCONJ
iajs-747	144	2	only	only	ADV
iajs-747	144	3	if	if	SCONJ
iajs-747	144	4	n	n	PRON
iajs-747	144	5	m	m	VERB
iajs-747	144	6	n	n	NOUN
iajs-747	144	7			ADJ
iajs-747	144	8	is	be	AUX
iajs-747	144	9	a	a	DET
iajs-747	144	10	faithful	faithful	ADJ
iajs-747	144	11	r	r	NOUN
iajs-747	144	12	-	-	PUNCT
iajs-747	144	13	module	module	NOUN
iajs-747	144	14	,	,	PUNCT
iajs-747	144	15	for	for	ADP
iajs-747	144	16	all	all	DET
iajs-747	144	17	mn	mn	PROPN
iajs-747	144	18			PROPN
iajs-747	144	19	.	.	PUNCT
iajs-747	145	1	proof	proof	NOUN
iajs-747	145	2	:	:	PUNCT
iajs-747	145	3	)	)	PUNCT
iajs-747	145	4			NOUN
iajs-747	145	5	suppose	suppose	VERB
iajs-747	145	6	that	that	SCONJ
iajs-747	145	7	r	r	NOUN
iajs-747	145	8	is	be	AUX
iajs-747	145	9	essentially	essentially	ADV
iajs-747	145	10	quasi	quasi	ADJ
iajs-747	145	11	-	-	ADJ
iajs-747	145	12	dedekind	dedekind	ADJ
iajs-747	145	13	,	,	PUNCT
iajs-747	145	14	so	so	CCONJ
iajs-747	145	15	by	by	ADP
iajs-747	145	16	(	(	PUNCT
iajs-747	145	17	(	(	PUNCT
iajs-747	145	18	prop	prop	NOUN
iajs-747	145	19	2.6	2.6	NUM
iajs-747	145	20	)	)	PUNCT
iajs-747	145	21	,	,	PUNCT
iajs-747	145	22	r	r	NOUN
iajs-747	145	23	is	be	AUX
iajs-747	145	24	semiprime	semiprime	NOUN
iajs-747	145	25	.	.	PUNCT
iajs-747	146	1	let	let	VERB
iajs-747	146	2	)	)	PUNCT
iajs-747	146	3	(	(	PUNCT
iajs-747	146	4	n	n	X
iajs-747	146	5	m	m	NOUN
iajs-747	146	6	nannr	nannr	NOUN
iajs-747	146	7	r	r	PROPN
iajs-747	146	8			PROPN
iajs-747	146	9	,	,	PUNCT
iajs-747	146	10	then	then	ADV
iajs-747	146	11	)	)	PUNCT
iajs-747	146	12	(	(	PUNCT
iajs-747	146	13	)	)	PUNCT
iajs-747	146	14	(	(	PUNCT
iajs-747	146	15	n	n	X
iajs-747	146	16	m	m	VERB
iajs-747	146	17	annnannr	annnannr	NOUN
iajs-747	146	18	rr	rr	PROPN
iajs-747	146	19			PROPN
iajs-747	146	20	;	;	PUNCT
iajs-747	146	21	that	that	PRON
iajs-747	146	22	is	be	AUX
iajs-747	146	23	rn	rn	PROPN
iajs-747	146	24	=	=	PROPN
iajs-747	146	25	0	0	PROPN
iajs-747	146	26	and	and	CCONJ
iajs-747	146	27	nrm	nrm	NOUN
iajs-747	146	28			PROPN
iajs-747	146	29	,	,	PUNCT
iajs-747	146	30	so	so	ADV
iajs-747	146	31	02	02	NUM
iajs-747	146	32			PROPN
iajs-747	146	33	rnmr	rnmr	NOUN
iajs-747	146	34	implies	imply	VERB
iajs-747	146	35	0)(2	0)(2	NUM
iajs-747	146	36			NOUN
iajs-747	146	37	mannr	mannr	NOUN
iajs-747	146	38	r	r	NOUN
iajs-747	146	39	then	then	ADV
iajs-747	146	40	02	02	NUM
iajs-747	146	41	r	r	PROPN
iajs-747	146	42	,	,	PUNCT
iajs-747	146	43	thus	thus	ADV
iajs-747	146	44	r	r	NOUN
iajs-747	146	45	=	=	SYM
iajs-747	146	46	0	0	NUM
iajs-747	146	47	,	,	PUNCT
iajs-747	146	48	since	since	SCONJ
iajs-747	146	49	r	r	NOUN
iajs-747	146	50	is	be	AUX
iajs-747	146	51	a	a	DET
iajs-747	146	52	semiprime	semiprime	NOUN
iajs-747	146	53	ring	ring	NOUN
iajs-747	146	54	.	.	PUNCT
iajs-747	147	1	therefore	therefore	ADV
iajs-747	147	2	n	n	ADV
iajs-747	147	3	m	m	VERB
iajs-747	147	4	n	n	NOUN
iajs-747	147	5			ADJ
iajs-747	147	6	is	be	AUX
iajs-747	147	7	a	a	DET
iajs-747	147	8	faithful	faithful	ADJ
iajs-747	147	9	r	r	NOUN
iajs-747	147	10	-	-	PUNCT
iajs-747	147	11	module	module	NOUN
iajs-747	147	12	for	for	ADP
iajs-747	147	13	all	all	DET
iajs-747	147	14	mn	mn	PROPN
iajs-747	147	15			PROPN
iajs-747	147	16	.	.	PUNCT
iajs-747	147	17	)	)	PUNCT
iajs-747	148	1			PROPN
iajs-747	148	2	suppose	suppose	VERB
iajs-747	148	3	that	that	SCONJ
iajs-747	148	4	n	n	PRON
iajs-747	148	5	m	m	VERB
iajs-747	148	6	n	n	NOUN
iajs-747	148	7			ADJ
iajs-747	148	8	is	be	AUX
iajs-747	148	9	a	a	DET
iajs-747	148	10	faithful	faithful	ADJ
iajs-747	148	11	r	r	NOUN
iajs-747	148	12	-	-	PUNCT
iajs-747	148	13	module	module	NOUN
iajs-747	148	14	,	,	PUNCT
iajs-747	148	15	for	for	ADP
iajs-747	148	16	all	all	DET
iajs-747	148	17	mn	mn	PROPN
iajs-747	148	18			PROPN
iajs-747	148	19	.	.	PUNCT
iajs-747	149	1	to	to	PART
iajs-747	149	2	prove	prove	VERB
iajs-747	149	3	that	that	SCONJ
iajs-747	149	4	r	r	NOUN
iajs-747	149	5	is	be	AUX
iajs-747	149	6	essentially	essentially	ADV
iajs-747	149	7	quasidedekind	quasidedekind	ADJ
iajs-747	149	8	.	.	PUNCT
iajs-747	150	1	we	we	PRON
iajs-747	150	2	shall	shall	AUX
iajs-747	150	3	prove	prove	VERB
iajs-747	150	4	that	that	SCONJ
iajs-747	150	5	r	r	NOUN
iajs-747	150	6	is	be	AUX
iajs-747	150	7	a	a	DET
iajs-747	150	8	semiprime	semiprime	NOUN
iajs-747	150	9	ring	ring	NOUN
iajs-747	150	10	.	.	PUNCT
iajs-747	151	1	let	let	VERB
iajs-747	151	2	rr	rr	VERB
iajs-747	151	3			NOUN
iajs-747	151	4	with	with	ADP
iajs-747	151	5	02	02	NUM
iajs-747	151	6	r	r	PROPN
iajs-747	151	7	,	,	PUNCT
iajs-747	151	8	suppose	suppose	VERB
iajs-747	151	9	that	that	SCONJ
iajs-747	151	10	0r	0r	X
iajs-747	151	11	,	,	PUNCT
iajs-747	151	12	so	so	ADV
iajs-747	151	13	)	)	PUNCT
iajs-747	151	14	(	(	PUNCT
iajs-747	151	15	mannr	mannr	NOUN
iajs-747	151	16	r	r	PROPN
iajs-747	151	17	,	,	PUNCT
iajs-747	151	18	since	since	SCONJ
iajs-747	151	19	m	m	PROPN
iajs-747	151	20	is	be	AUX
iajs-747	151	21	a	a	DET
iajs-747	151	22	faithful	faithful	ADJ
iajs-747	151	23	r	r	NOUN
iajs-747	151	24	-	-	PUNCT
iajs-747	151	25	module	module	NOUN
iajs-747	151	26	,	,	PUNCT
iajs-747	151	27	then	then	ADV
iajs-747	151	28	0rm	0rm	PROPN
iajs-747	151	29	.let	.let	VERB
iajs-747	151	30	mrmn	mrmn	ADJ
iajs-747	151	31			NOUN
iajs-747	151	32	,	,	PUNCT
iajs-747	151	33	hence	hence	ADV
iajs-747	151	34	rn	rn	PROPN
iajs-747	152	1	=	=	SYM
iajs-747	152	2	r	r	NOUN
iajs-747	152	3	2	2	NUM
iajs-747	152	4	m	m	NOUN
iajs-747	152	5	=	=	NOUN
iajs-747	152	6	0	0	NUM
iajs-747	152	7	,	,	PUNCT
iajs-747	152	8	so	so	ADV
iajs-747	152	9	)	)	PUNCT
iajs-747	152	10	(	(	PUNCT
iajs-747	152	11	nannr	nannr	PROPN
iajs-747	152	12	r	r	PROPN
iajs-747	152	13			NOUN
iajs-747	152	14	,	,	PUNCT
iajs-747	152	15	but	but	CCONJ
iajs-747	152	16	)	)	PUNCT
iajs-747	152	17	(	(	PUNCT
iajs-747	152	18	n	n	X
iajs-747	152	19	m	m	PROPN
iajs-747	152	20	annr	annr	NOUN
iajs-747	152	21	r	r	ADJ
iajs-747	152	22	(	(	PUNCT
iajs-747	152	23	since	since	SCONJ
iajs-747	152	24	nrmrm	nrmrm	PROPN
iajs-747	152	25			PROPN
iajs-747	152	26	)	)	PUNCT
iajs-747	152	27	,	,	PUNCT
iajs-747	152	28	so	so	ADV
iajs-747	152	29	0	0	NUM
iajs-747	152	30	)	)	PUNCT
iajs-747	152	31	(	(	PUNCT
iajs-747	152	32	)	)	PUNCT
iajs-747	152	33	(	(	PUNCT
iajs-747	152	34	)	)	PUNCT
iajs-747	152	35	(	(	PUNCT
iajs-747	152	36			PROPN
iajs-747	152	37	n	n	CCONJ
iajs-747	152	38	m	m	NOUN
iajs-747	152	39	nann	nann	NOUN
iajs-747	152	40	n	n	PROPN
iajs-747	152	41	m	m	PROPN
iajs-747	152	42	annnannr	annnannr	NOUN
iajs-747	152	43	rrr	rrr	NOUN
iajs-747	152	44	,	,	PUNCT
iajs-747	152	45	thus	thus	ADV
iajs-747	152	46	r	r	NOUN
iajs-747	152	47	=	=	SYM
iajs-747	152	48	0	0	NUM
iajs-747	152	49	which	which	PRON
iajs-747	152	50	is	be	AUX
iajs-747	152	51	a	a	DET
iajs-747	152	52	contradiction	contradiction	NOUN
iajs-747	152	53	.	.	PUNCT
iajs-747	153	1	hence	hence	ADV
iajs-747	153	2	r	r	NOUN
iajs-747	153	3	is	be	AUX
iajs-747	153	4	essentially	essentially	ADV
iajs-747	153	5	quasi	quasi	ADJ
iajs-747	153	6	-	-	NOUN
iajs-747	153	7	dedekind	dedekind	ADJ
iajs-747	153	8	.	.	PUNCT
iajs-747	154	1	ibn	ibn	PROPN
iajs-747	154	2	alhaitham	alhaitham	PROPN
iajs-747	154	3	j.	j.	PROPN
iajs-747	154	4	for	for	ADP
iajs-747	154	5	pure	pure	ADJ
iajs-747	154	6	&	&	CCONJ
iajs-747	154	7	appl	appl	PROPN
iajs-747	154	8	.	.	PUNCT
iajs-747	155	1	sci	sci	PROPN
iajs-747	155	2	.	.	PUNCT
iajs-747	155	3	vol.24	vol.24	NOUN
iajs-747	155	4	(	(	PUNCT
iajs-747	155	5	3	3	NUM
iajs-747	155	6	)	)	PUNCT
iajs-747	155	7	2011	2011	NUM
iajs-747	155	8	proposition	proposition	NOUN
iajs-747	155	9	(	(	PUNCT
iajs-747	155	10	2.9	2.9	NUM
iajs-747	155	11	)	)	PUNCT
iajs-747	155	12	let	let	VERB
iajs-747	155	13	m	m	PRON
iajs-747	155	14	be	be	AUX
iajs-747	155	15	an	an	DET
iajs-747	155	16	r	r	NOUN
iajs-747	155	17	-	-	PUNCT
iajs-747	155	18	module	module	NOUN
iajs-747	155	19	and	and	CCONJ
iajs-747	155	20	let	let	VERB
iajs-747	155	21	jrr	jrr	PROPN
iajs-747	155	22	,	,	PUNCT
iajs-747	155	23	where	where	SCONJ
iajs-747	155	24	j	j	PROPN
iajs-747	155	25	is	be	AUX
iajs-747	155	26	an	an	DET
iajs-747	155	27	ideal	ideal	NOUN
iajs-747	155	28	of	of	ADP
iajs-747	155	29	r	r	NOUN
iajs-747	155	30	such	such	ADJ
iajs-747	155	31	that	that	PRON
iajs-747	155	32	)	)	PUNCT
iajs-747	155	33	(	(	PUNCT
iajs-747	155	34	mannj	mannj	NOUN
iajs-747	155	35	r	r	NOUN
iajs-747	155	36	.	.	PUNCT
iajs-747	156	1	then	then	ADV
iajs-747	156	2	m	m	VERB
iajs-747	156	3	is	be	AUX
iajs-747	156	4	an	an	DET
iajs-747	156	5	essentially	essentially	ADV
iajs-747	156	6	quasi	quasi	ADJ
iajs-747	156	7	-	-	ADJ
iajs-747	156	8	dedekind	dedekind	ADJ
iajs-747	156	9	r	r	NOUN
iajs-747	156	10	-	-	PUNCT
iajs-747	156	11	module	module	NOUN
iajs-747	156	12	if	if	SCONJ
iajs-747	156	13	and	and	CCONJ
iajs-747	156	14	only	only	ADV
iajs-747	156	15	if	if	SCONJ
