id	sid	tid	token	lemma	pos
iajs-557	1	1	mathematics	mathematic	NOUN
iajs-557	1	2	393	393	NUM
iajs-557	1	3	مجلة	مجلة	NOUN
iajs-557	1	4	إبن	إبن	VERB
iajs-557	1	5	الهيثم	الهيثم	ADJ
iajs-557	1	6	للعلوم	للعلوم	NOUN
iajs-557	1	7	الصرفة	الصرفة	NOUN
iajs-557	2	1	و	و	PRON
iajs-557	2	2	التطبيقية	التطبيقية	ADJ
iajs-557	2	3	2012	2012	NUM
iajs-557	2	4	السنة	السنة	NOUN
iajs-557	2	5	25	25	NUM
iajs-557	2	6	المجلد	المجلد	NOUN
iajs-557	2	7	3	3	NUM
iajs-557	2	8	العدد	العدد	PROPN
iajs-557	2	9	ibn	ibn	PROPN
iajs-557	2	10	al	al	PROPN
iajs-557	2	11	-	-	PUNCT
iajs-557	2	12	haitham	haitham	PROPN
iajs-557	2	13	journal	journal	PROPN
iajs-557	2	14	for	for	ADP
iajs-557	2	15	pure	pure	ADJ
iajs-557	2	16	and	and	CCONJ
iajs-557	2	17	applied	apply	VERB
iajs-557	2	18	science	science	NOUN
iajs-557	2	19	no	no	NOUN
iajs-557	2	20	.	.	NOUN
iajs-557	2	21	3	3	NUM
iajs-557	2	22	vol	vol	NOUN
iajs-557	2	23	.	.	PUNCT
iajs-557	3	1	25	25	NUM
iajs-557	3	2	year	year	NOUN
iajs-557	3	3	2012	2012	NUM
iajs-557	3	4	strongly	strongly	ADV
iajs-557	3	5	(	(	PUNCT
iajs-557	3	6	comletely	comletely	ADV
iajs-557	3	7	)	)	PUNCT
iajs-557	3	8	hollow	hollow	ADJ
iajs-557	3	9	submodules	submodule	NOUN
iajs-557	3	10	i	i	PRON
iajs-557	3	11	i.	i.	PROPN
iajs-557	3	12	m.	m.	PROPN
iajs-557	3	13	a.	a.	PROPN
iajs-557	3	14	hadi	hadi	PROPN
iajs-557	3	15	and	and	CCONJ
iajs-557	3	16	gh	gh	PROPN
iajs-557	3	17	.	.	PUNCT
iajs-557	3	18	a.	a.	PROPN
iajs-557	3	19	humod	humod	PROPN
iajs-557	3	20	department	department	PROPN
iajs-557	3	21	of	of	ADP
iajs-557	3	22	mathematics	mathematics	PROPN
iajs-557	3	23	,	,	PUNCT
iajs-557	3	24	college	college	NOUN
iajs-557	3	25	of	of	ADP
iajs-557	3	26	education	education	PROPN
iajs-557	3	27	ibn	ibn	PROPN
iajs-557	3	28	-	-	PUNCT
iajs-557	3	29	al	al	PROPN
iajs-557	3	30	-	-	PUNCT
iajs-557	3	31	haitham	haitham	PROPN
iajs-557	3	32	,	,	PUNCT
iajs-557	3	33	university	university	PROPN
iajs-557	3	34	of	of	ADP
iajs-557	3	35	baghdad	baghdad	PROPN
iajs-557	3	36	received	receive	VERB
iajs-557	3	37	in	in	ADP
iajs-557	3	38	:	:	PUNCT
iajs-557	3	39	15	15	NUM
iajs-557	3	40	march	march	NOUN
iajs-557	3	41	2012	2012	NUM
iajs-557	3	42	accepted	accept	VERB
iajs-557	3	43	in	in	ADP
iajs-557	3	44	:	:	PUNCT
iajs-557	3	45	21	21	NUM
iajs-557	3	46	may	may	PROPN
iajs-557	3	47	2012	2012	NUM
iajs-557	3	48	abstract	abstract	ADV
iajs-557	3	49	let	let	VERB
iajs-557	3	50	r	r	PRON
iajs-557	3	51	be	be	AUX
iajs-557	3	52	a	a	DET
iajs-557	3	53	commutative	commutative	ADJ
iajs-557	3	54	ring	ring	NOUN
iajs-557	3	55	with	with	ADP
iajs-557	3	56	unity	unity	NOUN
iajs-557	3	57	and	and	CCONJ
iajs-557	3	58	let	let	VERB
iajs-557	3	59	m	m	PRON
iajs-557	3	60	be	be	AUX
iajs-557	3	61	an	an	DET
iajs-557	3	62	r	r	NOUN
iajs-557	3	63	-	-	PUNCT
iajs-557	3	64	module	module	NOUN
iajs-557	3	65	.	.	PUNCT
iajs-557	4	1	in	in	ADP
iajs-557	4	2	this	this	DET
iajs-557	4	3	paper	paper	NOUN
iajs-557	4	4	we	we	PRON
iajs-557	4	5	study	study	VERB
iajs-557	4	6	strongly	strongly	ADV
iajs-557	4	7	(	(	PUNCT
iajs-557	4	8	completely	completely	ADV
iajs-557	4	9	)	)	PUNCT
iajs-557	4	10	hollow	hollow	ADJ
iajs-557	4	11	submodules	submodule	NOUN
iajs-557	4	12	and	and	CCONJ
iajs-557	4	13	quasi	quasi	ADJ
iajs-557	4	14	-	-	ADJ
iajs-557	4	15	hollow	hollow	ADJ
iajs-557	4	16	submodules	submodule	NOUN
iajs-557	4	17	.	.	PUNCT
iajs-557	5	1	we	we	PRON
iajs-557	5	2	investigate	investigate	VERB
iajs-557	5	3	the	the	DET
iajs-557	5	4	basic	basic	ADJ
iajs-557	5	5	properties	property	NOUN
iajs-557	5	6	of	of	ADP
iajs-557	5	7	these	these	DET
iajs-557	5	8	submodules	submodule	NOUN
iajs-557	5	9	and	and	CCONJ
iajs-557	5	10	the	the	DET
iajs-557	5	11	relationships	relationship	NOUN
iajs-557	5	12	between	between	ADP
iajs-557	5	13	them	they	PRON
iajs-557	5	14	.	.	PUNCT
iajs-557	6	1	also	also	ADV
iajs-557	6	2	we	we	PRON
iajs-557	6	3	study	study	VERB
iajs-557	6	4	the	the	DET
iajs-557	6	5	be	be	NOUN
iajs-557	6	6	behavior	behavior	NOUN
iajs-557	6	7	of	of	ADP
iajs-557	6	8	these	these	DET
iajs-557	6	9	submodules	submodule	NOUN
iajs-557	6	10	under	under	ADP
iajs-557	6	11	certain	certain	ADJ
iajs-557	6	12	class	class	NOUN
iajs-557	6	13	of	of	ADP
iajs-557	6	14	modules	module	NOUN
iajs-557	6	15	such	such	ADJ
iajs-557	6	16	as	as	ADP
iajs-557	6	17	compultiplication	compultiplication	NOUN
iajs-557	6	18	,	,	PUNCT
iajs-557	6	19	distributive	distributive	ADJ
iajs-557	6	20	,	,	PUNCT
iajs-557	6	21	multiplication	multiplication	NOUN
iajs-557	6	22	and	and	CCONJ
iajs-557	6	23	scalar	scalar	ADJ
iajs-557	6	24	modules	module	NOUN
iajs-557	6	25	.	.	PUNCT
iajs-557	7	1	in	in	ADP
iajs-557	7	2	part	part	NOUN
iajs-557	7	3	ii	ii	NOUN
iajs-557	7	4	we	we	PRON
iajs-557	7	5	shall	shall	AUX
iajs-557	7	6	continue	continue	VERB
iajs-557	7	7	the	the	DET
iajs-557	7	8	study	study	NOUN
iajs-557	7	9	of	of	ADP
iajs-557	7	10	these	these	DET
iajs-557	7	11	submodules	submodule	NOUN
iajs-557	7	12	.	.	PUNCT
iajs-557	8	1	key	key	ADJ
iajs-557	8	2	words	word	NOUN
iajs-557	8	3	:	:	PUNCT
iajs-557	8	4	strongly	strongly	ADV
iajs-557	8	5	(	(	PUNCT
iajs-557	8	6	completely)-hollow	completely)-hollow	ADJ
iajs-557	8	7	submodules	submodule	NOUN
iajs-557	8	8	,	,	PUNCT
iajs-557	8	9	distributive	distributive	ADJ
iajs-557	8	10	modules	module	NOUN
iajs-557	8	11	,	,	PUNCT
iajs-557	8	12	multiplication	multiplication	NOUN
iajs-557	8	13	(	(	PUNCT
iajs-557	8	14	comultiplication	comultiplication	NOUN
iajs-557	8	15	)	)	PUNCT
iajs-557	8	16	modules	module	NOUN
iajs-557	8	17	,	,	PUNCT
iajs-557	8	18	scalar	scalar	ADJ
iajs-557	8	19	modules	module	NOUN
iajs-557	8	20	.	.	PUNCT
iajs-557	9	1	introduction	introduction	NOUN
iajs-557	9	2	throughout	throughout	ADP
iajs-557	9	3	this	this	DET
iajs-557	9	4	paper	paper	NOUN
iajs-557	9	5	,	,	PUNCT
iajs-557	9	6	all	all	DET
iajs-557	9	7	rings	ring	NOUN
iajs-557	9	8	are	be	AUX
iajs-557	9	9	commutative	commutative	ADJ
iajs-557	9	10	rings	ring	NOUN
iajs-557	9	11	with	with	ADP
iajs-557	9	12	identity	identity	NOUN
iajs-557	9	13	elements	element	NOUN
iajs-557	9	14	,	,	PUNCT
iajs-557	9	15	and	and	CCONJ
iajs-557	9	16	all	all	DET
iajs-557	9	17	modules	module	NOUN
iajs-557	9	18	are	be	AUX
iajs-557	9	19	unital	unital	ADJ
iajs-557	9	20	modules	module	NOUN
iajs-557	9	21	.	.	PUNCT
iajs-557	10	1	in	in	ADP
iajs-557	10	2	this	this	DET
iajs-557	10	3	article	article	NOUN
iajs-557	10	4	we	we	PRON
iajs-557	10	5	study	study	VERB
iajs-557	10	6	strongly	strongly	ADV
iajs-557	10	7	(	(	PUNCT
iajs-557	10	8	completely	completely	ADV
iajs-557	10	9	)	)	PUNCT
iajs-557	10	10	hollow	hollow	ADJ
iajs-557	10	11	submodules	submodule	NOUN
iajs-557	10	12	which	which	PRON
iajs-557	10	13	are	be	AUX
iajs-557	10	14	introduced	introduce	VERB
iajs-557	10	15	in	in	ADP
iajs-557	10	16	[	[	X
iajs-557	10	17	1	1	NUM
iajs-557	10	18	]	]	PUNCT
iajs-557	10	19	,	,	PUNCT
iajs-557	10	20	also	also	ADV
iajs-557	10	21	we	we	PRON
iajs-557	10	22	introduce	introduce	VERB
iajs-557	10	23	quasi	quasi	ADJ
iajs-557	10	24	-	-	ADJ
iajs-557	10	25	hollow	hollow	ADJ
iajs-557	10	26	submodules	submodule	NOUN
iajs-557	10	27	.	.	PUNCT
iajs-557	11	1	in	in	ADP
iajs-557	11	2	section	section	NOUN
iajs-557	11	3	one	one	NUM
iajs-557	11	4	of	of	ADP
iajs-557	11	5	this	this	DET
iajs-557	11	6	paper	paper	NOUN
iajs-557	11	7	we	we	PRON
iajs-557	11	8	give	give	VERB
iajs-557	11	9	the	the	DET
iajs-557	11	10	basic	basic	ADJ
iajs-557	11	11	properties	property	NOUN
iajs-557	11	12	of	of	ADP
iajs-557	11	13	these	these	DET
iajs-557	11	14	submodules	submodule	NOUN
iajs-557	11	15	.	.	PUNCT
iajs-557	12	1	also	also	ADV
iajs-557	12	2	we	we	PRON
iajs-557	12	3	give	give	VERB
iajs-557	12	4	some	some	DET
iajs-557	12	5	results	result	NOUN
iajs-557	12	6	under	under	ADP
iajs-557	12	7	the	the	DET
iajs-557	12	8	class	class	NOUN
iajs-557	12	9	of	of	ADP
iajs-557	12	10	distributive	distributive	ADJ
iajs-557	12	11	modules	module	NOUN
iajs-557	12	12	and	and	CCONJ
iajs-557	12	13	compultiplication	compultiplication	NOUN
iajs-557	12	14	modules	module	NOUN
iajs-557	12	15	.	.	PUNCT
iajs-557	13	1	in	in	ADP
iajs-557	13	2	section	section	NOUN
iajs-557	13	3	two	two	NUM
iajs-557	13	4	,	,	PUNCT
iajs-557	13	5	we	we	PRON
iajs-557	13	6	investigate	investigate	VERB
iajs-557	13	7	some	some	DET
iajs-557	13	8	properties	property	NOUN
iajs-557	13	9	of	of	ADP
iajs-557	13	10	strongly	strongly	ADV
iajs-557	13	11	,	,	PUNCT
iajs-557	13	12	completely	completely	ADV
iajs-557	13	13	and	and	CCONJ
iajs-557	13	14	quasi	quasi	ADJ
iajs-557	13	15	-	-	ADJ
iajs-557	13	16	hollow	hollow	ADJ
iajs-557	13	17	submodules	submodule	NOUN
iajs-557	13	18	under	under	ADP
iajs-557	13	19	the	the	DET
iajs-557	13	20	class	class	NOUN
iajs-557	13	21	of	of	ADP
iajs-557	13	22	multiplication	multiplication	NOUN
iajs-557	13	23	modules	module	NOUN
iajs-557	13	24	.	.	PUNCT
iajs-557	14	1	in	in	ADP
iajs-557	14	2	section	section	NOUN
iajs-557	14	3	three	three	NUM
iajs-557	14	4	we	we	PRON
iajs-557	14	5	introduce	introduce	VERB
iajs-557	14	6	some	some	DET
iajs-557	14	7	properties	property	NOUN
iajs-557	14	8	of	of	ADP
iajs-557	14	9	strongly	strongly	ADV
iajs-557	14	10	(	(	PUNCT
iajs-557	14	11	completely	completely	ADV
iajs-557	14	12	)	)	PUNCT
iajs-557	14	13	and	and	CCONJ
iajs-557	14	14	quasi	quasi	ADJ
iajs-557	14	15	-	-	ADJ
iajs-557	14	16	hollow	hollow	ADJ
iajs-557	14	17	submodules	submodule	NOUN
iajs-557	14	18	under	under	ADP
iajs-557	14	19	certain	certain	ADJ
iajs-557	14	20	class	class	NOUN
iajs-557	14	21	of	of	ADP
iajs-557	14	22	modules	module	NOUN
iajs-557	14	23	.	.	PUNCT
iajs-557	15	1	1strongly	1strongly	NUM
iajs-557	15	2	(	(	PUNCT
iajs-557	15	3	completely	completely	ADV
iajs-557	15	4	)	)	PUNCT
iajs-557	15	5	hollow	hollow	ADJ
iajs-557	15	6	and	and	CCONJ
iajs-557	15	7	quasi	quasi	ADJ
iajs-557	15	8	-	-	ADJ
iajs-557	15	9	hollow	hollow	ADJ
iajs-557	15	10	submodules	submodule	NOUN
iajs-557	15	11	we	we	PRON
iajs-557	15	12	begin	begin	VERB
iajs-557	15	13	this	this	DET
iajs-557	15	14	section	section	NOUN
iajs-557	15	15	with	with	ADP
iajs-557	15	16	the	the	DET
iajs-557	15	17	following	following	NOUN
iajs-557	15	18	:	:	PUNCT
iajs-557	15	19	definition	definition	NOUN
iajs-557	15	20	:	:	PUNCT
iajs-557	16	1	[	[	X
iajs-557	16	2	1	1	NUM
iajs-557	16	3	,	,	PUNCT
iajs-557	16	4	4.2	4.2	NUM
iajs-557	16	5	]	]	PUNCT
iajs-557	16	6	let	let	VERB
iajs-557	16	7	0	0	NUM
iajs-557	16	8	≠	≠	PROPN
iajs-557	16	9	l	l	NOUN
iajs-557	16	10	≤	≤	NUM
iajs-557	16	11	m	m	ADP
iajs-557	16	12	,	,	PUNCT
iajs-557	16	13	then	then	ADV
iajs-557	16	14	l	l	NOUN
iajs-557	16	15	is	be	AUX
iajs-557	16	16	called	call	VERB
iajs-557	16	17	a	a	DET
iajs-557	16	18	strongly	strongly	ADV
iajs-557	16	19	hollow	hollow	ADJ
iajs-557	16	20	submodule	submodule	NOUN
iajs-557	16	21	(	(	PUNCT
iajs-557	16	22	briefly	briefly	ADV
iajs-557	16	23	,	,	PUNCT
iajs-557	16	24	sh	sh	PROPN
iajs-557	16	25	-	-	PUNCT
iajs-557	16	26	submodule	submodule	NOUN
iajs-557	16	27	)	)	PUNCT
iajs-557	16	28	if	if	SCONJ
iajs-557	16	29	for	for	ADP
iajs-557	16	30	every	every	DET
iajs-557	16	31	l1	l1	PROPN
iajs-557	16	32	,	,	PUNCT
iajs-557	16	33	l2	l2	VERB
iajs-557	16	34	≤	≤	NOUN
iajs-557	16	35	m	m	VERB
iajs-557	16	36	with	with	ADP
iajs-557	16	37	l	l	NOUN
iajs-557	16	38	≤	≤	PROPN
iajs-557	16	39	l1	l1	PROPN
iajs-557	16	40	+	+	CCONJ
iajs-557	16	41	l2	l2	NOUN
iajs-557	16	42	implies	imply	VERB
iajs-557	16	43	l	l	NOUN
iajs-557	16	44	≤	≤	PROPN
iajs-557	16	45	l1	l1	PROPN
iajs-557	16	46	or	or	CCONJ
iajs-557	16	47	l	l	NOUN
iajs-557	16	48	≤	≤	NOUN
iajs-557	16	49	l2	l2	NOUN
iajs-557	16	50	,	,	PUNCT
iajs-557	16	51	we	we	PRON
iajs-557	16	52	say	say	VERB
iajs-557	16	53	that	that	SCONJ
iajs-557	16	54	an	an	DET
iajs-557	16	55	r	r	NOUN
iajs-557	16	56	-	-	PUNCT
iajs-557	16	57	module	module	NOUN
iajs-557	16	58	m	m	NOUN
iajs-557	16	59	is	be	AUX
iajs-557	16	60	a	a	DET
iajs-557	16	61	strongly	strongly	ADV
iajs-557	16	62	-	-	PUNCT
iajs-557	16	63	hollow	hollow	NOUN
iajs-557	16	64	module	module	NOUN
iajs-557	16	65	if	if	SCONJ
iajs-557	16	66	m	m	NOUN
iajs-557	16	67	is	be	AUX
iajs-557	16	68	a	a	DET
iajs-557	16	69	strongly	strongly	ADV
iajs-557	16	70	hollow	hollow	ADJ
iajs-557	16	71	submodule	submodule	NOUN
iajs-557	16	72	of	of	ADP
iajs-557	16	73	itself	itself	PRON
iajs-557	16	74	.	.	PUNCT
iajs-557	17	1	remark	remark	VERB
