id	sid	tid	token	lemma	pos
iajs-758	1	1	ibn	ibn	PROPN
iajs-758	1	2	alhaitham	alhaitham	NOUN
iajs-758	1	3	j.	j.	PROPN
iajs-758	1	4	for	for	ADP
iajs-758	1	5	pure	pure	ADJ
iajs-758	1	6	&	&	CCONJ
iajs-758	1	7	appl	appl	PROPN
iajs-758	1	8	.	.	PUNCT
iajs-758	2	1	sci	sci	PROPN
iajs-758	2	2	.	.	PUNCT
iajs-758	2	3	vol.24	vol.24	NOUN
iajs-758	2	4	(	(	PUNCT
iajs-758	2	5	2	2	NUM
iajs-758	2	6	)	)	PUNCT
iajs-758	2	7	2011	2011	NUM
iajs-758	2	8	on	on	ADP
iajs-758	2	9	max	max	PROPN
iajs-758	2	10	-	-	PUNCT
iajs-758	2	11	modules	module	NOUN
iajs-758	2	12	a.	a.	NOUN
iajs-758	2	13	j.	j.	PROPN
iajs-758	2	14	abdul	abdul	PROPN
iajs-758	2	15	–	–	PUNCT
iajs-758	2	16	al	al	PROPN
iajs-758	2	17	–	–	PUNCT
iajs-758	2	18	kalik	kalik	PROPN
iajs-758	2	19	ministry	ministry	PROPN
iajs-758	2	20	of	of	ADP
iajs-758	2	21	education	education	PROPN
iajs-758	2	22	vocational	vocational	ADJ
iajs-758	2	23	education	education	NOUN
iajs-758	2	24	,	,	PUNCT
iajs-758	2	25	khalis	khalis	PROPN
iajs-758	2	26	,	,	PUNCT
iajs-758	2	27	industrial	industrial	ADJ
iajs-758	2	28	school	school	NOUN
iajs-758	2	29	received	receive	VERB
iajs-758	2	30	in	in	ADP
iajs-758	2	31	:	:	PUNCT
iajs-758	2	32	15	15	NUM
iajs-758	2	33	,	,	PUNCT
iajs-758	2	34	december	december	PROPN
iajs-758	2	35	,	,	PUNCT
iajs-758	2	36	2009	2009	NUM
iajs-758	2	37	accepted	accept	VERB
iajs-758	2	38	in	in	ADP
iajs-758	2	39	:	:	PUNCT
iajs-758	2	40	17	17	NUM
iajs-758	2	41	,	,	PUNCT
iajs-758	2	42	june	june	PROPN
iajs-758	2	43	,	,	PUNCT
iajs-758	2	44	2010	2010	NUM
iajs-758	2	45	abstract	abstract	NOUN
iajs-758	2	46	in	in	ADP
iajs-758	2	47	this	this	DET
iajs-758	2	48	paper	paper	NOUN
iajs-758	2	49	,	,	PUNCT
iajs-758	2	50	we	we	PRON
iajs-758	2	51	introduce	introduce	VERB
iajs-758	2	52	a	a	DET
iajs-758	2	53	concept	concept	NOUN
iajs-758	2	54	of	of	ADP
iajs-758	2	55	max	max	PROPN
iajs-758	2	56	–	–	PUNCT
iajs-758	2	57	module	module	NOUN
iajs-758	2	58	as	as	SCONJ
iajs-758	2	59	follows	follow	VERB
iajs-758	2	60	:	:	PUNCT
iajs-758	2	61	m	m	VERB
iajs-758	2	62	is	be	AUX
iajs-758	2	63	called	call	VERB
iajs-758	2	64	a	a	DET
iajs-758	2	65	max	max	PROPN
iajs-758	2	66	module	module	NOUN
iajs-758	2	67	if	if	SCONJ
iajs-758	2	68	nann	nann	PROPN
iajs-758	2	69	r	r	NOUN
iajs-758	2	70	is	be	AUX
iajs-758	2	71	a	a	DET
iajs-758	2	72	maximal	maximal	ADJ
iajs-758	2	73	ideal	ideal	NOUN
iajs-758	2	74	of	of	ADP
iajs-758	2	75	r	r	NOUN
iajs-758	2	76	,	,	PUNCT
iajs-758	2	77	for	for	ADP
iajs-758	2	78	each	each	DET
iajs-758	2	79	non	non	ADJ
iajs-758	2	80	–	–	PUNCT
iajs-758	2	81	zero	zero	NUM
iajs-758	2	82	submodule	submodule	NOUN
iajs-758	2	83	n	n	PROPN
iajs-758	2	84	of	of	ADP
iajs-758	2	85	m	m	PRON
iajs-758	2	86	;	;	PUNCT
iajs-758	2	87	in	in	ADP
iajs-758	2	88	other	other	ADJ
iajs-758	2	89	words	word	NOUN
iajs-758	2	90	,	,	PUNCT
iajs-758	2	91	m	m	VERB
iajs-758	2	92	is	be	AUX
iajs-758	2	93	a	a	DET
iajs-758	2	94	max	max	PROPN
iajs-758	2	95	–	–	PUNCT
iajs-758	2	96	module	module	NOUN
iajs-758	2	97	iff	iff	PROPN
iajs-758	2	98	(	(	PUNCT
iajs-758	2	99	0	0	NUM
iajs-758	2	100	)	)	PUNCT
iajs-758	2	101	is	be	AUX
iajs-758	2	102	a	a	DET
iajs-758	2	103	*	*	ADJ
iajs-758	2	104	submodule	submodule	NOUN
iajs-758	2	105	,	,	PUNCT
iajs-758	2	106	where	where	SCONJ
iajs-758	2	107	a	a	DET
iajs-758	2	108	proper	proper	ADJ
iajs-758	2	109	submodule	submodule	NOUN
iajs-758	2	110	n	n	PROPN
iajs-758	2	111	of	of	ADP
iajs-758	2	112	m	m	PROPN
iajs-758	2	113	is	be	AUX
iajs-758	2	114	called	call	VERB
iajs-758	2	115	a	a	DET
iajs-758	2	116	*	*	ADJ
iajs-758	2	117	submodule	submodule	NOUN
iajs-758	2	118	if	if	SCONJ
iajs-758	2	119	]	]	X
iajs-758	2	120	[	[	PUNCT
iajs-758	2	121	:	:	PUNCT
iajs-758	2	122	kn	kn	NOUN
iajs-758	2	123	r	r	NOUN
iajs-758	2	124	is	be	AUX
iajs-758	2	125	a	a	DET
iajs-758	2	126	maximal	maximal	ADJ
iajs-758	2	127	ideal	ideal	NOUN
iajs-758	2	128	of	of	ADP
iajs-758	2	129	r	r	NOUN
iajs-758	2	130	,	,	PUNCT
iajs-758	2	131	for	for	SCONJ
iajs-758	2	132	each	each	DET
iajs-758	2	133	submodule	submodule	NOUN
iajs-758	2	134	k	k	PROPN
iajs-758	2	135	contains	contain	VERB
iajs-758	2	136	n	n	PRON
iajs-758	2	137	properly	properly	ADV
iajs-758	2	138	.	.	PUNCT
iajs-758	3	1	in	in	ADP
iajs-758	3	2	this	this	DET
iajs-758	3	3	paper	paper	NOUN
iajs-758	3	4	,	,	PUNCT
iajs-758	3	5	some	some	DET
iajs-758	3	6	properties	property	NOUN
iajs-758	3	7	and	and	CCONJ
iajs-758	3	8	characterizations	characterization	NOUN
iajs-758	3	9	of	of	ADP
iajs-758	3	10	max	max	PROPN
iajs-758	3	11	–	–	PUNCT
iajs-758	3	12	modules	module	NOUN
iajs-758	3	13	and	and	CCONJ
iajs-758	3	14	*	*	NUM
iajs-758	3	15	submodules	submodule	NOUN
iajs-758	3	16	are	be	AUX
iajs-758	3	17	given	give	VERB
iajs-758	3	18	.	.	PUNCT
iajs-758	4	1	also	also	ADV
iajs-758	4	2	,	,	PUNCT
iajs-758	4	3	various	various	ADJ
iajs-758	4	4	basic	basic	ADJ
iajs-758	4	5	results	result	NOUN
iajs-758	4	6	a	a	DET
iajs-758	4	7	bout	bout	NOUN
iajs-758	4	8	max	max	NOUN
iajs-758	4	9	–	–	PUNCT
iajs-758	4	10	modules	module	NOUN
iajs-758	4	11	are	be	AUX
iajs-758	4	12	considered	consider	VERB
iajs-758	4	13	.	.	PUNCT
iajs-758	5	1	moreover	moreover	ADV
iajs-758	5	2	,	,	PUNCT
iajs-758	5	3	some	some	DET
iajs-758	5	4	relations	relation	NOUN
iajs-758	5	5	between	between	ADP
iajs-758	5	6	maxmodules	maxmodule	NOUN
iajs-758	5	7	and	and	CCONJ
iajs-758	5	8	other	other	ADJ
iajs-758	5	9	types	type	NOUN
iajs-758	5	10	of	of	ADP
iajs-758	5	11	modules	module	NOUN
iajs-758	5	12	are	be	AUX
iajs-758	5	13	considered	consider	VERB
iajs-758	5	14	.	.	PUNCT
iajs-758	6	1	key	key	ADJ
iajs-758	6	2	word	word	NOUN
iajs-758	6	3	:	:	PUNCT
iajs-758	6	4	ring	ring	NOUN
iajs-758	6	5	,	,	PUNCT
iajs-758	6	6	module	module	NOUN
iajs-758	6	7	,	,	PUNCT
iajs-758	6	8	max	max	NOUN
iajs-758	6	9	-	-	PUNCT
iajs-758	6	10	module	module	NOUN
iajs-758	6	11	introduction	introduction	NOUN
iajs-758	6	12	every	every	DET
iajs-758	6	13	ring	ring	NOUN
iajs-758	6	14	considered	consider	VERB
iajs-758	6	15	in	in	ADP
iajs-758	6	16	this	this	DET
iajs-758	6	17	paper	paper	NOUN
iajs-758	6	18	will	will	AUX
iajs-758	6	19	be	be	AUX
iajs-758	6	20	assumed	assume	VERB
iajs-758	6	21	to	to	PART
iajs-758	6	22	be	be	AUX
iajs-758	6	23	commutative	commutative	ADJ
iajs-758	6	24	with	with	ADP
iajs-758	6	25	identity	identity	NOUN
iajs-758	6	26	and	and	CCONJ
iajs-758	6	27	every	every	DET
iajs-758	6	28	module	module	NOUN
iajs-758	6	29	is	be	AUX
iajs-758	6	30	unitary	unitary	ADJ
iajs-758	6	31	.	.	PUNCT
iajs-758	7	1	we	we	PRON
iajs-758	7	2	introduce	introduce	VERB
iajs-758	7	3	the	the	DET
iajs-758	7	4	following	following	NOUN
iajs-758	7	5	:	:	PUNCT
iajs-758	7	6	an	an	DET
iajs-758	7	7	r	r	NOUN
iajs-758	7	8	–	–	PUNCT
iajs-758	7	9	module	module	NOUN
iajs-758	7	10	m	m	NOUN
iajs-758	7	11	is	be	AUX
iajs-758	7	12	called	call	VERB
iajs-758	7	13	a	a	DET
iajs-758	7	14	maxmodule	maxmodule	NOUN
iajs-758	7	15	if	if	SCONJ
iajs-758	7	16	nann	nann	PROPN
iajs-758	7	17	r	r	NOUN
iajs-758	7	18	is	be	AUX
iajs-758	7	19	a	a	DET
iajs-758	7	20	maximal	maximal	ADJ
iajs-758	7	21	ideal	ideal	NOUN
iajs-758	7	22	of	of	ADP
iajs-758	7	23	r	r	NOUN
iajs-758	7	24	,	,	PUNCT
iajs-758	7	25	for	for	ADP
iajs-758	7	26	every	every	DET
iajs-758	7	27	non	non	ADJ
iajs-758	7	28	-	-	ADJ
iajs-758	7	29	zero	zero	NUM
iajs-758	7	30	submodule	submodule	NOUN
iajs-758	7	31	no	no	DET
iajs-758	7	32	f	f	NOUN
iajs-758	7	33	m	m	PROPN
iajs-758	7	34	,	,	PUNCT
iajs-758	7	35	where	where	SCONJ
iajs-758	7	36	annr	annr	NOUN
iajs-758	7	37	n	n	NOUN
iajs-758	7	38	=	=	SYM
iajs-758	7	39	{	{	PUNCT
iajs-758	7	40	r	r	NOUN
iajs-758	7	41	:	:	PUNCT
iajs-758	7	42	r	r	NOUN
iajs-758	7	43	∈r	∈r	NOUN
iajs-758	7	44	and	and	CCONJ
iajs-758	7	45	r	r	NOUN
iajs-758	7	46	n	n	NOUN
iajs-758	7	47	=	=	SYM
iajs-758	7	48	0	0	NUM
iajs-758	7	49	}	}	PUNCT
iajs-758	7	50	.	.	PUNCT
iajs-758	8	1	our	our	PRON
iajs-758	8	2	concern	concern	NOUN
iajs-758	8	3	in	in	ADP
iajs-758	8	4	this	this	DET
iajs-758	8	5	paper	paper	NOUN
iajs-758	8	6	is	be	AUX
iajs-758	8	7	to	to	PART
iajs-758	8	8	study	study	VERB
iajs-758	8	9	max	max	NOUN
iajs-758	8	10	-	-	PUNCT
iajs-758	8	11	modules	module	NOUN
iajs-758	8	12	and	and	CCONJ
iajs-758	8	13	to	to	PART
iajs-758	8	14	look	look	VERB
iajs-758	8	15	for	for	ADP
iajs-758	8	16	any	any	DET
iajs-758	8	17	relation	relation	NOUN
iajs-758	8	18	between	between	ADP
iajs-758	8	19	max	max	PROPN
iajs-758	8	20	–	–	PUNCT
iajs-758	8	21	modules	module	NOUN
iajs-758	8	22	and	and	CCONJ
iajs-758	8	23	certain	certain	ADJ
iajs-758	8	24	types	type	NOUN
iajs-758	8	25	of	of	ADP
iajs-758	8	26	well	well	ADV
iajs-758	8	27	–	–	PUNCT
iajs-758	8	28	known	know	VERB
iajs-758	8	29	modules	module	NOUN
iajs-758	8	30	specially	specially	ADV
iajs-758	8	31	with	with	ADP
iajs-758	8	32	primary	primary	ADJ
iajs-758	8	33	modules	module	NOUN
iajs-758	8	34	.	.	PUNCT
iajs-758	9	1	this	this	DET
iajs-758	9	2	paper	paper	NOUN
iajs-758	9	3	consists	consist	VERB
iajs-758	9	4	of	of	ADP
iajs-758	9	5	three	three	NUM
iajs-758	9	6	sections	section	NOUN
iajs-758	9	7	.	.	PUNCT
iajs-758	10	1	our	our	PRON
iajs-758	10	2	main	main	ADJ
iajs-758	10	3	concern	concern	NOUN
iajs-758	10	4	in	in	ADP
iajs-758	10	5	§	§	PROPN
iajs-758	10	6	1	1	NUM
iajs-758	10	7	,	,	PUNCT
iajs-758	10	8	is	be	AUX
iajs-758	10	9	to	to	PART
iajs-758	10	10	define	define	VERB
iajs-758	10	11	and	and	CCONJ
iajs-758	10	12	study	study	VERB
iajs-758	10	13	*	*	PUNCT
iajs-758	10	14	submodules	submodule	NOUN
iajs-758	10	15	.	.	PUNCT
iajs-758	11	1	also	also	ADV
iajs-758	11	2	we	we	PRON
iajs-758	11	3	study	study	VERB
iajs-758	11	4	the	the	DET
iajs-758	11	5	properties	property	NOUN
iajs-758	11	6	of	of	ADP
iajs-758	11	7	a	a	DET
iajs-758	11	8	multiplication	multiplication	NOUN
iajs-758	11	9	module	module	NOUN
iajs-758	11	10	that	that	PRON
iajs-758	11	11	contains	contain	VERB
iajs-758	11	12	*	*	PUNCT
iajs-758	11	13	submodules	submodule	NOUN
iajs-758	11	14	.	.	PUNCT
iajs-758	12	1	in	in	ADP
iajs-758	12	2	§	§	PROPN
iajs-758	12	3	2	2	NUM
iajs-758	12	4	,	,	PUNCT
iajs-758	12	5	we	we	PRON
iajs-758	12	6	study	study	VERB
iajs-758	12	7	max	max	PROPN
iajs-758	12	8	–	–	PUNCT
iajs-758	12	9	modules	module	NOUN
iajs-758	12	10	,	,	PUNCT
iajs-758	12	11	and	and	CCONJ
iajs-758	12	12	we	we	PRON
iajs-758	12	13	give	give	VERB
iajs-758	12	14	some	some	DET
iajs-758	12	15	characterizations	characterization	NOUN
iajs-758	12	16	for	for	ADP
iajs-758	12	17	this	this	DET
iajs-758	12	18	concept	concept	NOUN
iajs-758	12	19	.	.	PUNCT
iajs-758	13	1	also	also	ADV
iajs-758	13	2	other	other	ADJ
iajs-758	13	3	basic	basic	ADJ
iajs-758	13	4	results	result	NOUN
iajs-758	13	5	about	about	ADP
iajs-758	13	6	this	this	DET
iajs-758	13	7	concept	concept	NOUN
iajs-758	13	8	are	be	AUX
iajs-758	13	9	given	give	VERB
iajs-758	13	10	.	.	PUNCT
iajs-758	14	1	in	in	ADP
iajs-758	14	2	§	§	PROPN
iajs-758	14	3	3	3	NUM
iajs-758	14	4	,	,	PUNCT
iajs-758	14	5	we	we	PRON
iajs-758	14	6	study	study	VERB
iajs-758	14	7	the	the	DET
iajs-758	14	8	relation	relation	NOUN
iajs-758	14	9	between	between	ADP
iajs-758	14	10	max	max	PROPN
iajs-758	14	11	–	–	PUNCT
iajs-758	14	12	modules	module	NOUN
iajs-758	14	13	and	and	CCONJ
iajs-758	14	14	primary	primary	ADJ
iajs-758	14	15	modules	module	NOUN
iajs-758	14	16	.	.	PUNCT
iajs-758	15	1	it	it	PRON
iajs-758	15	2	is	be	AUX
iajs-758	15	3	clear	clear	ADJ
iajs-758	15	4	that	that	SCONJ
iajs-758	15	5	every	every	DET
iajs-758	15	6	max	max	PROPN
iajs-758	15	7	-	-	PUNCT
iajs-758	15	8	module	module	NOUN
iajs-758	15	9	is	be	AUX
iajs-758	15	10	primary	primary	ADJ
iajs-758	15	11	module	module	NOUN
iajs-758	15	12	,	,	PUNCT
iajs-758	15	13	but	but	CCONJ
iajs-758	15	14	the	the	DET
iajs-758	15	15	converse	converse	NOUN
iajs-758	15	16	is	be	AUX
iajs-758	15	17	not	not	PART
iajs-758	15	18	true	true	ADJ
iajs-758	15	19	in	in	ADP
iajs-758	15	20	general	general	ADJ
iajs-758	15	21	.	.	PUNCT
iajs-758	16	1	we	we	PRON
iajs-758	16	2	give	give	VERB
iajs-758	16	3	in	in	ADP
iajs-758	16	4	(	(	PUNCT
iajs-758	16	5	3.2	3.2	NUM
iajs-758	16	6	)	)	PUNCT
iajs-758	16	7	,	,	PUNCT
iajs-758	16	8	(	(	PUNCT
iajs-758	16	9	3.3	3.3	NUM
iajs-758	16	10	)	)	PUNCT
iajs-758	16	11	conditions	condition	NOUN
iajs-758	16	12	under	under	ADP
iajs-758	16	13	which	which	PRON
iajs-758	16	14	the	the	DET
iajs-758	16	15	two	two	NUM
iajs-758	16	16	concepts	concept	NOUN
iajs-758	16	17	are	be	AUX
iajs-758	16	18	equivalent	equivalent	ADJ
iajs-758	16	19	.	.	PUNCT
iajs-758	17	1	next	next	ADV
iajs-758	17	2	we	we	PRON
iajs-758	17	3	investigate	investigate	VERB
iajs-758	17	4	the	the	DET
iajs-758	17	5	relationships	relationship	NOUN
iajs-758	17	6	between	between	ADP
iajs-758	17	7	max	max	PROPN
iajs-758	17	8	,	,	PUNCT
iajs-758	17	9	prime	prime	ADJ
iajs-758	17	10	,	,	PUNCT
iajs-758	17	11	semi	semi	ADJ
iajs-758	17	12	–	–	PUNCT
iajs-758	17	13	primary	primary	ADJ
iajs-758	17	14	,	,	PUNCT
iajs-758	17	15	quasi	quasi	ADJ
iajs-758	17	16	-	-	ADJ
iajs-758	17	17	primary	primary	ADJ
iajs-758	17	18	finitely	finitely	ADV
iajs-758	17	19	generated	generate	VERB
iajs-758	17	20	and	and	CCONJ
iajs-758	17	21	uniform	uniform	ADJ
iajs-758	17	22	modules	module	NOUN
iajs-758	17	23	,	,	PUNCT
iajs-758	17	24	see	see	VERB
iajs-758	17	25	(	(	PUNCT
iajs-758	17	26	3.4	3.4	NUM
iajs-758	17	27	)	)	PUNCT
iajs-758	17	28	,	,	PUNCT
iajs-758	17	29	(	(	PUNCT
iajs-758	17	30	3.12	3.12	NUM
iajs-758	17	31	)	)	PUNCT
iajs-758	17	32	.	.	PUNCT
iajs-758	18	1	1	1	X
iajs-758	18	2	.	.	X
iajs-758	18	3	submodules	submodule	NOUN
iajs-758	18	4	in	in	ADP
iajs-758	18	5	this	this	DET
iajs-758	18	6	section	section	NOUN
iajs-758	18	7	,	,	PUNCT
iajs-758	18	8	we	we	PRON
iajs-758	18	9	introduce	introduce	VERB
iajs-758	18	10	the	the	DET
iajs-758	18	11	concept	concept	NOUN
iajs-758	18	12	of	of	ADP
iajs-758	18	13	*	*	PUNCT
iajs-758	18	14	submodule	submodule	NOUN
iajs-758	18	15	and	and	CCONJ
iajs-758	18	16	we	we	PRON
iajs-758	18	17	give	give	VERB
iajs-758	18	18	some	some	DET
iajs-758	18	19	characterizations	characterization	NOUN
iajs-758	18	20	for	for	ADP
iajs-758	18	21	this	this	DET
iajs-758	18	22	concept	concept	NOUN
iajs-758	18	23	.	.	PUNCT
iajs-758	19	1	and	and	CCONJ
iajs-758	19	2	we	we	PRON
iajs-758	19	3	end	end	VERB
iajs-758	19	4	this	this	DET
iajs-758	19	5	section	section	NOUN
iajs-758	19	6	by	by	ADP
iajs-758	19	7	studying	study	VERB
iajs-758	19	8	the	the	DET
iajs-758	19	9	properties	property	NOUN
iajs-758	19	10	of	of	ADP
iajs-758	19	11	a	a	DET
iajs-758	19	12	multiplication	multiplication	NOUN
iajs-758	19	13	module	module	NOUN
iajs-758	19	14	that	that	PRON
iajs-758	19	15	contains	contain	VERB
iajs-758	19	16	*	*	NOUN
iajs-758	19	17	submodules	submodules	NOUN
iajs-758	19	18	.	.	PUNCT
iajs-758	20	1	definition	definition	NOUN
iajs-758	20	2	1.1	1.1	NUM
iajs-758	20	3	:	:	PUNCT
iajs-758	20	4	a	a	DET
iajs-758	20	5	proper	proper	ADJ
iajs-758	20	6	submodule	submodule	NOUN
iajs-758	20	7	n	n	PROPN
iajs-758	20	8	of	of	ADP
iajs-758	20	9	an	an	DET
iajs-758	20	10	r	r	NOUN
iajs-758	20	11	-	-	PUNCT
iajs-758	20	12	module	module	NOUN
iajs-758	20	13	m	m	NOUN
iajs-758	20	14	is	be	AUX
iajs-758	20	15	said	say	VERB
iajs-758	20	16	to	to	PART
iajs-758	20	17	be	be	AUX
iajs-758	20	18	a	a	DET
iajs-758	20	19	*	*	ADJ
iajs-758	20	20	submodule	submodule	NOUN
iajs-758	20	21	if	if	SCONJ
iajs-758	20	22	]	]	X
iajs-758	20	23	[	[	PUNCT
iajs-758	20	24	:	:	PUNCT
iajs-758	20	25	kn	kn	NOUN
iajs-758	20	26	r	r	NOUN
iajs-758	20	27	is	be	AUX
iajs-758	20	28	a	a	DET
iajs-758	20	29	maximal	maximal	ADJ
iajs-758	20	30	ideal	ideal	NOUN
iajs-758	20	31	of	of	ADP
iajs-758	20	32	r	r	NOUN
iajs-758	20	33	for	for	ADP
iajs-758	20	34	each	each	DET
iajs-758	20	35	submodule	submodule	NOUN
iajs-758	21	1	ko	ko	PROPN
iajs-758	21	2	f	f	PROPN
iajs-758	21	3	m	m	VERB
iajs-758	21	4	such	such	ADJ
iajs-758	21	5	that	that	SCONJ
iajs-758	21	6	k	k	PROPN
iajs-758	21	7	n.	n.	PROPN
iajs-758	21	8	where	where	SCONJ
iajs-758	21	9	[	[	X
iajs-758	21	10	nr	nr	X
iajs-758	21	11	:	:	PUNCT
iajs-758	21	12	k	k	X
iajs-758	21	13	]	]	X
iajs-758	21	14	=	=	PUNCT
iajs-758	21	15	{	{	PUNCT
iajs-758	21	16	r	r	NOUN
iajs-758	21	17	∈	∈	PROPN
iajs-758	21	18	r	r	NOUN
iajs-758	21	19	:	:	PUNCT
iajs-758	21	20	rk	rk	PROPN
iajs-758	21	21	⊆	⊆	NUM
iajs-758	21	22	n	n	CCONJ
iajs-758	21	23	}	}	PUNCT
iajs-758	21	24	.	.	PUNCT
iajs-758	22	1	specially	specially	ADV
iajs-758	22	2	,	,	PUNCT
iajs-758	22	3	an	an	DET
iajs-758	22	4	ideal	ideal	NOUN
iajs-758	22	5	i	i	PRON
iajs-758	22	6	is	be	AUX
iajs-758	22	7	a	a	DET
iajs-758	22	8	*	*	ADJ
iajs-758	22	9	ideal	ideal	NOUN
iajs-758	22	10	of	of	ADP
iajs-758	22	11	r	r	NOUN
iajs-758	22	12	if	if	SCONJ
iajs-758	23	1	and	and	CCONJ
iajs-758	23	2	only	only	ADV
iajs-758	23	3	if	if	SCONJ
iajs-758	23	4	i	i	PRON
iajs-758	23	5	is	be	AUX
iajs-758	23	6	a	a	DET
iajs-758	23	7	*	*	ADJ
iajs-758	23	8	r	r	NOUN
iajs-758	23	9	–	–	PUNCT
iajs-758	23	10	submodule	submodule	NOUN
iajs-758	23	11	of	of	ADP
iajs-758	23	12	r	r	PROPN
iajs-758	23	13	–	–	PUNCT
iajs-758	23	14	module	module	NOUN
iajs-758	23	15	r.	r.	PROPN
iajs-758	23	16			PROPN
iajs-758	23	17			PROPN
iajs-758	23	18	ibn	ibn	PROPN
iajs-758	23	19	alhaitham	alhaitham	NOUN
iajs-758	23	20	j.	j.	PROPN
iajs-758	23	21	for	for	ADP
iajs-758	23	22	pure	pure	ADJ
iajs-758	23	23	&	&	CCONJ
iajs-758	23	24	appl	appl	PROPN
iajs-758	23	25	.	.	PUNCT
iajs-758	24	1	sci	sci	PROPN
iajs-758	24	2	.	.	PUNCT
iajs-758	24	3	vol.24	vol.24	NOUN
iajs-758	24	4	(	(	PUNCT
iajs-758	24	5	2	2	NUM
iajs-758	24	6	)	)	PUNCT
iajs-758	24	7	2011	2011	NUM
iajs-758	24	8	examples	example	NOUN
iajs-758	24	9	and	and	CCONJ
iajs-758	24	10	remarks	remark	NOUN
iajs-758	24	11	(	(	PUNCT
iajs-758	24	12	1.2	1.2	NUM
iajs-758	24	13	)	)	PUNCT
iajs-758	24	14	1recall	1recall	NUM
iajs-758	24	15	that	that	SCONJ
iajs-758	24	16	an	an	DET
iajs-758	24	17	r	r	NOUN
iajs-758	24	18	–	–	PUNCT
iajs-758	24	19	submodule	submodule	NOUN
iajs-758	24	20	n	n	PROPN
iajs-758	24	21	of	of	ADP
iajs-758	24	22	m	m	PROPN
iajs-758	24	23	is	be	AUX
iajs-758	24	24	a	a	DET
iajs-758	24	25	quasi	quasi	ADJ
iajs-758	24	26	–	–	PUNCT
iajs-758	24	27	primary	primary	ADJ
iajs-758	24	28	submodule	submodule	NOUN
iajs-758	24	29	of	of	ADP
iajs-758	24	30	m	m	PROPN
iajs-758	24	31	if	if	SCONJ
iajs-758	24	32	[	[	X
iajs-758	24	33	nr	nr	X
iajs-758	24	34	:	:	PUNCT
iajs-758	24	35	k	k	X
iajs-758	24	36	]	]	X
iajs-758	24	37	is	be	AUX
iajs-758	24	38	a	a	DET
iajs-758	24	39	primary	primary	ADJ
iajs-758	24	40	ideal	ideal	NOUN
iajs-758	24	41	of	of	ADP
iajs-758	24	42	r	r	NOUN
iajs-758	24	43	for	for	ADP
iajs-758	24	44	each	each	DET
iajs-758	24	45	submodule	submodule	NOUN
iajs-758	24	46	k	k	PROPN
iajs-758	24	47	of	of	ADP
iajs-758	24	48	m	m	PROPN
iajs-758	25	1	such	such	ADJ
iajs-758	25	2	that	that	SCONJ
iajs-758	25	3	k	k	PROPN
iajs-758	25	4	n,[2	n,[2	PROPN
iajs-758	25	5	]	]	X
iajs-758	25	6	.	.	PUNCT
iajs-758	26	1	it	it	PRON
iajs-758	26	2	is	be	AUX
iajs-758	26	3	well	well	ADV
iajs-758	26	4	–	–	PUNCT
iajs-758	26	5	known	know	VERB
iajs-758	26	6	that	that	SCONJ
iajs-758	26	7	if	if	SCONJ
iajs-758	26	8	]	]	X
iajs-758	26	9	[	[	PUNCT
iajs-758	26	10	:	:	PUNCT
iajs-758	26	11	kn	kn	NOUN
iajs-758	26	12	r	r	NOUN
iajs-758	26	13	is	be	AUX
iajs-758	26	14	a	a	DET
iajs-758	26	15	maximal	maximal	ADJ
iajs-758	26	16	ideal	ideal	NOUN
iajs-758	26	17	of	of	ADP
iajs-758	26	18	r	r	NOUN
iajs-758	26	19	,	,	PUNCT
iajs-758	26	20	then	then	ADV
iajs-758	26	21	[	[	X
iajs-758	26	22	nr	nr	X
iajs-758	26	23	:	:	PUNCT
iajs-758	26	24	k	k	X
iajs-758	26	25	]	]	X
iajs-758	26	26	is	be	AUX
iajs-758	26	27	a	a	DET
iajs-758	26	28	primary	primary	ADJ
iajs-758	26	29	ideal	ideal	NOUN
iajs-758	26	30	of	of	ADP
iajs-758	26	31	r	r	NOUN
iajs-758	26	32	,	,	PUNCT
iajs-758	26	33	[	[	X
iajs-758	26	34	1	1	NUM
iajs-758	26	35	,	,	PUNCT
iajs-758	26	36	prop	prop	NOUN
iajs-758	26	37	.	.	PUNCT
iajs-758	26	38	4.9	4.9	NUM
iajs-758	26	39	,	,	PUNCT
iajs-758	26	40	p.	p.	NOUN
iajs-758	26	41	64	64	NUM
iajs-758	26	42	]	]	PUNCT
iajs-758	26	43	.	.	PUNCT
iajs-758	27	1	thus	thus	ADV
iajs-758	27	2	every	every	DET
iajs-758	27	3	*	*	PUNCT
iajs-758	27	4	r	r	NOUN
iajs-758	27	5	-	-	PUNCT
iajs-758	27	6	submodule	submodule	NOUN
iajs-758	27	7	of	of	ADP
iajs-758	27	8	m	m	PROPN
iajs-758	27	9	is	be	AUX
iajs-758	27	10	a	a	DET
iajs-758	27	11	quasi	quasi	ADJ
iajs-758	27	12	–	–	PUNCT
iajs-758	27	13	primary	primary	ADJ
iajs-758	27	14	submodule	submodule	NOUN
iajs-758	27	15	.	.	PUNCT
iajs-758	28	1	2the	2the	PRON
iajs-758	28	2	submodule	submodule	PROPN
iajs-758	28	3	z	z	PROPN
iajs-758	28	4	of	of	ADP
iajs-758	28	5	the	the	DET
iajs-758	28	6	z	z	NOUN
iajs-758	28	7	-	-	PUNCT
iajs-758	28	8	module	module	NOUN
iajs-758	28	9	q	q	NOUN
iajs-758	28	10	is	be	AUX
iajs-758	28	11	not	not	PART