iajs-747	156	16	m	m	NOUN
iajs-747	156	17	is	be	AUX
iajs-747	156	18	an	an	DET
iajs-747	156	19	essentially	essentially	ADV
iajs-747	156	20	quasi	quasi	ADJ
iajs-747	156	21	-	-	ADJ
iajs-747	156	22	dedekind	dedekind	ADJ
iajs-747	156	23	r	r	NOUN
iajs-747	156	24	-module	-module	NOUN
iajs-747	156	25	.	.	PUNCT
iajs-747	157	1	proof	proof	NOUN
iajs-747	157	2	:	:	PUNCT
iajs-747	157	3	by	by	ADP
iajs-747	157	4	[	[	X
iajs-747	157	5	3	3	NUM
iajs-747	157	6	,	,	PUNCT
iajs-747	157	7	p.51	p.51	NOUN
iajs-747	157	8	]	]	X
iajs-747	157	9	,	,	PUNCT
iajs-747	157	10	we	we	PRON
iajs-747	157	11	have	have	VERB
iajs-747	157	12	)	)	PUNCT
iajs-747	157	13	,	,	PUNCT
iajs-747	157	14	(	(	PUNCT
iajs-747	157	15	)	)	PUNCT
iajs-747	157	16	,	,	PUNCT
iajs-747	157	17	(	(	PUNCT
iajs-747	157	18	mnmhommnmhom	mnmhommnmhom	PROPN
iajs-747	157	19	rr	rr	PROPN
iajs-747	157	20			PROPN
iajs-747	157	21	for	for	ADP
iajs-747	157	22	all	all	DET
iajs-747	157	23	mn	mn	PROPN
iajs-747	157	24			PROPN
iajs-747	157	25	.	.	PUNCT
iajs-747	158	1	suppose	suppose	VERB
iajs-747	158	2	that	that	SCONJ
iajs-747	158	3	m	m	PROPN
iajs-747	158	4	is	be	AUX
iajs-747	158	5	an	an	DET
iajs-747	158	6	essentially	essentially	ADV
iajs-747	158	7	quasi	quasi	ADJ
iajs-747	158	8	-	-	ADJ
iajs-747	158	9	dedekind	dedekind	ADJ
iajs-747	158	10	r	r	NOUN
iajs-747	158	11	-	-	NOUN
iajs-747	158	12	module	module	NOUN
iajs-747	158	13	,	,	PUNCT
iajs-747	158	14	then	then	ADV
iajs-747	158	15	0	0	NUM
iajs-747	158	16	)	)	PUNCT
iajs-747	158	17	,	,	PUNCT
iajs-747	158	18	(	(	PUNCT
iajs-747	158	19	)	)	PUNCT
iajs-747	158	20	,	,	PUNCT
iajs-747	158	21	(	(	PUNCT
iajs-747	158	22			NUM
iajs-747	158	23	mnmhommnmhom	mnmhommnmhom	NOUN
iajs-747	158	24	rr	rr	NOUN
iajs-747	158	25	for	for	ADP
iajs-747	158	26	all	all	DET
iajs-747	158	27	n	n	PRON
iajs-747	158	28	≤	≤	NOUN
iajs-747	158	29	e	e	X
iajs-747	158	30	m	m	PROPN
iajs-747	158	31	,	,	PUNCT
iajs-747	158	32	implies	imply	VERB
iajs-747	158	33	m	m	VERB
iajs-747	158	34	is	be	AUX
iajs-747	158	35	an	an	DET
iajs-747	158	36	essentially	essentially	ADV
iajs-747	158	37	quasi	quasi	ADJ
iajs-747	158	38	-	-	ADJ
iajs-747	158	39	dedekind	dedekind	ADJ
iajs-747	158	40	r	r	NOUN
iajs-747	158	41	-module	-module	NOUN
iajs-747	158	42	.	.	PUNCT
iajs-747	159	1	the	the	DET
iajs-747	159	2	converse	converse	NOUN
iajs-747	159	3	follows	follow	VERB
iajs-747	159	4	similarly	similarly	ADV
iajs-747	159	5	.	.	PUNCT
iajs-747	160	1	let	let	VERB
iajs-747	160	2	r	r	PRON
iajs-747	160	3	be	be	AUX
iajs-747	160	4	an	an	DET
iajs-747	160	5	integral	integral	ADJ
iajs-747	160	6	domain	domain	NOUN
iajs-747	160	7	,	,	PUNCT
iajs-747	160	8	and	and	CCONJ
iajs-747	160	9	let	let	VERB
iajs-747	160	10	m	m	PRON
iajs-747	160	11	be	be	AUX
iajs-747	160	12	an	an	DET
iajs-747	160	13	r	r	NOUN
iajs-747	160	14	-	-	PUNCT
iajs-747	160	15	module	module	NOUN
iajs-747	160	16	.	.	PUNCT
iajs-747	161	1	an	an	DET
iajs-747	161	2	element	element	NOUN
iajs-747	161	3	mx	mx	NOUN
iajs-747	161	4	is	be	AUX
iajs-747	161	5	called	call	VERB
iajs-747	161	6	a	a	DET
iajs-747	161	7	torsion	torsion	NOUN
iajs-747	161	8	element	element	NOUN
iajs-747	161	9	of	of	ADP
iajs-747	161	10	m	m	NOUN
iajs-747	161	11	if	if	SCONJ
iajs-747	161	12	,	,	PUNCT
iajs-747	161	13	0	0	NUM
iajs-747	161	14	)	)	PUNCT
iajs-747	161	15	(	(	PUNCT
iajs-747	161	16	xann	xann	PROPN
iajs-747	161	17	r	r	NOUN
iajs-747	161	18	.	.	PUNCT
iajs-747	162	1	the	the	DET
iajs-747	162	2	set	set	NOUN
iajs-747	162	3	of	of	ADP
iajs-747	162	4	all	all	DET
iajs-747	162	5	torsion	torsion	NOUN
iajs-747	162	6	elements	element	NOUN
iajs-747	162	7	of	of	ADP
iajs-747	162	8	m	m	AUX
iajs-747	162	9	denoted	denote	VERB
iajs-747	162	10	by	by	ADP
iajs-747	162	11	t(m	t(m	PROPN
iajs-747	162	12	)	)	PUNCT
iajs-747	162	13	and	and	CCONJ
iajs-747	162	14	it	it	PRON
iajs-747	162	15	is	be	AUX
iajs-747	162	16	a	a	DET
iajs-747	162	17	submodule	submodule	NOUN
iajs-747	162	18	of	of	ADP
iajs-747	162	19	m	m	PROPN
iajs-747	162	20	.	.	PUNCT
iajs-747	163	1	if	if	SCONJ
iajs-747	163	2	t(m	t(m	PROPN
iajs-747	163	3	)	)	PUNCT
iajs-747	164	1	=	=	PUNCT
iajs-747	164	2	0	0	PUNCT
iajs-747	165	1	the	the	DET
iajs-747	165	2	r	r	NOUN
iajs-747	165	3	-	-	PUNCT
iajs-747	165	4	module	module	NOUN
iajs-747	165	5	m	m	NOUN
iajs-747	165	6	is	be	AUX
iajs-747	165	7	said	say	VERB
iajs-747	165	8	to	to	PART
iajs-747	165	9	be	be	AUX
iajs-747	165	10	torsion	torsion	NOUN
iajs-747	165	11	-	-	PUNCT
iajs-747	165	12	free	free	ADJ
iajs-747	165	13	,	,	PUNCT
iajs-747	165	14	[	[	X
iajs-747	165	15	1	1	NUM
iajs-747	165	16	,	,	PUNCT
iajs-747	165	17	p.45	p.45	X
iajs-747	165	18	]	]	PUNCT
iajs-747	165	19	.	.	PUNCT
iajs-747	166	1	the	the	DET
iajs-747	166	2	following	follow	VERB
iajs-747	166	3	result	result	NOUN
iajs-747	166	4	shows	show	VERB
iajs-747	166	5	that	that	SCONJ
iajs-747	166	6	essentially	essentially	ADV
iajs-747	166	7	quasi	quasi	ADJ
iajs-747	166	8	-	-	ADJ
iajs-747	166	9	dedekind	dedekind	ADJ
iajs-747	166	10	preserves	preserve	NOUN
iajs-747	166	11	under	under	ADP
iajs-747	166	12	isomorphism	isomorphism	NOUN
iajs-747	166	13	.	.	PUNCT
iajs-747	167	1	proposition	proposition	NOUN
iajs-747	167	2	(	(	PUNCT
iajs-747	167	3	2.10	2.10	NUM
iajs-747	167	4	)	)	PUNCT
iajs-747	167	5	let	let	VERB
iajs-747	167	6	m	m	PROPN
iajs-747	167	7	1	1	NUM
iajs-747	167	8	,	,	PUNCT
iajs-747	167	9	m	m	VERB
iajs-747	167	10	2	2	NUM
iajs-747	167	11	be	be	VERB
iajs-747	167	12	r	r	NOUN
iajs-747	167	13	-	-	PUNCT
iajs-747	167	14	modules	module	NOUN
iajs-747	167	15	such	such	ADJ
iajs-747	167	16	that	that	SCONJ
iajs-747	167	17	21	21	NUM
iajs-747	167	18	mm	mm	PROPN
iajs-747	167	19			PROPN
iajs-747	167	20	.	.	PUNCT
iajs-747	168	1	then	then	ADV
iajs-747	168	2	m1	m1	PROPN
iajs-747	168	3	is	be	AUX
iajs-747	168	4	an	an	DET
iajs-747	168	5	essentially	essentially	ADV
iajs-747	168	6	quasidedekind	quasidedekind	ADJ
iajs-747	168	7	r	r	NOUN
iajs-747	168	8	-	-	PUNCT
iajs-747	168	9	module	module	NOUN
iajs-747	168	10	if	if	SCONJ
iajs-747	168	11	and	and	CCONJ
iajs-747	168	12	only	only	ADV
iajs-747	168	13	if	if	SCONJ
iajs-747	168	14	m2	m2	PROPN
iajs-747	168	15	is	be	AUX
iajs-747	168	16	an	an	DET
iajs-747	168	17	essentially	essentially	ADV
iajs-747	168	18	quasi	quasi	ADJ
iajs-747	168	19	-	-	ADJ
iajs-747	168	20	dedekind	dedekind	ADJ
iajs-747	168	21	r	r	NOUN
iajs-747	168	22	-	-	NOUN
iajs-747	168	23	module	module	NOUN
iajs-747	168	24	.	.	PUNCT
iajs-747	169	1	proof	proof	NOUN
iajs-747	169	2	:	:	PUNCT
iajs-747	169	3	)	)	PUNCT
iajs-747	169	4			NOUN
iajs-747	169	5	suppose	suppose	VERB
iajs-747	169	6	that	that	SCONJ
iajs-747	169	7	m	m	PROPN
iajs-747	169	8	1	1	NUM
iajs-747	169	9	is	be	AUX
iajs-747	169	10	an	an	DET
iajs-747	169	11	essentially	essentially	ADV
iajs-747	169	12	quasi	quasi	ADJ
iajs-747	169	13	-	-	ADJ
iajs-747	169	14	dedekind	dedekind	ADJ
iajs-747	169	15	r	r	NOUN
iajs-747	169	16	-	-	NOUN
iajs-747	169	17	module	module	NOUN
iajs-747	169	18	.	.	PUNCT
iajs-747	170	1	let	let	VERB
iajs-747	170	2	21	21	NUM
iajs-747	170	3	:	:	PUNCT
iajs-747	170	4	mm	mm	INTJ
iajs-747	170	5			PROPN
iajs-747	170	6	,	,	PUNCT
iajs-747	170	7			NOUN
iajs-747	170	8	is	be	AUX
iajs-747	170	9	an	an	DET
iajs-747	170	10	isomorphism	isomorphism	NOUN
iajs-747	170	11	.	.	PUNCT
iajs-747	171	1	to	to	PART
iajs-747	171	2	prove	prove	VERB
iajs-747	171	3	that	that	SCONJ
iajs-747	171	4	m	m	VERB
iajs-747	171	5	2	2	NUM
iajs-747	171	6	is	be	AUX
iajs-747	171	7	an	an	DET
iajs-747	171	8	essentially	essentially	ADV
iajs-747	171	9	quasi	quasi	ADJ
iajs-747	171	10	dedekind	dedekind	ADJ
iajs-747	171	11	r	r	NOUN
iajs-747	171	12	-	-	NOUN
iajs-747	171	13	module	module	NOUN
iajs-747	171	14	.	.	PUNCT
iajs-747	172	1	let	let	VERB
iajs-747	172	2	0	0	NUM
iajs-747	172	3	,	,	PUNCT
iajs-747	172	4	)	)	PUNCT
iajs-747	172	5	(	(	PUNCT
iajs-747	172	6	2	2	NUM
iajs-747	172	7			NOUN
iajs-747	172	8	fmendf	fmendf	ADJ
iajs-747	172	9	r	r	NOUN
iajs-747	172	10	.	.	PUNCT
iajs-747	173	1	we	we	PRON
iajs-747	173	2	have	have	VERB
iajs-747	173	3	1221	1221	NUM
iajs-747	173	4	1	1	NUM
iajs-747	173	5	mmmm	mmmm	NOUN
iajs-747	173	6	f	f	PROPN
iajs-747	173	7			PROPN
iajs-747	173	8			PROPN
iajs-747	173	9	,	,	PUNCT
iajs-747	173	10	let	let	VERB
iajs-747	173	11	)	)	PUNCT
iajs-747	173	12	(	(	PUNCT
iajs-747	173	13	1	1	NUM
iajs-747	173	14	1	1	NUM
iajs-747	173	15	mendofoh	mendofoh	NOUN
iajs-747	173	16	r	r	NOUN
iajs-747	173	17			PROPN
iajs-747	173	18			PROPN
iajs-747	173	19			PROPN
iajs-747	173	20	,	,	PUNCT
iajs-747	173	21	and	and	CCONJ
iajs-747	173	22	hence	hence	ADV
iajs-747	173	23	0h	0h	NUM
iajs-747	173	24	,	,	PUNCT
iajs-747	173	25	then	then	ADV
iajs-747	173	26	kerh	kerh	PROPN
iajs-747	173	27	≰e	≰e	PROPN
iajs-747	173	28	m	m	PROPN
iajs-747	173	29	1	1	NUM