iajs-557	17	2	:	:	PUNCT
iajs-557	17	3	let	let	VERB
iajs-557	17	4	0	0	NUM
iajs-557	17	5	≠	≠	PROPN
iajs-557	17	6	l	l	NOUN
iajs-557	17	7	≤	≤	NUM
iajs-557	17	8	m	m	NOUN
iajs-557	17	9	,	,	PUNCT
iajs-557	17	10	l	l	NOUN
iajs-557	17	11	is	be	AUX
iajs-557	17	12	a	a	DET
iajs-557	17	13	sh	sh	NOUN
iajs-557	17	14	-	-	PUNCT
iajs-557	17	15	submodule	submodule	NOUN
iajs-557	17	16	if	if	SCONJ
iajs-557	17	17	for	for	ADP
iajs-557	17	18	each	each	DET
iajs-557	17	19	l1	l1	PROPN
iajs-557	17	20	,	,	PUNCT
iajs-557	17	21	…	…	PUNCT
iajs-557	17	22	,	,	PUNCT
iajs-557	17	23	ln	ln	ADJ
iajs-557	17	24	≤	≤	NOUN
iajs-557	17	25	m	m	VERB
iajs-557	17	26	with	with	ADP
iajs-557	17	27	l	l	NOUN
iajs-557	17	28	≤	≤	PROPN
iajs-557	17	29	l1	l1	NOUN
iajs-557	17	30	+	+	CCONJ
iajs-557	17	31	l2	l2	NOUN
iajs-557	17	32	+	+	CCONJ
iajs-557	17	33	…	…	PUNCT
iajs-557	18	1	+	+	CCONJ
iajs-557	18	2	ln	ln	ADJ
iajs-557	18	3	,	,	PUNCT
iajs-557	18	4	implies	imply	VERB
iajs-557	18	5	l	l	NOUN
iajs-557	18	6	≤	≤	PROPN
iajs-557	18	7	l1	l1	PROPN
iajs-557	18	8	or	or	CCONJ
iajs-557	18	9	l	l	NOUN
iajs-557	18	10	≤	≤	NOUN
iajs-557	18	11	l2	l2	NOUN
iajs-557	18	12	…	…	PUNCT
iajs-557	18	13	or	or	CCONJ
iajs-557	18	14	l	l	NOUN
iajs-557	18	15	≤	≤	X
iajs-557	19	1	ln	ln	ADJ
iajs-557	19	2	.	.	PUNCT
iajs-557	20	1	definition	definition	NOUN
iajs-557	20	2	:	:	PUNCT
iajs-557	21	1	[	[	X
iajs-557	21	2	1	1	NUM
iajs-557	21	3	,	,	PUNCT
iajs-557	21	4	4.2	4.2	NUM
iajs-557	21	5	]	]	PUNCT
iajs-557	21	6	let	let	VERB
iajs-557	21	7	0	0	NUM
iajs-557	21	8	≠	≠	PROPN
iajs-557	21	9	l	l	NOUN
iajs-557	21	10	≤	≤	NUM
iajs-557	21	11	m	m	ADP
iajs-557	21	12	,	,	PUNCT
iajs-557	21	13	then	then	ADV
iajs-557	21	14	l	l	NOUN
iajs-557	21	15	is	be	AUX
iajs-557	21	16	called	call	VERB
iajs-557	21	17	a	a	DET
iajs-557	21	18	completely	completely	ADV
iajs-557	21	19	hollow	hollow	ADJ
iajs-557	21	20	submodule	submodule	NOUN
iajs-557	21	21	(	(	PUNCT
iajs-557	21	22	briefly	briefly	ADV
iajs-557	21	23	,	,	PUNCT
iajs-557	21	24	ch	ch	NOUN
iajs-557	21	25	-	-	PUNCT
iajs-557	21	26	submodule	submodule	NOUN
iajs-557	21	27	)	)	PUNCT
iajs-557	21	28	if	if	SCONJ
iajs-557	21	29	for	for	ADP
iajs-557	21	30	any	any	DET
iajs-557	21	31	collection	collection	NOUN
iajs-557	21	32	{	{	PUNCT
iajs-557	21	33	lλ}λ∈λ	lλ}λ∈λ	X
iajs-557	21	34	of	of	ADP
iajs-557	21	35	r	r	NOUN
iajs-557	21	36	-	-	PUNCT
iajs-557	21	37	submodules	submodule	NOUN
iajs-557	21	38	of	of	ADP
iajs-557	21	39	m	m	PROPN
iajs-557	21	40	with	with	ADP
iajs-557	21	41	l	l	NOUN
iajs-557	22	1	lλ	lλ	NOUN
iajs-557	22	2	λ∈λ	λ∈λ	NOUN
iajs-557	22	3	=	=	PUNCT
iajs-557	22	4	∑	∑	PROPN
iajs-557	22	5	,	,	PUNCT
iajs-557	22	6	implies	imply	VERB
iajs-557	22	7	l	l	NOUN
iajs-557	22	8	=	=	PUNCT
iajs-557	22	9	lλ	lλ	NOUN
iajs-557	22	10	for	for	ADP
iajs-557	22	11	some	some	DET
iajs-557	22	12	λ∈λ	λ∈λ	NOUN
iajs-557	22	13	.	.	PUNCT
iajs-557	23	1	we	we	PRON
iajs-557	23	2	say	say	VERB
iajs-557	23	3	that	that	SCONJ
iajs-557	23	4	an	an	DET
iajs-557	23	5	r	r	NOUN
iajs-557	23	6	-	-	PUNCT
iajs-557	23	7	module	module	NOUN
iajs-557	23	8	m	m	NOUN
iajs-557	23	9	is	be	AUX
iajs-557	23	10	completely	completely	ADV
iajs-557	23	11	hollow	hollow	ADJ
iajs-557	23	12	(	(	PUNCT
iajs-557	23	13	briefly	briefly	ADV
iajs-557	23	14	,	,	PUNCT
iajs-557	23	15	ch	ch	NOUN
iajs-557	23	16	-	-	PUNCT
iajs-557	23	17	module	module	NOUN
iajs-557	23	18	)	)	PUNCT
iajs-557	23	19	if	if	SCONJ
iajs-557	23	20	m	m	NOUN
iajs-557	23	21	is	be	AUX
iajs-557	23	22	completely	completely	ADV
iajs-557	23	23	hollow	hollow	ADJ
iajs-557	23	24	of	of	ADP
iajs-557	23	25	itself	itself	PRON
iajs-557	23	26	.	.	PUNCT
iajs-557	24	1	remarks	remark	NOUN
iajs-557	24	2	and	and	CCONJ
iajs-557	24	3	examples	example	NOUN
iajs-557	24	4	:	:	PUNCT
iajs-557	24	5	mathematics	mathematic	NOUN
iajs-557	24	6	394	394	NUM
iajs-557	24	7	مجلة	مجلة	NOUN
iajs-557	24	8	إبن	إبن	VERB
iajs-557	24	9	الهيثم	الهيثم	ADJ
iajs-557	24	10	للعلوم	للعلوم	NOUN
iajs-557	24	11	الصرفة	الصرفة	NOUN
iajs-557	25	1	و	و	PRON
iajs-557	25	2	التطبيقية	التطبيقية	ADJ
iajs-557	25	3	2012	2012	NUM
iajs-557	25	4	السنة	السنة	NOUN
iajs-557	25	5	25	25	NUM
iajs-557	25	6	المجلد	المجلد	NOUN
iajs-557	25	7	3	3	NUM
iajs-557	25	8	العدد	العدد	PROPN
iajs-557	25	9	ibn	ibn	PROPN
iajs-557	25	10	al	al	PROPN
iajs-557	25	11	-	-	PUNCT
iajs-557	25	12	haitham	haitham	PROPN
iajs-557	25	13	journal	journal	PROPN
iajs-557	25	14	for	for	ADP
iajs-557	25	15	pure	pure	ADJ
iajs-557	25	16	and	and	CCONJ
iajs-557	25	17	applied	apply	VERB
iajs-557	25	18	science	science	NOUN
iajs-557	25	19	no	no	NOUN
iajs-557	25	20	.	.	NOUN
iajs-557	25	21	3	3	NUM
iajs-557	25	22	vol	vol	NOUN
iajs-557	25	23	.	.	PUNCT
iajs-557	26	1	25	25	NUM
iajs-557	26	2	year	year	NOUN
iajs-557	26	3	2012	2012	NUM
iajs-557	26	4	the	the	DET
iajs-557	26	5	z	z	NOUN
iajs-557	26	6	as	as	ADP
iajs-557	26	7	z	z	NOUN
iajs-557	26	8	-	-	PUNCT
iajs-557	26	9	module	module	NOUN
iajs-557	26	10	is	be	AUX
iajs-557	26	11	not	not	PART
iajs-557	26	12	sh	sh	PROPN
iajs-557	26	13	,	,	PUNCT
iajs-557	26	14	not	not	PART
iajs-557	26	15	ch	ch	NOUN
iajs-557	26	16	,	,	PUNCT
iajs-557	26	17	and	and	CCONJ
iajs-557	26	18	every	every	DET
iajs-557	26	19	submodule	submodule	NOUN
iajs-557	26	20	is	be	AUX
iajs-557	26	21	not	not	PART
iajs-557	26	22	sh	sh	PROPN
iajs-557	26	23	,	,	PUNCT
iajs-557	26	24	not	not	PART
iajs-557	26	25	ch	ch	NOUN
iajs-557	26	26	.	.	PROPN
iajs-557	26	27	1	1	NUM
iajs-557	26	28	.	.	X
iajs-557	26	29	z6	z6	PROPN
iajs-557	26	30	as	as	SCONJ
iajs-557	26	31	z	z	NOUN
iajs-557	26	32	-	-	PUNCT
iajs-557	26	33	module	module	NOUN
iajs-557	26	34	is	be	AUX
iajs-557	26	35	not	not	PART
iajs-557	26	36	sh	sh	PROPN
iajs-557	26	37	,	,	PUNCT
iajs-557	26	38	and	and	CCONJ
iajs-557	26	39	every	every	DET
iajs-557	26	40	nonzero	nonzero	ADJ
iajs-557	26	41	proper	proper	ADJ
iajs-557	26	42	submodule	submodule	NOUN
iajs-557	26	43	is	be	AUX
iajs-557	26	44	sh	sh	PROPN
iajs-557	26	45	.	.	PROPN
iajs-557	26	46	2	2	NUM
iajs-557	26	47	.	.	X
iajs-557	26	48	q	q	PUNCT
iajs-557	27	1	as	as	SCONJ
iajs-557	27	2	z	z	NOUN
iajs-557	27	3	-	-	PUNCT
iajs-557	27	4	module	module	NOUN
iajs-557	27	5	is	be	AUX
iajs-557	27	6	not	not	PART
iajs-557	27	7	sh	sh	PROPN
iajs-557	27	8	,	,	PUNCT
iajs-557	27	9	since	since	SCONJ
iajs-557	27	10	there	there	PRON
iajs-557	27	11	exist	exist	VERB
iajs-557	27	12	two	two	NUM
iajs-557	27	13	proper	proper	ADJ
iajs-557	27	14	submodules	submodule	NOUN
iajs-557	27	15	a	a	PRON
iajs-557	27	16	,	,	PUNCT
iajs-557	27	17	b	b	PROPN
iajs-557	27	18	of	of	ADP
iajs-557	27	19	q	q	NOUN
iajs-557	28	1	such	such	ADJ
iajs-557	28	2	that	that	DET
iajs-557	28	3	q	q	NOUN
iajs-557	29	1	=	=	PUNCT
iajs-557	29	2	a	a	DET
iajs-557	29	3	+	+	NOUN
iajs-557	29	4	b	b	AUX
iajs-557	29	5	see	see	NOUN
iajs-557	29	6	[	[	X
iajs-557	29	7	2	2	NUM
iajs-557	29	8	,	,	PUNCT
iajs-557	29	9	p.187	p.187	NUM
iajs-557	29	10	,	,	PUNCT
iajs-557	29	11	exc.6(b	exc.6(b	NUM
iajs-557	29	12	)	)	PUNCT
iajs-557	29	13	]	]	PUNCT
iajs-557	29	14	.	.	PUNCT
iajs-557	30	1	3	3	X
iajs-557	30	2	.	.	X
iajs-557	30	3	let	let	VERB
iajs-557	30	4	m	m	PRON
iajs-557	30	5	be	be	AUX
iajs-557	30	6	an	an	DET
iajs-557	30	7	r	r	NOUN
iajs-557	30	8	-	-	PUNCT
iajs-557	30	9	module	module	NOUN
iajs-557	30	10	,	,	PUNCT
iajs-557	30	11	and	and	CCONJ
iajs-557	31	1	n	n	DET
iajs-557	31	2	≤	≤	NOUN
iajs-557	31	3	l	l	NOUN
iajs-557	31	4	≤	≤	ADJ
iajs-557	31	5	m.	m.	NOUN
iajs-557	31	6	if	if	SCONJ
iajs-557	31	7	l	l	NOUN
iajs-557	31	8	is	be	AUX
iajs-557	31	9	sh	sh	INTJ
iajs-557	31	10	then	then	ADV
iajs-557	31	11	n	n	PRON
iajs-557	31	12	need	need	AUX
iajs-557	31	13	not	not	PART
iajs-557	31	14	be	be	AUX
iajs-557	31	15	sh	sh	NOUN
iajs-557	31	16	-	-	PUNCT
iajs-557	31	17	submodule	submodule	NOUN
iajs-557	31	18	.	.	PUNCT
iajs-557	32	1	for	for	ADP
iajs-557	32	2	example	example	NOUN
iajs-557	32	3	,	,	PUNCT
iajs-557	32	4	2	2	X
iajs-557	32	5	<	<	X
iajs-557	32	6	>	>	X
iajs-557	32	7	is	be	AUX
iajs-557	32	8	sh	sh	INTJ
iajs-557	32	9	(	(	PUNCT
iajs-557	32	10	ch)-submodule	ch)-submodule	NOUN
iajs-557	32	11	of	of	ADP
iajs-557	32	12	z4	z4	PROPN
iajs-557	32	13	as	as	ADP
iajs-557	32	14	z	z	NOUN
iajs-557	32	15	-	-	NOUN
iajs-557	32	16	module	module	NOUN
iajs-557	32	17	.	.	PUNCT
iajs-557	33	1	but	but	CCONJ
iajs-557	33	2	0	0	X
iajs-557	33	3	<	<	X
iajs-557	33	4	>	>	X
iajs-557	33	5	is	be	AUX
iajs-557	33	6	not	not	PART
iajs-557	33	7	sh	sh	INTJ
iajs-557	33	8	(	(	PUNCT
iajs-557	33	9	not	not	PART
iajs-557	33	10	ch	ch	NOUN
iajs-557	33	11	)	)	PUNCT
iajs-557	33	12	.	.	PUNCT
iajs-557	34	1	4	4	X
iajs-557	34	2	.	.	X
iajs-557	34	3	let	let	VERB
iajs-557	34	4	m	m	PRON
iajs-557	34	5	be	be	AUX
iajs-557	34	6	an	an	DET
iajs-557	34	7	r	r	NOUN
iajs-557	34	8	-	-	PUNCT
iajs-557	34	9	module	module	NOUN
iajs-557	34	10	,	,	PUNCT
iajs-557	34	11	and	and	CCONJ
iajs-557	34	12	0	0	NUM
iajs-557	34	13	≠	≠	PROPN
iajs-557	34	14	l	l	NOUN
iajs-557	34	15	≤	≤	NUM
iajs-557	34	16	w	w	NOUN
iajs-557	34	17	≤	≤	NUM
iajs-557	34	18	m.	m.	NOUN
iajs-557	34	19	if	if	SCONJ
iajs-557	34	20	l	l	NOUN
iajs-557	34	21	is	be	AUX
iajs-557	34	22	sh	sh	PROPN
iajs-557	34	23	-	-	PUNCT
iajs-557	34	24	submodule	submodule	NOUN
iajs-557	34	25	,	,	PUNCT
iajs-557	34	26	then	then	ADV
iajs-557	34	27	w	w	NOUN
iajs-557	34	28	need	need	AUX
iajs-557	34	29	not	not	PART
iajs-557	34	30	be	be	AUX
iajs-557	34	31	sh	sh	NOUN
iajs-557	34	32	-	-	PUNCT
iajs-557	34	33	submodule	submodule	NOUN
iajs-557	34	34	.	.	PUNCT
iajs-557	35	1	for	for	ADP
iajs-557	35	2	example	example	NOUN
iajs-557	35	3	6	6	NUM
iajs-557	35	4	<	<	X
iajs-557	35	5	>	>	X
iajs-557	35	6	is	be	AUX
iajs-557	35	7	sh(ch)-submodule	sh(ch)-submodule	NOUN
iajs-557	35	8	of	of	ADP
iajs-557	35	9	z48	z48	NOUN
iajs-557	35	10	as	as	ADP
iajs-557	35	11	z	z	NOUN
iajs-557	35	12	-	-	NOUN
iajs-557	35	13	module	module	NOUN
iajs-557	35	14	.	.	PUNCT
iajs-557	36	1	but	but	CCONJ
iajs-557	36	2	2	2	NUM
iajs-557	36	3	<	<	X
iajs-557	36	4	>	>	X
iajs-557	36	5	is	be	AUX
iajs-557	36	6	not	not	PART
iajs-557	36	7	sh	sh	INTJ
iajs-557	36	8	(	(	PUNCT
iajs-557	36	9	not	not	PART
iajs-557	36	10	ch	ch	NOUN
iajs-557	36	11	)	)	PUNCT
iajs-557	36	12	,	,	PUNCT
iajs-557	36	13	since	since	SCONJ
iajs-557	36	14	2	2	NUM
iajs-557	36	15	<	<	X
iajs-557	36	16	>	>	X
iajs-557	36	17	⊆	⊆	NUM
iajs-557	36	18	8	8	NUM
iajs-557	36	19	<	<	X
iajs-557	36	20	>	>	X
iajs-557	37	1	+	+	NUM
iajs-557	37	2	6	6	NUM
iajs-557	37	3	<	<	X
iajs-557	37	4	>	>	X
iajs-557	37	5	,	,	PUNCT
iajs-557	37	6	and	and	CCONJ
iajs-557	37	7	2	2	NUM
iajs-557	37	8	<	<	X
iajs-557	37	9	>	>	X
iajs-557	37	10	⊈	⊈	PROPN
iajs-557	38	1	8	8	NUM
iajs-557	38	2	<	<	X
iajs-557	38	3	>	>	X
iajs-557	38	4	,	,	PUNCT
iajs-557	38	5	2	2	X
iajs-557	38	6	<	<	X
iajs-557	38	7	>	>	X
iajs-557	38	8	⊈	⊈	PROPN
iajs-557	38	9	6	6	NUM
iajs-557	38	10	<	<	X
iajs-557	38	11	>	>	X
iajs-557	38	12	.	.	PUNCT
iajs-557	39	1	5	5	X
iajs-557	39	2	.	.	X
iajs-557	39	3	let	let	VERB
iajs-557	39	4	m	m	PRON
iajs-557	39	5	be	be	AUX
iajs-557	39	6	an	an	DET
iajs-557	39	7	r	r	NOUN
iajs-557	39	8	-	-	PUNCT
iajs-557	39	9	module	module	NOUN
iajs-557	39	10	,	,	PUNCT
iajs-557	39	11	and	and	CCONJ
iajs-557	39	12	l1	l1	PROPN
iajs-557	39	13	,	,	PUNCT
iajs-557	39	14	l2	l2	NOUN
iajs-557	39	15	≤	≤	ADJ
iajs-557	39	16	m.	m.	NOUN
iajs-557	39	17	if	if	SCONJ
iajs-557	39	18	l1	l1	PROPN
iajs-557	39	19	and	and	CCONJ
iajs-557	39	20	l2	l2	NOUN
iajs-557	39	21	are	be	AUX
iajs-557	39	22	sh	sh	NOUN
iajs-557	39	23	-	-	PUNCT
iajs-557	39	24	submodule	submodule	NOUN
iajs-557	39	25	,	,	PUNCT
iajs-557	39	26	then	then	ADV
iajs-557	39	27	l1	l1	PROPN
iajs-557	39	28	+	+	CCONJ
iajs-557	39	29	l2	l2	NOUN
iajs-557	39	30	need	need	AUX
iajs-557	39	31	not	not	PART
iajs-557	39	32	be	be	AUX
iajs-557	39	33	sh	sh	PROPN
iajs-557	39	34	.	.	PUNCT
iajs-557	40	1	for	for	ADP
iajs-557	40	2	example	example	NOUN
iajs-557	40	3	:	:	PUNCT
iajs-557	40	4	in	in	ADP
iajs-557	40	5	z12	z12	PROPN
iajs-557	40	6	as	as	ADP
iajs-557	40	7	z	z	NOUN
iajs-557	40	8	-	-	PUNCT
iajs-557	40	9	module	module	NOUN
iajs-557	40	10	,	,	PUNCT
iajs-557	40	11	3	3	NUM
iajs-557	40	12	,	,	PUNCT
iajs-557	40	13	4	4	NUM
iajs-557	40	14	<	<	X
iajs-557	40	15	>	>	X
iajs-557	40	16	<	<	X
iajs-557	40	17	>	>	X
iajs-557	40	18	are	be	AUX
iajs-557	40	19	sh	sh	NOUN
iajs-557	40	20	-	-	PUNCT
iajs-557	40	21	submodules	submodule	NOUN
iajs-557	40	22	of	of	ADP