iajs-758	28	12	a	a	DET
iajs-758	28	13	*	*	NOUN
iajs-758	28	14	submodule	submodule	NOUN
iajs-758	28	15	since	since	SCONJ
iajs-758	28	16	zzzz	zzzz	PROPN
iajs-758	28	17	z	z	PROPN
iajs-758	28	18	66)6/1	66)6/1	NUM
iajs-758	28	19	(:	(:	VERB
iajs-758	28	20			NUM
iajs-758	28	21	is	be	AUX
iajs-758	28	22	not	not	PART
iajs-758	28	23	a	a	DET
iajs-758	28	24	maximal	maximal	ADJ
iajs-758	28	25	ideal	ideal	NOUN
iajs-758	28	26	of	of	ADP
iajs-758	28	27	z	z	NOUN
iajs-758	28	28	.	.	PUNCT
iajs-758	29	1	3the	3the	DET
iajs-758	29	2	intersection	intersection	NOUN
iajs-758	29	3	of	of	ADP
iajs-758	29	4	any	any	DET
iajs-758	29	5	two	two	NUM
iajs-758	29	6	*	*	ADJ
iajs-758	29	7	submodules	submodule	NOUN
iajs-758	29	8	of	of	ADP
iajs-758	29	9	an	an	DET
iajs-758	29	10	r	r	NOUN
iajs-758	29	11	–	–	PUNCT
iajs-758	29	12	module	module	NOUN
iajs-758	29	13	need	need	AUX
iajs-758	29	14	not	not	PART
iajs-758	29	15	be	be	AUX
iajs-758	29	16	*	*	PUNCT
iajs-758	29	17	-submodule	-submodule	NOUN
iajs-758	29	18	for	for	ADP
iajs-758	29	19	example	example	NOUN
iajs-758	29	20	.	.	PUNCT
iajs-758	30	1	the	the	DET
iajs-758	30	2	z	z	NOUN
iajs-758	30	3	–	–	PUNCT
iajs-758	30	4	module	module	NOUN
iajs-758	30	5	z6	z6	NOUN
iajs-758	30	6	has	have	VERB
iajs-758	30	7	two	two	NUM
iajs-758	30	8	*	*	ADJ
iajs-758	30	9	submodules	submodule	NOUN
iajs-758	30	10	,	,	PUNCT
iajs-758	30	11	)	)	PUNCT
iajs-758	30	12	2(1	2(1	NUM
iajs-758	30	13	n	n	NOUN
iajs-758	30	14	and	and	CCONJ
iajs-758	30	15	)	)	PUNCT
iajs-758	30	16	3(2	3(2	NUM
iajs-758	30	17	n	n	NOUN
iajs-758	30	18	,	,	PUNCT
iajs-758	30	19	but	but	CCONJ
iajs-758	30	20	)	)	PUNCT
iajs-758	31	1	0(21	0(21	NOUN
iajs-758	31	2	nn	nn	NOUN
iajs-758	31	3			NOUN
iajs-758	31	4	is	be	AUX
iajs-758	31	5	not	not	PART
iajs-758	31	6	a	a	DET
iajs-758	31	7	*	*	ADJ
iajs-758	31	8	submodule	submodule	NOUN
iajs-758	31	9	of	of	ADP
iajs-758	31	10	z6	z6	PROPN
iajs-758	31	11	,	,	PUNCT
iajs-758	31	12	since	since	SCONJ
iajs-758	31	13	zzzz	zzzz	NOUN
iajs-758	31	14	66])0	66])0	NUM
iajs-758	31	15	[	[	X
iajs-758	31	16	(	(	PUNCT
iajs-758	31	17	6	6	NUM
iajs-758	31	18	:	:	PUNCT
iajs-758	31	19			NUM
iajs-758	31	20	is	be	AUX
iajs-758	31	21	not	not	PART
iajs-758	31	22	a	a	DET
iajs-758	31	23	maximal	maximal	ADJ
iajs-758	31	24	ideal	ideal	NOUN
iajs-758	31	25	of	of	ADP
iajs-758	31	26	z	z	NOUN
iajs-758	31	27	.	.	PUNCT
iajs-758	32	1	4every	4every	NUM
iajs-758	33	1	*	*	PUNCT
iajs-758	33	2	submodule	submodule	PROPN
iajs-758	33	3	is	be	AUX
iajs-758	33	4	a	a	DET
iajs-758	33	5	semi	semi	ADJ
iajs-758	33	6	-	-	ADJ
iajs-758	33	7	primary	primary	ADJ
iajs-758	33	8	submodule	submodule	NOUN
iajs-758	33	9	.	.	PUNCT
iajs-758	34	1	proof	proof	NOUN
iajs-758	34	2	:	:	PUNCT
iajs-758	34	3	suppose	suppose	VERB
iajs-758	34	4	n	n	PRON
iajs-758	34	5	is	be	AUX
iajs-758	34	6	a	a	DET
iajs-758	34	7	*	*	ADJ
iajs-758	34	8	submodule	submodule	NOUN
iajs-758	34	9	of	of	ADP
iajs-758	34	10	an	an	DET
iajs-758	34	11	r	r	NOUN
iajs-758	34	12	-	-	PUNCT
iajs-758	34	13	module	module	NOUN
iajs-758	34	14	m.	m.	NOUN
iajs-758	34	15	hence	hence	ADV
iajs-758	34	16	]	]	PUNCT
iajs-758	34	17	[	[	PUNCT
iajs-758	34	18	:	:	PUNCT
iajs-758	34	19	kn	kn	NOUN
iajs-758	34	20	r	r	NOUN
iajs-758	34	21	is	be	AUX
iajs-758	34	22	a	a	DET
iajs-758	34	23	maximal	maximal	ADJ
iajs-758	34	24	ideal	ideal	NOUN
iajs-758	34	25	of	of	ADP
iajs-758	34	26	r.	r.	PROPN
iajs-758	34	27	therefore	therefore	ADV
iajs-758	34	28	]	]	X
iajs-758	34	29	[	[	PUNCT
iajs-758	34	30	:	:	PUNCT
iajs-758	34	31	kn	kn	NOUN
iajs-758	34	32	r	r	NOUN
iajs-758	34	33	is	be	AUX
iajs-758	34	34	a	a	DET
iajs-758	34	35	prime	prime	ADJ
iajs-758	34	36	ideal	ideal	NOUN
iajs-758	34	37	of	of	ADP
iajs-758	34	38	r	r	NOUN
iajs-758	34	39	,	,	PUNCT
iajs-758	34	40	which	which	PRON
iajs-758	34	41	implies	imply	VERB
iajs-758	34	42	that	that	SCONJ
iajs-758	34	43	n	n	X
iajs-758	34	44	is	be	AUX
iajs-758	34	45	a	a	DET
iajs-758	34	46	semi	semi	ADJ
iajs-758	34	47	–	–	PUNCT
iajs-758	34	48	primary	primary	ADJ
iajs-758	34	49	submodule	submodule	NOUN
iajs-758	34	50	of	of	ADP
iajs-758	34	51	m	m	PRON
iajs-758	34	52	by	by	ADP
iajs-758	34	53	[	[	PUNCT
iajs-758	34	54	2	2	NUM
iajs-758	34	55	,	,	PUNCT
iajs-758	34	56	definition	definition	NOUN
iajs-758	34	57	1.1	1.1	NUM
iajs-758	34	58	]	]	PUNCT
iajs-758	34	59	.	.	PUNCT
iajs-758	35	1	however	however	ADV
iajs-758	35	2	the	the	DET
iajs-758	35	3	converse	converse	NOUN
iajs-758	35	4	is	be	AUX
iajs-758	35	5	not	not	PART
iajs-758	35	6	true	true	ADJ
iajs-758	35	7	in	in	ADP
iajs-758	35	8	general	general	ADJ
iajs-758	35	9	as	as	SCONJ
iajs-758	35	10	the	the	DET
iajs-758	35	11	following	follow	VERB
iajs-758	35	12	example	example	NOUN
iajs-758	35	13	shows	show	VERB
iajs-758	35	14	:	:	PUNCT
iajs-758	35	15	let	let	VERB
iajs-758	35	16	m	m	VERB
iajs-758	35	17	=	=	ADJ
iajs-758	35	18	z	z	PROPN
iajs-758	35	19			PROPN
iajs-758	35	20	z12	z12	PROPN
iajs-758	35	21	as	as	ADP
iajs-758	35	22	a	a	DET
iajs-758	35	23	z	z	NOUN
iajs-758	35	24	–	–	PUNCT
iajs-758	35	25	module	module	NOUN
iajs-758	35	26	and	and	CCONJ
iajs-758	35	27	n=(0)=(0	n=(0)=(0	ADJ
iajs-758	35	28	)	)	PUNCT
iajs-758	35	29			NOUN
iajs-758	35	30	(	(	PUNCT
iajs-758	35	31	0	0	NUM
iajs-758	35	32	)	)	PUNCT
iajs-758	35	33	.	.	PUNCT
iajs-758	36	1	it	it	PRON
iajs-758	36	2	is	be	AUX
iajs-758	36	3	clear	clear	ADJ
iajs-758	36	4	that	that	SCONJ
iajs-758	36	5	n	n	X
iajs-758	36	6	is	be	AUX
iajs-758	36	7	a	a	DET
iajs-758	36	8	semi	semi	ADJ
iajs-758	36	9	–	–	PUNCT
iajs-758	36	10	primary	primary	ADJ
iajs-758	36	11	submodule	submodule	NOUN
iajs-758	36	12	of	of	ADP
iajs-758	36	13	m	m	PROPN
iajs-758	36	14	,	,	PUNCT
iajs-758	36	15	since	since	SCONJ
iajs-758	36	16	00])0	00])0	NOUN
iajs-758	36	17	[	[	X
iajs-758	36	18	(	(	PUNCT
iajs-758	36	19	:	:	PUNCT
iajs-758	36	20	mz	mz	NUM
iajs-758	36	21	is	be	AUX
iajs-758	36	22	a	a	DET
iajs-758	36	23	prime	prime	ADJ
iajs-758	36	24	ideal	ideal	NOUN
iajs-758	36	25	of	of	ADP
iajs-758	36	26	z.	z.	PROPN
iajs-758	36	27	but	but	CCONJ
iajs-758	36	28	(	(	PUNCT
iajs-758	36	29	0	0	X
iajs-758	36	30	)	)	PUNCT
iajs-758	36	31			NOUN
iajs-758	36	32	(	(	PUNCT
iajs-758	36	33	0	0	NUM
iajs-758	36	34	)	)	PUNCT
iajs-758	36	35	is	be	AUX
iajs-758	36	36	not	not	PART
iajs-758	36	37	a	a	DET
iajs-758	36	38	*	*	X
iajs-758	36	39	submodule	submodule	NOUN
iajs-758	36	40	of	of	ADP
iajs-758	36	41	m	m	PROPN
iajs-758	36	42	,	,	PUNCT
iajs-758	36	43	since	since	SCONJ
iajs-758	36	44	zzzz	zzzz	PROPN
iajs-758	36	45	612])0()0()0	612])0()0()0	NUM
iajs-758	36	46	[	[	X
iajs-758	36	47	(	(	PUNCT
iajs-758	36	48	12	12	NUM
iajs-758	36	49	:	:	PUNCT
iajs-758	36	50			VERB
iajs-758	36	51	which	which	PRON
iajs-758	36	52	is	be	AUX
iajs-758	36	53	not	not	PART
iajs-758	36	54	a	a	DET
iajs-758	36	55	maximal	maximal	ADJ
iajs-758	36	56	ideal	ideal	NOUN
iajs-758	36	57	of	of	ADP
iajs-758	36	58	z.	z.	PROPN
iajs-758	36	59	by	by	ADP
iajs-758	36	60	using	use	VERB
iajs-758	36	61	(	(	PUNCT
iajs-758	36	62	1.2	1.2	NUM
iajs-758	36	63	,	,	PUNCT
iajs-758	36	64	(	(	PUNCT
iajs-758	36	65	1	1	NUM
iajs-758	36	66	)	)	PUNCT
iajs-758	36	67	)	)	PUNCT
iajs-758	37	1	and	and	CCONJ
iajs-758	38	1	[	[	X
iajs-758	38	2	2	2	NUM
iajs-758	38	3	,	,	PUNCT
iajs-758	38	4	th	th	X
iajs-758	38	5	.	.	PUNCT
iajs-758	38	6	(	(	PUNCT
iajs-758	38	7	3.1.3	3.1.3	NUM
iajs-758	38	8	)	)	PUNCT
iajs-758	38	9	,	,	PUNCT
iajs-758	38	10	chapter	chapter	NOUN
iajs-758	38	11	3	3	X
iajs-758	38	12	]	]	PUNCT
iajs-758	38	13	we	we	PRON
iajs-758	38	14	can	can	AUX
iajs-758	38	15	give	give	VERB
iajs-758	38	16	the	the	DET
iajs-758	38	17	following	follow	VERB
iajs-758	38	18	characterization	characterization	NOUN
iajs-758	38	19	for	for	ADP
iajs-758	38	20	*	*	NOUN
iajs-758	38	21	submodule	submodule	NOUN
iajs-758	38	22	.	.	PUNCT
iajs-758	39	1	theorem	theorem	VERB
iajs-758	39	2	1.3	1.3	NUM
iajs-758	39	3	let	let	VERB
iajs-758	39	4	n	n	PRON
iajs-758	39	5	be	be	AUX
iajs-758	39	6	a	a	DET
iajs-758	39	7	proper	proper	ADJ
iajs-758	39	8	submodule	submodule	NOUN
iajs-758	39	9	of	of	ADP
iajs-758	39	10	an	an	DET
iajs-758	39	11	r	r	NOUN
iajs-758	39	12	-	-	PUNCT
iajs-758	39	13	module	module	NOUN
iajs-758	39	14	m.	m.	NOUN
iajs-758	39	15	if	if	SCONJ
iajs-758	39	16	n	n	PRON
iajs-758	39	17	is	be	AUX
iajs-758	39	18	a	a	DET
iajs-758	39	19	*	*	ADJ
iajs-758	39	20	submodule	submodule	NOUN
iajs-758	39	21	of	of	ADP
iajs-758	39	22	m	m	PROPN
iajs-758	39	23	,	,	PUNCT
iajs-758	39	24	then	then	ADV
iajs-758	39	25	]	]	PUNCT
iajs-758	39	26	[	[	PUNCT
iajs-758	39	27	:	:	PUNCT
iajs-758	39	28	kn	kn	NOUN
iajs-758	39	29	r	r	NOUN
iajs-758	39	30	=	=	PUNCT
iajs-758	39	31	]	]	X
iajs-758	39	32	[	[	PUNCT
iajs-758	39	33	:	:	PUNCT
iajs-758	39	34	rkn	rkn	NOUN
iajs-758	39	35	r	r	NOUN
iajs-758	39	36	for	for	ADP
iajs-758	39	37	each	each	DET
iajs-758	39	38	submodule	submodule	NOUN
iajs-758	39	39	k	k	PROPN
iajs-758	39	40	of	of	ADP
iajs-758	39	41	m	m	PROPN
iajs-758	39	42	such	such	ADJ
iajs-758	39	43	that	that	SCONJ
iajs-758	39	44	k	k	PROPN
iajs-758	39	45	n	n	CCONJ
iajs-758	39	46	,	,	PUNCT
iajs-758	39	47	rk	rk	PROPN
iajs-758	39	48	n	n	NOUN
iajs-758	39	49	and	and	CCONJ
iajs-758	39	50	r	r	NOUN
iajs-758	39	51			NOUN
iajs-758	39	52	r	r	NOUN
iajs-758	39	53	.	.	PUNCT
iajs-758	40	1	by	by	ADP
iajs-758	40	2	using	use	VERB
iajs-758	40	3	(	(	PUNCT
iajs-758	40	4	1.2	1.2	NUM
iajs-758	40	5	,	,	PUNCT
iajs-758	40	6	(	(	PUNCT
iajs-758	40	7	1	1	NUM
iajs-758	40	8	)	)	PUNCT
iajs-758	40	9	)	)	PUNCT
iajs-758	40	10	and	and	CCONJ
iajs-758	40	11	[	[	X
iajs-758	40	12	2	2	NUM
iajs-758	40	13	,	,	PUNCT
iajs-758	40	14	prop.(3.1.4	prop.(3.1.4	NUM
iajs-758	40	15	)	)	PUNCT
iajs-758	40	16	,	,	PUNCT
iajs-758	40	17	chapter	chapter	NOUN
iajs-758	40	18	3	3	X
iajs-758	40	19	]	]	PUNCT
iajs-758	40	20	we	we	PRON
iajs-758	40	21	can	can	AUX
iajs-758	40	22	give	give	VERB
iajs-758	40	23	the	the	DET
iajs-758	40	24	following	follow	VERB
iajs-758	40	25	result	result	NOUN
iajs-758	40	26	:	:	PUNCT
iajs-758	40	27	corollary	corollary	ADJ
iajs-758	40	28	1.4	1.4	NUM
iajs-758	40	29	let	let	VERB
iajs-758	40	30	n	n	PRON
iajs-758	40	31	be	be	AUX
iajs-758	40	32	a	a	DET
iajs-758	40	33	proper	proper	ADJ
iajs-758	40	34	submodule	submodule	NOUN
iajs-758	40	35	of	of	ADP
iajs-758	40	36	an	an	DET
iajs-758	40	37	rmodule	rmodule	NOUN
iajs-758	40	38	m	m	NOUN
iajs-758	40	39	.	.	PUNCT
iajs-758	41	1	if	if	SCONJ
iajs-758	41	2	n	n	PRON
iajs-758	41	3	is	be	AUX
iajs-758	41	4	a	a	DET
iajs-758	41	5	*	*	ADJ
iajs-758	41	6	submodule	submodule	NOUN
iajs-758	41	7	of	of	ADP
iajs-758	41	8	m	m	PROPN
iajs-758	41	9	,	,	PUNCT
iajs-758	41	10	then	then	ADV
iajs-758	41	11	)	)	PUNCT
iajs-758	41	12	]	]	PUNCT
iajs-758	41	13	(	(	PUNCT
iajs-758	41	14	[	[	X
iajs-758	41	15	)	)	PUNCT
iajs-758	41	16	]	]	PUNCT
iajs-758	41	17	(	(	PUNCT
iajs-758	41	18	[	[	PUNCT
iajs-758	41	19	:	:	PUNCT
iajs-758	41	20	:	:	PUNCT
iajs-758	41	21	mnrmn	mnrmn	PROPN
iajs-758	41	22	rr	rr	VERB
iajs-758	41	23			PROPN
iajs-758	41	24	for	for	ADP
iajs-758	41	25	each	each	DET
iajs-758	41	26	m	m	PROPN
iajs-758	41	27	∈	∈	NOUN
iajs-758	41	28	m\n	m\n	NOUN
iajs-758	41	29	,	,	PUNCT
iajs-758	41	30	r	r	NOUN
iajs-758	41	31	∈	∈	PROPN
iajs-758	41	32	r	r	NOUN
iajs-758	41	33	and	and	CCONJ
iajs-758	41	34	r	r	NOUN
iajs-758	41	35	∉	∉	PROPN
iajs-758	42	1	[	[	X
iajs-758	42	2	nr	nr	INTJ
iajs-758	42	3	:(	:(	PROPN
iajs-758	42	4	m	m	NOUN
iajs-758	42	5	)	)	PUNCT
iajs-758	42	6	]	]	PUNCT
iajs-758	42	7	.	.	PUNCT
iajs-758	43	1	the	the	DET
iajs-758	43	2	converse	converse	NOUN
iajs-758	43	3	of	of	ADP
iajs-758	43	4	corollary	corollary	ADJ
iajs-758	43	5	(	(	PUNCT
iajs-758	43	6	1.4	1.4	NUM
iajs-758	43	7	)	)	PUNCT
iajs-758	43	8	is	be	AUX
iajs-758	43	9	not	not	PART
iajs-758	43	10	true	true	ADJ
iajs-758	43	11	in	in	ADP
iajs-758	43	12	general	general	ADJ
iajs-758	43	13	for	for	ADP
iajs-758	43	14	example	example	NOUN
iajs-758	43	15	:	:	PUNCT
iajs-758	43	16	let	let	VERB
iajs-758	43	17	m	m	VERB
iajs-758	43	18	=	=	VERB
iajs-758	43	19	z	z	NOUN
iajs-758	43	20	as	as	ADP
iajs-758	43	21	a	a	DET
iajs-758	43	22	z	z	NOUN
iajs-758	43	23	–	–	PUNCT
iajs-758	43	24	module	module	NOUN
iajs-758	43	25	,	,	PUNCT
iajs-758	43	26	let	let	VERB
iajs-758	43	27	n	n	NOUN
iajs-758	43	28	=	=	NOUN
iajs-758	43	29	6z	6z	NOUN
iajs-758	43	30	,	,	PUNCT
iajs-758	43	31	r	r	NOUN
iajs-758	43	32	=	=	SYM
iajs-758	43	33	5	5	NUM
iajs-758	43	34	,	,	PUNCT
iajs-758	43	35	5	5	NUM
iajs-758	43	36	∉	∉	NOUN
iajs-758	43	37	[	[	X
iajs-758	43	38	6zz	6zz	ADJ
iajs-758	43	39	:(	:(	NOUN
iajs-758	43	40	1	1	NUM
iajs-758	43	41	)	)	PUNCT
iajs-758	43	42	]	]	PUNCT
iajs-758	44	1	=	=	SYM
iajs-758	44	2	6z	6z	NOUN
iajs-758	44	3	and	and	CCONJ
iajs-758	44	4			NUM
iajs-758	44	5	zzz	zzz	X
iajs-758	44	6	6)]1.5(6	6)]1.5(6	NUM
iajs-758	44	7	[	[	PUNCT
iajs-758	44	8	:	:	PUNCT
iajs-758	44	9	)	)	PUNCT
iajs-758	44	10	]	]	PUNCT
iajs-758	44	11	1(6[6	1(6[6	NUM
iajs-758	44	12	:	:	PUNCT
iajs-758	44	13	zzz	zzz	X
iajs-758	44	14			NOUN
iajs-758	44	15	.	.	PUNCT
iajs-758	45	1	but	but	CCONJ
iajs-758	45	2	n	n	PRON
iajs-758	45	3	is	be	AUX
iajs-758	45	4	not	not	PART
iajs-758	45	5	a	a	DET
iajs-758	45	6	*	*	ADJ
iajs-758	45	7	submodule	submodule	NOUN
iajs-758	45	8	of	of	ADP
iajs-758	45	9	z.	z.	PROPN
iajs-758	45	10	recall	recall	VERB
iajs-758	45	11	that	that	SCONJ
iajs-758	45	12	an	an	DET
iajs-758	45	13	r	r	NOUN
iajs-758	45	14	-	-	PUNCT
iajs-758	45	15	module	module	NOUN
iajs-758	45	16	m	m	NOUN
iajs-758	45	17	is	be	AUX
iajs-758	45	18	called	call	VERB
iajs-758	45	19	a	a	DET
iajs-758	45	20	multiplication	multiplication	NOUN
iajs-758	45	21	module	module	NOUN
iajs-758	45	22	,	,	PUNCT
iajs-758	45	23	if	if	SCONJ
iajs-758	45	24	for	for	ADP
iajs-758	45	25	every	every	DET
iajs-758	45	26	submodule	submodule	NOUN
iajs-758	45	27	n	n	PROPN
iajs-758	45	28	of	of	ADP
iajs-758	45	29	m	m	PRON
iajs-758	45	30	,	,	PUNCT
iajs-758	45	31	there	there	PRON
iajs-758	45	32	exists	exist	VERB
iajs-758	45	33	an	an	DET
iajs-758	45	34	ideal	ideal	NOUN
iajs-758	45	35	i	i	PRON
iajs-758	45	36	of	of	ADP
iajs-758	45	37	r	r	NOUN
iajs-758	46	1	such	such	ADJ
iajs-758	46	2	that	that	SCONJ
iajs-758	46	3	i	i	PRON
iajs-758	46	4	m	m	VERB
iajs-758	46	5	=	=	SYM
iajs-758	46	6	n	n	CCONJ
iajs-758	46	7	,	,	PUNCT
iajs-758	46	8	equivalenty	equivalenty	NOUN
iajs-758	46	9	;	;	PUNCT
iajs-758	46	10	for	for	ADP
iajs-758	46	11	every	every	DET
iajs-758	46	12	submodule	submodule	NOUN
iajs-758	46	13	n	n	PROPN
iajs-758	46	14	of	of	ADP
iajs-758	46	15	m	m	PROPN
iajs-758	46	16	,	,	PUNCT
iajs-758	46	17	n=	n=	X
iajs-758	47	1	[	[	X
iajs-758	47	2	nr	nr	X
iajs-758	47	3	:	:	PUNCT
iajs-758	47	4	m	m	VERB
iajs-758	47	5	]	]	X
iajs-758	47	6	m	m	VERB
iajs-758	47	7	,	,	PUNCT
iajs-758	47	8	see	see	VERB
iajs-758	47	9	[	[	X
iajs-758	47	10	3	3	X
iajs-758	47	11	]	]	PUNCT
iajs-758	47	12	.	.	PUNCT
iajs-758	48	1	an	an	DET
iajs-758	48	2	rsubmodule	rsubmodule	NOUN
iajs-758	48	3	n	n	PROPN
iajs-758	48	4	of	of	ADP
iajs-758	48	5	m	m	PROPN
iajs-758	48	6	is	be	AUX
iajs-758	48	7	called	call	VERB
iajs-758	48	8	a	a	DET
iajs-758	48	9	prime	prime	ADJ
iajs-758	48	10	r	r	NOUN
iajs-758	48	11	–	–	PUNCT
iajs-758	48	12	submodule	submodule	NOUN
iajs-758	48	13	if	if	SCONJ
iajs-758	48	14	and	and	CCONJ
iajs-758	48	15	only	only	ADV
iajs-758	48	16	if	if	SCONJ
iajs-758	48	17	n≠m	n≠m	NOUN
iajs-758	48	18	and	and	CCONJ
iajs-758	48	19	whenever	whenever	SCONJ
iajs-758	48	20	r	r	NOUN
iajs-758	48	21			NOUN
iajs-758	48	22	n	n	CCONJ
iajs-758	48	23	,	,	PUNCT
iajs-758	48	24	for	for	ADP
iajs-758	48	25	r	r	NOUN
iajs-758	48	26			NOUN
iajs-758	48	27	r	r	NOUN
iajs-758	48	28	and	and	CCONJ
iajs-758	48	29	x	x	PROPN
iajs-758	48	30			PROPN
iajs-758	48	31	m	m	PROPN
iajs-758	48	32	,	,	PUNCT
iajs-758	48	33	either	either	CCONJ
iajs-758	48	34	r	r	NOUN
iajs-758	48	35			NOUN
iajs-758	49	1	[	[	X
iajs-758	49	2	nr	nr	X
iajs-758	49	3	:	:	PUNCT
iajs-758	49	4	m	m	VERB
iajs-758	49	5	]	]	PUNCT
iajs-758	49	6	or	or	CCONJ
iajs-758	49	7	x	x	ADJ
iajs-758	49	8			NOUN
iajs-758	49	9	n	n	CCONJ
iajs-758	49	10	,	,	PUNCT
iajs-758	49	11	[	[	X
iajs-758	49	12	10	10	NUM
iajs-758	49	13	]	]	PUNCT
iajs-758	49	14	.	.	PUNCT
iajs-758	50	1	the	the	DET
iajs-758	50	2	prime	prime	ADJ
iajs-758	50	3	radical	radical	ADJ
iajs-758	50	4	p(n	p(n	PROPN
iajs-758	50	5	)	)	PUNCT
iajs-758	50	6	of	of	ADP
iajs-758	50	7	n	n	PROPN
iajs-758	50	8	in	in	ADP
iajs-758	50	9	m	m	PROPN
iajs-758	50	10	is	be	AUX
iajs-758	50	11	defined	define	VERB
iajs-758	50	12	to	to	PART
iajs-758	50	13	be	be	AUX
iajs-758	50	14	the	the	DET
iajs-758	50	15	intersection	intersection	NOUN
iajs-758	50	16	of	of	ADP
iajs-758	50	17	all	all	DET
iajs-758	50	18	prime	prime	ADJ
iajs-758	50	19	submodules	submodule	NOUN
iajs-758	50	20	p	p	NOUN
iajs-758	50	21	of	of	ADP
iajs-758	50	22	m	m	PRON
iajs-758	50	23	such	such	ADJ
iajs-758	50	24	that	that	SCONJ
iajs-758	50	25	n	n	PROPN
iajs-758	50	26	⊆	⊆	NUM
iajs-758	50	27	p	p	NOUN
iajs-758	50	28	i.e.	i.e.	X
iajs-758	50	29	p(n	p(n	NOUN
iajs-758	50	30	)	)	PUNCT
iajs-758	51	1	=	=	SYM
iajs-758	51	2	⋂	⋂	PROPN
iajs-758	51	3	{	{	PUNCT
iajs-758	51	4	p	p	NOUN
iajs-758	51	5	⊆	⊆	NUM
iajs-758	51	6	m	m	NOUN
iajs-758	51	7	:	:	PUNCT
iajs-758	51	8	pis	pis	ADJ
iajs-758	51	9	prime	prime	NOUN
iajs-758	51	10	and	and	CCONJ
iajs-758	51	11	n	n	PRON
iajs-758	51	12	⊆	⊆	NUM
iajs-758	51	13	p	p	NOUN
iajs-758	51	14	}	}	PUNCT
iajs-758	51	15	.	.	PUNCT
iajs-758	52	1	it	it	PRON
iajs-758	52	2	is	be	AUX
iajs-758	52	3	known	know	VERB
iajs-758	52	4	that	that	SCONJ
iajs-758	52	5	if	if	SCONJ
iajs-758	52	6	m	m	NOUN
iajs-758	52	7	is	be	AUX
iajs-758	52	8	multiplication	multiplication	NOUN
iajs-758	52	9	module	module	NOUN
iajs-758	52	10	and	and	CCONJ
iajs-758	52	11	n	n	NOUN
iajs-758	52	12	is	be	AUX
iajs-758	52	13	a	a	DET
iajs-758	52	14	submodule	submodule	NOUN
iajs-758	52	15	of	of	ADP
iajs-758	52	16	m	m	PROPN
iajs-758	52	17	,	,	PUNCT
iajs-758	52	18	then	then	ADV
iajs-758	52	19	]	]	PUNCT
iajs-758	52	20	[	[	X
iajs-758	52	21	)	)	PUNCT
iajs-758	52	22	(	(	PUNCT
iajs-758	52	23	:	:	PUNCT
iajs-758	52	24	mnnp	mnnp	X
iajs-758	52	25	r	r	VERB
iajs-758	52	26	m	m	PROPN
iajs-758	52	27	,	,	PUNCT
iajs-758	52	28	[	[	PUNCT
iajs-758	52	29	3	3	NUM
iajs-758	52	30	,	,	PUNCT
iajs-758	52	31	th	th	X
iajs-758	52	32	.	.	NOUN
iajs-758	52	33	2	2	NUM
iajs-758	52	34	.	.	NUM
iajs-758	52	35	12	12	NUM
iajs-758	52	36	]	]	PUNCT
iajs-758	52	37	.	.	PUNCT
iajs-758	53	1	the	the	DET
iajs-758	53	2	following	follow	VERB
iajs-758	53	3	remark	remark	NOUN
iajs-758	53	4	shows	show	VERB
iajs-758	53	5	that	that	SCONJ
iajs-758	53	6	a	a	DET
iajs-758	53	7	multiplication	multiplication	NOUN
iajs-758	53	8	r	r	NOUN
iajs-758	53	9	-	-	PUNCT
iajs-758	53	10	module	module	NOUN
iajs-758	53	11	which	which	PRON
iajs-758	53	12	has	have	VERB
iajs-758	53	13	a	a	DET
iajs-758	53	14	finitely	finitely	ADV
iajs-758	53	15	generated	generate	VERB
iajs-758	53	16	*	*	PUNCT
iajs-758	53	17	submodule	submodule	PROPN
iajs-758	53	18	is	be	AUX
iajs-758	53	19	finitely	finitely	ADV
iajs-758	53	20	generated	generate	VERB
iajs-758	53	21	r	r	NOUN
iajs-758	53	22	–	–	PUNCT
iajs-758	53	23	module	module	NOUN
iajs-758	53	24	.	.	PUNCT
iajs-758	54	1	remark	remark	NOUN
iajs-758	54	2	1.5	1.5	NUM
iajs-758	54	3	let	let	VERB
iajs-758	54	4	m	m	PRON
iajs-758	54	5	be	be	AUX
iajs-758	54	6	a	a	DET
iajs-758	54	7	multiplication	multiplication	NOUN
iajs-758	54	8	r	r	NOUN
iajs-758	54	9	-	-	NOUN
iajs-758	54	10	module	module	NOUN
iajs-758	54	11	.	.	PUNCT
iajs-758	55	1	if	if	SCONJ
iajs-758	55	2	m	m	NOUN
iajs-758	55	3	contains	contain	VERB
iajs-758	55	4	a	a	DET
iajs-758	55	5	finitely	finitely	ADV
iajs-758	55	6	generated	generate	VERB
iajs-758	55	7	*	*	PUNCT
iajs-758	55	8	submodule	submodule	PROPN
iajs-758	55	9	n	n	CCONJ
iajs-758	55	10	,	,	PUNCT
iajs-758	55	11	then	then	ADV
iajs-758	55	12	m	m	VERB
iajs-758	55	13	is	be	AUX
iajs-758	55	14	a	a	DET
iajs-758	55	15	finitely	finitely	ADV
iajs-758	55	16	generated	generate	VERB
iajs-758	55	17	r	r	NOUN
iajs-758	55	18	–	–	PUNCT
iajs-758	55	19	module	module	NOUN
iajs-758	55	20	.	.	PUNCT
iajs-758	56	1			PROPN
iajs-758	56	2			PROPN
iajs-758	56	3			PROPN
iajs-758	56	4			PROPN
iajs-758	56	5			PROPN
iajs-758	56	6	–	–	PUNCT
iajs-758	56	7	ibn	ibn	PROPN
iajs-758	56	8	alhaitham	alhaitham	NOUN
iajs-758	56	9	j.	j.	PROPN
iajs-758	56	10	for	for	ADP
iajs-758	56	11	pure	pure	ADJ
iajs-758	56	12	&	&	CCONJ
iajs-758	56	13	appl	appl	PROPN
iajs-758	56	14	.	.	PUNCT
iajs-758	57	1	sci	sci	PROPN
iajs-758	57	2	.	.	PUNCT
iajs-758	57	3	vol.24	vol.24	NOUN
iajs-758	57	4	(	(	PUNCT
iajs-758	57	5	2	2	NUM
iajs-758	57	6	)	)	PUNCT
iajs-758	57	7	2011	2011	NUM
iajs-758	57	8	proof	proof	NOUN
iajs-758	57	9	:	:	PUNCT
iajs-758	57	10	since	since	SCONJ
iajs-758	57	11	n	n	PRON
iajs-758	57	12	is	be	AUX
