id	sid	tid	token	lemma	pos
iajs-725	1	1	ibn	ibn	PROPN
iajs-725	1	2	alhaitham	alhaitham	NOUN
iajs-725	1	3	j.	j.	PROPN
iajs-725	1	4	for	for	ADP
iajs-725	1	5	pure	pure	ADJ
iajs-725	1	6	&	&	CCONJ
iajs-725	1	7	appl	appl	PROPN
iajs-725	1	8	.	.	PUNCT
iajs-725	2	1	sci	sci	PROPN
iajs-725	2	2	.	.	PUNCT
iajs-725	2	3	vol.24	vol.24	NOUN
iajs-725	2	4	(	(	PUNCT
iajs-725	2	5	3	3	NUM
iajs-725	2	6	)	)	PUNCT
iajs-725	2	7	2011	2011	NUM
iajs-725	2	8	annsemimaximal	annsemimaximal	ADJ
iajs-725	2	9	and	and	CCONJ
iajs-725	2	10	coannsemimaximal	coannsemimaximal	ADJ
iajs-725	2	11	modules	module	NOUN
iajs-725	2	12	i.	i.	PROPN
iajs-725	2	13	m.a.hadi	m.a.hadi	PROPN
iajs-725	2	14	,	,	PUNCT
iajs-725	2	15	h.	h.	PROPN
iajs-725	2	16	y.	y.	PROPN
iajs-725	2	17	khalaf	khalaf	PROPN
iajs-725	2	18	department	department	PROPN
iajs-725	2	19	of	of	ADP
iajs-725	2	20	mathematics	mathematics	PROPN
iajs-725	2	21	,	,	PUNCT
iajs-725	2	22	college	college	NOUN
iajs-725	2	23	of	of	ADP
iajs-725	2	24	education	education	PROPN
iajs-725	2	25	ibn	ibn	PROPN
iajs-725	2	26	-	-	PUNCT
iajs-725	2	27	al	al	PROPN
iajs-725	2	28	-	-	PUNCT
iajs-725	2	29	haitham	haitham	PROPN
iajs-725	2	30	university	university	PROPN
iajs-725	2	31	of	of	ADP
iajs-725	2	32	baghdad	baghdad	PROPN
iajs-725	2	33	received	receive	VERB
iajs-725	2	34	in	in	ADP
iajs-725	2	35	:	:	PUNCT
iajs-725	2	36	20	20	NUM
iajs-725	2	37	september	september	PROPN
iajs-725	2	38	2010	2010	NUM
iajs-725	2	39	accepted	accept	VERB
iajs-725	2	40	in	in	ADP
iajs-725	2	41	:	:	PUNCT
iajs-725	2	42	8	8	NUM
iajs-725	2	43	february	february	NOUN
iajs-725	2	44	2011	2011	NUM
iajs-725	2	45	abstract	abstract	ADV
iajs-725	2	46	some	some	DET
iajs-725	2	47	authors	author	NOUN
iajs-725	2	48	studied	study	VERB
iajs-725	2	49	modules	module	NOUN
iajs-725	2	50	with	with	ADP
iajs-725	2	51	annihilator	annihilator	NOUN
iajs-725	2	52	of	of	ADP
iajs-725	2	53	every	every	DET
iajs-725	2	54	nonzero	nonzero	PROPN
iajs-725	2	55	submodule	submodule	PROPN
iajs-725	2	56	is	be	AUX
iajs-725	2	57	p	p	NOUN
iajs-725	2	58	rime	rime	NOUN
iajs-725	2	59	,	,	PUNCT
iajs-725	2	60	primary	primary	ADJ
iajs-725	2	61	or	or	CCONJ
iajs-725	2	62	maximal	maximal	ADJ
iajs-725	2	63	.	.	PUNCT
iajs-725	3	1	in	in	ADP
iajs-725	3	2	this	this	DET
iajs-725	3	3	paper	paper	NOUN
iajs-725	3	4	,	,	PUNCT
iajs-725	3	5	we	we	PRON
iajs-725	3	6	introduce	introduce	VERB
iajs-725	3	7	and	and	CCONJ
iajs-725	3	8	study	study	VERB
iajs-725	3	9	annsemimaximal	annsemimaximal	ADJ
iajs-725	3	10	and	and	CCONJ
iajs-725	3	11	coannsemimaximal	coannsemimaximal	ADJ
iajs-725	3	12	modules	module	NOUN
iajs-725	3	13	,	,	PUNCT
iajs-725	3	14	where	where	SCONJ
iajs-725	3	15	an	an	DET
iajs-725	3	16	r	r	NOUN
iajs-725	3	17	-	-	PUNCT
iajs-725	3	18	module	module	NOUN
iajs-725	3	19	m	m	NOUN
iajs-725	3	20	is	be	AUX
iajs-725	3	21	called	call	VERB
iajs-725	3	22	annsemimaximal	annsemimaximal	ADJ
iajs-725	3	23	(	(	PUNCT
iajs-725	3	24	resp	resp	NOUN
iajs-725	3	25	.	.	PUNCT
iajs-725	4	1	coannsemimaximal	coannsemimaximal	ADJ
iajs-725	4	2	)	)	PUNCT
iajs-725	4	3	if	if	SCONJ
iajs-725	4	4	annrn	annrn	NOUN
iajs-725	4	5	(	(	PUNCT
iajs-725	4	6	resp	resp	NOUN
iajs-725	4	7	.	.	PUNCT
iajs-725	5	1	r	r	NOUN
iajs-725	5	2	m	m	VERB
iajs-725	5	3	ann	ann	PROPN
iajs-725	5	4	n	n	PROPN
iajs-725	5	5	)	)	PUNCT
iajs-725	5	6	is	be	AUX
iajs-725	5	7	semimaximal	semimaximal	ADJ
iajs-725	5	8	ideal	ideal	NOUN
iajs-725	5	9	of	of	ADP
iajs-725	5	10	r	r	NOUN
iajs-725	5	11	for	for	ADP
iajs-725	5	12	each	each	DET
iajs-725	5	13	nonzero	nonzero	PROPN
iajs-725	5	14	submodule	submodule	PROPN
iajs-725	5	15	n	n	PROPN
iajs-725	5	16	of	of	ADP
iajs-725	5	17	m	m	PROPN
iajs-725	5	18	.	.	PUNCT
iajs-725	6	1	keywords	keyword	NOUN
iajs-725	6	2	:	:	PUNCT
iajs-725	6	3	annsemimaximal	annsemimaximal	ADJ
iajs-725	6	4	module	module	NOUN
iajs-725	6	5	,	,	PUNCT
iajs-725	6	6	semisimple	semisimple	NOUN
iajs-725	6	7	module	module	NOUN
iajs-725	6	8	,	,	PUNCT
iajs-725	6	9	semisimple	semisimple	NOUN
iajs-725	6	10	ring	ring	NOUN
iajs-725	6	11	,	,	PUNCT
iajs-725	6	12	semiprime	semiprime	NOUN
iajs-725	6	13	module	module	NOUN
iajs-725	6	14	,	,	PUNCT
iajs-725	6	15	max	max	NOUN
iajs-725	6	16	-	-	PUNCT
iajs-725	6	17	module	module	NOUN
iajs-725	6	18	,	,	PUNCT
iajs-725	6	19	uniform	uniform	NOUN
iajs-725	6	20	module	module	NOUN
iajs-725	6	21	,	,	PUNCT
iajs-725	6	22	z	z	NOUN
iajs-725	6	23	-	-	PUNCT
iajs-725	6	24	regular	regular	ADJ
iajs-725	6	25	module	module	NOUN
iajs-725	6	26	,	,	PUNCT
iajs-725	6	27	f	f	X
iajs-725	6	28	-	-	PUNCT
iajs-725	6	29	regular	regular	ADJ
iajs-725	6	30	module	module	NOUN
iajs-725	6	31	,	,	PUNCT
iajs-725	6	32	artinian	artinian	ADJ
iajs-725	6	33	module	module	NOUN
iajs-725	6	34	,	,	PUNCT
iajs-725	6	35	flat	flat	ADJ
iajs-725	6	36	module	module	NOUN
iajs-725	6	37	,	,	PUNCT
iajs-725	6	38	coprime	coprime	NOUN
iajs-725	6	39	module	module	NOUN
iajs-725	6	40	,	,	PUNCT
iajs-725	6	41	coannsemimaximal	coannsemimaximal	ADJ
iajs-725	6	42	module	module	NOUN
iajs-725	6	43	.	.	PUNCT
iajs-725	7	1	introduction	introduction	NOUN
iajs-725	7	2	let	let	VERB
iajs-725	7	3	r	r	PRON
iajs-725	7	4	be	be	AUX
iajs-725	7	5	a	a	DET
iajs-725	7	6	commutative	commutative	ADJ
iajs-725	7	7	ring	ring	NOUN
iajs-725	7	8	with	with	ADP
iajs-725	7	9	unity	unity	NOUN
iajs-725	7	10	and	and	CCONJ
iajs-725	7	11	let	let	VERB
iajs-725	7	12	m	m	PRON
iajs-725	7	13	be	be	AUX
iajs-725	7	14	an	an	DET
iajs-725	7	15	r	r	NOUN
iajs-725	7	16	-	-	PUNCT
iajs-725	7	17	module	module	NOUN
iajs-725	7	18	.	.	PUNCT
iajs-725	8	1	muntaha	muntaha	PROPN
iajs-725	8	2	a.r.h	a.r.h	PROPN
iajs-725	8	3	.	.	PUNCT
iajs-725	9	1	in	in	ADP
iajs-725	9	2	[	[	X
iajs-725	9	3	1	1	X
iajs-725	9	4	]	]	PUNCT
iajs-725	9	5	introduced	introduce	VERB
iajs-725	9	6	and	and	CCONJ
iajs-725	9	7	studied	study	VERB
iajs-725	9	8	quasi	quasi	ADJ
iajs-725	9	9	-	-	ADJ
iajs-725	9	10	prime	prime	ADJ
iajs-725	9	11	modules	module	NOUN
iajs-725	9	12	where	where	SCONJ
iajs-725	9	13	an	an	DET
iajs-725	9	14	r	r	NOUN
iajs-725	9	15	-	-	PUNCT
iajs-725	9	16	module	module	NOUN
iajs-725	9	17	m	m	NOUN
iajs-725	9	18	is	be	AUX
iajs-725	9	19	quasi	quasi	ADJ
iajs-725	9	20	-	-	ADJ
iajs-725	9	21	prime	prime	ADJ
iajs-725	9	22	if	if	SCONJ
iajs-725	9	23	annrn	annrn	NOUN
iajs-725	9	24	is	be	AUX
iajs-725	9	25	a	a	DET
iajs-725	9	26	prime	prime	ADJ
iajs-725	9	27	ideal	ideal	NOUN
iajs-725	9	28	of	of	ADP
iajs-725	9	29	r	r	NOUN
iajs-725	9	30	for	for	ADP
iajs-725	9	31	every	every	DET
iajs-725	9	32	nonzero	nonzero	PROPN
iajs-725	9	33	submodule	submodule	PROPN
iajs-725	9	34	n	n	PROPN
iajs-725	9	35	of	of	ADP
iajs-725	9	36	m.	m.	NOUN
iajs-725	9	37	adwia	adwia	PROPN
iajs-725	9	38	j.a.a	j.a.a	PROPN
iajs-725	9	39	.	.	PUNCT
iajs-725	10	1	in	in	ADP
iajs-725	10	2	[	[	X
iajs-725	10	3	2	2	NUM
iajs-725	10	4	]	]	PUNCT
iajs-725	10	5	introduced	introduce	VERB
iajs-725	10	6	and	and	CCONJ
iajs-725	10	7	studied	study	VERB
iajs-725	10	8	quasi	quasi	ADJ
iajs-725	10	9	-	-	ADJ
iajs-725	10	10	primary	primary	ADJ
iajs-725	10	11	modules	module	NOUN
iajs-725	10	12	,	,	PUNCT
iajs-725	10	13	where	where	SCONJ
iajs-725	10	14	an	an	DET
iajs-725	10	15	r	r	NOUN
iajs-725	10	16	-	-	PUNCT
iajs-725	10	17	module	module	NOUN
iajs-725	10	18	m	m	NOUN
iajs-725	10	19	is	be	AUX
iajs-725	10	20	called	call	VERB
iajs-725	10	21	quasi	quasi	ADJ
iajs-725	10	22	-	-	NOUN
iajs-725	10	23	primary	primary	ADJ
iajs-725	10	24	if	if	SCONJ
iajs-725	10	25	annrn	annrn	NOUN
iajs-725	10	26	is	be	AUX
iajs-725	10	27	a	a	DET
iajs-725	10	28	primary	primary	ADJ
iajs-725	10	29	ideal	ideal	NOUN
iajs-725	10	30	of	of	ADP
iajs-725	10	31	r	r	NOUN
iajs-725	10	32	,	,	PUNCT
iajs-725	10	33	for	for	SCONJ
iajs-725	10	34	each	each	DET
iajs-725	10	35	nonzero	nonzero	PROPN
iajs-725	10	36	submodule	submodule	PROPN
iajs-725	10	37	n	n	PROPN
iajs-725	10	38	of	of	ADP
iajs-725	10	39	m.	m.	NOUN
iajs-725	10	40	adwia	adwia	PROPN
iajs-725	10	41	j.a.a	j.a.a	PROPN
iajs-725	10	42	.	.	PUNCT
iajs-725	11	1	in	in	ADP
iajs-725	11	2	[	[	X
iajs-725	11	3	3	3	NUM
iajs-725	11	4	]	]	PUNCT
iajs-725	11	5	introduced	introduce	VERB
iajs-725	11	6	and	and	CCONJ
iajs-725	11	7	studied	study	VERB
iajs-725	11	8	max	max	PROPN
iajs-725	11	9	modules	module	NOUN
iajs-725	11	10	,	,	PUNCT
iajs-725	11	11	where	where	SCONJ
iajs-725	11	12	an	an	DET
iajs-725	11	13	r	r	NOUN
iajs-725	11	14	-	-	PUNCT
iajs-725	11	15	module	module	NOUN
iajs-725	11	16	m	m	NOUN
iajs-725	11	17	is	be	AUX
iajs-725	11	18	said	say	VERB
iajs-725	11	19	to	to	PART
iajs-725	11	20	be	be	AUX
iajs-725	11	21	max	max	PROPN
iajs-725	11	22	module	module	NOUN
iajs-725	11	23	if	if	SCONJ
iajs-725	11	24	annrn	annrn	NOUN
iajs-725	11	25	is	be	AUX
iajs-725	11	26	a	a	DET
iajs-725	11	27	maximal	maximal	ADJ
iajs-725	11	28	ideal	ideal	NOUN
iajs-725	11	29	of	of	ADP
iajs-725	11	30	r	r	NOUN
iajs-725	11	31	,	,	PUNCT
iajs-725	11	32	for	for	ADP
iajs-725	11	33	each	each	DET
iajs-725	11	34	nonzero	nonzero	PROPN
iajs-725	11	35	submodule	submodule	PROPN
iajs-725	11	36	n	n	PROPN
iajs-725	11	37	of	of	ADP
iajs-725	11	38	m.	m.	NOUN
iajs-725	11	39	1	1	NUM
iajs-725	11	40	.	.	X
iajs-725	11	41	recall	recall	VERB
iajs-725	11	42	that	that	SCONJ
iajs-725	11	43	an	an	DET
iajs-725	11	44	ideal	ideal	NOUN
iajs-725	11	45	i	i	PRON
iajs-725	11	46	of	of	ADP
iajs-725	11	47	r	r	NOUN
iajs-725	11	48	is	be	AUX
iajs-725	11	49	called	call	VERB
iajs-725	11	50	semimaximal	semimaximal	ADJ
iajs-725	11	51	if	if	SCONJ
iajs-725	11	52	i	i	PRON
iajs-725	11	53	is	be	AUX
iajs-725	11	54	an	an	DET
iajs-725	11	55	intersection	intersection	NOUN
iajs-725	11	56	of	of	ADP
iajs-725	11	57	finitely	finitely	ADV
iajs-725	11	58	many	many	ADJ
iajs-725	11	59	maximal	maximal	ADJ
iajs-725	11	60	ideals	ideal	NOUN
iajs-725	11	61	of	of	ADP
iajs-725	11	62	r	r	NOUN
iajs-725	11	63	,	,	PUNCT
iajs-725	11	64	[	[	X
iajs-725	11	65	4	4	NUM
iajs-725	11	66	]	]	PUNCT
iajs-725	11	67	.	.	PUNCT
iajs-725	12	1	2	2	X
iajs-725	12	2	.	.	X
iajs-725	12	3	in	in	ADP
iajs-725	12	4	this	this	DET
iajs-725	12	5	paper	paper	NOUN
iajs-725	12	6	,	,	PUNCT
iajs-725	12	7	we	we	PRON
iajs-725	12	8	introduced	introduce	VERB
iajs-725	12	9	and	and	CCONJ
iajs-725	12	10	studied	study	VERB
iajs-725	12	11	annsemimaximal	annsemimaximal	ADJ
iajs-725	12	12	and	and	CCONJ
iajs-725	12	13	coannsemimaximal	coannsemimaximal	ADJ
iajs-725	12	14	modules	module	NOUN
iajs-725	12	15	where	where	SCONJ
iajs-725	12	16	an	an	DET
iajs-725	12	17	r	r	NOUN
iajs-725	12	18	-	-	PUNCT
iajs-725	12	19	module	module	NOUN
iajs-725	12	20	m	m	NOUN
iajs-725	12	21	is	be	AUX
iajs-725	12	22	called	call	VERB
iajs-725	12	23	annsemimaximal	annsemimaximal	ADJ
iajs-725	12	24	(	(	PUNCT
iajs-725	12	25	resp	resp	NOUN
iajs-725	12	26	.	.	PUNCT
iajs-725	13	1	coannsemimaximal	coannsemimaximal	ADJ
iajs-725	13	2	)	)	PUNCT
iajs-725	13	3	if	if	SCONJ
iajs-725	13	4	annrn	annrn	NOUN
iajs-725	13	5	(	(	PUNCT
iajs-725	13	6	resp	resp	NOUN
iajs-725	13	7	.	.	PUNCT
iajs-725	14	1	r	r	NOUN
iajs-725	14	2	m	m	VERB
iajs-725	14	3	ann	ann	PROPN
iajs-725	14	4	n	n	PROPN
iajs-725	14	5	)	)	PUNCT
iajs-725	14	6	is	be	AUX
iajs-725	14	7	a	a	DET
iajs-725	14	8	semimaximal	semimaximal	ADJ
iajs-725	14	9	ideal	ideal	NOUN
iajs-725	14	10	of	of	ADP
iajs-725	14	11	r.	r.	PROPN
iajs-725	14	12	1annsemimaximal	1annsemimaximal	NUM
iajs-725	14	13	modules	module	NOUN
iajs-725	14	14	in	in	ADP
iajs-725	14	15	this	this	DET
iajs-725	14	16	section	section	NOUN
iajs-725	14	17	,	,	PUNCT
iajs-725	14	18	we	we	PRON
iajs-725	14	19	introduce	introduce	VERB
iajs-725	14	20	the	the	DET
iajs-725	14	21	concept	concept	NOUN
iajs-725	14	22	of	of	ADP
iajs-725	14	23	annsemimaximal	annsemimaximal	ADJ
iajs-725	14	24	modules	module	NOUN
iajs-725	14	25	.	.	PUNCT
iajs-725	15	1	we	we	PRON
iajs-725	15	2	give	give	VERB
iajs-725	15	3	some	some	DET
iajs-725	15	4	characterizations	characterization	NOUN
iajs-725	15	5	to	to	ADP
iajs-725	15	6	this	this	DET
iajs-725	15	7	concept	concept	NOUN
iajs-725	15	8	and	and	CCONJ
iajs-725	15	9	establish	establish	VERB
iajs-725	15	10	some	some	DET
iajs-725	15	11	basic	basic	ADJ
iajs-725	15	12	properties	property	NOUN
iajs-725	15	13	of	of	ADP
iajs-725	15	14	this	this	DET
iajs-725	15	15	concept	concept	NOUN
iajs-725	15	16	.	.	PUNCT
iajs-725	16	1	1.1	1.1	NUM
iajs-725	16	2	definition	definition	NOUN
iajs-725	16	3	:	:	PUNCT
iajs-725	16	4	let	let	VERB
iajs-725	16	5	m	m	PRON
iajs-725	16	6	be	be	AUX
iajs-725	16	7	an	an	DET
iajs-725	16	8	r	r	NOUN
iajs-725	16	9	-	-	PUNCT
iajs-725	16	10	module	module	NOUN
iajs-725	16	11	.	.	PUNCT
iajs-725	17	1	m	m	PROPN
iajs-725	17	2	is	be	AUX
iajs-725	17	3	called	call	VERB
iajs-725	17	4	annsemimaximal	annsemimaximal	ADJ
iajs-725	17	5	module	module	NOUN
iajs-725	17	6	if	if	SCONJ
iajs-725	17	7	annrn	annrn	NOUN
iajs-725	17	8	is	be	AUX
iajs-725	17	9	a	a	DET
iajs-725	17	10	semimaximal	semimaximal	ADJ
iajs-725	17	11	ideal	ideal	NOUN
iajs-725	17	12	of	of	ADP
iajs-725	17	13	r	r	NOUN
iajs-725	17	14	for	for	ADP
iajs-725	17	15	each	each	DET
iajs-725	17	16	non	non	ADJ
iajs-725	17	17	-	-	ADJ
iajs-725	17	18	zero	zero	NUM
iajs-725	17	19	submodule	submodule	NOUN
iajs-725	17	20	n	n	PROPN
iajs-725	17	21	of	of	ADP
iajs-725	17	22	m	m	PROPN
iajs-725	17	23	.	.	PUNCT
iajs-725	18	1	ibn	ibn	PROPN
iajs-725	18	2	alhaitham	alhaitham	PROPN
iajs-725	18	3	j.	j.	PROPN
iajs-725	18	4	for	for	ADP
iajs-725	18	5	pure	pure	ADJ
iajs-725	18	6	&	&	CCONJ
iajs-725	18	7	appl	appl	PROPN
iajs-725	18	8	.	.	PUNCT
iajs-725	19	1	sci	sci	PROPN
iajs-725	19	2	.	.	PUNCT
iajs-725	19	3	vol.24	vol.24	NOUN
iajs-725	19	4	(	(	PUNCT
iajs-725	19	5	3	3	NUM
iajs-725	19	6	)	)	PUNCT
iajs-725	19	7	2011	2011	NUM
iajs-725	19	8	1.2	1.2	NUM
iajs-725	19	9	remarks	remark	NOUN
iajs-725	19	10	and	and	CCONJ
iajs-725	19	11	examples	example	NOUN
iajs-725	19	12	:	:	PUNCT
iajs-725	19	13	(	(	PUNCT
iajs-725	19	14	1	1	X
iajs-725	19	15	)	)	PUNCT
iajs-725	19	16	p	p	NOUN
iajs-725	19	17	z	z	NOUN
iajs-725	19	18			NOUN
iajs-725	19	19	is	be	AUX
iajs-725	19	20	not	not	PART
iajs-725	19	21	annsemimaximal	annsemimaximal	ADJ
iajs-725	19	22	z	z	NOUN
iajs-725	19	23	-	-	PUNCT
iajs-725	19	24	module	module	NOUN
iajs-725	19	25	.	.	PUNCT
iajs-725	20	1	(	(	PUNCT
iajs-725	20	2	2	2	X
iajs-725	20	3	)	)	PUNCT
iajs-725	20	4	z6	z6	PROPN
iajs-725	20	5	as	as	SCONJ
iajs-725	20	6	a	a	DET
iajs-725	20	7	z	z	NOUN
iajs-725	20	8	-	-	PUNCT
iajs-725	20	9	module	module	NOUN
iajs-725	20	10	is	be	AUX
iajs-725	20	11	annsemimaximal	annsemimaximal	ADJ
iajs-725	20	12	module	module	NOUN
iajs-725	20	13	.	.	PUNCT
iajs-725	21	1	(	(	PUNCT
iajs-725	21	2	3	3	X
iajs-725	21	3	)	)	PUNCT
iajs-725	21	4	z	z	NOUN
iajs-725	21	5	as	as	ADP
iajs-725	21	6	a	a	DET
iajs-725	21	7	z	z	NOUN
iajs-725	21	8	-	-	PUNCT
iajs-725	21	9	module	module	NOUN
iajs-725	21	10	is	be	AUX
iajs-725	21	11	not	not	PART
iajs-725	21	12	annsemimaximal	annsemimaximal	ADJ
iajs-725	21	13	module	module	NOUN
iajs-725	21	14	.	.	PUNCT
iajs-725	22	1	(	(	PUNCT
iajs-725	22	2	4	4	X
iajs-725	22	3	)	)	PUNCT
iajs-725	22	4	q	q	NOUN
iajs-725	22	5	as	as	SCONJ
iajs-725	22	6	a	a	DET
iajs-725	22	7	z	z	NOUN
iajs-725	22	8	-	-	PUNCT
iajs-725	22	9	module	module	NOUN
iajs-725	22	10	is	be	AUX
iajs-725	22	11	not	not	PART
iajs-725	22	12	annsemimaximal	annsemimaximal	ADJ
iajs-725	22	13	module	module	NOUN
iajs-725	22	14	.	.	PUNCT
iajs-725	23	1	(	(	PUNCT
iajs-725	23	2	5	5	X
iajs-725	23	3	)	)	PUNCT
iajs-725	23	4	zp	zp	NOUN
iajs-725	23	5	as	as	ADP
iajs-725	23	6	a	a	DET
iajs-725	23	7	z	z	NOUN
iajs-725	23	8	-	-	PUNCT
iajs-725	23	9	module	module	NOUN
iajs-725	23	10	is	be	AUX
iajs-725	23	11	annsemimaximal	annsemimaximal	ADJ
iajs-725	23	12	module	module	NOUN
iajs-725	23	13	.	.	PUNCT
iajs-725	24	1	(	(	PUNCT
iajs-725	24	2	6	6	NUM
iajs-725	24	3	)	)	PUNCT
iajs-725	24	4	for	for	ADP
iajs-725	24	5	each	each	DET
iajs-725	24	6	nz+	nz+	NOUN
iajs-725	24	7	,	,	PUNCT
iajs-725	24	8	zzn	zzn	NOUN
iajs-725	24	9	is	be	AUX
iajs-725	24	10	not	not	PART
iajs-725	24	11	annsemimaximal	annsemimaximal	ADJ
iajs-725	24	12	z	z	NOUN
iajs-725	24	13	-	-	PUNCT
iajs-725	24	14	module	module	NOUN
iajs-725	24	15	.	.	PUNCT
iajs-725	25	1	(	(	PUNCT
iajs-725	25	2	7	7	X
iajs-725	25	3	)	)	PUNCT
iajs-725	25	4	every	every	DET
iajs-725	25	5	submodule	submodule	NOUN
iajs-725	25	6	n	n	PROPN
iajs-725	25	7	of	of	ADP
iajs-725	25	8	an	an	DET
iajs-725	25	9	r	r	NOUN
iajs-725	25	10	-	-	PUNCT
iajs-725	25	11	module	module	NOUN
iajs-725	25	12	m	m	NOUN
iajs-725	25	13	(	(	PUNCT
iajs-725	25	14	where	where	SCONJ
iajs-725	25	15	m	m	PROPN
iajs-725	25	16	is	be	AUX
iajs-725	25	17	annsemimaximal	annsemimaximal	ADJ
iajs-725	25	18	module	module	NOUN
iajs-725	25	19	)	)	PUNCT
iajs-725	25	20	is	be	AUX
iajs-725	25	21	annsemimaximal	annsemimaximal	ADJ
iajs-725	25	22	module	module	NOUN
iajs-725	25	23	.	.	PUNCT
iajs-725	26	1	proof	proof	NOUN
iajs-725	26	2	:	:	PUNCT
iajs-725	26	3	let	let	VERB
iajs-725	26	4	k	k	PRON
iajs-725	26	5	be	be	AUX
iajs-725	26	6	a	a	DET
iajs-725	26	7	nonzero	nonzero	ADJ
iajs-725	26	8	submodule	submodule	NOUN
iajs-725	26	9	of	of	ADP
iajs-725	26	10	n.	n.	PROPN
iajs-725	26	11	then	then	ADV
iajs-725	26	12	k	k	PROPN
iajs-725	26	13	be	be	AUX
iajs-725	26	14	a	a	DET
iajs-725	26	15	non	non	ADJ
iajs-725	26	16	-	-	ADJ
iajs-725	26	17	zero	zero	NUM
iajs-725	26	18	submodule	submodule	NOUN
iajs-725	26	19	of	of	ADP
iajs-725	26	20	m	m	PROPN
iajs-725	26	21	and	and	CCONJ
iajs-725	26	22	so	so	ADV
iajs-725	26	23	that	that	SCONJ
iajs-725	26	24	annrk	annrk	PROPN
iajs-725	26	25	is	be	AUX
iajs-725	26	26	semimaximal	semimaximal	ADJ
iajs-725	26	27	ideal	ideal	NOUN
iajs-725	26	28	(	(	PUNCT
iajs-725	26	29	since	since	SCONJ
iajs-725	26	30	m	m	PROPN
iajs-725	26	31	is	be	AUX
iajs-725	26	32	annsemimaximal	annsemimaximal	ADJ
iajs-725	26	33	module	module	NOUN
iajs-725	26	34	)	)	PUNCT
iajs-725	26	35	.	.	PUNCT
iajs-725	27	1	(	(	PUNCT
iajs-725	27	2	8)	8)	NUM
iajs-725	27	3	let	let	VERB
iajs-725	27	4	m	m	PRON
iajs-725	27	5	be	be	AUX
iajs-725	27	6	annsemimaximal	annsemimaximal	ADJ
iajs-725	27	7	module	module	NOUN
iajs-725	27	8	and	and	CCONJ
iajs-725	27	9	let	let	VERB
iajs-725	27	10	n	n	PRON
iajs-725	27	11			NUM
iajs-725	27	12	m.	m.	NOUN
iajs-725	27	13	then	then	ADV
iajs-725	27	14			PROPN
iajs-725	27	15			PROPN
iajs-725	27	16	is	be	AUX
iajs-725	27	17	annsemimaximal	annsemimaximal	ADJ
iajs-725	27	18	module	module	NOUN
iajs-725	27	19	.	.	PUNCT
iajs-725	28	1	proof	proof	NOUN
iajs-725	28	2	:	:	PUNCT
iajs-725	28	3	let	let	AUX
iajs-725	28	4	:mm	:mm	NOUN
iajs-725	28	5	/	/	SYM
iajs-725	28	6	n	n	PRON
iajs-725	28	7	be	be	VERB
iajs-725	28	8	the	the	DET
iajs-725	28	9	natural	natural	ADJ
iajs-725	28	10	epimorphism	epimorphism	NOUN
iajs-725	28	11	and	and	CCONJ
iajs-725	28	12	m	m	PROPN
iajs-725	28	13	is	be	AUX
iajs-725	28	14	annsemimaximal	annsemimaximal	ADJ
iajs-725	28	15	module	module	NOUN
iajs-725	28	16	.	.	PUNCT
iajs-725	29	1	then	then	ADV
iajs-725	29	2	for	for	ADP
iajs-725	29	3	each	each	DET
iajs-725	29	4	non	non	ADJ
iajs-725	29	5	-	-	ADJ
iajs-725	29	6	zero	zero	NUM
iajs-725	29	7	submodule	submodule	NOUN
iajs-725	29	8	w	w	PROPN
iajs-725	29	9	of	of	ADP
iajs-725	29	10	m	m	PRON
iajs-725	29	11	,	,	PUNCT
iajs-725	29	12	annrw	annrw	PROPN
iajs-725	29	13	is	be	AUX
iajs-725	29	14	semimaximal	semimaximal	ADJ
iajs-725	29	15	ideal	ideal	NOUN
iajs-725	29	16	of	of	ADP
iajs-725	29	17	r.	r.	PROPN
iajs-725	29	18	but	but	CCONJ
iajs-725	29	19	annrwannrw	annrwannrw	PROPN
iajs-725	29	20	/	/	SYM
iajs-725	29	21	n.	n.	NOUN
iajs-725	29	22	hence	hence	ADV
iajs-725	29	23	annrw	annrw	VERB
iajs-725	29	24	/	/	SYM
iajs-725	29	25	n	n	PRON
iajs-725	29	26	is	be	AUX
iajs-725	29	27	semimaximal	semimaximal	ADJ
iajs-725	29	28	ideal	ideal	NOUN
iajs-725	29	29	by	by	ADP
iajs-725	29	30	[	[	X
iajs-725	29	31	5,prp.(1.2.11	5,prp.(1.2.11	NUM
iajs-725	29	32	)	)	PUNCT
iajs-725	29	33	]	]	PUNCT
iajs-725	29	34	.	.	PUNCT
iajs-725	30	1	thus	thus	ADV