iajs-557	40	23	z12	z12	PROPN
iajs-557	40	24	.	.	PUNCT
iajs-557	41	1	but	but	CCONJ
iajs-557	41	2	123	123	NUM
iajs-557	41	3	4	4	NUM
iajs-557	41	4	z	z	NOUN
iajs-557	41	5	<	<	X
iajs-557	41	6	>	>	X
iajs-557	41	7	+	+	X
iajs-557	41	8	<	<	X
iajs-557	41	9	>	>	X
iajs-557	41	10	=	=	PUNCT
iajs-557	41	11	is	be	AUX
iajs-557	41	12	not	not	PART
iajs-557	41	13	sh	sh	PROPN
iajs-557	41	14	.	.	PROPN
iajs-557	41	15	6	6	NUM
iajs-557	41	16	.	.	PUNCT
iajs-557	42	1	if	if	SCONJ
iajs-557	42	2	m	m	NOUN
iajs-557	42	3	is	be	AUX
iajs-557	42	4	a	a	DET
iajs-557	42	5	chained	chain	VERB
iajs-557	42	6	r	r	NOUN
iajs-557	42	7	-	-	PUNCT
iajs-557	42	8	module	module	NOUN
iajs-557	42	9	,	,	PUNCT
iajs-557	42	10	and	and	CCONJ
iajs-557	42	11	0	0	NUM
iajs-557	42	12	≠	≠	PROPN
iajs-557	42	13	n	n	PRON
iajs-557	42	14	≤	≤	NOUN
iajs-557	42	15	m.	m.	NOUN
iajs-557	42	16	then	then	ADV
iajs-557	42	17	n	n	X
iajs-557	42	18	is	be	AUX
iajs-557	42	19	sh	sh	PROPN
iajs-557	42	20	-	-	PUNCT
iajs-557	42	21	submodule	submodule	NOUN
iajs-557	42	22	,	,	PUNCT
iajs-557	42	23	where	where	SCONJ
iajs-557	42	24	m	m	PROPN
iajs-557	42	25	is	be	AUX
iajs-557	42	26	a	a	DET
iajs-557	42	27	chained	chain	VERB
iajs-557	42	28	module	module	NOUN
iajs-557	42	29	if	if	SCONJ
iajs-557	42	30	the	the	DET
iajs-557	42	31	lattic	lattic	ADJ
iajs-557	42	32	of	of	ADP
iajs-557	42	33	submodules	submodule	NOUN
iajs-557	42	34	are	be	AUX
iajs-557	42	35	linearly	linearly	ADV
iajs-557	42	36	ordered	order	VERB
iajs-557	42	37	by	by	ADP
iajs-557	42	38	inclusion	inclusion	NOUN
iajs-557	42	39	see	see	VERB
iajs-557	42	40	[	[	X
iajs-557	42	41	3	3	NUM
iajs-557	42	42	]	]	PUNCT
iajs-557	42	43	.	.	PUNCT
iajs-557	43	1	proof	proof	NOUN
iajs-557	43	2	:	:	PUNCT
iajs-557	43	3	let	let	VERB
iajs-557	43	4	0	0	NUM
iajs-557	43	5	≠	≠	PROPN
iajs-557	43	6	n	n	PRON
iajs-557	43	7	≤	≤	NOUN
iajs-557	43	8	m.	m.	NOUN
iajs-557	43	9	assume	assume	VERB
iajs-557	43	10	n	n	CCONJ
iajs-557	43	11	⊆	⊆	NUM
iajs-557	43	12	n1	n1	NOUN
iajs-557	43	13	+	+	CCONJ
iajs-557	43	14	n2	n2	ADJ
iajs-557	43	15	where	where	SCONJ
iajs-557	43	16	n1	n1	NOUN
iajs-557	43	17	,	,	PUNCT
iajs-557	43	18	n2	n2	ADJ
iajs-557	43	19	≤	≤	NUM
iajs-557	43	20	m.	m.	NOUN
iajs-557	43	21	since	since	SCONJ
iajs-557	43	22	m	m	PROPN
iajs-557	43	23	is	be	AUX
iajs-557	43	24	chained	chain	VERB
iajs-557	43	25	,	,	PUNCT
iajs-557	43	26	either	either	CCONJ
iajs-557	43	27	n1	n1	PROPN
iajs-557	43	28	⊆	⊆	NUM
iajs-557	43	29	n2	n2	NOUN
iajs-557	43	30	or	or	CCONJ
iajs-557	43	31	n2	n2	ADJ
iajs-557	43	32	⊆	⊆	NUM
iajs-557	43	33	n1	n1	NOUN
iajs-557	43	34	if	if	SCONJ
iajs-557	43	35	n1	n1	PROPN
iajs-557	43	36	⊆	⊆	NUM
iajs-557	43	37	n2	n2	NOUN
iajs-557	43	38	,	,	PUNCT
iajs-557	43	39	then	then	ADV
iajs-557	43	40	n1	n1	PROPN
iajs-557	43	41	+	+	CCONJ
iajs-557	43	42	n2	n2	ADJ
iajs-557	43	43	=	=	SYM
iajs-557	43	44	n2	n2	NOUN
iajs-557	43	45	,	,	PUNCT
iajs-557	43	46	so	so	ADV
iajs-557	43	47	n	n	CCONJ
iajs-557	43	48	⊆	⊆	NUM
iajs-557	43	49	n2	n2	NOUN
iajs-557	43	50	.	.	PUNCT
iajs-557	44	1	if	if	SCONJ
iajs-557	44	2	n2	n2	ADJ
iajs-557	44	3	⊆	⊆	NUM
iajs-557	44	4	n1	n1	NOUN
iajs-557	44	5	,	,	PUNCT
iajs-557	44	6	then	then	ADV
iajs-557	44	7	n1	n1	PROPN
iajs-557	44	8	+	+	CCONJ
iajs-557	44	9	n2	n2	ADJ
iajs-557	44	10	=	=	SYM
iajs-557	44	11	n1	n1	NOUN
iajs-557	44	12	,	,	PUNCT
iajs-557	44	13	so	so	ADV
iajs-557	44	14	n	n	CCONJ
iajs-557	44	15	⊆	⊆	NUM
iajs-557	44	16	n1	n1	NOUN
iajs-557	44	17	.	.	PUNCT
iajs-557	45	1	thus	thus	ADV
iajs-557	45	2	n	n	X
iajs-557	45	3	is	be	AUX
iajs-557	45	4	sh	sh	NOUN
iajs-557	45	5	-	-	PUNCT
iajs-557	45	6	submodule	submodule	NOUN
iajs-557	45	7	.	.	PUNCT
iajs-557	46	1	7	7	X
iajs-557	46	2	.	.	X
iajs-557	46	3	every	every	DET
iajs-557	46	4	simple	simple	ADJ
iajs-557	46	5	r	r	NOUN
iajs-557	46	6	-	-	PUNCT
iajs-557	46	7	module	module	NOUN
iajs-557	46	8	m	m	NOUN
iajs-557	46	9	is	be	AUX
iajs-557	46	10	sh	sh	INTJ
iajs-557	46	11	and	and	CCONJ
iajs-557	46	12	ch	ch	NOUN
iajs-557	46	13	.	.	PROPN
iajs-557	46	14	8	8	NUM
iajs-557	46	15	.	.	PUNCT
iajs-557	47	1	every	every	DET
iajs-557	47	2	simple	simple	ADJ
iajs-557	47	3	submodule	submodule	NOUN
iajs-557	47	4	n	n	PROPN
iajs-557	47	5	of	of	ADP
iajs-557	47	6	an	an	DET
iajs-557	47	7	r	r	NOUN
iajs-557	47	8	-	-	PUNCT
iajs-557	47	9	module	module	NOUN
iajs-557	47	10	is	be	AUX
iajs-557	47	11	ch	ch	NOUN
iajs-557	47	12	-	-	PUNCT
iajs-557	47	13	submodule	submodule	NOUN
iajs-557	47	14	.	.	PUNCT
iajs-557	48	1	9	9	X
iajs-557	48	2	.	.	X
iajs-557	49	1	every	every	DET
iajs-557	49	2	ch	ch	NOUN
iajs-557	49	3	-	-	PUNCT
iajs-557	49	4	module	module	NOUN
iajs-557	49	5	is	be	AUX
iajs-557	49	6	sh	sh	NOUN
iajs-557	49	7	-	-	PUNCT
iajs-557	49	8	module	module	NOUN
iajs-557	49	9	.	.	PUNCT
iajs-557	50	1	10	10	NUM
iajs-557	50	2	.	.	PUNCT
iajs-557	51	1	the	the	DET
iajs-557	51	2	concept	concept	NOUN
iajs-557	51	3	sh	sh	PROPN
iajs-557	51	4	-	-	PUNCT
iajs-557	51	5	submodule	submodule	PROPN
iajs-557	51	6	and	and	CCONJ
iajs-557	51	7	ch	ch	NOUN
iajs-557	51	8	-	-	PUNCT
iajs-557	51	9	submodule	submodule	NOUN
iajs-557	51	10	are	be	AUX
iajs-557	51	11	independent	independent	ADJ
iajs-557	51	12	for	for	ADP
iajs-557	51	13	examples	example	NOUN
iajs-557	51	14	:	:	PUNCT
iajs-557	51	15	(	(	PUNCT
iajs-557	51	16	a	a	X
iajs-557	51	17	)	)	PUNCT
iajs-557	51	18	the	the	DET
iajs-557	51	19	z	z	NOUN
iajs-557	51	20	-	-	PUNCT
iajs-557	51	21	module	module	NOUN
iajs-557	51	22	pz	pz	NOUN
iajs-557	51	23	∞	∞	PROPN
iajs-557	51	24	is	be	AUX
iajs-557	51	25	sh	sh	PROPN
iajs-557	51	26	-	-	PUNCT
iajs-557	51	27	submodule	submodule	NOUN
iajs-557	51	28	of	of	ADP
iajs-557	51	29	itself	itself	PRON
iajs-557	51	30	;	;	PUNCT
iajs-557	51	31	that	that	PRON
iajs-557	51	32	is	be	AUX
iajs-557	51	33	pz	pz	PROPN
iajs-557	51	34	∞	∞	PROPN
iajs-557	51	35	is	be	AUX
iajs-557	51	36	sh	sh	NOUN
iajs-557	51	37	-	-	PUNCT
iajs-557	51	38	module	module	NOUN
iajs-557	51	39	by	by	ADP
iajs-557	51	40	remark	remark	NOUN
iajs-557	51	41	1.4	1.4	NUM
iajs-557	51	42	(	(	PUNCT
iajs-557	51	43	7	7	NUM
iajs-557	51	44	)	)	PUNCT
iajs-557	51	45	,	,	PUNCT
iajs-557	51	46	pz	pz	NOUN
iajs-557	51	47	∞	∞	PROPN
iajs-557	51	48	is	be	AUX
iajs-557	51	49	not	not	PART
iajs-557	51	50	ch	ch	NOUN
iajs-557	51	51	-	-	PUNCT
iajs-557	51	52	module	module	NOUN
iajs-557	51	53	.	.	PUNCT
iajs-557	52	1	since	since	SCONJ
iajs-557	52	2	ip	ip	PROPN
iajs-557	52	3	i	i	NOUN
iajs-557	52	4	z	z	NOUN
iajs-557	52	5	1z	1z	NUM
iajs-557	52	6	z	z	NOUN
iajs-557	53	1	p	p	X
iajs-557	53	2	∞	∞	PROPN
iajs-557	53	3	+	+	NOUN
iajs-557	53	4	∈	∈	NOUN
iajs-557	53	5	=	=	X
iajs-557	53	6	<	<	X
iajs-557	53	7	+	+	X
iajs-557	53	8	>	>	PUNCT
iajs-557	53	9	∑	∑	PUNCT
iajs-557	53	10	,	,	PUNCT
iajs-557	53	11	and	and	CCONJ
iajs-557	53	12	ip	ip	VERB
iajs-557	53	13	1z	1z	NOUN
iajs-557	53	14	z	z	NOUN
iajs-557	53	15	p	p	NOUN
iajs-557	53	16	∞	∞	PROPN
iajs-557	53	17	≠	≠	X
iajs-557	53	18	<	<	X
iajs-557	53	19	+	+	X
iajs-557	53	20	>	>	X
iajs-557	53	21	for	for	ADP
iajs-557	53	22	any	any	DET
iajs-557	53	23	i	i	PROPN
iajs-557	53	24	∈	∈	PROPN
iajs-557	53	25	z+	z+	X
iajs-557	53	26	.	.	PUNCT
iajs-557	54	1	(	(	PUNCT
iajs-557	54	2	b	b	X
iajs-557	54	3	)	)	PUNCT
iajs-557	54	4	let	let	VERB
iajs-557	54	5	m	m	PRON
iajs-557	54	6	be	be	AUX
iajs-557	54	7	the	the	DET
iajs-557	54	8	vector	vector	NOUN
iajs-557	54	9	space	space	NOUN
iajs-557	54	10	ℝ2	ℝ2	VERB
iajs-557	54	11	over	over	ADP
iajs-557	54	12	ℝ.	ℝ.	PROPN
iajs-557	54	13	let	let	VERB
iajs-557	54	14	n	n	NOUN
iajs-557	54	15	=	=	SYM
iajs-557	54	16	ℝ	ℝ	PROPN
iajs-557	54	17	(	(	PUNCT
iajs-557	54	18	1,0	1,0	NUM
iajs-557	54	19	)	)	PUNCT
iajs-557	54	20	.	.	PUNCT
iajs-557	55	1	n	n	PRON
iajs-557	55	2	is	be	AUX
iajs-557	55	3	simple	simple	ADJ
iajs-557	55	4	submodule	submodule	NOUN
iajs-557	55	5	of	of	ADP
iajs-557	55	6	m.	m.	NOUN
iajs-557	55	7	since	since	SCONJ
iajs-557	55	8	dim	dim	ADJ
iajs-557	55	9	n	n	NOUN
iajs-557	55	10	=	=	SYM
iajs-557	55	11	1	1	NUM
iajs-557	55	12	.	.	PUNCT
iajs-557	56	1	so	so	ADV
iajs-557	56	2	by	by	ADP
iajs-557	56	3	remark	remark	NOUN
iajs-557	56	4	1.4	1.4	NUM
iajs-557	56	5	(	(	PUNCT
iajs-557	56	6	9	9	NUM
iajs-557	56	7	)	)	PUNCT
iajs-557	56	8	,	,	PUNCT
iajs-557	56	9	n	n	PRON
iajs-557	56	10	is	be	AUX
iajs-557	56	11	ch	ch	NOUN
iajs-557	56	12	.	.	PUNCT
iajs-557	57	1	on	on	ADP
iajs-557	57	2	the	the	DET
iajs-557	57	3	other	other	ADJ
iajs-557	57	4	hand	hand	NOUN
iajs-557	57	5	,	,	PUNCT
iajs-557	57	6	n⊆ℝ	n⊆ℝ	PROPN
iajs-557	57	7	(	(	PUNCT
iajs-557	57	8	1,1)+ℝ	1,1)+ℝ	NUM
iajs-557	57	9	(	(	PUNCT
iajs-557	57	10	1	1	NUM
iajs-557	57	11	,	,	PUNCT
iajs-557	57	12	1)=ℝ2	1)=ℝ2	NUM
iajs-557	57	13	=	=	NOUN
iajs-557	57	14	m	m	PROPN
iajs-557	57	15	,	,	PUNCT
iajs-557	57	16	and	and	CCONJ
iajs-557	57	17	n	n	PRON
iajs-557	57	18	⊈	⊈	PROPN
iajs-557	57	19	ℝ	ℝ	PROPN
iajs-557	57	20	(	(	PUNCT
iajs-557	57	21	1,1	1,1	NUM
iajs-557	57	22	)	)	PUNCT
iajs-557	57	23	,	,	PUNCT
iajs-557	57	24	n	n	PRON
iajs-557	57	25	⊈	⊈	PROPN
iajs-557	57	26	ℝ	ℝ	PROPN
iajs-557	57	27	(	(	PUNCT
iajs-557	57	28	1	1	NUM
iajs-557	57	29	,	,	PUNCT
iajs-557	57	30	1	1	NUM
iajs-557	57	31	)	)	PUNCT
iajs-557	57	32	.	.	PUNCT
iajs-557	58	1	that	that	PRON
iajs-557	58	2	is	be	AUX
iajs-557	58	3	n	n	PRON
iajs-557	58	4	is	be	AUX
iajs-557	58	5	not	not	PART
iajs-557	58	6	sh	sh	NOUN
iajs-557	58	7	-	-	PUNCT
iajs-557	58	8	submodule	submodule	NOUN
iajs-557	58	9	.	.	PUNCT
iajs-557	59	1	as	as	SCONJ
iajs-557	59	2	we	we	PRON
iajs-557	59	3	have	have	AUX
iajs-557	59	4	seen	see	VERB
iajs-557	59	5	by	by	ADP
iajs-557	59	6	example	example	NOUN
iajs-557	59	7	1.4	1.4	NUM
iajs-557	59	8	(	(	PUNCT
iajs-557	59	9	11	11	NUM
iajs-557	59	10	)	)	PUNCT
iajs-557	59	11	(	(	PUNCT
iajs-557	59	12	b	b	NOUN
iajs-557	59	13	)	)	PUNCT
iajs-557	59	14	,	,	PUNCT
iajs-557	59	15	simple	simple	ADJ
iajs-557	59	16	submodule	submodule	NOUN
iajs-557	59	17	need	need	AUX
iajs-557	59	18	not	not	PART
iajs-557	59	19	be	be	AUX
iajs-557	59	20	sh	sh	PROPN
iajs-557	59	21	.	.	PUNCT
iajs-557	60	1	however	however	ADV
iajs-557	60	2	under	under	ADP
iajs-557	60	3	the	the	DET
iajs-557	60	4	class	class	NOUN
iajs-557	60	5	of	of	ADP
iajs-557	60	6	distributive	distributive	ADJ
iajs-557	60	7	(	(	PUNCT
iajs-557	60	8	or	or	CCONJ
iajs-557	60	9	comultiplication	comultiplication	NOUN
iajs-557	60	10	)	)	PUNCT
iajs-557	60	11	modules	module	NOUN
iajs-557	60	12	,	,	PUNCT
iajs-557	60	13	every	every	DET
iajs-557	60	14	simple	simple	ADJ
iajs-557	60	15	submodule	submodule	NOUN
iajs-557	60	16	is	be	AUX
iajs-557	60	17	sh	sh	PROPN
iajs-557	60	18	.	.	PUNCT
iajs-557	61	1	before	before	ADP
iajs-557	61	2	proving	prove	VERB
iajs-557	61	3	this	this	DET
iajs-557	61	4	result	result	NOUN
iajs-557	61	5	,	,	PUNCT
iajs-557	61	6	recall	recall	VERB
iajs-557	61	7	that	that	SCONJ
iajs-557	61	8	the	the	DET
iajs-557	61	9	following	follow	VERB
iajs-557	61	10	definitions	definition	NOUN
iajs-557	61	11	an	an	DET
iajs-557	61	12	r	r	NOUN
iajs-557	61	13	-	-	PUNCT
iajs-557	61	14	module	module	NOUN
iajs-557	61	15	m	m	NOUN
iajs-557	61	16	is	be	AUX
iajs-557	61	17	called	call	VERB
iajs-557	61	18	distributive	distributive	ADJ
iajs-557	61	19	if	if	SCONJ
iajs-557	61	20	the	the	DET
iajs-557	61	21	lattic	lattic	ADJ
iajs-557	61	22	of	of	ADP
iajs-557	61	23	its	its	PRON
iajs-557	61	24	submodules	submodule	NOUN
iajs-557	61	25	is	be	AUX
iajs-557	61	26	distributive	distributive	ADJ
iajs-557	61	27	,	,	PUNCT
iajs-557	61	28	that	that	PRON
iajs-557	61	29	is	is	ADV
iajs-557	61	30	l	l	NOUN
iajs-557	61	31	∩	∩	X
iajs-557	61	32	(	(	PUNCT
iajs-557	61	33	n	n	PROPN
iajs-557	61	34	+	+	CCONJ
iajs-557	61	35	k	k	X
iajs-557	61	36	)	)	PUNCT
iajs-557	61	37	=	=	SYM
iajs-557	61	38	(	(	PUNCT
iajs-557	61	39	l	l	NOUN
iajs-557	61	40	∩	∩	X
iajs-557	61	41	n	n	CCONJ
iajs-557	61	42	)	)	PUNCT
iajs-557	61	43	+	+	CCONJ
iajs-557	61	44	(	(	PUNCT
iajs-557	61	45	l	l	X
iajs-557	61	46	∩	∩	X
iajs-557	61	47	k	k	NOUN
iajs-557	61	48	)	)	PUNCT
iajs-557	61	49	.	.	PUNCT
iajs-557	62	1	equivalently	equivalently	ADV
iajs-557	62	2	,	,	PUNCT
iajs-557	62	3	l	l	PROPN
iajs-557	62	4	+	+	CCONJ
iajs-557	62	5	(	(	PUNCT
iajs-557	62	6	n	n	X
iajs-557	62	7	∩	∩	X
iajs-557	62	8	k	k	NOUN
iajs-557	62	9	)	)	PUNCT
iajs-557	62	10	=	=	SYM
iajs-557	62	11	(	(	PUNCT
iajs-557	62	12	l	l	NOUN
iajs-557	62	13	+	+	CCONJ
iajs-557	62	14	n	n	CCONJ
iajs-557	62	15	)	)	PUNCT