iajs-758	57	13	a	a	DET
iajs-758	57	14	*	*	ADJ
iajs-758	57	15	submodule	submodule	NOUN
iajs-758	57	16	,	,	PUNCT
iajs-758	57	17	so	so	CCONJ
iajs-758	57	18	n	n	PRON
iajs-758	57	19	is	be	AUX
iajs-758	57	20	a	a	DET
iajs-758	57	21	semiprimary	semiprimary	ADJ
iajs-758	57	22	submodule	submodule	NOUN
iajs-758	57	23	of	of	ADP
iajs-758	57	24	m	m	PRON
iajs-758	57	25	by	by	ADP
iajs-758	57	26	(	(	PUNCT
iajs-758	57	27	1.2	1.2	NUM
iajs-758	57	28	,	,	PUNCT
iajs-758	57	29	(	(	PUNCT
iajs-758	57	30	3	3	NUM
iajs-758	57	31	)	)	PUNCT
iajs-758	57	32	)	)	PUNCT
iajs-758	57	33	.	.	PUNCT
iajs-758	58	1	therefore	therefore	ADV
iajs-758	58	2	,	,	PUNCT
iajs-758	58	3	m	m	VERB
iajs-758	58	4	is	be	AUX
iajs-758	58	5	finitely	finitely	ADV
iajs-758	58	6	generated	generate	VERB
iajs-758	58	7	by	by	ADP
iajs-758	58	8	[	[	X
iajs-758	58	9	2	2	NUM
iajs-758	58	10	,	,	PUNCT
iajs-758	58	11	proposition	proposition	NOUN
iajs-758	58	12	3.4	3.4	NUM
iajs-758	58	13	,	,	PUNCT
iajs-758	58	14	p.	p.	NOUN
iajs-758	58	15	135	135	NUM
iajs-758	58	16	]	]	PUNCT
iajs-758	58	17	.	.	PUNCT
iajs-758	59	1	corollary	corollary	ADJ
iajs-758	59	2	1.6	1.6	NUM
iajs-758	59	3	if	if	SCONJ
iajs-758	59	4	n	n	PRON
iajs-758	59	5	is	be	AUX
iajs-758	59	6	a	a	DET
iajs-758	59	7	*	*	ADJ
iajs-758	59	8	submodule	submodule	NOUN
iajs-758	59	9	of	of	ADP
iajs-758	59	10	a	a	DET
iajs-758	59	11	multiplication	multiplication	NOUN
iajs-758	59	12	r	r	NOUN
iajs-758	59	13	–	–	PUNCT
iajs-758	59	14	module	module	NOUN
iajs-758	59	15	m	m	NOUN
iajs-758	59	16	,	,	PUNCT
iajs-758	59	17	then	then	ADV
iajs-758	59	18	rad	rad	PROPN
iajs-758	59	19	(	(	PUNCT
iajs-758	59	20	n	n	CCONJ
iajs-758	59	21	)	)	PUNCT
iajs-758	59	22	is	be	AUX
iajs-758	59	23	a	a	DET
iajs-758	59	24	prime	prime	ADJ
iajs-758	59	25	submodule	submodule	NOUN
iajs-758	59	26	of	of	ADP
iajs-758	59	27	m.	m.	NOUN
iajs-758	59	28	proof	proof	NOUN
iajs-758	59	29	:	:	PUNCT
iajs-758	59	30	suppose	suppose	VERB
iajs-758	59	31	that	that	SCONJ
iajs-758	59	32	n	n	PRON
iajs-758	59	33	is	be	AUX
iajs-758	59	34	a	a	DET
iajs-758	59	35	*	*	ADJ
iajs-758	59	36	submodule	submodule	NOUN
iajs-758	59	37	.	.	PUNCT
iajs-758	60	1	hence	hence	ADV
iajs-758	60	2	,	,	PUNCT
iajs-758	60	3	n	n	PRON
iajs-758	60	4	is	be	AUX
iajs-758	60	5	a	a	DET
iajs-758	60	6	quasi	quasi	ADJ
iajs-758	60	7	–	–	PUNCT
iajs-758	60	8	primary	primary	ADJ
iajs-758	60	9	submodule	submodule	NOUN
iajs-758	60	10	by	by	ADP
iajs-758	60	11	(	(	PUNCT
iajs-758	60	12	1.2	1.2	NUM
iajs-758	60	13	,	,	PUNCT
iajs-758	60	14	(	(	PUNCT
iajs-758	60	15	1	1	NUM
iajs-758	60	16	)	)	PUNCT
iajs-758	60	17	)	)	PUNCT
iajs-758	60	18	.	.	PUNCT
iajs-758	61	1	but	but	CCONJ
iajs-758	61	2	m	m	PROPN
iajs-758	61	3	is	be	AUX
iajs-758	61	4	a	a	DET
iajs-758	61	5	multiplication	multiplication	NOUN
iajs-758	61	6	r	r	NOUN
iajs-758	61	7	–	–	PUNCT
iajs-758	61	8	module	module	NOUN
iajs-758	61	9	,	,	PUNCT
iajs-758	61	10	so	so	CCONJ
iajs-758	61	11	n	n	PROPN
iajs-758	61	12	is	be	AUX
iajs-758	61	13	a	a	DET
iajs-758	61	14	primary	primary	ADJ
iajs-758	61	15	submodule	submodule	NOUN
iajs-758	61	16	of	of	ADP
iajs-758	61	17	m	m	PRON
iajs-758	61	18	by	by	ADP
iajs-758	61	19	[	[	X
iajs-758	61	20	2	2	NUM
iajs-758	61	21	,	,	PUNCT
iajs-758	61	22	propostion	propostion	NOUN
iajs-758	61	23	(	(	PUNCT
iajs-758	61	24	3.1.5	3.1.5	NUM
iajs-758	61	25	)	)	PUNCT
iajs-758	61	26	,	,	PUNCT
iajs-758	61	27	chapter	chapter	NOUN
iajs-758	61	28	3	3	NUM
iajs-758	61	29	]	]	PUNCT
iajs-758	61	30	.	.	PUNCT
iajs-758	62	1	therefore	therefore	ADV
iajs-758	62	2	,	,	PUNCT
iajs-758	62	3	rad	rad	PROPN
iajs-758	62	4	(	(	PUNCT
iajs-758	62	5	n	n	CCONJ
iajs-758	62	6	)	)	PUNCT
iajs-758	62	7	is	be	AUX
iajs-758	62	8	a	a	DET
iajs-758	62	9	prime	prime	ADJ
iajs-758	62	10	submodule	submodule	NOUN
iajs-758	62	11	by	by	ADP
iajs-758	62	12	[	[	X
iajs-758	62	13	4	4	NUM
iajs-758	62	14	,	,	PUNCT
iajs-758	62	15	corollary	corollary	NOUN
iajs-758	62	16	2.13	2.13	NUM
iajs-758	62	17	,	,	PUNCT
iajs-758	62	18	chapter	chapter	NOUN
iajs-758	62	19	2	2	NUM
iajs-758	62	20	]	]	PUNCT
iajs-758	62	21	.	.	PUNCT
iajs-758	63	1	2	2	X
iajs-758	63	2	.	.	X
iajs-758	63	3	basic	basic	ADJ
iajs-758	63	4	properties	property	NOUN
iajs-758	63	5	of	of	ADP
iajs-758	63	6	max	max	NOUN
iajs-758	63	7	-	-	PUNCT
iajs-758	63	8	modules	module	NOUN
iajs-758	63	9	in	in	ADP
iajs-758	63	10	this	this	DET
iajs-758	63	11	section	section	NOUN
iajs-758	63	12	,	,	PUNCT
iajs-758	63	13	we	we	PRON
iajs-758	63	14	introduce	introduce	VERB
iajs-758	63	15	the	the	DET
iajs-758	63	16	concept	concept	NOUN
iajs-758	63	17	of	of	ADP
iajs-758	63	18	a	a	DET
iajs-758	63	19	max	max	PROPN
iajs-758	63	20	–	–	PUNCT
iajs-758	63	21	module	module	NOUN
iajs-758	63	22	and	and	CCONJ
iajs-758	63	23	give	give	VERB
iajs-758	63	24	some	some	DET
iajs-758	63	25	characterizations	characterization	NOUN
iajs-758	63	26	and	and	CCONJ
iajs-758	63	27	properties	property	NOUN
iajs-758	63	28	of	of	ADP
iajs-758	63	29	this	this	DET
iajs-758	63	30	concept	concept	NOUN
iajs-758	63	31	,	,	PUNCT
iajs-758	63	32	we	we	PRON
iajs-758	63	33	end	end	VERB
iajs-758	63	34	the	the	DET
iajs-758	63	35	section	section	NOUN
iajs-758	63	36	by	by	ADP
iajs-758	63	37	studying	study	VERB
iajs-758	63	38	the	the	DET
iajs-758	63	39	relationships	relationship	NOUN
iajs-758	63	40	between	between	ADP
iajs-758	63	41	maxrings	maxring	NOUN
iajs-758	63	42	and	and	CCONJ
iajs-758	63	43	max	max	PROPN
iajs-758	63	44	-	-	PUNCT
iajs-758	63	45	modules	module	NOUN
iajs-758	63	46	.	.	PUNCT
iajs-758	64	1	definition	definition	NOUN
iajs-758	64	2	2.1	2.1	NUM
iajs-758	64	3	an	an	DET
iajs-758	64	4	r	r	NOUN
iajs-758	64	5	–	–	PUNCT
iajs-758	64	6	module	module	NOUN
iajs-758	64	7	m	m	NOUN
iajs-758	64	8	is	be	AUX
iajs-758	64	9	said	say	VERB
iajs-758	64	10	to	to	PART
iajs-758	64	11	be	be	AUX
iajs-758	64	12	a	a	DET
iajs-758	64	13	max	max	NOUN
iajs-758	64	14	–	–	PUNCT
iajs-758	64	15	module	module	NOUN
iajs-758	64	16	if	if	SCONJ
iajs-758	64	17	nannr	nannr	NOUN
iajs-758	64	18	is	be	AUX
iajs-758	64	19	a	a	DET
iajs-758	64	20	maximal	maximal	ADJ
iajs-758	64	21	ideal	ideal	NOUN
iajs-758	64	22	of	of	ADP
iajs-758	64	23	r	r	NOUN
iajs-758	64	24	,	,	PUNCT
iajs-758	64	25	for	for	ADP
iajs-758	64	26	each	each	DET
iajs-758	64	27	non	non	ADJ
iajs-758	64	28	–	–	PUNCT
iajs-758	64	29	zero	zero	NUM
iajs-758	64	30	submodule	submodule	NOUN
iajs-758	64	31	n	n	PROPN
iajs-758	64	32	of	of	ADP
iajs-758	64	33	m.	m.	NOUN
iajs-758	64	34	specially	specially	ADV
iajs-758	64	35	,	,	PUNCT
iajs-758	64	36	a	a	DET
iajs-758	64	37	ring	ring	NOUN
iajs-758	64	38	r	r	NOUN
iajs-758	64	39	is	be	AUX
iajs-758	64	40	called	call	VERB
iajs-758	64	41	a	a	DET
iajs-758	64	42	max	max	PROPN
iajs-758	64	43	–	–	PUNCT
iajs-758	64	44	ring	ring	NOUN
iajs-758	64	45	if	if	SCONJ
iajs-758	64	46	and	and	CCONJ
iajs-758	64	47	only	only	ADV
iajs-758	64	48	if	if	SCONJ
iajs-758	64	49	r	r	NOUN
iajs-758	64	50	is	be	AUX
iajs-758	64	51	max	max	PROPN
iajs-758	64	52	–	–	PUNCT
iajs-758	64	53	r	r	NOUN
iajs-758	64	54	–	–	PUNCT
iajs-758	64	55	module	module	NOUN
iajs-758	64	56	.	.	PUNCT
iajs-758	65	1	we	we	PRON
iajs-758	65	2	give	give	VERB
iajs-758	65	3	some	some	DET
iajs-758	65	4	examples	example	NOUN
iajs-758	65	5	and	and	CCONJ
iajs-758	65	6	remarks	remark	VERB
iajs-758	65	7	:	:	PUNCT
iajs-758	65	8	remarks	remark	NOUN
iajs-758	65	9	and	and	CCONJ
iajs-758	65	10	examples	example	NOUN
iajs-758	65	11	2.2	2.2	NUM
iajs-758	65	12	1	1	NUM
iajs-758	65	13	p	p	NOUN
iajs-758	65	14	z	z	NOUN
iajs-758	66	1	as	as	PUNCT
iajs-758	66	2	z	z	X
iajs-758	66	3	–	–	PUNCT
iajs-758	66	4	module	module	NOUN
iajs-758	66	5	is	be	AUX
iajs-758	66	6	a	a	DET
iajs-758	66	7	max	max	NOUN
iajs-758	66	8	–	–	PUNCT
iajs-758	66	9	module	module	NOUN
iajs-758	66	10	.	.	PUNCT
iajs-758	67	1	n	n	NOUN
iajs-758	67	2	=	=	PUNCT
iajs-758	68	1	i	i	PRON
iajs-758	68	2	m	m	VERB
iajs-758	68	3	for	for	ADP
iajs-758	68	4	some	some	DET
iajs-758	68	5	ideal	ideal	NOUN
iajs-758	68	6	i	i	PRON
iajs-758	68	7	of	of	ADP
iajs-758	68	8	r.	r.	PROPN
iajs-758	69	1	but	but	CCONJ
iajs-758	69	2	m	m	PROPN
iajs-758	69	3	is	be	AUX
iajs-758	69	4	faithful	faithful	ADJ
iajs-758	69	5	,	,	PUNCT
iajs-758	69	6	annrn	annrn	NOUN
iajs-758	69	7	=	=	NOUN
iajs-758	69	8	annrim	annrim	NOUN
iajs-758	69	9	=	=	SYM
iajs-758	69	10	annri	annri	PROPN
iajs-758	69	11	and	and	CCONJ
iajs-758	69	12	so	so	ADV
iajs-758	69	13	iannimannnann	iannimannnann	PROPN
iajs-758	69	14	rrr	rrr	PROPN
iajs-758	69	15			NUM
iajs-758	69	16	which	which	PRON
iajs-758	69	17	is	be	AUX
iajs-758	69	18	a	a	DET
iajs-758	69	19	maximal	maximal	ADJ
iajs-758	69	20	ideal	ideal	NOUN
iajs-758	69	21	of	of	ADP
iajs-758	69	22	r.	r.	PROPN
iajs-758	69	23	therefore	therefore	ADV
iajs-758	69	24	m	m	PROPN
iajs-758	69	25	is	be	AUX
iajs-758	69	26	a	a	DET
iajs-758	69	27	max	max	PROPN
iajs-758	69	28	–	–	PUNCT
iajs-758	69	29	module	module	NOUN
iajs-758	69	30	.	.	PUNCT
iajs-758	70	1	proof	proof	NOUN
iajs-758	70	2	:	:	PUNCT
iajs-758	70	3	we	we	PRON
iajs-758	70	4	know	know	VERB
iajs-758	70	5	that	that	SCONJ
iajs-758	70	6	every	every	DET
iajs-758	70	7	submodule	submodule	NOUN
iajs-758	70	8	of	of	ADP
iajs-758	70	9	p	p	PROPN
iajs-758	70	10	z	z	PROPN
iajs-758	70	11	is	be	AUX
iajs-758	70	12	of	of	ADP
iajs-758	70	13	the	the	DET
iajs-758	70	14	form	form	NOUN
iajs-758	70	15			NOUN
iajs-758	70	16	z	z	PROPN
iajs-758	70	17	p	p	NOUN
iajs-758	70	18	n	n	NUM
iajs-758	70	19	1	1	NUM
iajs-758	70	20	,	,	PUNCT
iajs-758	70	21	where	where	SCONJ
iajs-758	70	22	n	n	PRON
iajs-758	70	23	be	be	VERB
iajs-758	70	24	a	a	DET
iajs-758	70	25	nonnegative	nonnegative	ADJ
iajs-758	70	26	integer	integer	NOUN
iajs-758	70	27	,	,	PUNCT
iajs-758	70	28	so	so	CCONJ
iajs-758	70	29	pzzpz	pzzpz	ADJ
iajs-758	70	30	p	p	PROPN
iajs-758	70	31	ann	ann	PROPN
iajs-758	70	32	n	n	CCONJ
iajs-758	70	33	nz	nz	PROPN
iajs-758	70	34			ADP
iajs-758	70	35	1	1	NUM
iajs-758	70	36	is	be	AUX
iajs-758	70	37	a	a	DET
iajs-758	70	38	maximal	maximal	ADJ
iajs-758	70	39	ideal	ideal	NOUN
iajs-758	70	40	of	of	ADP
iajs-758	70	41	z.	z.	PROPN
iajs-758	70	42	2z	2z	NUM
iajs-758	70	43	as	as	ADP
iajs-758	70	44	a	a	DET
iajs-758	70	45	z	z	NOUN
iajs-758	70	46	–	–	PUNCT
iajs-758	70	47	module	module	NOUN
iajs-758	70	48	is	be	AUX
iajs-758	70	49	not	not	PART
iajs-758	70	50	a	a	DET
iajs-758	70	51	max	max	PROPN
iajs-758	70	52	–	–	PUNCT
iajs-758	70	53	module	module	NOUN
iajs-758	70	54	,	,	PUNCT
iajs-758	70	55	since	since	SCONJ
iajs-758	70	56	00	00	NUM
iajs-758	70	57	zannz	zannz	NOUN
iajs-758	70	58	is	be	AUX
iajs-758	70	59	not	not	PART
iajs-758	70	60	a	a	DET
iajs-758	70	61	maximal	maximal	ADJ
iajs-758	70	62	ideal	ideal	NOUN
iajs-758	70	63	of	of	ADP
iajs-758	70	64	z.	z.	PROPN
iajs-758	70	65	3consider	3consider	NUM
iajs-758	70	66	,	,	PUNCT
iajs-758	70	67	the	the	DET
iajs-758	70	68	z	z	NOUN
iajs-758	70	69	–	–	PUNCT
iajs-758	70	70	module	module	NOUN
iajs-758	70	71	m	m	NOUN
iajs-758	70	72	=	=	SYM
iajs-758	70	73	z2	z2	PROPN
iajs-758	70	74			PROPN
iajs-758	70	75	z12	z12	PROPN
iajs-758	70	76	and	and	CCONJ
iajs-758	70	77	the	the	DET
iajs-758	70	78	z	z	PROPN
iajs-758	70	79	–	–	PUNCT
iajs-758	70	80	submodule	submodule	NOUN
iajs-758	70	81	)	)	PUNCT
iajs-758	70	82	2()0	2()0	PROPN
iajs-758	70	83	(	(	PUNCT
iajs-758	70	84	n	n	PROPN
iajs-758	70	85	.	.	PUNCT
iajs-758	71	1	then	then	ADV
iajs-758	71	2	,	,	PUNCT
iajs-758	71	3	zzzznannz	zzzznannz	NOUN
iajs-758	71	4	666	666	NUM
iajs-758	71	5			NOUN
iajs-758	71	6			PROPN
iajs-758	71	7	,	,	PUNCT
iajs-758	71	8	which	which	PRON
iajs-758	71	9	is	be	AUX
iajs-758	71	10	not	not	PART
iajs-758	71	11	a	a	DET
iajs-758	71	12	maximal	maximal	ADJ
iajs-758	71	13	ideal	ideal	NOUN
iajs-758	71	14	of	of	ADP
iajs-758	71	15	z.	z.	PROPN
iajs-758	71	16	therefore	therefore	ADV
iajs-758	71	17	,	,	PUNCT
iajs-758	71	18	m	m	VERB
iajs-758	71	19	is	be	AUX
iajs-758	71	20	not	not	PART
iajs-758	71	21	a	a	DET
iajs-758	71	22	max	max	NOUN
iajs-758	71	23	–	–	PUNCT
iajs-758	71	24	module	module	NOUN
iajs-758	71	25	.	.	PUNCT
iajs-758	72	1	4q	4q	NOUN
iajs-758	72	2	as	as	ADP
iajs-758	72	3	a	a	DET
iajs-758	72	4	z	z	NOUN
iajs-758	72	5	–	–	PUNCT
iajs-758	72	6	module	module	NOUN
iajs-758	72	7	is	be	AUX
iajs-758	72	8	not	not	PART
iajs-758	72	9	a	a	DET
iajs-758	72	10	max	max	NOUN
iajs-758	72	11	–	–	PUNCT
iajs-758	72	12	module	module	NOUN
iajs-758	72	13	.	.	PUNCT
iajs-758	73	1	5every	5every	NUM
iajs-758	73	2	non	non	ADJ
iajs-758	73	3	–	–	PUNCT
iajs-758	73	4	zero	zero	NUM
iajs-758	73	5	submodule	submodule	NOUN
iajs-758	73	6	of	of	ADP
iajs-758	73	7	a	a	DET
iajs-758	73	8	max	max	PROPN
iajs-758	73	9	–	–	PUNCT
iajs-758	73	10	module	module	NOUN
iajs-758	73	11	is	be	AUX
iajs-758	73	12	a	a	DET
iajs-758	73	13	max	max	PROPN
iajs-758	73	14	–	–	PUNCT
iajs-758	73	15	r	r	NOUN
iajs-758	73	16	-	-	PUNCT
iajs-758	73	17	module	module	NOUN
iajs-758	73	18	.	.	PUNCT
iajs-758	74	1	6let	6let	NUM
iajs-758	74	2	m	m	VERB
iajs-758	74	3	be	be	VERB
iajs-758	74	4	a	a	DET
iajs-758	74	5	max	max	PROPN
iajs-758	74	6	–	–	PUNCT
iajs-758	74	7	module	module	NOUN
iajs-758	74	8	,	,	PUNCT
iajs-758	74	9	then	then	ADV
iajs-758	74	10	mannr	mannr	NOUN
iajs-758	74	11	is	be	AUX
iajs-758	74	12	a	a	DET
iajs-758	74	13	maximal	maximal	ADJ
iajs-758	74	14	ideal	ideal	NOUN
iajs-758	74	15	of	of	ADP
iajs-758	74	16	r.	r.	PROPN
iajs-758	74	17	the	the	DET
iajs-758	74	18	following	follow	VERB
iajs-758	74	19	theorem	theorem	NOUN
iajs-758	74	20	gives	give	VERB
iajs-758	74	21	a	a	DET
iajs-758	74	22	characterization	characterization	NOUN
iajs-758	74	23	for	for	ADP
iajs-758	74	24	max	max	PROPN
iajs-758	74	25	–	–	PUNCT
iajs-758	74	26	modules	module	NOUN
iajs-758	74	27	.	.	PUNCT
iajs-758	75	1	theorem	theorem	VERB
iajs-758	75	2	2.3	2.3	NUM
iajs-758	75	3	let	let	VERB
iajs-758	75	4	m	m	PRON
iajs-758	75	5	be	be	AUX
iajs-758	75	6	an	an	DET
iajs-758	75	7	r	r	NOUN
iajs-758	75	8	-	-	PUNCT
iajs-758	75	9	module	module	NOUN
iajs-758	75	10	,	,	PUNCT
iajs-758	75	11	then	then	ADV
iajs-758	75	12	m	m	NOUN
iajs-758	75	13	is	be	AUX
iajs-758	75	14	a	a	DET
iajs-758	75	15	max	max	NOUN
iajs-758	75	16	–	–	PUNCT
iajs-758	75	17	module	module	NOUN
iajs-758	75	18	if	if	SCONJ
iajs-758	75	19	and	and	CCONJ
iajs-758	75	20	only	only	ADV
iajs-758	75	21	if	if	SCONJ
iajs-758	75	22	(	(	PUNCT
iajs-758	75	23	0	0	NUM
iajs-758	75	24	)	)	PUNCT
iajs-758	75	25	is	be	AUX
iajs-758	75	26	a	a	DET
iajs-758	75	27	*	*	ADJ
iajs-758	75	28	submodule	submodule	NOUN
iajs-758	75	29	.	.	PUNCT
iajs-758	76	1	proof	proof	NOUN
iajs-758	76	2	:	:	PUNCT
iajs-758	76	3	suppose	suppose	VERB
iajs-758	76	4	that	that	SCONJ
iajs-758	76	5	m	m	PROPN
iajs-758	76	6	is	be	AUX
iajs-758	76	7	a	a	DET
iajs-758	76	8	max	max	PROPN
iajs-758	76	9	–	–	PUNCT
iajs-758	76	10	module	module	NOUN
iajs-758	76	11	,	,	PUNCT
iajs-758	76	12	to	to	PART
iajs-758	76	13	prove	prove	VERB
iajs-758	76	14	(	(	PUNCT
iajs-758	76	15	0	0	NUM
iajs-758	76	16	)	)	PUNCT
iajs-758	76	17	is	be	AUX
iajs-758	76	18	a	a	DET
iajs-758	76	19	*	*	ADJ
iajs-758	76	20	submodule	submodule	NOUN
iajs-758	76	21	.	.	PUNCT
iajs-758	77	1	since	since	SCONJ
iajs-758	77	2	m	m	PROPN
iajs-758	77	3	is	be	AUX
iajs-758	77	4	a	a	DET
iajs-758	77	5	max	max	PROPN
iajs-758	77	6	,	,	PUNCT
iajs-758	77	7	then	then	ADV
iajs-758	77	8	nannr	nannr	NOUN
iajs-758	77	9	is	be	AUX
iajs-758	77	10	a	a	DET
iajs-758	77	11	maximal	maximal	ADJ
iajs-758	77	12	ideal	ideal	NOUN
iajs-758	77	13	of	of	ADP
iajs-758	77	14	r	r	NOUN
iajs-758	77	15	,	,	PUNCT
iajs-758	77	16	for	for	ADP
iajs-758	77	17	each	each	DET
iajs-758	77	18	non	non	ADJ
iajs-758	77	19	–	–	PUNCT
iajs-758	77	20	zero	zero	NUM
iajs-758	77	21	submodule	submodule	NOUN
iajs-758	77	22	n	n	PROPN
iajs-758	77	23	of	of	ADP
iajs-758	77	24	m.	m.	NOUN
iajs-758	77	25	but	but	CCONJ
iajs-758	77	26	]	]	PUNCT
iajs-758	77	27	)	)	PUNCT
iajs-758	77	28	0	0	PUNCT
iajs-758	78	1	[	[	X
iajs-758	78	2	(	(	PUNCT
iajs-758	78	3	:	:	PUNCT
iajs-758	78	4	nnann	nnann	PROPN
iajs-758	78	5	rr	rr	PROPN
iajs-758	78	6			PROPN
iajs-758	78	7	,	,	PUNCT
iajs-758	78	8	for	for	ADP
iajs-758	78	9	each	each	DET
iajs-758	78	10	non	non	ADJ
iajs-758	78	11	–	–	PUNCT
iajs-758	78	12	zero	zero	NUM
iajs-758	78	13	submodule	submodule	NOUN
iajs-758	78	14	n	n	PROPN
iajs-758	78	15	of	of	ADP
iajs-758	78	16	m	m	PRON
iajs-758	78	17	so	so	ADV
iajs-758	78	18	by	by	ADP
iajs-758	78	19	definition	definition	NOUN
iajs-758	78	20	(	(	PUNCT
iajs-758	78	21	1.1	1.1	NUM
iajs-758	78	22	)	)	PUNCT
iajs-758	78	23	,	,	PUNCT
iajs-758	78	24	(	(	PUNCT
iajs-758	78	25	0	0	X
iajs-758	78	26	)	)	PUNCT
iajs-758	78	27	is	be	AUX
iajs-758	78	28	a	a	DET
iajs-758	78	29	*	*	ADJ
iajs-758	78	30	submodule	submodule	NOUN
iajs-758	78	31	of	of	ADP
iajs-758	78	32	m.	m.	NOUN
iajs-758	78	33	conversely	conversely	ADV
iajs-758	78	34	,	,	PUNCT
iajs-758	78	35	if	if	SCONJ
iajs-758	78	36	(	(	PUNCT
iajs-758	78	37	0	0	NUM
iajs-758	78	38	)	)	PUNCT
iajs-758	78	39	is	be	AUX
iajs-758	78	40	a	a	DET
iajs-758	78	41	*	*	ADJ
iajs-758	78	42	submodule	submodule	NOUN
iajs-758	78	43	of	of	ADP
iajs-758	78	44	m	m	PROPN
iajs-758	78	45	,	,	PUNCT
iajs-758	78	46	to	to	PART
iajs-758	78	47	prove	prove	VERB
iajs-758	78	48	m	m	NOUN
iajs-758	78	49	is	be	AUX
iajs-758	78	50	a	a	DET
iajs-758	78	51	max	max	PROPN
iajs-758	78	52	–	–	PUNCT
iajs-758	78	53	module	module	NOUN
iajs-758	78	54	.	.	PUNCT
iajs-758	79	1	since	since	SCONJ
iajs-758	79	2	(	(	PUNCT
iajs-758	79	3	0	0	NUM
iajs-758	79	4	)	)	PUNCT
iajs-758	79	5	is	be	AUX
iajs-758	79	6	a	a	DET
iajs-758	79	7	*	*	PUNCT
iajs-758	79	8	submodule	submodule	NOUN
iajs-758	79	9	,	,	PUNCT
iajs-758	79	10	then	then	ADV
iajs-758	79	11	definition	definition	NOUN
iajs-758	79	12	(	(	PUNCT
iajs-758	79	13	1.1	1.1	NUM
iajs-758	79	14	)	)	PUNCT
iajs-758	79	15	implies	imply	VERB
iajs-758	79	16	that	that	SCONJ
iajs-758	79	17	]	]	X
iajs-758	79	18	)	)	PUNCT
iajs-758	79	19	0	0	PUNCT
iajs-758	80	1	[	[	X
iajs-758	80	2	(	(	PUNCT
iajs-758	80	3	:	:	PUNCT
iajs-758	80	4	nr	nr	PRON
iajs-758	80	5	is	be	AUX
iajs-758	80	6	a	a	DET
iajs-758	80	7	maximal	maximal	ADJ
iajs-758	80	8	ideal	ideal	NOUN
iajs-758	80	9	,	,	PUNCT
iajs-758	80	10	for	for	ADP
iajs-758	80	11	each	each	DET
iajs-758	80	12	non	non	ADJ
iajs-758	80	13	-	-	ADJ
iajs-758	80	14	zero	zero	NUM
iajs-758	80	15	submodule	submodule	NOUN
iajs-758	80	16	n	n	PROPN
iajs-758	80	17	of	of	ADP
iajs-758	80	18	m	m	PROPN
iajs-758	80	19	.	.	PUNCT
iajs-758	81	1	but	but	CCONJ
iajs-758	81	2	nannn	nannn	PROPN
iajs-758	81	3	rr	rr	PROPN
iajs-758	81	4	])0	])0	PROPN
iajs-758	81	5	[	[	X
iajs-758	81	6	(	(	PUNCT
iajs-758	81	7	:	:	PUNCT
iajs-758	81	8	,	,	PUNCT
iajs-758	81	9	so	so	ADV
iajs-758	81	10	m	m	VERB
iajs-758	81	11	is	be	AUX
iajs-758	81	12	a	a	DET
iajs-758	81	13	max	max	NOUN
iajs-758	81	14	–	–	PUNCT
iajs-758	81	15	module	module	NOUN
iajs-758	81	16	.	.	PUNCT
iajs-758	82	1	ibn	ibn	PROPN
iajs-758	82	2	alhaitham	alhaitham	PROPN
iajs-758	82	3	j.	j.	PROPN
iajs-758	82	4	for	for	ADP
iajs-758	82	5	pure	pure	ADJ
iajs-758	82	6	&	&	CCONJ
iajs-758	82	7	appl	appl	PROPN
iajs-758	82	8	.	.	PUNCT
iajs-758	83	1	sci	sci	PROPN
iajs-758	83	2	.	.	PUNCT
iajs-758	83	3	vol.24	vol.24	NOUN
iajs-758	83	4	(	(	PUNCT
iajs-758	83	5	2	2	NUM
iajs-758	83	6	)	)	PUNCT
iajs-758	83	7	2011	2011	NUM
iajs-758	83	8	by	by	ADP
iajs-758	83	9	using	use	VERB
iajs-758	83	10	(	(	PUNCT
iajs-758	83	11	1.2	1.2	NUM
iajs-758	83	12	,	,	PUNCT
iajs-758	83	13	(	(	PUNCT
iajs-758	83	14	1	1	NUM
iajs-758	83	15	)	)	PUNCT
iajs-758	83	16	)	)	PUNCT
iajs-758	83	17	and	and	CCONJ
iajs-758	83	18	[	[	X
iajs-758	83	19	2	2	NUM
iajs-758	83	20	,	,	PUNCT
iajs-758	83	21	theorem	theorem	ADJ
iajs-758	83	22	(	(	PUNCT
iajs-758	83	23	3.3.6	3.3.6	NUM
iajs-758	83	24	)	)	PUNCT
iajs-758	83	25	,	,	PUNCT
iajs-758	83	26	chapter	chapter	NOUN
iajs-758	83	27	3	3	NUM
iajs-758	83	28	]	]	PUNCT
iajs-758	83	29	,	,	PUNCT
iajs-758	83	30	we	we	PRON
iajs-758	83	31	can	can	AUX
iajs-758	83	32	give	give	VERB
iajs-758	83	33	the	the	DET
iajs-758	83	34	following	follow	VERB
iajs-758	83	35	characterization	characterization	NOUN
iajs-758	83	36	for	for	ADP
iajs-758	83	37	max	max	NOUN
iajs-758	83	38	-	-	PUNCT
iajs-758	83	39	module	module	NOUN
iajs-758	83	40	.	.	PUNCT
iajs-758	84	1	theorem	theorem	VERB
iajs-758	84	2	2.4	2.4	NUM
iajs-758	84	3	let	let	VERB
iajs-758	84	4	m	m	PRON
iajs-758	84	5	be	be	AUX
iajs-758	84	6	an	an	DET
iajs-758	84	7	r	r	NOUN
iajs-758	84	8	-	-	PUNCT
iajs-758	84	9	module	module	NOUN
iajs-758	84	10	,	,	PUNCT
iajs-758	84	11	if	if	SCONJ
iajs-758	84	12	m	m	NOUN
iajs-758	84	13	is	be	AUX
iajs-758	84	14	a	a	DET
iajs-758	84	15	max	max	PROPN
iajs-758	84	16	–	–	PUNCT
iajs-758	84	17	module	module	NOUN
iajs-758	84	18	then	then	ADV
iajs-758	84	19	rnannnann	rnannnann	PROPN
iajs-758	84	20	rr	rr	PROPN