iajs-725	30	2			VERB
iajs-725	30	3			NOUN
iajs-725	30	4	is	be	AUX
iajs-725	30	5	annsemimaximal	annsemimaximal	ADJ
iajs-725	30	6	module	module	NOUN
iajs-725	30	7	.	.	PUNCT
iajs-725	31	1	(	(	PUNCT
iajs-725	31	2	9	9	X
iajs-725	31	3	)	)	PUNCT
iajs-725	31	4	the	the	DET
iajs-725	31	5	homomorphic	homomorphic	ADJ
iajs-725	31	6	image	image	NOUN
iajs-725	31	7	of	of	ADP
iajs-725	31	8	annsemimaximal	annsemimaximal	ADJ
iajs-725	31	9	module	module	NOUN
iajs-725	31	10	is	be	AUX
iajs-725	31	11	annsemimaximal	annsemimaximal	ADJ
iajs-725	31	12	module	module	NOUN
iajs-725	31	13	.	.	PUNCT
iajs-725	32	1	proof	proof	NOUN
iajs-725	32	2	:	:	PUNCT
iajs-725	32	3	let	let	VERB
iajs-725	32	4	f	f	X
iajs-725	32	5	:	:	PUNCT
iajs-725	32	6	mm	mm	PROPN
iajs-725	32	7	'	'	PUNCT
iajs-725	32	8	be	be	AUX
iajs-725	32	9	an	an	DET
iajs-725	32	10	epimorphism	epimorphism	NOUN
iajs-725	32	11	such	such	ADJ
iajs-725	32	12	that	that	SCONJ
iajs-725	32	13	m	m	PROPN
iajs-725	32	14	is	be	AUX
iajs-725	32	15	annsemimaximal	annsemimaximal	ADJ
iajs-725	32	16	module	module	NOUN
iajs-725	32	17	.	.	PUNCT
iajs-725	33	1	then	then	ADV
iajs-725	33	2	by	by	ADP
iajs-725	33	3	the	the	DET
iajs-725	33	4	first	first	ADJ
iajs-725	33	5	fundamental	fundamental	ADJ
iajs-725	33	6	theorem	theorem	NOUN
iajs-725	33	7	of	of	ADP
iajs-725	33	8	homomorphisim	homomorphisim	NOUN
iajs-725	33	9	,	,	PUNCT
iajs-725	33	10	m	m	PROPN
iajs-725	33	11	'	'	PART
iajs-725	33	12	ker	ker	PROPN
iajs-725	33	13	f	f	PROPN
iajs-725	33	14			PROPN
iajs-725	33	15			PROPN
iajs-725	33	16	.	.	PUNCT
iajs-725	34	1	but	but	CCONJ
iajs-725	34	2	ker	ker	PROPN
iajs-725	34	3	f	f	PROPN
iajs-725	34	4			PROPN
iajs-725	34	5	is	be	AUX
iajs-725	34	6	annsemimaximal	annsemimaximal	ADJ
iajs-725	34	7	by	by	ADP
iajs-725	34	8	(	(	PUNCT
iajs-725	34	9	8)	8)	NUM
iajs-725	34	10	.	.	NOUN
iajs-725	35	1	hence	hence	ADV
iajs-725	35	2	m	m	NOUN
iajs-725	35	3	'	'	PUNCT
iajs-725	35	4	is	be	AUX
iajs-725	35	5	annsemimaximal	annsemimaximal	ADJ
iajs-725	35	6	module	module	NOUN
iajs-725	35	7	.	.	PUNCT
iajs-725	36	1	now	now	ADV
iajs-725	36	2	,	,	PUNCT
iajs-725	36	3	we	we	PRON
iajs-725	36	4	have	have	VERB
iajs-725	36	5	the	the	DET
iajs-725	36	6	following	follow	VERB
iajs-725	36	7	characterization	characterization	NOUN
iajs-725	36	8	of	of	ADP
iajs-725	36	9	annsemimaximal	annsemimaximal	ADJ
iajs-725	36	10	module	module	NOUN
iajs-725	36	11	.	.	PUNCT
iajs-725	37	1	1.3	1.3	NUM
iajs-725	37	2	proposition	proposition	NOUN
iajs-725	37	3	:	:	PUNCT
iajs-725	37	4	let	let	VERB
iajs-725	37	5	m	m	PRON
iajs-725	37	6	be	be	AUX
iajs-725	37	7	an	an	DET
iajs-725	37	8	r	r	NOUN
iajs-725	37	9	-	-	PUNCT
iajs-725	37	10	module	module	NOUN
iajs-725	37	11	.	.	PUNCT
iajs-725	38	1	then	then	ADV
iajs-725	38	2	m	m	PROPN
iajs-725	38	3	is	be	AUX
iajs-725	38	4	annsemimaximal	annsemimaximal	ADJ
iajs-725	38	5	module	module	NOUN
iajs-725	38	6	if	if	SCONJ
iajs-725	38	7	and	and	CCONJ
iajs-725	38	8	only	only	ADV
iajs-725	38	9	if	if	SCONJ
iajs-725	38	10	annrm	annrm	NOUN
iajs-725	38	11	is	be	AUX
iajs-725	38	12	a	a	DET
iajs-725	38	13	semimaximal	semimaximal	ADJ
iajs-725	38	14	ideal	ideal	NOUN
iajs-725	38	15	of	of	ADP
iajs-725	38	16	r.	r.	PROPN
iajs-725	38	17	proof	proof	NOUN
iajs-725	38	18	:	:	PUNCT
iajs-725	38	19	(	(	PUNCT
iajs-725	38	20			NOUN
iajs-725	38	21	)	)	PUNCT
iajs-725	38	22	it	it	PRON
iajs-725	38	23	follows	follow	VERB
iajs-725	38	24	directly	directly	ADV
iajs-725	38	25	by	by	ADP
iajs-725	38	26	definition	definition	NOUN
iajs-725	38	27	(	(	PUNCT
iajs-725	38	28	1.1	1.1	NUM
iajs-725	38	29	)	)	PUNCT
iajs-725	38	30	.	.	PUNCT
iajs-725	39	1	(	(	PUNCT
iajs-725	39	2			NOUN
iajs-725	39	3	)	)	PUNCT
iajs-725	39	4	let	let	VERB
iajs-725	39	5	(	(	PUNCT
iajs-725	39	6	0	0	NUM
iajs-725	39	7	)	)	PUNCT
iajs-725	39	8			NOUN
iajs-725	39	9	n	n	CCONJ
iajs-725	39	10	be	be	VERB
iajs-725	39	11	a	a	DET
iajs-725	39	12	submodule	submodule	NOUN
iajs-725	39	13	of	of	ADP
iajs-725	39	14	m.	m.	NOUN
iajs-725	40	1	then	then	ADV
iajs-725	40	2	annrn	annrn	VERB
iajs-725	40	3			PROPN
iajs-725	40	4	annrm	annrm	NOUN
iajs-725	40	5	.	.	PUNCT
iajs-725	41	1	but	but	CCONJ
iajs-725	41	2	annrm	annrm	NOUN
iajs-725	41	3	is	be	AUX
iajs-725	41	4	semimaximal	semimaximal	ADJ
iajs-725	41	5	,	,	PUNCT
iajs-725	41	6	so	so	ADV
iajs-725	41	7	by	by	ADP
iajs-725	41	8	[	[	X
iajs-725	41	9	5,prop	5,prop	NUM
iajs-725	41	10	.	.	PUNCT
iajs-725	42	1	(	(	PUNCT
iajs-725	42	2	1.2.11	1.2.11	NUM
iajs-725	42	3	)	)	PUNCT
iajs-725	42	4	]	]	PUNCT
iajs-725	42	5	,	,	PUNCT
iajs-725	42	6	annrn	annrn	NOUN
iajs-725	42	7	issemimaximal	issemimaximal	VERB
iajs-725	42	8	.	.	PUNCT
iajs-725	43	1	thus	thus	ADV
iajs-725	43	2	m	m	PROPN
iajs-725	43	3	is	be	AUX
iajs-725	43	4	annsemimaximal	annsemimaximal	ADJ
iajs-725	43	5	module	module	NOUN
iajs-725	43	6	.	.	PUNCT
iajs-725	44	1	1.4	1.4	NUM
iajs-725	44	2	corollary	corollary	NOUN
iajs-725	44	3	:	:	PUNCT
iajs-725	44	4	an	an	DET
iajs-725	44	5	r	r	NOUN
iajs-725	44	6	-	-	PUNCT
iajs-725	44	7	module	module	NOUN
iajs-725	44	8	m	m	NOUN
iajs-725	44	9	is	be	AUX
iajs-725	44	10	annsemimaximal	annsemimaximal	ADJ
iajs-725	44	11	if	if	SCONJ
iajs-725	44	12	and	and	CCONJ
iajs-725	44	13	only	only	ADV
iajs-725	44	14	if	if	SCONJ
iajs-725	44	15	r	r	NOUN
iajs-725	44	16	/	/	SYM
iajs-725	44	17	annrm	annrm	NOUN
iajs-725	44	18	is	be	AUX
iajs-725	44	19	semisimple	semisimple	ADJ
iajs-725	44	20	ring	ring	NOUN
iajs-725	44	21	.	.	PUNCT
iajs-725	45	1	proof	proof	NOUN
iajs-725	45	2	:	:	PUNCT
iajs-725	45	3	by	by	ADP
iajs-725	45	4	proposition	proposition	NOUN
iajs-725	45	5	(	(	PUNCT
iajs-725	45	6	1.3	1.3	NUM
iajs-725	45	7	)	)	PUNCT
iajs-725	45	8	m	m	VERB
iajs-725	45	9	is	be	AUX
iajs-725	45	10	annsemimaximal	annsemimaximal	ADJ
iajs-725	45	11	module	module	NOUN
iajs-725	45	12			PROPN
iajs-725	45	13	annrm	annrm	PROPN
iajs-725	45	14	is	be	AUX
iajs-725	45	15	a	a	DET
iajs-725	45	16	semimaximal	semimaximal	ADJ
iajs-725	45	17	ideal	ideal	NOUN
iajs-725	45	18	.	.	PUNCT
iajs-725	46	1			ADP
iajs-725	46	2	r	r	NOUN
iajs-725	46	3	/	/	SYM
iajs-725	46	4	annrm	annrm	NOUN
iajs-725	46	5	is	be	AUX
iajs-725	46	6	semisimple	semisimple	ADJ
iajs-725	46	7	ring	ring	NOUN
iajs-725	46	8	.	.	PUNCT
iajs-725	47	1	now	now	ADV
iajs-725	47	2	,	,	PUNCT
iajs-725	47	3	we	we	PRON
iajs-725	47	4	have	have	VERB
iajs-725	47	5	the	the	DET
iajs-725	47	6	following	follow	VERB
iajs-725	47	7	theorem	theorem	VERB
iajs-725	47	8	.	.	PROPN
iajs-725	48	1	1.5	1.5	NUM
iajs-725	48	2	theorem	theorem	NOUN
iajs-725	48	3	:	:	PUNCT
iajs-725	48	4	let	let	VERB
iajs-725	48	5	m	m	PRON
iajs-725	48	6	be	be	AUX
iajs-725	48	7	an	an	DET
iajs-725	48	8	r	r	NOUN
iajs-725	48	9	-	-	PUNCT
iajs-725	48	10	module	module	NOUN
iajs-725	48	11	.	.	PUNCT
iajs-725	49	1	then	then	ADV
iajs-725	49	2	(	(	PUNCT
iajs-725	49	3	1	1	X
iajs-725	49	4	)	)	PUNCT
iajs-725	49	5			NOUN
iajs-725	49	6	(	(	PUNCT
iajs-725	49	7	2	2	NUM
iajs-725	49	8	)	)	PUNCT
iajs-725	49	9	,	,	PUNCT
iajs-725	49	10	(	(	PUNCT
iajs-725	49	11	2	2	X
iajs-725	49	12	)	)	PUNCT
iajs-725	49	13			NOUN
iajs-725	49	14	(	(	PUNCT
iajs-725	49	15	3	3	NUM
iajs-725	49	16	)	)	PUNCT
iajs-725	49	17	,	,	PUNCT
iajs-725	49	18	(	(	PUNCT
iajs-725	49	19	3	3	X
iajs-725	49	20	)	)	PUNCT
iajs-725	49	21			NOUN
iajs-725	49	22	(	(	PUNCT
iajs-725	49	23	4	4	NUM
iajs-725	49	24	)	)	PUNCT
iajs-725	49	25	,	,	PUNCT
iajs-725	49	26	(	(	PUNCT
iajs-725	49	27	4	4	X
iajs-725	49	28	)	)	PUNCT
iajs-725	49	29			NOUN
iajs-725	49	30	(	(	PUNCT
iajs-725	49	31	1	1	X
iajs-725	49	32	)	)	PUNCT
iajs-725	49	33	if	if	SCONJ
iajs-725	49	34	m	m	NOUN
iajs-725	49	35	is	be	AUX
iajs-725	49	36	finitely	finitely	ADV
iajs-725	49	37	generated	generate	VERB
iajs-725	49	38	(	(	PUNCT
iajs-725	49	39	1	1	X
iajs-725	49	40	)	)	PUNCT
iajs-725	49	41	m	m	VERB
iajs-725	49	42	is	be	AUX
iajs-725	49	43	annsemimaximal	annsemimaximal	ADJ
iajs-725	49	44	module	module	NOUN
iajs-725	49	45	.	.	PUNCT
iajs-725	50	1	(	(	PUNCT
iajs-725	50	2	2	2	X
iajs-725	50	3	)	)	PUNCT
iajs-725	50	4	[	[	X
iajs-725	50	5	annrn	annrn	NOUN
iajs-725	50	6	r	r	NOUN
iajs-725	50	7	:	:	PUNCT
iajs-725	50	8	a	a	X
iajs-725	50	9	]	]	PUNCT
iajs-725	50	10	is	be	AUX
iajs-725	50	11	a	a	DET
iajs-725	50	12	semimaximal	semimaximal	NOUN
iajs-725	50	13	for	for	ADP
iajs-725	50	14	each	each	DET
iajs-725	50	15	non	non	ADJ
iajs-725	50	16	-	-	ADJ
iajs-725	50	17	zero	zero	NUM
iajs-725	50	18	submodule	submodule	NOUN
iajs-725	50	19	n	n	PROPN
iajs-725	50	20	of	of	ADP
iajs-725	50	21	m	m	PROPN
iajs-725	50	22	and	and	CCONJ
iajs-725	50	23	for	for	ADP
iajs-725	50	24	each	each	DET
iajs-725	50	25	ideal	ideal	NOUN
iajs-725	50	26	a	a	PRON
iajs-725	50	27	of	of	ADP
iajs-725	50	28	r	r	NOUN
iajs-725	50	29	such	such	ADJ
iajs-725	50	30	that	that	SCONJ
iajs-725	50	31	a	a	DET
iajs-725	50	32			PROPN
iajs-725	50	33	annrn	annrn	NOUN
iajs-725	50	34	.	.	PUNCT
iajs-725	50	35	.	.	PUNCT
iajs-725	51	1	ibn	ibn	PROPN
iajs-725	51	2	alhaitham	alhaitham	PROPN
iajs-725	51	3	j.	j.	PROPN
iajs-725	51	4	for	for	ADP
iajs-725	51	5	pure	pure	ADJ
iajs-725	51	6	&	&	CCONJ
iajs-725	51	7	appl	appl	PROPN
iajs-725	51	8	.	.	PUNCT
iajs-725	52	1	sci	sci	PROPN
iajs-725	52	2	.	.	PUNCT
iajs-725	52	3	vol.24	vol.24	NOUN
iajs-725	52	4	(	(	PUNCT
iajs-725	52	5	3	3	NUM
iajs-725	52	6	)	)	PUNCT
iajs-725	52	7	2011	2011	NUM
iajs-725	53	1	[	[	X
iajs-725	53	2	annn	annn	X
iajs-725	53	3	r	r	NOUN
iajs-725	53	4	:	:	PUNCT
iajs-725	54	1	r	r	X
iajs-725	54	2	]	]	X
iajs-725	54	3	is	be	AUX
iajs-725	54	4	a	a	DET
iajs-725	54	5	semimaximal	semimaximal	ADJ
iajs-725	54	6	ideal	ideal	NOUN
iajs-725	54	7	of	of	ADP
iajs-725	54	8	r	r	NOUN
iajs-725	54	9	for	for	ADP
iajs-725	54	10	each	each	DET
iajs-725	54	11	non	non	ADJ
iajs-725	54	12	-	-	ADJ
iajs-725	54	13	zero	zero	NUM
iajs-725	54	14	submodule	submodule	NOUN
iajs-725	54	15	n	n	PROPN
iajs-725	54	16	of	of	ADP
iajs-725	54	17	m	m	PRON
iajs-725	54	18	,	,	PUNCT
iajs-725	54	19	rr	rr	ADP
iajs-725	54	20	such	such	ADJ
iajs-725	54	21	that	that	SCONJ
iajs-725	54	22	(	(	PUNCT
iajs-725	54	23	r	r	NOUN
iajs-725	54	24	)	)	PUNCT
iajs-725	54	25			PROPN
iajs-725	54	26	annrn	annrn	NOUN
iajs-725	54	27	.	.	PUNCT
iajs-725	55	1	(	(	PUNCT
iajs-725	55	2	3	3	X
iajs-725	55	3	)	)	PUNCT
iajs-725	55	4	annr(m	annr(m	NOUN
iajs-725	55	5	)	)	PUNCT
iajs-725	55	6	is	be	AUX
iajs-725	55	7	a	a	DET
iajs-725	55	8	semimaximal	semimaximal	ADJ
iajs-725	55	9	ideal	ideal	NOUN
iajs-725	55	10	of	of	ADP
iajs-725	55	11	r	r	NOUN
iajs-725	55	12	for	for	ADP
iajs-725	55	13	each	each	DET
iajs-725	55	14	m0	m0	NOUN
iajs-725	55	15	,	,	PUNCT
iajs-725	55	16	mm	mm	PUNCT
iajs-725	55	17	.	.	PUNCT
iajs-725	56	1	proof	proof	NOUN
iajs-725	56	2	:	:	PUNCT
iajs-725	56	3	(	(	PUNCT
iajs-725	56	4	1	1	X
iajs-725	56	5	)	)	PUNCT
iajs-725	56	6			NOUN
iajs-725	56	7	(	(	PUNCT
iajs-725	56	8	2	2	NUM
iajs-725	56	9	)	)	PUNCT
iajs-725	56	10	,	,	PUNCT
iajs-725	56	11	suppose	suppose	VERB
iajs-725	56	12	that	that	SCONJ
iajs-725	56	13	m	m	PROPN
iajs-725	56	14	is	be	AUX
iajs-725	56	15	annsemimaximal	annsemimaximal	ADJ
iajs-725	56	16	module	module	NOUN
iajs-725	56	17	.	.	PUNCT
iajs-725	57	1	let	let	VERB
iajs-725	57	2	n	n	PRON
iajs-725	57	3	be	be	AUX
iajs-725	57	4	a	a	DET
iajs-725	57	5	non	non	ADJ
iajs-725	57	6	-	-	ADJ
iajs-725	57	7	zero	zero	NUM
iajs-725	57	8	submodule	submodule	NOUN
iajs-725	57	9	of	of	ADP
iajs-725	57	10	m.	m.	NOUN
iajs-725	57	11	then	then	ADV
iajs-725	57	12	annrn	annrn	NOUN
iajs-725	57	13	is	be	AUX
iajs-725	57	14	semimaximal	semimaximal	ADJ
iajs-725	57	15	ideal	ideal	NOUN
iajs-725	57	16	of	of	ADP
iajs-725	57	17	r.	r.	PROPN
iajs-725	57	18	assume	assume	VERB
iajs-725	57	19	that	that	SCONJ
iajs-725	57	20	a	a	PRON
iajs-725	57	21	is	be	AUX
iajs-725	57	22	an	an	DET
iajs-725	57	23	ideal	ideal	NOUN
iajs-725	57	24	of	of	ADP
iajs-725	57	25	r	r	NOUN
iajs-725	57	26	such	such	ADJ
iajs-725	57	27	that	that	SCONJ
iajs-725	57	28	a	a	DET
iajs-725	57	29			PROPN
iajs-725	57	30	annrn	annrn	NOUN
iajs-725	57	31	.	.	PUNCT
iajs-725	58	1	it	it	PRON
iajs-725	58	2	is	be	AUX
iajs-725	58	3	clear	clear	ADJ
iajs-725	58	4	that	that	SCONJ
iajs-725	58	5	annrn	annrn	NOUN
iajs-725	58	6			VERB
iajs-725	59	1	[	[	X
iajs-725	59	2	annrn	annrn	NOUN
iajs-725	59	3	r	r	NOUN
iajs-725	59	4	:	:	PUNCT
iajs-725	59	5	a	a	PRON
iajs-725	59	6	]	]	X
iajs-725	59	7	.	.	PUNCT
iajs-725	60	1	so	so	ADV
iajs-725	60	2	,	,	PUNCT
iajs-725	60	3	according	accord	VERB
iajs-725	60	4	to	to	ADP
iajs-725	60	5	[	[	X
iajs-725	60	6	5,coro.(1.2.12	5,coro.(1.2.12	NUM
iajs-725	60	7	)	)	PUNCT
iajs-725	60	8	]	]	PUNCT
iajs-725	60	9	,	,	PUNCT
iajs-725	60	10	[	[	X
iajs-725	60	11	annrn	annrn	NOUN
iajs-725	60	12	r	r	NOUN
iajs-725	60	13	:	:	PUNCT
iajs-725	60	14	a	a	X
iajs-725	60	15	]	]	PUNCT
iajs-725	60	16	is	be	AUX
iajs-725	60	17	semimaximal	semimaximal	ADJ
iajs-725	60	18	ideal	ideal	NOUN
iajs-725	60	19	of	of	ADP
iajs-725	60	20	r.	r.	PROPN
iajs-725	60	21	(	(	PUNCT
iajs-725	60	22	2	2	NUM
iajs-725	60	23	)	)	PUNCT
iajs-725	60	24			NOUN
iajs-725	60	25	(	(	PUNCT
iajs-725	60	26	3	3	NUM
iajs-725	60	27	)	)	PUNCT
iajs-725	60	28	,	,	PUNCT
iajs-725	60	29	take	take	VERB
iajs-725	60	30	a	a	DET
iajs-725	60	31	=	=	PUNCT
iajs-725	60	32	(	(	PUNCT
iajs-725	60	33	r	r	NOUN
iajs-725	60	34	)	)	PUNCT
iajs-725	60	35	the	the	DET
iajs-725	60	36	ideal	ideal	NOUN
iajs-725	60	37	of	of	ADP
iajs-725	60	38	r	r	NOUN
iajs-725	60	39	generated	generate	VERB
iajs-725	60	40	by	by	ADP
iajs-725	60	41	r	r	NOUN
iajs-725	60	42	,	,	PUNCT
iajs-725	60	43	the	the	DET
iajs-725	60	44	result	result	NOUN
iajs-725	60	45	follows	follow	VERB
iajs-725	60	46	by	by	ADP
iajs-725	60	47	(	(	PUNCT
iajs-725	60	48	2	2	NUM
iajs-725	60	49	)	)	PUNCT
iajs-725	60	50	.	.	PUNCT
iajs-725	61	1	(	(	PUNCT
iajs-725	61	2	3	3	X
iajs-725	61	3	)	)	PUNCT
iajs-725	61	4			NOUN
iajs-725	61	5	(	(	PUNCT
iajs-725	61	6	4	4	NUM
iajs-725	61	7	)	)	PUNCT
iajs-725	61	8	,	,	PUNCT
iajs-725	61	9	let	let	VERB
iajs-725	61	10	0m	0m	PUNCT
iajs-725	61	11	m	m	VERB
iajs-725	61	12	.	.	PUNCT
iajs-725	62	1	because	because	SCONJ
iajs-725	62	2	1annr(m	1annr(m	NUM
iajs-725	62	3	)	)	PUNCT
iajs-725	62	4	.	.	PUNCT
iajs-725	63	1	[	[	X
iajs-725	63	2	annr(m):r	annr(m):r	X
iajs-725	63	3	]	]	PUNCT
iajs-725	63	4	is	be	AUX
iajs-725	63	5	semimaximal	semimaximal	ADJ
iajs-725	63	6	ideal	ideal	NOUN
iajs-725	63	7	of	of	ADP
iajs-725	63	8	r	r	NOUN
iajs-725	63	9	by	by	ADP
iajs-725	63	10	(	(	PUNCT
iajs-725	63	11	3	3	NUM
iajs-725	63	12	)	)	PUNCT
iajs-725	63	13	.	.	PUNCT
iajs-725	64	1	but	but	CCONJ
iajs-725	65	1	[	[	X
iajs-725	65	2	annr(m):r]=annr(m	annr(m):r]=annr(m	NOUN
iajs-725	65	3	)	)	PUNCT
iajs-725	65	4	.	.	PUNCT
iajs-725	66	1	so	so	ADV
iajs-725	66	2	annr(m	annr(m	NOUN
iajs-725	66	3	)	)	PUNCT
iajs-725	66	4	is	be	AUX
iajs-725	66	5	semimaximal	semimaximal	ADJ
iajs-725	66	6	ideal	ideal	NOUN
iajs-725	66	7	of	of	ADP
iajs-725	66	8	r.	r.	PROPN
iajs-725	66	9	(	(	PUNCT
iajs-725	66	10	4	4	NUM
iajs-725	66	11	)	)	PUNCT
iajs-725	66	12			NOUN
iajs-725	66	13	(	(	PUNCT
iajs-725	66	14	1	1	NUM
iajs-725	66	15	)	)	PUNCT
iajs-725	66	16	,	,	PUNCT
iajs-725	66	17	since	since	SCONJ
iajs-725	66	18	m	m	PROPN
iajs-725	66	19	is	be	AUX
iajs-725	66	20	finitely	finitely	ADV
iajs-725	66	21	generated	generate	VERB
iajs-725	66	22	,	,	PUNCT
iajs-725	66	23	m=	m=	PRON
iajs-725	67	1	n	n	NOUN
iajs-725	68	1	i	i	PRON
iajs-725	68	2	i	i	PRON
iajs-725	68	3	1	1	NUM
iajs-725	68	4	rx	rx	VERB
iajs-725	68	5			NUM
iajs-725	68	6			X
iajs-725	68	7	,	,	PUNCT
iajs-725	68	8	xim	xim	PROPN
iajs-725	68	9	,	,	PUNCT
iajs-725	68	10	annrm=	annrm=	PROPN
iajs-725	68	11	n	n	ADV
iajs-725	69	1	i	i	PRON
iajs-725	69	2	i	i	VERB
iajs-725	69	3	1	1	NUM
iajs-725	69	4	annx	annx	PROPN
iajs-725	69	5			PROPN
iajs-725	69	6			PUNCT
iajs-725	69	7	.	.	PUNCT
iajs-725	70	1	but	but	CCONJ
iajs-725	70	2	ann(xi	ann(xi	NUM
iajs-725	70	3	)	)	PUNCT
iajs-725	70	4	for	for	ADP
iajs-725	70	5	all	all	DET
iajs-725	70	6	i=1,	i=1,	NOUN
iajs-725	70	7	…	…	PUNCT
iajs-725	70	8	,n	,n	PUNCT
iajs-725	70	9	is	be	AUX
iajs-725	70	10	semimaximal	semimaximal	ADJ
iajs-725	70	11	ideal	ideal	NOUN
iajs-725	70	12	.	.	PUNCT
iajs-725	71	1	so	so	ADV
iajs-725	71	2	,	,	PUNCT
iajs-725	71	3	by	by	ADP
iajs-725	71	4	[	[	X
iajs-725	71	5	5,coro.(1.2.15	5,coro.(1.2.15	NUM
iajs-725	71	6	)	)	PUNCT
iajs-725	71	7	]	]	PUNCT
iajs-725	71	8	,	,	PUNCT
iajs-725	71	9	annrm	annrm	PROPN
iajs-725	71	10	is	be	AUX
iajs-725	71	11	semimaximal	semimaximal	ADJ
iajs-725	71	12	.	.	PUNCT
iajs-725	72	1	thus	thus	ADV
iajs-725	72	2	m	m	PROPN
iajs-725	72	3	is	be	AUX
iajs-725	72	4	ansemimaximal	ansemimaximal	NOUN
iajs-725	72	5	by	by	ADP
iajs-725	72	6	prop	prop	NOUN
iajs-725	72	7	.	.	PUNCT
iajs-725	73	1	(	(	PUNCT
iajs-725	73	2	1.3	1.3	NUM
iajs-725	73	3	)	)	PUNCT
iajs-725	73	4	.	.	PUNCT
iajs-725	74	1	recall	recall	VERB
iajs-725	74	2	that	that	SCONJ
iajs-725	74	3	an	an	DET
iajs-725	74	4	r	r	NOUN
iajs-725	74	5	-	-	PUNCT
iajs-725	74	6	module	module	NOUN
iajs-725	74	7	m	m	NOUN
iajs-725	74	8	is	be	AUX
iajs-725	74	9	called	call	VERB
iajs-725	74	10	semisimple	semisimple	NOUN
iajs-725	74	11	if	if	SCONJ
iajs-725	74	12	every	every	DET
iajs-725	74	13	submodule	submodule	NOUN
iajs-725	74	14	of	of	ADP
iajs-725	74	15	m	m	PROPN
iajs-725	74	16	is	be	AUX
iajs-725	74	17	a	a	DET
iajs-725	74	18	direct	direct	ADJ
iajs-725	74	19	summand	summand	NOUN
iajs-725	74	20	of	of	ADP
iajs-725	74	21	m.	m.	NOUN
iajs-725	74	22	and	and	CCONJ
iajs-725	74	23	a	a	DET
iajs-725	74	24	ring	ring	NOUN
iajs-725	74	25	r	r	NOUN
iajs-725	74	26	is	be	AUX
iajs-725	74	27	said	say	VERB
iajs-725	74	28	to	to	PART
iajs-725	74	29	be	be	AUX
iajs-725	74	30	semisimple	semisimple	NOUN
iajs-725	74	31	ring	ring	NOUN
iajs-725	74	32	if	if	SCONJ
iajs-725	74	33	and	and	CCONJ
iajs-725	74	34	only	only	ADV
iajs-725	74	35	if	if	SCONJ
iajs-725	74	36	r	r	NOUN
iajs-725	74	37	is	be	AUX
iajs-725	74	38	a	a	DET
iajs-725	74	39	semisimple	semisimple	ADJ
iajs-725	74	40	rmodule	rmodule	NOUN
iajs-725	74	41	,	,	PUNCT
iajs-725	74	42	[	[	X
iajs-725	74	43	6	6	NUM
iajs-725	74	44	]	]	PUNCT
iajs-725	74	45	.	.	PUNCT
iajs-725	75	1	1.6	1.6	NUM
iajs-725	75	2	proposition	proposition	NOUN
iajs-725	75	3	:	:	PUNCT
iajs-725	75	4	every	every	DET
iajs-725	75	5	semisimple	semisimple	ADJ
iajs-725	75	6	r	r	NOUN
iajs-725	75	7	-	-	PUNCT
iajs-725	75	8	module	module	NOUN
iajs-725	75	9	m	m	NOUN
iajs-725	75	10	is	be	AUX
iajs-725	75	11	annsemimaximal	annsemimaximal	ADJ
iajs-725	75	12	.	.	PUNCT
iajs-725	76	1	proof	proof	NOUN
iajs-725	76	2	:	:	PUNCT
iajs-725	76	3	by	by	ADP
iajs-725	76	4	[	[	NOUN
iajs-725	76	5	6,prop.(1.1.46	6,prop.(1.1.46	NOUN
iajs-725	76	6	)	)	PUNCT
iajs-725	76	7	]	]	PUNCT
iajs-725	76	8	,	,	PUNCT
iajs-725	76	9	we	we	PRON
iajs-725	76	10	get	get	VERB
iajs-725	76	11	r	r	NOUN
iajs-725	76	12	/	/	SYM
iajs-725	76	13	annrm	annrm	NOUN
iajs-725	76	14	is	be	AUX
iajs-725	76	15	a	a	DET
iajs-725	76	16	semisimple	semisimple	NOUN
iajs-725	76	17	ring	ring	NOUN
iajs-725	76	18	.	.	PUNCT
iajs-725	77	1	therefore	therefore	ADV
iajs-725	77	2	annrm	annrm	PROPN
iajs-725	77	3	is	be	AUX
iajs-725	77	4	a	a	DET
iajs-725	77	5	semimaximal	semimaximal	ADJ
iajs-725	77	6	ideal	ideal	NOUN
iajs-725	77	7	by	by	ADP
iajs-725	77	8	[	[	X
iajs-725	77	9	4,prop.(1.2.5	4,prop.(1.2.5	PROPN
iajs-725	77	10	)	)	PUNCT
iajs-725	77	11	]	]	PUNCT
iajs-725	77	12	.	.	PUNCT
iajs-725	78	1	thus	thus	ADV
iajs-725	78	2	m	m	PROPN
iajs-725	78	3	is	be	AUX
iajs-725	78	4	annsemimaximal	annsemimaximal	ADJ
iajs-725	78	5	by	by	ADP
iajs-725	78	6	prop	prop	NOUN
iajs-725	78	7	.	.	PUNCT
iajs-725	79	1	(	(	PUNCT
iajs-725	79	2	1.3	1.3	NUM
iajs-725	79	3	)	)	PUNCT
iajs-725	79	4	.	.	PUNCT
iajs-725	80	1	the	the	DET
iajs-725	80	2	following	follow	VERB
iajs-725	80	3	corollary	corollary	NOUN
iajs-725	80	4	is	be	AUX
iajs-725	80	5	an	an	DET
iajs-725	80	6	application	application	NOUN
iajs-725	80	7	of	of	ADP
iajs-725	80	8	proposition	proposition	NOUN
iajs-725	80	9	(	(	PUNCT