iajs-557	62	16	∩	∩	NOUN
iajs-557	62	17	(	(	PUNCT
iajs-557	62	18	l	l	PROPN
iajs-557	62	19	+	+	CCONJ
iajs-557	62	20	k	k	NOUN
iajs-557	62	21	)	)	PUNCT
iajs-557	62	22	for	for	ADP
iajs-557	62	23	all	all	DET
iajs-557	62	24	submodules	submodule	NOUN
iajs-557	62	25	l	l	NOUN
iajs-557	62	26	,	,	PUNCT
iajs-557	62	27	n	n	CCONJ
iajs-557	62	28	,	,	PUNCT
iajs-557	62	29	and	and	CCONJ
iajs-557	62	30	k	k	PROPN
iajs-557	62	31	of	of	ADP
iajs-557	62	32	m	m	VERB
iajs-557	62	33	see	see	VERB
iajs-557	62	34	[	[	X
iajs-557	62	35	4	4	NUM
iajs-557	62	36	]	]	PUNCT
iajs-557	62	37	.	.	PUNCT
iajs-557	63	1	an	an	DET
iajs-557	63	2	r	r	NOUN
iajs-557	63	3	-	-	PUNCT
iajs-557	63	4	module	module	NOUN
iajs-557	63	5	m	m	NOUN
iajs-557	63	6	is	be	AUX
iajs-557	63	7	called	call	VERB
iajs-557	63	8	comultiplication	comultiplication	NOUN
iajs-557	63	9	if	if	SCONJ
iajs-557	63	10	every	every	DET
iajs-557	63	11	l	l	NOUN
iajs-557	63	12	≤	≤	NUM
iajs-557	63	13	m	m	VERB
iajs-557	63	14	is	be	AUX
iajs-557	63	15	of	of	ADP
iajs-557	63	16	the	the	DET
iajs-557	63	17	form	form	NOUN
iajs-557	63	18	l	l	NOUN
iajs-557	64	1	=	=	PUNCT
iajs-557	65	1	(	(	PUNCT
iajs-557	65	2	o	o	NOUN
iajs-557	65	3	m	m	VERB
iajs-557	65	4	:	:	PUNCT
iajs-557	65	5	i	i	NOUN
iajs-557	65	6	)	)	PUNCT
iajs-557	66	1	=	=	PUNCT
iajs-557	67	1	m	m	VERB
iajs-557	67	2	ann	ann	PROPN
iajs-557	67	3	i	i	PRON
iajs-557	67	4	for	for	ADP
iajs-557	67	5	some	some	DET
iajs-557	67	6	i	i	PROPN
iajs-557	67	7	≤	≤	PROPN
iajs-557	67	8	r.	r.	PROPN
iajs-557	67	9	equivalently	equivalently	PROPN
iajs-557	67	10	,	,	PUNCT
iajs-557	67	11	l	l	NOUN
iajs-557	67	12	=	=	PUNCT
iajs-557	67	13	(	(	PUNCT
iajs-557	67	14	o	o	NOUN
iajs-557	67	15	m	m	VERB
iajs-557	67	16	:	:	PUNCT
iajs-557	67	17	(	(	PUNCT
iajs-557	67	18	o	o	NOUN
iajs-557	67	19	r	r	NOUN
iajs-557	67	20	:	:	PUNCT
iajs-557	67	21	l	l	NOUN
iajs-557	67	22	)	)	PUNCT
iajs-557	67	23	)	)	PUNCT
iajs-557	68	1	=	=	PUNCT
iajs-557	68	2	m	m	VERB
iajs-557	68	3	ann	ann	PROPN
iajs-557	68	4	r	r	PROPN
iajs-557	68	5	ann	ann	PROPN
iajs-557	68	6	l	l	NOUN
iajs-557	68	7	,	,	PUNCT
iajs-557	68	8	see	see	VERB
iajs-557	68	9	[	[	X
iajs-557	68	10	5	5	NUM
iajs-557	68	11	]	]	PUNCT
iajs-557	68	12	.	.	PUNCT
iajs-557	69	1	where	where	SCONJ
iajs-557	69	2	(	(	PUNCT
iajs-557	69	3	o	o	NOUN
iajs-557	69	4	m	m	VERB
iajs-557	69	5	:	:	PUNCT
iajs-557	69	6	i	i	X
iajs-557	69	7	)	)	PUNCT
iajs-557	69	8	=	=	PRON
iajs-557	69	9	{	{	PUNCT
iajs-557	69	10	m	m	VERB
iajs-557	69	11	∈	∈	NOUN
iajs-557	69	12	m	m	NOUN
iajs-557	69	13	:	:	PUNCT
iajs-557	69	14	i	i	PRON
iajs-557	69	15	m	m	VERB
iajs-557	69	16	=	=	SYM
iajs-557	69	17	(	(	PUNCT
iajs-557	69	18	0	0	NUM
iajs-557	69	19	)	)	PUNCT
iajs-557	69	20	}	}	PUNCT
iajs-557	69	21	,	,	PUNCT
iajs-557	69	22	(	(	PUNCT
iajs-557	69	23	o	o	NOUN
iajs-557	69	24	r	r	NOUN
iajs-557	69	25	:	:	PUNCT
iajs-557	69	26	l	l	X
iajs-557	69	27	)	)	PUNCT
iajs-557	69	28	=	=	SYM
iajs-557	70	1	{	{	PUNCT
iajs-557	70	2	r	r	NOUN
iajs-557	70	3	∈	∈	PROPN
iajs-557	70	4	r	r	NOUN
iajs-557	70	5	:	:	PUNCT
iajs-557	70	6	rl	rl	NOUN
iajs-557	70	7	=	=	SYM
iajs-557	70	8	(	(	PUNCT
iajs-557	70	9	0	0	NUM
iajs-557	70	10	)	)	PUNCT
iajs-557	70	11	}	}	PUNCT
iajs-557	70	12	.	.	PUNCT
iajs-557	71	1	mathematics	mathematic	NOUN
iajs-557	71	2	395	395	NUM
iajs-557	71	3	مجلة	مجلة	NOUN
iajs-557	71	4	إبن	إبن	VERB
iajs-557	71	5	الهيثم	الهيثم	ADJ
iajs-557	71	6	للعلوم	للعلوم	NOUN
iajs-557	71	7	الصرفة	الصرفة	NOUN
iajs-557	71	8	و	و	PRON
iajs-557	71	9	التطبيقية	التطبيقية	ADJ
iajs-557	71	10	2012	2012	NUM
iajs-557	71	11	السنة	السنة	NOUN
iajs-557	72	1	25	25	NUM
iajs-557	72	2	المجلد	المجلد	NOUN
iajs-557	72	3	3	3	NUM
iajs-557	72	4	العدد	العدد	PROPN
iajs-557	72	5	ibn	ibn	PROPN
iajs-557	72	6	al	al	PROPN
iajs-557	72	7	-	-	PUNCT
iajs-557	72	8	haitham	haitham	PROPN
iajs-557	72	9	journal	journal	PROPN
iajs-557	72	10	for	for	ADP
iajs-557	72	11	pure	pure	ADJ
iajs-557	72	12	and	and	CCONJ
iajs-557	72	13	applied	apply	VERB
iajs-557	72	14	science	science	NOUN
iajs-557	72	15	no	no	NOUN
iajs-557	72	16	.	.	NOUN
iajs-557	72	17	3	3	NUM
iajs-557	72	18	vol	vol	NOUN
iajs-557	72	19	.	.	PUNCT
iajs-557	72	20	25	25	NUM
iajs-557	72	21	year	year	NOUN
iajs-557	72	22	2012	2012	NUM
iajs-557	72	23	examples	example	NOUN
iajs-557	72	24	:	:	PUNCT
iajs-557	72	25	1	1	X
iajs-557	72	26	.	.	X
iajs-557	72	27	pz	pz	NOUN
iajs-557	72	28	∞	∞	PROPN
iajs-557	72	29	as	as	SCONJ
iajs-557	72	30	z	z	NOUN
iajs-557	72	31	-	-	PUNCT
iajs-557	72	32	module	module	NOUN
iajs-557	72	33	is	be	AUX
iajs-557	72	34	comultiplication	comultiplication	NOUN
iajs-557	72	35	,	,	PUNCT
iajs-557	72	36	since	since	SCONJ
iajs-557	72	37	for	for	ADP
iajs-557	72	38	each	each	DET
iajs-557	72	39	l	l	NOUN
iajs-557	72	40	≤	≤	NOUN
iajs-557	73	1	pz	pz	NOUN
iajs-557	73	2	∞	∞	PROPN
iajs-557	73	3	l	l	NOUN
iajs-557	74	1	=	=	PUNCT
iajs-557	74	2	i	i	PRON
iajs-557	74	3	1	1	NUM
iajs-557	74	4	z	z	NOUN
iajs-557	74	5	p	p	X
iajs-557	74	6	<	<	X
iajs-557	74	7	+	+	X
iajs-557	74	8	>	>	X
iajs-557	74	9	,	,	PUNCT
iajs-557	74	10	then	then	ADV
iajs-557	74	11	p	p	X
iajs-557	74	12	z	z	PROPN
iajs-557	74	13	z	z	PROPN
iajs-557	74	14	ann	ann	PROPN
iajs-557	75	1	ann	ann	PROPN
iajs-557	75	2	l	l	PROPN
iajs-557	75	3	l	l	NOUN
iajs-557	75	4	∞	∞	NUM
iajs-557	75	5	=	=	PUNCT
iajs-557	75	6	for	for	ADP
iajs-557	75	7	some	some	DET
iajs-557	75	8	i	i	PROPN
iajs-557	75	9	∈	∈	PROPN
iajs-557	75	10	z+	z+	X
iajs-557	75	11	.	.	NOUN
iajs-557	75	12	2	2	NUM
iajs-557	75	13	.	.	X
iajs-557	75	14	z	z	NOUN
iajs-557	75	15	as	as	SCONJ
iajs-557	75	16	z	z	NOUN
iajs-557	75	17	-	-	PUNCT
iajs-557	75	18	module	module	NOUN
iajs-557	75	19	is	be	AUX
iajs-557	75	20	not	not	PART
iajs-557	75	21	comultiplication	comultiplication	NOUN
iajs-557	75	22	,	,	PUNCT
iajs-557	75	23	since	since	SCONJ
iajs-557	75	24	if	if	SCONJ
iajs-557	75	25	l	l	NOUN
iajs-557	75	26	=	=	SYM
iajs-557	75	27	3z	3z	NUM
iajs-557	75	28	,	,	PUNCT
iajs-557	75	29	then	then	ADV
iajs-557	75	30	z	z	PROPN
iajs-557	75	31	ann	ann	PROPN
iajs-557	75	32	z	z	PROPN
iajs-557	75	33	ann	ann	PROPN
iajs-557	75	34	3z	3z	NUM
iajs-557	75	35	=	=	SYM
iajs-557	75	36	z	z	SYM
iajs-557	75	37	≠	≠	PROPN
iajs-557	75	38	3z	3z	NUM
iajs-557	75	39	.	.	PUNCT
iajs-557	76	1	3	3	X
iajs-557	76	2	.	.	X
iajs-557	76	3	zn	zn	PROPN
iajs-557	76	4	as	as	SCONJ
iajs-557	76	5	z	z	NOUN
iajs-557	76	6	-	-	PUNCT
iajs-557	76	7	module	module	NOUN
iajs-557	76	8	is	be	AUX
iajs-557	76	9	comultiplication	comultiplication	NOUN
iajs-557	76	10	.	.	PUNCT
iajs-557	77	1	proof	proof	NOUN
iajs-557	77	2	:	:	PUNCT
iajs-557	77	3	let	let	VERB
iajs-557	77	4	l	l	NOUN
iajs-557	77	5	≤	≤	ADJ
iajs-557	77	6	m.	m.	NOUN
iajs-557	77	7	then	then	ADV
iajs-557	77	8	l	l	PROPN
iajs-557	78	1	=	=	PUNCT
iajs-557	78	2	m	m	VERB
iajs-557	78	3	<	<	X
iajs-557	78	4	>	>	X
iajs-557	78	5	and	and	CCONJ
iajs-557	78	6	m	m	PROPN
iajs-557	78	7	/	/	SYM
iajs-557	78	8	n	n	CCONJ
iajs-557	78	9	,	,	PUNCT
iajs-557	78	10	that	that	PRON
iajs-557	78	11	is	be	AUX
iajs-557	78	12	n	n	PRON
iajs-557	78	13	=	=	SYM
iajs-557	78	14	mk	mk	NOUN
iajs-557	78	15	for	for	ADP
iajs-557	78	16	some	some	DET
iajs-557	78	17	k	k	PROPN
iajs-557	78	18	∈	∈	PROPN
iajs-557	78	19	z.	z.	PROPN
iajs-557	79	1	hence	hence	ADV
iajs-557	79	2	z	z	PROPN
iajs-557	79	3	ann	ann	PROPN
iajs-557	79	4	m	m	SYM
iajs-557	79	5	<	<	X
iajs-557	79	6	>	>	X
iajs-557	79	7	=	=	SYM
iajs-557	79	8	<	<	X
iajs-557	79	9	k	k	X
iajs-557	79	10	>	>	X
iajs-557	79	11	and	and	CCONJ
iajs-557	79	12	z	z	PROPN
iajs-557	79	13	ann	ann	PROPN
iajs-557	79	14	k	k	PROPN
iajs-557	79	15	m	m	VERB
iajs-557	79	16	l	l	X
iajs-557	79	17	<	<	X
iajs-557	79	18	>	>	X
iajs-557	79	19	=	=	X
iajs-557	79	20	<	<	X
iajs-557	79	21	>	>	X
iajs-557	79	22	=	=	X
iajs-557	79	23	.	.	PUNCT
iajs-557	80	1	thus	thus	ADV
iajs-557	80	2	l	l	X
iajs-557	80	3	=	=	SYM
iajs-557	80	4	nz	nz	PROPN
iajs-557	80	5	z	z	PROPN
iajs-557	80	6	ann	ann	PROPN
iajs-557	80	7	ann	ann	PROPN
iajs-557	80	8	l	l	PROPN
iajs-557	80	9	.	.	PUNCT
iajs-557	81	1	recall	recall	VERB
iajs-557	81	2	that	that	SCONJ
iajs-557	81	3	a	a	DET
iajs-557	81	4	non	non	ADJ
iajs-557	81	5	-	-	ADJ
iajs-557	81	6	zero	zero	NUM
iajs-557	81	7	submodule	submodule	NOUN
iajs-557	81	8	n	n	PROPN
iajs-557	81	9	of	of	ADP
iajs-557	81	10	an	an	DET
iajs-557	81	11	r	r	NOUN
iajs-557	81	12	-	-	PUNCT
iajs-557	81	13	module	module	NOUN
iajs-557	81	14	m	m	NOUN
iajs-557	81	15	is	be	AUX
iajs-557	81	16	said	say	VERB
iajs-557	81	17	to	to	PART
iajs-557	81	18	be	be	AUX
iajs-557	81	19	second	second	ADJ
iajs-557	81	20	submodule	submodule	NOUN
iajs-557	81	21	of	of	ADP
iajs-557	81	22	m	m	PROPN
iajs-557	81	23	if	if	SCONJ
iajs-557	81	24	for	for	ADP
iajs-557	81	25	each	each	DET
iajs-557	81	26	r	r	NOUN
iajs-557	81	27	∈	∈	NOUN
iajs-557	81	28	r	r	NOUN
iajs-557	81	29	,	,	PUNCT
iajs-557	81	30	the	the	DET
iajs-557	81	31	homothety	homothety	NOUN
iajs-557	82	1	r	r	PROPN
iajs-557	82	2	*	*	NOUN
iajs-557	82	3	on	on	ADP
iajs-557	82	4	n	n	X
iajs-557	82	5	is	be	AUX
iajs-557	82	6	either	either	DET
iajs-557	82	7	zero	zero	NUM
iajs-557	82	8	or	or	CCONJ
iajs-557	82	9	surjective	surjective	NOUN
iajs-557	82	10	.	.	PUNCT
iajs-557	83	1	equivalently	equivalently	PROPN
iajs-557	83	2	,	,	PUNCT
iajs-557	83	3	rn	rn	PROPN
iajs-557	83	4	=	=	PUNCT
iajs-557	83	5	<	<	X
iajs-557	83	6	0	0	NUM
iajs-557	83	7	>	>	X
iajs-557	83	8	or	or	CCONJ
iajs-557	83	9	rn	rn	PROPN
iajs-557	83	10	=	=	PROPN
iajs-557	83	11	n	n	PROPN
iajs-557	83	12	for	for	ADP
iajs-557	83	13	each	each	DET
iajs-557	83	14	r	r	NOUN
iajs-557	83	15	∈	∈	NOUN
iajs-557	83	16	r	r	NOUN
iajs-557	83	17	,	,	PUNCT
iajs-557	83	18	see	see	VERB
iajs-557	83	19	[	[	X
iajs-557	83	20	6	6	NUM
iajs-557	83	21	]	]	PUNCT
iajs-557	83	22	.	.	PUNCT
iajs-557	84	1	where	where	SCONJ
iajs-557	84	2	the	the	DET
iajs-557	84	3	homothety	homothety	NOUN
iajs-557	84	4	r	r	PROPN
iajs-557	84	5	*	*	PROPN
iajs-557	84	6	is	be	AUX
iajs-557	84	7	an	an	DET
iajs-557	84	8	r	r	NOUN
iajs-557	84	9	-	-	PUNCT
iajs-557	84	10	endomorphism	endomorphism	NOUN
iajs-557	84	11	on	on	ADP
iajs-557	84	12	n	n	NUM
iajs-557	84	13	,	,	PUNCT
iajs-557	84	14	means	mean	VERB
iajs-557	84	15	r*(x	r*(x	NOUN
iajs-557	84	16	)	)	PUNCT
iajs-557	84	17	=	=	VERB
iajs-557	84	18	rx	rx	VERB
iajs-557	84	19	for	for	ADP
iajs-557	84	20	each	each	DET
iajs-557	84	21	x	x	SYM
iajs-557	84	22	∈	∈	PROPN
iajs-557	84	23	n.	n.	NOUN
iajs-557	84	24	a	a	DET
iajs-557	84	25	submodule	submodule	NOUN
iajs-557	84	26	n	n	PROPN
iajs-557	84	27	of	of	ADP
iajs-557	84	28	an	an	DET
iajs-557	84	29	r	r	NOUN
iajs-557	84	30	-	-	PUNCT
iajs-557	84	31	module	module	NOUN
iajs-557	84	32	m	m	NOUN
iajs-557	84	33	is	be	AUX
iajs-557	84	34	said	say	VERB
iajs-557	84	35	to	to	PART
iajs-557	84	36	be	be	AUX
iajs-557	84	37	strongly	strongly	ADV
iajs-557	84	38	irreducible	irreducible	ADJ
iajs-557	84	39	(	(	PUNCT
iajs-557	84	40	briefly	briefly	ADV
iajs-557	84	41	,	,	PUNCT
iajs-557	84	42	sisubmodule	sisubmodule	NOUN
iajs-557	84	43	)	)	PUNCT
iajs-557	84	44	if	if	SCONJ
iajs-557	84	45	for	for	ADP
iajs-557	84	46	any	any	DET
iajs-557	84	47	l1	l1	NOUN
iajs-557	84	48	,	,	PUNCT
iajs-557	84	49	l2	l2	VERB
iajs-557	84	50	≤	≤	NUM
iajs-557	84	51	m	m	PROPN
iajs-557	84	52	,	,	PUNCT
iajs-557	84	53	l1	l1	PROPN
iajs-557	84	54	∩	∩	NOUN
iajs-557	84	55	l2	l2	VERB
iajs-557	84	56	⊆	⊆	NUM
iajs-557	84	57	n	n	CCONJ
iajs-557	84	58	,	,	PUNCT
iajs-557	84	59	then	then	ADV
iajs-557	84	60	l1	l1	PROPN
iajs-557	84	61	⊆	⊆	NUM
iajs-557	84	62	n	n	NOUN
iajs-557	84	63	or	or	CCONJ
iajs-557	84	64	l2	l2	VERB
iajs-557	84	65	⊆	⊆	NUM
iajs-557	84	66	n	n	CCONJ
iajs-557	84	67	,	,	PUNCT
iajs-557	84	68	see	see	VERB
iajs-557	84	69	[	[	X
iajs-557	84	70	7	7	NUM
iajs-557	84	71	]	]	PUNCT
iajs-557	84	72	.	.	PUNCT
iajs-557	85	1	examples	example	NOUN
iajs-557	85	2	:	:	PUNCT
iajs-557	85	3	1	1	X
iajs-557	85	4	.	.	X
iajs-557	85	5	6z	6z	NOUN
iajs-557	85	6	is	be	AUX
iajs-557	85	7	not	not	PART
iajs-557	85	8	si	si	NOUN
iajs-557	85	9	-	-	PUNCT
iajs-557	85	10	submodule	submodule	NOUN
iajs-557	85	11	of	of	ADP
iajs-557	85	12	z	z	NOUN
iajs-557	85	13	as	as	ADP
iajs-557	85	14	z	z	NOUN
iajs-557	85	15	-	-	PUNCT
iajs-557	85	16	module	module	NOUN
iajs-557	85	17	since	since	SCONJ
iajs-557	85	18	6z	6z	NUM
iajs-557	85	19	⊇	⊇	ADJ
iajs-557	85	20	2z	2z	NUM
iajs-557	85	21	∩	∩	NOUN
iajs-557	85	22	3z	3z	NUM
iajs-557	85	23	,	,	PUNCT
iajs-557	85	24	but	but	CCONJ