iajs-758	84	21			PROPN
iajs-758	84	22	for	for	ADP
iajs-758	84	23	each	each	DET
iajs-758	84	24	non	non	ADJ
iajs-758	84	25	–	–	PUNCT
iajs-758	84	26	zero	zero	NUM
iajs-758	84	27	submodule	submodule	NOUN
iajs-758	84	28	n	n	PROPN
iajs-758	84	29	of	of	ADP
iajs-758	84	30	m	m	PRON
iajs-758	84	31	such	such	ADJ
iajs-758	84	32	that	that	SCONJ
iajs-758	84	33	rn	rn	PROPN
iajs-758	84	34	≠	≠	PROPN
iajs-758	84	35	(	(	PUNCT
iajs-758	84	36	0	0	NUM
iajs-758	84	37	)	)	PUNCT
iajs-758	84	38	,	,	PUNCT
iajs-758	84	39	r	r	NOUN
iajs-758	84	40			NOUN
iajs-758	84	41	r	r	NOUN
iajs-758	84	42	.	.	PUNCT
iajs-758	85	1	by	by	ADP
iajs-758	85	2	using	use	VERB
iajs-758	85	3	(	(	PUNCT
iajs-758	85	4	1.2	1.2	NUM
iajs-758	85	5	,	,	PUNCT
iajs-758	85	6	(	(	PUNCT
iajs-758	85	7	1	1	NUM
iajs-758	85	8	)	)	PUNCT
iajs-758	85	9	)	)	PUNCT
iajs-758	85	10	and	and	CCONJ
iajs-758	85	11	[	[	X
iajs-758	85	12	2	2	NUM
iajs-758	85	13	,	,	PUNCT
iajs-758	85	14	corollary	corollary	ADJ
iajs-758	85	15	(	(	PUNCT
iajs-758	85	16	3.3.7	3.3.7	NUM
iajs-758	85	17	)	)	PUNCT
iajs-758	85	18	,	,	PUNCT
iajs-758	85	19	chapter	chapter	NOUN
iajs-758	85	20	3	3	NUM
iajs-758	85	21	]	]	PUNCT
iajs-758	85	22	,	,	PUNCT
iajs-758	85	23	we	we	PRON
iajs-758	85	24	can	can	AUX
iajs-758	85	25	give	give	VERB
iajs-758	85	26	the	the	DET
iajs-758	85	27	following	follow	VERB
iajs-758	85	28	result	result	NOUN
iajs-758	85	29	:	:	PUNCT
iajs-758	85	30	corollary	corollary	ADJ
iajs-758	85	31	2.5	2.5	NUM
iajs-758	85	32	let	let	VERB
iajs-758	85	33	m	m	PRON
iajs-758	85	34	be	be	AUX
iajs-758	85	35	an	an	DET
iajs-758	85	36	r	r	NOUN
iajs-758	85	37	-	-	PUNCT
iajs-758	85	38	module	module	NOUN
iajs-758	85	39	,	,	PUNCT
iajs-758	85	40	if	if	SCONJ
iajs-758	85	41	m	m	NOUN
iajs-758	85	42	is	be	AUX
iajs-758	85	43	a	a	DET
iajs-758	85	44	max	max	PROPN
iajs-758	85	45	–	–	PUNCT
iajs-758	85	46	module	module	NOUN
iajs-758	85	47	then	then	ADV
iajs-758	85	48	)	)	PUNCT
iajs-758	85	49	(	(	PUNCT
iajs-758	85	50	)	)	PUNCT
iajs-758	85	51	(	(	PUNCT
iajs-758	85	52	rmannmann	rmannmann	PROPN
iajs-758	85	53	rr	rr	PROPN
iajs-758	85	54			PROPN
iajs-758	85	55	for	for	ADP
iajs-758	85	56	each	each	DET
iajs-758	85	57	0	0	NUM
iajs-758	85	58	≠	≠	PROPN
iajs-758	85	59	m	m	PROPN
iajs-758	85	60			NOUN
iajs-758	85	61	m	m	VERB
iajs-758	85	62	such	such	ADJ
iajs-758	85	63	that	that	SCONJ
iajs-758	85	64	rm	rm	PROPN
iajs-758	85	65	≠	≠	PROPN
iajs-758	85	66	0	0	NUM
iajs-758	85	67	,	,	PUNCT
iajs-758	85	68	r	r	NOUN
iajs-758	85	69			PROPN
iajs-758	85	70	r.	r.	PROPN
iajs-758	85	71	now	now	ADV
iajs-758	85	72	,	,	PUNCT
iajs-758	85	73	we	we	PRON
iajs-758	85	74	state	state	VERB
iajs-758	85	75	and	and	CCONJ
iajs-758	85	76	prove	prove	VERB
iajs-758	85	77	the	the	DET
iajs-758	85	78	following	follow	VERB
iajs-758	85	79	result	result	NOUN
iajs-758	85	80	.	.	PUNCT
iajs-758	86	1	proposition	proposition	NOUN
iajs-758	86	2	2.6	2.6	NUM
iajs-758	86	3	zm	zm	NOUN
iajs-758	86	4	as	as	ADP
iajs-758	86	5	a	a	DET
iajs-758	86	6	z	z	NOUN
iajs-758	86	7	–	–	PUNCT
iajs-758	86	8	module	module	NOUN
iajs-758	86	9	is	be	AUX
iajs-758	86	10	a	a	DET
iajs-758	86	11	max	max	NOUN
iajs-758	86	12	–	–	PUNCT
iajs-758	86	13	module	module	NOUN
iajs-758	86	14	if	if	SCONJ
iajs-758	86	15	and	and	CCONJ
iajs-758	86	16	only	only	ADV
iajs-758	86	17	if	if	SCONJ
iajs-758	86	18	m	m	VERB
iajs-758	86	19	=	=	VERB
iajs-758	86	20	p	p	NOUN
iajs-758	86	21	n	n	CCONJ
iajs-758	86	22	for	for	ADP
iajs-758	86	23	some	some	DET
iajs-758	86	24	prime	prime	ADJ
iajs-758	86	25	number	number	NOUN
iajs-758	86	26	p	p	NOUN
iajs-758	86	27	and	and	CCONJ
iajs-758	86	28	n	n	CCONJ
iajs-758	86	29			NOUN
iajs-758	86	30	z+	z+	NUM
iajs-758	86	31	.	.	PUNCT
iajs-758	87	1	proof	proof	NOUN
iajs-758	87	2	:	:	PUNCT
iajs-758	87	3	if	if	SCONJ
iajs-758	87	4	zm	zm	PROPN
iajs-758	87	5	is	be	AUX
iajs-758	87	6	a	a	DET
iajs-758	87	7	max	max	PROPN
iajs-758	87	8	–	–	PUNCT
iajs-758	87	9	z	z	NOUN
iajs-758	87	10	-	-	PUNCT
iajs-758	87	11	module	module	NOUN
iajs-758	87	12	,	,	PUNCT
iajs-758	87	13	to	to	PART
iajs-758	87	14	show	show	VERB
iajs-758	87	15	that	that	SCONJ
iajs-758	87	16	m	m	ADV
iajs-758	87	17	=	=	SYM
iajs-758	87	18	pn	pn	NOUN
iajs-758	87	19	for	for	ADP
iajs-758	87	20	some	some	DET
iajs-758	87	21	prime	prime	ADJ
iajs-758	87	22	number	number	NOUN
iajs-758	87	23	p	p	NOUN
iajs-758	87	24	and	and	CCONJ
iajs-758	87	25	n	n	CCONJ
iajs-758	87	26			NOUN
iajs-758	87	27	z+	z+	NUM
iajs-758	87	28	.	.	PUNCT
iajs-758	87	29	by	by	ADP
iajs-758	87	30	(	(	PUNCT
iajs-758	87	31	2.2	2.2	NUM
iajs-758	87	32	,	,	PUNCT
iajs-758	87	33	[	[	X
iajs-758	87	34	5	5	NUM
iajs-758	87	35	]	]	NUM
iajs-758	87	36	)	)	PUNCT
iajs-758	87	37	,	,	PUNCT
iajs-758	87	38	pzmzzmannz	pzmzzmannz	NOUN
iajs-758	87	39			NUM
iajs-758	87	40	is	be	AUX
iajs-758	87	41	a	a	DET
iajs-758	87	42	maximal	maximal	ADJ
iajs-758	87	43	ideal	ideal	NOUN
iajs-758	87	44	of	of	ADP
iajs-758	87	45	z	z	PROPN
iajs-758	87	46	,	,	PUNCT
iajs-758	87	47	therefore	therefore	ADV
iajs-758	87	48	m	m	VERB
iajs-758	87	49	=	=	ADJ
iajs-758	87	50	pn	pn	NOUN
iajs-758	87	51	for	for	ADP
iajs-758	87	52	some	some	DET
iajs-758	87	53	prime	prime	ADJ
iajs-758	87	54	number	number	NOUN
iajs-758	87	55	p	p	NOUN
iajs-758	87	56	and	and	CCONJ
iajs-758	87	57	n	n	CCONJ
iajs-758	87	58			NOUN
iajs-758	87	59	z+	z+	NUM
iajs-758	87	60	.	.	PUNCT
iajs-758	88	1	conversely	conversely	ADV
iajs-758	88	2	,	,	PUNCT
iajs-758	88	3	if	if	SCONJ
iajs-758	88	4	m	m	ADV
iajs-758	88	5	=	=	VERB
iajs-758	88	6	p	p	NOUN
iajs-758	88	7	n	n	CCONJ
iajs-758	88	8	for	for	ADP
iajs-758	88	9	some	some	DET
iajs-758	88	10	p	p	NOUN
iajs-758	88	11	(	(	PUNCT
iajs-758	88	12	prime	prime	ADJ
iajs-758	88	13	number	number	NOUN
iajs-758	88	14	)	)	PUNCT
iajs-758	88	15	and	and	CCONJ
iajs-758	88	16	n	n	DET
iajs-758	88	17			NOUN
iajs-758	88	18	z+	z+	NUM
iajs-758	88	19	,	,	PUNCT
iajs-758	88	20	to	to	PART
iajs-758	88	21	show	show	VERB
iajs-758	88	22	that	that	SCONJ
iajs-758	88	23	zm	zm	PROPN
iajs-758	88	24	a	a	DET
iajs-758	88	25	z	z	PROPN
iajs-758	88	26	–	–	PUNCT
iajs-758	88	27	module	module	NOUN
iajs-758	88	28	is	be	AUX
iajs-758	88	29	a	a	DET
iajs-758	88	30	max	max	PROPN
iajs-758	88	31	–	–	PUNCT
iajs-758	88	32	module	module	NOUN
iajs-758	88	33	.	.	PUNCT
iajs-758	89	1	let	let	VERB
iajs-758	89	2	n	n	PRON
iajs-758	89	3	be	be	AUX
iajs-758	89	4	anon	anon	ADJ
iajs-758	89	5	–	–	PUNCT
iajs-758	89	6	zero	zero	NUM
iajs-758	89	7	submodule	submodule	NOUN
iajs-758	89	8	of	of	ADP
iajs-758	89	9	zm	zm	PROPN
iajs-758	89	10	.	.	PUNCT
iajs-758	90	1	since	since	SCONJ
iajs-758	90	2	n	n	PROPN
iajs-758	90	3	⊆	⊆	NUM
iajs-758	90	4	zm	zm	PROPN
iajs-758	90	5	,	,	PUNCT
iajs-758	90	6	pzzpmzzmannnann	pzzpmzzmannnann	PROPN
iajs-758	90	7	n	n	CCONJ
iajs-758	90	8	zz	zz	NUM
iajs-758	90	9			NOUN
iajs-758	90	10	which	which	PRON
iajs-758	90	11	is	be	AUX
iajs-758	90	12	a	a	DET
iajs-758	90	13	maximal	maximal	ADJ
iajs-758	90	14	ideal	ideal	NOUN
iajs-758	90	15	,	,	PUNCT
iajs-758	90	16	then	then	ADV
iajs-758	90	17	pznannz	pznannz	VERB
iajs-758	90	18			NOUN
iajs-758	90	19	,	,	PUNCT
iajs-758	90	20	and	and	CCONJ
iajs-758	90	21	by	by	ADP
iajs-758	90	22	definition	definition	NOUN
iajs-758	90	23	(	(	PUNCT
iajs-758	90	24	2.1	2.1	NUM
iajs-758	90	25	)	)	PUNCT
iajs-758	90	26	,	,	PUNCT
iajs-758	90	27	zm	zm	PROPN
iajs-758	90	28	as	as	ADP
iajs-758	90	29	a	a	DET
iajs-758	90	30	zmodule	zmodule	NOUN
iajs-758	90	31	is	be	AUX
iajs-758	90	32	a	a	DET
iajs-758	90	33	max	max	PROPN
iajs-758	90	34	–	–	PUNCT
iajs-758	90	35	module	module	NOUN
iajs-758	90	36	.	.	PUNCT
iajs-758	91	1	in	in	ADP
iajs-758	91	2	the	the	DET
iajs-758	91	3	following	follow	VERB
iajs-758	91	4	result	result	NOUN
iajs-758	91	5	,	,	PUNCT
iajs-758	91	6	we	we	PRON
iajs-758	91	7	show	show	VERB
iajs-758	91	8	that	that	SCONJ
iajs-758	91	9	the	the	DET
iajs-758	91	10	converse	converse	NOUN
iajs-758	91	11	of	of	ADP
iajs-758	91	12	(	(	PUNCT
iajs-758	91	13	2.2	2.2	NUM
iajs-758	91	14	.	.	PUNCT
iajs-758	92	1	[	[	X
iajs-758	92	2	5	5	NUM
iajs-758	92	3	]	]	PUNCT
iajs-758	92	4	)	)	PUNCT
iajs-758	92	5	is	be	AUX
iajs-758	92	6	true	true	ADJ
iajs-758	92	7	.	.	PUNCT
iajs-758	93	1	proposition	proposition	NOUN
iajs-758	93	2	2.7	2.7	NUM
iajs-758	93	3	let	let	VERB
iajs-758	93	4	m	m	PRON
iajs-758	93	5	be	be	AUX
iajs-758	93	6	an	an	DET
iajs-758	93	7	r	r	NOUN
iajs-758	93	8	-	-	PUNCT
iajs-758	93	9	module	module	NOUN
iajs-758	93	10	that	that	PRON
iajs-758	93	11	satisfies	satisfy	VERB
iajs-758	93	12	✪	✪	PROPN
iajs-758	93	13	,	,	PUNCT
iajs-758	93	14	then	then	ADV
iajs-758	93	15	m	m	PROPN
iajs-758	93	16	is	be	AUX
iajs-758	93	17	max	max	PROPN
iajs-758	93	18	–	–	PUNCT
iajs-758	93	19	module	module	NOUN
iajs-758	93	20	if	if	SCONJ
iajs-758	93	21	and	and	CCONJ
iajs-758	93	22	only	only	ADV
iajs-758	93	23	if	if	SCONJ
iajs-758	93	24	mannr	mannr	NOUN
iajs-758	93	25	is	be	AUX
iajs-758	93	26	a	a	DET
iajs-758	93	27	maximal	maximal	ADJ
iajs-758	93	28	ideal	ideal	NOUN
iajs-758	93	29	of	of	ADP
iajs-758	93	30	r.	r.	PROPN
iajs-758	93	31	where	where	SCONJ
iajs-758	93	32	✪	✪	NOUN
iajs-758	93	33	:	:	PUNCT
iajs-758	93	34	]	]	X
iajs-758	93	35	[	[	PUNCT
iajs-758	93	36	:	:	PUNCT
iajs-758	93	37	mnmann	mnmann	PROPN
iajs-758	93	38	rr	rr	PROPN
iajs-758	93	39			PROPN
iajs-758	93	40			PROPN
iajs-758	93	41	,	,	PUNCT
iajs-758	93	42	for	for	ADP
iajs-758	93	43	each	each	DET
iajs-758	93	44	nonzero	nonzero	PROPN
iajs-758	93	45	submodule	submodule	PROPN
iajs-758	93	46	n	n	PROPN
iajs-758	93	47	of	of	ADP
iajs-758	93	48	m	m	PROPN
iajs-758	93	49	.	.	PUNCT
iajs-758	94	1	proof	proof	NOUN
iajs-758	94	2	:	:	PUNCT
iajs-758	94	3	if	if	SCONJ
iajs-758	94	4	m	m	NOUN
iajs-758	94	5	is	be	AUX
iajs-758	94	6	a	a	DET
iajs-758	94	7	max	max	PROPN
iajs-758	94	8	–	–	PUNCT
iajs-758	94	9	module	module	NOUN
iajs-758	94	10	,	,	PUNCT
iajs-758	94	11	then	then	ADV
iajs-758	94	12	by	by	ADP
iajs-758	94	13	(	(	PUNCT
iajs-758	94	14	2.2,[5	2.2,[5	NOUN
iajs-758	94	15	]	]	PUNCT
iajs-758	94	16	)	)	PUNCT
iajs-758	94	17	mannr	mannr	NOUN
iajs-758	94	18	is	be	AUX
iajs-758	94	19	a	a	DET
iajs-758	94	20	maximal	maximal	ADJ
iajs-758	94	21	ideal	ideal	NOUN
iajs-758	94	22	of	of	ADP
iajs-758	94	23	r.	r.	PROPN
iajs-758	94	24	conversely	conversely	ADV
iajs-758	94	25	,	,	PUNCT
iajs-758	94	26	if	if	SCONJ
iajs-758	94	27	mannr	mannr	NOUN
iajs-758	94	28	is	be	AUX
iajs-758	94	29	a	a	DET
iajs-758	94	30	maximal	maximal	ADJ
iajs-758	94	31	ideal	ideal	NOUN
iajs-758	94	32	of	of	ADP
iajs-758	94	33	r	r	NOUN
iajs-758	94	34	,	,	PUNCT
iajs-758	94	35	to	to	PART
iajs-758	94	36	prove	prove	VERB
iajs-758	94	37	that	that	SCONJ
iajs-758	94	38	m	m	PROPN
iajs-758	94	39	is	be	AUX
iajs-758	94	40	a	a	DET
iajs-758	94	41	max	max	PROPN
iajs-758	94	42	–	–	PUNCT
iajs-758	94	43	module	module	NOUN
iajs-758	94	44	,	,	PUNCT
iajs-758	94	45	(	(	PUNCT
iajs-758	94	46	nannr	nannr	NOUN
iajs-758	94	47	is	be	AUX
iajs-758	94	48	a	a	DET
iajs-758	94	49	maximal	maximal	ADJ
iajs-758	94	50	ideal	ideal	NOUN
iajs-758	94	51	of	of	ADP
iajs-758	94	52	r,∀0	r,∀0	ADJ
iajs-758	94	53	≠	≠	PROPN
iajs-758	94	54	n⊆	n⊆	ADV
iajs-758	94	55	m	m	NOUN
iajs-758	94	56	)	)	PUNCT
iajs-758	94	57	.	.	PUNCT
iajs-758	95	1	it	it	PRON
iajs-758	95	2	is	be	AUX
iajs-758	95	3	clear	clear	ADJ
iajs-758	95	4	that	that	SCONJ
iajs-758	95	5	mannnann	mannnann	PROPN
iajs-758	95	6	rr	rr	PROPN
iajs-758	95	7			PROPN
iajs-758	95	8	…	…	PUNCT
iajs-758	95	9	..	..	PUNCT
iajs-758	95	10	(	(	PUNCT
iajs-758	95	11	1	1	NUM
iajs-758	95	12	)	)	PUNCT
iajs-758	95	13	.	.	PUNCT
iajs-758	96	1	let	let	VERB
iajs-758	96	2	nannr	nannr	PROPN
iajs-758	96	3	r	r	PROPN
iajs-758	96	4	,	,	PUNCT
iajs-758	96	5	so	so	CCONJ
iajs-758	96	6	rn	rn	PROPN
iajs-758	96	7	n	n	PROPN
iajs-758	96	8	=	=	NOUN
iajs-758	96	9	0	0	NUM
iajs-758	96	10	for	for	ADP
iajs-758	96	11	some	some	DET
iajs-758	96	12	n	n	PRON
iajs-758	96	13			NOUN
iajs-758	96	14	z+	z+	NUM
iajs-758	96	15	.	.	PUNCT
iajs-758	97	1	by	by	ADP
iajs-758	97	2	✪	✪	PROPN
iajs-758	97	3	,	,	PUNCT
iajs-758	97	4	there	there	PRON
iajs-758	97	5	exists	exist	VERB
iajs-758	97	6	a	a	DET
iajs-758	97	7			PROPN
iajs-758	97	8	r	r	NOUN
iajs-758	97	9	,	,	PUNCT
iajs-758	97	10	a	a	DET
iajs-758	97	11	≠	≠	PROPN
iajs-758	97	12	0	0	NUM
iajs-758	97	13	such	such	ADJ
iajs-758	97	14	that	that	PRON
iajs-758	97	15	am	be	AUX
iajs-758	97	16	≠	≠	NOUN
iajs-758	97	17	0	0	NUM
iajs-758	97	18	and	and	CCONJ
iajs-758	97	19	am	be	AUX
iajs-758	97	20			PROPN
iajs-758	97	21	n.	n.	PROPN
iajs-758	97	22	hence	hence	ADV
iajs-758	97	23	rnam	rnam	PROPN
iajs-758	97	24			PROPN
iajs-758	97	25	rnn	rnn	PROPN
iajs-758	97	26	=	=	SYM
iajs-758	97	27	0	0	X
iajs-758	97	28	.	.	PUNCT
iajs-758	98	1	it	it	PRON
iajs-758	98	2	follows	follow	VERB
iajs-758	98	3	that	that	SCONJ
iajs-758	98	4	.mannar	.mannar	NUM
iajs-758	98	5	r	r	NOUN
iajs-758	98	6	n	n	PRON
iajs-758	98	7			NOUN
iajs-758	98	8	but	but	CCONJ
iajs-758	98	9	mannr	mannr	NOUN
iajs-758	98	10	is	be	AUX
iajs-758	98	11	a	a	DET
iajs-758	98	12	maximal	maximal	ADJ
iajs-758	98	13	ideal	ideal	NOUN
iajs-758	98	14	,	,	PUNCT
iajs-758	98	15	so	so	SCONJ
iajs-758	98	16	mannr	mannr	NOUN
iajs-758	98	17	is	be	AUX
iajs-758	98	18	a	a	DET
iajs-758	98	19	primary	primary	ADJ
iajs-758	98	20	ideal	ideal	NOUN
iajs-758	98	21	by	by	ADP
iajs-758	98	22	(	(	PUNCT
iajs-758	98	23	1	1	NUM
iajs-758	98	24	,	,	PUNCT
iajs-758	98	25	proposition	proposition	NOUN
iajs-758	98	26	4.6	4.6	NUM
iajs-758	98	27	,	,	PUNCT
iajs-758	98	28	p.	p.	NOUN
iajs-758	98	29	64	64	NUM
iajs-758	98	30	)	)	PUNCT
iajs-758	98	31	,	,	PUNCT
iajs-758	98	32	and	and	CCONJ
iajs-758	98	33	a	a	DET
iajs-758	98	34			NOUN
iajs-758	98	35	annrm	annrm	NOUN
iajs-758	98	36	(	(	PUNCT
iajs-758	98	37	since	since	SCONJ
iajs-758	98	38	am	be	AUX
iajs-758	98	39	≠	≠	NOUN
iajs-758	98	40	0	0	NUM
iajs-758	98	41	)	)	PUNCT
iajs-758	98	42	,	,	PUNCT
iajs-758	98	43	so	so	CCONJ
iajs-758	98	44	(	(	PUNCT
iajs-758	98	45	r	r	NOUN
iajs-758	98	46	n	n	NOUN
iajs-758	98	47	)	)	PUNCT
iajs-758	98	48	k	k	PROPN
iajs-758	98	49	annrm	annrm	PROPN
iajs-758	98	50	for	for	ADP
iajs-758	98	51	some	some	DET
iajs-758	98	52	k	k	PROPN
iajs-758	98	53			NOUN
iajs-758	98	54	z+	z+	NUM
iajs-758	98	55	and	and	CCONJ
iajs-758	98	56	hence	hence	ADV
iajs-758	98	57	mannr	mannr	NOUN
iajs-758	98	58	r	r	PROPN
iajs-758	98	59	.	.	PUNCT
iajs-758	99	1	thus	thus	ADV
iajs-758	99	2	,	,	PUNCT
iajs-758	99	3	mannnann	mannnann	PROPN
iajs-758	99	4	rr	rr	PROPN
iajs-758	99	5			PROPN
iajs-758	99	6	…	…	PUNCT
iajs-758	99	7	..	..	PUNCT
iajs-758	99	8	(	(	PUNCT
iajs-758	99	9	2	2	NUM
iajs-758	99	10	)	)	PUNCT
iajs-758	99	11	.	.	PUNCT
iajs-758	100	1	therefore	therefore	ADV
iajs-758	100	2	,	,	PUNCT
iajs-758	100	3	by	by	ADP
iajs-758	100	4	(	(	PUNCT
iajs-758	100	5	1	1	NUM
iajs-758	100	6	)	)	PUNCT
iajs-758	100	7	and	and	CCONJ
iajs-758	100	8	(	(	PUNCT
iajs-758	100	9	2	2	X
iajs-758	100	10	)	)	PUNCT
iajs-758	100	11	we	we	PRON
iajs-758	100	12	get	get	VERB
iajs-758	100	13	nannmann	nannmann	PROPN
iajs-758	100	14	rr	rr	NOUN
iajs-758	100	15			NOUN
iajs-758	100	16	.	.	PUNCT
iajs-758	101	1	thus	thus	ADV
iajs-758	101	2	nannr	nannr	NOUN
iajs-758	101	3	is	be	AUX
iajs-758	101	4	a	a	DET
iajs-758	101	5	maximal	maximal	ADJ
iajs-758	101	6	ideal	ideal	NOUN
iajs-758	101	7	and	and	CCONJ
iajs-758	101	8	so	so	ADV
iajs-758	101	9	by	by	ADP
iajs-758	101	10	definition	definition	NOUN
iajs-758	101	11	(	(	PUNCT
iajs-758	101	12	2.1	2.1	NUM
iajs-758	101	13	)	)	PUNCT
iajs-758	101	14	,	,	PUNCT
iajs-758	101	15	m	m	PROPN
iajs-758	101	16	is	be	AUX
iajs-758	101	17	a	a	DET
iajs-758	101	18	max	max	PROPN
iajs-758	101	19	–	–	PUNCT
iajs-758	101	20	module	module	NOUN
iajs-758	101	21	.	.	PUNCT
iajs-758	102	1	we	we	PRON
iajs-758	102	2	note	note	VERB
iajs-758	102	3	that	that	SCONJ
iajs-758	102	4	if	if	SCONJ
iajs-758	102	5	m	m	NOUN
iajs-758	102	6	is	be	AUX
iajs-758	102	7	a	a	DET
iajs-758	102	8	max	max	PROPN
iajs-758	102	9	–	–	PUNCT
iajs-758	102	10	module	module	NOUN
iajs-758	102	11	,	,	PUNCT
iajs-758	102	12	then	then	ADV
iajs-758	102	13	it	it	PRON
iajs-758	102	14	is	be	AUX
iajs-758	102	15	not	not	PART
iajs-758	102	16	necessary	necessary	ADJ
iajs-758	102	17	that	that	SCONJ
iajs-758	102	18	r	r	NOUN
iajs-758	102	19	is	be	AUX
iajs-758	102	20	a	a	DET
iajs-758	102	21	max	max	PROPN
iajs-758	102	22	–	–	PUNCT
iajs-758	102	23	ring	ring	NOUN
iajs-758	102	24	,	,	PUNCT
iajs-758	102	25	for	for	ADP
iajs-758	102	26	example	example	NOUN
iajs-758	102	27	:	:	PUNCT
iajs-758	102	28	the	the	DET
iajs-758	102	29	z	z	PROPN
iajs-758	102	30	–	–	PUNCT
iajs-758	102	31	module	module	NOUN
iajs-758	102	32	z2	z2	NOUN
iajs-758	102	33	is	be	AUX
iajs-758	102	34	max	max	PROPN
iajs-758	102	35	–	–	PUNCT
iajs-758	102	36	module	module	NOUN
iajs-758	102	37	,	,	PUNCT
iajs-758	102	38	but	but	CCONJ
iajs-758	102	39	z	z	NOUN
iajs-758	102	40	is	be	AUX
iajs-758	102	41	not	not	PART
iajs-758	102	42	max	max	PROPN
iajs-758	102	43	–	–	PUNCT
iajs-758	102	44	ring	ring	NOUN
iajs-758	102	45	.	.	PUNCT
iajs-758	103	1	moreover	moreover	ADV
iajs-758	103	2	,	,	PUNCT
iajs-758	103	3	if	if	SCONJ
iajs-758	103	4	r	r	NOUN
iajs-758	103	5	is	be	AUX
iajs-758	103	6	a	a	DET
iajs-758	103	7	max	max	PROPN
iajs-758	103	8	–	–	PUNCT
iajs-758	103	9	ring	ring	NOUN
iajs-758	103	10	and	and	CCONJ
iajs-758	103	11	m	m	NOUN
iajs-758	103	12	is	be	AUX
iajs-758	103	13	an	an	DET
iajs-758	103	14	r	r	NOUN
iajs-758	103	15	–	–	PUNCT
iajs-758	103	16	module	module	NOUN
iajs-758	103	17	,	,	PUNCT
iajs-758	103	18	then	then	ADV
iajs-758	103	19	m	m	NOUN
iajs-758	103	20	is	be	AUX
iajs-758	103	21	not	not	PART
iajs-758	103	22	necessarily	necessarily	ADV
iajs-758	103	23	max	max	NOUN
iajs-758	103	24	–	–	PUNCT
iajs-758	103	25	module	module	NOUN
iajs-758	103	26	,	,	PUNCT
iajs-758	103	27	for	for	ADP
iajs-758	103	28	example	example	NOUN
iajs-758	103	29	:	:	PUNCT
iajs-758	103	30	consider	consider	VERB
iajs-758	103	31	the	the	DET
iajs-758	103	32	z2	z2	NOUN
iajs-758	103	33	–	–	PUNCT
iajs-758	103	34	module	module	NOUN
iajs-758	103	35	z6	z6	NOUN
iajs-758	103	36	,	,	PUNCT
iajs-758	103	37	z2	z2	PROPN
iajs-758	103	38	is	be	AUX
iajs-758	103	39	a	a	DET
iajs-758	103	40	max	max	PROPN
iajs-758	103	41	–	–	PUNCT
iajs-758	103	42	ring	ring	NOUN
iajs-758	103	43	,	,	PUNCT
iajs-758	103	44	but	but	CCONJ
iajs-758	103	45	z6	z6	PROPN
iajs-758	103	46	,	,	PUNCT
iajs-758	103	47	is	be	AUX
iajs-758	103	48	not	not	PART
iajs-758	103	49	max	max	PROPN
iajs-758	103	50	–	–	PUNCT
iajs-758	103	51	module	module	NOUN
iajs-758	103	52	.	.	PUNCT
iajs-758	104	1	recall	recall	VERB
iajs-758	104	2	that	that	SCONJ
iajs-758	104	3	an	an	DET
iajs-758	104	4	r	r	NOUN
iajs-758	104	5	–	–	PUNCT
iajs-758	104	6	module	module	NOUN
iajs-758	104	7	m	m	NOUN
iajs-758	104	8	is	be	AUX
iajs-758	104	9	called	call	VERB
iajs-758	104	10	faithful	faithful	ADJ
iajs-758	104	11	r	r	NOUN
iajs-758	104	12	–	–	PUNCT
iajs-758	104	13	module	module	NOUN
iajs-758	104	14	if	if	SCONJ
iajs-758	104	15	annrm	annrm	NOUN
iajs-758	104	16	=	=	NOUN
iajs-758	104	17	0	0	X
iajs-758	104	18	.	.	PUNCT
iajs-758	105	1	ibn	ibn	PROPN
iajs-758	105	2	alhaitham	alhaitham	PROPN
iajs-758	105	3	j.	j.	PROPN
iajs-758	105	4	for	for	ADP
iajs-758	105	5	pure	pure	ADJ
iajs-758	105	6	&	&	CCONJ
iajs-758	105	7	appl	appl	PROPN
iajs-758	105	8	.	.	PUNCT
iajs-758	106	1	sci	sci	PROPN
iajs-758	106	2	.	.	PUNCT
iajs-758	106	3	vol.24	vol.24	NOUN
iajs-758	106	4	(	(	PUNCT
iajs-758	106	5	2	2	NUM
iajs-758	106	6	)	)	PUNCT
iajs-758	106	7	2011	2011	NUM
iajs-758	106	8	however	however	ADV
iajs-758	106	9	,	,	PUNCT
iajs-758	106	10	in	in	ADP
iajs-758	106	11	the	the	DET
iajs-758	106	12	class	class	NOUN
iajs-758	106	13	of	of	ADP
iajs-758	106	14	faithful	faithful	ADJ
iajs-758	106	15	multiplication	multiplication	NOUN
iajs-758	106	16	module	module	NOUN
iajs-758	106	17	,	,	PUNCT
iajs-758	106	18	they	they	PRON
iajs-758	106	19	are	be	AUX
iajs-758	106	20	equivalent	equivalent	ADJ
iajs-758	106	21	as	as	ADP
iajs-758	106	22	the	the	DET
iajs-758	106	23	following	following	ADJ
iajs-758	106	24	result	result	NOUN
iajs-758	106	25	shows	show	VERB
iajs-758	106	26	.	.	PUNCT
iajs-758	107	1	proposition	proposition	NOUN
iajs-758	107	2	2.8	2.8	NUM
iajs-758	107	3	if	if	SCONJ
iajs-758	107	4	m	m	NOUN
iajs-758	107	5	is	be	AUX
iajs-758	107	6	a	a	DET
iajs-758	107	7	faithful	faithful	ADJ
iajs-758	107	8	multiplication	multiplication	NOUN
iajs-758	107	9	r	r	NOUN
iajs-758	107	10	–	–	PUNCT
iajs-758	107	11	module	module	NOUN
iajs-758	107	12	,	,	PUNCT
iajs-758	107	13	then	then	ADV
iajs-758	107	14	m	m	NOUN
iajs-758	107	15	is	be	AUX
iajs-758	107	16	a	a	DET
iajs-758	107	17	max	max	NOUN
iajs-758	107	18	–	–	PUNCT
iajs-758	107	19	module	module	NOUN
iajs-758	107	20	if	if	SCONJ
iajs-758	107	21	and	and	CCONJ