iajs-725	80	10	1.6	1.6	NUM
iajs-725	80	11	)	)	PUNCT
iajs-725	80	12	.	.	PUNCT
iajs-725	81	1	1.7	1.7	NUM
iajs-725	81	2	corollary	corollary	NOUN
iajs-725	81	3	:	:	PUNCT
iajs-725	81	4	let	let	VERB
iajs-725	81	5	r	r	PRON
iajs-725	81	6	be	be	AUX
iajs-725	81	7	a	a	DET
iajs-725	81	8	semisimple	semisimple	NOUN
iajs-725	81	9	ring	ring	NOUN
iajs-725	81	10	.	.	PUNCT
iajs-725	82	1	then	then	ADV
iajs-725	82	2	every	every	DET
iajs-725	82	3	r	r	NOUN
iajs-725	82	4	-	-	PUNCT
iajs-725	82	5	module	module	NOUN
iajs-725	82	6	m	m	NOUN
iajs-725	82	7	is	be	AUX
iajs-725	82	8	annsemimaximal	annsemimaximal	ADJ
iajs-725	82	9	.	.	PUNCT
iajs-725	83	1	proof	proof	NOUN
iajs-725	83	2	:	:	PUNCT
iajs-725	83	3	it	it	PRON
iajs-725	83	4	is	be	AUX
iajs-725	83	5	known	know	VERB
iajs-725	83	6	that	that	SCONJ
iajs-725	83	7	if	if	SCONJ
iajs-725	83	8	r	r	NOUN
iajs-725	83	9	is	be	AUX
iajs-725	83	10	semisimple	semisimple	NOUN
iajs-725	83	11	ring	ring	NOUN
iajs-725	83	12	,	,	PUNCT
iajs-725	83	13	then	then	ADV
iajs-725	83	14	m	m	VERB
iajs-725	83	15	is	be	AUX
iajs-725	83	16	semisimple	semisimple	NOUN
iajs-725	83	17	module	module	NOUN
iajs-725	83	18	[	[	X
iajs-725	83	19	5,prop.(1.1.44	5,prop.(1.1.44	NUM
iajs-725	83	20	)	)	PUNCT
iajs-725	83	21	]	]	PUNCT
iajs-725	83	22	.	.	PUNCT
iajs-725	84	1	hence	hence	ADV
iajs-725	84	2	m	m	PROPN
iajs-725	84	3	is	be	AUX
iajs-725	84	4	annsemimaximal	annsemimaximal	ADJ
iajs-725	84	5	module	module	NOUN
iajs-725	84	6	by	by	ADP
iajs-725	84	7	previous	previous	ADJ
iajs-725	84	8	proposition	proposition	NOUN
iajs-725	84	9	.	.	PUNCT
iajs-725	85	1	next	next	ADV
iajs-725	85	2	,	,	PUNCT
iajs-725	85	3	we	we	PRON
iajs-725	85	4	have	have	VERB
iajs-725	85	5	the	the	DET
iajs-725	85	6	following	follow	VERB
iajs-725	85	7	proposition	proposition	NOUN
iajs-725	85	8	.	.	PUNCT
iajs-725	86	1	1.8	1.8	NUM
iajs-725	86	2	proposition	proposition	NOUN
iajs-725	86	3	:	:	PUNCT
iajs-725	86	4	if	if	SCONJ
iajs-725	86	5	m	m	NOUN
iajs-725	86	6	is	be	AUX
iajs-725	86	7	an	an	DET
iajs-725	86	8	artinian	artinian	NOUN
iajs-725	86	9	and	and	CCONJ
iajs-725	86	10	annsemimaximal	annsemimaximal	ADJ
iajs-725	86	11	r	r	NOUN
iajs-725	86	12	-	-	PUNCT
iajs-725	86	13	module	module	NOUN
iajs-725	86	14	,	,	PUNCT
iajs-725	86	15	then	then	ADV
iajs-725	86	16	m	m	NOUN
iajs-725	86	17	is	be	AUX
iajs-725	86	18	semisimple	semisimple	ADJ
iajs-725	86	19	.	.	PUNCT
iajs-725	87	1	proof	proof	NOUN
iajs-725	87	2	:	:	PUNCT
iajs-725	87	3	we	we	PRON
iajs-725	87	4	have	have	VERB
iajs-725	87	5	m	m	PROPN
iajs-725	87	6	is	be	AUX
iajs-725	87	7	annseimmaximal	annseimmaximal	ADJ
iajs-725	87	8	,	,	PUNCT
iajs-725	87	9	then	then	ADV
iajs-725	87	10	annrm	annrm	NOUN
iajs-725	87	11	is	be	AUX
iajs-725	87	12	semimaximal	semimaximal	ADJ
iajs-725	87	13	.	.	PUNCT
iajs-725	88	1	thus	thus	ADV
iajs-725	88	2	j(m	j(m	PROPN
iajs-725	88	3	)	)	PUNCT
iajs-725	89	1	=	=	SYM
iajs-725	89	2	0	0	NUM
iajs-725	89	3	by	by	ADP
iajs-725	89	4	[	[	X
iajs-725	89	5	5,coro,(1.3.6	5,coro,(1.3.6	NUM
iajs-725	89	6	)	)	PUNCT
iajs-725	89	7	]	]	PUNCT
iajs-725	89	8	.	.	PUNCT
iajs-725	90	1	but	but	CCONJ
iajs-725	90	2	m	m	PROPN
iajs-725	90	3	is	be	AUX
iajs-725	90	4	an	an	DET
iajs-725	90	5	artinian	artinian	NOUN
iajs-725	90	6	and	and	CCONJ
iajs-725	90	7	j(m	j(m	PROPN
iajs-725	90	8	)	)	PUNCT
iajs-725	91	1	=	=	SYM
iajs-725	91	2	0	0	NUM
iajs-725	91	3	,	,	PUNCT
iajs-725	91	4	then	then	ADV
iajs-725	91	5	m	m	NOUN
iajs-725	91	6	is	be	AUX
iajs-725	91	7	semisimple	semisimple	ADJ
iajs-725	91	8	,	,	PUNCT
iajs-725	91	9	[	[	X
iajs-725	91	10	5	5	NUM
iajs-725	91	11	]	]	PUNCT
iajs-725	91	12	.	.	PUNCT
iajs-725	92	1	1.9	1.9	NUM
iajs-725	92	2	example	example	NOUN
iajs-725	92	3	:	:	PUNCT
iajs-725	92	4	z24	z24	PROPN
iajs-725	92	5	as	as	ADP
iajs-725	92	6	a	a	DET
iajs-725	92	7	z	z	NOUN
iajs-725	92	8	-	-	PUNCT
iajs-725	92	9	module	module	NOUN
iajs-725	92	10	is	be	AUX
iajs-725	92	11	not	not	PART
iajs-725	92	12	annsemimaximal	annsemimaximal	ADJ
iajs-725	92	13	module	module	NOUN
iajs-725	92	14	and	and	CCONJ
iajs-725	92	15	z24	z24	NOUN
iajs-725	92	16	is	be	AUX
iajs-725	92	17	not	not	PART
iajs-725	92	18	semisimple	semisimple	ADJ
iajs-725	92	19	.	.	PUNCT
iajs-725	93	1	the	the	DET
iajs-725	93	2	following	following	ADJ
iajs-725	93	3	result	result	NOUN
iajs-725	93	4	is	be	AUX
iajs-725	93	5	consequence	consequence	NOUN
iajs-725	93	6	of	of	ADP
iajs-725	93	7	proposition	proposition	NOUN
iajs-725	93	8	(	(	PUNCT
iajs-725	93	9	1.8	1.8	NUM
iajs-725	93	10	)	)	PUNCT
iajs-725	93	11	.	.	PUNCT
iajs-725	94	1	1.10	1.10	NUM
iajs-725	94	2	corollary	corollary	NOUN
iajs-725	94	3	:	:	PUNCT
iajs-725	94	4	let	let	VERB
iajs-725	94	5	m	m	NOUN
iajs-725	94	6	is	be	AUX
iajs-725	94	7	an	an	DET
iajs-725	94	8	artinian	artinian	ADJ
iajs-725	94	9	r-module.then	r-module.then	NOUN
iajs-725	94	10	m	m	VERB
iajs-725	94	11	is	be	AUX
iajs-725	94	12	semisimple	semisimple	NOUN
iajs-725	94	13	module	module	NOUN
iajs-725	94	14	if	if	SCONJ
iajs-725	94	15	and	and	CCONJ
iajs-725	94	16	only	only	ADV
iajs-725	94	17	if	if	SCONJ
iajs-725	94	18	m	m	NOUN
iajs-725	94	19	is	be	AUX
iajs-725	94	20	annsemimaximal	annsemimaximal	ADJ
iajs-725	94	21	.	.	PUNCT
iajs-725	95	1	1.11	1.11	NUM
iajs-725	95	2	proposition	proposition	NOUN
iajs-725	95	3	:	:	PUNCT
iajs-725	95	4	if	if	SCONJ
iajs-725	95	5	m	m	NOUN
iajs-725	95	6	is	be	AUX
iajs-725	95	7	annsemimaximal	annsemimaximal	ADJ
iajs-725	95	8	r	r	NOUN
iajs-725	95	9	-	-	PUNCT
iajs-725	95	10	module	module	NOUN
iajs-725	95	11	,	,	PUNCT
iajs-725	95	12	then	then	ADV
iajs-725	95	13	every	every	DET
iajs-725	95	14	cyclic	cyclic	ADJ
iajs-725	95	15	submodule	submodule	NOUN
iajs-725	95	16	of	of	ADP
iajs-725	95	17	m	m	PROPN
iajs-725	95	18	is	be	AUX
iajs-725	95	19	semisimple	semisimple	ADJ
iajs-725	95	20	.	.	PUNCT
iajs-725	96	1	ibn	ibn	PROPN
iajs-725	96	2	alhaitham	alhaitham	PROPN
iajs-725	96	3	j.	j.	PROPN
iajs-725	96	4	for	for	ADP
iajs-725	96	5	pure	pure	ADJ
iajs-725	96	6	&	&	CCONJ
iajs-725	96	7	appl	appl	PROPN
iajs-725	96	8	.	.	PUNCT
iajs-725	97	1	sci	sci	PROPN
iajs-725	97	2	.	.	PUNCT
iajs-725	97	3	vol.24	vol.24	NOUN
iajs-725	97	4	(	(	PUNCT
iajs-725	97	5	3	3	NUM
iajs-725	97	6	)	)	PUNCT
iajs-725	97	7	2011	2011	NUM
iajs-725	97	8	proof	proof	NOUN
iajs-725	97	9	:	:	PUNCT
iajs-725	97	10	if	if	SCONJ
iajs-725	97	11	m	m	NOUN
iajs-725	97	12	is	be	AUX
iajs-725	97	13	annsemimaximal	annsemimaximal	ADJ
iajs-725	97	14	r	r	NOUN
iajs-725	97	15	-	-	PUNCT
iajs-725	97	16	module	module	NOUN
iajs-725	97	17	,	,	PUNCT
iajs-725	97	18	then	then	ADV
iajs-725	97	19	annr(x	annr(x	NOUN
iajs-725	97	20	)	)	PUNCT
iajs-725	97	21	is	be	AUX
iajs-725	97	22	semimaximal	semimaximal	ADJ
iajs-725	97	23	ideal	ideal	NOUN
iajs-725	97	24	by	by	ADP
iajs-725	97	25	th.((1.5),(4	th.((1.5),(4	NOUN
iajs-725	97	26	)	)	PUNCT
iajs-725	97	27	)	)	PUNCT
iajs-725	97	28	,	,	PUNCT
iajs-725	97	29	so	so	CCONJ
iajs-725	97	30	by	by	ADP
iajs-725	97	31	[	[	X
iajs-725	97	32	5,prop.(2.3.15	5,prop.(2.3.15	NUM
iajs-725	97	33	)	)	PUNCT
iajs-725	97	34	]	]	PUNCT
iajs-725	97	35	,	,	PUNCT
iajs-725	97	36	we	we	PRON
iajs-725	97	37	get	get	VERB
iajs-725	97	38	the	the	DET
iajs-725	97	39	result	result	NOUN
iajs-725	97	40	now	now	ADV
iajs-725	97	41	,	,	PUNCT
iajs-725	97	42	we	we	PRON
iajs-725	97	43	induced	induce	VERB
iajs-725	97	44	the	the	DET
iajs-725	97	45	following	follow	VERB
iajs-725	97	46	corollary	corollary	NOUN
iajs-725	97	47	.	.	PUNCT
iajs-725	98	1	1.12	1.12	NUM
iajs-725	98	2	corollary	corollary	NOUN
iajs-725	98	3	:	:	PUNCT
iajs-725	98	4	if	if	SCONJ
iajs-725	98	5	m	m	NOUN
iajs-725	98	6	is	be	AUX
iajs-725	98	7	finitely	finitely	ADV
iajs-725	98	8	generated	generate	VERB
iajs-725	98	9	and	and	CCONJ
iajs-725	98	10	annsemimaximal	annsemimaximal	ADJ
iajs-725	98	11	r	r	NOUN
iajs-725	98	12	-	-	PUNCT
iajs-725	98	13	module	module	NOUN
iajs-725	98	14	,	,	PUNCT
iajs-725	98	15	then	then	ADV
iajs-725	98	16	m	m	NOUN
iajs-725	98	17	is	be	AUX
iajs-725	98	18	semisimple	semisimple	ADJ
iajs-725	98	19	r	r	NOUN
iajs-725	98	20	-	-	PUNCT
iajs-725	98	21	module	module	NOUN
iajs-725	98	22	.	.	PUNCT
iajs-725	99	1	proof	proof	NOUN
iajs-725	99	2	:	:	PUNCT
iajs-725	99	3	let	let	VERB
iajs-725	99	4	m	m	PRON
iajs-725	99	5	=	=	VERB
iajs-725	99	6	rx1	rx1	PROPN
iajs-725	99	7	+	+	X
iajs-725	99	8	rx2	rx2	NOUN
iajs-725	99	9	+	+	CCONJ
iajs-725	99	10	…	…	PUNCT
iajs-725	99	11	+	+	NUM
iajs-725	99	12	rxn	rxn	NOUN
iajs-725	99	13	for	for	ADP
iajs-725	99	14	some	some	DET
iajs-725	99	15	x1	x1	PROPN
iajs-725	99	16	,	,	PUNCT
iajs-725	99	17	x2	x2	PROPN
iajs-725	99	18	,	,	PUNCT
iajs-725	99	19	…	…	PUNCT
iajs-725	99	20	,	,	PUNCT
iajs-725	99	21	xn	xn	PROPN
iajs-725	99	22	.	.	PUNCT
iajs-725	100	1	but	but	CCONJ
iajs-725	100	2	rxi	rxi	NOUN
iajs-725	100	3	is	be	AUX
iajs-725	100	4	semisimple	semisimple	NOUN
iajs-725	100	5	by	by	ADP
iajs-725	100	6	previous	previous	ADJ
iajs-725	100	7	proposition	proposition	NOUN
iajs-725	100	8	.	.	PUNCT
iajs-725	101	1	therefore	therefore	ADV
iajs-725	101	2	m=	m=	X
iajs-725	101	3	n	n	PROPN
iajs-725	102	1	i	i	PRON
iajs-725	102	2	i	i	PRON
iajs-725	102	3	1	1	NUM
iajs-725	102	4	rx	rx	VERB
iajs-725	102	5			PROPN
iajs-725	102	6			X
iajs-725	102	7	is	be	AUX
iajs-725	102	8	semisimple	semisimple	ADJ
iajs-725	102	9	by	by	ADP
iajs-725	102	10	combining	combine	VERB
iajs-725	102	11	corollary	corollary	ADJ
iajs-725	102	12	(	(	PUNCT
iajs-725	102	13	1.12	1.12	NUM
iajs-725	102	14	)	)	PUNCT
iajs-725	102	15	,	,	PUNCT
iajs-725	102	16	proposition	proposition	NOUN
iajs-725	102	17	(	(	PUNCT
iajs-725	102	18	1.6	1.6	NUM
iajs-725	102	19	)	)	PUNCT
iajs-725	102	20	,	,	PUNCT
iajs-725	102	21	we	we	PRON
iajs-725	102	22	get	get	VERB
iajs-725	102	23	the	the	DET
iajs-725	102	24	following	following	NOUN
iajs-725	102	25	:	:	PUNCT
iajs-725	102	26	1.13	1.13	NUM
iajs-725	102	27	corollary	corollary	NOUN
iajs-725	102	28	:	:	PUNCT
iajs-725	102	29	let	let	VERB
iajs-725	102	30	m	m	PRON
iajs-725	102	31	be	be	AUX
iajs-725	102	32	a	a	DET
iajs-725	102	33	finitely	finitely	ADV
iajs-725	102	34	generated	generate	VERB
iajs-725	102	35	r	r	NOUN
iajs-725	102	36	-	-	PUNCT
iajs-725	102	37	module	module	NOUN
iajs-725	102	38	.	.	PUNCT
iajs-725	103	1	then	then	ADV
iajs-725	103	2	m	m	PROPN
iajs-725	103	3	is	be	AUX
iajs-725	103	4	annsemimaximal	annsemimaximal	ADJ
iajs-725	103	5	module	module	NOUN
iajs-725	103	6	if	if	SCONJ
iajs-725	103	7	and	and	CCONJ
iajs-725	103	8	only	only	ADV
iajs-725	103	9	if	if	SCONJ
iajs-725	103	10	m	m	NOUN
iajs-725	103	11	is	be	AUX
iajs-725	103	12	semisimple	semisimple	ADJ
iajs-725	103	13	.	.	PUNCT
iajs-725	104	1	1.14	1.14	NUM
iajs-725	104	2	corollary	corollary	NOUN
iajs-725	104	3	:	:	PUNCT
iajs-725	104	4	r	r	NOUN
iajs-725	104	5	is	be	AUX
iajs-725	104	6	a	a	DET
iajs-725	104	7	semisimple	semisimple	NOUN
iajs-725	104	8	ring	ring	NOUN
iajs-725	104	9	if	if	SCONJ
iajs-725	104	10	and	and	CCONJ
iajs-725	104	11	only	only	ADV
iajs-725	104	12	if	if	SCONJ
iajs-725	104	13	r	r	NOUN
iajs-725	104	14	is	be	AUX
iajs-725	104	15	annsemimaximal	annsemimaximal	ADJ
iajs-725	104	16	ring	ring	NOUN
iajs-725	104	17	.	.	PUNCT
iajs-725	105	1	now	now	ADV
iajs-725	105	2	,	,	PUNCT
iajs-725	105	3	we	we	PRON
iajs-725	105	4	turn	turn	VERB
iajs-725	105	5	our	our	PRON
iajs-725	105	6	attention	attention	NOUN
iajs-725	105	7	to	to	ADP
iajs-725	105	8	direct	direct	ADJ
iajs-725	105	9	sum	sum	NOUN
iajs-725	105	10	of	of	ADP
iajs-725	105	11	annsemimaximal	annsemimaximal	ADJ
iajs-725	105	12	modules	module	NOUN
iajs-725	105	13	.	.	PUNCT
iajs-725	106	1	1.15	1.15	NUM
iajs-725	106	2	proposition	proposition	NOUN
iajs-725	106	3	:	:	PUNCT
iajs-725	106	4	let	let	VERB
iajs-725	106	5	m	m	PRON
iajs-725	106	6	be	be	AUX
iajs-725	106	7	a	a	DET
iajs-725	106	8	faithful	faithful	ADJ
iajs-725	106	9	r	r	NOUN
iajs-725	106	10	-	-	PUNCT
iajs-725	106	11	module	module	NOUN
iajs-725	106	12	.	.	PUNCT
iajs-725	107	1	then	then	ADV
iajs-725	107	2	r	r	NOUN
iajs-725	107	3	is	be	AUX
iajs-725	107	4	semisimple	semisimple	NOUN
iajs-725	107	5	if	if	SCONJ
iajs-725	107	6	and	and	CCONJ
iajs-725	107	7	only	only	ADV
iajs-725	107	8	if	if	SCONJ
iajs-725	107	9	m	m	NOUN
iajs-725	107	10	is	be	AUX
iajs-725	107	11	annsemimaximal	annsemimaximal	ADJ
iajs-725	107	12	.	.	PUNCT
iajs-725	108	1	proof	proof	NOUN
iajs-725	108	2	:	:	PUNCT
iajs-725	108	3	(	(	PUNCT
iajs-725	108	4			NOUN
iajs-725	108	5	)	)	PUNCT
iajs-725	108	6	directly	directly	ADV
iajs-725	108	7	from	from	ADP
iajs-725	108	8	[	[	X
iajs-725	108	9	5,prop.(1.1.44	5,prop.(1.1.44	NUM
iajs-725	108	10	)	)	PUNCT
iajs-725	108	11	]	]	PUNCT
iajs-725	108	12	and	and	CCONJ
iajs-725	108	13	proposition	proposition	NOUN
iajs-725	108	14	(	(	PUNCT
iajs-725	108	15	1.6	1.6	NUM
iajs-725	108	16	)	)	PUNCT
iajs-725	108	17	.	.	PUNCT
iajs-725	109	1	(	(	PUNCT
iajs-725	109	2			NOUN
iajs-725	109	3	)	)	PUNCT
iajs-725	109	4	if	if	SCONJ
iajs-725	109	5	m	m	NOUN
iajs-725	109	6	is	be	AUX
iajs-725	109	7	annsemimaxi	annsemimaxi	ADJ
iajs-725	109	8	,	,	PUNCT
iajs-725	109	9	then	then	ADV
iajs-725	109	10	annrm	annrm	NOUN
iajs-725	109	11	is	be	AUX
iajs-725	109	12	a	a	DET
iajs-725	109	13	semimaximal	semimaximal	ADJ
iajs-725	109	14	ideal	ideal	NOUN
iajs-725	109	15	;	;	PUNCT
iajs-725	109	16	that	that	ADV
iajs-725	109	17	is	is	ADV
iajs-725	109	18	(	(	PUNCT
iajs-725	109	19	0	0	NUM
iajs-725	109	20	)	)	PUNCT
iajs-725	109	21	is	be	AUX
iajs-725	109	22	a	a	DET
iajs-725	109	23	semimaximal	semimaximal	ADJ
iajs-725	109	24	ideal	ideal	NOUN
iajs-725	109	25	.	.	PUNCT
iajs-725	110	1	thus	thus	ADV
iajs-725	110	2	r/(0	r/(0	NOUN
iajs-725	110	3	)	)	PUNCT
iajs-725	110	4	�	�	PROPN
iajs-725	110	5	r	r	NOUN
iajs-725	110	6	is	be	AUX
iajs-725	110	7	semisimple	semisimple	ADJ
iajs-725	110	8	.	.	PUNCT
iajs-725	111	1	by	by	ADP
iajs-725	111	2	combining	combine	VERB
iajs-725	111	3	corollary	corollary	ADJ
iajs-725	111	4	(	(	PUNCT
iajs-725	111	5	1.13	1.13	NUM
iajs-725	111	6	)	)	PUNCT
iajs-725	111	7	,	,	PUNCT
iajs-725	111	8	proposition	proposition	NOUN
iajs-725	111	9	(	(	PUNCT
iajs-725	111	10	1.15	1.15	NUM
iajs-725	111	11	)	)	PUNCT
iajs-725	111	12	and	and	CCONJ
iajs-725	111	13	corollary	corollary	ADJ
iajs-725	111	14	(	(	PUNCT
iajs-725	111	15	1.14	1.14	NUM
iajs-725	111	16	)	)	PUNCT
iajs-725	111	17	,	,	PUNCT
iajs-725	111	18	we	we	PRON
iajs-725	111	19	get	get	VERB
iajs-725	111	20	the	the	DET
iajs-725	111	21	following	following	NOUN
iajs-725	111	22	:	:	PUNCT
iajs-725	111	23	1.16	1.16	NUM
iajs-725	111	24	corollary	corollary	NOUN
iajs-725	111	25	:	:	PUNCT
iajs-725	111	26	let	let	VERB
iajs-725	111	27	m	m	PRON
iajs-725	111	28	be	be	AUX
iajs-725	111	29	a	a	DET
iajs-725	111	30	faithful	faithful	ADJ
iajs-725	111	31	finitely	finitely	ADV
iajs-725	111	32	generated	generate	VERB
iajs-725	111	33	r	r	NOUN
iajs-725	111	34	-	-	PUNCT
iajs-725	111	35	module	module	NOUN
iajs-725	111	36	.	.	PUNCT
iajs-725	112	1	the	the	DET
iajs-725	112	2	following	follow	VERB
iajs-725	112	3	statements	statement	NOUN
iajs-725	112	4	are	be	AUX
iajs-725	112	5	equivalent	equivalent	ADJ
iajs-725	112	6	:	:	PUNCT
iajs-725	112	7	(	(	PUNCT
iajs-725	112	8	1	1	X
iajs-725	112	9	)	)	PUNCT
iajs-725	112	10	m	m	VERB
iajs-725	112	11	is	be	AUX
iajs-725	112	12	annsemimaximal	annsemimaximal	ADJ
iajs-725	112	13	.	.	PUNCT
iajs-725	113	1	(	(	PUNCT
iajs-725	113	2	2	2	X
iajs-725	113	3	)	)	PUNCT
iajs-725	113	4	m	m	VERB
iajs-725	113	5	is	be	AUX
iajs-725	113	6	semisimple	semisimple	ADJ
iajs-725	113	7	.	.	PUNCT
iajs-725	114	1	(	(	PUNCT
iajs-725	114	2	3	3	X
iajs-725	114	3	)	)	PUNCT
iajs-725	114	4	r	r	NOUN
iajs-725	114	5	is	be	AUX
iajs-725	114	6	semisimple	semisimple	ADJ
iajs-725	114	7	.	.	PUNCT
iajs-725	115	1	(	(	PUNCT
iajs-725	115	2	4	4	X
iajs-725	115	3	)	)	PUNCT
iajs-725	115	4	r	r	NOUN
iajs-725	115	5	is	be	AUX
iajs-725	115	6	annsemimaximal	annsemimaximal	ADJ
iajs-725	115	7	.	.	PUNCT
iajs-725	116	1	now	now	ADV
iajs-725	116	2	,	,	PUNCT
iajs-725	116	3	we	we	PRON
iajs-725	116	4	give	give	VERB
iajs-725	116	5	the	the	DET
iajs-725	116	6	following	follow	VERB
iajs-725	116	7	proposition	proposition	NOUN
iajs-725	116	8	.	.	PUNCT
iajs-725	117	1	1.17	1.17	NUM
iajs-725	117	2	proposition	proposition	NOUN
iajs-725	117	3	:	:	PUNCT
iajs-725	117	4	if	if	SCONJ
iajs-725	117	5	r	r	NOUN
iajs-725	117	6	is	be	AUX
iajs-725	117	7	a	a	DET
iajs-725	117	8	local	local	ADJ
iajs-725	117	9	ring	ring	NOUN
iajs-725	117	10	and	and	CCONJ
iajs-725	117	11	m	m	NOUN
iajs-725	117	12	is	be	AUX
iajs-725	117	13	annsemimaximal	annsemimaximal	ADJ
iajs-725	117	14	r	r	NOUN
iajs-725	117	15	-	-	PUNCT
iajs-725	117	16	module	module	NOUN
iajs-725	117	17	,	,	PUNCT
iajs-725	117	18	then	then	ADV
iajs-725	117	19	m	m	NOUN
iajs-725	117	20	is	be	AUX
iajs-725	117	21	semisimple	semisimple	ADJ
iajs-725	117	22	..	..	PUNCT
iajs-725	118	1	proof	proof	NOUN
iajs-725	118	2	:	:	PUNCT
iajs-725	118	3	m	m	VERB
iajs-725	118	4	is	be	AUX
iajs-725	118	5	annsemimaximal	annsemimaximal	ADJ
iajs-725	118	6	module	module	NOUN
iajs-725	118	7	.	.	PUNCT
iajs-725	119	1	then	then	ADV
iajs-725	119	2	annrm	annrm	PROPN
iajs-725	119	3	is	be	AUX
iajs-725	119	4	semimaximal	semimaximal	ADJ
iajs-725	119	5	ideal	ideal	ADJ
iajs-725	119	6	.	.	PUNCT
iajs-725	120	1	thus	thus	ADV
iajs-725	120	2	the	the	DET
iajs-725	120	3	result	result	NOUN
iajs-725	120	4	follows	follow	VERB
iajs-725	120	5	by	by	ADP
iajs-725	120	6	[	[	X
iajs-725	120	7	5,coro.(1.3.7	5,coro.(1.3.7	NOUN
iajs-725	120	8	)	)	PUNCT
iajs-725	120	9	]	]	PUNCT
iajs-725	120	10	.	.	PUNCT
iajs-725	121	1	1.18	1.18	NUM
iajs-725	121	2	proposition	proposition	NOUN
iajs-725	121	3	:	:	PUNCT
iajs-725	121	4	let	let	VERB
iajs-725	121	5	m	m	PROPN
iajs-725	121	6	1	1	NUM
iajs-725	121	7	,	,	PUNCT
iajs-725	121	8	m	m	VERB
iajs-725	121	9	2	2	NUM
iajs-725	121	10	be	be	VERB
iajs-725	121	11	two	two	NUM
iajs-725	121	12	r	r	NOUN
iajs-725	121	13	-	-	PUNCT
iajs-725	121	14	modules	module	NOUN
iajs-725	121	15	,	,	PUNCT
iajs-725	121	16	m	m	NOUN
iajs-725	121	17	=	=	NOUN
iajs-725	121	18	m	m	PROPN
iajs-725	121	19	1m	1m	NUM
iajs-725	121	20	2	2	NUM
iajs-725	121	21	.	.	PUNCT
iajs-725	122	1	then	then	ADV
iajs-725	122	2	m	m	PROPN
iajs-725	122	3	is	be	AUX
iajs-725	122	4	annsemimaximal	annsemimaximal	ADJ
iajs-725	122	5	if	if	SCONJ
iajs-725	122	6	and	and	CCONJ
iajs-725	122	7	only	only	ADV
iajs-725	122	8	if	if	SCONJ
iajs-725	122	9	m	m	PROPN
iajs-725	122	10	1	1	NUM
iajs-725	122	11	,	,	PUNCT
iajs-725	122	12	m	m	VERB
iajs-725	122	13	2	2	NUM
iajs-725	122	14	are	be	AUX
iajs-725	122	15	annsemimaximal	annsemimaximal	ADJ
iajs-725	122	16	r	r	NOUN
iajs-725	122	17	-	-	PUNCT
iajs-725	122	18	module	module	NOUN
iajs-725	122	19	.	.	PUNCT
iajs-725	123	1	proof	proof	NOUN
iajs-725	123	2	:	:	PUNCT
iajs-725	123	3	(	(	PUNCT
iajs-725	123	4			NOUN
iajs-725	123	5	)	)	PUNCT
iajs-725	123	6	let	let	VERB
iajs-725	123	7	1	1	NUM
iajs-725	123	8	:	:	PUNCT
iajs-725	123	9	m	m	PROPN
iajs-725	123	10			PROPN
iajs-725	123	11	m	m	VERB
iajs-725	123	12	1	1	NUM
iajs-725	123	13	,	,	PUNCT
iajs-725	123	14	2	2	PUNCT
iajs-725	123	15	:	:	PUNCT
iajs-725	123	16	m	m	PROPN
iajs-725	123	17			PROPN
iajs-725	123	18	m	m	VERB
iajs-725	123	19	2	2	NUM
iajs-725	123	20	be	be	VERB
iajs-725	123	21	the	the	DET
iajs-725	123	22	natural	natural	ADJ
iajs-725	123	23	projections	projection	NOUN
iajs-725	123	24	.	.	PUNCT
iajs-725	124	1	thus	thus	ADV
iajs-725	124	2	m	m	PROPN
iajs-725	124	3	1	1	NUM
iajs-725	124	4	and	and	CCONJ
iajs-725	124	5	m2	m2	PROPN
iajs-725	124	6	are	be	AUX
iajs-725	124	7	annsemimaximal	annsemimaximal	ADJ
iajs-725	124	8	modules	module	NOUN
iajs-725	124	9	by	by	ADP
iajs-725	124	10	remarks	remark	NOUN
iajs-725	124	11	and	and	CCONJ
iajs-725	124	12	examples	example	NOUN
iajs-725	124	13	(	(	PUNCT
iajs-725	124	14	(	(	PUNCT
iajs-725	124	15	1.2),(9	1.2),(9	NUM
iajs-725	124	16	)	)	PUNCT
iajs-725	124	17	)	)	PUNCT
iajs-725	124	18	.	.	PUNCT
iajs-725	125	1	(	(	PUNCT
iajs-725	125	2			NOUN