iajs-557	85	25	6z	6z	NUM
iajs-557	85	26	⊉	⊉	PROPN
iajs-557	85	27	2z	2z	NUM
iajs-557	85	28	,	,	PUNCT
iajs-557	85	29	6z	6z	NOUN
iajs-557	85	30	⊉	⊉	PROPN
iajs-557	85	31	3z	3z	NUM
iajs-557	85	32	.	.	PUNCT
iajs-557	86	1	2	2	X
iajs-557	86	2	.	.	X
iajs-557	86	3	it	it	PRON
iajs-557	86	4	is	be	AUX
iajs-557	86	5	clear	clear	ADJ
iajs-557	86	6	that	that	SCONJ
iajs-557	86	7	every	every	DET
iajs-557	86	8	submodule	submodule	NOUN
iajs-557	86	9	of	of	ADP
iajs-557	86	10	chained	chain	VERB
iajs-557	86	11	module	module	NOUN
iajs-557	86	12	is	be	AUX
iajs-557	86	13	si	si	NOUN
iajs-557	86	14	.	.	PUNCT
iajs-557	86	15	we	we	PRON
iajs-557	86	16	state	state	VERB
iajs-557	86	17	the	the	DET
iajs-557	86	18	following	follow	VERB
iajs-557	86	19	proposition	proposition	NOUN
iajs-557	86	20	which	which	PRON
iajs-557	86	21	is	be	AUX
iajs-557	86	22	needed	need	VERB
iajs-557	86	23	in	in	ADP
iajs-557	86	24	the	the	DET
iajs-557	86	25	next	next	ADJ
iajs-557	86	26	two	two	NUM
iajs-557	86	27	results	result	NOUN
iajs-557	86	28	.	.	PUNCT
iajs-557	87	1	proposition	proposition	NOUN
iajs-557	87	2	:	:	PUNCT
iajs-557	87	3	let	let	VERB
iajs-557	87	4	m	m	PRON
iajs-557	87	5	be	be	AUX
iajs-557	87	6	a	a	DET
iajs-557	87	7	comultiplication	comultiplication	NOUN
iajs-557	87	8	r	r	NOUN
iajs-557	87	9	-	-	PUNCT
iajs-557	87	10	module	module	NOUN
iajs-557	87	11	,	,	PUNCT
iajs-557	87	12	and	and	CCONJ
iajs-557	87	13	n	n	DET
iajs-557	87	14	≤	≤	NOUN
iajs-557	87	15	m	m	VERB
iajs-557	87	16	such	such	ADJ
iajs-557	87	17	that	that	SCONJ
iajs-557	87	18	r	r	PROPN
iajs-557	87	19	ann	ann	PROPN
iajs-557	87	20	n	n	PART
iajs-557	87	21	is	be	AUX
iajs-557	87	22	prime	prime	ADJ
iajs-557	87	23	ideal	ideal	NOUN
iajs-557	87	24	.	.	PUNCT
iajs-557	88	1	then	then	ADV
iajs-557	88	2	n	n	PRON
iajs-557	88	3	is	be	AUX
iajs-557	88	4	a	a	DET
iajs-557	88	5	sh	sh	NOUN
iajs-557	88	6	-	-	PUNCT
iajs-557	88	7	submodule	submodule	NOUN
iajs-557	88	8	.	.	PUNCT
iajs-557	89	1	proof	proof	NOUN
iajs-557	89	2	:	:	PUNCT
iajs-557	89	3	let	let	VERB
iajs-557	89	4	n	n	PRON
iajs-557	89	5	⊆	⊆	NUM
iajs-557	89	6	l1	l1	PROPN
iajs-557	89	7	+	+	CCONJ
iajs-557	89	8	l2	l2	NOUN
iajs-557	89	9	,	,	PUNCT
iajs-557	89	10	where	where	SCONJ
iajs-557	89	11	l1	l1	PROPN
iajs-557	89	12	,	,	PUNCT
iajs-557	89	13	l2	l2	NOUN
iajs-557	89	14	≤	≤	ADJ
iajs-557	89	15	m.	m.	NOUN
iajs-557	89	16	since	since	SCONJ
iajs-557	89	17	m	m	PROPN
iajs-557	89	18	is	be	AUX
iajs-557	89	19	comultiplication	comultiplication	NOUN
iajs-557	89	20	,	,	PUNCT
iajs-557	89	21	l1	l1	PROPN
iajs-557	89	22	=	=	PROPN
iajs-557	89	23	m	m	PROPN
iajs-557	89	24	ann	ann	PROPN
iajs-557	89	25	i1	i1	PROPN
iajs-557	89	26	,	,	PUNCT
iajs-557	89	27	l2	l2	NOUN
iajs-557	89	28	=	=	SYM
iajs-557	89	29	m	m	PROPN
iajs-557	89	30	ann	ann	PROPN
iajs-557	89	31	i2	i2	PROPN
iajs-557	89	32	for	for	ADP
iajs-557	89	33	some	some	DET
iajs-557	89	34	ideals	ideal	NOUN
iajs-557	89	35	i1	i1	PROPN
iajs-557	89	36	and	and	CCONJ
iajs-557	89	37	i2	i2	PROPN
iajs-557	89	38	of	of	ADP
iajs-557	89	39	r.	r.	PROPN
iajs-557	89	40	then	then	ADV
iajs-557	89	41	n	n	PROPN
iajs-557	89	42	⊆	⊆	NUM
iajs-557	89	43	m	m	NOUN
iajs-557	89	44	ann	ann	PROPN
iajs-557	89	45	i1	i1	PROPN
iajs-557	89	46	+	+	CCONJ
iajs-557	89	47	m	m	PROPN
iajs-557	89	48	ann	ann	PROPN
iajs-557	89	49	i2	i2	PROPN
iajs-557	89	50	⊆	⊆	NUM
iajs-557	89	51	m	m	NOUN
iajs-557	89	52	ann	ann	PROPN
iajs-557	89	53	(	(	PUNCT
iajs-557	89	54	i1	i1	PROPN
iajs-557	89	55	∩	∩	PROPN
iajs-557	89	56	i2	i2	PROPN
iajs-557	89	57	)	)	PUNCT
iajs-557	89	58	,	,	PUNCT
iajs-557	89	59	that	that	PRON
iajs-557	89	60	is	be	AUX
iajs-557	89	61	n	n	PRON
iajs-557	89	62	⊆	⊆	NUM
iajs-557	89	63	m	m	NOUN
iajs-557	89	64	ann	ann	PROPN
iajs-557	89	65	(	(	PUNCT
iajs-557	89	66	i1	i1	PROPN
iajs-557	89	67	∩	∩	PROPN
iajs-557	89	68	i2	i2	PROPN
iajs-557	89	69	)	)	PUNCT
iajs-557	89	70	.	.	PUNCT
iajs-557	90	1	so	so	ADV
iajs-557	90	2	r	r	PRON
iajs-557	90	3	ann	ann	PROPN
iajs-557	90	4	n	n	CCONJ
iajs-557	90	5	⊇	⊇	PROPN
iajs-557	90	6	r	r	PROPN
iajs-557	90	7	ann	ann	PROPN
iajs-557	90	8	m	m	PROPN
iajs-557	90	9	ann	ann	PROPN
iajs-557	90	10	(	(	PUNCT
iajs-557	90	11	i1	i1	PROPN
iajs-557	90	12	∩	∩	PROPN
iajs-557	90	13	i2	i2	PROPN
iajs-557	90	14	)	)	PUNCT
iajs-557	90	15	⊇	⊇	PROPN
iajs-557	90	16	i1	i1	PROPN
iajs-557	90	17	∩	∩	PROPN
iajs-557	90	18	i2	i2	PROPN
iajs-557	90	19	.	.	PUNCT
iajs-557	91	1	but	but	CCONJ
iajs-557	91	2	r	r	PROPN
iajs-557	91	3	ann	ann	PROPN
iajs-557	91	4	n	n	PART
iajs-557	91	5	is	be	AUX
iajs-557	91	6	prime	prime	ADJ
iajs-557	91	7	so	so	SCONJ
iajs-557	91	8	r	r	PRON
iajs-557	91	9	ann	ann	PROPN
iajs-557	91	10	n	n	X
iajs-557	91	11	is	be	AUX
iajs-557	91	12	si	si	ADJ
iajs-557	91	13	-	-	ADJ
iajs-557	91	14	ideal	ideal	ADJ
iajs-557	91	15	,	,	PUNCT
iajs-557	91	16	hence	hence	ADV
iajs-557	91	17	r	r	PROPN
iajs-557	91	18	ann	ann	PROPN
iajs-557	91	19	n	n	PROPN
iajs-557	91	20	⊇	⊇	PROPN
iajs-557	91	21	i1	i1	PROPN
iajs-557	91	22	or	or	CCONJ
iajs-557	91	23	r	r	PROPN
iajs-557	91	24	ann	ann	PROPN
iajs-557	91	25	n	n	PROPN
iajs-557	91	26	⊇	⊇	PROPN
iajs-557	91	27	i2	i2	PROPN
iajs-557	91	28	.	.	PUNCT
iajs-557	92	1	then	then	ADV
iajs-557	92	2	m	m	VERB
iajs-557	92	3	ann	ann	PROPN
iajs-557	92	4	r	r	PROPN
iajs-557	92	5	ann	ann	PROPN
iajs-557	92	6	n	n	PROPN
iajs-557	92	7	⊆	⊆	NUM
iajs-557	92	8	m	m	PROPN
iajs-557	92	9	ann	ann	PROPN
iajs-557	92	10	i1	i1	PROPN
iajs-557	92	11	=	=	PROPN
iajs-557	92	12	l1	l1	PROPN
iajs-557	92	13	or	or	CCONJ
iajs-557	92	14	m	m	PROPN
iajs-557	92	15	ann	ann	PROPN
iajs-557	92	16	r	r	PROPN
iajs-557	92	17	ann	ann	PROPN
iajs-557	92	18	n	n	PROPN
iajs-557	92	19	⊆	⊆	NUM
iajs-557	92	20	m	m	NOUN
iajs-557	92	21	ann	ann	PROPN
iajs-557	92	22	i2	i2	PROPN
iajs-557	92	23	=	=	PROPN
iajs-557	92	24	l2	l2	PROPN
iajs-557	92	25	.	.	PUNCT
iajs-557	93	1	so	so	ADV
iajs-557	93	2	n	n	PROPN
iajs-557	93	3	⊆	⊆	NUM
iajs-557	93	4	l1	l1	PROPN
iajs-557	93	5	or	or	CCONJ
iajs-557	93	6	n	n	CCONJ
iajs-557	93	7	⊆	⊆	NUM
iajs-557	93	8	l2	l2	NOUN
iajs-557	93	9	,	,	PUNCT
iajs-557	93	10	that	that	PRON
iajs-557	93	11	is	be	AUX
iajs-557	93	12	n	n	PRON
iajs-557	93	13	is	be	AUX
iajs-557	93	14	sh	sh	PROPN
iajs-557	93	15	.	.	PUNCT
iajs-557	94	1	the	the	DET
iajs-557	94	2	following	following	ADJ
iajs-557	94	3	result	result	NOUN
iajs-557	94	4	is	be	AUX
iajs-557	94	5	given	give	VERB
iajs-557	94	6	in	in	ADP
iajs-557	94	7	[	[	X
iajs-557	94	8	1	1	NUM
iajs-557	94	9	]	]	PUNCT
iajs-557	94	10	.	.	PUNCT
iajs-557	95	1	however	however	ADV
iajs-557	95	2	we	we	PRON
iajs-557	95	3	get	get	VERB
iajs-557	95	4	it	it	PRON
iajs-557	95	5	directly	directly	ADV
iajs-557	95	6	by	by	ADP
iajs-557	95	7	proposition	proposition	NOUN
iajs-557	95	8	1.7	1.7	NUM
iajs-557	95	9	.	.	PUNCT
iajs-557	96	1	corollary	corollary	ADJ
iajs-557	96	2	:	:	PUNCT
iajs-557	96	3	let	let	VERB
iajs-557	96	4	m	m	PRON
iajs-557	96	5	be	be	AUX
iajs-557	96	6	a	a	DET
iajs-557	96	7	comultiplication	comultiplication	NOUN
iajs-557	96	8	r	r	NOUN
iajs-557	96	9	-	-	PUNCT
iajs-557	96	10	module	module	NOUN
iajs-557	96	11	,	,	PUNCT
iajs-557	96	12	and	and	CCONJ
iajs-557	96	13	n	n	DET
iajs-557	96	14	≤	≤	NOUN
iajs-557	96	15	m.	m.	NOUN
iajs-557	96	16	then	then	ADV
iajs-557	96	17	1	1	X
iajs-557	96	18	.	.	PUNCT
iajs-557	97	1	n	n	PRON
iajs-557	97	2	is	be	AUX
iajs-557	97	3	a	a	DET
iajs-557	97	4	second	second	ADJ
iajs-557	97	5	submodule	submodule	NOUN
iajs-557	97	6	implies	imply	VERB
iajs-557	97	7	n	n	AUX
iajs-557	97	8	is	be	AUX
iajs-557	97	9	sh	sh	PROPN
iajs-557	97	10	.	.	PROPN
iajs-557	97	11	2	2	NUM
iajs-557	97	12	.	.	X
iajs-557	98	1	n	n	PRON
iajs-557	98	2	is	be	AUX
iajs-557	98	3	a	a	DET
iajs-557	98	4	finitely	finitely	ADV
iajs-557	98	5	generated	generate	VERB
iajs-557	98	6	second	second	ADJ
iajs-557	98	7	submodule	submodule	NOUN
iajs-557	98	8	,	,	PUNCT
iajs-557	98	9	implies	imply	VERB
iajs-557	98	10	n	n	ADV
iajs-557	98	11	is	be	AUX
iajs-557	98	12	ch	ch	NOUN
iajs-557	98	13	.	.	PUNCT
iajs-557	99	1	proof	proof	NOUN
iajs-557	99	2	:	:	PUNCT
iajs-557	99	3	(	(	PUNCT
iajs-557	99	4	1	1	X
iajs-557	99	5	)	)	PUNCT
iajs-557	99	6	since	since	SCONJ
iajs-557	99	7	n	n	ADV
iajs-557	99	8	is	be	AUX
iajs-557	99	9	second	second	ADJ
iajs-557	99	10	,	,	PUNCT
iajs-557	99	11	then	then	ADV
iajs-557	99	12	r	r	PROPN
iajs-557	99	13	ann	ann	PROPN
iajs-557	99	14	n	n	PART
iajs-557	99	15	is	be	AUX
iajs-557	99	16	a	a	DET
iajs-557	99	17	prime	prime	ADJ
iajs-557	99	18	ideal	ideal	NOUN
iajs-557	99	19	by	by	ADP
iajs-557	99	20	[	[	X
iajs-557	99	21	6	6	NUM
iajs-557	99	22	]	]	PUNCT
iajs-557	99	23	.	.	PUNCT
iajs-557	100	1	hence	hence	ADV
iajs-557	100	2	the	the	DET
iajs-557	100	3	result	result	NOUN
iajs-557	100	4	is	be	AUX
iajs-557	100	5	obtained	obtain	VERB
iajs-557	100	6	by	by	ADP
iajs-557	100	7	proposition	proposition	NOUN
iajs-557	100	8	1.7	1.7	NUM
iajs-557	100	9	.	.	PUNCT
iajs-557	101	1	(	(	PUNCT
iajs-557	101	2	2	2	NUM
iajs-557	101	3	)	)	PUNCT
iajs-557	101	4	by	by	ADP
iajs-557	101	5	part	part	NOUN
iajs-557	101	6	(	(	PUNCT
iajs-557	101	7	1	1	NUM
iajs-557	101	8	)	)	PUNCT
iajs-557	102	1	n	n	PRON
iajs-557	102	2	is	be	AUX
iajs-557	102	3	sh	sh	PROPN
iajs-557	102	4	.	.	PUNCT
iajs-557	103	1	but	but	CCONJ
iajs-557	103	2	n	n	PRON
iajs-557	103	3	is	be	AUX
iajs-557	103	4	finitely	finitely	ADV
iajs-557	103	5	generated	generate	VERB
iajs-557	103	6	,	,	PUNCT
iajs-557	103	7	so	so	CCONJ
iajs-557	104	1	n	n	CCONJ
iajs-557	104	2	i	i	PRON
iajs-557	104	3	i	i	VERB
iajs-557	104	4	1	1	NUM
iajs-557	104	5	n	n	PRON
iajs-557	104	6	rx	rx	VERB
iajs-557	104	7	=	=	PUNCT
iajs-557	104	8	=	=	PUNCT
iajs-557	104	9	∑	∑	PROPN
iajs-557	104	10	for	for	ADP
iajs-557	104	11	some	some	DET
iajs-557	104	12	x1	x1	PROPN
iajs-557	104	13	,	,	PUNCT
iajs-557	104	14	…	…	PUNCT
iajs-557	104	15	,	,	PUNCT
iajs-557	104	16	xn	xn	PROPN
iajs-557	104	17	.	.	PUNCT
iajs-557	105	1	hence	hence	ADV
iajs-557	105	2	n	n	CCONJ
iajs-557	105	3	⊆	⊆	NUM
iajs-557	105	4	rxi	rxi	NOUN
iajs-557	105	5	for	for	ADP
iajs-557	105	6	some	some	DET
iajs-557	105	7	i	i	NOUN
iajs-557	105	8	=	=	NOUN
iajs-557	105	9	1	1	NUM
iajs-557	105	10	,	,	PUNCT
iajs-557	105	11	…	…	PUNCT
iajs-557	105	12	,	,	PUNCT
iajs-557	105	13	n.	n.	NOUN
iajs-557	105	14	but	but	CCONJ
iajs-557	105	15	rxi	rxi	PROPN
iajs-557	105	16	⊆	⊆	NUM
iajs-557	105	17	n.	n.	NOUN
iajs-557	105	18	thus	thus	ADV
iajs-557	105	19	n	n	PROPN
iajs-557	105	20	=	=	SYM
iajs-557	105	21	rxi	rxi	NOUN
iajs-557	105	22	.	.	PUNCT
iajs-557	106	1	mathematics	mathematic	NOUN
iajs-557	106	2	396	396	NUM
iajs-557	106	3	مجلة	مجلة	PROPN
iajs-557	106	4	إبن	إبن	VERB
iajs-557	106	5	الهيثم	الهيثم	ADJ
iajs-557	106	6	للعلوم	للعلوم	NOUN
iajs-557	106	7	الصرفة	الصرفة	NOUN
iajs-557	106	8	و	و	PRON
iajs-557	106	9	التطبيقية	التطبيقية	ADJ
iajs-557	106	10	2012	2012	NUM
iajs-557	106	11	السنة	السنة	NOUN
iajs-557	106	12	25	25	NUM
iajs-557	106	13	المجلد	المجلد	NOUN
iajs-557	106	14	3	3	NUM
iajs-557	106	15	العدد	العدد	PROPN
iajs-557	106	16	ibn	ibn	PROPN
iajs-557	106	17	al	al	PROPN
iajs-557	106	18	-	-	PUNCT
iajs-557	106	19	haitham	haitham	PROPN
iajs-557	106	20	journal	journal	PROPN
iajs-557	106	21	for	for	ADP
iajs-557	106	22	pure	pure	ADJ
iajs-557	106	23	and	and	CCONJ
iajs-557	106	24	applied	apply	VERB
iajs-557	106	25	science	science	NOUN
iajs-557	106	26	no	no	NOUN
iajs-557	106	27	.	.	NOUN
iajs-557	106	28	3	3	NUM
iajs-557	106	29	vol	vol	NOUN
iajs-557	106	30	.	.	PUNCT
iajs-557	107	1	25	25	NUM
iajs-557	107	2	year	year	NOUN
iajs-557	107	3	2012	2012	NUM
iajs-557	107	4	corollary	corollary	NOUN
iajs-557	107	5	:	:	PUNCT
iajs-557	107	6	let	let	VERB
iajs-557	107	7	m	m	PRON
iajs-557	107	8	be	be	AUX
iajs-557	107	9	a	a	DET
iajs-557	107	10	comultiplication	comultiplication	NOUN
iajs-557	107	11	r	r	NOUN
iajs-557	107	12	-	-	PUNCT
iajs-557	107	13	module	module	NOUN
iajs-557	107	14	,	,	PUNCT
iajs-557	107	15	and	and	CCONJ
iajs-557	107	16	let	let	VERB
iajs-557	107	17	n	n	PRON
iajs-557	107	18	be	be	AUX
iajs-557	107	19	a	a	DET
iajs-557	107	20	simple	simple	ADJ
iajs-557	107	21	submodule	submodule	NOUN
iajs-557	107	22	.	.	PUNCT
iajs-557	108	1	then	then	ADV
iajs-557	108	2	n	n	PRON
iajs-557	108	3	is	be	AUX
iajs-557	108	4	sh	sh	PROPN
iajs-557	108	5	.	.	PUNCT
iajs-557	108	6	proof	proof	NOUN
iajs-557	108	7	:	:	PUNCT
iajs-557	108	8	it	it	PRON
iajs-557	108	9	is	be	AUX