iajs-758	107	22	only	only	ADV
iajs-758	107	23	if	if	SCONJ
iajs-758	107	24	r	r	NOUN
iajs-758	107	25	is	be	AUX
iajs-758	107	26	a	a	DET
iajs-758	107	27	max	max	PROPN
iajs-758	107	28	–	–	PUNCT
iajs-758	107	29	ring	ring	NOUN
iajs-758	107	30	.	.	PUNCT
iajs-758	108	1	proof	proof	NOUN
iajs-758	108	2	:	:	PUNCT
iajs-758	108	3	if	if	SCONJ
iajs-758	108	4	m	m	NOUN
iajs-758	108	5	is	be	AUX
iajs-758	108	6	a	a	DET
iajs-758	108	7	max	max	PROPN
iajs-758	108	8	–	–	PUNCT
iajs-758	108	9	module	module	NOUN
iajs-758	108	10	.	.	PUNCT
iajs-758	109	1	to	to	PART
iajs-758	109	2	prove	prove	VERB
iajs-758	109	3	r	r	NOUN
iajs-758	109	4	is	be	AUX
iajs-758	109	5	a	a	DET
iajs-758	109	6	max	max	PROPN
iajs-758	109	7	–	–	PUNCT
iajs-758	109	8	ring	ring	NOUN
iajs-758	109	9	.	.	PUNCT
iajs-758	110	1	let	let	VERB
iajs-758	110	2	i	i	PRON
iajs-758	110	3	be	be	AUX
iajs-758	110	4	a	a	DET
iajs-758	110	5	non	non	ADJ
iajs-758	110	6	–	–	PUNCT
iajs-758	110	7	zero	zero	NUM
iajs-758	110	8	ideal	ideal	NOUN
iajs-758	110	9	of	of	ADP
iajs-758	110	10	r.	r.	PROPN
iajs-758	110	11	then	then	ADV
iajs-758	110	12	n	n	PROPN
iajs-758	110	13	=	=	SYM
iajs-758	111	1	i	i	PRON
iajs-758	111	2	m	m	VERB
iajs-758	111	3	is	be	AUX
iajs-758	111	4	a	a	DET
iajs-758	111	5	non	non	ADJ
iajs-758	111	6	–	–	PUNCT
iajs-758	111	7	zero	zero	NUM
iajs-758	111	8	submodule	submodule	NOUN
iajs-758	111	9	of	of	ADP
iajs-758	111	10	m.	m.	NOUN
iajs-758	111	11	hence	hence	ADV
iajs-758	111	12	nannr	nannr	PROPN
iajs-758	111	13	is	be	AUX
iajs-758	111	14	a	a	DET
iajs-758	111	15	maximal	maximal	ADJ
iajs-758	111	16	ideal	ideal	NOUN
iajs-758	111	17	of	of	ADP
iajs-758	111	18	r	r	NOUN
iajs-758	111	19	because	because	SCONJ
iajs-758	111	20	m	m	PROPN
iajs-758	111	21	is	be	AUX
iajs-758	111	22	a	a	DET
iajs-758	111	23	max	max	PROPN
iajs-758	111	24	–	–	PUNCT
iajs-758	111	25	module	module	NOUN
iajs-758	111	26	.	.	PUNCT
iajs-758	112	1	on	on	ADP
iajs-758	112	2	the	the	DET
iajs-758	112	3	other	other	ADJ
iajs-758	112	4	hand	hand	NOUN
iajs-758	112	5	,	,	PUNCT
iajs-758	112	6	since	since	SCONJ
iajs-758	112	7	m	m	PROPN
iajs-758	112	8	is	be	AUX
iajs-758	112	9	a	a	DET
iajs-758	112	10	faithful	faithful	ADJ
iajs-758	112	11	multiplication	multiplication	NOUN
iajs-758	112	12	r	r	NOUN
iajs-758	112	13	–	–	PUNCT
iajs-758	112	14	module	module	NOUN
iajs-758	112	15	,	,	PUNCT
iajs-758	112	16	then	then	ADV
iajs-758	112	17	annrn	annrn	NOUN
iajs-758	112	18	=	=	SYM
iajs-758	112	19	annri	annri	PROPN
iajs-758	112	20	,	,	PUNCT
iajs-758	112	21	so	so	ADV
iajs-758	112	22	iannnann	iannnann	PROPN
iajs-758	112	23	rr	rr	PROPN
iajs-758	113	1			NOUN
iajs-758	113	2	.	.	PUNCT
iajs-758	114	1	thus	thus	ADV
iajs-758	114	2	iannr	iannr	NOUN
iajs-758	114	3	is	be	AUX
iajs-758	114	4	a	a	DET
iajs-758	114	5	maximal	maximal	ADJ
iajs-758	114	6	ideal	ideal	NOUN
iajs-758	114	7	and	and	CCONJ
iajs-758	114	8	r	r	NOUN
iajs-758	114	9	is	be	AUX
iajs-758	114	10	a	a	DET
iajs-758	114	11	max	max	PROPN
iajs-758	114	12	–	–	PUNCT
iajs-758	114	13	ring	ring	NOUN
iajs-758	114	14	.	.	PUNCT
iajs-758	115	1	conversely	conversely	ADV
iajs-758	115	2	,	,	PUNCT
iajs-758	115	3	if	if	SCONJ
iajs-758	115	4	r	r	NOUN
iajs-758	115	5	is	be	AUX
iajs-758	115	6	a	a	DET
iajs-758	115	7	max	max	PROPN
iajs-758	115	8	–	–	PUNCT
iajs-758	115	9	ring	ring	NOUN
iajs-758	115	10	,	,	PUNCT
iajs-758	115	11	to	to	PART
iajs-758	115	12	prove	prove	VERB
iajs-758	115	13	m	m	NOUN
iajs-758	115	14	is	be	AUX
iajs-758	115	15	a	a	DET
iajs-758	115	16	max	max	PROPN
iajs-758	115	17	–	–	PUNCT
iajs-758	115	18	module	module	NOUN
iajs-758	115	19	.	.	PUNCT
iajs-758	116	1	let	let	VERB
iajs-758	116	2	n	n	PRON
iajs-758	116	3	be	be	AUX
iajs-758	116	4	a	a	DET
iajs-758	116	5	non	non	ADJ
iajs-758	116	6	–	–	PUNCT
iajs-758	116	7	zero	zero	NUM
iajs-758	116	8	submodule	submodule	NOUN
iajs-758	116	9	of	of	ADP
iajs-758	116	10	m.	m.	NOUN
iajs-758	116	11	since	since	SCONJ
iajs-758	116	12	m	m	PROPN
iajs-758	116	13	is	be	AUX
iajs-758	116	14	a	a	DET
iajs-758	116	15	multiplication	multiplication	NOUN
iajs-758	116	16	r	r	NOUN
iajs-758	116	17	–	–	PUNCT
iajs-758	116	18	module	module	NOUN
iajs-758	116	19	,	,	PUNCT
iajs-758	116	20	3	3	NUM
iajs-758	116	21	.	.	X
iajs-758	117	1	some	some	DET
iajs-758	117	2	relations	relation	NOUN
iajs-758	117	3	between	between	ADP
iajs-758	117	4	max	max	PROPN
iajs-758	117	5	–	–	PUNCT
iajs-758	117	6	modules	module	NOUN
iajs-758	117	7	and	and	CCONJ
iajs-758	117	8	other	other	ADJ
iajs-758	117	9	modules	module	NOUN
iajs-758	117	10	in	in	ADP
iajs-758	117	11	this	this	DET
iajs-758	117	12	section	section	NOUN
iajs-758	117	13	,	,	PUNCT
iajs-758	117	14	we	we	PRON
iajs-758	117	15	study	study	VERB
iajs-758	117	16	the	the	DET
iajs-758	117	17	relationships	relationship	NOUN
iajs-758	117	18	between	between	ADP
iajs-758	117	19	max	max	PROPN
iajs-758	117	20	-	-	PUNCT
iajs-758	117	21	modules	module	NOUN
iajs-758	117	22	and	and	CCONJ
iajs-758	117	23	primary	primary	ADJ
iajs-758	117	24	modules	module	NOUN
iajs-758	117	25	and	and	CCONJ
iajs-758	117	26	prime	prime	ADJ
iajs-758	117	27	modules	module	NOUN
iajs-758	117	28	,	,	PUNCT
iajs-758	117	29	semi	semi	ADJ
iajs-758	117	30	-	-	ADJ
iajs-758	117	31	primary	primary	ADJ
iajs-758	117	32	,	,	PUNCT
iajs-758	117	33	quasi	quasi	ADJ
iajs-758	117	34	-	-	ADJ
iajs-758	117	35	primary	primary	ADJ
iajs-758	117	36	,	,	PUNCT
iajs-758	117	37	finitely	finitely	ADV
iajs-758	117	38	generated	generate	VERB
iajs-758	117	39	and	and	CCONJ
iajs-758	117	40	uniform	uniform	ADJ
iajs-758	117	41	modules	module	NOUN
iajs-758	117	42	.	.	PUNCT
iajs-758	118	1	we	we	PRON
iajs-758	118	2	start	start	VERB
iajs-758	118	3	with	with	ADP
iajs-758	118	4	the	the	DET
iajs-758	118	5	following	follow	VERB
iajs-758	118	6	definitions	definition	NOUN
iajs-758	118	7	which	which	PRON
iajs-758	118	8	are	be	AUX
iajs-758	118	9	needed	need	VERB
iajs-758	118	10	.	.	PUNCT
iajs-758	119	1	recall	recall	VERB
iajs-758	119	2	that	that	SCONJ
iajs-758	119	3	an	an	DET
iajs-758	119	4	r	r	NOUN
iajs-758	119	5	-	-	PUNCT
iajs-758	119	6	module	module	NOUN
iajs-758	119	7	m	m	NOUN
iajs-758	119	8	is	be	AUX
iajs-758	119	9	said	say	VERB
iajs-758	119	10	to	to	PART
iajs-758	119	11	be	be	AUX
iajs-758	119	12	a	a	DET
iajs-758	119	13	primary	primary	ADJ
iajs-758	119	14	module	module	NOUN
iajs-758	119	15	if	if	SCONJ
iajs-758	119	16	(	(	PUNCT
iajs-758	119	17	0	0	NUM
iajs-758	119	18	)	)	PUNCT
iajs-758	119	19	is	be	AUX
iajs-758	119	20	a	a	DET
iajs-758	119	21	primary	primary	ADJ
iajs-758	119	22	r	r	NOUN
iajs-758	119	23	–	–	PUNCT
iajs-758	119	24	submodule	submodule	NOUN
iajs-758	119	25	of	of	ADP
iajs-758	119	26	m	m	PRON
iajs-758	119	27	,	,	PUNCT
iajs-758	120	1	[	[	X
iajs-758	120	2	2	2	NUM
iajs-758	120	3	]	]	PUNCT
iajs-758	120	4	.	.	PUNCT
iajs-758	121	1	where	where	SCONJ
iajs-758	121	2	a	a	DET
iajs-758	121	3	submodule	submodule	NOUN
iajs-758	121	4	n	n	PROPN
iajs-758	121	5	of	of	ADP
iajs-758	121	6	an	an	DET
iajs-758	121	7	r	r	NOUN
iajs-758	121	8	–	–	PUNCT
iajs-758	121	9	module	module	NOUN
iajs-758	121	10	m	m	NOUN
iajs-758	121	11	is	be	AUX
iajs-758	121	12	called	call	VERB
iajs-758	121	13	a	a	DET
iajs-758	121	14	primary	primary	ADJ
iajs-758	121	15	submodule	submodule	NOUN
iajs-758	121	16	if	if	SCONJ
iajs-758	121	17	n	n	PROPN
iajs-758	121	18	≠	≠	PROPN
iajs-758	121	19	m	m	VERB
iajs-758	121	20	and	and	CCONJ
iajs-758	121	21	whenever	whenever	SCONJ
iajs-758	121	22	rx	rx	VERB
iajs-758	121	23			NOUN
iajs-758	121	24	n	n	CCONJ
iajs-758	121	25	for	for	ADP
iajs-758	121	26	r	r	NOUN
iajs-758	121	27			NOUN
iajs-758	121	28	r	r	NOUN
iajs-758	121	29	and	and	CCONJ
iajs-758	121	30	x	x	NOUN
iajs-758	121	31			NOUN
iajs-758	121	32	m	m	VERB
iajs-758	121	33	we	we	PRON
iajs-758	121	34	have	have	VERB
iajs-758	121	35	either	either	CCONJ
iajs-758	121	36	x	x	SYM
iajs-758	121	37			NOUN
iajs-758	121	38	n	n	CCONJ
iajs-758	121	39	or	or	CCONJ
iajs-758	121	40	r	r	NOUN
iajs-758	121	41	n	n	NOUN
iajs-758	121	42			NOUN
iajs-758	122	1	[	[	X
iajs-758	122	2	nr	nr	X
iajs-758	122	3	:	:	PUNCT
iajs-758	122	4	m	m	VERB
iajs-758	122	5	]	]	X
iajs-758	122	6	for	for	ADP
iajs-758	122	7	some	some	DET
iajs-758	122	8	n	n	ADJ
iajs-758	122	9			NOUN
iajs-758	122	10	z+	z+	NUM
iajs-758	122	11	,	,	PUNCT
iajs-758	122	12	where	where	SCONJ
iajs-758	122	13	[	[	X
iajs-758	122	14	nr	nr	X
iajs-758	122	15	:	:	PUNCT
iajs-758	122	16	m	m	VERB
iajs-758	122	17	]	]	X
iajs-758	122	18	=	=	X
iajs-758	122	19	{	{	PUNCT
iajs-758	122	20	r	r	NOUN
iajs-758	122	21	:	:	PUNCT
iajs-758	122	22	r	r	NOUN
iajs-758	122	23			NOUN
iajs-758	122	24	r	r	NOUN
iajs-758	122	25	^	^	PUNCT
iajs-758	122	26	rm	rm	PROPN
iajs-758	122	27			PROPN
iajs-758	122	28	n	n	CCONJ
iajs-758	122	29	}	}	PUNCT
iajs-758	122	30	,	,	PUNCT
iajs-758	122	31	[	[	X
iajs-758	122	32	8	8	NUM
iajs-758	122	33	]	]	PUNCT
iajs-758	122	34	.	.	PUNCT
iajs-758	123	1	by	by	ADP
iajs-758	123	2	using	use	VERB
iajs-758	123	3	this	this	DET
iajs-758	123	4	concept	concept	NOUN
iajs-758	123	5	,	,	PUNCT
iajs-758	123	6	we	we	PRON
iajs-758	123	7	have	have	VERB
iajs-758	123	8	the	the	DET
iajs-758	123	9	following	follow	VERB
iajs-758	123	10	:	:	PUNCT
iajs-758	123	11	remark	remark	VERB
iajs-758	123	12	3.1	3.1	NUM
iajs-758	123	13	every	every	DET
iajs-758	123	14	max	max	PROPN
iajs-758	123	15	–	–	PUNCT
iajs-758	123	16	module	module	NOUN
iajs-758	123	17	is	be	AUX
iajs-758	123	18	a	a	DET
iajs-758	123	19	primary	primary	ADJ
iajs-758	123	20	module	module	NOUN
iajs-758	123	21	.	.	PUNCT
iajs-758	124	1	proof	proof	NOUN
iajs-758	124	2	:	:	PUNCT
iajs-758	124	3	let	let	VERB
iajs-758	124	4	n	n	PRON
iajs-758	124	5	be	be	AUX
iajs-758	124	6	a	a	DET
iajs-758	124	7	non	non	ADJ
iajs-758	124	8	–	–	PUNCT
iajs-758	124	9	zero	zero	NUM
iajs-758	124	10	submodule	submodule	NOUN
iajs-758	124	11	of	of	ADP
iajs-758	124	12	an	an	DET
iajs-758	124	13	r	r	NOUN
iajs-758	124	14	–	–	PUNCT
iajs-758	124	15	module	module	NOUN
iajs-758	124	16	m.	m.	NOUN
iajs-758	124	17	suppose	suppose	VERB
iajs-758	124	18	that	that	SCONJ
iajs-758	124	19	m	m	PROPN
iajs-758	124	20	is	be	AUX
iajs-758	124	21	a	a	DET
iajs-758	124	22	max	max	PROPN
iajs-758	124	23	–	–	PUNCT
iajs-758	124	24	module	module	NOUN
iajs-758	124	25	,	,	PUNCT
iajs-758	124	26	to	to	PART
iajs-758	124	27	prove	prove	VERB
iajs-758	124	28	m	m	NOUN
iajs-758	124	29	is	be	AUX
iajs-758	124	30	a	a	DET
iajs-758	124	31	primary	primary	ADJ
iajs-758	124	32	module	module	NOUN
iajs-758	124	33	.	.	PUNCT
iajs-758	125	1	since	since	SCONJ
iajs-758	125	2	m	m	PROPN
iajs-758	125	3	is	be	AUX
iajs-758	125	4	a	a	DET
iajs-758	125	5	max	max	PROPN
iajs-758	125	6	–	–	PUNCT
iajs-758	125	7	module	module	NOUN
iajs-758	125	8	,	,	PUNCT
iajs-758	125	9	then	then	ADV
iajs-758	125	10	nannr	nannr	NOUN
iajs-758	125	11	is	be	AUX
iajs-758	125	12	a	a	DET
iajs-758	125	13	maximal	maximal	ADJ
iajs-758	125	14	ideal	ideal	NOUN
iajs-758	125	15	of	of	ADP
iajs-758	125	16	r	r	NOUN
iajs-758	125	17	,	,	PUNCT
iajs-758	125	18	for	for	ADP
iajs-758	125	19	each	each	DET
iajs-758	125	20	non	non	ADJ
iajs-758	125	21	–	–	PUNCT
iajs-758	125	22	zero	zero	NUM
iajs-758	125	23	submodule	submodule	NOUN
iajs-758	125	24	n	n	PROPN
iajs-758	125	25	of	of	ADP
iajs-758	125	26	m	m	VERB
iajs-758	125	27	by	by	ADP
iajs-758	125	28	definition	definition	NOUN
iajs-758	125	29	(	(	PUNCT
iajs-758	125	30	2.1	2.1	NUM
iajs-758	125	31	)	)	PUNCT
iajs-758	125	32	and	and	CCONJ
iajs-758	125	33	so	so	ADV
iajs-758	125	34	mannr	mannr	NOUN
iajs-758	125	35	is	be	AUX
iajs-758	125	36	a	a	DET
iajs-758	125	37	maximal	maximal	ADJ
iajs-758	125	38	ideal	ideal	NOUN
iajs-758	125	39	of	of	ADP
iajs-758	125	40	r	r	NOUN
iajs-758	125	41	by	by	ADP
iajs-758	125	42	(	(	PUNCT
iajs-758	125	43	2.2,6	2.2,6	NUM
iajs-758	125	44	)	)	PUNCT
iajs-758	125	45	.	.	PUNCT
iajs-758	126	1	but	but	CCONJ
iajs-758	126	2	mannnann	mannnann	PROPN
iajs-758	126	3	rr	rr	PROPN
iajs-758	126	4			PROPN
iajs-758	126	5	so	so	ADV
iajs-758	126	6	mannnann	mannnann	PROPN
iajs-758	126	7	rr	rr	PROPN
iajs-758	126	8			PROPN
iajs-758	126	9	.	.	PUNCT
iajs-758	127	1	therefore	therefore	ADV
iajs-758	127	2	m	m	PROPN
iajs-758	127	3	is	be	AUX
iajs-758	127	4	a	a	DET
iajs-758	127	5	primary	primary	ADJ
iajs-758	127	6	r	r	NOUN
iajs-758	127	7	–	–	PUNCT
iajs-758	127	8	module	module	NOUN
iajs-758	127	9	by	by	ADP
iajs-758	127	10	(	(	PUNCT
iajs-758	127	11	2,theorem	2,theorem	NUM
iajs-758	127	12	(	(	PUNCT
iajs-758	127	13	2.1.3	2.1.3	NUM
iajs-758	127	14	)	)	PUNCT
iajs-758	127	15	,	,	PUNCT
iajs-758	127	16	chapter	chapter	NOUN
iajs-758	127	17	2	2	NUM
iajs-758	127	18	)	)	PUNCT
iajs-758	127	19	.	.	PUNCT
iajs-758	128	1	note	note	VERB
iajs-758	128	2	that	that	SCONJ
iajs-758	128	3	,	,	PUNCT
iajs-758	128	4	the	the	DET
iajs-758	128	5	converse	converse	NOUN
iajs-758	128	6	of	of	ADP
iajs-758	128	7	(	(	PUNCT
iajs-758	128	8	3.1	3.1	NUM
iajs-758	128	9	)	)	PUNCT
iajs-758	128	10	is	be	AUX
iajs-758	128	11	not	not	PART
iajs-758	128	12	true	true	ADJ
iajs-758	128	13	in	in	ADP
iajs-758	128	14	general	general	ADJ
iajs-758	128	15	.	.	PUNCT
iajs-758	129	1	for	for	ADP
iajs-758	129	2	example	example	NOUN
iajs-758	129	3	,	,	PUNCT
iajs-758	129	4	the	the	DET
iajs-758	129	5	z	z	NOUN
iajs-758	129	6	–	–	PUNCT
iajs-758	129	7	module	module	NOUN
iajs-758	129	8	m	m	NOUN
iajs-758	129	9	=	=	ADJ
iajs-758	129	10	z	z	NOUN
iajs-758	129	11			PROPN
iajs-758	129	12	z	z	NOUN
iajs-758	129	13	is	be	AUX
iajs-758	129	14	a	a	DET
iajs-758	129	15	primary	primary	NOUN
iajs-758	129	16	by	by	ADP
iajs-758	129	17	[	[	X
iajs-758	129	18	2	2	NUM
iajs-758	129	19	,	,	PUNCT
iajs-758	129	20	(	(	PUNCT
iajs-758	129	21	2.1.2	2.1.2	NUM
iajs-758	129	22	,	,	PUNCT
iajs-758	129	23	(	(	PUNCT
iajs-758	129	24	2	2	NUM
iajs-758	129	25	)	)	PUNCT
iajs-758	129	26	)	)	PUNCT
iajs-758	129	27	,	,	PUNCT
iajs-758	129	28	chapter	chapter	NOUN
iajs-758	129	29	2	2	NUM
iajs-758	129	30	]	]	PUNCT
iajs-758	129	31	,	,	PUNCT
iajs-758	129	32	but	but	CCONJ
iajs-758	129	33	it	it	PRON
iajs-758	129	34	is	be	AUX
iajs-758	129	35	not	not	PART
iajs-758	129	36	a	a	DET
iajs-758	129	37	max	max	PROPN
iajs-758	129	38	–	–	PUNCT
iajs-758	129	39	module	module	NOUN
iajs-758	129	40	.	.	PUNCT
iajs-758	130	1	in	in	ADP
iajs-758	130	2	the	the	DET
iajs-758	130	3	following	follow	VERB
iajs-758	130	4	proposition	proposition	NOUN
iajs-758	130	5	,	,	PUNCT
iajs-758	130	6	we	we	PRON
iajs-758	130	7	give	give	VERB
iajs-758	130	8	a	a	DET
iajs-758	130	9	sufficient	sufficient	ADJ
iajs-758	130	10	condition	condition	NOUN
iajs-758	130	11	under	under	ADP
iajs-758	130	12	which	which	PRON
iajs-758	130	13	the	the	DET
iajs-758	130	14	converse	converse	NOUN
iajs-758	130	15	of	of	ADP
iajs-758	130	16	(	(	PUNCT
iajs-758	130	17	3.1	3.1	NUM
iajs-758	130	18	)	)	PUNCT
iajs-758	130	19	is	be	AUX
iajs-758	130	20	true	true	ADJ
iajs-758	130	21	.	.	PUNCT
iajs-758	131	1	proposition	proposition	NOUN
iajs-758	131	2	3.2	3.2	NUM
iajs-758	131	3	let	let	VERB
iajs-758	131	4	m	m	NOUN
iajs-758	131	5	is	be	AUX
iajs-758	131	6	a	a	DET
iajs-758	131	7	module	module	NOUN
iajs-758	131	8	over	over	ADP
iajs-758	131	9	a	a	DET
iajs-758	131	10	pid	pid	NOUN
iajs-758	131	11	,	,	PUNCT
iajs-758	131	12	and	and	CCONJ
iajs-758	131	13	0	0	NUM
iajs-758	131	14	≠	≠	PROPN
iajs-758	131	15	annrm	annrm	NOUN
iajs-758	131	16	is	be	AUX
iajs-758	131	17	a	a	DET
iajs-758	131	18	primary	primary	ADJ
iajs-758	131	19	ideal	ideal	NOUN
iajs-758	131	20	of	of	ADP
iajs-758	131	21	r.	r.	PROPN
iajs-758	131	22	if	if	SCONJ
iajs-758	131	23	m	m	PROPN
iajs-758	131	24	is	be	AUX
iajs-758	131	25	a	a	DET
iajs-758	131	26	primary	primary	ADJ
iajs-758	131	27	r	r	NOUN
iajs-758	131	28	–	–	PUNCT
iajs-758	131	29	module	module	NOUN
iajs-758	131	30	,	,	PUNCT
iajs-758	131	31	then	then	ADV
iajs-758	131	32	m	m	NOUN
iajs-758	131	33	is	be	AUX
iajs-758	131	34	a	a	DET
iajs-758	131	35	max	max	PROPN
iajs-758	131	36	–	–	PUNCT
iajs-758	131	37	module	module	NOUN
iajs-758	131	38	.	.	PUNCT
iajs-758	132	1	proof	proof	NOUN
iajs-758	132	2	:	:	PUNCT
iajs-758	132	3	let	let	VERB
iajs-758	132	4	n	n	PRON
iajs-758	132	5	be	be	AUX
iajs-758	132	6	a	a	DET
iajs-758	132	7	non	non	ADJ
iajs-758	132	8	-	-	ADJ
iajs-758	132	9	zero	zero	NUM
iajs-758	132	10	r	r	NOUN
iajs-758	132	11	–	–	PUNCT
iajs-758	132	12	submodule	submodule	NOUN
iajs-758	132	13	of	of	ADP
iajs-758	132	14	m	m	PROPN
iajs-758	132	15	,	,	PUNCT
iajs-758	132	16	to	to	PART
iajs-758	132	17	prove	prove	VERB
iajs-758	132	18	nannr	nannr	NOUN
iajs-758	132	19	is	be	AUX
iajs-758	132	20	a	a	DET
iajs-758	132	21	maximal	maximal	ADJ
iajs-758	132	22	ideal	ideal	NOUN
iajs-758	132	23	.	.	PUNCT
iajs-758	133	1	since	since	SCONJ
iajs-758	133	2	m	m	PROPN
iajs-758	133	3	is	be	AUX
iajs-758	133	4	a	a	DET
iajs-758	133	5	module	module	NOUN
iajs-758	133	6	over	over	ADP
iajs-758	133	7	a	a	DET
iajs-758	133	8	pid	pid	NOUN
iajs-758	133	9	,	,	PUNCT
iajs-758	133	10	then	then	ADV
iajs-758	133	11	the	the	DET
iajs-758	133	12	only	only	ADJ
iajs-758	133	13	primary	primary	ADJ
iajs-758	133	14	ideals	ideal	NOUN
iajs-758	133	15	in	in	ADP
iajs-758	133	16	r	r	NOUN
iajs-758	133	17	are	be	AUX
iajs-758	133	18	(	(	PUNCT
iajs-758	133	19	0	0	NUM
iajs-758	133	20	)	)	PUNCT
iajs-758	133	21	and	and	CCONJ
iajs-758	133	22	<	<	X
iajs-758	133	23	pn	pn	X
iajs-758	133	24	>	>	X
iajs-758	133	25	for	for	ADP
iajs-758	133	26	some	some	DET
iajs-758	133	27	a	a	DET
iajs-758	133	28	prime	prime	ADJ
iajs-758	133	29	element	element	NOUN
iajs-758	133	30	p	p	NOUN
iajs-758	133	31	and	and	CCONJ
iajs-758	133	32	n	n	CCONJ
iajs-758	133	33			NOUN
iajs-758	133	34	z+	z+	NUM
iajs-758	133	35	.	.	PUNCT
iajs-758	134	1	but	but	CCONJ
iajs-758	134	2	0	0	NUM
iajs-758	134	3	≠	≠	PROPN
iajs-758	134	4	annrm	annrm	NOUN
iajs-758	134	5	is	be	AUX
iajs-758	134	6	a	a	DET
iajs-758	134	7	primary	primary	ADJ
iajs-758	134	8	ideal	ideal	NOUN
iajs-758	134	9	,	,	PUNCT
iajs-758	134	10	so	so	SCONJ
iajs-758	134	11	annrm	annrm	PROPN
iajs-758	134	12	=	=	PUNCT
iajs-758	135	1	<	<	X
iajs-758	135	2	p	p	X
iajs-758	135	3	n	n	X
iajs-758	135	4	>	>	PUNCT
iajs-758	135	5	,	,	PUNCT
iajs-758	135	6	and	and	CCONJ
iajs-758	135	7	this	this	PRON
iajs-758	135	8	implies	imply	VERB
iajs-758	135	9			PROPN
iajs-758	135	10	ppmann	ppmann	NOUN
iajs-758	135	11	n	n	PRON
iajs-758	135	12	r	r	NOUN
iajs-758	135	13	which	which	PRON
iajs-758	135	14	is	be	AUX
iajs-758	135	15	a	a	DET
iajs-758	135	16	maximal	maximal	ADJ
iajs-758	135	17	ideal	ideal	NOUN
iajs-758	135	18	.	.	PUNCT
iajs-758	136	1	but	but	CCONJ
iajs-758	136	2	m	m	PROPN
iajs-758	136	3	is	be	AUX
iajs-758	136	4	a	a	DET
iajs-758	136	5	primary	primary	NOUN
iajs-758	136	6	,	,	PUNCT
iajs-758	136	7	then	then	ADV
iajs-758	136	8	mannnann	mannnann	PROPN
iajs-758	136	9	rr	rr	PROPN
iajs-758	136	10			NOUN
iajs-758	136	11	by	by	ADP
iajs-758	136	12	[	[	X
iajs-758	136	13	2	2	NUM
iajs-758	136	14	,	,	PUNCT
iajs-758	136	15	theorem	theorem	ADJ
iajs-758	136	16	(	(	PUNCT
iajs-758	136	17	2.1.3	2.1.3	NUM
iajs-758	136	18	)	)	PUNCT
iajs-758	136	19	,	,	PUNCT
iajs-758	136	20	chapter	chapter	NOUN
iajs-758	136	21	2	2	NUM
iajs-758	136	22	]	]	PUNCT
iajs-758	136	23	.	.	PUNCT
iajs-758	137	1	hence	hence	ADV
iajs-758	137	2	nannr	nannr	PROPN
iajs-758	137	3	is	be	AUX
iajs-758	137	4	a	a	DET
iajs-758	137	5	maximal	maximal	ADJ
iajs-758	137	6	ideal	ideal	NOUN
iajs-758	137	7	and	and	CCONJ
iajs-758	137	8	so	so	ADV
iajs-758	137	9	by	by	ADP
iajs-758	137	10	definition	definition	NOUN
iajs-758	137	11	(	(	PUNCT
iajs-758	137	12	2.1	2.1	NUM
iajs-758	137	13	)	)	PUNCT
iajs-758	138	1	m	m	VERB
iajs-758	138	2	is	be	AUX
iajs-758	138	3	a	a	DET
iajs-758	138	4	max	max	PROPN
iajs-758	138	5	–	–	PUNCT
iajs-758	138	6	module	module	NOUN
iajs-758	138	7	.	.	PUNCT
iajs-758	139	1	in	in	ADP
iajs-758	139	2	the	the	DET
iajs-758	139	3	following	follow	VERB
iajs-758	139	4	result	result	NOUN
iajs-758	139	5	,	,	PUNCT
iajs-758	139	6	we	we	PRON
iajs-758	139	7	give	give	VERB
iajs-758	139	8	another	another	DET
iajs-758	139	9	condition	condition	NOUN
iajs-758	139	10	for	for	ADP
iajs-758	139	11	which	which	PRON
iajs-758	139	12	a	a	DET
iajs-758	139	13	primary	primary	ADJ
iajs-758	139	14	module	module	NOUN
iajs-758	139	15	be	be	AUX
iajs-758	139	16	a	a	DET
iajs-758	139	17	max	max	NOUN
iajs-758	139	18	–	–	PUNCT
iajs-758	139	19	module	module	NOUN
iajs-758	139	20	.	.	PUNCT
iajs-758	140	1	but	but	CCONJ
iajs-758	140	2	first	first	ADV
iajs-758	140	3	we	we	PRON
iajs-758	140	4	need	need	VERB
iajs-758	140	5	the	the	DET
iajs-758	140	6	following	follow	VERB
iajs-758	140	7	definition	definition	NOUN
iajs-758	140	8	.	.	PUNCT
iajs-758	141	1	the	the	DET
iajs-758	141	2	dimension	dimension	NOUN
iajs-758	141	3	of	of	ADP
iajs-758	141	4	r	r	NOUN
iajs-758	141	5	,	,	PUNCT
iajs-758	141	6	denoted	denote	VERB