iajs-725	125	3	)	)	PUNCT
iajs-725	125	4	we	we	PRON
iajs-725	125	5	have	have	VERB
iajs-725	125	6	annrm	annrm	NOUN
iajs-725	125	7	1	1	NUM
iajs-725	125	8	is	be	AUX
iajs-725	125	9	semimaximal	semimaximal	ADJ
iajs-725	125	10	ideal	ideal	NOUN
iajs-725	125	11	and	and	CCONJ
iajs-725	125	12	annrm	annrm	NOUN
iajs-725	125	13	2	2	NUM
iajs-725	125	14	is	be	AUX
iajs-725	125	15	semimaximal	semimaximal	ADJ
iajs-725	125	16	by	by	ADP
iajs-725	125	17	proposition	proposition	NOUN
iajs-725	125	18	(	(	PUNCT
iajs-725	125	19	1.3	1.3	NUM
iajs-725	125	20	)	)	PUNCT
iajs-725	125	21	.	.	PUNCT
iajs-725	126	1	on	on	ADP
iajs-725	126	2	the	the	DET
iajs-725	126	3	other	other	ADJ
iajs-725	126	4	hand	hand	NOUN
iajs-725	126	5	annr(m	annr(m	NOUN
iajs-725	126	6	1m	1m	NOUN
iajs-725	126	7	2	2	NUM
iajs-725	126	8	)	)	PUNCT
iajs-725	126	9	=	=	NOUN
iajs-725	126	10	annrm	annrm	NOUN
iajs-725	126	11	1	1	NUM
iajs-725	126	12	annrm	annrm	NOUN
iajs-725	126	13	2	2	NUM
iajs-725	126	14	.	.	PUNCT
iajs-725	127	1	but	but	CCONJ
iajs-725	127	2	by	by	ADP
iajs-725	127	3	[	[	X
iajs-725	127	4	5	5	NUM
iajs-725	127	5	,	,	PUNCT
iajs-725	127	6	coro.(1.2.15	coro.(1.2.15	PROPN
iajs-725	127	7	)	)	PUNCT
iajs-725	127	8	]	]	PUNCT
iajs-725	127	9	,	,	PUNCT
iajs-725	127	10	ibn	ibn	PROPN
iajs-725	127	11	alhaitham	alhaitham	NOUN
iajs-725	127	12	j.	j.	PROPN
iajs-725	127	13	for	for	ADP
iajs-725	127	14	pure	pure	ADJ
iajs-725	127	15	&	&	CCONJ
iajs-725	127	16	appl	appl	PROPN
iajs-725	127	17	.	.	PUNCT
iajs-725	128	1	sci	sci	PROPN
iajs-725	128	2	.	.	PUNCT
iajs-725	128	3	vol.24	vol.24	NOUN
iajs-725	128	4	(	(	PUNCT
iajs-725	128	5	3	3	NUM
iajs-725	128	6	)	)	PUNCT
iajs-725	128	7	2011	2011	NUM
iajs-725	128	8	annrm	annrm	NOUN
iajs-725	128	9	1	1	NUM
iajs-725	128	10	annrm	annrm	NOUN
iajs-725	128	11	2	2	NUM
iajs-725	128	12	is	be	AUX
iajs-725	128	13	semimaximal	semimaximal	ADJ
iajs-725	128	14	.	.	PUNCT
iajs-725	129	1	therefore	therefore	ADV
iajs-725	129	2	annr(m	annr(m	ADV
iajs-725	129	3	1m	1m	NOUN
iajs-725	129	4	2	2	NUM
iajs-725	129	5	)	)	PUNCT
iajs-725	129	6	is	be	AUX
iajs-725	129	7	semimaximal	semimaximal	ADJ
iajs-725	129	8	.	.	PUNCT
iajs-725	130	1	thus	thus	ADV
iajs-725	130	2	m	m	VERB
iajs-725	130	3	1m	1m	NUM
iajs-725	130	4	2	2	NUM
iajs-725	130	5	is	be	AUX
iajs-725	130	6	annsemimaximal	annsemimaximal	ADJ
iajs-725	130	7	module	module	NOUN
iajs-725	130	8	,	,	PUNCT
iajs-725	130	9	by	by	ADP
iajs-725	130	10	prop.(1.3	prop.(1.3	NOUN
iajs-725	130	11	)	)	PUNCT
iajs-725	130	12	.	.	PUNCT
iajs-725	131	1	..	..	PUNCT
iajs-725	131	2	recall	recall	VERB
iajs-725	131	3	that	that	SCONJ
iajs-725	131	4	an	an	DET
iajs-725	131	5	r	r	NOUN
iajs-725	131	6	-	-	PUNCT
iajs-725	131	7	module	module	NOUN
iajs-725	131	8	m	m	NOUN
iajs-725	131	9	is	be	AUX
iajs-725	131	10	called	call	VERB
iajs-725	131	11	semiprime	semiprime	NOUN
iajs-725	131	12	if	if	SCONJ
iajs-725	131	13	and	and	CCONJ
iajs-725	131	14	only	only	ADV
iajs-725	131	15	if	if	SCONJ
iajs-725	131	16	annrn	annrn	NOUN
iajs-725	131	17	is	be	AUX
iajs-725	131	18	a	a	DET
iajs-725	131	19	semiprime	semiprime	NOUN
iajs-725	131	20	ideal	ideal	NOUN
iajs-725	131	21	of	of	ADP
iajs-725	131	22	r	r	NOUN
iajs-725	131	23	,	,	PUNCT
iajs-725	131	24	for	for	ADP
iajs-725	131	25	each	each	DET
iajs-725	131	26	non	non	ADJ
iajs-725	131	27	-	-	ADJ
iajs-725	131	28	zero	zero	ADJ
iajs-725	131	29	r	r	NOUN
iajs-725	131	30	-	-	PUNCT
iajs-725	131	31	submodule	submodule	NOUN
iajs-725	131	32	n	n	PROPN
iajs-725	131	33	of	of	ADP
iajs-725	131	34	m	m	PRON
iajs-725	131	35	,	,	PUNCT
iajs-725	132	1	[	[	X
iajs-725	132	2	7,def.(4.1.1	7,def.(4.1.1	NUM
iajs-725	132	3	)	)	PUNCT
iajs-725	132	4	]	]	PUNCT
iajs-725	132	5	.	.	PUNCT
iajs-725	133	1	by	by	ADP
iajs-725	133	2	using	use	VERB
iajs-725	133	3	this	this	DET
iajs-725	133	4	concept	concept	NOUN
iajs-725	133	5	,	,	PUNCT
iajs-725	133	6	we	we	PRON
iajs-725	133	7	have	have	VERB
iajs-725	133	8	the	the	DET
iajs-725	133	9	following	following	NOUN
iajs-725	133	10	.	.	PUNCT
iajs-725	134	1	1.19	1.19	NUM
iajs-725	134	2	proposition	proposition	NOUN
iajs-725	134	3	:	:	PUNCT
iajs-725	134	4	every	every	DET
iajs-725	134	5	annsemimaximal	annsemimaximal	ADJ
iajs-725	134	6	r	r	NOUN
iajs-725	134	7	-	-	PUNCT
iajs-725	134	8	module	module	NOUN
iajs-725	134	9	is	be	AUX
iajs-725	134	10	semiprime	semiprime	NOUN
iajs-725	134	11	r	r	NOUN
iajs-725	134	12	-	-	PUNCT
iajs-725	134	13	module	module	NOUN
iajs-725	134	14	.	.	PUNCT
iajs-725	135	1	proof	proof	NOUN
iajs-725	135	2	:	:	PUNCT
iajs-725	135	3	let	let	VERB
iajs-725	135	4	m	m	PRON
iajs-725	135	5	be	be	AUX
iajs-725	135	6	an	an	DET
iajs-725	135	7	annsemimaximal	annsemimaximal	ADJ
iajs-725	135	8	module	module	NOUN
iajs-725	135	9	.	.	PUNCT
iajs-725	136	1	then	then	ADV
iajs-725	136	2	for	for	ADP
iajs-725	136	3	each	each	DET
iajs-725	136	4	non	non	ADJ
iajs-725	136	5	-	-	ADJ
iajs-725	136	6	zero	zero	NUM
iajs-725	136	7	submodule	submodule	NOUN
iajs-725	136	8	n	n	PROPN
iajs-725	136	9	of	of	ADP
iajs-725	136	10	m	m	PROPN
iajs-725	136	11	,	,	PUNCT
iajs-725	136	12	annrn	annrn	NOUN
iajs-725	136	13	is	be	AUX
iajs-725	136	14	semimaximal	semimaximal	ADJ
iajs-725	136	15	ideal	ideal	NOUN
iajs-725	136	16	of	of	ADP
iajs-725	136	17	r.	r.	PROPN
iajs-725	136	18	thus	thus	ADV
iajs-725	136	19	by	by	ADP
iajs-725	136	20	[	[	X
iajs-725	136	21	5,prop.(1.2.21	5,prop.(1.2.21	NUM
iajs-725	136	22	)	)	PUNCT
iajs-725	136	23	]	]	PUNCT
iajs-725	136	24	,	,	PUNCT
iajs-725	136	25	annrn	annrn	NOUN
iajs-725	136	26	is	be	AUX
iajs-725	136	27	semiprime	semiprime	NOUN
iajs-725	136	28	and	and	CCONJ
iajs-725	136	29	hence	hence	ADV
iajs-725	136	30	m	m	VERB
iajs-725	136	31	is	be	AUX
iajs-725	136	32	a	a	DET
iajs-725	136	33	semiprime	semiprime	NOUN
iajs-725	136	34	module	module	NOUN
iajs-725	136	35	.	.	PUNCT
iajs-725	137	1	the	the	DET
iajs-725	137	2	converse	converse	NOUN
iajs-725	137	3	of	of	ADP
iajs-725	137	4	this	this	DET
iajs-725	137	5	proposition	proposition	NOUN
iajs-725	137	6	is	be	AUX
iajs-725	137	7	not	not	PART
iajs-725	137	8	true	true	ADJ
iajs-725	137	9	in	in	ADP
iajs-725	137	10	general	general	ADJ
iajs-725	137	11	.	.	PUNCT
iajs-725	138	1	for	for	ADP
iajs-725	138	2	example	example	NOUN
iajs-725	138	3	:	:	PUNCT
iajs-725	138	4	z	z	NOUN
iajs-725	138	5	as	as	ADP
iajs-725	138	6	a	a	DET
iajs-725	138	7	z	z	NOUN
iajs-725	138	8	-	-	PUNCT
iajs-725	138	9	module	module	NOUN
iajs-725	138	10	is	be	AUX
iajs-725	138	11	semiprime	semiprime	NOUN
iajs-725	138	12	module	module	NOUN
iajs-725	138	13	,	,	PUNCT
iajs-725	138	14	but	but	CCONJ
iajs-725	138	15	it	it	PRON
iajs-725	138	16	is	be	AUX
iajs-725	138	17	not	not	PART
iajs-725	138	18	annsemimaximal	annsemimaximal	ADJ
iajs-725	138	19	module	module	NOUN
iajs-725	138	20	by	by	ADP
iajs-725	138	21	remarks	remark	NOUN
iajs-725	138	22	and	and	CCONJ
iajs-725	138	23	examples	example	NOUN
iajs-725	138	24	(	(	PUNCT
iajs-725	138	25	(	(	PUNCT
iajs-725	138	26	1.2),(3	1.2),(3	NUM
iajs-725	138	27	)	)	PUNCT
iajs-725	138	28	)	)	PUNCT
iajs-725	138	29	.	.	PUNCT
iajs-725	139	1	for	for	ADP
iajs-725	139	2	our	our	PRON
iajs-725	139	3	next	next	ADJ
iajs-725	139	4	corollary	corollary	NOUN
iajs-725	139	5	the	the	DET
iajs-725	139	6	following	follow	VERB
iajs-725	139	7	definitions	definition	NOUN
iajs-725	139	8	are	be	AUX
iajs-725	139	9	needed	need	VERB
iajs-725	139	10	.	.	PUNCT
iajs-725	140	1	an	an	DET
iajs-725	140	2	r	r	NOUN
iajs-725	140	3	-	-	PUNCT
iajs-725	140	4	module	module	NOUN
iajs-725	140	5	m	m	NOUN
iajs-725	140	6	is	be	AUX
iajs-725	140	7	said	say	VERB
iajs-725	140	8	to	to	PART
iajs-725	140	9	be	be	AUX
iajs-725	140	10	serial	serial	ADJ
iajs-725	140	11	(	(	PUNCT
iajs-725	140	12	chain	chain	NOUN
iajs-725	140	13	)	)	PUNCT
iajs-725	140	14	r	r	NOUN
iajs-725	140	15	-	-	PUNCT
iajs-725	140	16	module	module	NOUN
iajs-725	140	17	if	if	SCONJ
iajs-725	140	18	the	the	DET
iajs-725	140	19	r	r	NOUN
iajs-725	140	20	-	-	PUNCT
iajs-725	140	21	submodules	submodule	NOUN
iajs-725	140	22	of	of	ADP
iajs-725	140	23	m	m	NOUN
iajs-725	140	24	are	be	AUX
iajs-725	140	25	linearly	linearly	ADV
iajs-725	140	26	orderd	orderd	ADJ
iajs-725	140	27	with	with	ADP
iajs-725	140	28	respect	respect	NOUN
iajs-725	140	29	to	to	ADP
iajs-725	140	30	inclusion	inclusion	NOUN
iajs-725	140	31	,	,	PUNCT
iajs-725	141	1	[	[	X
iajs-725	141	2	6	6	NUM
iajs-725	141	3	]	]	PUNCT
iajs-725	141	4	,	,	PUNCT
iajs-725	141	5	[	[	X
iajs-725	141	6	7	7	NUM
iajs-725	141	7	]	]	PUNCT
iajs-725	141	8	.	.	PUNCT
iajs-725	142	1	an	an	DET
iajs-725	142	2	r	r	NOUN
iajs-725	142	3	-	-	PUNCT
iajs-725	142	4	module	module	NOUN
iajs-725	142	5	m	m	NOUN
iajs-725	142	6	is	be	AUX
iajs-725	142	7	said	say	VERB
iajs-725	142	8	to	to	PART
iajs-725	142	9	be	be	AUX
iajs-725	142	10	a	a	DET
iajs-725	142	11	prime	prime	ADJ
iajs-725	142	12	module	module	NOUN
iajs-725	142	13	if	if	SCONJ
iajs-725	142	14	annrm	annrm	NOUN
iajs-725	142	15	=	=	NOUN
iajs-725	142	16	annrn	annrn	NOUN
iajs-725	142	17	for	for	ADP
iajs-725	142	18	every	every	DET
iajs-725	142	19	non	non	ADJ
iajs-725	142	20	-	-	ADJ
iajs-725	142	21	zero	zero	NUM
iajs-725	142	22	submodule	submodule	NOUN
iajs-725	142	23	n	n	PROPN
iajs-725	142	24	of	of	ADP
iajs-725	142	25	m	m	PRON
iajs-725	142	26	,	,	PUNCT
iajs-725	143	1	[	[	X
iajs-725	143	2	8	8	NUM
iajs-725	143	3	]	]	PUNCT
iajs-725	143	4	,	,	PUNCT
iajs-725	143	5	[	[	X
iajs-725	143	6	9	9	NUM
iajs-725	143	7	]	]	PUNCT
iajs-725	143	8	.	.	PUNCT
iajs-725	144	1	as	as	ADP
iajs-725	144	2	an	an	DET
iajs-725	144	3	application	application	NOUN
iajs-725	144	4	of	of	ADP
iajs-725	144	5	proposition	proposition	NOUN
iajs-725	144	6	(	(	PUNCT
iajs-725	144	7	1.19	1.19	NUM
iajs-725	144	8	)	)	PUNCT
iajs-725	144	9	,	,	PUNCT
iajs-725	144	10	we	we	PRON
iajs-725	144	11	give	give	VERB
iajs-725	144	12	the	the	DET
iajs-725	144	13	following	follow	VERB
iajs-725	144	14	corollary	corollary	NOUN
iajs-725	144	15	.	.	PUNCT
iajs-725	145	1	1.20	1.20	NUM
iajs-725	145	2	corollary	corollary	NOUN
iajs-725	145	3	:	:	PUNCT
iajs-725	145	4	let	let	VERB
iajs-725	145	5	m	m	PRON
iajs-725	145	6	be	be	AUX
iajs-725	145	7	a	a	DET
iajs-725	145	8	serial	serial	ADJ
iajs-725	145	9	annsemimaximal	annsemimaximal	ADJ
iajs-725	145	10	module	module	NOUN
iajs-725	145	11	.	.	PUNCT
iajs-725	146	1	then	then	ADV
iajs-725	146	2	m	m	PROPN
iajs-725	146	3	is	be	AUX
iajs-725	146	4	prime	prime	ADJ
iajs-725	146	5	r	r	NOUN
iajs-725	146	6	-	-	PUNCT
iajs-725	146	7	module	module	NOUN
iajs-725	146	8	.	.	PUNCT
iajs-725	147	1	proof	proof	NOUN
iajs-725	147	2	:	:	PUNCT
iajs-725	147	3	from	from	ADP
iajs-725	147	4	proposition	proposition	NOUN
iajs-725	147	5	(	(	PUNCT
iajs-725	147	6	1.19	1.19	NUM
iajs-725	147	7	)	)	PUNCT
iajs-725	147	8	,	,	PUNCT
iajs-725	147	9	m	m	VERB
iajs-725	147	10	is	be	AUX
iajs-725	147	11	semiprime	semiprime	NOUN
iajs-725	147	12	module	module	NOUN
iajs-725	147	13	and	and	CCONJ
iajs-725	147	14	from	from	ADP
iajs-725	147	15	[	[	X
iajs-725	147	16	7,prop.(4.2.1	7,prop.(4.2.1	NOUN
iajs-725	147	17	)	)	PUNCT
iajs-725	147	18	]	]	PUNCT
iajs-725	147	19	,	,	PUNCT
iajs-725	147	20	we	we	PRON
iajs-725	147	21	get	get	VERB
iajs-725	147	22	the	the	DET
iajs-725	147	23	result	result	NOUN
iajs-725	147	24	.	.	PUNCT
iajs-725	148	1	recall	recall	VERB
iajs-725	148	2	that	that	SCONJ
iajs-725	148	3	an	an	DET
iajs-725	148	4	r	r	NOUN
iajs-725	148	5	-	-	PUNCT
iajs-725	148	6	module	module	NOUN
iajs-725	148	7	m	m	NOUN
iajs-725	148	8	is	be	AUX
iajs-725	148	9	said	say	VERB
iajs-725	148	10	to	to	PART
iajs-725	148	11	be	be	AUX
iajs-725	148	12	a	a	DET
iajs-725	148	13	max	max	NOUN
iajs-725	148	14	-	-	PUNCT
iajs-725	148	15	module	module	NOUN
iajs-725	148	16	if	if	SCONJ
iajs-725	148	17	rann	rann	PROPN
iajs-725	148	18	n	n	ADV
iajs-725	148	19	is	be	AUX
iajs-725	148	20	maximal	maximal	ADJ
iajs-725	148	21	ideal	ideal	NOUN
iajs-725	148	22	of	of	ADP
iajs-725	148	23	r	r	NOUN
iajs-725	148	24	for	for	ADP
iajs-725	148	25	each	each	DET
iajs-725	148	26	non	non	ADJ
iajs-725	148	27	-	-	ADJ
iajs-725	148	28	zero	zero	NUM
iajs-725	148	29	submodule	submodule	NOUN
iajs-725	148	30	n	n	PROPN
iajs-725	148	31	of	of	ADP
iajs-725	148	32	m	m	PRON
iajs-725	148	33	,	,	PUNCT
iajs-725	149	1	[	[	X
iajs-725	149	2	3	3	NUM
iajs-725	149	3	]	]	PUNCT
iajs-725	149	4	.	.	PUNCT
iajs-725	150	1	in	in	ADP
iajs-725	150	2	the	the	DET
iajs-725	150	3	class	class	NOUN
iajs-725	150	4	of	of	ADP
iajs-725	150	5	max	max	PROPN
iajs-725	150	6	-	-	PUNCT
iajs-725	150	7	module	module	NOUN
iajs-725	150	8	.	.	PUNCT
iajs-725	151	1	the	the	DET
iajs-725	151	2	two	two	NUM
iajs-725	151	3	concept	concept	NOUN
iajs-725	151	4	of	of	ADP
iajs-725	151	5	annsemimaximal	annsemimaximal	ADJ
iajs-725	151	6	module	module	NOUN
iajs-725	151	7	and	and	CCONJ
iajs-725	151	8	semiprime	semiprime	NOUN
iajs-725	151	9	module	module	NOUN
iajs-725	151	10	are	be	AUX
iajs-725	151	11	equivalent	equivalent	ADJ
iajs-725	151	12	.	.	PUNCT
iajs-725	152	1	1.21	1.21	NUM
iajs-725	152	2	proposition	proposition	NOUN
iajs-725	152	3	:	:	PUNCT
iajs-725	152	4	let	let	VERB
iajs-725	152	5	m	m	PRON
iajs-725	152	6	be	be	AUX
iajs-725	152	7	a	a	DET
iajs-725	152	8	max	max	NOUN
iajs-725	152	9	-	-	PUNCT
iajs-725	152	10	module	module	NOUN
iajs-725	152	11	.	.	PUNCT
iajs-725	153	1	then	then	ADV
iajs-725	153	2	m	m	PROPN
iajs-725	153	3	is	be	AUX
iajs-725	153	4	annsemimaximal	annsemimaximal	ADJ
iajs-725	153	5	module	module	NOUN
iajs-725	153	6	if	if	SCONJ
iajs-725	153	7	and	and	CCONJ
iajs-725	153	8	only	only	ADV
iajs-725	153	9	if	if	SCONJ
iajs-725	153	10	m	m	NOUN
iajs-725	153	11	is	be	AUX
iajs-725	153	12	semiprime	semiprime	NOUN
iajs-725	153	13	module	module	NOUN
iajs-725	153	14	.	.	PUNCT
iajs-725	154	1	proof	proof	NOUN
iajs-725	154	2	:	:	PUNCT
iajs-725	154	3	suppose	suppose	VERB
iajs-725	154	4	that	that	SCONJ
iajs-725	154	5	m	m	PROPN
iajs-725	154	6	is	be	AUX
iajs-725	154	7	semiprime	semiprime	NOUN
iajs-725	154	8	r	r	NOUN
iajs-725	154	9	-	-	PUNCT
iajs-725	154	10	module	module	NOUN
iajs-725	154	11	.	.	PUNCT
iajs-725	155	1	then	then	ADV
iajs-725	155	2	for	for	ADP
iajs-725	155	3	each	each	DET
iajs-725	155	4	a	a	DET
iajs-725	155	5	non	non	ADJ
iajs-725	155	6	-	-	ADJ
iajs-725	155	7	zero	zero	NUM
iajs-725	155	8	submodule	submodule	NOUN
iajs-725	155	9	n	n	PROPN
iajs-725	155	10	of	of	ADP
iajs-725	155	11	m	m	PROPN
iajs-725	155	12	,	,	PUNCT
iajs-725	155	13	annrn	annrn	NOUN
iajs-725	155	14	is	be	AUX
iajs-725	155	15	semiprime	semiprime	NOUN
iajs-725	155	16	ideal	ideal	NOUN
iajs-725	155	17	of	of	ADP
iajs-725	155	18	r	r	NOUN
iajs-725	155	19	,	,	PUNCT
iajs-725	155	20	that	that	PRON
iajs-725	155	21	is	be	AUX
iajs-725	155	22	annrn=	annrn=	NUM
iajs-725	155	23	rann	rann	PROPN
iajs-725	155	24	n	n	PROPN
iajs-725	155	25	for	for	ADP
iajs-725	155	26	each	each	DET
iajs-725	155	27	non	non	ADJ
iajs-725	155	28	-	-	ADJ
iajs-725	155	29	zero	zero	NUM
iajs-725	155	30	submodule	submodule	NOUN
iajs-725	155	31	n	n	PROPN
iajs-725	155	32	of	of	ADP
iajs-725	155	33	m.	m.	NOUN
iajs-725	155	34	but	but	CCONJ
iajs-725	155	35	m	m	PROPN
iajs-725	155	36	is	be	AUX
iajs-725	155	37	max	max	NOUN
iajs-725	155	38	-	-	PUNCT
iajs-725	155	39	module	module	NOUN
iajs-725	155	40	which	which	PRON
iajs-725	155	41	implies	imply	VERB
iajs-725	155	42	that	that	SCONJ
iajs-725	155	43	rann	rann	PROPN
iajs-725	155	44	n	n	ADV
iajs-725	155	45	is	be	AUX
iajs-725	155	46	maximal	maximal	ADJ
iajs-725	155	47	ideal	ideal	NOUN
iajs-725	155	48	of	of	ADP
iajs-725	155	49	r	r	NOUN
iajs-725	155	50	for	for	ADP
iajs-725	155	51	each	each	DET
iajs-725	155	52	non	non	ADJ
iajs-725	155	53	-	-	ADJ
iajs-725	155	54	zero	zero	NUM
iajs-725	155	55	submodule	submodule	NOUN
iajs-725	155	56	n	n	PROPN
iajs-725	155	57	of	of	ADP
iajs-725	155	58	m	m	PRON
iajs-725	155	59	and	and	CCONJ
iajs-725	155	60	hence	hence	ADV
iajs-725	155	61	annrn	annrn	NOUN
iajs-725	155	62	is	be	AUX
iajs-725	155	63	maximal	maximal	ADJ
iajs-725	155	64	ideal	ideal	NOUN
iajs-725	155	65	for	for	ADP
iajs-725	155	66	each	each	DET
iajs-725	155	67	non	non	ADJ
iajs-725	155	68	-	-	ADJ
iajs-725	155	69	zero	zero	NUM
iajs-725	155	70	submodule	submodule	NOUN
iajs-725	155	71	n	n	PROPN
iajs-725	155	72	of	of	ADP
iajs-725	155	73	m	m	VERB
iajs-725	155	74	by	by	ADP
iajs-725	155	75	[	[	X
iajs-725	155	76	5,rem.(1.2.2),(2	5,rem.(1.2.2),(2	NUM
iajs-725	155	77	)	)	PUNCT
iajs-725	155	78	]	]	PUNCT
iajs-725	155	79	,	,	PUNCT
iajs-725	155	80	annrn	annrn	NOUN
iajs-725	155	81	is	be	AUX
iajs-725	155	82	semimaximal	semimaximal	ADJ
iajs-725	155	83	ideal	ideal	NOUN
iajs-725	155	84	of	of	ADP
iajs-725	155	85	r	r	NOUN
iajs-725	155	86	and	and	CCONJ
iajs-725	155	87	hence	hence	ADV
iajs-725	155	88	m	m	VERB
iajs-725	155	89	is	be	AUX
iajs-725	155	90	annsemimaximal	annsemimaximal	ADJ
iajs-725	155	91	module	module	NOUN
iajs-725	155	92	.	.	PUNCT
iajs-725	156	1	conversely	conversely	ADV
iajs-725	156	2	:	:	PUNCT
iajs-725	156	3	it	it	PRON
iajs-725	156	4	follows	follow	VERB
iajs-725	156	5	by	by	ADP
iajs-725	156	6	proposition	proposition	NOUN
iajs-725	156	7	(	(	PUNCT
iajs-725	156	8	1.19	1.19	NUM
iajs-725	156	9	)	)	PUNCT
iajs-725	156	10	.	.	PUNCT
iajs-725	157	1	now	now	ADV
iajs-725	157	2	,	,	PUNCT
iajs-725	157	3	the	the	DET
iajs-725	157	4	following	follow	VERB
iajs-725	157	5	results	result	NOUN
iajs-725	157	6	are	be	AUX
iajs-725	157	7	other	other	ADJ
iajs-725	157	8	consequences	consequence	NOUN
iajs-725	157	9	of	of	ADP
iajs-725	157	10	proposition	proposition	NOUN
iajs-725	157	11	(	(	PUNCT
iajs-725	157	12	1.21	1.21	NUM
iajs-725	157	13	)	)	PUNCT
iajs-725	157	14	,	,	PUNCT
iajs-725	157	15	but	but	CCONJ
iajs-725	157	16	first	first	ADV
iajs-725	157	17	we	we	PRON
iajs-725	157	18	need	need	VERB
iajs-725	157	19	to	to	PART
iajs-725	157	20	recall	recall	VERB
iajs-725	157	21	some	some	DET
iajs-725	157	22	definitions	definition	NOUN
iajs-725	157	23	.	.	PUNCT
iajs-725	158	1	an	an	DET
iajs-725	158	2	r	r	NOUN
iajs-725	158	3	-	-	PUNCT
iajs-725	158	4	module	module	NOUN
iajs-725	158	5	m	m	NOUN
iajs-725	158	6	is	be	AUX
iajs-725	158	7	called	call	VERB
iajs-725	158	8	z	z	ADJ
iajs-725	158	9	-	-	PUNCT
iajs-725	158	10	regular	regular	ADJ
iajs-725	158	11	module	module	NOUN
iajs-725	158	12	if	if	SCONJ
iajs-725	158	13	for	for	ADP
iajs-725	158	14	all	all	DET
iajs-725	158	15	mm	mm	NUM
iajs-725	158	16	,	,	PUNCT
iajs-725	158	17	there	there	PRON
iajs-725	158	18	exists	exist	VERB
iajs-725	158	19	fhomr(m	fhomr(m	X
iajs-725	158	20	,	,	PUNCT
iajs-725	158	21	r)=m	r)=m	NOUN
iajs-725	158	22	*	*	PUNCT
iajs-725	158	23	such	such	ADJ
iajs-725	158	24	that	that	SCONJ
iajs-725	158	25	f(m)m	f(m)m	PROPN
iajs-725	158	26	=	=	SYM
iajs-725	158	27	m	m	PROPN
iajs-725	158	28	,	,	PUNCT
iajs-725	158	29	[	[	X
iajs-725	158	30	10	10	NUM
iajs-725	158	31	]	]	PUNCT
iajs-725	158	32	.	.	PUNCT
iajs-725	159	1	an	an	DET
iajs-725	159	2	r	r	NOUN
iajs-725	159	3	-	-	PUNCT
iajs-725	159	4	submodule	submodule	NOUN
iajs-725	159	5	n	n	PROPN
iajs-725	159	6	of	of	ADP
iajs-725	159	7	m	m	PROPN
iajs-725	159	8	is	be	AUX
iajs-725	159	9	called	call	VERB
iajs-725	159	10	essential	essential	ADJ
iajs-725	159	11	in	in	ADP
iajs-725	159	12	m	m	PROPN
iajs-725	159	13	if	if	SCONJ
iajs-725	159	14	for	for	ADP
iajs-725	159	15	each	each	DET
iajs-725	159	16	non	non	ADJ
iajs-725	159	17	-	-	ADJ
iajs-725	159	18	zero	zero	ADJ
iajs-725	159	19	r	r	NOUN
iajs-725	159	20	-	-	PUNCT
iajs-725	159	21	submodule	submodule	NOUN
iajs-725	159	22	l	l	NOUN
iajs-725	159	23	of	of	ADP
iajs-725	159	24	m	m	PROPN
iajs-725	159	25	,	,	PUNCT
iajs-725	159	26	nl0	nl0	INTJ
iajs-725	159	27	,	,	PUNCT
iajs-725	159	28	[	[	X
iajs-725	159	29	6].and	6].and	NUM
iajs-725	159	30	an	an	DET
iajs-725	159	31	r	r	NOUN
iajs-725	159	32	-	-	PUNCT
iajs-725	159	33	module	module	NOUN
iajs-725	159	34	m	m	NOUN
iajs-725	159	35	is	be	AUX
iajs-725	159	36	called	call	VERB
iajs-725	159	37	uniform	uniform	ADJ
iajs-725	159	38	if	if	SCONJ