iajs-557	108	10	clear	clear	ADJ
iajs-557	108	11	that	that	SCONJ
iajs-557	108	12	every	every	DET
iajs-557	108	13	simple	simple	ADJ
iajs-557	108	14	submodule	submodule	NOUN
iajs-557	108	15	is	be	AUX
iajs-557	108	16	second	second	ADJ
iajs-557	108	17	,	,	PUNCT
iajs-557	108	18	hence	hence	ADV
iajs-557	108	19	the	the	DET
iajs-557	108	20	result	result	NOUN
iajs-557	108	21	follows	follow	VERB
iajs-557	108	22	by	by	ADP
iajs-557	108	23	corollary	corollary	ADJ
iajs-557	108	24	1.8	1.8	NUM
iajs-557	108	25	(	(	PUNCT
iajs-557	108	26	1	1	NUM
iajs-557	108	27	)	)	PUNCT
iajs-557	108	28	.	.	PUNCT
iajs-557	109	1	recall	recall	VERB
iajs-557	109	2	that	that	SCONJ
iajs-557	109	3	an	an	DET
iajs-557	109	4	r	r	NOUN
iajs-557	109	5	-	-	PUNCT
iajs-557	109	6	module	module	NOUN
iajs-557	109	7	m	m	NOUN
iajs-557	109	8	is	be	AUX
iajs-557	109	9	said	say	VERB
iajs-557	109	10	to	to	PART
iajs-557	109	11	be	be	AUX
iajs-557	109	12	prime	prime	ADJ
iajs-557	109	13	if	if	SCONJ
iajs-557	109	14	r	r	NOUN
iajs-557	109	15	ann	ann	PROPN
iajs-557	109	16	m	m	NOUN
iajs-557	109	17	=	=	SYM
iajs-557	109	18	r	r	NOUN
iajs-557	109	19	ann	ann	PROPN
iajs-557	109	20	n	n	PROPN
iajs-557	109	21	for	for	ADP
iajs-557	109	22	every	every	DET
iajs-557	109	23	non	non	ADJ
iajs-557	109	24	-	-	ADJ
iajs-557	109	25	zero	zero	NUM
iajs-557	109	26	submodule	submodule	NOUN
iajs-557	109	27	n	n	PROPN
iajs-557	109	28	of	of	ADP
iajs-557	109	29	m.	m.	NOUN
iajs-557	109	30	,	,	PUNCT
iajs-557	109	31	see	see	VERB
iajs-557	109	32	[	[	X
iajs-557	109	33	8	8	NUM
iajs-557	109	34	]	]	PUNCT
iajs-557	109	35	.	.	PUNCT
iajs-557	110	1	if	if	SCONJ
iajs-557	110	2	m	m	NOUN
iajs-557	110	3	is	be	AUX
iajs-557	110	4	a	a	DET
iajs-557	110	5	prime	prime	ADJ
iajs-557	110	6	r	r	NOUN
iajs-557	110	7	-	-	PUNCT
iajs-557	110	8	module	module	NOUN
iajs-557	110	9	,	,	PUNCT
iajs-557	110	10	then	then	ADV
iajs-557	110	11	r	r	PROPN
iajs-557	110	12	ann	ann	PROPN
iajs-557	110	13	m	m	NOUN
iajs-557	110	14	is	be	AUX
iajs-557	110	15	prime	prime	ADJ
iajs-557	110	16	by	by	ADP
iajs-557	110	17	[	[	X
iajs-557	110	18	8	8	NUM
iajs-557	110	19	]	]	PUNCT
iajs-557	110	20	.	.	PUNCT
iajs-557	111	1	an	an	DET
iajs-557	111	2	r	r	NOUN
iajs-557	111	3	-	-	PUNCT
iajs-557	111	4	module	module	NOUN
iajs-557	111	5	m	m	NOUN
iajs-557	111	6	is	be	AUX
iajs-557	111	7	called	call	VERB
iajs-557	111	8	a	a	DET
iajs-557	111	9	quasi	quasi	NOUN
iajs-557	111	10	-	-	NOUN
iajs-557	111	11	prime	prime	ADJ
iajs-557	111	12	if	if	SCONJ
iajs-557	111	13	r	r	PROPN
iajs-557	111	14	ann	ann	PROPN
iajs-557	111	15	n	n	PART
iajs-557	111	16	is	be	AUX
iajs-557	111	17	a	a	DET
iajs-557	111	18	prime	prime	NOUN
iajs-557	111	19	for	for	ADP
iajs-557	111	20	each	each	DET
iajs-557	111	21	non	non	ADJ
iajs-557	111	22	-	-	ADJ
iajs-557	111	23	zero	zero	NUM
iajs-557	111	24	submodule	submodule	NOUN
iajs-557	111	25	n	n	PROPN
iajs-557	111	26	of	of	ADP
iajs-557	111	27	m	m	PRON
iajs-557	111	28	,	,	PUNCT
iajs-557	111	29	see	see	VERB
iajs-557	111	30	[	[	X
iajs-557	111	31	9	9	NUM
iajs-557	111	32	,	,	PUNCT
iajs-557	111	33	definition	definition	NOUN
iajs-557	111	34	1.2.1	1.2.1	NUM
iajs-557	111	35	]	]	PUNCT
iajs-557	111	36	.	.	PUNCT
iajs-557	112	1	notice	notice	VERB
iajs-557	112	2	that	that	SCONJ
iajs-557	112	3	every	every	DET
iajs-557	112	4	prime	prime	ADJ
iajs-557	112	5	r	r	NOUN
iajs-557	112	6	-	-	PUNCT
iajs-557	112	7	module	module	NOUN
iajs-557	112	8	m	m	NOUN
iajs-557	112	9	is	be	AUX
iajs-557	112	10	quasi	quasi	ADJ
iajs-557	112	11	-	-	NOUN
iajs-557	112	12	prime	prime	ADJ
iajs-557	112	13	by	by	ADP
iajs-557	112	14	[	[	PUNCT
iajs-557	112	15	9	9	NUM
iajs-557	112	16	,	,	PUNCT
iajs-557	112	17	remark	remark	NOUN
iajs-557	112	18	1.2.2	1.2.2	NUM
iajs-557	112	19	]	]	PUNCT
iajs-557	112	20	.	.	PUNCT
iajs-557	113	1	corollary	corollary	ADJ
iajs-557	113	2	:	:	PUNCT
iajs-557	113	3	let	let	VERB
iajs-557	113	4	m	m	PRON
iajs-557	113	5	be	be	AUX
iajs-557	113	6	a	a	DET
iajs-557	113	7	comultiplication	comultiplication	NOUN
iajs-557	113	8	prime	prime	NOUN
iajs-557	113	9	(	(	PUNCT
iajs-557	113	10	or	or	CCONJ
iajs-557	113	11	quasi	quasi	ADJ
iajs-557	113	12	-	-	ADJ
iajs-557	113	13	prime	prime	ADJ
iajs-557	113	14	)	)	PUNCT
iajs-557	113	15	r	r	NOUN
iajs-557	113	16	-	-	PUNCT
iajs-557	113	17	module	module	NOUN
iajs-557	113	18	.	.	PUNCT
iajs-557	114	1	then	then	ADV
iajs-557	114	2	every	every	DET
iajs-557	114	3	non	non	ADJ
iajs-557	114	4	-	-	ADJ
iajs-557	114	5	zero	zero	NUM
iajs-557	114	6	submodule	submodule	NOUN
iajs-557	114	7	is	be	AUX
iajs-557	114	8	sh	sh	PROPN
iajs-557	114	9	.	.	PUNCT
iajs-557	114	10	proof	proof	NOUN
iajs-557	114	11	:	:	PUNCT
iajs-557	114	12	since	since	SCONJ
iajs-557	114	13	m	m	PROPN
iajs-557	114	14	is	be	AUX
iajs-557	114	15	prime	prime	ADJ
iajs-557	114	16	(	(	PUNCT
iajs-557	114	17	or	or	CCONJ
iajs-557	114	18	quasi	quasi	ADJ
iajs-557	114	19	-	-	ADJ
iajs-557	114	20	prime	prime	ADJ
iajs-557	114	21	)	)	PUNCT
iajs-557	114	22	implies	imply	VERB
iajs-557	114	23	r	r	PROPN
iajs-557	114	24	ann	ann	PROPN
iajs-557	114	25	n	n	PART
iajs-557	114	26	is	be	AUX
iajs-557	114	27	prime	prime	ADJ
iajs-557	114	28	ideal	ideal	NOUN
iajs-557	114	29	for	for	ADP
iajs-557	114	30	each	each	DET
iajs-557	114	31	non	non	ADJ
iajs-557	114	32	-	-	ADJ
iajs-557	114	33	zero	zero	NUM
iajs-557	114	34	submodule	submodule	NOUN
iajs-557	114	35	n	n	PROPN
iajs-557	114	36	of	of	ADP
iajs-557	114	37	m.	m.	NOUN
iajs-557	114	38	hence	hence	ADV
iajs-557	114	39	the	the	DET
iajs-557	114	40	result	result	NOUN
iajs-557	114	41	follows	follow	VERB
iajs-557	114	42	from	from	ADP
iajs-557	114	43	proposition	proposition	NOUN
iajs-557	114	44	1.7	1.7	NUM
iajs-557	114	45	.	.	PUNCT
iajs-557	115	1	proposition	proposition	NOUN
iajs-557	115	2	:	:	PUNCT
iajs-557	115	3	let	let	VERB
iajs-557	115	4	m	m	PRON
iajs-557	115	5	be	be	AUX
iajs-557	115	6	a	a	DET
iajs-557	115	7	distributive	distributive	ADJ
iajs-557	115	8	r	r	NOUN
iajs-557	115	9	-	-	PUNCT
iajs-557	115	10	module	module	NOUN
iajs-557	115	11	,	,	PUNCT
iajs-557	115	12	and	and	CCONJ
iajs-557	115	13	<	<	X
iajs-557	115	14	0	0	NUM
iajs-557	115	15	>	>	X
iajs-557	115	16	≠	≠	PROPN
iajs-557	115	17	n	n	PRON
iajs-557	115	18	≤	≤	NOUN
iajs-557	115	19	m.	m.	NOUN
iajs-557	115	20	if	if	SCONJ
iajs-557	115	21	n	n	PRON
iajs-557	115	22	is	be	AUX
iajs-557	115	23	a	a	DET
iajs-557	115	24	simple	simple	ADJ
iajs-557	115	25	submodule	submodule	NOUN
iajs-557	115	26	of	of	ADP
iajs-557	115	27	m	m	PROPN
iajs-557	115	28	,	,	PUNCT
iajs-557	115	29	then	then	ADV
iajs-557	115	30	n	n	PROPN
iajs-557	115	31	is	be	AUX
iajs-557	115	32	sh	sh	PROPN
iajs-557	115	33	.	.	PUNCT
iajs-557	115	34	proof	proof	NOUN
iajs-557	115	35	:	:	PUNCT
iajs-557	115	36	assume	assume	VERB
iajs-557	115	37	n	n	PRON
iajs-557	115	38	is	be	AUX
iajs-557	115	39	simple	simple	ADJ
iajs-557	115	40	,	,	PUNCT
iajs-557	115	41	n	n	PRON
iajs-557	115	42	≤	≤	NOUN
iajs-557	115	43	l1	l1	NOUN
iajs-557	115	44	+	+	CCONJ
iajs-557	115	45	l2	l2	PROPN
iajs-557	115	46	where	where	SCONJ
iajs-557	115	47	l1	l1	PROPN
iajs-557	115	48	,	,	PUNCT
iajs-557	115	49	l2	l2	NOUN
iajs-557	115	50	≤	≤	ADJ
iajs-557	115	51	m.	m.	NOUN
iajs-557	115	52	hence	hence	ADV
iajs-557	115	53	n	n	NOUN
iajs-557	115	54	=	=	SYM
iajs-557	115	55	n	n	NOUN
iajs-557	115	56	∩	∩	NOUN
iajs-557	115	57	(	(	PUNCT
iajs-557	115	58	l1	l1	PROPN
iajs-557	115	59	+	+	CCONJ
iajs-557	115	60	l2	l2	NOUN
iajs-557	115	61	)	)	PUNCT
iajs-557	115	62	=	=	PUNCT
iajs-557	115	63	(	(	PUNCT
iajs-557	115	64	n	n	X
iajs-557	115	65	∩	∩	X
iajs-557	115	66	l1	l1	PROPN
iajs-557	115	67	)	)	PUNCT
iajs-557	116	1	+	+	CCONJ
iajs-557	116	2	(	(	PUNCT
iajs-557	116	3	n	n	CCONJ
iajs-557	116	4	∩	∩	ADJ
iajs-557	116	5	l2	l2	NOUN
iajs-557	116	6	)	)	PUNCT
iajs-557	116	7	,	,	PUNCT
iajs-557	116	8	since	since	SCONJ
iajs-557	116	9	m	m	PROPN
iajs-557	116	10	is	be	AUX
iajs-557	116	11	distributive	distributive	ADJ
iajs-557	116	12	.	.	PUNCT
iajs-557	117	1	then	then	ADV
iajs-557	117	2	(	(	PUNCT
iajs-557	117	3	n	n	X
iajs-557	117	4	∩	∩	X
iajs-557	117	5	l1	l1	NOUN
iajs-557	117	6	=	=	PUNCT
iajs-557	117	7	<	<	X
iajs-557	117	8	0	0	NUM
iajs-557	117	9	>	>	X
iajs-557	117	10	or	or	CCONJ
iajs-557	117	11	n	n	CCONJ
iajs-557	117	12	∩	∩	ADJ
iajs-557	117	13	l1	l1	PROPN
iajs-557	117	14	=	=	PUNCT
iajs-557	117	15	n	n	CCONJ
iajs-557	117	16	)	)	PUNCT
iajs-557	117	17	and	and	CCONJ
iajs-557	117	18	(	(	PUNCT
iajs-557	117	19	n	n	CCONJ
iajs-557	117	20	∩	∩	ADJ
iajs-557	117	21	l2	l2	NOUN
iajs-557	117	22	=	=	PUNCT
iajs-557	117	23	<	<	X
iajs-557	117	24	0	0	NUM
iajs-557	117	25	>	>	X
iajs-557	117	26	or	or	CCONJ
iajs-557	117	27	n	n	CCONJ
iajs-557	117	28	∩	∩	ADJ
iajs-557	117	29	l2	l2	NOUN
iajs-557	117	30	=	=	SYM
iajs-557	117	31	n	n	CCONJ
iajs-557	117	32	)	)	PUNCT
iajs-557	117	33	.	.	PUNCT
iajs-557	118	1	but	but	CCONJ
iajs-557	118	2	n	n	CCONJ
iajs-557	118	3	≠	≠	PROPN
iajs-557	118	4	0	0	NUM
iajs-557	118	5	.	.	PUNCT
iajs-557	119	1	so	so	ADV
iajs-557	119	2	we	we	PRON
iajs-557	119	3	have	have	VERB
iajs-557	119	4	only	only	ADV
iajs-557	119	5	three	three	NUM
iajs-557	119	6	possible	possible	ADJ
iajs-557	119	7	cases	case	NOUN
iajs-557	119	8	(	(	PUNCT
iajs-557	119	9	1	1	NUM
iajs-557	119	10	)	)	PUNCT
iajs-557	119	11	n	n	NOUN
iajs-557	119	12	∩	∩	X
iajs-557	119	13	l1	l1	PROPN
iajs-557	119	14	=	=	PUNCT
iajs-557	119	15	<	<	X
iajs-557	119	16	0	0	NUM
iajs-557	119	17	>	>	X
iajs-557	119	18	,	,	PUNCT
iajs-557	119	19	n	n	CCONJ
iajs-557	119	20	⊆	⊆	NUM
iajs-557	119	21	l2	l2	NOUN
iajs-557	119	22	.	.	PUNCT
iajs-557	120	1	(	(	PUNCT
iajs-557	120	2	2	2	X
iajs-557	120	3	)	)	PUNCT
iajs-557	120	4	n	n	NOUN
iajs-557	120	5	∩	∩	ADJ
iajs-557	120	6	l2	l2	NOUN
iajs-557	120	7	=	=	PUNCT
iajs-557	120	8	<	<	X
iajs-557	120	9	0	0	NUM
iajs-557	120	10	>	>	X
iajs-557	120	11	,	,	PUNCT
iajs-557	120	12	n	n	PROPN
iajs-557	120	13	⊆	⊆	NUM
iajs-557	120	14	l1	l1	PROPN
iajs-557	120	15	.	.	PUNCT
iajs-557	121	1	(	(	PUNCT
iajs-557	121	2	3	3	X
iajs-557	121	3	)	)	PUNCT
iajs-557	121	4	n	n	CCONJ
iajs-557	121	5	⊆	⊆	NUM
iajs-557	121	6	l1	l1	PROPN
iajs-557	121	7	,	,	PUNCT
iajs-557	121	8	n	n	CCONJ
iajs-557	121	9	⊆	⊆	NUM
iajs-557	121	10	l2	l2	NOUN
iajs-557	121	11	.	.	PUNCT
iajs-557	122	1	thus	thus	ADV
iajs-557	122	2	either	either	CCONJ
iajs-557	122	3	n	n	PROPN
iajs-557	122	4	⊆	⊆	NUM
iajs-557	122	5	l1	l1	PROPN
iajs-557	122	6	or	or	CCONJ
iajs-557	122	7	n	n	CCONJ
iajs-557	122	8	⊆	⊆	NUM
iajs-557	122	9	l2	l2	NOUN
iajs-557	122	10	;	;	PUNCT
iajs-557	122	11	that	that	PRON
iajs-557	122	12	is	be	AUX
iajs-557	122	13	n	n	PRON
iajs-557	122	14	is	be	AUX
iajs-557	122	15	sh	sh	PROPN
iajs-557	122	16	.	.	PROPN
iajs-557	122	17	remark	remark	PROPN
iajs-557	122	18	:	:	PUNCT
iajs-557	122	19	the	the	DET
iajs-557	122	20	condition	condition	NOUN
iajs-557	122	21	m	m	VERB
iajs-557	122	22	is	be	AUX
iajs-557	122	23	distributive	distributive	ADJ
iajs-557	122	24	or	or	CCONJ
iajs-557	122	25	comultiplication	comultiplication	NOUN
iajs-557	122	26	is	be	AUX
iajs-557	122	27	necessary	necessary	ADJ
iajs-557	122	28	condition	condition	NOUN
iajs-557	122	29	in	in	ADP
iajs-557	122	30	proposition	proposition	NOUN
iajs-557	122	31	1.11	1.11	NUM
iajs-557	122	32	and	and	CCONJ
iajs-557	122	33	corollary	corollary	ADJ
iajs-557	122	34	1.9	1.9	NUM
iajs-557	122	35	.	.	PUNCT
iajs-557	123	1	as	as	SCONJ
iajs-557	123	2	we	we	PRON
iajs-557	123	3	have	have	AUX
iajs-557	123	4	seen	see	VERB
iajs-557	123	5	in	in	ADP
iajs-557	123	6	remark	remark	NOUN
iajs-557	123	7	1.4(11)(b	1.4(11)(b	NUM
iajs-557	123	8	)	)	PUNCT
iajs-557	123	9	,	,	PUNCT
iajs-557	123	10	n	n	NOUN
iajs-557	123	11	=	=	SYM
iajs-557	123	12	ℝ(1,0	ℝ(1,0	PROPN
iajs-557	123	13	)	)	PUNCT
iajs-557	123	14	and	and	CCONJ
iajs-557	123	15	n	n	PRON
iajs-557	123	16	is	be	AUX
iajs-557	123	17	simple	simple	ADJ
iajs-557	123	18	but	but	CCONJ
iajs-557	123	19	not	not	PART
iajs-557	123	20	sh	sh	PROPN
iajs-557	123	21	.	.	PUNCT
iajs-557	124	1	moreover	moreover	ADV
iajs-557	124	2	the	the	DET
iajs-557	124	3	vector	vector	NOUN
iajs-557	124	4	space	space	NOUN
iajs-557	124	5	ℝ2	ℝ2	VERB
iajs-557	124	6	over	over	ADP
iajs-557	124	7	ℝ	ℝ	PROPN
iajs-557	124	8	is	be	AUX
iajs-557	124	9	not	not	PART
iajs-557	124	10	distributive	distributive	ADJ
iajs-557	124	11	since	since	SCONJ
iajs-557	124	12	ℝ2	ℝ2	NOUN
iajs-557	124	13	=	=	SYM
iajs-557	124	14	ℝ(1,1	ℝ(1,1	NOUN