iajs-758	141	7	by	by	ADP
iajs-758	141	8	dim	dim	ADJ
iajs-758	141	9	r	r	NOUN
iajs-758	141	10	,	,	PUNCT
iajs-758	141	11	is	be	AUX
iajs-758	141	12	defined	define	VERB
iajs-758	141	13	to	to	PART
iajs-758	141	14	be	be	AUX
iajs-758	141	15	:	:	PUNCT
iajs-758	141	16	sup	sup	NUM
iajs-758	141	17	{	{	PUNCT
iajs-758	141	18	n	n	CCONJ
iajs-758	141	19			NOUN
iajs-758	141	20	n	n	CCONJ
iajs-758	141	21	:	:	PUNCT
iajs-758	141	22	there	there	PRON
iajs-758	141	23	exists	exist	VERB
iajs-758	141	24	a	a	DET
iajs-758	141	25	chain	chain	NOUN
iajs-758	141	26	of	of	ADP
iajs-758	141	27	prime	prime	ADJ
iajs-758	141	28	ideals	ideal	NOUN
iajs-758	141	29	of	of	ADP
iajs-758	141	30	r	r	NOUN
iajs-758	141	31	of	of	ADP
iajs-758	141	32	length	length	NOUN
iajs-758	141	33	n	n	CCONJ
iajs-758	141	34	,	,	PUNCT
iajs-758	141	35	if	if	SCONJ
iajs-758	141	36	the	the	DET
iajs-758	141	37	supremum	supremum	ADJ
iajs-758	141	38	exists	exist	VERB
iajs-758	141	39	,	,	PUNCT
iajs-758	141	40	and	and	CCONJ
iajs-758	141	41	∞	∞	NUM
iajs-758	141	42	,	,	PUNCT
iajs-758	141	43	otherwise	otherwise	ADV
iajs-758	141	44	}	}	PUNCT
iajs-758	141	45	,	,	PUNCT
iajs-758	141	46	[	[	X
iajs-758	141	47	1	1	NUM
iajs-758	141	48	]	]	PUNCT
iajs-758	141	49	.	.	PUNCT
iajs-758	142	1	ibn	ibn	PROPN
iajs-758	142	2	alhaitham	alhaitham	PROPN
iajs-758	142	3	j.	j.	PROPN
iajs-758	142	4	for	for	ADP
iajs-758	142	5	pure	pure	ADJ
iajs-758	142	6	&	&	CCONJ
iajs-758	142	7	appl	appl	PROPN
iajs-758	142	8	.	.	PUNCT
iajs-758	143	1	sci	sci	PROPN
iajs-758	143	2	.	.	PUNCT
iajs-758	143	3	vol.24	vol.24	NOUN
iajs-758	143	4	(	(	PUNCT
iajs-758	143	5	2	2	NUM
iajs-758	143	6	)	)	PUNCT
iajs-758	143	7	2011	2011	NUM
iajs-758	143	8	proposition	proposition	NOUN
iajs-758	143	9	3.3	3.3	NUM
iajs-758	143	10	let	let	VERB
iajs-758	143	11	r	r	PRON
iajs-758	143	12	be	be	AUX
iajs-758	143	13	a	a	DET
iajs-758	143	14	0	0	NUM
iajs-758	143	15	–	–	PUNCT
iajs-758	143	16	dimensional	dimensional	ADJ
iajs-758	143	17	ring	ring	NOUN
iajs-758	143	18	.	.	PUNCT
iajs-758	144	1	then	then	ADV
iajs-758	144	2	a	a	DET
iajs-758	144	3	primary	primary	ADJ
iajs-758	144	4	r	r	NOUN
iajs-758	144	5	–	–	PUNCT
iajs-758	144	6	module	module	NOUN
iajs-758	144	7	m	m	NOUN
iajs-758	144	8	is	be	AUX
iajs-758	144	9	a	a	DET
iajs-758	144	10	max	max	PROPN
iajs-758	144	11	–	–	PUNCT
iajs-758	144	12	module	module	NOUN
iajs-758	144	13	.	.	PUNCT
iajs-758	145	1	proof	proof	NOUN
iajs-758	145	2	:	:	PUNCT
iajs-758	145	3	since	since	SCONJ
iajs-758	145	4	m	m	PROPN
iajs-758	145	5	is	be	AUX
iajs-758	145	6	a	a	DET
iajs-758	145	7	primary	primary	ADJ
iajs-758	145	8	module	module	NOUN
iajs-758	145	9	,	,	PUNCT
iajs-758	145	10	so	so	SCONJ
iajs-758	145	11	annrm	annrm	NOUN
iajs-758	145	12	is	be	AUX
iajs-758	145	13	a	a	DET
iajs-758	145	14	primary	primary	ADJ
iajs-758	145	15	ideal	ideal	NOUN
iajs-758	145	16	of	of	ADP
iajs-758	145	17	r	r	NOUN
iajs-758	145	18	by	by	ADP
iajs-758	145	19	(	(	PUNCT
iajs-758	145	20	2	2	NUM
iajs-758	145	21	,	,	PUNCT
iajs-758	145	22	corollary	corollary	ADJ
iajs-758	145	23	2.1.7	2.1.7	NUM
iajs-758	145	24	,	,	PUNCT
iajs-758	145	25	chapter	chapter	NOUN
iajs-758	145	26	2	2	NUM
iajs-758	145	27	)	)	PUNCT
iajs-758	145	28	and	and	CCONJ
iajs-758	145	29	hence	hence	ADV
iajs-758	145	30	mannp	mannp	NOUN
iajs-758	145	31	r	r	VERB
iajs-758	145	32	is	be	AUX
iajs-758	145	33	a	a	DET
iajs-758	145	34	prime	prime	ADJ
iajs-758	145	35	ideal	ideal	NOUN
iajs-758	145	36	.	.	PUNCT
iajs-758	146	1	but	but	CCONJ
iajs-758	146	2	dim	dim	ADJ
iajs-758	146	3	r	r	NOUN
iajs-758	146	4	=	=	SYM
iajs-758	146	5	0	0	NUM
iajs-758	146	6	implies	imply	VERB
iajs-758	146	7	that	that	SCONJ
iajs-758	146	8	p	p	PROPN
iajs-758	146	9	is	be	AUX
iajs-758	146	10	a	a	DET
iajs-758	146	11	maximal	maximal	ADJ
iajs-758	146	12	ideal	ideal	NOUN
iajs-758	146	13	.	.	PUNCT
iajs-758	147	1	on	on	ADP
iajs-758	147	2	the	the	DET
iajs-758	147	3	other	other	ADJ
iajs-758	147	4	hand	hand	NOUN
iajs-758	147	5	,	,	PUNCT
iajs-758	147	6	nannmann	nannmann	PROPN
iajs-758	147	7	rr	rr	NOUN
iajs-758	147	8			PROPN
iajs-758	147	9	for	for	ADP
iajs-758	147	10	every	every	DET
iajs-758	147	11	non	non	ADJ
iajs-758	147	12	–	–	PUNCT
iajs-758	147	13	zero	zero	NUM
iajs-758	147	14	submodule	submodule	NOUN
iajs-758	147	15	n	n	PROPN
iajs-758	147	16	of	of	ADP
iajs-758	147	17	m	m	PROPN
iajs-758	147	18	(	(	PUNCT
iajs-758	147	19	since	since	SCONJ
iajs-758	147	20	m	m	PROPN
iajs-758	147	21	is	be	AUX
iajs-758	147	22	a	a	DET
iajs-758	147	23	primary	primary	ADJ
iajs-758	147	24	module	module	NOUN
iajs-758	147	25	)	)	PUNCT
iajs-758	147	26	,	,	PUNCT
iajs-758	147	27	so	so	SCONJ
iajs-758	147	28	that	that	SCONJ
iajs-758	147	29	nannr	nannr	NOUN
iajs-758	147	30	is	be	AUX
iajs-758	147	31	a	a	DET
iajs-758	147	32	maximal	maximal	ADJ
iajs-758	147	33	ideal	ideal	NOUN
iajs-758	147	34	.	.	PUNCT
iajs-758	148	1	therefore	therefore	ADV
iajs-758	148	2	m	m	PROPN
iajs-758	148	3	is	be	AUX
iajs-758	148	4	a	a	DET
iajs-758	148	5	max	max	PROPN
iajs-758	148	6	–	–	PUNCT
iajs-758	148	7	module	module	NOUN
iajs-758	148	8	.	.	PUNCT
iajs-758	149	1	now	now	ADV
iajs-758	149	2	,	,	PUNCT
iajs-758	149	3	we	we	PRON
iajs-758	149	4	study	study	VERB
iajs-758	149	5	the	the	DET
iajs-758	149	6	relation	relation	NOUN
iajs-758	149	7	between	between	ADP
iajs-758	149	8	max	max	PROPN
iajs-758	149	9	-	-	PUNCT
iajs-758	149	10	modules	module	NOUN
iajs-758	149	11	and	and	CCONJ
iajs-758	149	12	prime	prime	ADJ
iajs-758	149	13	modules	module	NOUN
iajs-758	149	14	.	.	PUNCT
iajs-758	150	1	but	but	CCONJ
iajs-758	150	2	first	first	ADV
iajs-758	150	3	we	we	PRON
iajs-758	150	4	need	need	VERB
iajs-758	150	5	the	the	DET
iajs-758	150	6	following	follow	VERB
iajs-758	150	7	definitions	definition	NOUN
iajs-758	150	8	:	:	PUNCT
iajs-758	150	9	recall	recall	VERB
iajs-758	150	10	an	an	DET
iajs-758	150	11	r	r	NOUN
iajs-758	150	12	–	–	PUNCT
iajs-758	150	13	module	module	NOUN
iajs-758	150	14	m	m	NOUN
iajs-758	150	15	is	be	AUX
iajs-758	150	16	said	say	VERB
iajs-758	150	17	to	to	PART
iajs-758	150	18	be	be	AUX
iajs-758	150	19	a	a	DET
iajs-758	150	20	prime	prime	ADJ
iajs-758	150	21	module	module	NOUN
iajs-758	150	22	if	if	SCONJ
iajs-758	150	23	(	(	PUNCT
iajs-758	150	24	0	0	NUM
iajs-758	150	25	)	)	PUNCT
iajs-758	150	26	is	be	AUX
iajs-758	150	27	a	a	DET
iajs-758	150	28	prime	prime	ADJ
iajs-758	150	29	r	r	NOUN
iajs-758	150	30	–	–	PUNCT
iajs-758	150	31	submodule	submodule	NOUN
iajs-758	150	32	of	of	ADP
iajs-758	150	33	m	m	PROPN
iajs-758	150	34	,	,	PUNCT
iajs-758	150	35	see	see	VERB
iajs-758	150	36	[	[	X
iajs-758	150	37	9	9	NUM
iajs-758	150	38	]	]	PUNCT
iajs-758	150	39	.	.	PUNCT
iajs-758	151	1	we	we	PRON
iajs-758	151	2	notice	notice	VERB
iajs-758	151	3	that	that	SCONJ
iajs-758	151	4	not	not	PART
iajs-758	151	5	every	every	DET
iajs-758	151	6	max	max	NOUN
iajs-758	151	7	–	–	PUNCT
iajs-758	151	8	module	module	NOUN
iajs-758	151	9	is	be	AUX
iajs-758	151	10	a	a	DET
iajs-758	151	11	prime	prime	ADJ
iajs-758	151	12	–	–	PUNCT
iajs-758	151	13	module	module	NOUN
iajs-758	151	14	,	,	PUNCT
iajs-758	151	15	for	for	ADP
iajs-758	151	16	example	example	NOUN
iajs-758	151	17	:	:	PUNCT
iajs-758	151	18	the	the	DET
iajs-758	151	19	z	z	NOUN
iajs-758	151	20	–	–	PUNCT
iajs-758	151	21	module	module	NOUN
iajs-758	151	22	z4	z4	PROPN
iajs-758	151	23	is	be	AUX
iajs-758	151	24	max	max	NOUN
iajs-758	151	25	by	by	ADP
iajs-758	151	26	proposition	proposition	NOUN
iajs-758	151	27	(	(	PUNCT
iajs-758	151	28	2.6	2.6	NUM
iajs-758	151	29	)	)	PUNCT
iajs-758	151	30	,	,	PUNCT
iajs-758	151	31	but	but	CCONJ
iajs-758	151	32	it	it	PRON
iajs-758	151	33	is	be	AUX
iajs-758	151	34	not	not	PART
iajs-758	151	35	a	a	DET
iajs-758	151	36	prime	prime	ADJ
iajs-758	151	37	z	z	NOUN
iajs-758	151	38	-	-	PUNCT
iajs-758	151	39	module	module	NOUN
iajs-758	151	40	by	by	ADP
iajs-758	151	41	[	[	X
iajs-758	151	42	5	5	NUM
iajs-758	151	43	,	,	PUNCT
iajs-758	151	44	(	(	PUNCT
iajs-758	151	45	1.1.3	1.1.3	NUM
iajs-758	151	46	(	(	PUNCT
iajs-758	151	47	3	3	NUM
iajs-758	151	48	)	)	PUNCT
iajs-758	151	49	)	)	PUNCT
iajs-758	151	50	,	,	PUNCT
iajs-758	151	51	chapter	chapter	NOUN
iajs-758	151	52	1	1	NUM
iajs-758	151	53	]	]	PUNCT
iajs-758	151	54	.	.	PUNCT
iajs-758	152	1	the	the	DET
iajs-758	152	2	following	follow	VERB
iajs-758	152	3	proposition	proposition	NOUN
iajs-758	152	4	shows	show	VERB
iajs-758	152	5	that	that	SCONJ
iajs-758	152	6	(	(	PUNCT
iajs-758	152	7	annrm	annrm	NOUN
iajs-758	152	8	is	be	AUX
iajs-758	152	9	a	a	DET
iajs-758	152	10	semi	semi	ADJ
iajs-758	152	11	–	–	PUNCT
iajs-758	152	12	prime	prime	ADJ
iajs-758	152	13	ideal	ideal	NOUN
iajs-758	152	14	)	)	PUNCT
iajs-758	152	15	is	be	AUX
iajs-758	152	16	a	a	DET
iajs-758	152	17	sufficient	sufficient	ADJ
iajs-758	152	18	condition	condition	NOUN
iajs-758	152	19	for	for	ADP
iajs-758	152	20	max	max	PROPN
iajs-758	152	21	–	–	PUNCT
iajs-758	152	22	module	module	NOUN
iajs-758	152	23	to	to	PART
iajs-758	152	24	be	be	AUX
iajs-758	152	25	prime	prime	ADJ
iajs-758	152	26	.	.	PUNCT
iajs-758	153	1	proposition	proposition	NOUN
iajs-758	153	2	3.4	3.4	NUM
iajs-758	153	3	if	if	SCONJ
iajs-758	153	4	m	m	NOUN
iajs-758	153	5	is	be	AUX
iajs-758	153	6	a	a	DET
iajs-758	153	7	max	max	NOUN
iajs-758	153	8	-	-	PUNCT
iajs-758	153	9	module	module	NOUN
iajs-758	153	10	and	and	CCONJ
iajs-758	153	11	annrm	annrm	NOUN
iajs-758	153	12	is	be	AUX
iajs-758	153	13	a	a	DET
iajs-758	153	14	semi	semi	ADJ
iajs-758	153	15	–	–	PUNCT
iajs-758	153	16	prime	prime	ADJ
iajs-758	153	17	ideal	ideal	NOUN
iajs-758	153	18	of	of	ADP
iajs-758	153	19	r	r	NOUN
iajs-758	153	20	,	,	PUNCT
iajs-758	153	21	then	then	ADV
iajs-758	153	22	m	m	NOUN
iajs-758	153	23	is	be	AUX
iajs-758	153	24	a	a	DET
iajs-758	153	25	prime	prime	ADJ
iajs-758	153	26	r	r	NOUN
iajs-758	153	27	–	–	PUNCT
iajs-758	153	28	module	module	NOUN
iajs-758	153	29	.	.	PUNCT
iajs-758	154	1	proof	proof	NOUN
iajs-758	154	2	:	:	PUNCT
iajs-758	154	3	since	since	SCONJ
iajs-758	154	4	m	m	PROPN
iajs-758	154	5	is	be	AUX
iajs-758	154	6	a	a	DET
iajs-758	154	7	max	max	PROPN
iajs-758	154	8	–	–	PUNCT
iajs-758	154	9	module	module	NOUN
iajs-758	154	10	,	,	PUNCT
iajs-758	154	11	then	then	ADV
iajs-758	154	12	m	m	NOUN
iajs-758	154	13	is	be	AUX
iajs-758	154	14	a	a	DET
iajs-758	154	15	primary	primary	ADJ
iajs-758	154	16	r	r	NOUN
iajs-758	154	17	–	–	PUNCT
iajs-758	154	18	module	module	NOUN
iajs-758	154	19	by	by	ADP
iajs-758	154	20	(	(	PUNCT
iajs-758	154	21	3.1	3.1	NUM
iajs-758	154	22	)	)	PUNCT
iajs-758	154	23	.	.	PUNCT
iajs-758	155	1	but	but	CCONJ
iajs-758	155	2	annrm	annrm	NOUN
iajs-758	155	3	is	be	AUX
iajs-758	155	4	a	a	DET
iajs-758	155	5	semi	semi	ADJ
iajs-758	155	6	–	–	PUNCT
iajs-758	155	7	prime	prime	ADJ
iajs-758	155	8	ideal	ideal	NOUN
iajs-758	155	9	of	of	ADP
iajs-758	155	10	r	r	NOUN
iajs-758	155	11	,	,	PUNCT
iajs-758	155	12	hence	hence	ADV
iajs-758	155	13	by	by	ADP
iajs-758	155	14	[	[	X
iajs-758	155	15	2	2	NUM
iajs-758	155	16	,	,	PUNCT
iajs-758	155	17	proposition	proposition	NOUN
iajs-758	155	18	(	(	PUNCT
iajs-758	155	19	2.3.2	2.3.2	NUM
iajs-758	155	20	)	)	PUNCT
iajs-758	155	21	,	,	PUNCT
iajs-758	155	22	chapter	chapter	NOUN
iajs-758	155	23	2	2	NUM
iajs-758	155	24	]	]	PUNCT
iajs-758	155	25	,	,	PUNCT
iajs-758	155	26	m	m	VERB
iajs-758	155	27	is	be	AUX
iajs-758	155	28	a	a	DET
iajs-758	155	29	prime	prime	ADJ
iajs-758	155	30	r	r	NOUN
iajs-758	155	31	–	–	PUNCT
iajs-758	155	32	module	module	NOUN
iajs-758	155	33	.	.	PUNCT
iajs-758	156	1	next	next	ADJ
iajs-758	156	2	,	,	PUNCT
iajs-758	156	3	a	a	DET
iajs-758	156	4	proper	proper	ADJ
iajs-758	156	5	submodule	submodule	NOUN
iajs-758	156	6	n	n	PROPN
iajs-758	156	7	of	of	ADP
iajs-758	156	8	m	m	PROPN
iajs-758	156	9	is	be	AUX
iajs-758	156	10	called	call	VERB
iajs-758	156	11	semi	semi	ADJ
iajs-758	156	12	–	–	PUNCT
iajs-758	156	13	prime	prime	ADJ
iajs-758	156	14	submodule	submodule	NOUN
iajs-758	156	15	if	if	SCONJ
iajs-758	156	16	for	for	ADP
iajs-758	156	17	every	every	DET
iajs-758	156	18	r	r	NOUN
iajs-758	156	19			NOUN
iajs-758	156	20	r	r	NOUN
iajs-758	156	21	,	,	PUNCT
iajs-758	156	22	x	x	SYM
iajs-758	156	23			PROPN
iajs-758	156	24	m	m	PROPN
iajs-758	156	25	,	,	PUNCT
iajs-758	156	26	k	k	PROPN
iajs-758	156	27			PROPN
iajs-758	156	28	z+	z+	NUM
iajs-758	156	29	,	,	PUNCT
iajs-758	156	30	such	such	ADJ
iajs-758	156	31	that	that	SCONJ
iajs-758	156	32	r	r	NOUN
iajs-758	156	33	k	k	PROPN
iajs-758	156	34	x	x	X
iajs-758	156	35			NOUN
iajs-758	156	36	n	n	CCONJ
iajs-758	156	37	,	,	PUNCT
iajs-758	156	38	than	than	SCONJ
iajs-758	156	39	rx	rx	VERB
iajs-758	156	40			PROPN
iajs-758	156	41	n	n	CCONJ
iajs-758	156	42	,	,	PUNCT
iajs-758	156	43	see	see	VERB
iajs-758	156	44	[	[	X
iajs-758	156	45	7	7	X
iajs-758	156	46	]	]	PUNCT
iajs-758	156	47	.	.	PUNCT
iajs-758	157	1	by	by	ADP
iajs-758	157	2	using	use	VERB
iajs-758	157	3	this	this	DET
iajs-758	157	4	concept	concept	NOUN
iajs-758	157	5	,	,	PUNCT
iajs-758	157	6	we	we	PRON
iajs-758	157	7	have	have	VERB
iajs-758	157	8	the	the	DET
iajs-758	157	9	following	follow	VERB
iajs-758	157	10	:	:	PUNCT
iajs-758	157	11	corollary	corollary	ADJ
iajs-758	157	12	3.5	3.5	NUM
iajs-758	157	13	if	if	SCONJ
iajs-758	157	14	m	m	NOUN
iajs-758	157	15	is	be	AUX
iajs-758	157	16	a	a	DET
iajs-758	157	17	max	max	PROPN
iajs-758	157	18	–	–	PUNCT
iajs-758	157	19	module	module	NOUN
iajs-758	157	20	and	and	CCONJ
iajs-758	157	21	(	(	PUNCT
iajs-758	157	22	0	0	NUM
iajs-758	157	23	)	)	PUNCT
iajs-758	157	24	is	be	AUX
iajs-758	157	25	a	a	DET
iajs-758	157	26	semi	semi	ADJ
iajs-758	157	27	–	–	PUNCT
iajs-758	157	28	prime	prime	ADJ
iajs-758	157	29	submodule	submodule	NOUN
iajs-758	157	30	,	,	PUNCT
iajs-758	157	31	then	then	ADV
iajs-758	157	32	m	m	VERB
iajs-758	157	33	is	be	AUX
iajs-758	157	34	a	a	DET
iajs-758	157	35	prime	prime	ADJ
iajs-758	157	36	r	r	NOUN
iajs-758	157	37	-	-	PUNCT
iajs-758	157	38	module	module	NOUN
iajs-758	157	39	.	.	PUNCT
iajs-758	158	1	proof	proof	NOUN
iajs-758	158	2	:	:	PUNCT
iajs-758	158	3	since	since	SCONJ
iajs-758	158	4	(	(	PUNCT
iajs-758	158	5	0	0	NUM
iajs-758	158	6	)	)	PUNCT
iajs-758	158	7	is	be	AUX
iajs-758	158	8	a	a	DET
iajs-758	158	9	semi	semi	ADJ
iajs-758	158	10	-	-	ADJ
iajs-758	158	11	prime	prime	ADJ
iajs-758	158	12	submodule	submodule	NOUN
iajs-758	158	13	,	,	PUNCT
iajs-758	158	14	so	so	SCONJ
iajs-758	158	15	annrm	annrm	NOUN
iajs-758	158	16	is	be	AUX
iajs-758	158	17	a	a	DET
iajs-758	158	18	semi	semi	ADJ
iajs-758	158	19	–	–	PUNCT
iajs-758	158	20	prime	prime	ADJ
iajs-758	158	21	ideal	ideal	NOUN
iajs-758	158	22	by	by	ADP
iajs-758	158	23	[	[	X
iajs-758	158	24	8	8	NUM
iajs-758	158	25	,	,	PUNCT
iajs-758	158	26	proposition	proposition	NOUN
iajs-758	158	27	(	(	PUNCT
iajs-758	158	28	1	1	NUM
iajs-758	158	29	-	-	SYM
iajs-758	158	30	5	5	NUM
iajs-758	158	31	)	)	PUNCT
iajs-758	158	32	,	,	PUNCT
iajs-758	158	33	chapter	chapter	NOUN
iajs-758	158	34	2	2	NUM
iajs-758	158	35	]	]	PUNCT
iajs-758	158	36	,	,	PUNCT
iajs-758	158	37	hence	hence	ADV
iajs-758	158	38	the	the	DET
iajs-758	158	39	result	result	NOUN
iajs-758	158	40	follows	follow	VERB
iajs-758	158	41	by	by	ADP
iajs-758	158	42	(	(	PUNCT
iajs-758	158	43	3.4	3.4	NUM
iajs-758	158	44	)	)	PUNCT
iajs-758	158	45	.	.	PUNCT
iajs-758	159	1	recall	recall	VERB
iajs-758	159	2	an	an	DET
iajs-758	159	3	r	r	NOUN
iajs-758	159	4	–	–	PUNCT
iajs-758	159	5	module	module	NOUN
iajs-758	159	6	m	m	NOUN
iajs-758	159	7	is	be	AUX
iajs-758	159	8	said	say	VERB
iajs-758	159	9	to	to	PART
iajs-758	159	10	be	be	AUX
iajs-758	159	11	a	a	DET
iajs-758	159	12	semi	semi	ADJ
iajs-758	159	13	–	–	PUNCT
iajs-758	159	14	primary	primary	ADJ
iajs-758	159	15	if	if	SCONJ
iajs-758	159	16	(	(	PUNCT
iajs-758	159	17	0	0	NUM
iajs-758	159	18	)	)	PUNCT
iajs-758	159	19	is	be	AUX
iajs-758	159	20	a	a	DET
iajs-758	159	21	semi	semi	ADJ
iajs-758	159	22	–	–	PUNCT
iajs-758	159	23	primary	primary	ADJ
iajs-758	159	24	r	r	NOUN
iajs-758	159	25	–	–	PUNCT
iajs-758	159	26	submodule	submodule	NOUN
iajs-758	159	27	,	,	PUNCT
iajs-758	159	28	(	(	PUNCT
iajs-758	159	29	2	2	NUM
iajs-758	159	30	)	)	PUNCT
iajs-758	159	31	.	.	PUNCT
iajs-758	160	1	it	it	PRON
iajs-758	160	2	is	be	AUX
iajs-758	160	3	well	well	ADV
iajs-758	160	4	known	know	VERB
iajs-758	160	5	that	that	SCONJ
iajs-758	160	6	every	every	DET
iajs-758	160	7	primary	primary	ADJ
iajs-758	160	8	r	r	NOUN
iajs-758	160	9	–	–	PUNCT
iajs-758	160	10	module	module	NOUN
iajs-758	160	11	is	be	AUX
iajs-758	160	12	a	a	DET
iajs-758	160	13	semi	semi	ADJ
iajs-758	160	14	–	–	PUNCT
iajs-758	160	15	primary	primary	ADJ
iajs-758	160	16	module	module	NOUN
iajs-758	160	17	[	[	X
iajs-758	160	18	2	2	NUM
iajs-758	160	19	,	,	PUNCT
iajs-758	160	20	(	(	PUNCT
iajs-758	160	21	3.5.3	3.5.3	NUM
iajs-758	160	22	,	,	PUNCT
iajs-758	160	23	(	(	PUNCT
iajs-758	160	24	2	2	NUM
iajs-758	160	25	)	)	PUNCT
iajs-758	160	26	)	)	PUNCT
iajs-758	160	27	,	,	PUNCT
iajs-758	160	28	chapter	chapter	NOUN
iajs-758	160	29	3	3	NUM
iajs-758	160	30	]	]	PUNCT
iajs-758	160	31	.	.	PUNCT
iajs-758	161	1	so	so	ADV
iajs-758	161	2	that	that	SCONJ
iajs-758	161	3	following	follow	VERB
iajs-758	161	4	result	result	NOUN
iajs-758	161	5	follows	follow	VERB
iajs-758	161	6	immediately	immediately	ADV
iajs-758	161	7	from	from	ADP
iajs-758	161	8	(	(	PUNCT
iajs-758	161	9	3.1	3.1	NUM
iajs-758	161	10	)	)	PUNCT
iajs-758	161	11	.	.	PUNCT
iajs-758	162	1	corollary	corollary	ADJ
iajs-758	162	2	3.6	3.6	NUM
iajs-758	162	3	every	every	DET
iajs-758	162	4	max	max	PROPN
iajs-758	162	5	-	-	PUNCT
iajs-758	162	6	module	module	NOUN
iajs-758	162	7	is	be	AUX
iajs-758	162	8	a	a	DET
iajs-758	162	9	semi	semi	ADJ
iajs-758	162	10	–	–	PUNCT
iajs-758	162	11	primary	primary	ADJ
iajs-758	162	12	r	r	NOUN
iajs-758	162	13	–	–	PUNCT
iajs-758	162	14	module	module	NOUN
iajs-758	162	15	.	.	PUNCT
iajs-758	163	1	note	note	VERB
iajs-758	163	2	that	that	SCONJ
iajs-758	163	3	the	the	DET
iajs-758	163	4	converse	converse	NOUN
iajs-758	163	5	of	of	ADP
iajs-758	163	6	(	(	PUNCT
iajs-758	163	7	3.6	3.6	NUM
iajs-758	163	8	)	)	PUNCT
iajs-758	163	9	is	be	AUX
iajs-758	163	10	not	not	PART
iajs-758	163	11	true	true	ADJ
iajs-758	163	12	in	in	ADP
iajs-758	163	13	general	general	ADJ
iajs-758	163	14	.	.	PUNCT
iajs-758	164	1	for	for	ADP
iajs-758	164	2	example	example	NOUN
iajs-758	164	3	,	,	PUNCT
iajs-758	164	4	the	the	DET
iajs-758	164	5	z	z	NOUN
iajs-758	164	6	–	–	PUNCT
iajs-758	164	7	module	module	NOUN
iajs-758	164	8	m	m	NOUN
iajs-758	164	9	=	=	NOUN
iajs-758	164	10	z	z	PROPN
iajs-758	164	11			PROPN
iajs-758	164	12	z12	z12	PROPN
iajs-758	164	13	is	be	AUX
iajs-758	164	14	a	a	DET
iajs-758	164	15	semi	semi	ADJ
iajs-758	164	16	–	–	NOUN
iajs-758	164	17	primary	primary	ADJ
iajs-758	164	18	,	,	PUNCT
iajs-758	164	19	but	but	CCONJ
iajs-758	164	20	not	not	PART
iajs-758	164	21	a	a	DET
iajs-758	164	22	max	max	PROPN
iajs-758	164	23	–	–	PUNCT
iajs-758	164	24	module	module	NOUN
iajs-758	164	25	.	.	PUNCT
iajs-758	165	1	recall	recall	VERB
iajs-758	165	2	that	that	SCONJ
iajs-758	165	3	an	an	DET
iajs-758	165	4	r	r	NOUN
iajs-758	165	5	-	-	PUNCT
iajs-758	165	6	module	module	NOUN
iajs-758	165	7	m	m	NOUN
iajs-758	165	8	is	be	AUX
iajs-758	165	9	said	say	VERB
iajs-758	165	10	to	to	PART
iajs-758	165	11	be	be	AUX
iajs-758	165	12	a	a	DET
iajs-758	165	13	quasi	quasi	ADJ
iajs-758	165	14	–	–	PUNCT
iajs-758	165	15	primary	primary	ADJ
iajs-758	165	16	module	module	NOUN
iajs-758	165	17	if	if	SCONJ
iajs-758	165	18	annrn	annrn	NOUN
iajs-758	165	19	is	be	AUX
iajs-758	165	20	a	a	DET
iajs-758	165	21	primary	primary	ADJ
iajs-758	165	22	ideal	ideal	NOUN
iajs-758	165	23	of	of	ADP
iajs-758	165	24	r	r	NOUN
iajs-758	165	25	,	,	PUNCT
iajs-758	165	26	for	for	ADP
iajs-758	165	27	each	each	DET
iajs-758	165	28	non	non	ADJ
iajs-758	165	29	–	–	PUNCT
iajs-758	165	30	zero	zero	NUM
iajs-758	165	31	submodule	submodule	NOUN
iajs-758	165	32	n	n	PROPN
iajs-758	165	33	of	of	ADP
iajs-758	165	34	m	m	PRON
iajs-758	165	35	,	,	PUNCT
iajs-758	165	36	[	[	X
iajs-758	165	37	2	2	NUM
iajs-758	165	38	]	]	PUNCT
iajs-758	165	39	.	.	PUNCT
iajs-758	166	1	however	however	ADV
iajs-758	166	2	,	,	PUNCT
iajs-758	166	3	we	we	PRON
iajs-758	166	4	have	have	VERB
iajs-758	166	5	the	the	DET
iajs-758	166	6	following	follow	VERB
iajs-758	166	7	:	:	PUNCT
iajs-758	166	8	remark	remark	VERB
iajs-758	166	9	3.7	3.7	NUM
iajs-758	166	10	every	every	DET
iajs-758	166	11	max	max	PROPN
iajs-758	166	12	-	-	PUNCT
iajs-758	166	13	module	module	NOUN
iajs-758	166	14	is	be	AUX
iajs-758	166	15	a	a	DET
iajs-758	166	16	quasi	quasi	ADJ
iajs-758	166	17	–	–	PUNCT
iajs-758	166	18	primary	primary	ADJ
iajs-758	166	19	module	module	NOUN
iajs-758	166	20	.	.	PUNCT
iajs-758	167	1	proof	proof	NOUN