iajs-725	159	39	every	every	DET
iajs-725	159	40	non	non	ADJ
iajs-725	159	41	-	-	ADJ
iajs-725	159	42	zero	zero	ADJ
iajs-725	159	43	r	r	NOUN
iajs-725	159	44	-	-	PUNCT
iajs-725	159	45	submodule	submodule	NOUN
iajs-725	159	46	of	of	ADP
iajs-725	159	47	m	m	PROPN
iajs-725	159	48	is	be	AUX
iajs-725	159	49	essential	essential	ADJ
iajs-725	159	50	.	.	PUNCT
iajs-725	160	1	an	an	DET
iajs-725	160	2	r	r	NOUN
iajs-725	160	3	-	-	PUNCT
iajs-725	160	4	submodule	submodule	NOUN
iajs-725	160	5	n	n	PROPN
iajs-725	160	6	of	of	ADP
iajs-725	160	7	m	m	PROPN
iajs-725	160	8	is	be	AUX
iajs-725	160	9	called	call	VERB
iajs-725	160	10	quasi	quasi	ADJ
iajs-725	160	11	-	-	ADJ
iajs-725	160	12	invertible	invertible	ADJ
iajs-725	160	13	if	if	SCONJ
iajs-725	160	14	hom	hom	X
iajs-725	160	15	(	(	PUNCT
iajs-725	160	16			PROPN
iajs-725	160	17			NOUN
iajs-725	160	18	,	,	PUNCT
iajs-725	160	19	m)=0	m)=0	PROPN
iajs-725	160	20	.	.	PROPN
iajs-725	161	1	and	and	CCONJ
iajs-725	161	2	an	an	DET
iajs-725	161	3	r	r	NOUN
iajs-725	161	4	-	-	PUNCT
iajs-725	161	5	module	module	NOUN
iajs-725	161	6	m	m	NOUN
iajs-725	161	7	is	be	AUX
iajs-725	161	8	called	call	VERB
iajs-725	161	9	quasi	quasi	ADJ
iajs-725	161	10	-	-	NOUN
iajs-725	161	11	dedekind	dedekind	ADJ
iajs-725	161	12	if	if	SCONJ
iajs-725	161	13	every	every	DET
iajs-725	161	14	non	non	ADJ
iajs-725	161	15	-	-	ADJ
iajs-725	161	16	zero	zero	ADJ
iajs-725	161	17	r	r	NOUN
iajs-725	161	18	-	-	PUNCT
iajs-725	161	19	submodule	submodule	NOUN
iajs-725	161	20	of	of	ADP
iajs-725	161	21	m	m	PROPN
iajs-725	161	22	is	be	AUX
iajs-725	161	23	quasi	quasi	ADJ
iajs-725	161	24	-	-	ADJ
iajs-725	161	25	invertible	invertible	ADJ
iajs-725	161	26	,	,	PUNCT
iajs-725	161	27	[	[	X
iajs-725	161	28	11	11	NUM
iajs-725	161	29	]	]	PUNCT
iajs-725	161	30	.	.	PUNCT
iajs-725	162	1	ibn	ibn	PROPN
iajs-725	162	2	alhaitham	alhaitham	PROPN
iajs-725	162	3	j.	j.	PROPN
iajs-725	162	4	for	for	ADP
iajs-725	162	5	pure	pure	ADJ
iajs-725	162	6	&	&	CCONJ
iajs-725	162	7	appl	appl	PROPN
iajs-725	162	8	.	.	PUNCT
iajs-725	163	1	sci	sci	PROPN
iajs-725	163	2	.	.	PUNCT
iajs-725	163	3	vol.24	vol.24	NOUN
iajs-725	163	4	(	(	PUNCT
iajs-725	163	5	3	3	NUM
iajs-725	163	6	)	)	PUNCT
iajs-725	163	7	2011	2011	NUM
iajs-725	163	8	hence	hence	ADV
iajs-725	163	9	,	,	PUNCT
iajs-725	163	10	we	we	PRON
iajs-725	163	11	have	have	VERB
iajs-725	163	12	the	the	DET
iajs-725	163	13	following	follow	VERB
iajs-725	163	14	consequences	consequence	NOUN
iajs-725	163	15	of	of	ADP
iajs-725	163	16	(	(	PUNCT
iajs-725	163	17	1.21	1.21	NUM
iajs-725	163	18	)	)	PUNCT
iajs-725	163	19	.	.	PUNCT
iajs-725	164	1	1.22	1.22	NUM
iajs-725	164	2	corollary	corollary	NOUN
iajs-725	164	3	:	:	PUNCT
iajs-725	164	4	if	if	SCONJ
iajs-725	164	5	m	m	NOUN
iajs-725	164	6	is	be	AUX
iajs-725	164	7	max	max	NOUN
iajs-725	164	8	-	-	PUNCT
iajs-725	164	9	module	module	NOUN
iajs-725	164	10	and	and	CCONJ
iajs-725	164	11	z	z	NOUN
iajs-725	164	12	-	-	PUNCT
iajs-725	164	13	regular	regular	ADJ
iajs-725	164	14	module	module	NOUN
iajs-725	164	15	.	.	PUNCT
iajs-725	165	1	thus	thus	ADV
iajs-725	165	2	m	m	PROPN
iajs-725	165	3	is	be	AUX
iajs-725	165	4	annsemimaximal	annsemimaximal	ADJ
iajs-725	165	5	module	module	NOUN
iajs-725	165	6	.	.	PUNCT
iajs-725	166	1	proof	proof	NOUN
iajs-725	166	2	:	:	PUNCT
iajs-725	166	3	it	it	PRON
iajs-725	166	4	follows	follow	VERB
iajs-725	166	5	directly	directly	ADV
iajs-725	166	6	from	from	ADP
iajs-725	166	7	proposition	proposition	NOUN
iajs-725	166	8	(	(	PUNCT
iajs-725	166	9	1.21	1.21	NUM
iajs-725	166	10	)	)	PUNCT
iajs-725	166	11	and	and	CCONJ
iajs-725	167	1	[	[	X
iajs-725	167	2	7,prop.(4.2.2	7,prop.(4.2.2	NOUN
iajs-725	167	3	)	)	PUNCT
iajs-725	167	4	]	]	PUNCT
iajs-725	167	5	.	.	PUNCT
iajs-725	168	1	1.23	1.23	NUM
iajs-725	168	2	corollary	corollary	NOUN
iajs-725	168	3	:	:	PUNCT
iajs-725	168	4	let	let	VERB
iajs-725	168	5	m	m	PRON
iajs-725	168	6	be	be	AUX
iajs-725	168	7	a	a	DET
iajs-725	168	8	uniform	uniform	ADJ
iajs-725	168	9	annsemimaximal	annsemimaximal	ADJ
iajs-725	168	10	r	r	NOUN
iajs-725	168	11	-	-	PUNCT
iajs-725	168	12	module	module	NOUN
iajs-725	168	13	.	.	PUNCT
iajs-725	169	1	then	then	ADV
iajs-725	169	2	m	m	PROPN
iajs-725	169	3	is	be	AUX
iajs-725	169	4	quasi	quasi	ADJ
iajs-725	169	5	-	-	ADJ
iajs-725	169	6	dedekind	dedekind	ADJ
iajs-725	169	7	.	.	PUNCT
iajs-725	170	1	proof	proof	NOUN
iajs-725	170	2	:	:	PUNCT
iajs-725	170	3	m	m	VERB
iajs-725	170	4	is	be	AUX
iajs-725	170	5	annsemimaximal	annsemimaximal	ADJ
iajs-725	170	6	module	module	NOUN
iajs-725	170	7	,	,	PUNCT
iajs-725	170	8	then	then	ADV
iajs-725	170	9	m	m	NOUN
iajs-725	170	10	is	be	AUX
iajs-725	170	11	semiprime	semiprime	NOUN
iajs-725	170	12	by	by	ADP
iajs-725	170	13	proposition	proposition	NOUN
iajs-725	170	14	(	(	PUNCT
iajs-725	170	15	1.21	1.21	NUM
iajs-725	170	16	)	)	PUNCT
iajs-725	170	17	and	and	CCONJ
iajs-725	170	18	by	by	ADP
iajs-725	170	19	[	[	X
iajs-725	170	20	7,prop.(4.2.4	7,prop.(4.2.4	NUM
iajs-725	170	21	)	)	PUNCT
iajs-725	170	22	]	]	PUNCT
iajs-725	170	23	,	,	PUNCT
iajs-725	170	24	we	we	PRON
iajs-725	170	25	get	get	VERB
iajs-725	170	26	the	the	DET
iajs-725	170	27	result	result	NOUN
iajs-725	170	28	.	.	PUNCT
iajs-725	171	1	now	now	ADV
iajs-725	171	2	,	,	PUNCT
iajs-725	171	3	we	we	PRON
iajs-725	171	4	can	can	AUX
iajs-725	171	5	give	give	VERB
iajs-725	171	6	the	the	DET
iajs-725	171	7	following	follow	VERB
iajs-725	171	8	proposition	proposition	NOUN
iajs-725	171	9	.	.	PUNCT
iajs-725	172	1	1.24	1.24	NUM
iajs-725	172	2	proposition	proposition	NOUN
iajs-725	172	3	:	:	PUNCT
iajs-725	172	4	let	let	VERB
iajs-725	172	5	m	m	PRON
iajs-725	172	6	be	be	AUX
iajs-725	172	7	a	a	DET
iajs-725	172	8	uniform	uniform	ADJ
iajs-725	172	9	max	max	PROPN
iajs-725	172	10	-	-	PUNCT
iajs-725	172	11	r	r	NOUN
iajs-725	172	12	-	-	PUNCT
iajs-725	172	13	module	module	NOUN
iajs-725	172	14	.	.	PUNCT
iajs-725	173	1	then	then	ADV
iajs-725	173	2	the	the	DET
iajs-725	173	3	following	follow	VERB
iajs-725	173	4	statements	statement	NOUN
iajs-725	173	5	are	be	AUX
iajs-725	173	6	equivalent	equivalent	ADJ
iajs-725	173	7	.	.	PUNCT
iajs-725	174	1	(	(	PUNCT
iajs-725	174	2	1	1	X
iajs-725	174	3	)	)	PUNCT
iajs-725	174	4	m	m	VERB
iajs-725	174	5	is	be	AUX
iajs-725	174	6	annsemimaximal	annsemimaximal	ADJ
iajs-725	174	7	module	module	NOUN
iajs-725	174	8	.	.	PUNCT
iajs-725	175	1	(	(	PUNCT
iajs-725	175	2	2	2	X
iajs-725	175	3	)	)	PUNCT
iajs-725	175	4	m	m	VERB
iajs-725	175	5	is	be	AUX
iajs-725	175	6	seiprime	seiprime	ADJ
iajs-725	175	7	module	module	NOUN
iajs-725	175	8	.	.	PUNCT
iajs-725	176	1	(	(	PUNCT
iajs-725	176	2	3	3	X
iajs-725	176	3	)	)	PUNCT
iajs-725	176	4	m	m	VERB
iajs-725	176	5	is	be	AUX
iajs-725	176	6	quasi	quasi	ADJ
iajs-725	176	7	-	-	ADJ
iajs-725	176	8	dedekind	dedekind	ADJ
iajs-725	176	9	.	.	PUNCT
iajs-725	177	1	(	(	PUNCT
iajs-725	177	2	4	4	X
iajs-725	177	3	)	)	PUNCT
iajs-725	177	4	m	m	VERB
iajs-725	177	5	is	be	AUX
iajs-725	177	6	prime	prime	ADJ
iajs-725	177	7	.	.	PUNCT
iajs-725	178	1	proof	proof	NOUN
iajs-725	178	2	:	:	PUNCT
iajs-725	178	3	(	(	PUNCT
iajs-725	178	4	1	1	X
iajs-725	178	5	)	)	PUNCT
iajs-725	178	6			NOUN
iajs-725	178	7	(	(	PUNCT
iajs-725	178	8	2	2	NUM
iajs-725	178	9	)	)	PUNCT
iajs-725	178	10	by	by	ADP
iajs-725	178	11	proposition	proposition	NOUN
iajs-725	178	12	(	(	PUNCT
iajs-725	178	13	1.19	1.19	NUM
iajs-725	178	14	)	)	PUNCT
iajs-725	178	15	.	.	PUNCT
iajs-725	179	1	(	(	PUNCT
iajs-725	179	2	2	2	X
iajs-725	179	3	)	)	PUNCT
iajs-725	179	4			NOUN
iajs-725	179	5	(	(	PUNCT
iajs-725	179	6	3	3	NUM
iajs-725	179	7	)	)	PUNCT
iajs-725	179	8	by	by	ADP
iajs-725	179	9	[	[	X
iajs-725	179	10	7,prop.(4.2.4	7,prop.(4.2.4	NUM
iajs-725	179	11	)	)	PUNCT
iajs-725	179	12	]	]	PUNCT
iajs-725	179	13	.	.	PUNCT
iajs-725	180	1	(	(	PUNCT
iajs-725	180	2	3	3	X
iajs-725	180	3	)	)	PUNCT
iajs-725	180	4			NOUN
iajs-725	180	5	(	(	PUNCT
iajs-725	180	6	4	4	NUM
iajs-725	180	7	)	)	PUNCT
iajs-725	180	8	by	by	ADP
iajs-725	180	9	[	[	X
iajs-725	180	10	11,prop.(1.7	11,prop.(1.7	NUM
iajs-725	180	11	)	)	PUNCT
iajs-725	180	12	,	,	PUNCT
iajs-725	180	13	ch.2	ch.2	PROPN
iajs-725	180	14	]	]	PUNCT
iajs-725	180	15	.	.	PUNCT
iajs-725	181	1	(	(	PUNCT
iajs-725	181	2	4	4	NUM
iajs-725	181	3	)	)	PUNCT
iajs-725	181	4			NOUN
iajs-725	181	5	(	(	PUNCT
iajs-725	181	6	1	1	X
iajs-725	181	7	)	)	PUNCT
iajs-725	181	8	it	it	PRON
iajs-725	181	9	is	be	AUX
iajs-725	181	10	clear	clear	ADJ
iajs-725	181	11	that	that	SCONJ
iajs-725	181	12	every	every	DET
iajs-725	181	13	prime	prime	ADJ
iajs-725	181	14	module	module	NOUN
iajs-725	181	15	is	be	AUX
iajs-725	181	16	semiprime	semiprime	NOUN
iajs-725	181	17	module	module	NOUN
iajs-725	181	18	and	and	CCONJ
iajs-725	181	19	hence	hence	ADV
iajs-725	181	20	by	by	ADP
iajs-725	181	21	proposition	proposition	NOUN
iajs-725	181	22	(	(	PUNCT
iajs-725	181	23	1.21	1.21	NUM
iajs-725	181	24	)	)	PUNCT
iajs-725	181	25	we	we	PRON
iajs-725	181	26	get	get	VERB
iajs-725	181	27	the	the	DET
iajs-725	181	28	result	result	NOUN
iajs-725	181	29	.	.	PUNCT
iajs-725	182	1	recall	recall	VERB
iajs-725	182	2	that	that	SCONJ
iajs-725	182	3	an	an	DET
iajs-725	182	4	r	r	NOUN
iajs-725	182	5	-	-	PUNCT
iajs-725	182	6	module	module	NOUN
iajs-725	182	7	m	m	NOUN
iajs-725	182	8	is	be	AUX
iajs-725	182	9	said	say	VERB
iajs-725	182	10	to	to	PART
iajs-725	182	11	be	be	AUX
iajs-725	182	12	regular	regular	ADJ
iajs-725	182	13	module	module	NOUN
iajs-725	182	14	if	if	SCONJ
iajs-725	182	15	r	r	NOUN
iajs-725	182	16	/	/	SYM
iajs-725	182	17	annr(x	annr(x	NOUN
iajs-725	182	18	)	)	PUNCT
iajs-725	182	19	is	be	AUX
iajs-725	182	20	regular	regular	ADJ
iajs-725	182	21	ring	ring	NOUN
iajs-725	182	22	for	for	ADP
iajs-725	182	23	all	all	DET
iajs-725	182	24	0	0	NUM
iajs-725	182	25			NOUN
iajs-725	182	26	x	x	SYM
iajs-725	182	27			NOUN
iajs-725	182	28	m	m	VERB
iajs-725	182	29	,	,	PUNCT
iajs-725	182	30	[	[	X
iajs-725	182	31	5	5	NUM
iajs-725	182	32	]	]	PUNCT
iajs-725	182	33	.	.	PUNCT
iajs-725	183	1	by	by	ADP
iajs-725	183	2	using	use	VERB
iajs-725	183	3	this	this	DET
iajs-725	183	4	concept	concept	NOUN
iajs-725	183	5	,	,	PUNCT
iajs-725	183	6	we	we	PRON
iajs-725	183	7	have	have	VERB
iajs-725	183	8	the	the	DET
iajs-725	183	9	following	following	NOUN
iajs-725	183	10	.	.	PUNCT
iajs-725	184	1	1.25	1.25	NUM
iajs-725	184	2	remark	remark	NOUN
iajs-725	184	3	:	:	PUNCT
iajs-725	184	4	every	every	DET
iajs-725	184	5	annsemimaximal	annsemimaximal	ADJ
iajs-725	184	6	module	module	NOUN
iajs-725	184	7	is	be	AUX
iajs-725	184	8	regular	regular	ADJ
iajs-725	184	9	module	module	NOUN
iajs-725	184	10	.	.	PUNCT
iajs-725	185	1	proof	proof	NOUN
iajs-725	185	2	:	:	PUNCT
iajs-725	185	3	let	let	VERB
iajs-725	185	4	m	m	PRON
iajs-725	185	5	be	be	AUX
iajs-725	185	6	annsemimaximal	annsemimaximal	ADJ
iajs-725	185	7	r	r	NOUN
iajs-725	185	8	-	-	PUNCT
iajs-725	185	9	module	module	NOUN
iajs-725	185	10	.	.	PUNCT
iajs-725	186	1	then	then	ADV
iajs-725	186	2	annrm	annrm	PROPN
iajs-725	186	3	is	be	AUX
iajs-725	186	4	semimaximal	semimaximal	ADJ
iajs-725	186	5	ideal	ideal	NOUN
iajs-725	186	6	and	and	CCONJ
iajs-725	186	7	by	by	ADP
iajs-725	186	8	[	[	X
iajs-725	186	9	5,prop.(1.3.5	5,prop.(1.3.5	PROPN
iajs-725	186	10	)	)	PUNCT
iajs-725	186	11	]	]	PUNCT
iajs-725	186	12	,	,	PUNCT
iajs-725	186	13	m	m	VERB
iajs-725	186	14	is	be	AUX
iajs-725	186	15	regular	regular	ADJ
iajs-725	186	16	module	module	NOUN
iajs-725	186	17	.	.	PUNCT
iajs-725	187	1	1.26	1.26	NUM
iajs-725	187	2	proposition	proposition	NOUN
iajs-725	187	3	:	:	PUNCT
iajs-725	187	4	if	if	SCONJ
iajs-725	187	5	m	m	NOUN
iajs-725	187	6	is	be	AUX
iajs-725	187	7	annsemimaximal	annsemimaximal	ADJ
iajs-725	187	8	r	r	NOUN
iajs-725	187	9	-	-	PUNCT
iajs-725	187	10	module	module	NOUN
iajs-725	187	11	,	,	PUNCT
iajs-725	187	12	then	then	ADV
iajs-725	187	13	m	m	PROPN
iajs-725	187	14	/	/	SYM
iajs-725	187	15	n	n	PROPN
iajs-725	187	16	is	be	AUX
iajs-725	187	17	regular	regular	ADJ
iajs-725	187	18	r	r	NOUN
iajs-725	187	19	-	-	PUNCT
iajs-725	187	20	module	module	NOUN
iajs-725	187	21	for	for	ADP
iajs-725	187	22	all	all	DET
iajs-725	187	23	submodules	submodule	NOUN
iajs-725	187	24	n	n	INTJ
iajs-725	187	25	of	of	ADP
iajs-725	187	26	m.	m.	NOUN
iajs-725	187	27	proof	proof	NOUN
iajs-725	187	28	:	:	PUNCT
iajs-725	187	29	let	let	VERB
iajs-725	187	30	m	m	NOUN
iajs-725	187	31	is	be	AUX
iajs-725	187	32	annsemimaximal	annsemimaximal	ADJ
iajs-725	187	33	module	module	NOUN
iajs-725	187	34	.	.	PUNCT
iajs-725	188	1	then	then	ADV
iajs-725	188	2	annrm	annrm	PROPN
iajs-725	188	3	is	be	AUX
iajs-725	188	4	semimaximal	semimaximal	ADJ
iajs-725	188	5	.	.	PUNCT
iajs-725	189	1	but	but	CCONJ
iajs-725	189	2	annrm	annrm	PROPN
iajs-725	189	3			PROPN
iajs-725	190	1	[	[	X
iajs-725	190	2	n	n	X
iajs-725	190	3	r	r	NOUN
iajs-725	190	4	:	:	PUNCT
iajs-725	190	5	m	m	X
iajs-725	190	6	]	]	X
iajs-725	190	7	for	for	ADP
iajs-725	190	8	all	all	DET
iajs-725	190	9	submodule	submodule	NOUN
iajs-725	190	10	n	n	PROPN
iajs-725	190	11	of	of	ADP
iajs-725	190	12	m	m	PRON
iajs-725	190	13	,	,	PUNCT
iajs-725	190	14	so	so	SCONJ
iajs-725	191	1	[	[	X
iajs-725	191	2	n	n	X
iajs-725	191	3	r	r	NOUN
iajs-725	191	4	:	:	PUNCT
iajs-725	191	5	m	m	VERB
iajs-725	191	6	]	]	X
iajs-725	191	7	is	be	AUX
iajs-725	191	8	semimaximal	semimaximal	ADJ
iajs-725	191	9	ideal	ideal	NOUN
iajs-725	191	10	by	by	ADP
iajs-725	191	11	[	[	X
iajs-725	191	12	5,prop.(1.2.11	5,prop.(1.2.11	NUM
iajs-725	191	13	)	)	PUNCT
iajs-725	191	14	]	]	PUNCT
iajs-725	191	15	.	.	PUNCT
iajs-725	192	1	hence	hence	ADV
iajs-725	192	2	m	m	PROPN
iajs-725	192	3	/	/	SYM
iajs-725	192	4	n	n	PROPN
iajs-725	192	5	is	be	AUX
iajs-725	192	6	regular	regular	ADJ
iajs-725	192	7	r	r	NOUN
iajs-725	192	8	-	-	PUNCT
iajs-725	192	9	module	module	NOUN
iajs-725	192	10	by	by	ADP
iajs-725	192	11	[	[	X
iajs-725	192	12	5,prop.(1.3.8	5,prop.(1.3.8	NUM
iajs-725	192	13	)	)	PUNCT
iajs-725	192	14	]	]	PUNCT
iajs-725	192	15	.	.	PUNCT
iajs-725	193	1	the	the	DET
iajs-725	193	2	jacobson	jacobson	PROPN
iajs-725	193	3	radical	radical	PROPN
iajs-725	193	4	of	of	ADP
iajs-725	193	5	an	an	DET
iajs-725	193	6	r	r	NOUN
iajs-725	193	7	-	-	PUNCT
iajs-725	193	8	module	module	NOUN
iajs-725	193	9	m	m	NOUN
iajs-725	193	10	denoted	denote	VERB
iajs-725	193	11	by	by	ADP
iajs-725	193	12	j(m	j(m	PROPN
iajs-725	193	13	)	)	PUNCT
iajs-725	193	14	,	,	PUNCT
iajs-725	193	15	is	be	AUX
iajs-725	193	16	defined	define	VERB
iajs-725	193	17	to	to	PART
iajs-725	193	18	be	be	AUX
iajs-725	193	19	the	the	DET
iajs-725	193	20	intersection	intersection	NOUN
iajs-725	193	21	of	of	ADP
iajs-725	193	22	all	all	DET
iajs-725	193	23	maximal	maximal	ADJ
iajs-725	193	24	submodules	submodule	NOUN
iajs-725	193	25	of	of	ADP
iajs-725	193	26	m	m	PRON
iajs-725	193	27	,	,	PUNCT
iajs-725	193	28	in	in	ADP
iajs-725	193	29	case	case	NOUN
iajs-725	193	30	m	m	NOUN
iajs-725	193	31	has	have	VERB
iajs-725	193	32	maximal	maximal	ADJ
iajs-725	193	33	submodules	submodule	NOUN
iajs-725	193	34	and	and	CCONJ
iajs-725	193	35	j(m	j(m	PROPN
iajs-725	193	36	)	)	PUNCT
iajs-725	194	1	=	=	X
iajs-725	194	2	m	m	VERB
iajs-725	194	3	in	in	ADP
iajs-725	194	4	case	case	NOUN
iajs-725	194	5	m	m	NOUN
iajs-725	194	6	has	have	VERB
iajs-725	194	7	no	no	DET
iajs-725	194	8	maximal	maximal	ADJ
iajs-725	194	9	submodule	submodule	NOUN
iajs-725	194	10	,	,	PUNCT
iajs-725	194	11	[	[	X
iajs-725	194	12	6	6	NUM
iajs-725	194	13	]	]	PUNCT
iajs-725	194	14	.	.	PUNCT
iajs-725	195	1	1.27	1.27	NUM
iajs-725	195	2	remark	remark	NOUN
iajs-725	195	3	:	:	PUNCT
iajs-725	195	4	let	let	VERB
iajs-725	195	5	m	m	PRON
iajs-725	195	6	be	be	AUX
iajs-725	195	7	an	an	DET
iajs-725	195	8	annsemimaximal	annsemimaximal	ADJ
iajs-725	195	9	r	r	NOUN
iajs-725	195	10	-	-	PUNCT
iajs-725	195	11	module	module	NOUN
iajs-725	195	12	.	.	PUNCT
iajs-725	196	1	then	then	ADV
iajs-725	196	2	j(m)=0	j(m)=0	X
iajs-725	196	3	.	.	PUNCT
iajs-725	197	1	proof	proof	NOUN
iajs-725	197	2	:	:	PUNCT
iajs-725	197	3	it	it	PRON
iajs-725	197	4	is	be	AUX
iajs-725	197	5	abvious	abvious	ADJ
iajs-725	197	6	according	accord	VERB
iajs-725	197	7	to	to	ADP
iajs-725	197	8	[	[	X
iajs-725	197	9	5,coro.(1.3.6	5,coro.(1.3.6	NUM
iajs-725	197	10	)	)	PUNCT
iajs-725	197	11	]	]	PUNCT
iajs-725	197	12	.	.	PUNCT
iajs-725	198	1	recall	recall	VERB
iajs-725	198	2	that	that	SCONJ
iajs-725	198	3	an	an	DET
iajs-725	198	4	r	r	NOUN
iajs-725	198	5	-	-	PUNCT
iajs-725	198	6	module	module	NOUN
iajs-725	198	7	m	m	NOUN
iajs-725	198	8	is	be	AUX
iajs-725	198	9	called	call	VERB
iajs-725	198	10	f	f	X
iajs-725	198	11	-	-	PUNCT
iajs-725	198	12	regular	regular	ADJ
iajs-725	198	13	if	if	SCONJ
iajs-725	198	14	every	every	DET
iajs-725	198	15	submodule	submodule	NOUN
iajs-725	198	16	of	of	ADP
iajs-725	198	17	m	m	PROPN
iajs-725	198	18	is	be	AUX
iajs-725	198	19	pure	pure	ADJ
iajs-725	199	1	[	[	X
iajs-725	199	2	12,ch.2	12,ch.2	NUM
iajs-725	199	3	]	]	X
iajs-725	199	4	.	.	PUNCT
iajs-725	200	1	by	by	ADP
iajs-725	200	2	using	use	VERB
iajs-725	200	3	this	this	DET
iajs-725	200	4	concept	concept	NOUN
iajs-725	200	5	,	,	PUNCT
iajs-725	200	6	we	we	PRON
iajs-725	200	7	give	give	VERB
iajs-725	200	8	the	the	DET
iajs-725	200	9	following	follow	VERB
iajs-725	200	10	proposition	proposition	NOUN
iajs-725	200	11	.	.	PUNCT
iajs-725	201	1	1.28	1.28	NUM
iajs-725	201	2	proposition	proposition	NOUN
iajs-725	201	3	:	:	PUNCT
iajs-725	201	4	if	if	SCONJ
iajs-725	201	5	m	m	NOUN
iajs-725	201	6	is	be	AUX
iajs-725	201	7	annsemimaximal	annsemimaximal	ADJ
iajs-725	201	8	r	r	NOUN
iajs-725	201	9	-	-	PUNCT
iajs-725	201	10	module	module	NOUN
iajs-725	201	11	,	,	PUNCT
iajs-725	201	12	then	then	ADV
iajs-725	201	13	m	m	PROPN
iajs-725	201	14	is	be	AUX
iajs-725	201	15	f	f	NOUN
iajs-725	201	16	-	-	PUNCT
iajs-725	201	17	regular	regular	ADJ
iajs-725	201	18	.	.	PUNCT
iajs-725	202	1	ibn	ibn	PROPN
iajs-725	202	2	alhaitham	alhaitham	PROPN
iajs-725	202	3	j.	j.	PROPN
iajs-725	202	4	for	for	ADP
iajs-725	202	5	pure	pure	ADJ
iajs-725	202	6	&	&	CCONJ
iajs-725	202	7	appl	appl	PROPN
iajs-725	202	8	.	.	PUNCT
iajs-725	203	1	sci	sci	PROPN
iajs-725	203	2	.	.	PUNCT
iajs-725	203	3	vol.24	vol.24	NOUN
iajs-725	203	4	(	(	PUNCT
iajs-725	203	5	3	3	NUM
iajs-725	203	6	)	)	PUNCT
iajs-725	203	7	2011	2011	NUM
iajs-725	203	8	proof	proof	NOUN
iajs-725	203	9	:	:	PUNCT
iajs-725	203	10	we	we	PRON
iajs-725	203	11	have	have	VERB
iajs-725	203	12	m	m	PROPN
iajs-725	203	13	is	be	AUX
iajs-725	203	14	annsemimaximal	annsemimaximal	ADJ
iajs-725	203	15	,	,	PUNCT
iajs-725	203	16	then	then	ADV
iajs-725	203	17	annrm	annrm	NOUN
iajs-725	203	18	is	be	AUX
iajs-725	203	19	semimaximal	semimaximal	ADJ
iajs-725	203	20	.	.	PUNCT
iajs-725	204	1	thus	thus	ADV
iajs-725	204	2	every	every	DET
iajs-725	204	3	cyclic	cyclic	ADJ
iajs-725	204	4	submodule	submodule	NOUN
iajs-725	204	5	is	be	AUX
iajs-725	204	6	pure	pure	ADJ
iajs-725	204	7	by	by	ADP
iajs-725	204	8	[	[	X
iajs-725	204	9	5,prop.(1.3.9	5,prop.(1.3.9	NUM
iajs-725	204	10	)	)	PUNCT
iajs-725	204	11	]	]	PUNCT
iajs-725	204	12	.	.	PUNCT
iajs-725	205	1	hence	hence	ADV
iajs-725	205	2	m	m	PROPN
iajs-725	205	3	is	be	AUX
iajs-725	205	4	f	f	NOUN
iajs-725	205	5	-	-	PUNCT
iajs-725	205	6	regular	regular	ADJ
iajs-725	205	7	.	.	PUNCT
iajs-725	206	1	1.29	1.29	NUM
iajs-725	206	2	proposition	proposition	NOUN
iajs-725	206	3	:	:	PUNCT
iajs-725	206	4	let	let	VERB
iajs-725	206	5	r	r	PRON
iajs-725	206	6	be	be	AUX
iajs-725	206	7	a	a	DET
iajs-725	206	8	pid	pid	NOUN
iajs-725	206	9	,	,	PUNCT
iajs-725	206	10	annrm0	annrm0	NOUN
iajs-725	206	11	,	,	PUNCT
iajs-725	206	12	m	m	VERB
iajs-725	206	13	is	be	AUX
iajs-725	206	14	prime	prime	ADJ
iajs-725	206	15	r	r	NOUN
iajs-725	206	16	-	-	PUNCT
iajs-725	206	17	module	module	NOUN
iajs-725	206	18	.	.	PUNCT
iajs-725	207	1	then	then	ADV
iajs-725	207	2	m	m	PROPN
iajs-725	207	3	is	be	AUX