iajs-747	173	30	.	.	PUNCT
iajs-747	174	1	to	to	PART
iajs-747	174	2	prove	prove	VERB
iajs-747	174	3	kerf	kerf	NOUN
iajs-747	174	4	≰e	≰e	PROPN
iajs-747	174	5	m	m	NOUN
iajs-747	174	6	2	2	NUM
iajs-747	174	7	,	,	PUNCT
iajs-747	174	8	we	we	PRON
iajs-747	174	9	cliam	cliam	VERB
iajs-747	174	10	that	that	PRON
iajs-747	174	11	}	}	PUNCT
iajs-747	174	12	)	)	PUNCT
iajs-747	174	13	(:	(:	NOUN
iajs-747	174	14	{	{	PUNCT
iajs-747	174	15	1	1	NUM
iajs-747	174	16	2	2	NUM
iajs-747	174	17	kerhymykerf	kerhymykerf	NOUN
iajs-747	174	18			PROPN
iajs-747	174	19			PROPN
iajs-747	174	20	,	,	PUNCT
iajs-747	174	21	to	to	PART
iajs-747	174	22	prove	prove	VERB
iajs-747	174	23	our	our	PRON
iajs-747	174	24	a	a	DET
iajs-747	174	25	sseration	sseration	NOUN
iajs-747	174	26	.	.	PUNCT
iajs-747	175	1	let	let	VERB
iajs-747	175	2	0	0	NUM
iajs-747	175	3	)	)	PUNCT
iajs-747	175	4	(	(	PUNCT
iajs-747	175	5	,	,	PUNCT
iajs-747	175	6			NOUN
iajs-747	175	7	yfkerfy	yfkerfy	ADV
iajs-747	175	8	,	,	PUNCT
iajs-747	175	9			VERB
iajs-747	175	10			NOUN
iajs-747	175	11	)	)	PUNCT
iajs-747	175	12	)	)	PUNCT
iajs-747	175	13	(	(	PUNCT
iajs-747	175	14	(	(	PUNCT
iajs-747	175	15	)	)	PUNCT
iajs-747	175	16	)	)	PUNCT
iajs-747	175	17	(	(	PUNCT
iajs-747	175	18	)	)	PUNCT
iajs-747	175	19	(	(	PUNCT
iajs-747	175	20	(	(	PUNCT
iajs-747	175	21	)	)	PUNCT
iajs-747	175	22	)	)	PUNCT
iajs-747	175	23	(	(	PUNCT
iajs-747	175	24	(	(	PUNCT
iajs-747	175	25	1111	1111	NUM
iajs-747	175	26	yofyofoyh	yofyofoyh	NOUN
iajs-747	175	27			VERB
iajs-747	175	28	0)0	0)0	PROPN
iajs-747	175	29	(	(	PUNCT
iajs-747	175	30	)	)	PUNCT
iajs-747	175	31	)	)	PUNCT
iajs-747	175	32	(	(	PUNCT
iajs-747	175	33	(	(	PUNCT
iajs-747	175	34	11	11	NUM
iajs-747	175	35			NUM
iajs-747	175	36			NUM
iajs-747	175	37			NOUN
iajs-747	175	38	yf	yf	NOUN
iajs-747	175	39	.then	.then	X
iajs-747	175	40	for	for	ADP
iajs-747	175	41	all	all	DET
iajs-747	175	42	kerfy	kerfy	PROPN
iajs-747	175	43			NOUN
iajs-747	175	44	,	,	PUNCT
iajs-747	175	45	kerhy	kerhy	ADV
iajs-747	175	46			NUM
iajs-747	175	47	)	)	PUNCT
iajs-747	175	48	(	(	PUNCT
iajs-747	175	49	1	1	NUM
iajs-747	175	50	,	,	PUNCT
iajs-747	175	51	so	so	ADV
iajs-747	175	52	kerhkerf	kerhkerf	NOUN
iajs-747	175	53			PUNCT
iajs-747	175	54	)	)	PUNCT
iajs-747	176	1	(	(	PUNCT
iajs-747	176	2	1	1	NUM
iajs-747	176	3	≰e	≰e	NOUN
iajs-747	176	4	m	m	NOUN
iajs-747	176	5	1	1	NUM
iajs-747	176	6	which	which	PRON
iajs-747	176	7	implies	imply	VERB
iajs-747	176	8	)	)	PUNCT
iajs-747	176	9	(	(	PUNCT
iajs-747	176	10	1	1	NUM
iajs-747	176	11	kerf	kerf	NOUN
iajs-747	176	12	≰e	≰e	PROPN
iajs-747	176	13	m	m	VERB
iajs-747	176	14	1	1	NUM
iajs-747	176	15	,	,	PUNCT
iajs-747	176	16	so	so	ADV
iajs-747	176	17	kerf	kerf	NOUN
iajs-747	176	18	≰e	≰e	PROPN
iajs-747	176	19	m	m	NOUN
iajs-747	176	20	2	2	NUM
iajs-747	176	21	.	.	PUNCT
iajs-747	177	1	thus	thus	ADV
iajs-747	177	2	m	m	VERB
iajs-747	177	3	2	2	NUM
iajs-747	177	4	is	be	AUX
iajs-747	177	5	an	an	DET
iajs-747	177	6	essentially	essentially	ADV
iajs-747	177	7	quasi	quasi	ADJ
iajs-747	177	8	-	-	ADJ
iajs-747	177	9	dedekind	dedekind	ADJ
iajs-747	177	10	r	r	NOUN
iajs-747	177	11	-	-	NOUN
iajs-747	177	12	module	module	NOUN
iajs-747	177	13	.	.	PUNCT
iajs-747	177	14	)	)	PUNCT
iajs-747	178	1			PROPN
iajs-747	178	2	the	the	DET
iajs-747	178	3	proof	proof	NOUN
iajs-747	178	4	is	be	AUX
iajs-747	178	5	similarly	similarly	ADV
iajs-747	178	6	.	.	PUNCT
iajs-747	179	1	remark	remark	NOUN
iajs-747	179	2	(	(	PUNCT
iajs-747	179	3	2.11	2.11	NUM
iajs-747	179	4	)	)	PUNCT
iajs-747	179	5	let	let	VERB
iajs-747	179	6	m	m	PRON
iajs-747	179	7	be	be	AUX
iajs-747	179	8	an	an	DET
iajs-747	179	9	r	r	NOUN
iajs-747	179	10	-	-	PUNCT
iajs-747	179	11	module	module	NOUN
iajs-747	179	12	and	and	CCONJ
iajs-747	179	13	let	let	VERB
iajs-747	179	14	mn	mn	PROPN
iajs-747	179	15			PROPN
iajs-747	179	16	.	.	PUNCT
iajs-747	180	1	if	if	SCONJ
iajs-747	180	2	nm	nm	PRON
iajs-747	180	3	is	be	AUX
iajs-747	180	4	an	an	DET
iajs-747	180	5	essentially	essentially	ADV
iajs-747	180	6	quasidedekind	quasidedekind	ADJ
iajs-747	180	7	r	r	NOUN
iajs-747	180	8	-	-	PUNCT
iajs-747	180	9	module	module	NOUN
iajs-747	180	10	.	.	PUNCT
iajs-747	181	1	then	then	ADV
iajs-747	181	2	m	m	PROPN
iajs-747	181	3	is	be	AUX
iajs-747	181	4	not	not	PART
iajs-747	181	5	necessarily	necessarily	ADV
iajs-747	181	6	an	an	DET
iajs-747	181	7	essentially	essentially	ADV
iajs-747	181	8	quasi	quasi	ADJ
iajs-747	181	9	-	-	ADJ
iajs-747	181	10	dedekind	dedekind	ADJ
iajs-747	181	11	r	r	NOUN
iajs-747	181	12	-	-	NOUN
iajs-747	181	13	module	module	NOUN
iajs-747	181	14	,	,	PUNCT
iajs-747	181	15	as	as	SCONJ
iajs-747	181	16	we	we	PRON
iajs-747	181	17	can	can	AUX
iajs-747	181	18	see	see	VERB
iajs-747	181	19	by	by	ADP
iajs-747	181	20	the	the	DET
iajs-747	181	21	following	following	ADJ
iajs-747	181	22	example	example	NOUN
iajs-747	181	23	.	.	PUNCT
iajs-747	182	1	example	example	NOUN
iajs-747	182	2	(	(	PUNCT
iajs-747	182	3	2.12	2.12	NUM
iajs-747	182	4	)	)	PUNCT
iajs-747	182	5	let	let	VERB
iajs-747	182	6	m	m	NOUN
iajs-747	182	7	=	=	VERB
iajs-747	182	8	z4	z4	PROPN
iajs-747	182	9	as	as	ADP
iajs-747	182	10	a	a	DET
iajs-747	182	11	z	z	NOUN
iajs-747	182	12	-	-	PUNCT
iajs-747	182	13	module	module	NOUN
iajs-747	182	14	,	,	PUNCT
iajs-747	182	15	and	and	CCONJ
iajs-747	182	16	)	)	PUNCT
iajs-747	183	1	2(n	2(n	PROPN
iajs-747	183	2	≤	≤	NUM
iajs-747	183	3	z4	z4	X
iajs-747	183	4	,	,	PUNCT
iajs-747	183	5	then	then	ADV
iajs-747	183	6	24	24	NUM
iajs-747	183	7	)	)	SYM
iajs-747	183	8	2	2	NUM
iajs-747	183	9	(	(	PUNCT
iajs-747	183	10	zz	zz	PROPN
iajs-747	183	11			PROPN
iajs-747	183	12	is	be	AUX
iajs-747	183	13	an	an	DET
iajs-747	183	14	essentially	essentially	ADV
iajs-747	183	15	quasi	quasi	ADJ
iajs-747	183	16	-	-	ADJ
iajs-747	183	17	dedekind	dedekind	ADJ
iajs-747	183	18	z	z	NOUN
iajs-747	183	19	-	-	NOUN
iajs-747	183	20	module	module	NOUN
iajs-747	183	21	,	,	PUNCT
iajs-747	183	22	but	but	CCONJ
iajs-747	183	23	m	m	NOUN
iajs-747	183	24	=	=	NOUN
iajs-747	183	25	z4	z4	PROPN
iajs-747	183	26	is	be	AUX
iajs-747	183	27	not	not	PART
iajs-747	183	28	an	an	DET
iajs-747	183	29	essentially	essentially	ADV
iajs-747	183	30	quasi	quasi	ADJ
iajs-747	183	31	-	-	ADJ
iajs-747	183	32	dedekind	dedekind	ADJ
iajs-747	183	33	z	z	NOUN
iajs-747	183	34	-	-	NOUN
iajs-747	183	35	module	module	NOUN
iajs-747	183	36	.	.	PUNCT
iajs-747	184	1	now	now	ADV
iajs-747	184	2	,	,	PUNCT
iajs-747	184	3	we	we	PRON
iajs-747	184	4	turn	turn	VERB
iajs-747	184	5	our	our	PRON
iajs-747	184	6	attention	attention	NOUN
iajs-747	184	7	to	to	ADP
iajs-747	184	8	a	a	DET
iajs-747	184	9	submodule	submodule	NOUN
iajs-747	184	10	of	of	ADP
iajs-747	184	11	essentially	essentially	ADV
iajs-747	184	12	quasi	quasi	ADJ
iajs-747	184	13	-	-	ADJ
iajs-747	184	14	dedekind	dedekind	ADJ
iajs-747	184	15	.	.	PUNCT
iajs-747	185	1	first	first	ADV
iajs-747	185	2	consider	consider	VERB
iajs-747	185	3	the	the	DET
iajs-747	185	4	following	follow	VERB
iajs-747	185	5	remark	remark	NOUN
iajs-747	185	6	:	:	PUNCT
iajs-747	185	7	ibn	ibn	PROPN
iajs-747	185	8	alhaitham	alhaitham	NOUN
iajs-747	185	9	j.	j.	PROPN
iajs-747	185	10	for	for	ADP
iajs-747	185	11	pure	pure	ADJ
iajs-747	185	12	&	&	CCONJ
iajs-747	185	13	appl	appl	PROPN
iajs-747	185	14	.	.	PUNCT
iajs-747	186	1	sci	sci	PROPN
iajs-747	186	2	.	.	PUNCT
iajs-747	186	3	vol.24	vol.24	NOUN
iajs-747	186	4	(	(	PUNCT
iajs-747	186	5	3	3	NUM
iajs-747	186	6	)	)	PUNCT
iajs-747	186	7	2011	2011	NUM
iajs-747	186	8	remark	remark	NOUN
iajs-747	186	9	(	(	PUNCT
iajs-747	186	10	2.13	2.13	NUM
iajs-747	186	11	)	)	PUNCT
iajs-747	186	12	let	let	VERB
iajs-747	186	13	m	m	PRON
iajs-747	186	14	be	be	AUX
iajs-747	186	15	an	an	DET
iajs-747	186	16	essentially	essentially	ADV
iajs-747	186	17	quasidedekind	quasidedekind	ADJ
iajs-747	186	18	r	r	NOUN
iajs-747	186	19	-	-	PUNCT
iajs-747	186	20	module	module	NOUN
iajs-747	186	21	,	,	PUNCT
iajs-747	186	22	mn	mn	PROPN
iajs-747	186	23			PROPN
iajs-747	186	24	.	.	PUNCT
iajs-747	187	1	then	then	ADV
iajs-747	187	2	it	it	PRON
iajs-747	187	3	is	be	AUX
iajs-747	187	4	not	not	PART
iajs-747	187	5	necessarily	necessarily	ADV
iajs-747	187	6	that	that	PRON
iajs-747	187	7	n	n	AUX
iajs-747	187	8	be	be	AUX
iajs-747	187	9	an	an	DET
iajs-747	187	10	essentially	essentially	ADV
iajs-747	187	11	quasi	quasi	ADJ
iajs-747	187	12	-	-	ADJ
iajs-747	187	13	dedekind	dedekind	ADJ
iajs-747	187	14	r	r	NOUN
iajs-747	187	15	-	-	PUNCT
iajs-747	187	16	module	module	NOUN
iajs-747	187	17	.to	.to	PUNCT
iajs-747	187	18	show	show	VERB
iajs-747	187	19	this	this	PRON
iajs-747	187	20	,	,	PUNCT
iajs-747	187	21	consider	consider	VERB
iajs-747	187	22	the	the	DET
iajs-747	187	23	following	follow	VERB