iajs-758	167	2	:	:	PUNCT
iajs-758	167	3	since	since	SCONJ
iajs-758	167	4	m	m	PROPN
iajs-758	167	5	is	be	AUX
iajs-758	167	6	a	a	DET
iajs-758	167	7	max	max	PROPN
iajs-758	167	8	–	–	PUNCT
iajs-758	167	9	module	module	NOUN
iajs-758	167	10	,	,	PUNCT
iajs-758	167	11	then	then	ADV
iajs-758	167	12	nannr	nannr	NOUN
iajs-758	167	13	is	be	AUX
iajs-758	167	14	a	a	DET
iajs-758	167	15	maximal	maximal	ADJ
iajs-758	167	16	ideal	ideal	NOUN
iajs-758	167	17	of	of	ADP
iajs-758	167	18	r	r	NOUN
iajs-758	167	19	for	for	ADP
iajs-758	167	20	each	each	DET
iajs-758	167	21	non	non	ADJ
iajs-758	167	22	–	–	PUNCT
iajs-758	167	23	zero	zero	NUM
iajs-758	167	24	submodule	submodule	NOUN
iajs-758	167	25	n	n	PROPN
iajs-758	167	26	of	of	ADP
iajs-758	167	27	m	m	PROPN
iajs-758	167	28	.	.	PUNCT
iajs-758	168	1	hence	hence	ADV
iajs-758	168	2	annrn	annrn	NOUN
iajs-758	168	3	is	be	AUX
iajs-758	168	4	a	a	DET
iajs-758	168	5	primary	primary	ADJ
iajs-758	168	6	ideal	ideal	NOUN
iajs-758	168	7	by	by	ADP
iajs-758	168	8	[	[	X
iajs-758	168	9	1	1	NUM
iajs-758	168	10	,	,	PUNCT
iajs-758	168	11	proposition	proposition	NOUN
iajs-758	168	12	4.9	4.9	NUM
iajs-758	168	13	,	,	PUNCT
iajs-758	168	14	p.	p.	NOUN
iajs-758	168	15	64	64	NUM
iajs-758	168	16	]	]	PUNCT
iajs-758	168	17	,	,	PUNCT
iajs-758	168	18	and	and	CCONJ
iajs-758	168	19	so	so	ADV
iajs-758	168	20	m	m	VERB
iajs-758	168	21	is	be	AUX
iajs-758	168	22	a	a	DET
iajs-758	168	23	quasi	quasi	ADJ
iajs-758	168	24	–	–	NOUN
iajs-758	168	25	primary	primary	ADJ
iajs-758	168	26	.	.	PUNCT
iajs-758	169	1	note	note	VERB
iajs-758	169	2	that	that	SCONJ
iajs-758	169	3	,	,	PUNCT
iajs-758	169	4	the	the	DET
iajs-758	169	5	converse	converse	NOUN
iajs-758	169	6	of	of	ADP
iajs-758	169	7	(	(	PUNCT
iajs-758	169	8	3.7	3.7	NUM
iajs-758	169	9	)	)	PUNCT
iajs-758	169	10	is	be	AUX
iajs-758	169	11	not	not	PART
iajs-758	169	12	true	true	ADJ
iajs-758	169	13	in	in	ADP
iajs-758	169	14	general	general	ADJ
iajs-758	169	15	,	,	PUNCT
iajs-758	169	16	for	for	ADP
iajs-758	169	17	example	example	NOUN
iajs-758	169	18	,	,	PUNCT
iajs-758	169	19	the	the	DET
iajs-758	169	20	z	z	NOUN
iajs-758	169	21	–	–	PUNCT
iajs-758	169	22	module	module	NOUN
iajs-758	169	23	z	z	NOUN
iajs-758	169	24	is	be	AUX
iajs-758	169	25	a	a	DET
iajs-758	169	26	quasi	quasi	NOUN
iajs-758	169	27	–	–	NOUN
iajs-758	169	28	primary	primary	ADJ
iajs-758	169	29	since	since	SCONJ
iajs-758	169	30	annz(n	annz(n	NOUN
iajs-758	169	31	)	)	PUNCT
iajs-758	169	32	=	=	SYM
iajs-758	169	33	0	0	NUM
iajs-758	169	34	is	be	AUX
iajs-758	169	35	a	a	DET
iajs-758	169	36	prime	prime	ADJ
iajs-758	169	37	ideal	ideal	NOUN
iajs-758	169	38	,	,	PUNCT
iajs-758	169	39	for	for	ADP
iajs-758	169	40	each	each	DET
iajs-758	169	41	non	non	ADJ
iajs-758	169	42	–	–	PUNCT
iajs-758	169	43	zero	zero	NUM
iajs-758	169	44	n	n	NOUN
iajs-758	169	45	of	of	ADP
iajs-758	169	46	z	z	PROPN
iajs-758	169	47	,	,	PUNCT
iajs-758	169	48	so	so	ADV
iajs-758	169	49	it	it	PRON
iajs-758	169	50	is	be	AUX
iajs-758	169	51	a	a	DET
iajs-758	169	52	primary	primary	ADJ
iajs-758	169	53	ideal	ideal	NOUN
iajs-758	169	54	.	.	PUNCT
iajs-758	170	1	but	but	CCONJ
iajs-758	170	2	it	it	PRON
iajs-758	170	3	is	be	AUX
iajs-758	170	4	not	not	PART
iajs-758	170	5	a	a	DET
iajs-758	170	6	max	max	NOUN
iajs-758	170	7	–	–	PUNCT
iajs-758	170	8	module	module	NOUN
iajs-758	170	9	by	by	ADP
iajs-758	170	10	[	[	X
iajs-758	170	11	2.2	2.2	NUM
iajs-758	170	12	,	,	PUNCT
iajs-758	170	13	(	(	PUNCT
iajs-758	170	14	2	2	NUM
iajs-758	170	15	)	)	PUNCT
iajs-758	170	16	]	]	PUNCT
iajs-758	170	17	.	.	PUNCT
iajs-758	171	1	we	we	PRON
iajs-758	171	2	notice	notice	VERB
iajs-758	171	3	that	that	SCONJ
iajs-758	171	4	not	not	PART
iajs-758	171	5	every	every	DET
iajs-758	171	6	max	max	NOUN
iajs-758	171	7	-	-	PUNCT
iajs-758	171	8	module	module	NOUN
iajs-758	171	9	is	be	AUX
iajs-758	171	10	finitely	finitely	ADV
iajs-758	171	11	generated	generate	VERB
iajs-758	171	12	,	,	PUNCT
iajs-758	171	13	for	for	ADP
iajs-758	171	14	example	example	NOUN
iajs-758	171	15	:	:	PUNCT
iajs-758	171	16	z	z	NOUN
iajs-758	171	17	as	as	SCONJ
iajs-758	171	18	a	a	DET
iajs-758	171	19	z	z	NOUN
iajs-758	171	20	-	-	PUNCT
iajs-758	171	21	module	module	NOUN
iajs-758	171	22	is	be	AUX
iajs-758	171	23	a	a	DET
iajs-758	171	24	max	max	NOUN
iajs-758	171	25	–	–	PUNCT
iajs-758	171	26	module	module	NOUN
iajs-758	171	27	but	but	CCONJ
iajs-758	171	28	not	not	PART
iajs-758	171	29	finitely	finitely	ADV
iajs-758	171	30	generated	generate	VERB
iajs-758	171	31	.	.	PUNCT
iajs-758	172	1	ibn	ibn	PROPN
iajs-758	172	2	alhaitham	alhaitham	PROPN
iajs-758	172	3	j.	j.	PROPN
iajs-758	172	4	for	for	ADP
iajs-758	172	5	pure	pure	ADJ
iajs-758	172	6	&	&	CCONJ
iajs-758	172	7	appl	appl	PROPN
iajs-758	172	8	.	.	PUNCT
iajs-758	173	1	sci	sci	PROPN
iajs-758	173	2	.	.	PUNCT
iajs-758	173	3	vol.24	vol.24	NOUN
iajs-758	173	4	(	(	PUNCT
iajs-758	173	5	2	2	NUM
iajs-758	173	6	)	)	PUNCT
iajs-758	173	7	2011	2011	NUM
iajs-758	173	8	however	however	ADV
iajs-758	173	9	,	,	PUNCT
iajs-758	173	10	we	we	PRON
iajs-758	173	11	have	have	VERB
iajs-758	173	12	the	the	DET
iajs-758	173	13	following	follow	VERB
iajs-758	173	14	proposition	proposition	NOUN
iajs-758	173	15	:	:	PUNCT
iajs-758	173	16	proposition	proposition	NOUN
iajs-758	173	17	3.8	3.8	NUM
iajs-758	173	18	if	if	SCONJ
iajs-758	173	19	m	m	NOUN
iajs-758	173	20	is	be	AUX
iajs-758	173	21	a	a	DET
iajs-758	173	22	multiplication	multiplication	NOUN
iajs-758	173	23	max	max	NOUN
iajs-758	173	24	-	-	PUNCT
iajs-758	173	25	module	module	NOUN
iajs-758	173	26	,	,	PUNCT
iajs-758	173	27	then	then	ADV
iajs-758	173	28	m	m	NOUN
iajs-758	173	29	is	be	AUX
iajs-758	173	30	a	a	DET
iajs-758	173	31	finitely	finitely	ADV
iajs-758	173	32	generated	generate	VERB
iajs-758	173	33	module	module	NOUN
iajs-758	173	34	.	.	PUNCT
iajs-758	174	1	proof	proof	NOUN
iajs-758	174	2	:	:	PUNCT
iajs-758	174	3	since	since	SCONJ
iajs-758	174	4	m	m	PROPN
iajs-758	174	5	is	be	AUX
iajs-758	174	6	a	a	DET
iajs-758	174	7	max	max	PROPN
iajs-758	174	8	–	–	PUNCT
iajs-758	174	9	module	module	NOUN
iajs-758	174	10	,	,	PUNCT
iajs-758	174	11	then	then	ADV
iajs-758	174	12	mannr	mannr	NOUN
iajs-758	174	13	is	be	AUX
iajs-758	174	14	a	a	DET
iajs-758	174	15	maximal	maximal	ADJ
iajs-758	174	16	ideal	ideal	NOUN
iajs-758	174	17	by	by	ADP
iajs-758	174	18	[	[	X
iajs-758	174	19	2.2	2.2	NUM
iajs-758	174	20	,	,	PUNCT
iajs-758	174	21	(	(	PUNCT
iajs-758	174	22	6	6	NUM
iajs-758	174	23	)	)	PUNCT
iajs-758	174	24	]	]	PUNCT
iajs-758	174	25	and	and	CCONJ
iajs-758	174	26	so	so	ADV
iajs-758	174	27	annrm	annrm	NOUN
iajs-758	174	28	is	be	AUX
iajs-758	174	29	a	a	DET
iajs-758	174	30	primary	primary	ADJ
iajs-758	174	31	ideal	ideal	NOUN
iajs-758	174	32	by	by	ADP
iajs-758	174	33	[	[	X
iajs-758	174	34	1	1	NUM
iajs-758	174	35	,	,	PUNCT
iajs-758	174	36	prop	prop	NOUN
iajs-758	174	37	.	.	PUNCT
iajs-758	174	38	4.9	4.9	NUM
iajs-758	174	39	,	,	PUNCT
iajs-758	174	40	p.	p.	NOUN
iajs-758	174	41	64	64	NUM
iajs-758	174	42	]	]	PUNCT
iajs-758	174	43	.	.	PUNCT
iajs-758	175	1	on	on	ADP
iajs-758	175	2	the	the	DET
iajs-758	175	3	other	other	ADJ
iajs-758	175	4	hand	hand	NOUN
iajs-758	175	5	m	m	NOUN
iajs-758	175	6	is	be	AUX
iajs-758	175	7	a	a	DET
iajs-758	175	8	multiplication	multiplication	NOUN
iajs-758	175	9	imply	imply	VERB
iajs-758	175	10	,	,	PUNCT
iajs-758	175	11	m	m	VERB
iajs-758	175	12	is	be	AUX
iajs-758	175	13	a	a	DET
iajs-758	175	14	finitely	finitely	ADV
iajs-758	175	15	generated	generate	VERB
iajs-758	175	16	by	by	ADP
iajs-758	175	17	[	[	X
iajs-758	175	18	5	5	NUM
iajs-758	175	19	,	,	PUNCT
iajs-758	175	20	prop.(2.7	prop.(2.7	NOUN
iajs-758	175	21	)	)	PUNCT
iajs-758	175	22	,	,	PUNCT
iajs-758	175	23	chapter	chapter	NOUN
iajs-758	175	24	2	2	NUM
iajs-758	175	25	]	]	PUNCT
iajs-758	175	26	.	.	PUNCT
iajs-758	176	1	now	now	ADV
iajs-758	176	2	,	,	PUNCT
iajs-758	176	3	we	we	PRON
iajs-758	176	4	study	study	VERB
iajs-758	176	5	the	the	DET
iajs-758	176	6	relation	relation	NOUN
iajs-758	176	7	between	between	ADP
iajs-758	176	8	max	max	PROPN
iajs-758	176	9	-	-	PUNCT
iajs-758	176	10	modules	module	NOUN
iajs-758	176	11	and	and	CCONJ
iajs-758	176	12	uniform	uniform	ADJ
iajs-758	176	13	modules	module	NOUN
iajs-758	176	14	.	.	PUNCT
iajs-758	177	1	but	but	CCONJ
iajs-758	177	2	first	first	ADV
iajs-758	177	3	we	we	PRON
iajs-758	177	4	need	need	VERB
iajs-758	177	5	the	the	DET
iajs-758	177	6	following	follow	VERB
iajs-758	177	7	definition	definition	NOUN
iajs-758	177	8	:	:	PUNCT
iajs-758	177	9	recall	recall	VERB
iajs-758	177	10	that	that	SCONJ
iajs-758	177	11	an	an	DET
iajs-758	177	12	r	r	NOUN
iajs-758	177	13	–	–	PUNCT
iajs-758	177	14	module	module	NOUN
iajs-758	177	15	m	m	NOUN
iajs-758	177	16	is	be	AUX
iajs-758	177	17	said	say	VERB
iajs-758	177	18	to	to	PART
iajs-758	177	19	be	be	AUX
iajs-758	177	20	uniform	uniform	ADJ
iajs-758	177	21	module	module	NOUN
iajs-758	177	22	if	if	SCONJ
iajs-758	177	23	every	every	DET
iajs-758	177	24	non	non	ADJ
iajs-758	177	25	–	–	PUNCT
iajs-758	177	26	zero	zero	NUM
iajs-758	177	27	submodule	submodule	NOUN
iajs-758	177	28	of	of	ADP
iajs-758	177	29	m	m	PROPN
iajs-758	177	30	is	be	AUX
iajs-758	177	31	essential	essential	ADJ
iajs-758	177	32	,	,	PUNCT
iajs-758	177	33	[	[	X
iajs-758	177	34	11	11	NUM
iajs-758	177	35	]	]	PUNCT
iajs-758	177	36	.	.	PUNCT
iajs-758	178	1	where	where	SCONJ
iajs-758	178	2	a	a	DET
iajs-758	178	3	submodule	submodule	NOUN
iajs-758	178	4	n	n	PROPN
iajs-758	178	5	of	of	ADP
iajs-758	178	6	an	an	DET
iajs-758	178	7	r	r	NOUN
iajs-758	178	8	–	–	PUNCT
iajs-758	178	9	module	module	NOUN
iajs-758	178	10	m	m	NOUN
iajs-758	178	11	is	be	AUX
iajs-758	178	12	called	call	VERB
iajs-758	178	13	essential	essential	ADJ
iajs-758	178	14	proved	prove	VERB
iajs-758	178	15	that	that	SCONJ
iajs-758	178	16	n	n	ADP
iajs-758	178	17	∩	∩	NOUN
iajs-758	178	18	k	k	PROPN
iajs-758	178	19	≠	≠	PROPN
iajs-758	178	20	0	0	NUM
iajs-758	178	21	for	for	ADP
iajs-758	178	22	every	every	DET
iajs-758	178	23	non	non	ADJ
iajs-758	178	24	–	–	PUNCT
iajs-758	178	25	zero	zero	NUM
iajs-758	178	26	submodule	submodule	NOUN
iajs-758	178	27	k	k	PROPN
iajs-758	178	28	of	of	ADP
iajs-758	178	29	m	m	PRON
iajs-758	178	30	,	,	PUNCT
iajs-758	178	31	[	[	X
iajs-758	178	32	11	11	NUM
iajs-758	178	33	]	]	PUNCT
iajs-758	178	34	.	.	PUNCT
iajs-758	179	1	note	note	VERB
iajs-758	179	2	that	that	SCONJ
iajs-758	179	3	,	,	PUNCT
iajs-758	179	4	it	it	PRON
iajs-758	179	5	is	be	AUX
iajs-758	179	6	not	not	PART
iajs-758	179	7	necessary	necessary	ADJ
iajs-758	179	8	that	that	SCONJ
iajs-758	179	9	every	every	DET
iajs-758	179	10	uniform	uniform	ADJ
iajs-758	179	11	r	r	NOUN
iajs-758	179	12	–	–	PUNCT
iajs-758	179	13	module	module	NOUN
iajs-758	179	14	is	be	AUX
iajs-758	179	15	a	a	DET
iajs-758	179	16	max	max	NOUN
iajs-758	179	17	–	–	PUNCT
iajs-758	179	18	module	module	NOUN
iajs-758	179	19	for	for	ADP
iajs-758	179	20	example	example	NOUN
iajs-758	180	1	=	=	PUNCT
iajs-758	180	2	q	q	X
iajs-758	180	3	as	as	ADP
iajs-758	180	4	a	a	DET
iajs-758	180	5	z	z	NOUN
iajs-758	180	6	–	–	PUNCT
iajs-758	180	7	module	module	NOUN
iajs-758	180	8	is	be	AUX
iajs-758	180	9	uniform	uniform	ADJ
iajs-758	180	10	.	.	PUNCT
iajs-758	181	1	but	but	CCONJ
iajs-758	181	2	it	it	PRON
iajs-758	181	3	is	be	AUX
iajs-758	181	4	not	not	PART
iajs-758	181	5	a	a	DET
iajs-758	181	6	max	max	NOUN
iajs-758	181	7	–	–	PUNCT
iajs-758	181	8	module	module	NOUN
iajs-758	181	9	by	by	ADP
iajs-758	181	10	[	[	X
iajs-758	181	11	2.2	2.2	NUM
iajs-758	181	12	,	,	PUNCT
iajs-758	181	13	(	(	PUNCT
iajs-758	181	14	4	4	NUM
iajs-758	181	15	)	)	PUNCT
iajs-758	181	16	]	]	PUNCT
iajs-758	181	17	.	.	PUNCT
iajs-758	182	1	however	however	ADV
iajs-758	182	2	,	,	PUNCT
iajs-758	182	3	we	we	PRON
iajs-758	182	4	have	have	VERB
iajs-758	182	5	the	the	DET
iajs-758	182	6	following	follow	VERB
iajs-758	182	7	result	result	NOUN
iajs-758	182	8	.	.	PUNCT
iajs-758	183	1	proposition	proposition	NOUN
iajs-758	183	2	3.9	3.9	NUM
iajs-758	183	3	if	if	SCONJ
iajs-758	183	4	m	m	NOUN
iajs-758	183	5	is	be	AUX
iajs-758	183	6	a	a	DET
iajs-758	183	7	max	max	PROPN
iajs-758	183	8	–	–	PUNCT
iajs-758	183	9	module	module	NOUN
iajs-758	183	10	such	such	ADJ
iajs-758	183	11	that	that	DET
iajs-758	183	12	annr(n	annr(n	ADJ
iajs-758	183	13	∩	∩	ADJ
iajs-758	183	14	u	u	NOUN
iajs-758	183	15	)	)	PUNCT
iajs-758	183	16	=	=	SYM
iajs-758	183	17	annrn	annrn	NOUN
iajs-758	183	18	+	+	CCONJ
iajs-758	183	19	annru	annru	NOUN
iajs-758	183	20	,	,	PUNCT
iajs-758	183	21	for	for	ADP
iajs-758	183	22	every	every	DET
iajs-758	183	23	non	non	ADJ
iajs-758	183	24	–	–	PUNCT
iajs-758	183	25	zero	zero	NUM
iajs-758	183	26	submodules	submodule	NOUN
iajs-758	183	27	n	n	NOUN
iajs-758	183	28	and	and	CCONJ
iajs-758	183	29	u	u	PROPN
iajs-758	183	30	of	of	ADP
iajs-758	183	31	m	m	PROPN
iajs-758	183	32	,	,	PUNCT
iajs-758	183	33	then	then	ADV
iajs-758	183	34	uniform	uniform	ADJ
iajs-758	183	35	.	.	PUNCT
iajs-758	184	1	proof	proof	NOUN
iajs-758	184	2	:	:	PUNCT
iajs-758	184	3	since	since	SCONJ
iajs-758	184	4	m	m	PROPN
iajs-758	184	5	is	be	AUX
iajs-758	184	6	a	a	DET
iajs-758	184	7	max	max	PROPN
iajs-758	184	8	–	–	PUNCT
iajs-758	184	9	module	module	NOUN
iajs-758	184	10	,	,	PUNCT
iajs-758	184	11	so	so	SCONJ
iajs-758	184	12	m	m	NOUN
iajs-758	184	13	is	be	AUX
iajs-758	184	14	a	a	DET
iajs-758	184	15	primary	primary	ADJ
iajs-758	184	16	module	module	NOUN
iajs-758	184	17	by	by	ADP
iajs-758	184	18	(	(	PUNCT
iajs-758	184	19	3.1	3.1	NUM
iajs-758	184	20	)	)	PUNCT
iajs-758	184	21	,	,	PUNCT
iajs-758	184	22	hence	hence	ADV
iajs-758	184	23	the	the	DET
iajs-758	184	24	result	result	NOUN
iajs-758	184	25	follows	follow	VERB
iajs-758	184	26	by	by	ADP
iajs-758	184	27	[	[	X
iajs-758	184	28	2	2	NUM
iajs-758	184	29	,	,	PUNCT
iajs-758	184	30	proposition	proposition	NOUN
iajs-758	184	31	(	(	PUNCT
iajs-758	184	32	2.3.7	2.3.7	NUM
iajs-758	184	33	)	)	PUNCT
iajs-758	184	34	,	,	PUNCT
iajs-758	184	35	chapter	chapter	NOUN
iajs-758	184	36	2	2	NUM
iajs-758	184	37	]	]	PUNCT
iajs-758	184	38	.	.	PUNCT
iajs-758	185	1	now	now	ADV
iajs-758	185	2	we	we	PRON
iajs-758	185	3	can	can	AUX
iajs-758	185	4	give	give	VERB
iajs-758	185	5	the	the	DET
iajs-758	185	6	following	follow	VERB
iajs-758	185	7	result	result	NOUN
iajs-758	185	8	:	:	PUNCT
iajs-758	185	9	proposition	proposition	NOUN
iajs-758	185	10	3.10	3.10	NUM
iajs-758	185	11	let	let	VERB
iajs-758	185	12	m	m	PRON
iajs-758	185	13	be	be	AUX
iajs-758	185	14	an	an	DET
iajs-758	185	15	r	r	NOUN
iajs-758	185	16	-	-	PUNCT
iajs-758	185	17	module	module	NOUN
iajs-758	185	18	and	and	CCONJ
iajs-758	185	19	let	let	VERB
iajs-758	185	20	0	0	NUM
iajs-758	185	21	≠	≠	PROPN
iajs-758	185	22	x	x	X
iajs-758	185	23			NOUN
iajs-758	185	24	m	m	VERB
iajs-758	185	25	such	such	ADJ
iajs-758	185	26	that	that	SCONJ
iajs-758	185	27	:	:	PUNCT
iajs-758	185	28	1	1	X
iajs-758	185	29	.	.	X
iajs-758	185	30	rx	rx	NOUN
iajs-758	185	31	is	be	AUX
iajs-758	185	32	an	an	DET
iajs-758	185	33	essential	essential	ADJ
iajs-758	185	34	submodule	submodule	NOUN
iajs-758	185	35	of	of	ADP
iajs-758	185	36	m.	m.	NOUN
iajs-758	185	37	2	2	NUM
iajs-758	185	38	.	.	PUNCT
iajs-758	185	39	)	)	PUNCT
iajs-758	186	1	(	(	PUNCT
iajs-758	186	2	xannr	xannr	NOUN
iajs-758	186	3	is	be	AUX
iajs-758	186	4	a	a	DET
iajs-758	186	5	maximal	maximal	ADJ
iajs-758	186	6	ideal	ideal	NOUN
iajs-758	186	7	of	of	ADP
iajs-758	186	8	r	r	NOUN
iajs-758	186	9	,	,	PUNCT
iajs-758	186	10	and	and	CCONJ
iajs-758	186	11	3	3	X
iajs-758	186	12	.	.	NUM
iajs-758	186	13	)	)	PUNCT
iajs-758	187	1	(	(	PUNCT
iajs-758	187	2	xannmann	xannmann	PROPN
iajs-758	187	3	rr	rr	PROPN
iajs-758	187	4			PROPN
iajs-758	187	5	.	.	PUNCT
iajs-758	188	1	then	then	ADV
iajs-758	188	2	m	m	PROPN
iajs-758	188	3	is	be	AUX
iajs-758	188	4	a	a	DET
iajs-758	188	5	max	max	PROPN
iajs-758	188	6	–	–	PUNCT
iajs-758	188	7	module	module	NOUN
iajs-758	188	8	.	.	PUNCT
iajs-758	189	1	proof	proof	NOUN
iajs-758	189	2	:	:	PUNCT
iajs-758	189	3	let	let	VERB
iajs-758	189	4	n	n	PRON
iajs-758	189	5	be	be	AUX
iajs-758	189	6	a	a	DET
iajs-758	189	7	non	non	ADJ
iajs-758	189	8	–	–	PUNCT
iajs-758	189	9	zero	zero	NUM
iajs-758	189	10	submodule	submodule	NOUN
iajs-758	189	11	of	of	ADP
iajs-758	189	12	m.	m.	NOUN
iajs-758	189	13	since	since	SCONJ
iajs-758	189	14	rx	rx	NOUN
iajs-758	189	15	is	be	AUX
iajs-758	189	16	an	an	DET
iajs-758	189	17	essential	essential	ADJ
iajs-758	189	18	submodule	submodule	NOUN
iajs-758	189	19	of	of	ADP
iajs-758	189	20	m	m	PROPN
iajs-758	189	21	,	,	PUNCT
iajs-758	189	22	there	there	PRON
iajs-758	189	23	exists	exist	VERB
iajs-758	189	24	0	0	NUM
iajs-758	189	25	≠	≠	PROPN
iajs-758	189	26	t	t	NOUN
iajs-758	189	27			NOUN
iajs-758	189	28	r	r	NOUN
iajs-758	189	29	such	such	ADJ
iajs-758	189	30	that	that	PRON
iajs-758	189	31	0	0	NUM
iajs-758	189	32	≠	≠	PROPN
iajs-758	189	33	tx	tx	VERB
iajs-758	189	34			NOUN
iajs-758	189	35	n	n	CCONJ
iajs-758	189	36	and	and	CCONJ
iajs-758	189	37	hence	hence	ADV
iajs-758	189	38	(	(	PUNCT
iajs-758	189	39	tx	tx	PROPN
iajs-758	189	40	)	)	PUNCT
iajs-758	189	41			PROPN
iajs-758	189	42	n.	n.	PROPN
iajs-758	189	43	this	this	PRON
iajs-758	189	44	implies	imply	VERB
iajs-758	189	45	that	that	SCONJ
iajs-758	189	46	annrn	annrn	NOUN
iajs-758	189	47			PROPN
iajs-758	189	48	annr(tx	annr(tx	PROPN
iajs-758	189	49	)	)	PUNCT
iajs-758	189	50	and	and	CCONJ
iajs-758	189	51	so	so	ADV
iajs-758	189	52	,	,	PUNCT
iajs-758	189	53	)	)	PUNCT
iajs-758	189	54	(	(	PUNCT
iajs-758	189	55	txannnann	txannnann	PROPN
iajs-758	189	56	rr	rr	PROPN
iajs-758	189	57			PROPN
iajs-758	189	58	.	.	PUNCT
iajs-758	190	1	but	but	CCONJ
iajs-758	190	2	n	n	CCONJ
iajs-758	190	3			PROPN
iajs-758	190	4	m	m	PROPN
iajs-758	190	5	,	,	PUNCT
iajs-758	190	6	then	then	ADV
iajs-758	190	7	nannmann	nannmann	PROPN
iajs-758	190	8	rr	rr	PROPN
iajs-758	190	9			PROPN
iajs-758	190	10	and	and	CCONJ
iajs-758	190	11	hence	hence	ADV
iajs-758	190	12	nannxann	nannxann	VERB
iajs-758	190	13	rr	rr	PROPN
iajs-758	190	14			PROPN
iajs-758	190	15	)	)	PUNCT
iajs-758	190	16	(	(	PUNCT
iajs-758	190	17	(	(	PUNCT
iajs-758	190	18	by	by	ADP
iajs-758	190	19	condition	condition	NOUN
iajs-758	190	20	3	3	NUM
iajs-758	190	21	)	)	PUNCT
iajs-758	190	22	.	.	PUNCT
iajs-758	191	1	thus	thus	ADV
iajs-758	191	2	,	,	PUNCT
iajs-758	191	3	)	)	PUNCT
iajs-758	191	4	(	(	PUNCT
iajs-758	191	5	)	)	PUNCT
iajs-758	191	6	(	(	PUNCT
iajs-758	191	7	xannnannxann	xannnannxann	PROPN
iajs-758	191	8	rrr	rrr	PROPN
iajs-758	191	9			VERB
iajs-758	191	10	…	…	PUNCT
iajs-758	191	11	…	…	PUNCT
iajs-758	191	12	(	(	PUNCT
iajs-758	191	13	1	1	NUM
iajs-758	191	14	)	)	PUNCT
iajs-758	191	15	.	.	PUNCT
iajs-758	192	1	let	let	VERB
iajs-758	192	2	)	)	PUNCT
iajs-758	192	3	(	(	PUNCT
iajs-758	192	4	txannr	txannr	NOUN
iajs-758	192	5	r	r	PROPN
iajs-758	192	6	,	,	PUNCT
iajs-758	192	7	then	then	ADV
iajs-758	192	8	rntx	rntx	VERB
iajs-758	192	9	=	=	SYM
iajs-758	192	10	0	0	NUM
iajs-758	192	11	for	for	ADP
iajs-758	192	12	some	some	DET
iajs-758	192	13	n	n	PRON
iajs-758	192	14			NOUN
iajs-758	192	15	z+	z+	NUM
iajs-758	192	16	and	and	CCONJ
iajs-758	192	17	rnt	rnt	VERB
iajs-758	192	18			PROPN
iajs-758	192	19	annr(x	annr(x	NOUN
iajs-758	192	20	)	)	PUNCT
iajs-758	192	21	.	.	PUNCT
iajs-758	193	1	but	but	CCONJ
iajs-758	193	2	tx	tx	VERB
iajs-758	193	3	≠	≠	PROPN
iajs-758	193	4	0	0	NUM
iajs-758	193	5	;	;	PUNCT
iajs-758	193	6	that	that	PRON
iajs-758	193	7	is	be	AUX
iajs-758	193	8	t	t	PROPN
iajs-758	193	9			NUM
iajs-758	193	10	annr(x	annr(x	NOUN
iajs-758	193	11	)	)	PUNCT
iajs-758	193	12	and	and	CCONJ
iajs-758	193	13	by	by	ADP
iajs-758	193	14	condition	condition	NOUN
iajs-758	193	15	(	(	PUNCT
iajs-758	193	16	2	2	NUM
iajs-758	193	17	)	)	PUNCT
iajs-758	193	18	)	)	PUNCT
iajs-758	193	19	(	(	PUNCT
iajs-758	193	20	xannr	xannr	NOUN
iajs-758	193	21	is	be	AUX
iajs-758	193	22	a	a	DET
iajs-758	193	23	maximal	maximal	ADJ
iajs-758	193	24	ideal	ideal	NOUN
iajs-758	193	25	of	of	ADP
iajs-758	193	26	r	r	NOUN
iajs-758	193	27	,	,	PUNCT
iajs-758	193	28	so	so	ADV
iajs-758	193	29	annr(x	annr(x	NOUN
iajs-758	193	30	)	)	PUNCT
iajs-758	193	31	is	be	AUX
iajs-758	193	32	a	a	DET
iajs-758	193	33	primary	primary	ADJ
iajs-758	193	34	ideal	ideal	NOUN
iajs-758	193	35	of	of	ADP
iajs-758	193	36	r	r	NOUN
iajs-758	193	37	,	,	PUNCT
iajs-758	193	38	by	by	ADP
iajs-758	193	39	[	[	X
iajs-758	193	40	1	1	NUM
iajs-758	193	41	,	,	PUNCT
iajs-758	193	42	proposition	proposition	NOUN
iajs-758	193	43	4.9	4.9	NUM
iajs-758	193	44	,	,	PUNCT
iajs-758	193	45	p.	p.	NOUN
iajs-758	193	46	64	64	NUM
iajs-758	193	47	]	]	PUNCT
iajs-758	193	48	.	.	PUNCT
iajs-758	194	1	then	then	ADV
iajs-758	194	2	)	)	PUNCT
iajs-758	194	3	(	(	PUNCT