iajs-725	207	4	annsemimaximal	annsemimaximal	ADJ
iajs-725	207	5	module	module	NOUN
iajs-725	207	6	.	.	PUNCT
iajs-725	208	1	proof	proof	NOUN
iajs-725	208	2	:	:	PUNCT
iajs-725	208	3	since	since	SCONJ
iajs-725	208	4	m	m	PROPN
iajs-725	208	5	is	be	AUX
iajs-725	208	6	prime	prime	ADJ
iajs-725	208	7	r	r	NOUN
iajs-725	208	8	-	-	PUNCT
iajs-725	208	9	module	module	NOUN
iajs-725	208	10	.	.	PUNCT
iajs-725	209	1	then	then	ADV
iajs-725	209	2	annrm	annrm	PROPN
iajs-725	209	3	is	be	AUX
iajs-725	209	4	prime	prime	ADJ
iajs-725	209	5	ideal	ideal	NOUN
iajs-725	209	6	which	which	PRON
iajs-725	209	7	implies	imply	VERB
iajs-725	209	8	that	that	SCONJ
iajs-725	209	9	annrm	annrm	NOUN
iajs-725	209	10	is	be	AUX
iajs-725	209	11	maximal	maximal	ADJ
iajs-725	209	12	ideal	ideal	ADJ
iajs-725	209	13	(	(	PUNCT
iajs-725	209	14	since	since	SCONJ
iajs-725	209	15	r	r	NOUN
iajs-725	209	16	is	be	AUX
iajs-725	209	17	pid	pid	NOUN
iajs-725	209	18	)	)	PUNCT
iajs-725	209	19	.	.	PUNCT
iajs-725	210	1	thus	thus	ADV
iajs-725	210	2	annrm	annrm	NOUN
iajs-725	210	3	is	be	AUX
iajs-725	210	4	semimaximal	semimaximal	ADJ
iajs-725	210	5	ideal	ideal	NOUN
iajs-725	210	6	of	of	ADP
iajs-725	210	7	r.	r.	PROPN
iajs-725	210	8	hence	hence	ADV
iajs-725	210	9	m	m	VERB
iajs-725	210	10	is	be	AUX
iajs-725	210	11	annsemimaximal	annsemimaximal	ADJ
iajs-725	210	12	r	r	NOUN
iajs-725	210	13	-	-	PUNCT
iajs-725	210	14	module	module	NOUN
iajs-725	210	15	,	,	PUNCT
iajs-725	210	16	by	by	ADP
iajs-725	210	17	proposition	proposition	NOUN
iajs-725	210	18	(	(	PUNCT
iajs-725	210	19	1.3	1.3	NUM
iajs-725	210	20	)	)	PUNCT
iajs-725	210	21	.	.	PUNCT
iajs-725	211	1	the	the	DET
iajs-725	211	2	converse	converse	NOUN
iajs-725	211	3	of	of	ADP
iajs-725	211	4	proposition	proposition	NOUN
iajs-725	211	5	(	(	PUNCT
iajs-725	211	6	1.29	1.29	NUM
iajs-725	211	7	)	)	PUNCT
iajs-725	211	8	is	be	AUX
iajs-725	211	9	not	not	PART
iajs-725	211	10	true	true	ADJ
iajs-725	211	11	,	,	PUNCT
iajs-725	211	12	for	for	ADP
iajs-725	211	13	example	example	NOUN
iajs-725	211	14	:	:	PUNCT
iajs-725	211	15	z6	z6	PROPN
iajs-725	211	16	as	as	SCONJ
iajs-725	211	17	z	z	NOUN
iajs-725	211	18	-	-	PUNCT
iajs-725	211	19	module	module	NOUN
iajs-725	211	20	is	be	AUX
iajs-725	211	21	annsemimaximal	annsemimaximal	ADJ
iajs-725	211	22	.	.	PUNCT
iajs-725	212	1	but	but	CCONJ
iajs-725	212	2	m	m	PROPN
iajs-725	212	3	is	be	AUX
iajs-725	212	4	not	not	PART
iajs-725	212	5	prime	prime	ADJ
iajs-725	212	6	.	.	PUNCT
iajs-725	213	1	recall	recall	VERB
iajs-725	213	2	that	that	SCONJ
iajs-725	213	3	an	an	DET
iajs-725	213	4	r	r	NOUN
iajs-725	213	5	-	-	PUNCT
iajs-725	213	6	module	module	NOUN
iajs-725	213	7	m	m	NOUN
iajs-725	213	8	is	be	AUX
iajs-725	213	9	flat	flat	ADJ
iajs-725	213	10	if	if	SCONJ
iajs-725	213	11	for	for	ADP
iajs-725	213	12	each	each	DET
iajs-725	213	13	injective	injective	ADJ
iajs-725	213	14	homomorphisim	homomorphisim	NOUN
iajs-725	214	1	f	f	X
iajs-725	214	2	:	:	PUNCT
iajs-725	214	3	n	n	CCONJ
iajs-725	214	4	'	'	CCONJ
iajs-725	214	5			PROPN
iajs-725	214	6	n	n	X
iajs-725	214	7	from	from	ADP
iajs-725	214	8	one	one	NUM
iajs-725	214	9	r	r	NOUN
iajs-725	214	10	-	-	PUNCT
iajs-725	214	11	module	module	NOUN
iajs-725	214	12	into	into	ADP
iajs-725	214	13	another	another	PRON
iajs-725	214	14	,	,	PUNCT
iajs-725	214	15	the	the	DET
iajs-725	214	16	homomorphisim	homomorphisim	NOUN
iajs-725	214	17	1mf	1mf	NUM
iajs-725	214	18	:	:	PUNCT
iajs-725	214	19	m	m	VERB
iajs-725	214	20	r	r	NOUN
iajs-725	214	21	n'	n'	PROPN
iajs-725	214	22	m	m	PROPN
iajs-725	214	23	r	r	NOUN
iajs-725	214	24	n	n	NOUN
iajs-725	214	25	is	be	AUX
iajs-725	214	26	injective	injective	ADJ
iajs-725	214	27	,	,	PUNCT
iajs-725	214	28	where	where	SCONJ
iajs-725	214	29	1	1	NUM
iajs-725	214	30	m	m	NOUN
iajs-725	214	31	is	be	AUX
iajs-725	214	32	the	the	DET
iajs-725	214	33	identity	identity	NOUN
iajs-725	214	34	isomorphisim	isomorphisim	NOUN
iajs-725	214	35	of	of	ADP
iajs-725	214	36	m	m	PRON
iajs-725	214	37	,	,	PUNCT
iajs-725	214	38	[	[	X
iajs-725	214	39	6	6	NUM
iajs-725	214	40	]	]	PUNCT
iajs-725	214	41	.	.	PUNCT
iajs-725	215	1	1.30	1.30	NUM
iajs-725	215	2	proposition	proposition	NOUN
iajs-725	215	3	:	:	PUNCT
iajs-725	215	4	if	if	SCONJ
iajs-725	215	5	m	m	NOUN
iajs-725	215	6	is	be	AUX
iajs-725	215	7	flat	flat	ADJ
iajs-725	215	8	annsemimaximal	annsemimaximal	ADJ
iajs-725	215	9	r	r	NOUN
iajs-725	215	10	-	-	PUNCT
iajs-725	215	11	module	module	NOUN
iajs-725	215	12	,	,	PUNCT
iajs-725	215	13	then	then	ADV
iajs-725	215	14	every	every	DET
iajs-725	215	15	homomorphic	homomorphic	ADJ
iajs-725	215	16	image	image	NOUN
iajs-725	215	17	of	of	ADP
iajs-725	215	18	m	m	PROPN
iajs-725	215	19	is	be	AUX
iajs-725	215	20	flat	flat	ADJ
iajs-725	215	21	.	.	PUNCT
iajs-725	216	1	proof	proof	NOUN
iajs-725	216	2	:	:	PUNCT
iajs-725	216	3	we	we	PRON
iajs-725	216	4	have	have	VERB
iajs-725	216	5	m	m	PROPN
iajs-725	216	6	is	be	AUX
iajs-725	216	7	annsemimaximal	annsemimaximal	ADJ
iajs-725	216	8	,	,	PUNCT
iajs-725	216	9	then	then	ADV
iajs-725	216	10	annrm	annrm	NOUN
iajs-725	216	11	is	be	AUX
iajs-725	216	12	semimaximal	semimaximal	ADJ
iajs-725	216	13	ideal	ideal	ADJ
iajs-725	216	14	.	.	PUNCT
iajs-725	217	1	thus	thus	ADV
iajs-725	217	2	by	by	ADP
iajs-725	217	3	[	[	X
iajs-725	217	4	5,prop.(1.3.10	5,prop.(1.3.10	NUM
iajs-725	217	5	)	)	PUNCT
iajs-725	217	6	]	]	PUNCT
iajs-725	217	7	,	,	PUNCT
iajs-725	217	8	we	we	PRON
iajs-725	217	9	get	get	VERB
iajs-725	217	10	the	the	DET
iajs-725	217	11	result	result	NOUN
iajs-725	217	12	.	.	PUNCT
iajs-725	218	1	next	next	ADV
iajs-725	218	2	,	,	PUNCT
iajs-725	218	3	we	we	PRON
iajs-725	218	4	introduce	introduce	VERB
iajs-725	218	5	the	the	DET
iajs-725	218	6	following	following	ADJ
iajs-725	218	7	definition	definition	NOUN
iajs-725	218	8	.	.	PUNCT
iajs-725	219	1	1.31	1.31	NUM
iajs-725	219	2	definition	definition	NOUN
iajs-725	219	3	:	:	PUNCT
iajs-725	219	4	let	let	VERB
iajs-725	219	5	n	n	PRON
iajs-725	219	6	be	be	AUX
iajs-725	219	7	a	a	DET
iajs-725	219	8	proper	proper	ADJ
iajs-725	219	9	submodule	submodule	NOUN
iajs-725	219	10	of	of	ADP
iajs-725	219	11	an	an	DET
iajs-725	219	12	r	r	NOUN
iajs-725	219	13	-	-	PUNCT
iajs-725	219	14	module	module	NOUN
iajs-725	219	15	m.	m.	NOUN
iajs-725	219	16	n	n	CCONJ
iajs-725	219	17	is	be	AUX
iajs-725	219	18	called	call	VERB
iajs-725	219	19	quasi	quasi	ADJ
iajs-725	219	20	-	-	ADJ
iajs-725	219	21	semimaximal	semimaximal	ADJ
iajs-725	219	22	if	if	SCONJ
iajs-725	219	23	[	[	X
iajs-725	219	24	n	n	X
iajs-725	219	25	r	r	NOUN
iajs-725	219	26	:	:	PUNCT
iajs-725	219	27	(	(	PUNCT
iajs-725	219	28	m	m	NOUN
iajs-725	219	29	)	)	PUNCT
iajs-725	219	30	]	]	PUNCT
iajs-725	219	31	is	be	AUX
iajs-725	219	32	a	a	DET
iajs-725	219	33	semimaximal	semimaximal	ADJ
iajs-725	219	34	ideal	ideal	NOUN
iajs-725	219	35	for	for	ADP
iajs-725	219	36	each	each	DET
iajs-725	219	37	m	m	PROPN
iajs-725	219	38			ADJ
iajs-725	219	39	n.	n.	NOUN
iajs-725	219	40	1.32	1.32	NUM
iajs-725	219	41	remark	remark	NOUN
iajs-725	219	42	:	:	PUNCT
iajs-725	219	43	let	let	VERB
iajs-725	219	44	m	m	PRON
iajs-725	219	45	be	be	AUX
iajs-725	219	46	a	a	DET
iajs-725	219	47	finitely	finitely	ADV
iajs-725	219	48	generated	generate	VERB
iajs-725	219	49	r	r	NOUN
iajs-725	219	50	-	-	PUNCT
iajs-725	219	51	module	module	NOUN
iajs-725	219	52	,	,	PUNCT
iajs-725	219	53	n	n	PRON
iajs-725	219	54	be	be	AUX
iajs-725	219	55	semimaximal	semimaximal	ADJ
iajs-725	219	56	submodule	submodule	NOUN
iajs-725	219	57	of	of	ADP
iajs-725	219	58	m.	m.	NOUN
iajs-725	220	1	then	then	ADV
iajs-725	220	2	[	[	X
iajs-725	220	3	n	n	X
iajs-725	220	4	r	r	NOUN
iajs-725	220	5	:	:	PUNCT
iajs-725	220	6	m	m	VERB
iajs-725	220	7	]	]	X
iajs-725	220	8	is	be	AUX
iajs-725	220	9	semimaximal	semimaximal	ADJ
iajs-725	220	10	ideal	ideal	NOUN
iajs-725	220	11	.	.	PUNCT
iajs-725	221	1	1.33	1.33	NUM
iajs-725	221	2	remark	remark	NOUN
iajs-725	221	3	:	:	PUNCT
iajs-725	221	4	let	let	VERB
iajs-725	221	5	m	m	PRON
iajs-725	221	6	be	be	AUX
iajs-725	221	7	a	a	DET
iajs-725	221	8	finitely	finitely	ADV
iajs-725	221	9	generated	generate	VERB
iajs-725	221	10	r	r	NOUN
iajs-725	221	11	-	-	PUNCT
iajs-725	221	12	module	module	NOUN
iajs-725	221	13	,	,	PUNCT
iajs-725	221	14	n	n	PRON
iajs-725	221	15	be	be	AUX
iajs-725	221	16	semimaximal	semimaximal	ADJ
iajs-725	221	17	submodule	submodule	NOUN
iajs-725	221	18	of	of	ADP
iajs-725	221	19	m.	m.	NOUN
iajs-725	221	20	then	then	ADV
iajs-725	221	21	n	n	PRON
iajs-725	221	22	is	be	AUX
iajs-725	221	23	quasi	quasi	ADJ
iajs-725	221	24	-	-	ADJ
iajs-725	221	25	semimaximal	semimaximal	ADJ
iajs-725	221	26	submodule	submodule	NOUN
iajs-725	221	27	.	.	PUNCT
iajs-725	222	1	proof	proof	NOUN
iajs-725	222	2	:	:	PUNCT
iajs-725	222	3	by	by	ADP
iajs-725	222	4	remark	remark	NOUN
iajs-725	222	5	(	(	PUNCT
iajs-725	222	6	1.32	1.32	NUM
iajs-725	222	7	)	)	PUNCT
iajs-725	222	8	,	,	PUNCT
iajs-725	223	1	[	[	X
iajs-725	223	2	n	n	X
iajs-725	223	3	r	r	NOUN
iajs-725	223	4	:	:	PUNCT
iajs-725	223	5	m	m	VERB
iajs-725	223	6	]	]	X
iajs-725	223	7	is	be	AUX
iajs-725	223	8	semimaximal	semimaximal	ADJ
iajs-725	223	9	ideal	ideal	NOUN
iajs-725	223	10	.	.	PUNCT
iajs-725	224	1	but	but	CCONJ
iajs-725	224	2	for	for	ADP
iajs-725	224	3	each	each	DET
iajs-725	224	4	mn	mn	PROPN
iajs-725	224	5	,	,	PUNCT
iajs-725	224	6	[	[	X
iajs-725	224	7	n	n	X
iajs-725	224	8	r	r	NOUN
iajs-725	224	9	:	:	PUNCT
iajs-725	224	10	(	(	PUNCT
iajs-725	224	11	m	m	NOUN
iajs-725	224	12	)	)	PUNCT
iajs-725	224	13	]	]	PUNCT
iajs-725	225	1			PROPN
iajs-725	226	1	[	[	X
iajs-725	226	2	n	n	ADP
iajs-725	226	3	r	r	NOUN
iajs-725	226	4	:	:	PUNCT
iajs-725	226	5	m	m	VERB
iajs-725	226	6	]	]	X
iajs-725	226	7	.	.	PUNCT
iajs-725	227	1	thus	thus	ADV
iajs-725	227	2	by	by	ADP
iajs-725	227	3	[	[	X
iajs-725	227	4	5,prop.(1.2.11	5,prop.(1.2.11	NUM
iajs-725	227	5	)	)	PUNCT
iajs-725	227	6	]	]	PUNCT
iajs-725	227	7	,	,	PUNCT
iajs-725	227	8	we	we	PRON
iajs-725	227	9	get	get	VERB
iajs-725	227	10	[	[	X
iajs-725	227	11	n	n	NOUN
iajs-725	227	12	r	r	NOUN
iajs-725	227	13	:	:	PUNCT
iajs-725	227	14	(	(	PUNCT
iajs-725	227	15	m	m	NOUN
iajs-725	227	16	)	)	PUNCT
iajs-725	227	17	]	]	PUNCT
iajs-725	228	1	is	be	AUX
iajs-725	228	2	semimaximal	semimaximal	ADJ
iajs-725	228	3	ideal	ideal	NOUN
iajs-725	228	4	of	of	ADP
iajs-725	228	5	r.	r.	PROPN
iajs-725	228	6	we	we	PRON
iajs-725	228	7	end	end	VERB
iajs-725	228	8	this	this	DET
iajs-725	228	9	section	section	NOUN
iajs-725	228	10	by	by	ADP
iajs-725	228	11	the	the	DET
iajs-725	228	12	following	follow	VERB
iajs-725	228	13	result	result	NOUN
iajs-725	228	14	.	.	PUNCT
iajs-725	229	1	1.34	1.34	NUM
iajs-725	229	2	proposition	proposition	NOUN
iajs-725	229	3	:	:	PUNCT
iajs-725	229	4	let	let	VERB
iajs-725	229	5	m	m	PRON
iajs-725	229	6	be	be	AUX
iajs-725	229	7	a	a	DET
iajs-725	229	8	finitely	finitely	ADV
iajs-725	229	9	generated	generate	VERB
iajs-725	229	10	r	r	NOUN
iajs-725	229	11	-	-	PUNCT
iajs-725	229	12	module	module	NOUN
iajs-725	229	13	.	.	PUNCT
iajs-725	230	1	then	then	ADV
iajs-725	230	2	m	m	PROPN
iajs-725	230	3	is	be	AUX
iajs-725	230	4	annsemimaximal	annsemimaximal	ADJ
iajs-725	230	5	module	module	NOUN
iajs-725	230	6	if	if	SCONJ
iajs-725	230	7	and	and	CCONJ
iajs-725	230	8	only	only	ADV
iajs-725	230	9	if	if	SCONJ
iajs-725	230	10	(	(	PUNCT
iajs-725	230	11	0	0	X
iajs-725	230	12	)	)	PUNCT
iajs-725	230	13	is	be	AUX
iajs-725	230	14	quasi	quasi	ADJ
iajs-725	230	15	-	-	ADJ
iajs-725	230	16	semimaximal	semimaximal	ADJ
iajs-725	230	17	submodule	submodule	NOUN
iajs-725	230	18	of	of	ADP
iajs-725	230	19	m.	m.	NOUN
iajs-725	230	20	proof	proof	NOUN
iajs-725	230	21	:	:	PUNCT
iajs-725	230	22	suppose	suppose	VERB
iajs-725	230	23	that	that	SCONJ
iajs-725	230	24	m	m	PROPN
iajs-725	230	25	is	be	AUX
iajs-725	230	26	annsemimaximal	annsemimaximal	ADJ
iajs-725	230	27	module	module	NOUN
iajs-725	230	28	.	.	PUNCT
iajs-725	231	1	then	then	ADV
iajs-725	231	2	annr(m	annr(m	NOUN
iajs-725	231	3	)	)	PUNCT
iajs-725	231	4	is	be	AUX
iajs-725	231	5	semimaximal	semimaximal	ADJ
iajs-725	231	6	ideal	ideal	NOUN
iajs-725	231	7	for	for	ADP
iajs-725	231	8	each	each	DET
iajs-725	231	9	mm	mm	PROPN
iajs-725	231	10	.	.	PUNCT
iajs-725	232	1	thus	thus	ADV
iajs-725	232	2	[	[	X
iajs-725	232	3	0	0	X
iajs-725	232	4	r	r	NOUN
iajs-725	232	5	:	:	PUNCT
iajs-725	232	6	m	m	VERB
iajs-725	232	7	]	]	X
iajs-725	232	8	is	be	AUX
iajs-725	232	9	semimaximal	semimaximal	ADJ
iajs-725	232	10	ideal	ideal	NOUN
iajs-725	232	11	for	for	ADP
iajs-725	232	12	each	each	DET
iajs-725	232	13	mm	mm	PROPN
iajs-725	232	14	.	.	PUNCT
iajs-725	233	1	hence	hence	ADV
iajs-725	233	2	(	(	PUNCT
iajs-725	233	3	0	0	NUM
iajs-725	233	4	)	)	PUNCT
iajs-725	233	5	is	be	AUX
iajs-725	233	6	semimaximal	semimaximal	ADJ
iajs-725	233	7	ideal	ideal	NOUN
iajs-725	233	8	.	.	PUNCT
iajs-725	234	1	conversely	conversely	ADV
iajs-725	234	2	:	:	PUNCT
iajs-725	234	3	if	if	SCONJ
iajs-725	234	4	(	(	PUNCT
iajs-725	234	5	0	0	X
iajs-725	234	6	)	)	PUNCT
iajs-725	234	7	is	be	AUX
iajs-725	234	8	quasi	quasi	ADJ
iajs-725	234	9	-	-	ADJ
iajs-725	234	10	semimaximal	semimaximal	ADJ
iajs-725	234	11	submodule	submodule	NOUN
iajs-725	234	12	of	of	ADP
iajs-725	234	13	m	m	PROPN
iajs-725	234	14	,	,	PUNCT
iajs-725	234	15	then	then	ADV
iajs-725	234	16	[	[	X
iajs-725	234	17	0	0	X
iajs-725	234	18	r	r	NOUN
iajs-725	234	19	:	:	PUNCT
iajs-725	234	20	m	m	VERB
iajs-725	234	21	]	]	X
iajs-725	234	22	is	be	AUX
iajs-725	234	23	semimaximal	semimaximal	ADJ
iajs-725	234	24	ideal	ideal	NOUN
iajs-725	234	25	for	for	ADP
iajs-725	234	26	each	each	DET
iajs-725	234	27	mm	mm	PROPN
iajs-725	234	28	.	.	PUNCT
iajs-725	235	1	therefore	therefore	ADV
iajs-725	235	2	m	m	PROPN
iajs-725	235	3	is	be	AUX
iajs-725	235	4	annsemmaximal	annsemmaximal	ADJ
iajs-725	235	5	by	by	ADP
iajs-725	235	6	theorem	theorem	NOUN
iajs-725	235	7	(	(	PUNCT
iajs-725	235	8	(	(	PUNCT
iajs-725	235	9	1.5),(4	1.5),(4	NUM
iajs-725	235	10	)	)	PUNCT
iajs-725	235	11	)	)	PUNCT
iajs-725	235	12	.	.	PUNCT
iajs-725	236	1	ibn	ibn	PROPN
iajs-725	236	2	alhaitham	alhaitham	PROPN
iajs-725	236	3	j.	j.	PROPN
iajs-725	236	4	for	for	ADP
iajs-725	236	5	pure	pure	ADJ
iajs-725	236	6	&	&	CCONJ
iajs-725	236	7	appl	appl	PROPN
iajs-725	236	8	.	.	PUNCT
iajs-725	237	1	sci	sci	PROPN
iajs-725	237	2	.	.	PUNCT
iajs-725	237	3	vol.24	vol.24	NOUN
iajs-725	237	4	(	(	PUNCT
iajs-725	237	5	3	3	NUM
iajs-725	237	6	)	)	PUNCT
iajs-725	237	7	2011	2011	NUM
iajs-725	237	8	2coannsemimaximal	2coannsemimaximal	NUM
iajs-725	237	9	modules	module	NOUN
iajs-725	237	10	:	:	PUNCT
iajs-725	237	11	in	in	ADP
iajs-725	237	12	this	this	DET
iajs-725	237	13	section	section	NOUN
iajs-725	237	14	,	,	PUNCT
iajs-725	237	15	we	we	PRON
iajs-725	237	16	introduce	introduce	VERB
iajs-725	237	17	the	the	DET
iajs-725	237	18	concept	concept	NOUN
iajs-725	237	19	of	of	ADP
iajs-725	237	20	coannsemimaximal	coannsemimaximal	ADJ
iajs-725	237	21	module	module	NOUN
iajs-725	237	22	which	which	PRON
iajs-725	237	23	is	be	AUX
iajs-725	237	24	strongly	strongly	ADV
iajs-725	237	25	from	from	ADP
iajs-725	237	26	the	the	DET
iajs-725	237	27	concept	concept	NOUN
iajs-725	237	28	of	of	ADP
iajs-725	237	29	annsemimaximal	annsemimaximal	ADJ
iajs-725	237	30	module	module	NOUN
iajs-725	237	31	in	in	ADP
iajs-725	237	32	section	section	NOUN
iajs-725	237	33	one	one	NUM
iajs-725	237	34	.	.	PUNCT
iajs-725	238	1	we	we	PRON
iajs-725	238	2	give	give	VERB
iajs-725	238	3	some	some	DET
iajs-725	238	4	characterizations	characterization	NOUN
iajs-725	238	5	about	about	ADP
iajs-725	238	6	this	this	DET
iajs-725	238	7	concept	concept	NOUN
iajs-725	238	8	and	and	CCONJ
iajs-725	238	9	many	many	ADJ
iajs-725	238	10	results	result	NOUN
iajs-725	238	11	are	be	AUX
iajs-725	238	12	studied	study	VERB
iajs-725	238	13	.	.	PUNCT
iajs-725	239	1	we	we	PRON
iajs-725	239	2	start	start	VERB
iajs-725	239	3	with	with	ADP
iajs-725	239	4	the	the	DET
iajs-725	239	5	following	follow	VERB
iajs-725	239	6	definition	definition	NOUN
iajs-725	239	7	.	.	PUNCT
iajs-725	240	1	2.1	2.1	NUM
iajs-725	240	2	definition	definition	NOUN
iajs-725	240	3	:	:	PUNCT
iajs-725	240	4	an	an	DET
iajs-725	240	5	r	r	NOUN
iajs-725	240	6	-	-	PUNCT
iajs-725	240	7	module	module	NOUN
iajs-725	240	8	m	m	NOUN
iajs-725	240	9	is	be	AUX
iajs-725	240	10	called	call	VERB
iajs-725	240	11	coannsemimaximal	coannsemimaximal	ADJ
iajs-725	240	12	module	module	NOUN
iajs-725	240	13	if	if	SCONJ
iajs-725	240	14	annr	annr	NOUN
iajs-725	240	15	m	m	VERB
iajs-725	240	16	n	n	VERB
iajs-725	240	17	is	be	AUX
iajs-725	240	18	semimaximal	semimaximal	ADJ
iajs-725	240	19	ideal	ideal	NOUN
iajs-725	240	20	of	of	ADP
iajs-725	240	21	r	r	NOUN
iajs-725	240	22	for	for	ADP
iajs-725	240	23	each	each	DET
iajs-725	240	24	non	non	ADJ
iajs-725	240	25	-	-	ADJ
iajs-725	240	26	zero	zero	ADJ
iajs-725	240	27	proper	proper	ADJ
iajs-725	240	28	submodule	submodule	NOUN
iajs-725	240	29	n	n	PROPN
iajs-725	240	30	of	of	ADP
iajs-725	240	31	m.	m.	NOUN
iajs-725	240	32	equivalently	equivalently	ADV
iajs-725	240	33	,	,	PUNCT
iajs-725	240	34	m	m	VERB
iajs-725	240	35	is	be	AUX
iajs-725	240	36	coannsemimaximal	coannsemimaximal	ADJ
iajs-725	240	37	if	if	SCONJ
iajs-725	240	38	m	m	VERB
iajs-725	240	39	n	n	VERB
iajs-725	240	40	is	be	AUX
iajs-725	240	41	annsemimaximal	annsemimaximal	ADJ
iajs-725	240	42	module	module	NOUN
iajs-725	240	43	for	for	ADP
iajs-725	240	44	each	each	DET
iajs-725	240	45	non	non	ADJ
iajs-725	240	46	-	-	ADJ
iajs-725	240	47	zero	zero	ADJ
iajs-725	240	48	proper	proper	ADJ
iajs-725	240	49	submodule	submodule	NOUN
iajs-725	240	50	n	n	PROPN
iajs-725	240	51	of	of	ADP
iajs-725	240	52	m.	m.	NOUN
iajs-725	240	53	2.2	2.2	NUM
iajs-725	240	54	examples	example	NOUN
iajs-725	240	55	:	:	PUNCT
iajs-725	240	56	(	(	PUNCT
iajs-725	240	57	1	1	X
iajs-725	240	58	)	)	PUNCT
iajs-725	240	59	z12	z12	PROPN
iajs-725	240	60	is	be	AUX
iajs-725	240	61	not	not	PART
iajs-725	240	62	coannsemimaximal	coannsemimaximal	ADJ
iajs-725	240	63	z	z	NOUN
iajs-725	240	64	-	-	PUNCT
iajs-725	240	65	module	module	NOUN
iajs-725	240	66	,	,	PUNCT
iajs-725	240	67	since	since	SCONJ
iajs-725	240	68	if	if	SCONJ
iajs-725	240	69	n	n	ADV
iajs-725	240	70	=	=	PUNCT
iajs-725	240	71	<	<	X
iajs-725	240	72	4	4	NUM
iajs-725	240	73	>	>	PUNCT
iajs-725	240	74	,	,	PUNCT
iajs-725	240	75	then	then	ADV
iajs-725	240	76	annz	annz	VERB
iajs-725	240	77	12z	12z	PROPN
iajs-725	240	78	n	n	CCONJ
iajs-725	240	79	=	=	SYM
iajs-725	240	80	annzz4	annzz4	NOUN
iajs-725	240	81	=	=	NOUN
iajs-725	240	82	4z	4z	NOUN
iajs-725	240	83	which	which	PRON
iajs-725	240	84	is	be	AUX
iajs-725	240	85	not	not	PART
iajs-725	240	86	semimaximal	semimaximal	ADJ
iajs-725	240	87	ideal	ideal	NOUN
iajs-725	240	88	.	.	PUNCT
iajs-725	241	1	(	(	PUNCT
iajs-725	241	2	2	2	X
iajs-725	241	3	)	)	PUNCT
iajs-725	241	4	2p	2p	NUM
iajs-725	241	5	z	z	NOUN
iajs-725	241	6	as	as	ADP
iajs-725	241	7	a	a	DET
iajs-725	241	8	z	z	NOUN
iajs-725	241	9	-	-	PUNCT
iajs-725	241	10	module	module	NOUN
iajs-725	241	11	is	be	AUX
iajs-725	241	12	coannsemimaximal	coannsemimaximal	ADJ
iajs-725	241	13	,	,	PUNCT
iajs-725	241	14	where	where	SCONJ
iajs-725	241	15	p	p	NOUN
iajs-725	241	16	is	be	AUX
iajs-725	241	17	a	a	DET
iajs-725	241	18	prime	prime	ADJ
iajs-725	241	19	number	number	NOUN
iajs-725	241	20	.	.	PUNCT
iajs-725	242	1	proof	proof	NOUN
iajs-725	242	2	:	:	PUNCT
iajs-725	242	3	since	since	SCONJ
iajs-725	242	4	<	<	X
iajs-725	242	5	p	p	X
iajs-725	242	6	>	>	X
iajs-725	242	7	is	be	AUX
iajs-725	242	8	only	only	ADV
iajs-725	242	9	non	non	ADJ
iajs-725	242	10	-	-	ADJ
iajs-725	242	11	zero	zero	ADJ
iajs-725	242	12	proper	proper	ADJ
iajs-725	242	13	submodule	submodule	NOUN
iajs-725	242	14	of	of	ADP
iajs-725	242	15	2p	2p	NUM
iajs-725	242	16	z	z	NOUN
iajs-725	242	17	.	.	PUNCT
iajs-725	243	1	2p	2p	NUM
iajs-725	243	2	z	z	NOUN
iajs-725	243	3	/	/	SYM
iajs-725	243	4	<	<	X