iajs-747	187	24	example	example	NOUN
iajs-747	187	25	which	which	PRON
iajs-747	187	26	appeared	appear	VERB
iajs-747	187	27	in	in	ADP
iajs-747	187	28	[	[	X
iajs-747	187	29	7	7	NUM
iajs-747	187	30	]	]	PUNCT
iajs-747	187	31	.	.	PUNCT
iajs-747	188	1	let	let	VERB
iajs-747	188	2	2zqm	2zqm	NUM
iajs-747	188	3			NOUN
iajs-747	188	4	as	as	ADP
iajs-747	188	5	a	a	DET
iajs-747	188	6	z	z	NOUN
iajs-747	188	7	-	-	PUNCT
iajs-747	188	8	module	module	NOUN
iajs-747	188	9	is	be	AUX
iajs-747	188	10	essentially	essentially	ADV
iajs-747	188	11	quasi	quasi	ADJ
iajs-747	188	12	-	-	NOUN
iajs-747	188	13	dedekind	dedekind	ADJ
iajs-747	188	14	.	.	PUNCT
iajs-747	189	1	take	take	VERB
iajs-747	189	2	22	22	NUM
iajs-747	189	3	zqzzn	zqzzn	NOUN
iajs-747	189	4			PUNCT
iajs-747	189	5	as	as	ADP
iajs-747	189	6	a	a	DET
iajs-747	189	7	z	z	NOUN
iajs-747	189	8	-	-	PUNCT
iajs-747	189	9	module	module	NOUN
iajs-747	189	10	,	,	PUNCT
iajs-747	189	11	then	then	ADV
iajs-747	189	12	n	n	PRON
iajs-747	189	13	is	be	AUX
iajs-747	189	14	not	not	PART
iajs-747	189	15	essentially	essentially	ADV
iajs-747	189	16	quasidedekind	quasidedekind	ADJ
iajs-747	189	17	as	as	ADP
iajs-747	189	18	a	a	DET
iajs-747	189	19	z	z	NOUN
iajs-747	189	20	-	-	PUNCT
iajs-747	189	21	module	module	NOUN
iajs-747	189	22	,	,	PUNCT
iajs-747	189	23	since	since	SCONJ
iajs-747	189	24	if	if	SCONJ
iajs-747	189	25	nnf	nnf	NOUN
iajs-747	189	26			NOUN
iajs-747	189	27	:	:	PUNCT
iajs-747	189	28	define	define	VERB
iajs-747	189	29	by	by	ADP
iajs-747	189	30	)	)	PUNCT
iajs-747	189	31	,	,	PUNCT
iajs-747	189	32	0	0	NUM
iajs-747	189	33	(	(	PUNCT
iajs-747	189	34	)	)	PUNCT
iajs-747	189	35	,	,	PUNCT
iajs-747	189	36	(	(	PUNCT
iajs-747	189	37	xyxf	xyxf	VERB
iajs-747	189	38			NUM
iajs-747	189	39	,	,	PUNCT
iajs-747	189	40	2	2	NUM
iajs-747	189	41	,	,	PUNCT
iajs-747	189	42	zyzx	zyzx	X
iajs-747	189	43			X
iajs-747	189	44	,	,	PUNCT
iajs-747	189	45	then	then	ADV
iajs-747	189	46	0f	0f	NUM
iajs-747	189	47	and	and	CCONJ
iajs-747	189	48	22}0:),{()}0,0	22}0:),{()}0,0	NUM
iajs-747	189	49	(	(	PUNCT
iajs-747	189	50	)	)	PUNCT
iajs-747	189	51	,	,	PUNCT
iajs-747	189	52	(:	(:	PROPN
iajs-747	189	53	)	)	PUNCT
iajs-747	189	54	,	,	PUNCT
iajs-747	189	55	{	{	PUNCT
iajs-747	189	56	(	(	PUNCT
iajs-747	189	57	zzxnyxyxfnyxkerf	zzxnyxyxfnyxkerf	NOUN
iajs-747	189	58			PROPN
iajs-747	189	59	.	.	PUNCT
iajs-747	190	1	hence	hence	ADV
iajs-747	190	2	kerf	kerf	VERB
iajs-747	190	3	≤	≤	NUM
iajs-747	190	4	e	e	NOUN
iajs-747	190	5	n	n	NOUN
iajs-747	190	6	.	.	PUNCT
iajs-747	191	1	thus	thus	ADV
iajs-747	191	2	2	2	NUM
iajs-747	191	3	zzn	zzn	NOUN
iajs-747	191	4			NOUN
iajs-747	191	5	is	be	AUX
iajs-747	191	6	not	not	PART
iajs-747	191	7	an	an	DET
iajs-747	191	8	essentially	essentially	ADV
iajs-747	191	9	quasi	quasi	ADJ
iajs-747	191	10	-	-	NOUN
iajs-747	191	11	dedekind	dedekind	ADJ
iajs-747	191	12	as	as	ADP
iajs-747	191	13	a	a	DET
iajs-747	191	14	z	z	NOUN
iajs-747	191	15	-	-	PUNCT
iajs-747	191	16	module	module	NOUN
iajs-747	191	17	.	.	PUNCT
iajs-747	192	1	now	now	ADV
iajs-747	192	2	,	,	PUNCT
iajs-747	192	3	in	in	ADP
iajs-747	192	4	the	the	DET
iajs-747	192	5	next	next	ADJ
iajs-747	192	6	proposition	proposition	NOUN
iajs-747	192	7	we	we	PRON
iajs-747	192	8	give	give	VERB
iajs-747	192	9	a	a	DET
iajs-747	192	10	condition	condition	NOUN
iajs-747	192	11	which	which	PRON
iajs-747	192	12	makes	make	VERB
iajs-747	192	13	r	r	NOUN
iajs-747	192	14	-	-	PUNCT
iajs-747	192	15	submodule	submodule	NOUN
iajs-747	192	16	of	of	ADP
iajs-747	192	17	an	an	DET
iajs-747	192	18	essentially	essentially	ADV
iajs-747	192	19	quasi	quasi	ADJ
iajs-747	192	20	-	-	ADJ
iajs-747	192	21	dedekind	dedekind	ADJ
iajs-747	192	22	r	r	NOUN
iajs-747	192	23	-	-	PUNCT
iajs-747	192	24	module	module	NOUN
iajs-747	192	25	is	be	AUX
iajs-747	192	26	essentially	essentially	ADV
iajs-747	192	27	quasi	quasi	ADJ
iajs-747	192	28	-	-	ADJ
iajs-747	192	29	dedekind	dedekind	ADJ
iajs-747	192	30	.	.	PUNCT
iajs-747	193	1	proposition	proposition	NOUN
iajs-747	193	2	(	(	PUNCT
iajs-747	193	3	2.14	2.14	NUM
iajs-747	193	4	)	)	PUNCT
iajs-747	193	5	let	let	VERB
iajs-747	193	6	m	m	PRON
iajs-747	193	7	be	be	AUX
iajs-747	193	8	an	an	DET
iajs-747	193	9	essentially	essentially	ADV
iajs-747	193	10	quasi	quasi	ADJ
iajs-747	193	11	-	-	ADJ
iajs-747	193	12	dedekind	dedekind	ADJ
iajs-747	193	13	r	r	NOUN
iajs-747	193	14	-	-	NOUN
iajs-747	193	15	module	module	NOUN
iajs-747	193	16	,	,	PUNCT
iajs-747	193	17	and	and	CCONJ
iajs-747	193	18	m	m	PROPN
iajs-747	193	19	is	be	AUX
iajs-747	193	20	quasi	quasi	NOUN
iajs-747	193	21	–	–	PUNCT
iajs-747	193	22	injective	injective	ADJ
iajs-747	193	23	.	.	PUNCT
iajs-747	194	1	if	if	SCONJ
iajs-747	194	2	n	n	NOUN
iajs-747	194	3	≤	≤	X
iajs-747	194	4	e	e	X
iajs-747	194	5	m	m	VERB
iajs-747	194	6	then	then	ADV
iajs-747	194	7	n	n	PRON
iajs-747	194	8	is	be	AUX
iajs-747	194	9	an	an	DET
iajs-747	194	10	essentially	essentially	ADV
iajs-747	194	11	quasi	quasi	ADJ
iajs-747	194	12	-	-	ADJ
iajs-747	194	13	dedekind	dedekind	ADJ
iajs-747	194	14	r	r	NOUN
iajs-747	194	15	-	-	PUNCT
iajs-747	194	16	module	module	NOUN
iajs-747	194	17	.	.	PUNCT
iajs-747	195	1	proof	proof	NOUN
iajs-747	195	2	:	:	PUNCT
iajs-747	195	3	let	let	VERB
iajs-747	195	4	)	)	PUNCT
iajs-747	195	5	(	(	PUNCT
iajs-747	195	6	nendf	nendf	ADJ
iajs-747	195	7	r	r	ADJ
iajs-747	195	8	,	,	PUNCT
iajs-747	195	9	0f	0f	NUM
iajs-747	195	10	,	,	PUNCT
iajs-747	195	11	to	to	PART
iajs-747	195	12	prove	prove	VERB
iajs-747	195	13	that	that	SCONJ
iajs-747	195	14	kerf	kerf	NOUN
iajs-747	195	15	≰e	≰e	PROPN
iajs-747	195	16	n	n	NUM
iajs-747	195	17	.	.	PUNCT
iajs-747	196	1	assume	assume	VERB
iajs-747	196	2	that	that	SCONJ
iajs-747	196	3	kerf	kerf	NOUN
iajs-747	196	4	≤	≤	NUM
iajs-747	196	5	e	e	NOUN
iajs-747	196	6	n	n	NOUN
iajs-747	196	7	.	.	PUNCT
iajs-747	197	1	since	since	SCONJ
iajs-747	197	2	m	m	PROPN
iajs-747	197	3	is	be	AUX
iajs-747	197	4	quasi	quasi	ADJ
iajs-747	197	5	–	–	PUNCT
iajs-747	197	6	injective	injective	ADJ
iajs-747	197	7	,	,	PUNCT
iajs-747	197	8	then	then	ADV
iajs-747	197	9	there	there	PRON
iajs-747	197	10	exists	exist	VERB
iajs-747	197	11	)	)	PUNCT
iajs-747	197	12	(	(	PUNCT
iajs-747	197	13	mendg	mendg	NOUN
iajs-747	197	14	r	r	VERB
iajs-747	197	15	such	such	ADJ
iajs-747	197	16	that	that	PRON
iajs-747	197	17	goi	goi	PROPN
iajs-747	197	18	=	=	SYM
iajs-747	197	19	iof	iof	PROPN
iajs-747	197	20	,	,	PUNCT
iajs-747	197	21	(	(	PUNCT
iajs-747	197	22	where	where	SCONJ
iajs-747	197	23	i	i	PRON
iajs-747	197	24	is	be	AUX
iajs-747	197	25	the	the	DET
iajs-747	197	26	inclusion	inclusion	NOUN
iajs-747	197	27	mapping	mapping	NOUN
iajs-747	197	28	)	)	PUNCT
iajs-747	197	29	.	.	PUNCT
iajs-747	198	1	it	it	PRON
iajs-747	198	2	follows	follow	VERB
iajs-747	198	3	that	that	PRON
iajs-747	198	4	0g	0g	NUM
iajs-747	198	5	,	,	PUNCT
iajs-747	198	6	and	and	CCONJ
iajs-747	198	7	this	this	PRON
iajs-747	198	8	implies	imply	VERB
iajs-747	198	9	kerg	kerg	PROPN
iajs-747	198	10	≰e	≰e	PROPN
iajs-747	198	11	m	m	PROPN
iajs-747	198	12	,	,	PUNCT
iajs-747	198	13	since	since	SCONJ
iajs-747	198	14	m	m	PROPN
iajs-747	198	15	is	be	AUX
iajs-747	198	16	essentially	essentially	ADV
iajs-747	198	17	quasidedekind	quasidedekind	ADJ
iajs-747	198	18	.	.	PUNCT
iajs-747	199	1	but	but	CCONJ
iajs-747	199	2	kergkerf	kergkerf	NOUN
iajs-747	199	3			PROPN
iajs-747	199	4	,	,	PUNCT
iajs-747	199	5	so	so	CCONJ
iajs-747	199	6	kerf	kerf	NOUN
iajs-747	199	7	≰e	≰e	NOUN
iajs-747	199	8	m	m	NOUN
iajs-747	199	9	.	.	PUNCT
iajs-747	200	1	on	on	ADP
iajs-747	200	2	the	the	DET
iajs-747	200	3	other	other	ADJ
iajs-747	200	4	hand	hand	NOUN
iajs-747	200	5	n	n	CCONJ
iajs-747	200	6	≤	≤	NUM
iajs-747	200	7	e	e	X
iajs-747	200	8	m	m	NOUN
iajs-747	200	9	and	and	CCONJ
iajs-747	200	10	by	by	ADP
iajs-747	200	11	assumption	assumption	NOUN
iajs-747	200	12	kerf	kerf	NOUN
iajs-747	200	13	≤	≤	NUM
iajs-747	200	14	e	e	NOUN
iajs-747	200	15	n	n	PRON
iajs-747	200	16	imply	imply	VERB
iajs-747	200	17	kerf	kerf	NOUN
iajs-747	200	18	≤	≤	NUM
iajs-747	200	19	e	e	NOUN
iajs-747	200	20	m	m	NOUN
iajs-747	200	21	.	.	PUNCT
iajs-747	201	1	to	to	PART
iajs-747	201	2	show	show	VERB
iajs-747	201	3	this	this	PRON
iajs-747	201	4	,	,	PUNCT
iajs-747	201	5	since	since	SCONJ
iajs-747	201	6	n	n	ADV
iajs-747	201	7	≤	≤	NOUN
iajs-747	202	1	e	e	X
iajs-747	202	2	m	m	VERB
iajs-747	202	3	then	then	ADV
iajs-747	202	4	for	for	ADP
iajs-747	202	5	all	all	DET
iajs-747	202	6	mu	mu	NOUN
iajs-747	202	7			NOUN
iajs-747	202	8	,	,	PUNCT
iajs-747	202	9	0u	0u	NUM
iajs-747	202	10	then	then	ADV
iajs-747	202	11	0un	0un	PROPN
iajs-747	202	12	and	and	CCONJ
iajs-747	202	13	nun	nun	PROPN
iajs-747	202	14			PROPN
iajs-747	202	15	.but	.but	PUNCT
iajs-747	203	1	kerf	kerf	NOUN