iajs-758	194	4	xannr	xannr	PROPN
iajs-758	194	5	r	r	PROPN
iajs-758	194	6	and	and	CCONJ
iajs-758	194	7	hence	hence	ADV
iajs-758	194	8	)	)	PUNCT
iajs-758	194	9	(	(	PUNCT
iajs-758	194	10	)	)	PUNCT
iajs-758	194	11	(	(	PUNCT
iajs-758	194	12	xanntxnann	xanntxnann	PROPN
iajs-758	194	13	rr	rr	PROPN
iajs-758	194	14			PROPN
iajs-758	194	15	..	..	PUNCT
iajs-758	194	16	…	…	PUNCT
iajs-758	194	17	(	(	PUNCT
iajs-758	194	18	2	2	NUM
iajs-758	194	19	)	)	PUNCT
iajs-758	194	20	.	.	PUNCT
iajs-758	195	1	thus	thus	ADV
iajs-758	195	2	by	by	ADP
iajs-758	195	3	(	(	PUNCT
iajs-758	195	4	1	1	NUM
iajs-758	195	5	)	)	PUNCT
iajs-758	195	6	and	and	CCONJ
iajs-758	195	7	(	(	PUNCT
iajs-758	195	8	2	2	NUM
iajs-758	195	9	)	)	PUNCT
iajs-758	195	10	,	,	PUNCT
iajs-758	195	11	)	)	PUNCT
iajs-758	195	12	(	(	PUNCT
iajs-758	195	13	)	)	PUNCT
iajs-758	195	14	(	(	PUNCT
iajs-758	195	15	txannxann	txannxann	NOUN
iajs-758	195	16	rr	rr	NOUN
iajs-758	195	17			PROPN
iajs-758	195	18	and	and	CCONJ
iajs-758	195	19	so	so	ADV
iajs-758	195	20	)	)	PUNCT
iajs-758	195	21	(	(	PUNCT
iajs-758	195	22	xannnann	xannnann	PROPN
iajs-758	195	23	rr	rr	PROPN
iajs-758	196	1			PROPN
iajs-758	196	2	.	.	PUNCT
iajs-758	197	1	therefore	therefore	ADV
iajs-758	197	2	(	(	PUNCT
iajs-758	197	3	by	by	ADP
iajs-758	197	4	condition	condition	NOUN
iajs-758	197	5	2	2	NUM
iajs-758	197	6	)	)	PUNCT
iajs-758	197	7	nannr	nannr	NOUN
iajs-758	197	8	is	be	AUX
iajs-758	197	9	a	a	DET
iajs-758	197	10	maximal	maximal	ADJ
iajs-758	197	11	ideal	ideal	NOUN
iajs-758	197	12	of	of	ADP
iajs-758	197	13	r	r	NOUN
iajs-758	197	14	and	and	CCONJ
iajs-758	197	15	m	m	PROPN
iajs-758	197	16	is	be	AUX
iajs-758	197	17	a	a	DET
iajs-758	197	18	max	max	NOUN
iajs-758	197	19	–	–	PUNCT
iajs-758	197	20	module	module	NOUN
iajs-758	197	21	by	by	ADP
iajs-758	197	22	definition	definition	NOUN
iajs-758	197	23	(	(	PUNCT
iajs-758	197	24	2.1	2.1	NUM
iajs-758	197	25	)	)	PUNCT
iajs-758	197	26	.	.	PUNCT
iajs-758	198	1	the	the	DET
iajs-758	198	2	following	follow	VERB
iajs-758	198	3	result	result	NOUN
iajs-758	198	4	is	be	AUX
iajs-758	198	5	a	a	DET
iajs-758	198	6	consequence	consequence	NOUN
iajs-758	198	7	of	of	ADP
iajs-758	198	8	proposition	proposition	NOUN
iajs-758	198	9	(	(	PUNCT
iajs-758	198	10	3.10	3.10	NUM
iajs-758	198	11	)	)	PUNCT
iajs-758	198	12	.	.	PUNCT
iajs-758	199	1	corollary	corollary	NOUN
iajs-758	199	2	3.11	3.11	NUM
iajs-758	199	3	let	let	VERB
iajs-758	199	4	m	m	PRON
iajs-758	199	5	be	be	AUX
iajs-758	199	6	uniform	uniform	ADJ
iajs-758	199	7	r	r	NOUN
iajs-758	199	8	-	-	NOUN
iajs-758	199	9	module	module	NOUN
iajs-758	199	10	such	such	ADJ
iajs-758	199	11	that	that	PRON
iajs-758	199	12	)	)	PUNCT
iajs-758	199	13	(	(	PUNCT
iajs-758	199	14	xannr	xannr	NOUN
iajs-758	199	15	is	be	AUX
iajs-758	199	16	a	a	DET
iajs-758	199	17	maximal	maximal	ADJ
iajs-758	199	18	ideal	ideal	NOUN
iajs-758	199	19	of	of	ADP
iajs-758	199	20	r	r	NOUN
iajs-758	199	21	and	and	CCONJ
iajs-758	199	22	)	)	PUNCT
iajs-758	199	23	(	(	PUNCT
iajs-758	199	24	xannmann	xannmann	PROPN
iajs-758	199	25	rr	rr	PROPN
iajs-758	199	26			PROPN
iajs-758	199	27	for	for	ADP
iajs-758	199	28	some	some	DET
iajs-758	199	29	x	x	SYM
iajs-758	199	30	≠	≠	PROPN
iajs-758	199	31	0	0	NUM
iajs-758	199	32	.	.	PUNCT
iajs-758	200	1	then	then	ADV
iajs-758	200	2	m	m	PROPN
iajs-758	200	3	is	be	AUX
iajs-758	200	4	a	a	DET
iajs-758	200	5	max	max	PROPN
iajs-758	200	6	–	–	PUNCT
iajs-758	200	7	module	module	NOUN
iajs-758	200	8	.	.	PUNCT
iajs-758	201	1	in	in	ADP
iajs-758	201	2	the	the	DET
iajs-758	201	3	following	follow	VERB
iajs-758	201	4	corollary	corollary	NOUN
iajs-758	201	5	,	,	PUNCT
iajs-758	201	6	we	we	PRON
iajs-758	201	7	give	give	VERB
iajs-758	201	8	a	a	DET
iajs-758	201	9	condition	condition	NOUN
iajs-758	201	10	under	under	ADP
iajs-758	201	11	which	which	PRON
iajs-758	201	12	the	the	DET
iajs-758	201	13	converse	converse	NOUN
iajs-758	201	14	of	of	ADP
iajs-758	201	15	proposition	proposition	NOUN
iajs-758	201	16	(	(	PUNCT
iajs-758	201	17	3.9	3.9	NUM
iajs-758	201	18	)	)	PUNCT
iajs-758	201	19	is	be	AUX
iajs-758	201	20	true	true	ADJ
iajs-758	201	21	.	.	PUNCT
iajs-758	202	1	ibn	ibn	PROPN
iajs-758	202	2	alhaitham	alhaitham	PROPN
iajs-758	202	3	j.	j.	PROPN
iajs-758	202	4	for	for	ADP
iajs-758	202	5	pure	pure	ADJ
iajs-758	202	6	&	&	CCONJ
iajs-758	202	7	appl	appl	PROPN
iajs-758	202	8	.	.	PUNCT
iajs-758	203	1	sci	sci	PROPN
iajs-758	203	2	.	.	PUNCT
iajs-758	203	3	vol.24	vol.24	NOUN
iajs-758	203	4	(	(	PUNCT
iajs-758	203	5	2	2	NUM
iajs-758	203	6	)	)	PUNCT
iajs-758	203	7	2011	2011	NUM
iajs-758	203	8	corollary	corollary	NOUN
iajs-758	203	9	3.12	3.12	NUM
iajs-758	203	10	if	if	SCONJ
iajs-758	203	11	m	m	NOUN
iajs-758	203	12	is	be	AUX
iajs-758	203	13	a	a	DET
iajs-758	203	14	uniform	uniform	ADJ
iajs-758	203	15	r	r	NOUN
iajs-758	203	16	–	–	PUNCT
iajs-758	203	17	module	module	NOUN
iajs-758	203	18	such	such	ADJ
iajs-758	203	19	that	that	PRON
iajs-758	203	20	)	)	PUNCT
iajs-758	203	21	(	(	PUNCT
iajs-758	203	22	xannr	xannr	NOUN
iajs-758	203	23	is	be	AUX
iajs-758	203	24	a	a	DET
iajs-758	203	25	maximal	maximal	ADJ
iajs-758	203	26	ideal	ideal	NOUN
iajs-758	203	27	of	of	ADP
iajs-758	203	28	r	r	NOUN
iajs-758	203	29	for	for	ADP
iajs-758	203	30	some	some	DET
iajs-758	203	31	x	x	SYM
iajs-758	203	32			NOUN
iajs-758	203	33	m.	m.	NOUN
iajs-758	203	34	then	then	ADV
iajs-758	203	35	the	the	DET
iajs-758	203	36	following	follow	VERB
iajs-758	203	37	statements	statement	NOUN
iajs-758	203	38	are	be	AUX
iajs-758	203	39	equivalent	equivalent	ADJ
iajs-758	203	40	.	.	PUNCT
iajs-758	204	1	1	1	X
iajs-758	204	2	.	.	NUM
iajs-758	204	3	)	)	PUNCT
iajs-758	205	1	(	(	PUNCT
iajs-758	205	2	xannmann	xannmann	PROPN
iajs-758	205	3	rr	rr	PROPN
iajs-758	205	4			PROPN
iajs-758	205	5	for	for	ADP
iajs-758	205	6	some	some	DET
iajs-758	205	7	x	x	SYM
iajs-758	205	8			NOUN
iajs-758	205	9	m.	m.	NOUN
iajs-758	205	10	2	2	NUM
iajs-758	205	11	.	.	PUNCT
iajs-758	206	1	m	m	PROPN
iajs-758	206	2	is	be	AUX
iajs-758	206	3	a	a	DET
iajs-758	206	4	max	max	PROPN
iajs-758	206	5	–	–	PUNCT
iajs-758	206	6	module	module	NOUN
iajs-758	206	7	.	.	PUNCT
iajs-758	207	1	references	reference	NOUN
iajs-758	207	2	1	1	NUM
iajs-758	207	3	.	.	PUNCT
iajs-758	207	4	sharp	sharp	ADJ
iajs-758	207	5	,	,	PUNCT
iajs-758	207	6	r.y	r.y	PROPN
iajs-758	207	7	.	.	PROPN
iajs-758	208	1	(	(	PUNCT
iajs-758	208	2	1990	1990	NUM
iajs-758	208	3	)	)	PUNCT
iajs-758	208	4	steps	step	NOUN
iajs-758	208	5	in	in	ADP
iajs-758	208	6	commutative	commutative	ADJ
iajs-758	208	7	algebra	algebra	PROPN
iajs-758	208	8	london	london	PROPN
iajs-758	208	9	mathematical	mathematical	ADJ
iajs-758	208	10	society	society	NOUN
iajs-758	208	11	student	student	NOUN
iajs-758	208	12	texts	text	NOUN
iajs-758	208	13	no	no	DET
iajs-758	208	14	19	19	NUM
iajs-758	208	15	(	(	PUNCT
iajs-758	208	16	cambridge	cambridge	PROPN
iajs-758	208	17	university	university	PROPN
iajs-758	208	18	press	press	PROPN
iajs-758	208	19	)	)	PUNCT
iajs-758	208	20	.	.	PUNCT
iajs-758	209	1	2	2	X
iajs-758	209	2	.	.	X
iajs-758	209	3	mijbass	mijbass	PROPN
iajs-758	209	4	,	,	PUNCT
iajs-758	209	5	a.s.(2000	a.s.(2000	NUM
iajs-758	209	6	)	)	PUNCT
iajs-758	209	7	"	"	PUNCT
iajs-758	209	8	semi	semi	ADJ
iajs-758	209	9	–	–	PUNCT
iajs-758	209	10	primary	primary	ADJ
iajs-758	209	11	submodules	submodules	NOUN
iajs-758	209	12	"	"	PUNCT
iajs-758	209	13	,	,	PUNCT
iajs-758	209	14	scince	scince	NOUN
iajs-758	209	15	journal	journal	NOUN
iajs-758	209	16	,	,	PUNCT
iajs-758	209	17	university	university	NOUN
iajs-758	209	18	of	of	ADP
iajs-758	209	19	tikrit	tikrit	NOUN
iajs-758	209	20	,	,	PUNCT
iajs-758	209	21	j.6	j.6	NOUN
iajs-758	209	22	.	.	PUNCT
iajs-758	210	1	no.1	no.1	X
iajs-758	210	2	3	3	X
iajs-758	210	3	.	.	PUNCT
iajs-758	211	1	elbast	elbast	ADJ
iajs-758	211	2	,	,	PUNCT
iajs-758	211	3	z.a	z.a	PROPN
iajs-758	211	4	.	.	PROPN
iajs-758	211	5	and	and	CCONJ
iajs-758	211	6	smith	smith	PROPN
iajs-758	211	7	,	,	PUNCT
iajs-758	211	8	p.f.(1988	p.f.(1988	PROPN
iajs-758	211	9	)	)	PUNCT
iajs-758	211	10	multiplication	multiplication	NOUN
iajs-758	211	11	modules	module	NOUN
iajs-758	211	12	,	,	PUNCT
iajs-758	211	13	comm	comm	NOUN
iajs-758	211	14	.	.	PUNCT
iajs-758	212	1	in	in	ADP
iajs-758	212	2	algebra	algebra	NOUN
iajs-758	212	3	,	,	PUNCT
iajs-758	212	4	16	16	NUM
iajs-758	212	5	:	:	SYM
iajs-758	212	6	755779	755779	NUM
iajs-758	212	7	.	.	PUNCT
iajs-758	212	8	.معة	.معة	VERB
iajs-758	212	9	بغدادجاحول	بغدادجاحول	ADJ
iajs-758	212	10	مودیوالت	مودیوالت	ADJ
iajs-758	212	11	جزئیة	جزئیة	NOUN
iajs-758	213	1	في	في	X
iajs-758	213	2	مودیوالت	مودیوالت	ADJ
iajs-758	213	3	جدائیة	جدائیة	PROPN
iajs-758	213	4	،	،	PROPN
iajs-758	213	5	رسالة	رسالة	PROPN
iajs-758	213	6	ماجستیر	ماجستیر	PROPN
iajs-758	213	7	،	،	X
iajs-758	213	8	)	)	PUNCT
iajs-758	213	9	1992	1992	NUM
iajs-758	213	10	(	(	PUNCT
iajs-758	213	11	عبد	عبد	PROPN
iajs-758	213	12	الرحمن	الرحمن	PROPN
iajs-758	213	13	عبود	عبود	PROPN
iajs-758	213	14	احمد	احمد	VERB
iajs-758	213	15	.4	.4	NUM
iajs-758	213	16	5	5	NUM
iajs-758	213	17	.	.	PUNCT
iajs-758	213	18	abdulrazak	abdulrazak	PROPN
iajs-758	213	19	,	,	PUNCT
iajs-758	213	20	h.m	h.m	PROPN
iajs-758	213	21	,	,	PUNCT
iajs-758	213	22	quasi	quasi	ADJ
iajs-758	213	23	-	-	ADJ
iajs-758	213	24	prime	prime	ADJ
iajs-758	213	25	modules	module	NOUN
iajs-758	213	26	and	and	CCONJ
iajs-758	213	27	'	'	PUNCT
iajs-758	213	28	quasi	quasi	ADJ
iajs-758	213	29	-	-	ADJ
iajs-758	213	30	prime	prime	ADJ
iajs-758	213	31	submodules	submodule	NOUN
iajs-758	213	32	'	'	PART
iajs-758	213	33	m.	m.	NOUN
iajs-758	213	34	d.	d.	PROPN
iajs-758	213	35	thesis	thesis	PROPN
iajs-758	213	36	,	,	PUNCT
iajs-758	213	37	univ	univ	PROPN
iajs-758	213	38	.	.	PROPN
iajs-758	213	39	of	of	ADP
iajs-758	213	40	baghdad	baghdad	PROPN
iajs-758	213	41	.	.	PUNCT
iajs-758	214	1	6	6	NUM
iajs-758	214	2	.	.	PUNCT
iajs-758	214	3	dauns	daun	NOUN
iajs-758	214	4	,	,	PUNCT
iajs-758	214	5	j.	j.	PROPN
iajs-758	214	6	(	(	PUNCT
iajs-758	214	7	1980	1980	NUM
iajs-758	214	8	)	)	PUNCT
iajs-758	214	9	prime	prime	ADJ
iajs-758	214	10	modules	module	NOUN
iajs-758	214	11	and	and	CCONJ
iajs-758	214	12	one	one	NUM
iajs-758	214	13	–	–	PUNCT
iajs-758	214	14	sided	side	VERB
iajs-758	214	15	ideals	ideal	NOUN
iajs-758	214	16	in	in	ADP
iajs-758	214	17	"	"	PUNCT
iajs-758	214	18	ring	ring	NOUN
iajs-758	214	19	theory	theory	NOUN
iajs-758	214	20	and	and	CCONJ
iajs-758	214	21	algebra	algebra	NOUN
iajs-758	214	22	iii	iii	PROPN
iajs-758	214	23	"	"	PUNCT
iajs-758	214	24	,	,	PUNCT
iajs-758	214	25	(	(	PUNCT
iajs-758	214	26	proceedings	proceeding	NOUN
iajs-758	214	27	of	of	ADP
iajs-758	214	28	third	third	PROPN
iajs-758	214	29	oklahoma	oklahoma	PROPN
iajs-758	214	30	conference	conference	PROPN
iajs-758	214	31	)	)	PUNCT
iajs-758	214	32	,	,	PUNCT
iajs-758	214	33	b.	b.	PROPN
iajs-758	214	34	r.	r.	PROPN
iajs-758	214	35	mc	mc	PROPN
iajs-758	214	36	donald	donald	PROPN
iajs-758	214	37	(	(	PUNCT
iajs-758	214	38	editor	editor	NOUN
iajs-758	214	39	)	)	PUNCT
iajs-758	214	40	,	,	PUNCT
iajs-758	214	41	dekker	dekker	PROPN
iajs-758	214	42	,	,	PUNCT
iajs-758	214	43	new	new	PROPN
iajs-758	214	44	york	york	PROPN
iajs-758	214	45	,	,	PUNCT
iajs-758	214	46	301	301	NUM
iajs-758	214	47	-	-	SYM
iajs-758	214	48	344	344	NUM
iajs-758	214	49	.	.	PUNCT
iajs-758	215	1	.المودیوالت	.المودیوالت	PROPN
iajs-758	215	2	الجزئیة	الجزئیة	PROPN
iajs-758	215	3	االولیة	االولیة	PROPN
iajs-758	215	4	والمودیوالت	والمودیوالت	PROPN
iajs-758	215	5	الجزئیة	الجزئیة	PROPN
iajs-758	215	6	شبه	شبه	VERB
iajs-758	215	7	االولیة	االولیة	PROPN
iajs-758	215	8	،	،	PROPN
iajs-758	215	9	رسالة	رسالة	PROPN
iajs-758	215	10	ماجستیر	ماجستیر	PROPN
iajs-758	215	11	،	،	PROPN
iajs-758	215	12	جامعة	جامعة	PROPN
iajs-758	215	13	بغداد	بغداد	PROPN
iajs-758	215	14	)	)	PUNCT
iajs-758	215	15	1996	1996	NUM
iajs-758	215	16	(	(	PUNCT
iajs-758	215	17	ایمان	ایمان	NOUN
iajs-758	215	18	علي	علي	PROPN
iajs-758	215	19	عذاب	عذاب	PROPN
iajs-758	215	20	.7	.7	PROPN
iajs-758	215	21	8	8	NUM
iajs-758	215	22	.	.	PUNCT
iajs-758	216	1	lu	lu	PROPN
iajs-758	216	2	,	,	PUNCT
iajs-758	216	3	c.p.(1989	c.p.(1989	PROPN
iajs-758	216	4	)	)	PUNCT
iajs-758	217	1	m	m	NOUN
iajs-758	217	2	-	-	NOUN
iajs-758	217	3	radicals	radical	NOUN
iajs-758	217	4	of	of	ADP
iajs-758	217	5	submodules	submodule	NOUN
iajs-758	217	6	in	in	ADP
iajs-758	217	7	modules	module	NOUN
iajs-758	217	8	,	,	PUNCT
iajs-758	217	9	math	math	NOUN
iajs-758	217	10	.	.	PUNCT
iajs-758	218	1	japon	japon	PROPN
iajs-758	218	2	,	,	PUNCT
iajs-758	218	3	34	34	NUM
iajs-758	218	4	:	:	SYM
iajs-758	218	5	211	211	NUM
iajs-758	218	6	-	-	SYM
iajs-758	218	7	219	219	NUM
iajs-758	218	8	.	.	PUNCT
iajs-758	219	1	9	9	X
iajs-758	219	2	.	.	X
iajs-758	219	3	saymach	saymach	PROPN
iajs-758	219	4	,	,	PUNCT
iajs-758	219	5	s.a	s.a	PROPN
iajs-758	219	6	.	.	PROPN
iajs-758	220	1	(	(	PUNCT
iajs-758	220	2	1979	1979	NUM
iajs-758	220	3	)	)	PUNCT
iajs-758	220	4	on	on	ADP
iajs-758	220	5	prime	prime	ADJ
iajs-758	220	6	submodules	submodule	NOUN
iajs-758	220	7	,	,	PUNCT
iajs-758	221	1	university	university	NOUN
iajs-758	221	2	noc	noc	PROPN
iajs-758	221	3	.	.	PROPN
iajs-758	221	4	tucumare	tucumare	PROPN
iajs-758	221	5	.	.	PUNCT
iajs-758	222	1	ser	ser	PROPN
iajs-758	222	2	.	.	PUNCT
iajs-758	223	1	a.	a.	PROPN
iajs-758	223	2	29	29	NUM
iajs-758	223	3	:	:	PUNCT
iajs-758	223	4	121136	121136	NUM
iajs-758	223	5	.	.	PUNCT
iajs-758	224	1	10	10	NUM
iajs-758	224	2	.	.	PUNCT
iajs-758	225	1	ahmed	ahmed	PROPN
iajs-758	225	2	abdul	abdul	PROPN
iajs-758	225	3	–	–	PUNCT
iajs-758	225	4	rahman	rahman	PROPN
iajs-758	225	5	,	,	PUNCT
iajs-758	225	6	a.	a.	NOUN
iajs-758	225	7	and	and	CCONJ
iajs-758	225	8	al	al	PROPN
iajs-758	225	9	-	-	PUNCT
iajs-758	225	10	hashimi	hashimi	PROPN
iajs-758	225	11	,	,	PUNCT
iajs-758	225	12	b.	b.	PROPN
iajs-758	225	13	(	(	PUNCT
iajs-758	225	14	1994	1994	NUM
iajs-758	225	15	)	)	PUNCT
iajs-758	225	16	"	"	PUNCT
iajs-758	225	17	on	on	ADP
iajs-758	225	18	submodules	submodule	NOUN
iajs-758	225	19	of	of	ADP
iajs-758	225	20	multiplication	multiplication	NOUN
iajs-758	225	21	modules	module	NOUN
iajs-758	225	22	"	"	PUNCT
iajs-758	225	23	,	,	PUNCT
iajs-758	225	24	iraqi	iraqi	PROPN
iajs-758	225	25	.	.	PUNCT
iajs-758	226	1	j.	j.	PROPN
iajs-758	226	2	sci	sci	PROPN
iajs-758	226	3	.	.	PROPN
iajs-758	226	4	,	,	PUNCT
iajs-758	226	5	35	35	NUM
iajs-758	226	6	:	:	SYM
iajs-758	226	7	4	4	NUM
iajs-758	226	8	.	.	NOUN
iajs-758	226	9	2011	2011	NUM
iajs-758	226	10	)	)	PUNCT
iajs-758	226	11	2	2	NUM
iajs-758	226	12	(	(	PUNCT
iajs-758	226	13	24المجلد	24المجلد	NUM
iajs-758	226	14	مجلة	مجلة	VERB
iajs-758	226	15	ابن	ابن	PROPN
iajs-758	226	16	الهیثم	الهیثم	PROPN
iajs-758	226	17	للعلوم	للعلوم	PROPN
iajs-758	226	18	الصرفة	الصرفة	PROPN
iajs-758	226	19	والتطبیقیة	والتطبیقیة	PROPN
iajs-758	226	20	حـول	حـول	PROPN
iajs-758	226	21	مـقـاس	مـقـاس	PROPN
iajs-758	226	22	أعـظـم	أعـظـم	NOUN
iajs-758	226	23	عدویه	عدویه	NOUN
iajs-758	226	24	جاسم	جاسم	NOUN
iajs-758	226	25	عبد	عبد	NOUN
iajs-758	226	26	الخالق	الخالق	PROPN
iajs-758	226	27	إعدادیة	إعدادیة	NOUN
iajs-758	226	28	الخالص	الخالص	PROPN
iajs-758	226	29	الصناعیة	الصناعیة	VERB
iajs-758	226	30	-التعلیم	-التعلیم	PROPN
iajs-758	226	31	المهني	المهني	ADJ
iajs-758	226	32	-وزارة	-وزارة	PROPN
iajs-758	226	33	التربیة	التربیة	NOUN
iajs-758	226	34	2009	2009	NUM
iajs-758	226	35	،	،	NOUN
iajs-758	226	36	كانون	كانون	NOUN
iajs-758	226	37	االول	االول	PROPN
iajs-758	226	38	،	،	PROPN
iajs-758	226	39	15	15	NUM
iajs-758	226	40	:	:	PUNCT
iajs-758	226	41	استلم	استلم	PROPN
iajs-758	226	42	البحث	البحث	VERB
iajs-758	226	43	في	في	X
iajs-758	226	44	,	,	PUNCT
iajs-758	226	45	2010حزیران	2010حزیران	NUM
iajs-758	226	46	،	،	NOUN
iajs-758	226	47	17	17	NUM
iajs-758	226	48	:	:	PUNCT
iajs-758	226	49	قبل	قبل	NOUN
iajs-758	226	50	البحث	البحث	VERB
iajs-758	226	51	في	في	ADP
iajs-758	226	52	الخالصة	الخالصة	PROPN
iajs-758	226	53	maxفي	maxفي	PROPN
iajs-758	226	54	هذا	هذا	PROPN
iajs-758	226	55	البحث	البحث	PROPN
iajs-758	226	56	قدمنا	قدمنا	PROPN
iajs-758	226	57	مفهوم	مفهوم	PROPN
iajs-758	226	58	مقاس	مقاس	PROPN
iajs-758	226	59	مـن	مـن	PROPN
iajs-758	226	60	النـوع	النـوع	PROPN
iajs-758	226	61	.	.	PUNCT
iajs-758	227	1	rمقاسا	rمقاسا	NOUN
iajs-758	228	1	أحادیًا	أحادیًا	PROPN
iajs-758	228	2	على	على	NOUN
iajs-758	228	3	mحلقة	mحلقة	PROPN
iajs-758	228	4	أبدالیة	أبدالیة	PROPN
iajs-758	228	5	ذات	ذات	NOUN
iajs-758	228	6	محاید	محاید	PROPN
iajs-758	228	7	،	،	PROPN
iajs-758	228	8	ولیكن	ولیكن	PROPN
iajs-758	228	9	rلتكن	rلتكن	PROPN
iajs-758	228	10	nannnannradإذا	nannnannradإذا	NOUN
iajs-758	228	11	كان	كان	NOUN
iajs-758	228	12	)	)	PUNCT
iajs-758	228	13	max(مقاسًا	max(مقاسًا	VERB
iajs-758	228	14	mیطلق	mیطلق	NOUN
iajs-758	228	15	على	على	NOUN
iajs-758	228	16	:	:	PUNCT
iajs-758	228	17	يكما	يكما	VERB
iajs-758	228	18	یأت	یأت	VERB
iajs-758	228	19	rr	rr	PRON
iajs-758	228	20			PROPN
iajs-758	228	21	)	)	PUNCT
iajs-758	228	22	(	(	PUNCT
iajs-758	228	23	مثالیًا	مثالیًا	NOUN
iajs-758	228	24	أعظمیا	أعظمیا	VERB
iajs-758	228	25	فيr	فيr	PROPN
iajs-758	228	26	لكل	لكل	NOUN
iajs-758	228	27	مقاس	مقاس	PROPN
iajs-758	228	28	جزئي	جزئي	NOUN
iajs-758	228	29	،	،	PROPN
iajs-758	228	30	وقد	وقد	PROPN
iajs-758	228	31	أطلقنا	أطلقنا	PROPN
iajs-758	228	32	على	على	NOUN
iajs-758	228	33	أي	أي	VERB
iajs-758	228	34	مقـاس	مقـاس	PROPN
iajs-758	228	35	*	*	PUNCT
iajs-758	228	36	مقاسًا	مقاسًا	VERB
iajs-758	228	37	من	من	PRON
iajs-758	228	38	النوع	النوع	PROPN
iajs-758	228	39	)	)	PUNCT
iajs-758	228	40	0(إذا	0(إذا	VERB
iajs-758	228	41	كان	كان	PROPN
iajs-758	228	42	)	)	PUNCT
iajs-758	228	43	max(مقاسًا	max(مقاسًا	PROPN
iajs-758	228	44	m	m	PROPN
iajs-758	228	45	،	،	PROPN
iajs-758	228	46	بعبارة	بعبارة	PROPN
iajs-758	228	47	مكافئة	مكافئة	PROPN
iajs-758	228	48	،	،	PROPN
iajs-758	228	49	یكونmفي	یكونmفي	NOUN
iajs-758	228	50	nغیر	nغیر	ADJ
iajs-758	228	51	صفري	صفري	NOUN
iajs-758	228	52	]	]	X
iajs-758	228	53	[	[	X
iajs-758	228	54	إذا	إذا	NUM
iajs-758	228	55	كان	كان	NOUN
iajs-758	228	56	*	*	PUNCT
iajs-758	228	57	مقاسًا	مقاسًا	VERB
iajs-758	228	58	من	من	PRON
iajs-758	228	59	النوع	النوع	NOUN
iajs-758	228	60	mفي	mفي	ADJ
iajs-758	228	61	nجزئي	nجزئي	ADJ
iajs-758	228	62	فعلي	فعلي	NOUN
iajs-758	228	63	:	:	PUNCT
iajs-758	228	64	kn	kn	NOUN
iajs-758	228	65	r	r	NOUN
iajs-758	228	66	یحتـوي	یحتـوي	NOUN
iajs-758	228	67	mفي	mفي	NOUN
iajs-758	228	68	k	k	X
iajs-758	228	69	،	،	PROPN
iajs-758	228	70	لكل	لكل	PROPN
iajs-758	228	71	مقاس	مقاس	PROPN
iajs-758	228	72	جزئي	جزئي	NOUN
iajs-758	228	73	rمثالیا	rمثالیا	NOUN
iajs-758	228	74	أعظمیا	أعظمیا	PROPN
iajs-758	228	75	في	في	ADP
iajs-758	228	76	n	n	CCONJ
iajs-758	228	77	ًفي	ًفي	NUM
iajs-758	228	78	هذا	هذا	NOUN
iajs-758	228	79	البحث	البحث	NOUN
iajs-758	228	80	،	،	NOUN
iajs-758	228	81	أعطیت	أعطیت	PROPN
iajs-758	228	82	بعض	بعض	NOUN
iajs-758	229	1	الخواص	الخواص	NOUN
iajs-758	229	2	و	و	PRON
iajs-758	229	3	التمیزات	التمیزات	PROPN
iajs-758	229	4	وكذلك	وكذلك	PROPN
iajs-758	229	5	ُدرست	ُدرست	PROPN
iajs-758	229	6	العدید	العدید	VERB
iajs-758	229	7	من	من	NOUN
iajs-758	229	8	النتـائج	النتـائج	NOUN
iajs-758	229	9	األساسـیة	األساسـیة	PROPN
iajs-758	229	10	حـول	حـول	PROPN
iajs-758	229	11	المقاسـات	المقاسـات	PROPN
iajs-758	229	12	مـن	مـن	PROPN
iajs-758	229	13	.	.	PUNCT
iajs-758	230	1	فعلیا	فعلیا	PROPN
iajs-758	230	2	والمخطــط	والمخطــط	PROPN
iajs-758	230	3	اآلتــي	اآلتــي	PROPN
iajs-758	230	4	یوضـــح	یوضـــح	PROPN
iajs-758	230	5	.	.	PUNCT
iajs-758	231	1	هــذا	هــذا	PROPN
iajs-758	231	2	ُدرســت	ُدرســت	PROPN
iajs-758	231	3	بعــض	بعــض	PROPN
iajs-758	231	4	العالقــات	العالقــات	PROPN
iajs-758	231	5	بینـــه	بینـــه	PROPN
iajs-758	231	6	وبــین	وبــین	PROPN
iajs-758	231	7	أنــواع	أنــواع	PROPN
iajs-758	231	8	أخــرى	أخــرى	NOUN
iajs-758	231	9	مــن	مــن	PROPN
iajs-758	231	10	المقاســات	المقاســات	PROPN
iajs-758	231	11	فضــال	فضــال	NOUN
iajs-758	231	12	عــن	عــن	NOUN
iajs-758	231	13	)	)	PUNCT
iajs-758	231	14	.max(النــوع	.max(النــوع	PROPN
iajs-758	231	15	:	:	PUNCT
iajs-758	231	16	ملخص	ملخص	ADJ
iajs-758	231	17	لما	لما	VERB
iajs-758	231	18	حصلت	حصلت	PROPN
iajs-758	231	19	علیه	علیه	PROPN
iajs-758	231	20	مودیول	مودیول	PROPN
iajs-758	231	21	،	،	PROPN
iajs-758	231	22	اكبر	اكبر	PROPN
iajs-758	231	23	،	،	PROPN
iajs-758	231	24	المودیل	المودیل	PROPN
iajs-758	231	25	،	،	PROPN
iajs-758	231	26	الحلقة	الحلقة	PROPN
iajs-758	231	27	:	:	PUNCT
iajs-758	231	28	الكلمات	الكلمات	VERB
iajs-758	231	29	المفتاحیة	المفتاحیة	PROPN
iajs-758	231	30	max	max	PROPN
iajs-758	231	31	primary	primary	ADJ
iajs-758	231	32	quasi	quasi	NOUN
iajs-758	231	33	-	-	ADJ
iajs-758	231	34	primary	primary	ADJ
iajs-758	231	35	semi	semi	ADJ
iajs-758	231	36	-	-	ADJ
iajs-758	231	37	primary	primary	ADJ