iajs-725	243	5	p	p	X
iajs-725	243	6	>	>	X
iajs-725	243	7	�	�	PROPN
iajs-725	243	8	zp	zp	PROPN
iajs-725	243	9	and	and	CCONJ
iajs-725	243	10	annzzp	annzzp	PROPN
iajs-725	243	11	=	=	PROPN
iajs-725	243	12	pz	pz	NOUN
iajs-725	243	13	which	which	PRON
iajs-725	243	14	is	be	AUX
iajs-725	243	15	clear	clear	ADJ
iajs-725	243	16	semimaximal	semimaximal	ADJ
iajs-725	243	17	ideal	ideal	NOUN
iajs-725	243	18	.	.	PUNCT
iajs-725	244	1	next	next	ADV
iajs-725	244	2	,	,	PUNCT
iajs-725	244	3	we	we	PRON
iajs-725	244	4	have	have	VERB
iajs-725	244	5	the	the	DET
iajs-725	244	6	following	follow	VERB
iajs-725	244	7	proposition	proposition	NOUN
iajs-725	244	8	.	.	PUNCT
iajs-725	245	1	2.3	2.3	NUM
iajs-725	245	2	proposition	proposition	NOUN
iajs-725	245	3	:	:	PUNCT
iajs-725	245	4	let	let	VERB
iajs-725	245	5	m	m	PRON
iajs-725	245	6	be	be	AUX
iajs-725	245	7	an	an	DET
iajs-725	245	8	r	r	NOUN
iajs-725	245	9	-	-	PUNCT
iajs-725	245	10	module	module	NOUN
iajs-725	245	11	.	.	PUNCT
iajs-725	246	1	then	then	ADV
iajs-725	246	2	every	every	DET
iajs-725	246	3	annsemimaximal	annsemimaximal	ADJ
iajs-725	246	4	module	module	NOUN
iajs-725	246	5	is	be	AUX
iajs-725	246	6	coannsemimaximal	coannsemimaximal	ADJ
iajs-725	246	7	module	module	NOUN
iajs-725	246	8	.	.	PUNCT
iajs-725	247	1	proof	proof	NOUN
iajs-725	247	2	:	:	PUNCT
iajs-725	247	3	let	let	VERB
iajs-725	247	4	m	m	PRON
iajs-725	247	5	be	be	AUX
iajs-725	247	6	an	an	DET
iajs-725	247	7	annsemimaximal	annsemimaximal	ADJ
iajs-725	247	8	module	module	NOUN
iajs-725	247	9	.	.	PUNCT
iajs-725	248	1	then	then	ADV
iajs-725	248	2	m	m	VERB
iajs-725	248	3	n	n	VERB
iajs-725	248	4	is	be	AUX
iajs-725	248	5	annsemimaximal	annsemimaximal	ADJ
iajs-725	248	6	module	module	NOUN
iajs-725	248	7	by	by	ADP
iajs-725	248	8	remarks	remark	NOUN
iajs-725	248	9	and	and	CCONJ
iajs-725	248	10	examples	example	NOUN
iajs-725	248	11	(	(	PUNCT
iajs-725	248	12	(	(	PUNCT
iajs-725	248	13	1.2),(9	1.2),(9	NUM
iajs-725	248	14	)	)	PUNCT
iajs-725	248	15	)	)	PUNCT
iajs-725	248	16	.	.	PUNCT
iajs-725	249	1	thus	thus	ADV
iajs-725	249	2	annr	annr	NOUN
iajs-725	249	3	m	m	VERB
iajs-725	249	4	n	n	VERB
iajs-725	249	5	is	be	AUX
iajs-725	249	6	semimaximal	semimaximal	ADJ
iajs-725	249	7	which	which	PRON
iajs-725	249	8	implies	imply	VERB
iajs-725	249	9	that	that	SCONJ
iajs-725	249	10	m	m	PROPN
iajs-725	249	11	is	be	AUX
iajs-725	249	12	coannsemimaximal	coannsemimaximal	ADJ
iajs-725	249	13	module	module	NOUN
iajs-725	249	14	.	.	PUNCT
iajs-725	250	1	the	the	DET
iajs-725	250	2	converse	converse	NOUN
iajs-725	250	3	of	of	ADP
iajs-725	250	4	proposition	proposition	NOUN
iajs-725	250	5	(	(	PUNCT
iajs-725	250	6	2.3	2.3	NUM
iajs-725	250	7	)	)	PUNCT
iajs-725	250	8	is	be	AUX
iajs-725	250	9	not	not	PART
iajs-725	250	10	true	true	ADJ
iajs-725	250	11	in	in	ADP
iajs-725	250	12	general	general	ADJ
iajs-725	250	13	.	.	PUNCT
iajs-725	251	1	for	for	ADP
iajs-725	251	2	example	example	NOUN
iajs-725	251	3	:	:	PUNCT
iajs-725	251	4	let	let	VERB
iajs-725	251	5	z9	z9	PROPN
iajs-725	251	6	be	be	AUX
iajs-725	251	7	a	a	DET
iajs-725	251	8	zmodule	zmodule	NOUN
iajs-725	251	9	.	.	PUNCT
iajs-725	252	1	then	then	ADV
iajs-725	252	2	z9	z9	PROPN
iajs-725	252	3	is	be	AUX
iajs-725	252	4	coannsemimaximal	coannsemimaximal	ADJ
iajs-725	252	5	module	module	NOUN
iajs-725	252	6	but	but	CCONJ
iajs-725	252	7	not	not	PART
iajs-725	252	8	annsemimaximal	annsemimaximal	ADJ
iajs-725	252	9	.	.	PUNCT
iajs-725	253	1	and	and	CCONJ
iajs-725	253	2	z4	z4	PROPN
iajs-725	253	3	as	as	ADP
iajs-725	253	4	a	a	DET
iajs-725	253	5	zmodule	zmodule	NOUN
iajs-725	253	6	is	be	AUX
iajs-725	253	7	coannsemimaximal	coannsemimaximal	ADJ
iajs-725	253	8	module	module	NOUN
iajs-725	253	9	but	but	CCONJ
iajs-725	253	10	it	it	PRON
iajs-725	253	11	is	be	AUX
iajs-725	253	12	not	not	PART
iajs-725	253	13	annsemimaximal	annsemimaximal	ADJ
iajs-725	253	14	module	module	NOUN
iajs-725	253	15	.	.	PUNCT
iajs-725	254	1	the	the	DET
iajs-725	254	2	following	follow	VERB
iajs-725	254	3	proposition	proposition	NOUN
iajs-725	254	4	proves	prove	VERB
iajs-725	254	5	that	that	SCONJ
iajs-725	254	6	the	the	DET
iajs-725	254	7	converse	converse	NOUN
iajs-725	254	8	of	of	ADP
iajs-725	254	9	(	(	PUNCT
iajs-725	254	10	2.3	2.3	NUM
iajs-725	254	11	)	)	PUNCT
iajs-725	254	12	is	be	AUX
iajs-725	254	13	true	true	ADJ
iajs-725	254	14	under	under	ADP
iajs-725	254	15	the	the	DET
iajs-725	254	16	condition	condition	NOUN
iajs-725	254	17	that	that	PRON
iajs-725	254	18	m	m	NOUN
iajs-725	254	19	is	be	AUX
iajs-725	254	20	coprime	coprime	ADJ
iajs-725	254	21	module	module	NOUN
iajs-725	254	22	,	,	PUNCT
iajs-725	254	23	but	but	CCONJ
iajs-725	254	24	first	first	ADV
iajs-725	254	25	we	we	PRON
iajs-725	254	26	need	need	VERB
iajs-725	254	27	to	to	PART
iajs-725	254	28	recall	recall	VERB
iajs-725	254	29	the	the	DET
iajs-725	254	30	definition	definition	NOUN
iajs-725	254	31	of	of	ADP
iajs-725	254	32	coprime	coprime	NOUN
iajs-725	254	33	module	module	NOUN
iajs-725	254	34	.	.	PUNCT
iajs-725	255	1	an	an	DET
iajs-725	255	2	r	r	NOUN
iajs-725	255	3	-	-	PUNCT
iajs-725	255	4	module	module	NOUN
iajs-725	255	5	m	m	NOUN
iajs-725	255	6	is	be	AUX
iajs-725	255	7	called	call	VERB
iajs-725	255	8	coprime	coprime	NOUN
iajs-725	255	9	module	module	NOUN
iajs-725	255	10	if	if	SCONJ
iajs-725	255	11	annrm	annrm	NOUN
iajs-725	255	12	=	=	PUNCT
iajs-725	255	13	annr	annr	NOUN
iajs-725	255	14	m	m	VERB
iajs-725	255	15	n	n	NOUN
iajs-725	255	16	for	for	ADP
iajs-725	255	17	every	every	DET
iajs-725	255	18	proper	proper	ADJ
iajs-725	255	19	submodule	submodule	NOUN
iajs-725	255	20	n	n	PROPN
iajs-725	255	21	of	of	ADP
iajs-725	255	22	m	m	PRON
iajs-725	255	23	,	,	PUNCT
iajs-725	255	24	[	[	X
iajs-725	255	25	13	13	NUM
iajs-725	255	26	]	]	PUNCT
iajs-725	255	27	.	.	PUNCT
iajs-725	256	1	2.4	2.4	NUM
iajs-725	256	2	proposition	proposition	NOUN
iajs-725	256	3	:	:	PUNCT
iajs-725	256	4	let	let	VERB
iajs-725	256	5	m	m	PRON
iajs-725	256	6	be	be	AUX
iajs-725	256	7	a	a	DET
iajs-725	256	8	coprime	coprime	NOUN
iajs-725	256	9	and	and	CCONJ
iajs-725	256	10	coannsemimaximal	coannsemimaximal	ADJ
iajs-725	256	11	r	r	NOUN
iajs-725	256	12	-	-	PUNCT
iajs-725	256	13	module	module	NOUN
iajs-725	256	14	.	.	PUNCT
iajs-725	257	1	then	then	ADV
iajs-725	257	2	m	m	PROPN
iajs-725	257	3	is	be	AUX
iajs-725	257	4	annsemimaximal	annsemimaximal	ADJ
iajs-725	257	5	r	r	NOUN
iajs-725	257	6	-	-	PUNCT
iajs-725	257	7	module	module	NOUN
iajs-725	257	8	.	.	PUNCT
iajs-725	258	1	proof	proof	NOUN
iajs-725	258	2	:	:	PUNCT
iajs-725	258	3	since	since	SCONJ
iajs-725	258	4	m	m	PROPN
iajs-725	258	5	is	be	AUX
iajs-725	258	6	coprime	coprime	NOUN
iajs-725	258	7	module	module	NOUN
iajs-725	258	8	.	.	PUNCT
iajs-725	259	1	then	then	ADV
iajs-725	259	2	annrm	annrm	NOUN
iajs-725	259	3	=	=	SYM
iajs-725	259	4	annr	annr	NOUN
iajs-725	259	5	m	m	VERB
iajs-725	259	6	n	n	NOUN
iajs-725	259	7	for	for	ADP
iajs-725	259	8	every	every	DET
iajs-725	259	9	proper	proper	ADJ
iajs-725	259	10	submodule	submodule	NOUN
iajs-725	259	11	n	n	PROPN
iajs-725	259	12	of	of	ADP
iajs-725	259	13	m.	m.	NOUN
iajs-725	259	14	but	but	CCONJ
iajs-725	259	15	annr	annr	NOUN
iajs-725	259	16	m	m	VERB
iajs-725	259	17	n	n	VERB
iajs-725	259	18	is	be	AUX
iajs-725	259	19	semimaximal	semimaximal	ADJ
iajs-725	259	20	ideal	ideal	NOUN
iajs-725	259	21	of	of	ADP
iajs-725	259	22	r	r	NOUN
iajs-725	259	23	for	for	ADP
iajs-725	259	24	each	each	DET
iajs-725	259	25	non	non	ADJ
iajs-725	259	26	-	-	ADJ
iajs-725	259	27	zero	zero	ADJ
iajs-725	259	28	proper	proper	ADJ
iajs-725	259	29	submodule	submodule	NOUN
iajs-725	259	30	n	n	PROPN
iajs-725	259	31	of	of	ADP
iajs-725	259	32	m	m	PROPN
iajs-725	259	33	(	(	PUNCT
iajs-725	259	34	since	since	SCONJ
iajs-725	259	35	m	m	PROPN
iajs-725	259	36	is	be	AUX
iajs-725	259	37	coannsemimaximal	coannsemimaximal	ADJ
iajs-725	259	38	)	)	PUNCT
iajs-725	259	39	.	.	PUNCT
iajs-725	260	1	thus	thus	ADV
iajs-725	260	2	annnrm	annnrm	PROPN
iajs-725	260	3	is	be	AUX
iajs-725	260	4	semimaximal	semimaximal	ADJ
iajs-725	260	5	ideal	ideal	NOUN
iajs-725	260	6	of	of	ADP
iajs-725	260	7	r	r	NOUN
iajs-725	260	8	and	and	CCONJ
iajs-725	260	9	hence	hence	ADV
iajs-725	260	10	m	m	VERB
iajs-725	260	11	is	be	AUX
iajs-725	260	12	annsemimaximal	annsemimaximal	ADJ
iajs-725	260	13	module	module	NOUN
iajs-725	260	14	by	by	ADP
iajs-725	260	15	proposition	proposition	NOUN
iajs-725	260	16	(	(	PUNCT
iajs-725	260	17	1.3	1.3	NUM
iajs-725	260	18	)	)	PUNCT
iajs-725	260	19	.	.	PUNCT
iajs-725	261	1	ibn	ibn	PROPN
iajs-725	261	2	alhaitham	alhaitham	PROPN
iajs-725	261	3	j.	j.	PROPN
iajs-725	261	4	for	for	ADP
iajs-725	261	5	pure	pure	ADJ
iajs-725	261	6	&	&	CCONJ
iajs-725	261	7	appl	appl	PROPN
iajs-725	261	8	.	.	PUNCT
iajs-725	262	1	sci	sci	PROPN
iajs-725	262	2	.	.	PUNCT
iajs-725	262	3	vol.24	vol.24	NOUN
iajs-725	262	4	(	(	PUNCT
iajs-725	262	5	3	3	NUM
iajs-725	262	6	)	)	PUNCT
iajs-725	262	7	2011	2011	NUM
iajs-725	262	8	as	as	ADP
iajs-725	262	9	an	an	DET
iajs-725	262	10	application	application	NOUN
iajs-725	262	11	of	of	ADP
iajs-725	262	12	(	(	PUNCT
iajs-725	262	13	2.4	2.4	NUM
iajs-725	262	14	)	)	PUNCT
iajs-725	262	15	,	,	PUNCT
iajs-725	262	16	we	we	PRON
iajs-725	262	17	have	have	VERB
iajs-725	262	18	the	the	DET
iajs-725	262	19	following	following	NOUN
iajs-725	262	20	.	.	PUNCT
iajs-725	263	1	2.5	2.5	NUM
iajs-725	263	2	corollary	corollary	NOUN
iajs-725	263	3	:	:	PUNCT
iajs-725	263	4	let	let	VERB
iajs-725	263	5	m	m	PRON
iajs-725	263	6	be	be	AUX
iajs-725	263	7	a	a	DET
iajs-725	263	8	coannsemimaximal	coannsemimaximal	ADJ
iajs-725	263	9	r	r	NOUN
iajs-725	263	10	-	-	PUNCT
iajs-725	263	11	module	module	NOUN
iajs-725	263	12	and	and	CCONJ
iajs-725	263	13	m	m	NOUN
iajs-725	263	14	is	be	AUX
iajs-725	263	15	a	a	DET
iajs-725	263	16	coprime	coprime	ADJ
iajs-725	263	17	e	e	NOUN
iajs-725	263	18	-	-	NOUN
iajs-725	263	19	module	module	NOUN
iajs-725	263	20	,	,	PUNCT
iajs-725	263	21	where	where	SCONJ
iajs-725	263	22	e	e	NOUN
iajs-725	263	23	=	=	NOUN
iajs-725	263	24	endr(m	endr(m	NOUN
iajs-725	263	25	)	)	PUNCT
iajs-725	263	26	.	.	PUNCT
iajs-725	264	1	then	then	ADV
iajs-725	264	2	m	m	PROPN
iajs-725	264	3	is	be	AUX
iajs-725	264	4	annsemimaximal	annsemimaximal	ADJ
iajs-725	264	5	r	r	NOUN
iajs-725	264	6	-	-	PUNCT
iajs-725	264	7	module	module	NOUN
iajs-725	264	8	.	.	PUNCT
iajs-725	265	1	proof	proof	NOUN
iajs-725	265	2	:	:	PUNCT
iajs-725	265	3	m	m	VERB
iajs-725	265	4	is	be	AUX
iajs-725	265	5	coprime	coprime	ADJ
iajs-725	265	6	e	e	NOUN
iajs-725	265	7	-	-	NOUN
iajs-725	265	8	module	module	NOUN
iajs-725	265	9	,	,	PUNCT
iajs-725	265	10	then	then	ADV
iajs-725	265	11	m	m	NOUN
iajs-725	265	12	is	be	AUX
iajs-725	265	13	coprime	coprime	ADJ
iajs-725	265	14	r	r	NOUN
iajs-725	265	15	-	-	PUNCT
iajs-725	265	16	module	module	NOUN
iajs-725	265	17	by	by	ADP
iajs-725	265	18	[	[	X
iajs-725	265	19	14,coro.(2.2.3	14,coro.(2.2.3	NUM
iajs-725	265	20	)	)	PUNCT
iajs-725	265	21	]	]	PUNCT
iajs-725	265	22	and	and	CCONJ
iajs-725	265	23	from	from	ADP
iajs-725	265	24	proposition	proposition	NOUN
iajs-725	265	25	(	(	PUNCT
iajs-725	265	26	2.4	2.4	NUM
iajs-725	265	27	)	)	PUNCT
iajs-725	265	28	,	,	PUNCT
iajs-725	265	29	we	we	PRON
iajs-725	265	30	get	get	VERB
iajs-725	265	31	the	the	DET
iajs-725	265	32	result	result	NOUN
iajs-725	265	33	.	.	PUNCT
iajs-725	265	34	.	.	PUNCT
iajs-725	266	1	recall	recall	VERB
iajs-725	266	2	that	that	SCONJ
iajs-725	266	3	a	a	DET
iajs-725	266	4	non	non	ADJ
iajs-725	266	5	-	-	ADJ
iajs-725	266	6	simple	simple	ADJ
iajs-725	266	7	r	r	NOUN
iajs-725	266	8	-	-	PUNCT
iajs-725	266	9	module	module	NOUN
iajs-725	266	10	m	m	NOUN
iajs-725	266	11	is	be	AUX
iajs-725	266	12	called	call	VERB
iajs-725	266	13	antihopfian	antihopfian	ADJ
iajs-725	266	14	if	if	SCONJ
iajs-725	266	15	mm	mm	NOUN
iajs-725	266	16	/	/	SYM
iajs-725	266	17	n	n	NOUN
iajs-725	266	18	for	for	ADP
iajs-725	266	19	all	all	DET
iajs-725	266	20	proper	proper	ADJ
iajs-725	266	21	submodules	submodule	NOUN
iajs-725	266	22	n	n	PROPN
iajs-725	266	23	of	of	ADP
iajs-725	266	24	m	m	PRON
iajs-725	266	25	,	,	PUNCT
iajs-725	266	26	[	[	X
iajs-725	266	27	15	15	NUM
iajs-725	266	28	]	]	PUNCT
iajs-725	266	29	.	.	PUNCT
iajs-725	267	1	by	by	ADP
iajs-725	267	2	using	use	VERB
iajs-725	267	3	this	this	DET
iajs-725	267	4	concept	concept	NOUN
iajs-725	267	5	we	we	PRON
iajs-725	267	6	get	get	VERB
iajs-725	267	7	the	the	DET
iajs-725	267	8	following	following	NOUN
iajs-725	267	9	.	.	PUNCT
iajs-725	268	1	2.6	2.6	NUM
iajs-725	268	2	proposition	proposition	NOUN
iajs-725	268	3	:	:	PUNCT
iajs-725	268	4	let	let	VERB
iajs-725	268	5	m	m	PRON
iajs-725	268	6	be	be	AUX
iajs-725	268	7	an	an	DET
iajs-725	268	8	antihopfian	antihopfian	NOUN
iajs-725	268	9	,	,	PUNCT
iajs-725	268	10	n	n	PRON
iajs-725	268	11	is	be	AUX
iajs-725	268	12	semimaximal	semimaximal	ADJ
iajs-725	268	13	submodule	submodule	NOUN
iajs-725	268	14	of	of	ADP
iajs-725	268	15	m.	m.	NOUN
iajs-725	268	16	then	then	ADV
iajs-725	268	17	m	m	VERB
iajs-725	268	18	is	be	AUX
iajs-725	268	19	coannsemimaximal	coannsemimaximal	ADJ
iajs-725	268	20	r	r	NOUN
iajs-725	268	21	-	-	PUNCT
iajs-725	268	22	module	module	NOUN
iajs-725	268	23	.	.	PUNCT
iajs-725	269	1	proof	proof	NOUN
iajs-725	269	2	:	:	PUNCT
iajs-725	269	3	we	we	PRON
iajs-725	269	4	have	have	VERB
iajs-725	269	5	n	n	ADV
iajs-725	269	6	is	be	AUX
iajs-725	269	7	semimaximal	semimaximal	ADJ
iajs-725	269	8	submodule	submodule	NOUN
iajs-725	269	9	.	.	PUNCT
iajs-725	270	1	then	then	ADV
iajs-725	270	2	m	m	VERB
iajs-725	270	3	n	n	VERB
iajs-725	270	4	is	be	AUX
iajs-725	270	5	semisimple	semisimple	ADJ
iajs-725	270	6	by	by	ADP
iajs-725	270	7	[	[	X
iajs-725	270	8	5,def.(2.1.1	5,def.(2.1.1	NUM
iajs-725	270	9	)	)	PUNCT
iajs-725	270	10	]	]	PUNCT
iajs-725	270	11	.	.	PUNCT
iajs-725	271	1	thus	thus	ADV
iajs-725	271	2	m	m	VERB
iajs-725	271	3	n	n	VERB
iajs-725	271	4	is	be	AUX
iajs-725	271	5	annsemimaximal	annsemimaximal	ADJ
iajs-725	271	6	module	module	NOUN
iajs-725	271	7	by	by	ADP
iajs-725	271	8	prop.(1.5	prop.(1.5	NOUN
iajs-725	271	9	)	)	PUNCT
iajs-725	271	10	.	.	PUNCT
iajs-725	272	1	but	but	CCONJ
iajs-725	272	2	m	m	PROPN
iajs-725	272	3	m	m	VERB
iajs-725	272	4	w	w	PROPN
iajs-725	272	5	n	n	X
iajs-725	272	6	�	�	PROPN
iajs-725	272	7	for	for	ADP
iajs-725	272	8	each	each	DET
iajs-725	272	9	proper	proper	ADJ
iajs-725	272	10	submodule	submodule	PROPN
iajs-725	272	11	w	w	PROPN
iajs-725	272	12	of	of	ADP
iajs-725	272	13	m	m	PROPN
iajs-725	272	14	,	,	PUNCT
iajs-725	272	15	since	since	SCONJ
iajs-725	272	16	m	m	PROPN
iajs-725	272	17	is	be	AUX
iajs-725	272	18	antihopfian	antihopfian	ADJ
iajs-725	272	19	.	.	PUNCT
iajs-725	273	1	that	that	PRON
iajs-725	273	2	means	mean	VERB
iajs-725	273	3	m	m	PROPN
iajs-725	273	4	m	m	PROPN
iajs-725	273	5	n	n	NOUN
iajs-725	273	6	.	.	PUNCT
iajs-725	274	1	thus	thus	ADV
iajs-725	274	2	m	m	PRON
iajs-725	274	3	w	w	PROPN
iajs-725	274	4	is	be	AUX
iajs-725	274	5	annsemimaximal	annsemimaximal	ADJ
iajs-725	274	6	for	for	ADP
iajs-725	274	7	all	all	DET
iajs-725	274	8	proper	proper	ADJ
iajs-725	274	9	submodule	submodule	NOUN
iajs-725	274	10	w	w	PROPN
iajs-725	274	11	of	of	ADP
iajs-725	274	12	m.	m.	NOUN
iajs-725	274	13	therefore	therefore	ADV
iajs-725	274	14	m	m	PROPN
iajs-725	274	15	is	be	AUX
iajs-725	274	16	coannsemimaximal	coannsemimaximal	ADJ
iajs-725	274	17	module	module	NOUN
iajs-725	274	18	.	.	PUNCT
iajs-725	275	1	now	now	ADV
iajs-725	275	2	,	,	PUNCT
iajs-725	275	3	we	we	PRON
iajs-725	275	4	prove	prove	VERB
iajs-725	275	5	the	the	DET
iajs-725	275	6	following	follow	VERB
iajs-725	275	7	lemma	lemma	PROPN
iajs-725	275	8	.	.	PROPN
iajs-725	276	1	2.7	2.7	NUM
iajs-725	276	2	lemma	lemma	PROPN
iajs-725	276	3	:	:	PUNCT
iajs-725	276	4	let	let	VERB
iajs-725	276	5	m	m	PRON
iajs-725	276	6	be	be	AUX
iajs-725	276	7	an	an	DET
iajs-725	276	8	r	r	NOUN
iajs-725	276	9	-	-	PUNCT
iajs-725	276	10	module	module	NOUN
iajs-725	276	11	.	.	PUNCT
iajs-725	277	1	if	if	SCONJ
iajs-725	277	2	n	n	PRON
iajs-725	277	3	is	be	AUX
iajs-725	277	4	a	a	DET
iajs-725	277	5	semimaximal	semimaximal	ADJ
iajs-725	277	6	submodule	submodule	NOUN
iajs-725	277	7	,	,	PUNCT
iajs-725	277	8	then	then	ADV
iajs-725	277	9	[	[	X
iajs-725	277	10	n	n	X
iajs-725	277	11	r	r	NOUN
iajs-725	277	12	:	:	PUNCT
iajs-725	277	13	m	m	VERB
iajs-725	277	14	]	]	X
iajs-725	277	15	is	be	AUX
iajs-725	277	16	semimaximal	semimaximal	ADJ
iajs-725	277	17	ideal	ideal	ADJ
iajs-725	277	18	.	.	PUNCT
iajs-725	278	1	proof	proof	NOUN
iajs-725	278	2	:	:	PUNCT
iajs-725	278	3	suppose	suppose	VERB
iajs-725	278	4	that	that	SCONJ
iajs-725	278	5	n	n	PRON
iajs-725	278	6	is	be	AUX
iajs-725	278	7	a	a	DET
iajs-725	278	8	semimaximal	semimaximal	ADJ
iajs-725	278	9	submodule	submodule	NOUN
iajs-725	278	10	.	.	PUNCT
iajs-725	279	1	then	then	ADV
iajs-725	279	2	by	by	ADP
iajs-725	279	3	[	[	X
iajs-725	279	4	5,def.(2.1.1	5,def.(2.1.1	NUM
iajs-725	279	5	)	)	PUNCT
iajs-725	279	6	]	]	PUNCT
iajs-725	279	7	,	,	PUNCT
iajs-725	279	8	m	m	VERB
iajs-725	279	9	n	n	VERB
iajs-725	279	10	is	be	AUX
iajs-725	279	11	semisimple	semisimple	ADJ
iajs-725	279	12	r	r	NOUN
iajs-725	279	13	-	-	PUNCT
iajs-725	279	14	module	module	NOUN
iajs-725	279	15	and	and	CCONJ
iajs-725	279	16	hence	hence	ADV
iajs-725	279	17	by	by	ADP
iajs-725	279	18	proposition	proposition	NOUN
iajs-725	279	19	(	(	PUNCT
iajs-725	279	20	1.6	1.6	NUM
iajs-725	279	21	)	)	PUNCT
iajs-725	279	22	,	,	PUNCT
iajs-725	279	23	m	m	VERB
iajs-725	279	24	n	n	VERB
iajs-725	279	25	is	be	AUX
iajs-725	279	26	annsemimaximal	annsemimaximal	ADJ
iajs-725	279	27	module	module	NOUN
iajs-725	279	28	.	.	PUNCT
iajs-725	280	1	then	then	ADV
iajs-725	280	2	by	by	ADP
iajs-725	280	3	proposition	proposition	NOUN
iajs-725	280	4	(	(	PUNCT
iajs-725	280	5	1.3	1.3	NUM
iajs-725	280	6	)	)	PUNCT
iajs-725	280	7	,	,	PUNCT
iajs-725	280	8	annr	annr	NOUN
iajs-725	280	9	m	m	VERB
iajs-725	280	10	n	n	VERB
iajs-725	280	11	is	be	AUX
iajs-725	280	12	semimaximal	semimaximal	ADJ
iajs-725	280	13	ideal	ideal	ADJ
iajs-725	280	14	.	.	PUNCT
iajs-725	281	1	but	but	CCONJ
iajs-725	281	2	[	[	X
iajs-725	281	3	n	n	X
iajs-725	281	4	r	r	NOUN
iajs-725	281	5	:	:	PUNCT
iajs-725	281	6	m	m	VERB
iajs-725	281	7	]	]	X
iajs-725	281	8	=	=	PUNCT
iajs-725	281	9	annr	annr	NOUN
iajs-725	281	10	m	m	VERB
iajs-725	281	11	n	n	NOUN
iajs-725	281	12	,	,	PUNCT
iajs-725	281	13	thus	thus	ADV
iajs-725	281	14	[	[	X
iajs-725	281	15	n	n	X
iajs-725	281	16	r	r	NOUN
iajs-725	281	17	:	:	PUNCT
iajs-725	281	18	m	m	VERB
iajs-725	281	19	]	]	X
iajs-725	281	20	is	be	AUX
iajs-725	281	21	a	a	DET
iajs-725	281	22	semimaximal	semimaximal	ADJ
iajs-725	281	23	ideal	ideal	NOUN
iajs-725	281	24	.	.	PUNCT
iajs-725	282	1	the	the	DET
iajs-725	282	2	following	following	ADJ
iajs-725	282	3	result	result	NOUN
iajs-725	282	4	follows	follow	VERB
iajs-725	282	5	immediately	immediately	ADV
iajs-725	282	6	by	by	ADP
iajs-725	282	7	lemma	lemma	PROPN
iajs-725	282	8	(	(	PUNCT
iajs-725	282	9	2.7	2.7	NUM
iajs-725	282	10	)	)	PUNCT
iajs-725	282	11	.	.	PUNCT
iajs-725	283	1	2.8	2.8	NUM
iajs-725	283	2	proposition	proposition	NOUN
iajs-725	283	3	:	:	PUNCT
iajs-725	283	4	if	if	SCONJ
iajs-725	283	5	every	every	DET
iajs-725	283	6	submodule	submodule	NOUN
iajs-725	283	7	n	n	PROPN
iajs-725	283	8	of	of	ADP
iajs-725	283	9	an	an	DET
iajs-725	283	10	r	r	NOUN
iajs-725	283	11	-	-	PUNCT
iajs-725	283	12	module	module	NOUN
iajs-725	283	13	m	m	NOUN
iajs-725	283	14	is	be	AUX
iajs-725	283	15	semimaximal	semimaximal	ADJ
iajs-725	283	16	,	,	PUNCT
iajs-725	283	17	then	then	ADV
iajs-725	283	18	m	m	VERB
iajs-725	283	19	is	be	AUX
iajs-725	283	20	coannsemimaximal	coannsemimaximal	ADJ
iajs-725	283	21	.	.	PUNCT
iajs-725	284	1	..	..	PUNCT
iajs-725	284	2	next	next	ADV
iajs-725	284	3	,	,	PUNCT
iajs-725	284	4	we	we	PRON
iajs-725	284	5	have	have	VERB