iajs-747	203	2	≤	≤	NUM
iajs-747	203	3	e	e	NOUN
iajs-747	203	4	n	n	NOUN
iajs-747	203	5	,	,	PUNCT
iajs-747	203	6	hence	hence	ADV
iajs-747	203	7	0	0	NUM
iajs-747	203	8	)	)	PUNCT
iajs-747	203	9	(	(	PUNCT
iajs-747	203	10			VERB
iajs-747	203	11	unkerf	unkerf	NOUN
iajs-747	203	12	;	;	PUNCT
iajs-747	203	13	that	that	PRON
iajs-747	203	14	is	be	AUX
iajs-747	203	15	0	0	NUM
iajs-747	203	16	)	)	PUNCT
iajs-747	203	17	(	(	PUNCT
iajs-747	203	18			INTJ
iajs-747	203	19	nukerf	nukerf	INTJ
iajs-747	203	20	which	which	PRON
iajs-747	203	21	implies	imply	VERB
iajs-747	203	22	that	that	SCONJ
iajs-747	203	23	0ukerf	0ukerf	NOUN
iajs-747	203	24	which	which	PRON
iajs-747	203	25	is	be	AUX
iajs-747	203	26	a	a	DET
iajs-747	203	27	contradiction	contradiction	NOUN
iajs-747	203	28	.	.	PUNCT
iajs-747	204	1	thus	thus	ADV
iajs-747	204	2	kerf	kerf	VERB
iajs-747	204	3	≰e	≰e	PROPN
iajs-747	204	4	n	n	NOUN
iajs-747	204	5	and	and	CCONJ
iajs-747	204	6	hence	hence	ADV
iajs-747	204	7	n	n	PRON
iajs-747	204	8	is	be	AUX
iajs-747	204	9	an	an	DET
iajs-747	204	10	essentially	essentially	ADV
iajs-747	204	11	quasi	quasi	ADJ
iajs-747	204	12	-	-	ADJ
iajs-747	204	13	dedekind	dedekind	ADJ
iajs-747	204	14	r	r	NOUN
iajs-747	204	15	-	-	PUNCT
iajs-747	204	16	module	module	NOUN
iajs-747	204	17	.	.	PUNCT
iajs-747	205	1	n	n	CCONJ
iajs-747	205	2	n	n	PRON
iajs-747	205	3	m	m	VERB
iajs-747	205	4	m	m	VERB
iajs-747	205	5	i	i	INTJ
iajs-747	205	6	f	f	NOUN
iajs-747	205	7	g	g	NOUN
iajs-747	206	1	i	i	PRON
iajs-747	206	2	ibn	ibn	PROPN
iajs-747	206	3	alhaitham	alhaitham	PROPN
iajs-747	206	4	j.	j.	PROPN
iajs-747	206	5	for	for	ADP
iajs-747	206	6	pure	pure	ADJ
iajs-747	206	7	&	&	CCONJ
iajs-747	206	8	appl	appl	PROPN
iajs-747	206	9	.	.	PUNCT
iajs-747	207	1	sci	sci	PROPN
iajs-747	207	2	.	.	PUNCT
iajs-747	207	3	vol.24	vol.24	NOUN
iajs-747	207	4	(	(	PUNCT
iajs-747	207	5	3	3	NUM
iajs-747	207	6	)	)	PUNCT
iajs-747	207	7	2011	2011	NUM
iajs-747	207	8	corollary	corollary	NOUN
iajs-747	207	9	(	(	PUNCT
iajs-747	207	10	2.15	2.15	NUM
iajs-747	207	11	)	)	PUNCT
iajs-747	207	12	let	let	VERB
iajs-747	207	13	m	m	PRON
iajs-747	207	14	be	be	AUX
iajs-747	207	15	an	an	DET
iajs-747	207	16	r	r	NOUN
iajs-747	207	17	-	-	PUNCT
iajs-747	207	18	module	module	NOUN
iajs-747	207	19	.	.	PUNCT
iajs-747	208	1	if	if	SCONJ
iajs-747	208	2	m	m	NOUN
iajs-747	208	3	is	be	AUX
iajs-747	208	4	an	an	DET
iajs-747	208	5	essentially	essentially	ADV
iajs-747	208	6	quasi	quasi	ADJ
iajs-747	208	7	-	-	ADJ
iajs-747	208	8	dedekind	dedekind	ADJ
iajs-747	208	9	r	r	NOUN
iajs-747	208	10	-	-	PUNCT
iajs-747	208	11	module	module	NOUN
iajs-747	208	12	then	then	ADV
iajs-747	208	13	m	m	VERB
iajs-747	208	14	is	be	AUX
iajs-747	208	15	an	an	DET
iajs-747	208	16	essentially	essentially	ADV
iajs-747	208	17	quasi	quasi	ADJ
iajs-747	208	18	-	-	ADJ
iajs-747	208	19	dedekind	dedekind	ADJ
iajs-747	208	20	r	r	NOUN
iajs-747	208	21	-	-	NOUN
iajs-747	208	22	module	module	NOUN
iajs-747	208	23	.	.	PUNCT
iajs-747	209	1	proof	proof	NOUN
iajs-747	209	2	:	:	PUNCT
iajs-747	209	3	suppose	suppose	VERB
iajs-747	209	4	that	that	SCONJ
iajs-747	209	5	m	m	PROPN
iajs-747	209	6	is	be	AUX
iajs-747	209	7	an	an	DET
iajs-747	209	8	essentially	essentially	ADV
iajs-747	209	9	quasi	quasi	ADJ
iajs-747	209	10	-	-	ADJ
iajs-747	209	11	dedekind	dedekind	ADJ
iajs-747	209	12	r	r	NOUN
iajs-747	209	13	-	-	NOUN
iajs-747	209	14	module	module	NOUN
iajs-747	209	15	,	,	PUNCT
iajs-747	209	16	and	and	CCONJ
iajs-747	209	17	since	since	SCONJ
iajs-747	209	18	m	m	PROPN
iajs-747	209	19	is	be	AUX
iajs-747	209	20	a	a	DET
iajs-747	209	21	quasi	quasi	NOUN
iajs-747	209	22	–	–	PUNCT
iajs-747	209	23	injective	injective	ADJ
iajs-747	209	24	r	r	NOUN
iajs-747	209	25	-	-	PUNCT
iajs-747	209	26	module	module	NOUN
iajs-747	209	27	and	and	CCONJ
iajs-747	209	28	m	m	NOUN
iajs-747	209	29	≤	≤	NOUN
iajs-747	209	30	e	e	X
iajs-747	209	31	m	m	NOUN
iajs-747	209	32	,	,	PUNCT
iajs-747	209	33	so	so	ADV
iajs-747	209	34	by	by	ADP
iajs-747	209	35	(	(	PUNCT
iajs-747	209	36	prop	prop	NOUN
iajs-747	209	37	2.14	2.14	NUM
iajs-747	209	38	)	)	PUNCT
iajs-747	209	39	,	,	PUNCT
iajs-747	209	40	m	m	VERB
iajs-747	209	41	is	be	AUX
iajs-747	209	42	an	an	DET
iajs-747	209	43	essentially	essentially	ADV
iajs-747	209	44	quasidedekind	quasidedekind	ADJ
iajs-747	209	45	r	r	NOUN
iajs-747	209	46	-	-	PUNCT
iajs-747	209	47	module	module	NOUN
iajs-747	209	48	.	.	PUNCT
iajs-747	210	1	corollary	corollary	ADJ
iajs-747	210	2	(	(	PUNCT
iajs-747	210	3	2.16	2.16	NUM
iajs-747	210	4	)	)	PUNCT
iajs-747	210	5	let	let	VERB
iajs-747	210	6	m	m	PRON
iajs-747	210	7	be	be	AUX
iajs-747	210	8	an	an	DET
iajs-747	210	9	r	r	NOUN
iajs-747	210	10	-	-	PUNCT
iajs-747	210	11	module	module	NOUN
iajs-747	210	12	.	.	PUNCT
iajs-747	211	1	if	if	SCONJ
iajs-747	211	2	e(m	e(m	PROPN
iajs-747	211	3	)	)	PUNCT
iajs-747	211	4	is	be	AUX
iajs-747	211	5	an	an	DET
iajs-747	211	6	essentially	essentially	ADV
iajs-747	211	7	quasi	quasi	ADJ
iajs-747	211	8	-	-	ADJ
iajs-747	211	9	dedekind	dedekind	ADJ
iajs-747	211	10	r	r	NOUN
iajs-747	211	11	-	-	PUNCT
iajs-747	211	12	module	module	NOUN
iajs-747	211	13	then	then	ADV
iajs-747	211	14	m	m	VERB
iajs-747	211	15	is	be	AUX
iajs-747	211	16	an	an	DET
iajs-747	211	17	essentially	essentially	ADV
iajs-747	211	18	quasi	quasi	ADJ
iajs-747	211	19	-	-	ADJ
iajs-747	211	20	dedekind	dedekind	ADJ
iajs-747	211	21	r	r	NOUN
iajs-747	211	22	-	-	NOUN
iajs-747	211	23	module	module	NOUN
iajs-747	211	24	.	.	PUNCT
iajs-747	212	1	proof	proof	NOUN
iajs-747	212	2	:	:	PUNCT
iajs-747	212	3	it	it	PRON
iajs-747	212	4	is	be	AUX
iajs-747	212	5	clear	clear	ADJ
iajs-747	212	6	.	.	PUNCT
iajs-747	213	1	the	the	DET
iajs-747	213	2	converse	converse	NOUN
iajs-747	213	3	of	of	ADP
iajs-747	213	4	(	(	PUNCT
iajs-747	213	5	coro2.16	coro2.16	PROPN
iajs-747	213	6	)	)	PUNCT
iajs-747	213	7	is	be	AUX
iajs-747	213	8	not	not	PART
iajs-747	213	9	true	true	ADJ
iajs-747	213	10	in	in	ADP
iajs-747	213	11	general	general	ADJ
iajs-747	213	12	,	,	PUNCT
iajs-747	213	13	consider	consider	VERB
iajs-747	213	14	the	the	DET
iajs-747	213	15	following	follow	VERB
iajs-747	213	16	example	example	NOUN
iajs-747	213	17	.	.	PUNCT
iajs-747	214	1	example	example	NOUN
iajs-747	214	2	(	(	PUNCT
iajs-747	214	3	2.17	2.17	NUM
iajs-747	214	4	)	)	PUNCT
iajs-747	214	5	let	let	VERB
iajs-747	214	6	m	m	NOUN
iajs-747	214	7	=	=	ADJ
iajs-747	214	8	z2	z2	PROPN
iajs-747	214	9	as	as	ADP
iajs-747	214	10	a	a	DET
iajs-747	214	11	z	z	NOUN
iajs-747	214	12	-	-	PUNCT
iajs-747	214	13	module	module	NOUN
iajs-747	214	14	.	.	PUNCT
iajs-747	215	1	m	m	PROPN
iajs-747	215	2	is	be	AUX
iajs-747	215	3	an	an	DET
iajs-747	215	4	essentially	essentially	ADV
iajs-747	215	5	quasi	quasi	ADJ
iajs-747	215	6	-	-	ADJ
iajs-747	215	7	dedekind	dedekind	ADJ
iajs-747	215	8	z	z	NOUN
iajs-747	215	9	-	-	PUNCT
iajs-747	215	10	module	module	NOUN
iajs-747	215	11	.	.	PUNCT
iajs-747	216	1	but	but	CCONJ
iajs-747	216	2	e(z2	e(z2	NOUN
iajs-747	216	3	)	)	PUNCT
iajs-747	217	1	=	=	SYM
iajs-747	217	2	z2	z2	PROPN
iajs-747	217	3	∞	∞	PROPN
iajs-747	217	4	is	be	AUX
iajs-747	217	5	not	not	PART
iajs-747	217	6	an	an	DET
iajs-747	217	7	essentially	essentially	ADV
iajs-747	217	8	quasi	quasi	ADJ
iajs-747	217	9	-	-	ADJ
iajs-747	217	10	dedekind	dedekind	ADJ
iajs-747	217	11	z	z	NOUN
iajs-747	217	12	-	-	NOUN
iajs-747	217	13	module	module	NOUN
iajs-747	217	14	,	,	PUNCT
iajs-747	217	15	(	(	PUNCT
iajs-747	217	16	see	see	VERB
iajs-747	217	17	rem.and.ex	rem.and.ex	PROPN
iajs-747	217	18	2.2(4	2.2(4	NUM
iajs-747	217	19	)	)	PUNCT
iajs-747	217	20	)	)	PUNCT
iajs-747	217	21	.	.	PUNCT
iajs-747	218	1	now	now	ADV
iajs-747	218	2	we	we	PRON
iajs-747	218	3	prove	prove	VERB
iajs-747	218	4	the	the	DET
iajs-747	218	5	following	follow	VERB
iajs-747	218	6	proposition	proposition	NOUN
iajs-747	218	7	:	:	PUNCT
iajs-747	218	8	proposition	proposition	NOUN
iajs-747	218	9	(	(	PUNCT
iajs-747	218	10	2.18	2.18	NUM
iajs-747	218	11	)	)	PUNCT
iajs-747	218	12	let	let	VERB
iajs-747	218	13	m	m	PRON
iajs-747	218	14	be	be	AUX
iajs-747	218	15	an	an	DET
iajs-747	218	16	r	r	NOUN
iajs-747	218	17	-	-	PUNCT
iajs-747	218	18	module	module	NOUN
iajs-747	218	19	such	such	ADJ
iajs-747	218	20	that	that	PRON
iajs-747	218	21	,	,	PUNCT
iajs-747	218	22	for	for	ADP
iajs-747	218	23	each	each	DET
iajs-747	218	24	0	0	NUM
iajs-747	218	25	,	,	PUNCT
iajs-747	218	26	)	)	PUNCT
iajs-747	218	27	)	)	PUNCT
iajs-747	218	28	(	(	PUNCT
iajs-747	218	29	,	,	PUNCT
iajs-747	218	30	(	(	PUNCT
iajs-747	218	31			NOUN
iajs-747	218	32	fmemhomf	fmemhomf	NOUN
iajs-747	218	33	implies	imply	VERB
iajs-747	218	34	kerf	kerf	NOUN
iajs-747	218	35	≰e	≰e	PROPN
iajs-747	218	36	m	m	NOUN
iajs-747	218	37	.	.	PUNCT
iajs-747	219	1	then	then	ADV