iajs-557	124	15	)	)	PUNCT
iajs-557	124	16	+	+	NUM
iajs-557	124	17	ℝ(1,–1	ℝ(1,–1	NOUN
iajs-557	124	18	)	)	PUNCT
iajs-557	124	19	and	and	CCONJ
iajs-557	124	20	n	n	NOUN
iajs-557	124	21	∩	∩	NOUN
iajs-557	124	22	ℝ2=	ℝ2=	VERB
iajs-557	124	23	n	n	CCONJ
iajs-557	124	24	,	,	PUNCT
iajs-557	124	25	but	but	CCONJ
iajs-557	124	26	(	(	PUNCT
iajs-557	124	27	n	n	CCONJ
iajs-557	124	28	∩	∩	X
iajs-557	124	29	ℝ(1,1	ℝ(1,1	ADJ
iajs-557	124	30	)	)	PUNCT
iajs-557	124	31	)	)	PUNCT
iajs-557	125	1	+	+	CCONJ
iajs-557	125	2	(	(	PUNCT
iajs-557	125	3	n	n	CCONJ
iajs-557	125	4	∩	∩	NOUN
iajs-557	125	5	ℝ(1	ℝ(1	ADP
iajs-557	125	6	,	,	PUNCT
iajs-557	125	7	–	–	PUNCT
iajs-557	125	8	1	1	NUM
iajs-557	125	9	)	)	PUNCT
iajs-557	125	10	)	)	PUNCT
iajs-557	126	1	=	=	PRON
iajs-557	126	2	{	{	PUNCT
iajs-557	126	3	(	(	PUNCT
iajs-557	126	4	0,0	0,0	NOUN
iajs-557	126	5	)	)	PUNCT
iajs-557	126	6	}	}	PUNCT
iajs-557	126	7	.	.	PUNCT
iajs-557	127	1	thus	thus	ADV
iajs-557	127	2	ℝ2	ℝ2	PROPN
iajs-557	127	3	is	be	AUX
iajs-557	127	4	not	not	PART
iajs-557	127	5	distributive	distributive	ADJ
iajs-557	127	6	.	.	PUNCT
iajs-557	128	1	also	also	ADV
iajs-557	128	2	ℝ2	ℝ2	PROPN
iajs-557	128	3	is	be	AUX
iajs-557	128	4	not	not	PART
iajs-557	128	5	comultiplication	comultiplication	NOUN
iajs-557	128	6	r	r	NOUN
iajs-557	128	7	-	-	NOUN
iajs-557	128	8	module	module	NOUN
iajs-557	128	9	.	.	PUNCT
iajs-557	129	1	for	for	ADP
iajs-557	129	2	if	if	SCONJ
iajs-557	129	3	l	l	NOUN
iajs-557	129	4	=	=	SYM
iajs-557	129	5	ℝ(1,1	ℝ(1,1	ADJ
iajs-557	129	6	)	)	PUNCT
iajs-557	129	7	,	,	PUNCT
iajs-557	129	8	then	then	ADV
iajs-557	129	9	ann	ann	PROPN
iajs-557	129	10			PUNCT
iajs-557	129	11	l	l	X
iajs-557	129	12	=	=	PUNCT
iajs-557	129	13	{	{	PUNCT
iajs-557	129	14	0	0	NUM
iajs-557	129	15	}	}	PUNCT
iajs-557	129	16	and	and	CCONJ
iajs-557	129	17	2	2	NUM
iajs-557	129	18	2ann{0}=	2ann{0}=	NUM
iajs-557	129	19			PUNCT
iajs-557	129	20			PUNCT
iajs-557	129	21	,	,	PUNCT
iajs-557	129	22	thus	thus	ADV
iajs-557	129	23	l	l	NOUN
iajs-557	129	24	≠	≠	PROPN
iajs-557	129	25	2	2	NUM
iajs-557	129	26	ann	ann	PROPN
iajs-557	129	27	ann	ann	PROPN
iajs-557	129	28	l	l	PROPN
iajs-557	129	29			PROPN
iajs-557	129	30	.	.	PUNCT
iajs-557	130	1	now	now	ADV
iajs-557	130	2	we	we	PRON
iajs-557	130	3	introduce	introduce	VERB
iajs-557	130	4	the	the	DET
iajs-557	130	5	following	follow	VERB
iajs-557	130	6	concept	concept	NOUN
iajs-557	130	7	.	.	PUNCT
iajs-557	131	1	definition	definition	NOUN
iajs-557	131	2	:	:	PUNCT
iajs-557	131	3	let	let	VERB
iajs-557	131	4	<	<	X
iajs-557	131	5	0	0	NUM
iajs-557	131	6	>	>	X
iajs-557	131	7	≠	≠	PROPN
iajs-557	131	8	l	l	NOUN
iajs-557	131	9	≤	≤	NUM
iajs-557	131	10	m	m	NOUN
iajs-557	131	11	,	,	PUNCT
iajs-557	131	12	l	l	PROPN
iajs-557	131	13	is	be	AUX
iajs-557	131	14	called	call	VERB
iajs-557	131	15	a	a	DET
iajs-557	131	16	quasi	quasi	ADJ
iajs-557	131	17	-	-	ADJ
iajs-557	131	18	hollow	hollow	ADJ
iajs-557	131	19	submodule	submodule	NOUN
iajs-557	131	20	(	(	PUNCT
iajs-557	131	21	briefly	briefly	NOUN
iajs-557	131	22	qh	qh	NOUN
iajs-557	131	23	-	-	NOUN
iajs-557	131	24	submodule	submodule	NOUN
iajs-557	131	25	)	)	PUNCT
iajs-557	131	26	if	if	SCONJ
iajs-557	131	27	for	for	ADP
iajs-557	131	28	each	each	DET
iajs-557	131	29	l1	l1	NOUN
iajs-557	131	30	,	,	PUNCT
iajs-557	131	31	l2	l2	VERB
iajs-557	131	32	≤	≤	NOUN
iajs-557	131	33	m	m	VERB
iajs-557	131	34	with	with	ADP
iajs-557	131	35	l	l	NOUN
iajs-557	131	36	=	=	PROPN
iajs-557	131	37	l1	l1	PROPN
iajs-557	131	38	+	+	CCONJ
iajs-557	131	39	l2	l2	NOUN
iajs-557	131	40	,	,	PUNCT
iajs-557	131	41	then	then	ADV
iajs-557	131	42	l	l	PROPN
iajs-557	131	43	=	=	PROPN
iajs-557	131	44	l1	l1	PROPN
iajs-557	131	45	or	or	CCONJ
iajs-557	131	46	l	l	NOUN
iajs-557	131	47	=	=	NOUN
iajs-557	131	48	l2	l2	NOUN
iajs-557	131	49	.	.	PUNCT
iajs-557	132	1	an	an	DET
iajs-557	132	2	r	r	NOUN
iajs-557	132	3	-	-	PUNCT
iajs-557	132	4	module	module	NOUN
iajs-557	132	5	m	m	NOUN
iajs-557	132	6	is	be	AUX
iajs-557	132	7	said	say	VERB
iajs-557	132	8	a	a	DET
iajs-557	132	9	quasi	quasi	ADJ
iajs-557	132	10	-	-	ADJ
iajs-557	132	11	hollow	hollow	ADJ
iajs-557	132	12	module	module	NOUN
iajs-557	132	13	if	if	SCONJ
iajs-557	132	14	m	m	NOUN
iajs-557	132	15	is	be	AUX
iajs-557	132	16	a	a	DET
iajs-557	132	17	quasi	quasi	ADJ
iajs-557	132	18	-	-	ADJ
iajs-557	132	19	hollow	hollow	ADJ
iajs-557	132	20	submodule	submodule	NOUN
iajs-557	132	21	.	.	PUNCT
iajs-557	133	1	remark	remark	NOUN
iajs-557	133	2	:	:	PUNCT
iajs-557	133	3	mathematics	mathematic	NOUN
iajs-557	133	4	397	397	NUM
iajs-557	133	5	مجلة	مجلة	NOUN
iajs-557	133	6	إبن	إبن	VERB
iajs-557	133	7	الهيثم	الهيثم	ADJ
iajs-557	133	8	للعلوم	للعلوم	NOUN
iajs-557	133	9	الصرفة	الصرفة	NOUN
iajs-557	133	10	و	و	PRON
iajs-557	133	11	التطبيقية	التطبيقية	ADJ
iajs-557	133	12	2012	2012	NUM
iajs-557	133	13	السنة	السنة	NOUN
iajs-557	134	1	25	25	NUM
iajs-557	134	2	المجلد	المجلد	NOUN
iajs-557	134	3	3	3	NUM
iajs-557	134	4	العدد	العدد	PROPN
iajs-557	134	5	ibn	ibn	PROPN
iajs-557	134	6	al	al	PROPN
iajs-557	134	7	-	-	PUNCT
iajs-557	134	8	haitham	haitham	PROPN
iajs-557	134	9	journal	journal	PROPN
iajs-557	134	10	for	for	ADP
iajs-557	134	11	pure	pure	ADJ
iajs-557	134	12	and	and	CCONJ
iajs-557	134	13	applied	apply	VERB
iajs-557	134	14	science	science	NOUN
iajs-557	134	15	no	no	NOUN
iajs-557	134	16	.	.	NOUN
iajs-557	134	17	3	3	NUM
iajs-557	134	18	vol	vol	NOUN
iajs-557	134	19	.	.	PUNCT
iajs-557	135	1	25	25	NUM
iajs-557	135	2	year	year	NOUN
iajs-557	135	3	2012	2012	NUM
iajs-557	135	4	let	let	VERB
iajs-557	135	5	<	<	X
iajs-557	135	6	0	0	NUM
iajs-557	135	7	>	>	X
iajs-557	135	8	≠	≠	PROPN
iajs-557	135	9	l	l	NOUN
iajs-557	135	10	≤	≤	NUM
iajs-557	135	11	m	m	NOUN
iajs-557	135	12	,	,	PUNCT
iajs-557	135	13	l	l	NOUN
iajs-557	135	14	is	be	AUX
iajs-557	135	15	a	a	DET
iajs-557	135	16	quasi	quasi	ADJ
iajs-557	135	17	-	-	ADJ
iajs-557	135	18	hollow	hollow	ADJ
iajs-557	135	19	submodule	submodule	NOUN
iajs-557	135	20	if	if	SCONJ
iajs-557	135	21	for	for	ADP
iajs-557	135	22	each	each	DET
iajs-557	135	23	l1	l1	PROPN
iajs-557	135	24	,	,	PUNCT
iajs-557	135	25	…	…	PUNCT
iajs-557	135	26	,	,	PUNCT
iajs-557	135	27	ln	ln	ADJ
iajs-557	135	28	with	with	ADP
iajs-557	135	29	l	l	NOUN
iajs-557	135	30	=	=	PROPN
iajs-557	135	31	l1	l1	PROPN
iajs-557	135	32	+	+	CCONJ
iajs-557	135	33	…	…	PUNCT
iajs-557	135	34	+	+	ADJ
iajs-557	135	35	ln	ln	ADJ
iajs-557	135	36	,	,	PUNCT
iajs-557	135	37	then	then	ADV
iajs-557	135	38	l	l	PROPN
iajs-557	135	39	=	=	PROPN
iajs-557	135	40	l1	l1	PROPN
iajs-557	135	41	or	or	CCONJ
iajs-557	135	42	…	…	PUNCT
iajs-557	135	43	or	or	CCONJ
iajs-557	135	44	l	l	NOUN
iajs-557	136	1	=	=	SYM
iajs-557	136	2	ln	ln	ADJ
iajs-557	136	3	.	.	PUNCT
iajs-557	136	4	remarks	remark	NOUN
iajs-557	136	5	and	and	CCONJ
iajs-557	136	6	examples	example	NOUN
iajs-557	136	7	:	:	PUNCT
iajs-557	137	1	1	1	X
iajs-557	137	2	.	.	X
iajs-557	137	3	it	it	PRON
iajs-557	137	4	is	be	AUX
iajs-557	137	5	clear	clear	ADJ
iajs-557	137	6	that	that	SCONJ
iajs-557	137	7	every	every	DET
iajs-557	137	8	ch	ch	NOUN
iajs-557	137	9	-	-	PUNCT
iajs-557	137	10	submodule	submodule	NOUN
iajs-557	137	11	is	be	AUX
iajs-557	137	12	qh	qh	NOUN
iajs-557	137	13	-	-	NOUN
iajs-557	137	14	submodule	submodule	NOUN
iajs-557	137	15	.	.	PUNCT
iajs-557	138	1	the	the	DET
iajs-557	138	2	converse	converse	NOUN
iajs-557	138	3	is	be	AUX
iajs-557	138	4	not	not	PART
iajs-557	138	5	true	true	ADJ
iajs-557	138	6	.	.	PUNCT
iajs-557	139	1	for	for	ADP
iajs-557	139	2	example	example	NOUN
iajs-557	139	3	the	the	DET
iajs-557	139	4	z	z	NOUN
iajs-557	139	5	-	-	PUNCT
iajs-557	139	6	module	module	NOUN
iajs-557	139	7	pz	pz	NOUN
iajs-557	139	8	∞	∞	PROPN
iajs-557	139	9	is	be	AUX
iajs-557	139	10	qh	qh	NOUN
iajs-557	139	11	-	-	NOUN
iajs-557	139	12	module	module	NOUN
iajs-557	139	13	(	(	PUNCT
iajs-557	139	14	qh	qh	NOUN
iajs-557	139	15	-	-	NOUN
iajs-557	139	16	submodule	submodule	NOUN
iajs-557	139	17	of	of	ADP
iajs-557	139	18	itself	itself	PRON
iajs-557	139	19	)	)	PUNCT
iajs-557	139	20	since	since	SCONJ
iajs-557	139	21	there	there	PRON
iajs-557	139	22	is	be	VERB
iajs-557	139	23	no	no	DET
iajs-557	139	24	l1≨m	l1≨m	NOUN
iajs-557	139	25	and	and	CCONJ
iajs-557	139	26	l2≨m	l2≨m	NOUN
iajs-557	139	27	such	such	ADJ
iajs-557	139	28	that	that	DET
iajs-557	139	29	pz	pz	NOUN
iajs-557	139	30	∞	∞	PROPN
iajs-557	139	31	=	=	PROPN
iajs-557	139	32	l1	l1	PROPN
iajs-557	139	33	+	+	CCONJ
iajs-557	139	34	l2	l2	NOUN
iajs-557	139	35	.	.	PUNCT
iajs-557	140	1	but	but	CCONJ
iajs-557	140	2	by	by	ADP
iajs-557	140	3	remark	remark	NOUN
iajs-557	140	4	1.4(11)(b	1.4(11)(b	NUM
iajs-557	140	5	)	)	PUNCT
iajs-557	140	6	pz	pz	NOUN
iajs-557	140	7	∞	∞	PROPN
iajs-557	140	8	is	be	AUX
iajs-557	140	9	not	not	PART
iajs-557	140	10	ch	ch	NOUN
iajs-557	140	11	.	.	PROPN
iajs-557	140	12	2	2	NUM
iajs-557	140	13	.	.	X
iajs-557	141	1	every	every	DET
iajs-557	141	2	simple	simple	ADJ
iajs-557	141	3	submodule	submodule	NOUN
iajs-557	141	4	of	of	ADP
iajs-557	141	5	an	an	DET
iajs-557	141	6	r	r	NOUN
iajs-557	141	7	-	-	PUNCT
iajs-557	141	8	module	module	NOUN
iajs-557	141	9	is	be	AUX
iajs-557	141	10	qh	qh	NOUN
iajs-557	141	11	-	-	NOUN
iajs-557	141	12	submodule	submodule	NOUN
iajs-557	141	13	.	.	PUNCT
iajs-557	142	1	3	3	X
iajs-557	142	2	.	.	X
iajs-557	143	1	every	every	DET
iajs-557	143	2	sh	sh	PROPN
iajs-557	143	3	-	-	PUNCT
iajs-557	143	4	submodule	submodule	PROPN
iajs-557	143	5	is	be	AUX
iajs-557	143	6	qh	qh	NOUN
iajs-557	143	7	-	-	NOUN
iajs-557	143	8	submodule	submodule	NOUN
iajs-557	143	9	.	.	PUNCT
iajs-557	144	1	the	the	DET
iajs-557	144	2	converse	converse	NOUN
iajs-557	144	3	is	be	AUX
iajs-557	144	4	not	not	PART
iajs-557	144	5	true	true	ADJ
iajs-557	144	6	in	in	ADP
iajs-557	144	7	general	general	ADJ
iajs-557	144	8	,	,	PUNCT
iajs-557	144	9	for	for	ADP
iajs-557	144	10	example	example	NOUN
iajs-557	144	11	in	in	ADP
iajs-557	144	12	the	the	DET
iajs-557	144	13	vector	vector	NOUN
iajs-557	144	14	space	space	NOUN
iajs-557	144	15	ℝ2	ℝ2	VERB
iajs-557	144	16	over	over	ADP
iajs-557	144	17	ℝ	ℝ	PROPN
iajs-557	144	18	,	,	PUNCT
iajs-557	144	19	n	n	NOUN
iajs-557	144	20	=	=	SYM
iajs-557	144	21	ℝ(1,0	ℝ(1,0	PROPN
iajs-557	144	22	)	)	PUNCT
iajs-557	144	23	is	be	AUX
iajs-557	144	24	simple	simple	ADJ
iajs-557	144	25	submodule	submodule	NOUN
iajs-557	144	26	,	,	PUNCT
iajs-557	144	27	so	so	ADV
iajs-557	144	28	by	by	ADP
iajs-557	144	29	remark	remark	NOUN
iajs-557	144	30	1.15(2	1.15(2	NUM
iajs-557	144	31	)	)	PUNCT
iajs-557	144	32	,	,	PUNCT
iajs-557	144	33	n	n	PROPN
iajs-557	144	34	is	be	AUX
iajs-557	144	35	qh	qh	NOUN
iajs-557	144	36	,	,	PUNCT
iajs-557	144	37	but	but	CCONJ
iajs-557	144	38	it	it	PRON
iajs-557	144	39	is	be	AUX
iajs-557	144	40	not	not	PART
iajs-557	144	41	sh	sh	PRON
iajs-557	144	42	by	by	ADP
iajs-557	144	43	remark	remark	NOUN
iajs-557	144	44	1.4(11)(b	1.4(11)(b	NUM
iajs-557	144	45	)	)	PUNCT
iajs-557	144	46	.	.	PUNCT
iajs-557	145	1	4	4	X
iajs-557	145	2	.	.	X
iajs-557	145	3	if	if	SCONJ
iajs-557	145	4	m	m	NOUN
iajs-557	145	5	is	be	AUX
iajs-557	145	6	chained	chain	VERB
iajs-557	145	7	,	,	PUNCT
iajs-557	145	8	then	then	ADV
iajs-557	145	9	every	every	DET
iajs-557	145	10	submodule	submodule	NOUN
iajs-557	145	11	is	be	AUX
iajs-557	145	12	qh	qh	NOUN
iajs-557	145	13	.	.	PROPN
iajs-557	145	14	proof	proof	NOUN
iajs-557	145	15	:	:	PUNCT
iajs-557	145	16	it	it	PRON
iajs-557	145	17	follows	follow	VERB
iajs-557	145	18	by	by	ADP
iajs-557	145	19	remark	remark	NOUN
iajs-557	145	20	1.4(7	1.4(7	NUM
iajs-557	145	21	)	)	PUNCT
iajs-557	145	22	and	and	CCONJ
iajs-557	145	23	remark	remark	NOUN
iajs-557	145	24	1.15(3	1.15(3	NUM
iajs-557	145	25	)	)	PUNCT
iajs-557	145	26	.	.	PUNCT
iajs-557	146	1	5	5	X
iajs-557	146	2	.	.	X
iajs-557	146	3	let	let	VERB
iajs-557	146	4	m	m	PRON
iajs-557	146	5	be	be	AUX
iajs-557	146	6	an	an	DET
iajs-557	146	7	r	r	NOUN
iajs-557	146	8	-	-	PUNCT
iajs-557	146	9	module	module	NOUN
iajs-557	146	10	.	.	PUNCT
iajs-557	147	1	then	then	ADV
iajs-557	147	2	m	m	PROPN
iajs-557	147	3	is	be	AUX
iajs-557	147	4	a	a	DET
iajs-557	147	5	qh	qh	NOUN
iajs-557	147	6	-	-	NOUN
iajs-557	147	7	module	module	NOUN
iajs-557	147	8	if	if	SCONJ
iajs-557	147	9	and	and	CCONJ
iajs-557	147	10	only	only	ADV
iajs-557	147	11	if	if	SCONJ
iajs-557	147	12	m	m	NOUN