iajs-725	284	6	the	the	DET
iajs-725	284	7	following	follow	VERB
iajs-725	284	8	remark	remark	NOUN
iajs-725	284	9	.	.	PUNCT
iajs-725	285	1	2.9	2.9	NUM
iajs-725	285	2	remark	remark	NOUN
iajs-725	285	3	:	:	PUNCT
iajs-725	285	4	the	the	DET
iajs-725	285	5	direct	direct	ADJ
iajs-725	285	6	sum	sum	NOUN
iajs-725	285	7	of	of	ADP
iajs-725	285	8	coannsemimaximal	coannsemimaximal	ADJ
iajs-725	285	9	modules	module	NOUN
iajs-725	285	10	need	need	AUX
iajs-725	285	11	not	not	PART
iajs-725	285	12	be	be	AUX
iajs-725	285	13	coannsemimaximal	coannsemimaximal	ADJ
iajs-725	285	14	.	.	PUNCT
iajs-725	286	1	for	for	ADP
iajs-725	286	2	example	example	NOUN
iajs-725	286	3	:	:	PUNCT
iajs-725	286	4	let	let	VERB
iajs-725	286	5	m	m	PRON
iajs-725	286	6	=	=	VERB
iajs-725	286	7	z4z3	z4z3	PROPN
iajs-725	286	8	be	be	AUX
iajs-725	286	9	a	a	DET
iajs-725	286	10	z	z	NOUN
iajs-725	286	11	-	-	PUNCT
iajs-725	286	12	module	module	NOUN
iajs-725	286	13	.	.	PUNCT
iajs-725	287	1	z4	z4	PROPN
iajs-725	287	2	and	and	CCONJ
iajs-725	287	3	z3	z3	PROPN
iajs-725	287	4	are	be	AUX
iajs-725	287	5	two	two	NUM
iajs-725	287	6	coannsemimaximal	coannsemimaximal	ADJ
iajs-725	287	7	z	z	NOUN
iajs-725	287	8	-	-	PUNCT
iajs-725	287	9	modules	module	NOUN
iajs-725	287	10	.	.	PUNCT
iajs-725	288	1	but	but	CCONJ
iajs-725	288	2	m	m	PROPN
iajs-725	288	3	�	�	PROPN
iajs-725	288	4	z12	z12	PROPN
iajs-725	288	5	which	which	PRON
iajs-725	288	6	is	be	AUX
iajs-725	288	7	not	not	PART
iajs-725	288	8	coannsemimaximal	coannsemimaximal	ADJ
iajs-725	288	9	.	.	PUNCT
iajs-725	289	1	ibn	ibn	PROPN
iajs-725	289	2	alhaitham	alhaitham	PROPN
iajs-725	289	3	j.	j.	PROPN
iajs-725	289	4	for	for	ADP
iajs-725	289	5	pure	pure	ADJ
iajs-725	289	6	&	&	CCONJ
iajs-725	289	7	appl	appl	PROPN
iajs-725	289	8	.	.	PUNCT
iajs-725	290	1	sci	sci	PROPN
iajs-725	290	2	.	.	PUNCT
iajs-725	290	3	vol.24	vol.24	NOUN
iajs-725	290	4	(	(	PUNCT
iajs-725	290	5	3	3	NUM
iajs-725	290	6	)	)	PUNCT
iajs-725	290	7	2011	2011	NUM
iajs-725	290	8	references	reference	NOUN
iajs-725	290	9	1	1	NUM
iajs-725	290	10	.	.	PUNCT
iajs-725	290	11	abdul	abdul	PROPN
iajs-725	290	12	-	-	PUNCT
iajs-725	290	13	razak	razak	PROPN
iajs-725	290	14	,	,	PUNCT
iajs-725	290	15	h.m	h.m	PROPN
iajs-725	290	16	.	.	PROPN
iajs-725	290	17	,	,	PUNCT
iajs-725	290	18	(	(	PUNCT
iajs-725	290	19	1999	1999	NUM
iajs-725	290	20	)	)	PUNCT
iajs-725	290	21	,	,	PUNCT
iajs-725	290	22	quasi	quasi	ADJ
iajs-725	290	23	-	-	ADJ
iajs-725	290	24	prime	prime	ADJ
iajs-725	290	25	modules	module	NOUN
iajs-725	290	26	and	and	CCONJ
iajs-725	290	27	quasi	quasi	ADJ
iajs-725	290	28	-	-	ADJ
iajs-725	290	29	prime	prime	ADJ
iajs-725	290	30	submodules	submodule	NOUN
iajs-725	290	31	,	,	PUNCT
iajs-725	290	32	m.d	m.d	PROPN
iajs-725	290	33	.	.	PROPN
iajs-725	290	34	thesis	thesis	PROPN
iajs-725	290	35	,	,	PUNCT
iajs-725	290	36	univ	univ	PROPN
iajs-725	290	37	.	.	PROPN
iajs-725	290	38	of	of	ADP
iajs-725	290	39	baghdad	baghdad	PROPN
iajs-725	290	40	.	.	PUNCT
iajs-725	291	1	2	2	X
iajs-725	291	2	.	.	X
iajs-725	291	3	abdul	abdul	PROPN
iajs-725	291	4	-	-	PUNCT
iajs-725	291	5	al	al	PROPN
iajs-725	291	6	-	-	PUNCT
iajs-725	291	7	kalik	kalik	NOUN
iajs-725	291	8	,	,	PUNCT
iajs-725	291	9	j.a	j.a	PROPN
iajs-725	291	10	.	.	PROPN
iajs-725	291	11	,	,	PUNCT
iajs-725	291	12	(	(	PUNCT
iajs-725	291	13	2005	2005	NUM
iajs-725	291	14	)	)	PUNCT
iajs-725	291	15	,	,	PUNCT
iajs-725	291	16	primary	primary	ADJ
iajs-725	291	17	modules	module	NOUN
iajs-725	291	18	,	,	PUNCT
iajs-725	291	19	m	m	VERB
iajs-725	291	20	.d	.d	ADJ
iajs-725	291	21	.	.	PUNCT
iajs-725	292	1	thesis	thesis	NOUN
iajs-725	292	2	,	,	PUNCT
iajs-725	292	3	univ	univ	PROPN
iajs-725	292	4	.	.	PROPN
iajs-725	292	5	of	of	ADP
iajs-725	292	6	baghdad	baghdad	PROPN
iajs-725	292	7	.	.	PUNCT
iajs-725	293	1	3	3	X
iajs-725	293	2	.	.	X
iajs-725	293	3	abdul	abdul	PROPN
iajs-725	293	4	-	-	PUNCT
iajs-725	293	5	al	al	PROPN
iajs-725	293	6	-	-	PUNCT
iajs-725	293	7	kalik	kalik	NOUN
iajs-725	293	8	,	,	PUNCT
iajs-725	293	9	j.a	j.a	PROPN
iajs-725	293	10	.	.	PROPN
iajs-725	293	11	,	,	PUNCT
iajs-725	293	12	(	(	PUNCT
iajs-725	293	13	2009	2009	NUM
iajs-725	293	14	)	)	PUNCT
iajs-725	293	15	,	,	PUNCT
iajs-725	293	16	on	on	ADP
iajs-725	293	17	m	m	NOUN
iajs-725	293	18	ax	ax	NOUN
iajs-725	293	19	-	-	PUNCT
iajs-725	293	20	modules	module	NOUN
iajs-725	293	21	,	,	PUNCT
iajs-725	293	22	to	to	PART
iajs-725	293	23	appear	appear	VERB
iajs-725	293	24	.	.	PUNCT
iajs-725	294	1	4	4	X
iajs-725	294	2	.	.	X
iajs-725	294	3	coodreal	coodreal	PROPN
iajs-725	294	4	,	,	PUNCT
iajs-725	294	5	k.r	k.r	PROPN
iajs-725	294	6	.	.	PROPN
iajs-725	294	7	,	,	PUNCT
iajs-725	294	8	(	(	PUNCT
iajs-725	294	9	1976	1976	NUM
iajs-725	294	10	)	)	PUNCT
iajs-725	294	11	,	,	PUNCT
iajs-725	294	12	ring	ring	NOUN
iajs-725	294	13	theory	theory	NOUN
iajs-725	294	14	-	-	PUNCT
iajs-725	294	15	non	non	ADJ
iajs-725	294	16	singular	singular	PROPN
iajs-725	294	17	rings	ring	NOUN
iajs-725	294	18	and	and	CCONJ
iajs-725	294	19	modules	module	NOUN
iajs-725	294	20	,	,	PUNCT
iajs-725	294	21	marcei	marcei	NOUN
iajs-725	294	22	-	-	PUNCT
iajs-725	294	23	dekker	dekker	NOUN
iajs-725	294	24	,	,	PUNCT
iajs-725	294	25	new	new	PROPN
iajs-725	294	26	york	york	PROPN
iajs-725	294	27	and	and	CCONJ
iajs-725	294	28	basel	basel	PROPN
iajs-725	294	29	.	.	PUNCT
iajs-725	295	1	5	5	X
iajs-725	295	2	.	.	X
iajs-725	295	3	khalaf	khalaf	PROPN
iajs-725	295	4	,	,	PUNCT
iajs-725	295	5	y.h	y.h	PROPN
iajs-725	295	6	.	.	PROPN
iajs-725	295	7	,	,	PUNCT
iajs-725	295	8	(	(	PUNCT
iajs-725	295	9	2007	2007	NUM
iajs-725	295	10	)	)	PUNCT
iajs-725	295	11	,	,	PUNCT
iajs-725	295	12	semimaximal	semimaximal	NOUN
iajs-725	295	13	submodules	submodule	NOUN
iajs-725	295	14	,	,	PUNCT
iajs-725	295	15	ph.d	ph.d	PROPN
iajs-725	295	16	.	.	PUNCT
iajs-725	296	1	thesis	thesis	NOUN
iajs-725	296	2	,	,	PUNCT
iajs-725	296	3	university	university	NOUN
iajs-725	296	4	of	of	ADP
iajs-725	296	5	baghdad	baghdad	PROPN
iajs-725	296	6	.	.	PUNCT
iajs-725	297	1	6	6	NUM
iajs-725	297	2	.	.	X
iajs-725	297	3	kasch	kasch	PROPN
iajs-725	297	4	,	,	PUNCT
iajs-725	297	5	f.	f.	PROPN
iajs-725	297	6	,	,	PUNCT
iajs-725	297	7	(	(	PUNCT
iajs-725	297	8	1982	1982	NUM
iajs-725	297	9	)	)	PUNCT
iajs-725	297	10	,	,	PUNCT
iajs-725	297	11	m	m	VERB
iajs-725	297	12	odules	odule	NOUN
iajs-725	297	13	and	and	CCONJ
iajs-725	297	14	rings	ring	NOUN
iajs-725	297	15	,	,	PUNCT
iajs-725	297	16	academic	academic	ADJ
iajs-725	297	17	press	press	NOUN
iajs-725	297	18	,	,	PUNCT
iajs-725	297	19	london	london	PROPN
iajs-725	297	20	.	.	PUNCT
iajs-725	298	1	7	7	X
iajs-725	298	2	.	.	X
iajs-725	298	3	al	al	PROPN
iajs-725	298	4	-	-	PUNCT
iajs-725	298	5	sharide	sharide	NOUN
iajs-725	298	6	,	,	PUNCT
iajs-725	298	7	f.a.f	f.a.f	NOUN
iajs-725	298	8	.	.	PROPN
iajs-725	298	9	,	,	PUNCT
iajs-725	298	10	(	(	PUNCT
iajs-725	298	11	2008	2008	NUM
iajs-725	298	12	)	)	PUNCT
iajs-725	298	13	,	,	PUNCT
iajs-725	298	14	s	s	X
iajs-725	298	15	-	-	PUNCT
iajs-725	298	16	compactly	compactly	ADV
iajs-725	298	17	packed	pack	VERB
iajs-725	298	18	submodules	submodule	NOUN
iajs-725	298	19	and	and	CCONJ
iajs-725	298	20	semiprime	semiprime	NOUN
iajs-725	298	21	modules	module	NOUN
iajs-725	298	22	,	,	PUNCT
iajs-725	298	23	ms.c.thesis	ms.c.thesis	PROPN
iajs-725	298	24	,	,	PUNCT
iajs-725	298	25	university	university	NOUN
iajs-725	298	26	of	of	ADP
iajs-725	298	27	tikrit	tikrit	NOUN
iajs-725	298	28	.	.	PUNCT
iajs-725	299	1	8	8	NUM
iajs-725	299	2	.	.	X
iajs-725	299	3	farzalipour	farzalipour	PROPN
iajs-725	299	4	,	,	PUNCT
iajs-725	299	5	f.	f.	PROPN
iajs-725	299	6	and	and	CCONJ
iajs-725	299	7	ghiasvand	ghiasvand	PROPN
iajs-725	299	8	,	,	PUNCT
iajs-725	299	9	p.	p.	PROPN
iajs-725	299	10	,	,	PUNCT
iajs-725	299	11	(	(	PUNCT
iajs-725	299	12	2009	2009	NUM
iajs-725	299	13	)	)	PUNCT
iajs-725	299	14	,	,	PUNCT
iajs-725	299	15	quasi	quasi	ADJ
iajs-725	299	16	-	-	ADJ
iajs-725	299	17	multiplication	multiplication	NOUN
iajs-725	299	18	modules	module	NOUN
iajs-725	299	19	,	,	PUNCT
iajs-725	299	20	thai	thai	PROPN
iajs-725	299	21	j.	j.	PROPN
iajs-725	299	22	of	of	ADP
iajs-725	299	23	math	math	PROPN
iajs-725	299	24	.	.	PUNCT
iajs-725	299	25	,	,	PUNCT
iajs-725	299	26	vol.7(2	vol.7(2	NOUN
iajs-725	299	27	)	)	PUNCT
iajs-725	299	28	,	,	PUNCT
iajs-725	299	29	pp.361	pp.361	NOUN
iajs-725	299	30	-	-	PUNCT
iajs-725	299	31	366	366	NUM
iajs-725	299	32	.	.	PUNCT
iajs-725	300	1	9	9	X
iajs-725	300	2	.	.	X
iajs-725	300	3	desale	desale	NOUN
iajs-725	300	4	,	,	PUNCT
iajs-725	300	5	g.	g.	PROPN
iajs-725	300	6	and	and	CCONJ
iajs-725	300	7	nicholson	nicholson	PROPN
iajs-725	300	8	,	,	PUNCT
iajs-725	300	9	w.k	w.k	PROPN
iajs-725	300	10	.	.	PROPN
iajs-725	300	11	,	,	PUNCT
iajs-725	300	12	(	(	PUNCT
iajs-725	300	13	1981	1981	NUM
iajs-725	300	14	)	)	PUNCT
iajs-725	300	15	,	,	PUNCT
iajs-725	300	16	endoprimitive	endoprimitive	ADJ
iajs-725	300	17	rings	ring	NOUN
iajs-725	300	18	,	,	PUNCT
iajs-725	300	19	j.algebra	j.algebra	PROPN
iajs-725	300	20	,	,	PUNCT
iajs-725	300	21	vol.70	vol.70	NOUN
iajs-725	300	22	,	,	PUNCT
iajs-725	300	23	pp.548	pp.548	NOUN
iajs-725	300	24	-	-	SYM
iajs-725	300	25	560	560	NUM
iajs-725	300	26	.	.	NOUN
iajs-725	300	27	10	10	NUM
iajs-725	300	28	.	.	PUNCT
iajs-725	301	1	zelmanowitz	zelmanowitz	PROPN
iajs-725	301	2	,	,	PUNCT
iajs-725	301	3	j.	j.	PROPN
iajs-725	301	4	,	,	PUNCT
iajs-725	301	5	(	(	PUNCT
iajs-725	301	6	1972	1972	NUM
iajs-725	301	7	)	)	PUNCT
iajs-725	301	8	,	,	PUNCT
iajs-725	301	9	regular	regular	ADJ
iajs-725	301	10	modules	module	NOUN
iajs-725	301	11	,	,	PUNCT
iajs-725	301	12	trans.amer.math	trans.amer.math	PROPN
iajs-725	301	13	.	.	PUNCT
iajs-725	302	1	soc	soc	PROPN
iajs-725	302	2	.	.	PROPN
iajs-725	302	3	,	,	PUNCT
iajs-725	302	4	vol.163	vol.163	NOUN
iajs-725	302	5	,	,	PUNCT
iajs-725	302	6	pp.341	pp.341	NOUN
iajs-725	302	7	-	-	SYM
iajs-725	302	8	355	355	NUM
iajs-725	302	9	.	.	PUNCT
iajs-725	303	1	11	11	NUM
iajs-725	303	2	.	.	X
iajs-725	304	1	mijbass	mijbass	PROPN
iajs-725	304	2	,	,	PUNCT
iajs-725	304	3	s.a	s.a	PROPN
iajs-725	304	4	.	.	PROPN
iajs-725	304	5	,	,	PUNCT
iajs-725	304	6	(	(	PUNCT
iajs-725	304	7	1997	1997	NUM
iajs-725	304	8	)	)	PUNCT
iajs-725	304	9	,	,	PUNCT
iajs-725	304	10	quasi	quasi	ADJ
iajs-725	304	11	-	-	ADJ
iajs-725	304	12	dedekind	dedekind	ADJ
iajs-725	304	13	modules	module	NOUN
iajs-725	304	14	,	,	PUNCT
iajs-725	304	15	ph.d	ph.d	PROPN
iajs-725	304	16	.	.	PUNCT
iajs-725	305	1	thesis	thesis	PROPN
iajs-725	305	2	,	,	PUNCT
iajs-725	305	3	univ	univ	PROPN
iajs-725	305	4	.	.	PROPN
iajs-725	305	5	of	of	ADP
iajs-725	305	6	baghdad	baghdad	PROPN
iajs-725	305	7	.	.	PUNCT
iajs-725	306	1	12	12	NUM
iajs-725	306	2	.	.	PUNCT
iajs-725	307	1	atani	atani	PROPN
iajs-725	307	2	,	,	PUNCT
iajs-725	307	3	e.sh	e.sh	PROPN
iajs-725	307	4	.	.	PROPN
iajs-725	307	5	and	and	CCONJ
iajs-725	307	6	darani	darani	PROPN
iajs-725	307	7	,	,	PUNCT
iajs-725	307	8	y.a	y.a	PROPN
iajs-725	307	9	.	.	PROPN
iajs-725	307	10	,	,	PUNCT
iajs-725	307	11	(	(	PUNCT
iajs-725	307	12	2006	2006	NUM
iajs-725	307	13	)	)	PUNCT
iajs-725	307	14	,	,	PUNCT
iajs-725	307	15	on	on	ADP
iajs-725	307	16	quasi	quasi	ADJ
iajs-725	307	17	-	-	ADJ
iajs-725	307	18	primary	primary	ADJ
iajs-725	307	19	submodules	submodule	NOUN
iajs-725	307	20	,	,	PUNCT
iajs-725	307	21	chiange	chiange	PROPN
iajs-725	307	22	mai	mai	PROPN
iajs-725	307	23	j.	j.	PROPN
iajs-725	307	24	sci	sci	PROPN
iajs-725	307	25	,	,	PUNCT
iajs-725	307	26	vol.33(3	vol.33(3	NOUN
iajs-725	307	27	)	)	PUNCT
iajs-725	307	28	,	,	PUNCT
iajs-725	307	29	pp.249	pp.249	NOUN
iajs-725	307	30	-	-	SYM
iajs-725	307	31	254	254	NUM
iajs-725	307	32	.	.	PUNCT
iajs-725	308	1	13	13	NUM
iajs-725	308	2	.	.	PUNCT
iajs-725	309	1	annin	annin	PROPN
iajs-725	309	2	,	,	PUNCT
iajs-725	309	3	s.	s.	PROPN
iajs-725	309	4	,	,	PUNCT
iajs-725	309	5	(	(	PUNCT
iajs-725	309	6	2002	2002	NUM
iajs-725	309	7	)	)	PUNCT
iajs-725	309	8	,	,	PUNCT
iajs-725	309	9	associated	associate	VERB
iajs-725	309	10	and	and	CCONJ
iajs-725	309	11	attached	attach	VERB
iajs-725	309	12	primes	prime	NOUN
iajs-725	309	13	over	over	ADP
iajs-725	309	14	non	non	ADJ
iajs-725	309	15	commutative	commutative	ADJ
iajs-725	309	16	rings	ring	NOUN
iajs-725	309	17	,	,	PUNCT
iajs-725	309	18	ph.d	ph.d	PROPN
iajs-725	309	19	.	.	PUNCT
iajs-725	310	1	thesis	thesis	PROPN
iajs-725	310	2	,	,	PUNCT
iajs-725	310	3	univ	univ	PROPN
iajs-725	310	4	.	.	PROPN
iajs-725	310	5	of	of	ADP
iajs-725	310	6	berkeley	berkeley	PROPN
iajs-725	310	7	.	.	PUNCT
iajs-725	311	1	14	14	NUM
iajs-725	311	2	.	.	PUNCT
iajs-725	312	1	khalaf	khalaf	PROPN
iajs-725	312	2	,	,	PUNCT
iajs-725	312	3	i.r	i.r	PROPN
iajs-725	312	4	.	.	PROPN
iajs-725	312	5	,	,	PUNCT
iajs-725	312	6	(	(	PUNCT
iajs-725	312	7	2009	2009	NUM
iajs-725	312	8	)	)	PUNCT
iajs-725	312	9	,	,	PUNCT
iajs-725	312	10	dual	dual	ADJ
iajs-725	312	11	notions	notion	NOUN
iajs-725	312	12	of	of	ADP
iajs-725	312	13	prime	prime	ADJ
iajs-725	312	14	submodules	submodule	NOUN
iajs-725	312	15	and	and	CCONJ
iajs-725	312	16	prime	prime	ADJ
iajs-725	312	17	modules	module	NOUN
iajs-725	312	18	,	,	PUNCT
iajs-725	312	19	m.d	m.d	PROPN
iajs-725	312	20	.	.	PROPN
iajs-725	312	21	thesis	thesis	PROPN
iajs-725	312	22	,	,	PUNCT
iajs-725	312	23	univ	univ	PROPN
iajs-725	312	24	.	.	PROPN
iajs-725	312	25	of	of	ADP
iajs-725	312	26	baghdad	baghdad	PROPN
iajs-725	312	27	.	.	PUNCT
iajs-725	313	1	15	15	NUM
iajs-725	313	2	.	.	PUNCT
iajs-725	314	1	hirano	hirano	PROPN
iajs-725	314	2	,	,	PUNCT
iajs-725	314	3	y.	y.	PROPN
iajs-725	314	4	and	and	CCONJ
iajs-725	314	5	mogani	mogani	PROPN
iajs-725	314	6	,	,	PUNCT
iajs-725	314	7	i.	i.	NOUN
iajs-725	314	8	,	,	PUNCT
iajs-725	314	9	(	(	PUNCT
iajs-725	314	10	1986	1986	NUM
iajs-725	314	11	)	)	PUNCT
iajs-725	314	12	,	,	PUNCT
iajs-725	314	13	on	on	ADP
iajs-725	314	14	restricted	restrict	VERB
iajs-725	314	15	anti	anti	ADJ
iajs-725	314	16	-	-	ADJ
iajs-725	314	17	hopfian	hopfian	ADJ
iajs-725	314	18	modules	module	NOUN
iajs-725	314	19	,	,	PUNCT
iajs-725	314	20	math	math	NOUN
iajs-725	314	21	.	.	PUNCT
iajs-725	315	1	j.	j.	PROPN
iajs-725	315	2	okayama	okayama	PROPN
iajs-725	315	3	univ	univ	PROPN
iajs-725	315	4	.	.	PROPN
iajs-725	315	5	,	,	PUNCT
iajs-725	315	6	vol.28	vol.28	PROPN
iajs-725	315	7	,	,	PUNCT
iajs-725	315	8	pp.119	pp.119	PROPN
iajs-725	315	9	-	-	PUNCT
iajs-725	315	10	131	131	NUM
iajs-725	315	11	.	.	NOUN
iajs-725	315	12	2011	2011	NUM
iajs-725	315	13	)	)	PUNCT
iajs-725	315	14	3	3	NUM
iajs-725	315	15	(	(	PUNCT
iajs-725	315	16	24المجلد	24المجلد	NUM
iajs-725	315	17	مجلة	مجلة	VERB
iajs-725	315	18	ابن	ابن	PROPN
iajs-725	315	19	الهیثم	الهیثم	PROPN
iajs-725	315	20	للعلوم	للعلوم	PROPN
iajs-725	315	21	الصرفة	الصرفة	NOUN
iajs-725	315	22	والتطبیقیة	والتطبیقیة	PROPN
iajs-725	315	23	وشبھ	وشبھ	PROPN
iajs-725	315	24	االعظمیة	االعظمیة	PROPN
iajs-725	315	25	التالفة	التالفة	VERB
iajs-725	315	26	المضادة	المضادة	PROPN
iajs-725	315	27	المقاسات	المقاسات	PROPN
iajs-725	315	28	شبھ	شبھ	PROPN
iajs-725	315	29	االعظمیة	االعظمیة	NOUN
iajs-725	315	30	التالفة	التالفة	VERB
iajs-725	315	31	حاتم	حاتم	PROPN
iajs-725	315	32	یحیى	یحیى	NOUN
iajs-725	315	33	خلف	خلف	PROPN
iajs-725	315	34	،	،	PROPN
iajs-725	315	35	أنعام	أنعام	PROPN
iajs-725	315	36	محمد	محمد	PROPN
iajs-725	315	37	علي	علي	NOUN
iajs-725	315	38	هادي	هادي	NOUN
iajs-725	315	39	جامعة	جامعة	PROPN
iajs-725	315	40	بغداد	بغداد	PROPN
iajs-725	315	41	،	،	PROPN
iajs-725	316	1	ابن	ابن	X
iajs-725	316	2	الهیثم	الهیثم	PROPN
iajs-725	316	3	-كلیة	-كلیة	PROPN
iajs-725	316	4	التربیة،قسم	التربیة،قسم	PROPN
iajs-725	316	5	الریاضیات	الریاضیات	NOUN
iajs-725	316	6	یلول	یلول	NOUN
iajs-725	316	7	20	20	NUM
iajs-725	316	8	:	:	PUNCT
iajs-725	316	9	استلم	استلم	PROPN
iajs-725	316	10	البحث	البحث	VERB
iajs-725	316	11	في	في	ADP
iajs-725	316	12	2010	2010	NUM
iajs-725	316	13	أ	أ	SYM
iajs-725	316	14	2011	2011	NUM
iajs-725	316	15	شباط	شباط	ADV
iajs-725	316	16	8	8	NUM
iajs-725	316	17	:	:	PUNCT
iajs-725	316	18	قبل	قبل	NOUN
iajs-725	316	19	البحث	البحث	VERB
iajs-725	316	20	في	في	ADP
iajs-725	316	21	الخالصة	الخالصة	NOUN
iajs-725	316	22	في	في	INTJ
iajs-725	316	23	.	.	PUNCT
iajs-725	317	1	ن	ن	PROPN
iajs-725	317	2	درسوا	درسوا	PROPN
iajs-725	317	3	المقاسات	المقاسات	PROPN
iajs-725	317	4	التي	التي	PROPN
iajs-725	317	5	تالف	تالف	PROPN
iajs-725	317	6	كل	كل	PROPN
iajs-725	317	7	مقاس	مقاس	PROPN
iajs-725	317	8	جزئي	جزئي	NOUN
iajs-725	317	9	غیر	غیر	ADJ
iajs-725	317	10	صفري	صفري	NOUN
iajs-725	317	11	منها	منها	NOUN
iajs-725	317	12	هو	هو	PROPN
iajs-725	317	13	أولي	أولي	PROPN
iajs-725	317	14	،	،	PROPN
iajs-725	317	15	ابتدائي	ابتدائي	PROPN
iajs-725	317	16	أو	أو	PROPN
iajs-725	317	17	اعظميبعض	اعظميبعض	PROPN
iajs-725	317	18	الباحثی	الباحثی	PROPN
iajs-725	318	1	mهذا	mهذا	AUX
iajs-725	318	2	البحث	البحث	PROPN
iajs-725	318	3	قدمنا	قدمنا	PROPN
iajs-725	318	4	ودرسنا	ودرسنا	PROPN
iajs-725	318	5	المقاسات	المقاسات	PROPN
iajs-725	318	6	شبه	شبه	NOUN
iajs-725	318	7	االعظمیة	االعظمیة	NOUN
iajs-725	318	8	التالفة	التالفة	VERB
iajs-725	318	9	والمقاسات	والمقاسات	PROPN
iajs-725	318	10	شبه	شبه	NOUN
iajs-725	318	11	االعظمیة	االعظمیة	NOUN
iajs-725	318	12	التالفة	التالفة	VERB
iajs-725	318	13	المضادة	المضادة	PROPN
iajs-725	318	14	،	،	PROPN
iajs-725	318	15	حیث	حیث	PROPN
iajs-725	318	16	یدعى	یدعى	PROPN
iajs-725	318	17	المقاس	المقاس	X
iajs-725	318	18	على	على	NOUN
iajs-725	318	19	التوالي	التوالي	NOUN
iajs-725	318	20	(	(	PUNCT
iajs-725	318	21	rعلى	rعلى	NOUN
iajs-725	318	22	الحلقة	الحلقة	PROPN
iajs-725	318	23	nاذا	nاذا	VERB
iajs-725	318	24	كان	كان	NOUN
iajs-725	318	25	تالف	تالف	NOUN
iajs-725	318	26	)	)	PUNCT
iajs-725	318	27	على	على	NOUN
iajs-725	318	28	التوالي	التوالي	NOUN
iajs-725	318	29	شبه	شبه	NOUN
iajs-725	318	30	اعظمي	اعظمي	ADJ
iajs-725	318	31	تالف	تالف	NOUN
iajs-725	318	32	مضاد(شبه	مضاد(شبه	VERB
iajs-725	318	33	اعظمي	اعظمي	ADJ
iajs-725	318	34	تالف	تالف	NOUN
iajs-725	318	35	rعلى	rعلى	PROPN
iajs-725	318	36	الحلقة	الحلقة	PROPN
iajs-725	318	37	تالف	تالف	PROPN
iajs-725	318	38	m	m	VERB
iajs-725	318	39	n	n	ADP
iajs-725	318	40	.mفي	.mفي	PUNCT
iajs-725	318	41	nلكل	nلكل	PROPN
iajs-725	318	42	مقاس	مقاس	PROPN
iajs-725	319	1	جزئي	جزئي	NOUN
iajs-725	319	2	غیر	غیر	ADJ
iajs-725	319	3	صفري	صفري	NOUN
iajs-725	319	4	rهو	rهو	NOUN
iajs-725	319	5	مثالي	مثالي	ADJ
iajs-725	319	6	شبه	شبه	NOUN
iajs-725	319	7	اعظمي	اعظمي	ADJ
iajs-725	319	8	في	في	PRON
iajs-725	319	9	)	)	PUNCT
iajs-725	319	10	rعلى	rعلى	PROPN
iajs-725	319	11	الحلقة	الحلقة	NOUN
iajs-725	319	12	:	:	PUNCT
iajs-725	319	13	الكلمات	الكلمات	VERB
iajs-725	319	14	المفتاحیة	المفتاحیة	ADJ
iajs-725	319	15	annsemimaximal	annsemimaximal	ADJ
iajs-725	319	16	module	module	NOUN
iajs-725	319	17	,	,	PUNCT
iajs-725	319	18	semisimple	semisimple	NOUN
iajs-725	319	19	module	module	NOUN
iajs-725	319	20	,	,	PUNCT
iajs-725	319	21	semisimple	semisimple	NOUN
iajs-725	319	22	ring	ring	NOUN
iajs-725	319	23	,	,	PUNCT
iajs-725	319	24	semiprime	semiprime	NOUN
iajs-725	319	25	module	module	NOUN
iajs-725	319	26	,	,	PUNCT
iajs-725	319	27	maxmodule	maxmodule	NOUN
iajs-725	319	28	,	,	PUNCT
iajs-725	319	29	uniform	uniform	NOUN
iajs-725	319	30	module	module	NOUN
iajs-725	319	31	,	,	PUNCT
iajs-725	319	32	z	z	NOUN
iajs-725	319	33	-	-	PUNCT
iajs-725	319	34	regular	regular	ADJ
iajs-725	319	35	module	module	NOUN
iajs-725	319	36	,	,	PUNCT
iajs-725	319	37	f	f	X
iajs-725	319	38	-	-	PUNCT
iajs-725	319	39	regular	regular	ADJ
iajs-725	319	40	module	module	NOUN
iajs-725	319	41	,	,	PUNCT
iajs-725	319	42	artinian	artinian	ADJ
iajs-725	319	43	module	module	NOUN
iajs-725	319	44	,	,	PUNCT
iajs-725	319	45	flat	flat	ADJ
iajs-725	319	46	module	module	NOUN
iajs-725	319	47	,	,	PUNCT
iajs-725	319	48	coprime	coprime	NOUN
iajs-725	319	49	module	module	NOUN
iajs-725	319	50	,	,	PUNCT
iajs-725	319	51	coannsemimaximal	coannsemimaximal	ADJ
iajs-725	319	52	module	module	NOUN
iajs-725	319	53	.	.	PUNCT