iajs-747	219	2	m	m	PROPN
iajs-747	219	3	is	be	AUX
iajs-747	219	4	essentially	essentially	ADV
iajs-747	219	5	quasi	quasi	ADJ
iajs-747	219	6	-	-	NOUN
iajs-747	219	7	dedekind	dedekind	ADJ
iajs-747	219	8	.	.	PUNCT
iajs-747	220	1	proof	proof	NOUN
iajs-747	220	2	:	:	PUNCT
iajs-747	220	3	let	let	VERB
iajs-747	220	4	)	)	PUNCT
iajs-747	220	5	(	(	PUNCT
iajs-747	220	6	mendg	mendg	NOUN
iajs-747	220	7	r	r	ADJ
iajs-747	220	8	,	,	PUNCT
iajs-747	220	9	0g	0g	NUM
iajs-747	220	10	.	.	PUNCT
iajs-747	221	1	then	then	ADV
iajs-747	221	2	)	)	PUNCT
iajs-747	221	3	)	)	PUNCT
iajs-747	222	1	(	(	PUNCT
iajs-747	222	2	,	,	PUNCT
iajs-747	222	3	(	(	PUNCT
iajs-747	222	4	memhomiog	memhomiog	NOUN
iajs-747	222	5	,	,	PUNCT
iajs-747	222	6	and	and	CCONJ
iajs-747	222	7	0iog	0iog	NOUN
iajs-747	222	8	,	,	PUNCT
iajs-747	222	9	where	where	SCONJ
iajs-747	222	10	i	i	PRON
iajs-747	222	11	is	be	AUX
iajs-747	222	12	the	the	DET
iajs-747	222	13	inclusion	inclusion	NOUN
iajs-747	222	14	mapping	mapping	NOUN
iajs-747	222	15	.	.	PUNCT
iajs-747	223	1	hence	hence	ADV
iajs-747	223	2	ker(iog	ker(iog	PROPN
iajs-747	223	3	)	)	PUNCT
iajs-747	224	1	≰e	≰e	PROPN
iajs-747	224	2	m	m	NOUN
iajs-747	224	3	.	.	PUNCT
iajs-747	225	1	but	but	CCONJ
iajs-747	225	2	kerg	kerg	PROPN
iajs-747	225	3	=	=	PROPN
iajs-747	225	4	ker(iog	ker(iog	PROPN
iajs-747	225	5	)	)	PUNCT
iajs-747	225	6	.	.	PUNCT
iajs-747	226	1	thus	thus	ADV
iajs-747	226	2	kerg	kerg	PROPN
iajs-747	226	3	≰e	≰e	PROPN
iajs-747	226	4	m	m	PROPN
iajs-747	226	5	and	and	CCONJ
iajs-747	226	6	m	m	PROPN
iajs-747	226	7	is	be	AUX
iajs-747	226	8	essentially	essentially	ADV
iajs-747	226	9	quasi	quasi	ADJ
iajs-747	226	10	-	-	NOUN
iajs-747	226	11	dedekind	dedekind	ADJ
iajs-747	226	12	.	.	PUNCT
iajs-747	227	1	next	next	ADV
iajs-747	227	2	we	we	PRON
iajs-747	227	3	study	study	VERB
iajs-747	227	4	the	the	DET
iajs-747	227	5	behavior	behavior	NOUN
iajs-747	227	6	of	of	ADP
iajs-747	227	7	the	the	DET
iajs-747	227	8	quotient	quotient	NOUN
iajs-747	227	9	module	module	NOUN
iajs-747	227	10	of	of	ADP
iajs-747	227	11	essentially	essentially	ADV
iajs-747	227	12	quasi	quasi	ADJ
iajs-747	227	13	-	-	ADJ
iajs-747	227	14	dedekind	dedekind	ADJ
iajs-747	227	15	module	module	NOUN
iajs-747	227	16	.	.	PUNCT
iajs-747	228	1	first	first	ADV
iajs-747	228	2	we	we	PRON
iajs-747	228	3	have	have	VERB
iajs-747	228	4	the	the	DET
iajs-747	228	5	following	following	NOUN
iajs-747	228	6	.	.	PUNCT
iajs-747	229	1	remark	remark	NOUN
iajs-747	229	2	(	(	PUNCT
iajs-747	229	3	2.19	2.19	NUM
iajs-747	229	4	)	)	PUNCT
iajs-747	229	5	let	let	VERB
iajs-747	229	6	m	m	PRON
iajs-747	229	7	be	be	AUX
iajs-747	229	8	an	an	DET
iajs-747	229	9	r	r	NOUN
iajs-747	229	10	-	-	PUNCT
iajs-747	229	11	module	module	NOUN
iajs-747	229	12	,	,	PUNCT
iajs-747	229	13	mn	mn	PROPN
iajs-747	229	14			PROPN
iajs-747	229	15	.	.	PUNCT
iajs-747	230	1	if	if	SCONJ
iajs-747	230	2	m	m	NOUN
iajs-747	230	3	is	be	AUX
iajs-747	230	4	an	an	DET
iajs-747	230	5	essentially	essentially	ADV
iajs-747	230	6	quasidedekind	quasidedekind	ADJ
iajs-747	230	7	r	r	NOUN
iajs-747	230	8	-	-	PUNCT
iajs-747	230	9	module	module	NOUN
iajs-747	230	10	,	,	PUNCT
iajs-747	230	11	then	then	ADV
iajs-747	230	12	nm	nm	ADV
iajs-747	230	13	is	be	AUX
iajs-747	230	14	not	not	PART
iajs-747	230	15	necessarily	necessarily	ADV
iajs-747	230	16	essentially	essentially	ADV
iajs-747	230	17	quasidedekind	quasidedekind	VERB
iajs-747	230	18	r	r	NOUN
iajs-747	230	19	-	-	PUNCT
iajs-747	230	20	module	module	NOUN
iajs-747	230	21	,	,	PUNCT
iajs-747	230	22	consider	consider	VERB
iajs-747	230	23	the	the	DET
iajs-747	230	24	following	follow	VERB
iajs-747	230	25	example	example	NOUN
iajs-747	230	26	.	.	PUNCT
iajs-747	231	1	example(2.20	example(2.20	X
iajs-747	231	2	)	)	PUNCT
iajs-747	232	1	it	it	PRON
iajs-747	232	2	is	be	AUX
iajs-747	232	3	well	well	ADV
iajs-747	232	4	-	-	PUNCT
iajs-747	232	5	known	know	VERB
iajs-747	232	6	that	that	SCONJ
iajs-747	232	7	z	z	NOUN
iajs-747	232	8	as	as	ADP
iajs-747	232	9	a	a	DET
iajs-747	232	10	z	z	NOUN
iajs-747	232	11	-	-	PUNCT
iajs-747	232	12	module	module	NOUN
iajs-747	232	13	is	be	AUX
iajs-747	232	14	essentially	essentially	ADV
iajs-747	232	15	quasidedekind	quasidedekind	ADJ
iajs-747	232	16	.	.	PUNCT
iajs-747	233	1	let	let	VERB
iajs-747	233	2	(	(	PUNCT
iajs-747	233	3	4)n	4)n	NUM
iajs-747	233	4	z	z	PROPN
iajs-747	233	5			NOUN
iajs-747	233	6	,	,	PUNCT
iajs-747	233	7	4	4	NUM
iajs-747	233	8	(	(	PUNCT
iajs-747	233	9	4)z	4)z	NUM
iajs-747	233	10	n	n	NOUN
iajs-747	233	11	z	z	NOUN
iajs-747	233	12	z	z	NOUN
iajs-747	233	13			PROPN
iajs-747	233	14	is	be	AUX
iajs-747	233	15	not	not	PART
iajs-747	233	16	essentially	essentially	ADV
iajs-747	233	17	quasi	quasi	ADJ
iajs-747	233	18	-	-	ADJ
iajs-747	233	19	dedekind	dedekind	ADJ
iajs-747	233	20	as	as	ADP
iajs-747	233	21	a	a	DET
iajs-747	233	22	z	z	NOUN
iajs-747	233	23	-	-	PUNCT
iajs-747	233	24	module	module	NOUN
iajs-747	233	25	,	,	PUNCT
iajs-747	233	26	(	(	PUNCT
iajs-747	233	27	see	see	VERB
iajs-747	233	28	rem.and.ex	rem.and.ex	PROPN
iajs-747	233	29	2.2(3	2.2(3	NUM
iajs-747	233	30	)	)	PUNCT
iajs-747	233	31	)	)	PUNCT
iajs-747	233	32	.	.	PUNCT
iajs-747	234	1	we	we	PRON
iajs-747	234	2	need	need	VERB
iajs-747	234	3	to	to	PART
iajs-747	234	4	recall	recall	VERB
iajs-747	234	5	that	that	SCONJ
iajs-747	234	6	an	an	DET
iajs-747	234	7	r	r	NOUN
iajs-747	234	8	-	-	PUNCT
iajs-747	234	9	module	module	NOUN
iajs-747	234	10	p	p	NOUN
iajs-747	234	11	is	be	AUX
iajs-747	234	12	projective	projective	ADJ
iajs-747	234	13	if	if	SCONJ
iajs-747	234	14	and	and	CCONJ
iajs-747	234	15	only	only	ADV
iajs-747	234	16	if	if	SCONJ
iajs-747	234	17	,	,	PUNCT
iajs-747	234	18	for	for	ADP
iajs-747	234	19	any	any	DET
iajs-747	234	20	rmodules	rmodule	NOUN
iajs-747	234	21	a	a	DET
iajs-747	234	22	,	,	PUNCT
iajs-747	234	23	b	b	NOUN
iajs-747	234	24	and	and	CCONJ
iajs-747	234	25	for	for	ADP
iajs-747	234	26	any	any	DET
iajs-747	234	27	epimorphism	epimorphism	NOUN
iajs-747	234	28	baf	baf	PROPN
iajs-747	234	29			PROPN
iajs-747	234	30	:	:	PUNCT
iajs-747	234	31	and	and	CCONJ
iajs-747	234	32	for	for	ADP
iajs-747	234	33	any	any	DET
iajs-747	234	34	homomorphism	homomorphism	NOUN
iajs-747	234	35	bpg	bpg	PROPN
iajs-747	234	36			PROPN
iajs-747	234	37	:	:	PUNCT
iajs-747	234	38	,	,	PUNCT
iajs-747	234	39	there	there	PRON
iajs-747	234	40	exists	exist	VERB
iajs-747	234	41	a	a	DET
iajs-747	234	42	homomorphism	homomorphism	PROPN
iajs-747	234	43	aph	aph	PROPN
iajs-747	234	44			NOUN
iajs-747	234	45	:	:	PUNCT
iajs-747	234	46	such	such	ADJ
iajs-747	234	47	that	that	SCONJ
iajs-747	234	48	foh	foh	PROPN
iajs-747	234	49	=	=	PROPN
iajs-747	234	50	g	g	PROPN
iajs-747	234	51	(	(	PUNCT
iajs-747	234	52	i.e	i.e	X
iajs-747	234	53	the	the	DET
iajs-747	234	54	following	follow	VERB
iajs-747	234	55	diagram	diagram	NOUN
iajs-747	234	56	is	be	AUX
iajs-747	234	57	a	a	DET
iajs-747	234	58	commutative	commutative	ADJ
iajs-747	234	59	)	)	PUNCT
iajs-747	234	60	,	,	PUNCT
iajs-747	235	1	[	[	X
iajs-747	235	2	3	3	NUM
iajs-747	235	3	,	,	PUNCT
iajs-747	235	4	p.117	p.117	VERB
iajs-747	235	5	]	]	PUNCT
iajs-747	235	6	.	.	PUNCT
iajs-747	236	1	ibn	ibn	PROPN
iajs-747	236	2	alhaitham	alhaitham	PROPN
iajs-747	236	3	j.	j.	PROPN
iajs-747	236	4	for	for	ADP
iajs-747	236	5	pure	pure	ADJ
iajs-747	236	6	&	&	CCONJ
iajs-747	236	7	appl	appl	PROPN
iajs-747	236	8	.	.	PUNCT
iajs-747	237	1	sci	sci	PROPN
iajs-747	237	2	.	.	PUNCT
iajs-747	237	3	vol.24	vol.24	NOUN
iajs-747	237	4	(	(	PUNCT
iajs-747	237	5	3	3	NUM
iajs-747	237	6	)	)	PUNCT
iajs-747	237	7	2011	2011	NUM
iajs-747	237	8	now	now	ADV
iajs-747	237	9	,	,	PUNCT
iajs-747	237	10	in	in	ADP
iajs-747	237	11	the	the	DET
iajs-747	237	12	next	next	ADJ
iajs-747	237	13	proposition	proposition	NOUN
iajs-747	237	14	we	we	PRON
iajs-747	237	15	give	give	VERB
iajs-747	237	16	a	a	DET
iajs-747	237	17	condition	condition	NOUN
iajs-747	237	18	under	under	ADP
iajs-747	237	19	which	which	PRON
iajs-747	237	20	the	the	DET
iajs-747	237	21	(	(	PUNCT
iajs-747	237	22	remark	remark	NOUN
iajs-747	237	23	2.19	2.19	NUM
iajs-747	237	24	)	)	PUNCT
iajs-747	237	25	is	be	AUX
iajs-747	237	26	true	true	ADJ
iajs-747	237	27	.	.	PUNCT
iajs-747	238	1	proposition	proposition	NOUN
iajs-747	238	2	(	(	PUNCT
iajs-747	238	3	2.21	2.21	NUM
iajs-747	238	4	)	)	PUNCT
iajs-747	238	5	let	let	VERB
iajs-747	238	6	m	m	PRON
iajs-747	238	7	be	be	AUX
iajs-747	238	8	an	an	DET
iajs-747	238	9	r	r	NOUN
iajs-747	238	10	-	-	PUNCT
iajs-747	238	11	module	module	NOUN
iajs-747	238	12	such	such	ADJ
iajs-747	238	13	that	that	DET
iajs-747	238	14	km	km	PROPN
iajs-747	238	15	is	be	AUX
iajs-747	238	16	a	a	DET
iajs-747	238	17	projective	projective	ADJ
iajs-747	238	18	r	r	NOUN
iajs-747	238	19	-	-	PUNCT
iajs-747	238	20	module	module	NOUN
iajs-747	238	21	for	for	ADP
iajs-747	238	22	all	all	DET
iajs-747	238	23	k	k	NOUN
iajs-747	238	24	≤	≤	X
iajs-747	239	1	e	e	X
iajs-747	239	2	m.	m.	NOUN
iajs-747	239	3	if	if	SCONJ
iajs-747	239	4	m	m	NOUN
iajs-747	239	5	is	be	AUX
iajs-747	239	6	an	an	DET
iajs-747	239	7	essentially	essentially	ADV
iajs-747	239	8	quasi	quasi	ADJ
iajs-747	239	9	-	-	ADJ
iajs-747	239	10	dedekind	dedekind	ADJ
iajs-747	239	11	r	r	NOUN
iajs-747	239	12	-	-	NOUN
iajs-747	239	13	module	module	NOUN
iajs-747	239	14	,	,	PUNCT
iajs-747	239	15	then	then	ADV
iajs-747	239	16	nm	nm	PRON
iajs-747	239	17	is	be	AUX
iajs-747	239	18	an	an	DET
iajs-747	239	19	essentially	essentially	ADV
iajs-747	239	20	quasi	quasi	ADJ
iajs-747	239	21	-	-	ADJ
iajs-747	239	22	dedekind	dedekind	ADJ
iajs-747	239	23	r	r	NOUN
iajs-747	239	24	-	-	PUNCT
iajs-747	239	25	module	module	NOUN
iajs-747	239	26	for	for	ADP
iajs-747	239	27	all	all	DET
iajs-747	239	28	mn	mn	PROPN
iajs-747	239	29			PROPN
iajs-747	239	30	.	.	PUNCT
iajs-747	240	1	proof	proof	NOUN
iajs-747	240	2	:	:	PUNCT
iajs-747	240	3	let	let	VERB
iajs-747	240	4	nu	nu	PRON
iajs-747	240	5	≤	≤	NUM
iajs-747	240	6	e	e	X
iajs-747	240	7	nm	nm	NOUN
iajs-747	240	8	.	.	PUNCT
iajs-747	241	1	then	then	ADV
iajs-747	241	2	u	u	X
iajs-747	241	3	≤	≤	X
iajs-747	241	4	e	e	NOUN
iajs-747	241	5	m	m	NOUN
iajs-747	241	6	and	and	CCONJ
iajs-747	241	7	hence	hence	ADV
iajs-747	241	8	by	by	ADP
iajs-747	241	9	hypothesis	hypothesis	NOUN
iajs-747	241	10	um	um	INTJ
iajs-747	241	11	is	be	AUX
iajs-747	241	12	a	a	DET
iajs-747	241	13	projective	projective	ADJ
iajs-747	241	14	r	r	NOUN
iajs-747	241	15	-	-	PUNCT
iajs-747	241	16	module	module	NOUN
iajs-747	241	17	.	.	PUNCT
iajs-747	242	1	suppose	suppose	VERB
iajs-747	242	2	that	that	SCONJ
iajs-747	242	3	there	there	PRON
iajs-747	242	4	exists	exist	VERB
iajs-747	242	5	0	0	NUM
iajs-747	242	6	,	,	PUNCT
iajs-747	242	7	)	)	PUNCT
iajs-747	242	8	,	,	PUNCT
iajs-747	242	9	(	(	PUNCT
iajs-747	242	10			NOUN
iajs-747	242	11	f	f	PROPN
iajs-747	242	12	n	n	PROPN
iajs-747	242	13	m	m	PROPN
iajs-747	242	14	nu	nu	INTJ
iajs-747	242	15	nm	nm	ADJ
iajs-747	242	16	homf	homf	NOUN
iajs-747	242	17	.	.	PUNCT
iajs-747	243	1	but	but	CCONJ
iajs-747	243	2	)	)	PUNCT
iajs-747	243	3	,	,	PUNCT
iajs-747	243	4	(	(	PUNCT
iajs-747	243	5	)	)	PUNCT
iajs-747	243	6	,	,	PUNCT
iajs-747	243	7	(	(	PUNCT
iajs-747	243	8	n	n	X
iajs-747	243	9	m	m	VERB
iajs-747	243	10	u	u	NOUN
iajs-747	243	11	m	m	NOUN
iajs-747	243	12	hom	hom	NOUN
iajs-747	244	1	n	n	ADV
iajs-747	244	2	m	m	PROPN
iajs-747	244	3	nu	nu	VERB
iajs-747	244	4	nm	nm	INTJ
iajs-747	244	5	hom	hom	NOUN
iajs-747	244	6			PROPN
iajs-747	245	1	and	and	CCONJ
iajs-747	245	2	since	since	SCONJ
iajs-747	245	3	um	um	INTJ
iajs-747	245	4	is	be	AUX
iajs-747	245	5	projective	projective	ADJ
iajs-747	245	6	,	,	PUNCT
iajs-747	245	7	so	so	CCONJ
iajs-747	245	8	there	there	PRON
iajs-747	245	9	exists	exist	VERB
iajs-747	245	10	m	m	VERB
iajs-747	245	11	u	u	NOUN
iajs-747	245	12	m	m	NOUN
iajs-747	245	13	g	g	NOUN
iajs-747	245	14			NOUN
iajs-747	245	15	:	:	PUNCT
iajs-747	245	16	such	such	ADJ
iajs-747	245	17	that	that	PRON
iajs-747	245	18	πog	πog	NOUN
iajs-747	245	19	=	=	SYM
iajs-747	245	20	f	f	PROPN
iajs-747	245	21	,	,	PUNCT
iajs-747	245	22	where	where	SCONJ
iajs-747	245	23	π	π	PROPN
iajs-747	245	24	is	be	AUX
iajs-747	245	25	the	the	DET
iajs-747	245	26	canonical	canonical	ADJ
iajs-747	245	27	projection	projection	NOUN
iajs-747	245	28	mapping	mapping	NOUN
iajs-747	245	29	.	.	PUNCT
iajs-747	246	1	since	since	SCONJ
iajs-747	246	2	0f	0f	NUM
iajs-747	246	3	then	then	ADV
iajs-747	246	4	0g	0g	NUM
iajs-747	246	5	,	,	PUNCT
iajs-747	246	6	thus	thus	ADV
iajs-747	246	7	0	0	NUM
iajs-747	246	8	)	)	PUNCT
iajs-747	246	9	,	,	PUNCT
iajs-747	246	10	(	(	PUNCT
iajs-747	246	11	m	m	PROPN
iajs-747	246	12	u	u	NOUN
iajs-747	246	13	m	m	VERB
iajs-747	246	14	hom	hom	NOUN
iajs-747	246	15	,	,	PUNCT
iajs-747	246	16	u	u	NOUN
iajs-747	246	17	≤	≤	X
iajs-747	246	18	e	e	NOUN
iajs-747	246	19	m	m	NOUN
iajs-747	246	20	;	;	PUNCT
iajs-747	246	21	that	that	PRON
iajs-747	246	22	is	is	ADV
iajs-747	246	23	m	m	VERB
iajs-747	246	24	is	be	AUX
iajs-747	246	25	not	not	PART
iajs-747	246	26	an	an	DET
iajs-747	246	27	essentially	essentially	ADV
iajs-747	246	28	quasi	quasi	ADJ
iajs-747	246	29	-	-	ADJ
iajs-747	246	30	dedekind	dedekind	ADJ
iajs-747	246	31	r	r	NOUN
iajs-747	246	32	-	-	PUNCT
iajs-747	246	33	module	module	NOUN
iajs-747	246	34	,	,	PUNCT
iajs-747	246	35	which	which	PRON
iajs-747	246	36	is	be	AUX
iajs-747	246	37	a	a	DET
iajs-747	246	38	contradiction	contradiction	NOUN
iajs-747	246	39	.	.	PUNCT
iajs-747	247	1	thus	thus	ADV
iajs-747	247	2	nm	nm	PRON
iajs-747	247	3	is	be	AUX
iajs-747	247	4	an	an	DET
iajs-747	247	5	essentially	essentially	ADV
iajs-747	247	6	quasi	quasi	ADJ
iajs-747	247	7	-	-	ADJ
iajs-747	247	8	dedekind	dedekind	ADJ
iajs-747	247	9	r	r	NOUN
iajs-747	247	10	-	-	PUNCT
iajs-747	247	11	module	module	NOUN
iajs-747	247	12	for	for	ADP
iajs-747	247	13	all	all	DET
iajs-747	247	14	mn	mn	PROPN
iajs-747	247	15			PROPN
iajs-747	247	16	.	.	PUNCT
iajs-747	248	1	a	a	DET
iajs-747	248	2	b	b	X
iajs-747	248	3	p	p	NOUN
iajs-747	248	4	0	0	NUM
iajs-747	249	1	f	f	NOUN
iajs-747	250	1	h	h	NOUN
iajs-747	251	1	g	g	PROPN
iajs-747	251	2	m	m	PROPN
iajs-747	251	3	0	0	NUM
iajs-747	252	1	π	π	NOUN
iajs-747	252	2	g	g	PROPN
iajs-747	252	3	f	f	PROPN
iajs-747	252	4	u	u	PROPN
iajs-747	252	5	m	m	VERB
iajs-747	252	6	n	n	NUM
iajs-747	252	7	m	m	PROPN
iajs-747	252	8	ibn	ibn	PROPN
iajs-747	252	9	alhaitham	alhaitham	NOUN
iajs-747	252	10	j.	j.	PROPN
iajs-747	252	11	for	for	ADP
iajs-747	252	12	pure	pure	ADJ
iajs-747	252	13	&	&	CCONJ
iajs-747	252	14	appl	appl	PROPN
iajs-747	252	15	.	.	PUNCT
iajs-747	253	1	sci	sci	PROPN
iajs-747	253	2	.	.	PUNCT
iajs-747	253	3	vol.24	vol.24	NOUN
iajs-747	253	4	(	(	PUNCT
iajs-747	253	5	3	3	NUM
iajs-747	253	6	)	)	PUNCT
iajs-747	253	7	2011	2011	NUM
iajs-747	253	8	references	reference	NOUN
iajs-747	253	9	1	1	NUM
iajs-747	253	10	.	.	PUNCT
iajs-747	253	11	atiyah	atiyah	PROPN
iajs-747	253	12	,	,	PUNCT
iajs-747	253	13	m	m	VERB
iajs-747	253	14	.f	.f	NOUN
iajs-747	253	15	.	.	PUNCT
iajs-747	254	1	and	and	CCONJ
iajs-747	254	2	macdonald	macdonald	PROPN
iajs-747	254	3	,	,	PUNCT
iajs-747	254	4	i.g	i.g	PROPN
iajs-747	254	5	.	.	PROPN
iajs-747	254	6	(	(	PUNCT
iajs-747	254	7	1969	1969	NUM
iajs-747	254	8	)	)	PUNCT
iajs-747	254	9	"	"	PUNCT
iajs-747	254	10	introduction	introduction	NOUN
iajs-747	254	11	to	to	ADP
iajs-747	254	12	commutative	commutative	ADJ
iajs-747	254	13	algebra	algebra	NOUN
iajs-747	254	14	"	"	PUNCT
iajs-747	254	15	,	,	PUNCT
iajs-747	254	16	university	university	NOUN
iajs-747	254	17	of	of	ADP
iajs-747	254	18	oxford	oxford	PROPN
iajs-747	254	19	.	.	PUNCT
iajs-747	255	1	2	2	X
iajs-747	255	2	.	.	X
iajs-747	255	3	goodearl	goodearl	PROPN
iajs-747	255	4	,	,	PUNCT
iajs-747	255	5	k.r	k.r	PROPN
iajs-747	255	6	.	.	PROPN
iajs-747	256	1	(	(	PUNCT
iajs-747	256	2	1976	1976	NUM
iajs-747	256	3	)	)	PUNCT
iajs-747	256	4	"	"	PUNCT
iajs-747	256	5	ring	ring	NOUN
iajs-747	256	6	theory	theory	NOUN
iajs-747	256	7	"	"	PUNCT
iajs-747	256	8	maracel	maracel	PROPN
iajs-747	256	9	dekker	dekker	PROPN
iajs-747	256	10	,	,	PUNCT
iajs-747	256	11	newyork	newyork	PROPN
iajs-747	256	12	.	.	PUNCT
iajs-747	257	1	3	3	X
iajs-747	257	2	.	.	X
iajs-747	257	3	kasch	kasch	PROPN
iajs-747	257	4	,	,	PUNCT
iajs-747	257	5	f.	f.	PROPN
iajs-747	257	6	(	(	PUNCT
iajs-747	257	7	1982	1982	NUM
iajs-747	257	8	)	)	PUNCT
iajs-747	257	9	"	"	PUNCT
iajs-747	257	10	m	m	NOUN
iajs-747	257	11	odules	odule	NOUN
iajs-747	257	12	and	and	CCONJ
iajs-747	257	13	rings	ring	NOUN
iajs-747	257	14	"	"	PUNCT
iajs-747	257	15	,	,	PUNCT
iajs-747	257	16	academic	academic	ADJ
iajs-747	257	17	press	press	NOUN
iajs-747	257	18	,	,	PUNCT
iajs-747	257	19	london	london	PROPN
iajs-747	257	20	.	.	PUNCT
iajs-747	258	1	4	4	X
iajs-747	258	2	.	.	X
iajs-747	258	3	larsen	larsen	PROPN
iajs-747	258	4	,	,	PUNCT
iajs-747	258	5	m	m	VERB
iajs-747	258	6	.d	.d	ADJ
iajs-747	258	7	.	.	PUNCT
iajs-747	259	1	and	and	CCONJ
iajs-747	259	2	mc	mc	PROPN
iajs-747	259	3	carthy	carthy	PROPN
iajs-747	259	4	,	,	PUNCT
iajs-747	259	5	p.	p.	PROPN
iajs-747	259	6	j.	j.	PROPN
iajs-747	260	1	(	(	PUNCT
iajs-747	260	2	1971	1971	NUM
iajs-747	260	3	)	)	PUNCT
iajs-747	260	4	"	"	PUNCT
iajs-747	260	5	multiplication	multiplication	NOUN
iajs-747	260	6	theory	theory	NOUN
iajs-747	260	7	of	of	ADP
iajs-747	260	8	ideals	ideal	NOUN
iajs-747	260	9	"	"	PUNCT
iajs-747	260	10	,	,	PUNCT
iajs-747	260	11	academic	academic	ADJ
iajs-747	260	12	press	press	NOUN
iajs-747	260	13	newyork	newyork	NOUN
iajs-747	260	14	and	and	CCONJ
iajs-747	260	15	london	london	PROPN
iajs-747	260	16	.	.	PUNCT
iajs-747	261	1	5	5	X
iajs-747	261	2	.	.	X
iajs-747	261	3	mijbass	mijbass	PROPN
iajs-747	261	4	,	,	PUNCT
iajs-747	261	5	a	a	DET
iajs-747	261	6	.s	.s	NOUN
iajs-747	261	7	.	.	PUNCT
iajs-747	262	1	(	(	PUNCT
iajs-747	262	2	1997	1997	NUM
iajs-747	262	3	)	)	PUNCT
iajs-747	262	4	"	"	PUNCT
iajs-747	262	5	quasi	quasi	X
iajs-747	262	6	–	–	PUNCT
iajs-747	262	7	dedekind	dedekind	ADJ
iajs-747	262	8	modules	module	NOUN
iajs-747	262	9	"	"	PUNCT
iajs-747	262	10	,	,	PUNCT
iajs-747	262	11	ph	ph	PROPN
iajs-747	262	12	.	.	PROPN
iajs-747	262	13	d	d	NOUN
iajs-747	262	14	.thesis	.thesis	NOUN
iajs-747	262	15	,	,	PUNCT
iajs-747	262	16	college	college	NOUN
iajs-747	262	17	of	of	ADP
iajs-747	262	18	science	science	NOUN
iajs-747	262	19	,	,	PUNCT
iajs-747	262	20	university	university	NOUN
iajs-747	262	21	of	of	ADP
iajs-747	262	22	baghdad	baghdad	PROPN
iajs-747	262	23	.	.	PUNCT
iajs-747	263	1	6	6	X
iajs-747	263	2	.	.	X
iajs-747	263	3	naoum	naoum	PROPN
iajs-747	263	4	,	,	PUNCT
iajs-747	263	5	a	a	DET
iajs-747	263	6	.g	.g	NOUN
iajs-747	263	7	.	.	PUNCT
iajs-747	264	1	and	and	CCONJ
iajs-747	264	2	hadi	hadi	PROPN
iajs-747	264	3	,	,	PUNCT
iajs-747	264	4	i.	i.	PROPN
iajs-747	264	5	m	m	PROPN
iajs-747	264	6	-	-	PROPN
iajs-747	264	7	a	a	PRON
iajs-747	264	8	.(2002	.(2002	NOUN
iajs-747	264	9	)	)	PUNCT
iajs-747	264	10	"	"	PUNCT
iajs-747	264	11	sqi	sqi	NOUN
iajs-747	264	12	submodules	submodule	NOUN
iajs-747	264	13	and	and	CCONJ
iajs-747	264	14	sqd	sqd	PROPN
iajs-747	264	15	modules	module	NOUN
iajs-747	264	16	"	"	PUNCT
iajs-747	264	17	,	,	PUNCT
iajs-747	264	18	iraqi	iraqi	ADJ
iajs-747	264	19	j.	j.	PROPN
iajs-747	264	20	sci	sci	PROPN
iajs-747	264	21	,	,	PUNCT
iajs-747	264	22	1	1	NUM
iajs-747	264	23	.43.d	.43.d	PROPN
iajs-747	264	24	(	(	PUNCT
iajs-747	264	25	2	2	NUM
iajs-747	264	26	):	):	PUNCT
iajs-747	264	27	43	43	NUM
iajs-747	264	28	–	–	SYM
iajs-747	264	29	54	54	NUM
iajs-747	264	30	.	.	PUNCT
iajs-747	265	1	7	7	X
iajs-747	265	2	.	.	X
iajs-747	266	1	rizvi	rizvi	PROPN
iajs-747	266	2	,	,	PUNCT
iajs-747	266	3	s.t	s.t	PROPN
iajs-747	266	4	.	.	PROPN
iajs-747	266	5	and	and	CCONJ
iajs-747	266	6	roman	roman	PROPN
iajs-747	266	7	,	,	PUNCT
iajs-747	266	8	c.s	c.s	PROPN
iajs-747	266	9	.	.	PROPN
iajs-747	266	10	(	(	PUNCT
iajs-747	266	11	2007	2007	NUM
iajs-747	266	12	)	)	PUNCT
iajs-747	266	13	"	"	PUNCT
iajs-747	266	14	on	on	ADP
iajs-747	266	15	knonsingular	knonsingular	ADJ
iajs-747	266	16	modules	module	NOUN
iajs-747	266	17	and	and	CCONJ
iajs-747	266	18	applications	application	NOUN
iajs-747	266	19	"	"	PUNCT
iajs-747	266	20	,	,	PUNCT
iajs-747	266	21	comm	comm	NOUN
iajs-747	266	22	.	.	PUNCT
iajs-747	267	1	in	in	ADP
iajs-747	267	2	algebra	algebra	PROPN
iajs-747	267	3	,	,	PUNCT
iajs-747	267	4	no.35	no.35	PROPN
iajs-747	267	5	:	:	PUNCT
iajs-747	267	6	2960	2960	NUM
iajs-747	267	7	–	–	PUNCT
iajs-747	267	8	2982	2982	NUM
iajs-747	267	9	.	.	PUNCT
iajs-747	267	10	8	8	X
iajs-747	267	11	.	.	X
iajs-747	267	12	roman	roman	PROPN
iajs-747	267	13	,	,	PUNCT
iajs-747	267	14	c.	c.	PROPN
iajs-747	267	15	s	s	PROPN
iajs-747	267	16	.	.	PUNCT
iajs-747	268	1	(	(	PUNCT
iajs-747	268	2	2004	2004	NUM
iajs-747	268	3	)	)	PUNCT
iajs-747	268	4	"	"	PUNCT
iajs-747	269	1	baer	baer	PROPN
iajs-747	269	2	and	and	CCONJ
iajs-747	269	3	quasi	quasi	PROPN
iajs-747	269	4	baer	baer	PROPN
iajs-747	269	5	modules	modules	PROPN
iajs-747	269	6	"	"	PUNCT
iajs-747	269	7	,	,	PUNCT
iajs-747	269	8	ph	ph	NOUN
iajs-747	269	9	.	.	PUNCT
iajs-747	270	1	d	d	NOUN
iajs-747	270	2	.thesis	.thesis	NOUN
iajs-747	270	3	,	,	PUNCT
iajs-747	270	4	graduate	graduate	NOUN
iajs-747	270	5	,	,	PUNCT
iajs-747	270	6	school	school	NOUN
iajs-747	270	7	of	of	ADP
iajs-747	270	8	ohio	ohio	PROPN
iajs-747	270	9	,	,	PUNCT
iajs-747	270	10	state	state	NOUN
iajs-747	270	11	university	university	NOUN
iajs-747	270	12	.	.	PUNCT
iajs-747	271	1	2011	2011	NUM
iajs-747	271	2	)	)	PUNCT
iajs-747	272	1	3	3	NUM
iajs-747	272	2	(	(	PUNCT
iajs-747	272	3	24بن	24بن	ADJ
iajs-747	272	4	الھیثم	الھیثم	NOUN
iajs-747	272	5	للعلوم	للعلوم	NOUN
iajs-747	272	6	الصرفة	الصرفة	PROPN
iajs-747	272	7	والتطبیقیة	والتطبیقیة	PROPN
iajs-747	272	8	المجلدمجلة	المجلدمجلة	VERB
iajs-747	272	9	ا	ا	PROPN
iajs-747	272	10	الواسعة	الواسعة	PROPN
iajs-747	272	11	معكوسة	معكوسة	NOUN
iajs-747	272	12	-	-	PUNCT
iajs-747	272	13	المقاسات	المقاسات	PROPN
iajs-747	272	14	الجزئیة	الجزئیة	NOUN
iajs-747	272	15	شبه	شبه	VERB
iajs-747	272	16	الواسعةدیدیكاندیة	الواسعةدیدیكاندیة	NOUN
iajs-747	272	17	-و	-و	PART
iajs-747	272	18	المقاسات	المقاسات	PROPN
iajs-747	272	19	شبه	شبه	VERB
iajs-747	272	20	ثائر	ثائر	NOUN
iajs-747	272	21	یونس	یونس	NOUN
iajs-747	272	22	غاوي	غاوي	PROPN
iajs-747	272	23	،	،	X
iajs-747	272	24	نعام	نعام	VERB
iajs-747	272	25	محمد	محمد	PROPN
iajs-747	272	26	عليأ	عليأ	NOUN
iajs-747	272	27	جامعة	جامعة	NOUN
iajs-747	272	28	بغداد	بغداد	PROPN
iajs-747	272	29	كلیة	كلیة	PROPN
iajs-747	272	30	التربیة	التربیة	NOUN
iajs-747	272	31	أبن	أبن	NOUN
iajs-747	272	32	الهیثم	الهیثم	VERB
iajs-747	272	33	،	،	X
iajs-747	272	34	لریاضیاتقسم	لریاضیاتقسم	PROPN
iajs-747	272	35	ا	ا	X
iajs-747	272	36	جامعة	جامعة	PROPN
iajs-747	272	37	القادســیة	القادســیة	PROPN
iajs-747	272	38	،	،	NOUN
iajs-747	272	39	كلیة	كلیة	PROPN
iajs-747	272	40	التربیــة	التربیــة	PROPN
iajs-747	272	41	،	،	NOUN
iajs-747	272	42	قسم	قسم	PROPN
iajs-747	272	43	الّریاضیات	الّریاضیات	NOUN
iajs-747	272	44	2011حزیران	2011حزیران	NUM
iajs-747	272	45	6	6	NUM
iajs-747	272	46	:	:	PUNCT
iajs-747	272	47	استلم	استلم	PROPN
iajs-747	272	48	البحث	البحث	VERB
iajs-747	272	49	في	في	ADP
iajs-747	272	50	2011	2011	NUM
iajs-747	272	51	شباط	شباط	ADJ
iajs-747	272	52	8	8	NUM
iajs-747	272	53	:	:	PUNCT
iajs-747	272	54	قبل	قبل	NOUN
iajs-747	272	55	البحث	البحث	VERB
iajs-747	272	56	في	في	ADP
iajs-747	272	57	ةصالخال	ةصالخال	ADV
iajs-747	272	58	ـة	ـة	AUX
iajs-747	272	59	ذا	ذا	PROPN
iajs-747	272	60	عنصـر	عنصـر	NOUN
iajs-747	272	61	محایـد	محایـد	NOUN
iajs-747	272	62	rلـتكن	rلـتكن	NOUN
iajs-747	272	63	ـة	ـة	AUX
iajs-747	272	64	المقاسـات	المقاسـات	PROPN
iajs-747	272	65	يمفهـوم	يمفهـوم	PROPN
iajs-747	272	66	االبحـث	االبحـث	VERB
iajs-747	272	67	درســنفــي	درســنفــي	NOUN
iajs-747	272	68	هـذا	هـذا	NOUN
iajs-747	272	69	.	.	PUNCT
iajs-747	273	1	حلقـة	حلقـة	PROPN
iajs-747	273	2	أبدالیـ	أبدالیـ	PROPN
iajs-747	273	3	الواســعة	الواســعة	PROPN
iajs-747	273	4	معكوسـة	معكوسـة	PROPN
iajs-747	273	5	-	-	PUNCT
iajs-747	273	6	شـبه	شـبه	PROPN
iajs-747	273	7	الجزئیـ	الجزئیـ	NOUN
iajs-747	273	8	ومـن	ومـن	NOUN
iajs-747	273	9	بـین	بـین	NOUN
iajs-747	273	10	.	.	PUNCT
iajs-747	274	1	دیدیكاندیـة	دیدیكاندیـة	ADJ
iajs-747	274	2	-شـبه	-شـبه	PROPN
iajs-747	274	3	و	و	PRON
iajs-747	274	4	المقاسـات	المقاسـات	PROPN
iajs-747	274	5	معكوسـة	معكوسـة	PROPN
iajs-747	274	6	-	-	PUNCT
iajs-747	274	7	شـبهالجزئیـة	شـبهالجزئیـة	VERB
iajs-747	274	8	أعمام	أعمام	X
iajs-747	274	9	إلـى	إلـى	NOUN
iajs-747	274	10	المقاسـات	المقاسـات	PROPN
iajs-747	274	11	الواسعةدیدیكاندیة	الواسعةدیدیكاندیة	NOUN
iajs-747	274	12	-شبه	-شبه	PUNCT
iajs-747	274	13	اسات	اسات	PROPN
iajs-747	274	14	والمق	والمق	NOUN
iajs-747	274	15	مقاس	مقاس	PROPN
iajs-747	274	16	غیر	غیر	PROPN
iajs-747	274	17	منفرد	منفرد	PROPN
iajs-747	275	1	من	من	PRON
iajs-747	275	2	النمط	النمط	PROPN
iajs-747	275	3	m	m	PROPN
iajs-747	275	4	اذا	اذا	PROPN
iajs-747	275	5	كان	كان	PROPN
iajs-747	275	6	واسع	واسع	PROPN
iajs-747	275	7	دیدیكاندي	دیدیكاندي	PROPN
iajs-747	275	8	-مقاس	-مقاس	PROPN
iajs-747	275	9	شبه	شبه	VERB
iajs-747	275	10	m	m	NOUN
iajs-747	275	11	"	"	PUNCT
iajs-747	275	12	یة	یة	ADV
iajs-747	275	13	تالنتائج	تالنتائج	PROPN
iajs-747	275	14	التي	التي	PROPN
iajs-747	275	15	حصلنا	حصلنا	PROPN
iajs-747	275	16	علیها	علیها	PROPN
iajs-747	275	17	النتیجة	النتیجة	ADJ
iajs-747	275	18	اال	اال	PROPN
iajs-747	275	19	k	k	X
iajs-747	275	20	"	"	PUNCT
iajs-747	275	21	،	،	X
iajs-747	276	1	المقاس	المقاس	NOUN
iajs-747	276	2	اذm	اذm	PROPN
iajs-747	276	3	لـنمطهو	لـنمطهو	PROPN
iajs-747	276	4	مقاس	مقاس	PROPN
iajs-747	276	5	غیر	غیر	PROPN
iajs-747	276	6	منفرد	منفرد	PROPN
iajs-747	276	7	مـن	مـن	PROPN
iajs-747	276	8	ا	ا	X
iajs-747	277	1	k	k	PROPN
iajs-747	277	2	تشـاكل	تشـاكل	PROPN
iajs-747	277	3	ذا	ذا	PROPN
iajs-747	277	4	كـان	كـان	PROPN
iajs-747	278	1	لكـل	لكـل	PROPN
iajs-747	278	2	اf	اf	PROPN
iajs-747	278	3	مـنm	مـنm	PROPN
iajs-747	278	4	إلـىm	إلـىm	PROPN
iajs-747	278	5	علـى	علـى	PROPN
iajs-747	278	6	الحلقـةr	الحلقـةr	NOUN
iajs-747	278	7	.	.	PUNCT
iajs-747	279	1	f	f	X
iajs-747	279	2	=	=	SYM
iajs-747	279	3	0یؤدي	0یؤدي	NUM
iajs-747	279	4	إلى	إلى	NOUN
iajs-747	279	5	أن	أن	ADP
iajs-747	279	6	kerf	kerf	NOUN
iajs-747	279	7	≤e	≤e	NOUN
iajs-747	279	8	mبحیث	mبحیث	NOUN