iajs-557	147	13	is	be	AUX
iajs-557	147	14	sh	sh	INTJ
iajs-557	147	15	if	if	SCONJ
iajs-557	147	16	and	and	CCONJ
iajs-557	147	17	only	only	ADV
iajs-557	147	18	if	if	SCONJ
iajs-557	147	19	m	m	NOUN
iajs-557	147	20	is	be	AUX
iajs-557	147	21	hollow	hollow	ADJ
iajs-557	147	22	.	.	PUNCT
iajs-557	148	1	where	where	SCONJ
iajs-557	148	2	m	m	NOUN
iajs-557	148	3	is	be	AUX
iajs-557	148	4	hollow	hollow	ADJ
iajs-557	148	5	if	if	SCONJ
iajs-557	148	6	every	every	DET
iajs-557	148	7	proper	proper	ADJ
iajs-557	148	8	submodule	submodule	NOUN
iajs-557	148	9	n	n	PROPN
iajs-557	148	10	of	of	ADP
iajs-557	148	11	m	m	PROPN
iajs-557	148	12	is	be	AUX
iajs-557	148	13	small	small	ADJ
iajs-557	148	14	.	.	PUNCT
iajs-557	149	1	that	that	PRON
iajs-557	149	2	is	be	AUX
iajs-557	149	3	there	there	PRON
iajs-557	149	4	is	be	VERB
iajs-557	149	5	no	no	DET
iajs-557	149	6	proper	proper	ADJ
iajs-557	149	7	submodule	submodule	NOUN
iajs-557	149	8	w	w	PROPN
iajs-557	149	9	of	of	ADP
iajs-557	149	10	m	m	PRON
iajs-557	150	1	such	such	ADJ
iajs-557	150	2	that	that	SCONJ
iajs-557	150	3	n	n	PROPN
iajs-557	150	4	+	+	CCONJ
iajs-557	150	5	w	w	NOUN
iajs-557	150	6	=	=	VERB
iajs-557	150	7	m.	m.	NOUN
iajs-557	150	8	equivalently	equivalently	ADV
iajs-557	150	9	,	,	PUNCT
iajs-557	150	10	for	for	ADP
iajs-557	150	11	every	every	DET
iajs-557	150	12	submodules	submodule	NOUN
iajs-557	150	13	n	n	CCONJ
iajs-557	150	14	,	,	PUNCT
iajs-557	150	15	w	w	ADP
iajs-557	150	16	such	such	ADJ
iajs-557	150	17	that	that	SCONJ
iajs-557	150	18	n	n	CCONJ
iajs-557	150	19	≨	≨	PROPN
iajs-557	150	20	m	m	PROPN
iajs-557	150	21	,	,	PUNCT
iajs-557	150	22	w	w	PROPN
iajs-557	150	23	≨	≨	PROPN
iajs-557	150	24	m	m	VERB
iajs-557	150	25	implies	imply	VERB
iajs-557	150	26	n	n	X
iajs-557	150	27	+	+	CCONJ
iajs-557	150	28	w	w	PROPN
iajs-557	150	29	≨	≨	PROPN
iajs-557	150	30	m.	m.	NOUN
iajs-557	150	31	6	6	NUM
iajs-557	150	32	.	.	PUNCT
iajs-557	151	1	let	let	VERB
iajs-557	151	2	m	m	PRON
iajs-557	151	3	be	be	AUX
iajs-557	151	4	ch	ch	PROPN
iajs-557	151	5	(	(	PUNCT
iajs-557	151	6	qh	qh	NOUN
iajs-557	151	7	or	or	CCONJ
iajs-557	151	8	sh	sh	PROPN
iajs-557	151	9	)	)	PUNCT
iajs-557	151	10	r	r	NOUN
iajs-557	151	11	-	-	PUNCT
iajs-557	151	12	module	module	NOUN
iajs-557	151	13	,	,	PUNCT
iajs-557	151	14	then	then	ADV
iajs-557	151	15	there	there	PRON
iajs-557	151	16	is	be	VERB
iajs-557	151	17	no	no	DET
iajs-557	151	18	submodules	submodule	NOUN
iajs-557	151	19	n	n	CCONJ
iajs-557	151	20	,	,	PUNCT
iajs-557	151	21	w	w	PROPN
iajs-557	151	22	of	of	ADP
iajs-557	151	23	m	m	PRON
iajs-557	151	24	such	such	ADJ
iajs-557	151	25	that	that	SCONJ
iajs-557	151	26	m	m	VERB
iajs-557	151	27	=	=	SYM
iajs-557	152	1	n	n	ADJ
iajs-557	152	2	⊕w	⊕w	NOUN
iajs-557	152	3	.	.	PUNCT
iajs-557	153	1	7	7	X
iajs-557	153	2	.	.	X
iajs-557	153	3	consider	consider	VERB
iajs-557	153	4	z48	z48	NOUN
iajs-557	153	5	as	as	ADP
iajs-557	153	6	z	z	NOUN
iajs-557	153	7	-	-	NOUN
iajs-557	153	8	module	module	NOUN
iajs-557	153	9	.	.	PUNCT
iajs-557	154	1	each	each	PRON
iajs-557	154	2	of	of	ADP
iajs-557	154	3	2	2	NUM
iajs-557	154	4	,	,	PUNCT
iajs-557	154	5	4	4	NUM
iajs-557	154	6	,	,	PUNCT
iajs-557	154	7	8	8	NUM
iajs-557	154	8	<	<	X
iajs-557	154	9	>	>	X
iajs-557	154	10	<	<	X
iajs-557	154	11	>	>	X
iajs-557	154	12	<	<	X
iajs-557	154	13	>	>	X
iajs-557	154	14	and	and	CCONJ
iajs-557	154	15	z48	z48	PROPN
iajs-557	154	16	is	be	AUX
iajs-557	154	17	not	not	PART
iajs-557	154	18	qh	qh	NOUN
iajs-557	154	19	,	,	PUNCT
iajs-557	154	20	not	not	PART
iajs-557	154	21	sh	sh	PROPN
iajs-557	154	22	.	.	PUNCT
iajs-557	155	1	each	each	PRON
iajs-557	155	2	of	of	ADP
iajs-557	155	3	3	3	NUM
iajs-557	155	4	,	,	PUNCT
iajs-557	155	5	6	6	NUM
iajs-557	155	6	,	,	PUNCT
iajs-557	155	7	12	12	NUM
iajs-557	155	8	,	,	PUNCT
iajs-557	155	9	24	24	NUM
iajs-557	155	10	<	<	X
iajs-557	155	11	>	>	X
iajs-557	155	12	<	<	X
iajs-557	155	13	>	>	X
iajs-557	155	14	<	<	X
iajs-557	155	15	>	>	X
iajs-557	155	16	<	<	X
iajs-557	155	17	>	>	X
iajs-557	155	18	is	be	AUX
iajs-557	155	19	qh	qh	PROPN
iajs-557	155	20	and	and	CCONJ
iajs-557	155	21	sh	sh	PROPN
iajs-557	155	22	.	.	PROPN
iajs-557	155	23	8	8	NUM
iajs-557	155	24	.	.	X
iajs-557	156	1	consider	consider	VERB
iajs-557	156	2	m	m	VERB
iajs-557	156	3	=	=	ADJ
iajs-557	156	4	z4⊕z2	z4⊕z2	PROPN
iajs-557	156	5	as	as	ADP
iajs-557	156	6	z	z	NOUN
iajs-557	156	7	-	-	NOUN
iajs-557	156	8	module	module	NOUN
iajs-557	156	9	.	.	PUNCT
iajs-557	157	1	each	each	PRON
iajs-557	157	2	of	of	ADP
iajs-557	157	3	2	2	NUM
iajs-557	157	4	40	40	NUM
iajs-557	157	5	z	z	NOUN
iajs-557	157	6	,	,	PUNCT
iajs-557	157	7	z	z	NOUN
iajs-557	157	8	0	0	NUM
iajs-557	157	9	,	,	PUNCT
iajs-557	157	10	2	2	NUM
iajs-557	157	11	0	0	NUM
iajs-557	157	12	<	<	X
iajs-557	157	13	>	>	X
iajs-557	157	14	⊕	⊕	PROPN
iajs-557	157	15	⊕	⊕	PROPN
iajs-557	157	16	<	<	X
iajs-557	157	17	>	>	X
iajs-557	157	18	<	<	X
iajs-557	157	19	>	>	X
iajs-557	157	20	⊕	⊕	PROPN
iajs-557	157	21	<	<	X
iajs-557	157	22	>	>	X
iajs-557	157	23	is	be	AUX
iajs-557	157	24	qh	qh	NOUN
iajs-557	157	25	and	and	CCONJ
iajs-557	157	26	sh	sh	PROPN
iajs-557	157	27	,	,	PUNCT
iajs-557	157	28	and	and	CCONJ
iajs-557	157	29	each	each	PRON
iajs-557	157	30	of	of	ADP
iajs-557	157	31	2	2	NUM
iajs-557	157	32	4	4	NUM
iajs-557	157	33	2z	2z	NUM
iajs-557	157	34	z	z	NOUN
iajs-557	157	35	,	,	PUNCT
iajs-557	157	36	2	2	NUM
iajs-557	157	37	z⊕	z⊕	VERB
iajs-557	157	38	<	<	X
iajs-557	157	39	>	>	X
iajs-557	157	40	⊕	⊕	PROPN
iajs-557	157	41	is	be	AUX
iajs-557	157	42	not	not	PART
iajs-557	157	43	qh	qh	NOUN
iajs-557	157	44	,	,	PUNCT
iajs-557	157	45	not	not	PART
iajs-557	157	46	sh	sh	PROPN
iajs-557	157	47	.	.	PROPN
iajs-557	157	48	9	9	NUM
iajs-557	157	49	.	.	X
iajs-557	158	1	let	let	VERB
iajs-557	158	2	<	<	X
iajs-557	158	3	0	0	NUM
iajs-557	158	4	>	>	X
iajs-557	158	5	≠	≠	PROPN
iajs-557	158	6	l	l	NOUN
iajs-557	158	7	≤	≤	NUM
iajs-557	158	8	m	m	VERB
iajs-557	158	9	as	as	ADP
iajs-557	158	10	r	r	NOUN
iajs-557	158	11	-	-	PUNCT
iajs-557	158	12	module	module	NOUN
iajs-557	158	13	.	.	PUNCT
iajs-557	159	1	let	let	VERB
iajs-557	159	2	n	n	PRON
iajs-557	159	3	≤	≤	X
iajs-557	159	4	l.	l.	NOUN
iajs-557	159	5	if	if	SCONJ
iajs-557	159	6	l	l	NOUN
iajs-557	159	7	is	be	AUX
iajs-557	159	8	qh	qh	NOUN
iajs-557	159	9	-	-	NOUN
iajs-557	159	10	submodule	submodule	NOUN
iajs-557	159	11	,	,	PUNCT
iajs-557	159	12	then	then	ADV
iajs-557	159	13	n	n	PRON
iajs-557	159	14	need	need	AUX
iajs-557	159	15	not	not	PART
iajs-557	159	16	be	be	AUX
iajs-557	159	17	qh	qh	NOUN
iajs-557	159	18	.	.	PROPN
iajs-557	160	1	for	for	ADP
iajs-557	160	2	example	example	NOUN
iajs-557	160	3	,	,	PUNCT
iajs-557	160	4	z	z	NOUN
iajs-557	160	5	-	-	PUNCT
iajs-557	160	6	module	module	NOUN
iajs-557	160	7	z48	z48	NOUN
iajs-557	160	8	where	where	SCONJ
iajs-557	160	9	3	3	X
iajs-557	160	10	<	<	X
iajs-557	160	11	>	>	X
iajs-557	160	12	is	be	AUX
iajs-557	160	13	qh	qh	NOUN
iajs-557	160	14	,	,	PUNCT
iajs-557	160	15	but	but	CCONJ
iajs-557	160	16	0	0	NUM
iajs-557	160	17	<	<	X
iajs-557	160	18	>	>	X
iajs-557	160	19	is	be	AUX
iajs-557	160	20	not	not	PART
iajs-557	160	21	qh	qh	NOUN
iajs-557	160	22	.	.	PROPN
iajs-557	160	23	10	10	NUM
iajs-557	160	24	.	.	PUNCT
iajs-557	161	1	let	let	VERB
iajs-557	161	2	<	<	X
iajs-557	161	3	0	0	NUM
iajs-557	161	4	>	>	X
iajs-557	161	5	≠	≠	PROPN
iajs-557	161	6	l	l	NOUN
iajs-557	161	7	≤	≤	NUM
iajs-557	161	8	w	w	NOUN
iajs-557	161	9	≤	≤	NUM
iajs-557	161	10	m	m	NOUN
iajs-557	161	11	as	as	ADP
iajs-557	161	12	r	r	NOUN
iajs-557	161	13	-	-	NOUN
iajs-557	161	14	module	module	NOUN
iajs-557	161	15	.	.	PUNCT
iajs-557	162	1	if	if	SCONJ
iajs-557	162	2	l	l	NOUN
iajs-557	162	3	is	be	AUX
iajs-557	162	4	qh	qh	NOUN
iajs-557	162	5	,	,	PUNCT
iajs-557	162	6	then	then	ADV
iajs-557	162	7	w	w	NOUN
iajs-557	162	8	need	need	AUX
iajs-557	162	9	not	not	PART
iajs-557	162	10	be	be	AUX
iajs-557	162	11	qh	qh	NOUN
iajs-557	162	12	.	.	PROPN
iajs-557	163	1	for	for	ADP
iajs-557	163	2	example	example	NOUN
iajs-557	163	3	,	,	PUNCT
iajs-557	163	4	m	m	VERB
iajs-557	163	5	=	=	ADJ
iajs-557	163	6	z4⊕z2	z4⊕z2	PROPN
iajs-557	163	7	as	as	ADP
iajs-557	163	8	z	z	NOUN
iajs-557	163	9	-	-	PUNCT
iajs-557	163	10	module	module	NOUN
iajs-557	163	11	,	,	PUNCT
iajs-557	163	12	where	where	SCONJ
iajs-557	163	13	20	20	NUM
iajs-557	163	14	z	z	NOUN
iajs-557	163	15	<	<	X
iajs-557	163	16	>	>	X
iajs-557	163	17	⊕	⊕	PROPN
iajs-557	163	18	is	be	AUX
iajs-557	163	19	qh	qh	NOUN
iajs-557	163	20	and	and	CCONJ
iajs-557	163	21	2	2	NUM
iajs-557	163	22	20	20	NUM
iajs-557	163	23	z	z	NOUN
iajs-557	163	24	2	2	NUM
iajs-557	163	25	z	z	NOUN
iajs-557	163	26	<	<	X
iajs-557	163	27	>	>	X
iajs-557	163	28	⊕	⊕	PROPN
iajs-557	163	29	⊆	⊆	SYM
iajs-557	163	30	<	<	X
iajs-557	163	31	>	>	X
iajs-557	163	32	⊕	⊕	PROPN
iajs-557	163	33	.	.	PUNCT
iajs-557	164	1	but	but	CCONJ
iajs-557	164	2	22	22	NUM
iajs-557	164	3	z	z	NOUN
iajs-557	164	4	<	<	X
iajs-557	164	5	>	>	X
iajs-557	164	6	⊕	⊕	PROPN
iajs-557	164	7	is	be	AUX
iajs-557	164	8	not	not	PART
iajs-557	164	9	qh	qh	NOUN
iajs-557	164	10	.	.	PROPN
iajs-557	164	11	11	11	NUM
iajs-557	164	12	.	.	PUNCT
iajs-557	165	1	if	if	SCONJ
iajs-557	165	2	l1	l1	PROPN
iajs-557	165	3	,	,	PUNCT
iajs-557	165	4	l2	l2	NOUN
iajs-557	165	5	are	be	AUX
iajs-557	165	6	qh	qh	NOUN
iajs-557	165	7	of	of	ADP
iajs-557	165	8	an	an	DET
iajs-557	165	9	r	r	NOUN
iajs-557	165	10	-	-	PUNCT
iajs-557	165	11	module	module	NOUN
iajs-557	165	12	m	m	NOUN
iajs-557	165	13	,	,	PUNCT
iajs-557	165	14	then	then	ADV
iajs-557	165	15	l1	l1	PROPN
iajs-557	165	16	+	+	CCONJ
iajs-557	165	17	l2	l2	NOUN
iajs-557	165	18	need	need	AUX
iajs-557	165	19	not	not	PART
iajs-557	165	20	be	be	AUX
iajs-557	165	21	qh	qh	NOUN
iajs-557	165	22	.	.	PROPN
iajs-557	166	1	for	for	ADP
iajs-557	166	2	example	example	NOUN
iajs-557	166	3	,	,	PUNCT
iajs-557	166	4	1	1	NUM
iajs-557	166	5	2	2	NUM
iajs-557	166	6	2	2	NUM
iajs-557	166	7	4l	4l	NOUN
iajs-557	166	8	0	0	NUM
iajs-557	166	9	z	z	NOUN
iajs-557	166	10	,	,	PUNCT
iajs-557	166	11	l	l	NOUN
iajs-557	167	1	z	z	NOUN
iajs-557	167	2	0=	0=	NOUN
iajs-557	167	3	<	<	X
iajs-557	167	4	>	>	X
iajs-557	167	5	⊕	⊕	PROPN
iajs-557	167	6	=	=	PUNCT
iajs-557	167	7	⊕	⊕	PROPN
iajs-557	167	8	<	<	X
iajs-557	167	9	>	>	X
iajs-557	167	10	are	be	AUX
iajs-557	167	11	qh	qh	NOUN
iajs-557	167	12	of	of	ADP
iajs-557	167	13	m	m	PROPN
iajs-557	167	14	=	=	NOUN
iajs-557	167	15	z4⊕z2	z4⊕z2	PROPN
iajs-557	167	16	as	as	ADP
iajs-557	167	17	z	z	NOUN
iajs-557	167	18	-	-	NOUN
iajs-557	167	19	module	module	NOUN
iajs-557	167	20	.	.	PUNCT
iajs-557	168	1	but	but	CCONJ
iajs-557	168	2	l1	l1	PROPN
iajs-557	168	3	+	+	CCONJ
iajs-557	168	4	l2	l2	NOUN
iajs-557	169	1	=	=	VERB
iajs-557	169	2	m	m	VERB
iajs-557	169	3	is	be	AUX
iajs-557	169	4	not	not	PART
iajs-557	169	5	qh	qh	NOUN
iajs-557	169	6	.	.	PROPN
iajs-557	169	7	12	12	NUM
iajs-557	169	8	.	.	PUNCT
iajs-557	170	1	let	let	VERB
iajs-557	170	2	r	r	PRON
iajs-557	170	3	be	be	AUX
iajs-557	170	4	a	a	DET
iajs-557	170	5	ring	ring	NOUN
iajs-557	170	6	.	.	PUNCT
iajs-557	171	1	if	if	SCONJ
iajs-557	171	2	a	a	PRON
iajs-557	171	3	and	and	CCONJ
iajs-557	171	4	b	b	NOUN
iajs-557	171	5	are	be	AUX
iajs-557	171	6	qh(sh)-ideals	qh(sh)-ideal	NOUN
iajs-557	171	7	.	.	PUNCT
iajs-557	172	1	then	then	ADV
iajs-557	172	2	ab	ab	PROPN
iajs-557	172	3	need	need	AUX
iajs-557	172	4	not	not	PART
iajs-557	172	5	be	be	AUX
iajs-557	172	6	qh(sh)-ideals	qh(sh)-ideal	NOUN
iajs-557	172	7	of	of	ADP
iajs-557	172	8	r.	r.	PROPN
iajs-557	172	9	for	for	ADP
iajs-557	172	10	example	example	NOUN
iajs-557	172	11	2	2	NUM
iajs-557	172	12	<	<	X
iajs-557	172	13	>	>	X
iajs-557	172	14	and	and	CCONJ
iajs-557	172	15	3	3	NUM
iajs-557	172	16	<	<	X
iajs-557	172	17	>	>	X
iajs-557	172	18	are	be	AUX
iajs-557	172	19	qh(sh)-ideals	qh(sh)-ideal	NOUN
iajs-557	172	20	of	of	ADP
iajs-557	172	21	the	the	DET
iajs-557	172	22	ring	ring	NOUN
iajs-557	172	23	z6	z6	PROPN
iajs-557	172	24	.	.	PUNCT
iajs-557	173	1	but	but	CCONJ
iajs-557	173	2	2	2	NUM
iajs-557	173	3	3	3	NUM
iajs-557	173	4	0	0	NUM
iajs-557	173	5	<	<	X
iajs-557	173	6	>	>	X
iajs-557	173	7	⋅	⋅	X
iajs-557	173	8	<	<	X
iajs-557	173	9	>	>	X
iajs-557	173	10	=	=	X
iajs-557	173	11	<	<	X
iajs-557	173	12	>	>	X
iajs-557	173	13	is	be	AUX
iajs-557	173	14	not	not	PART
iajs-557	173	15	sh	sh	PROPN
iajs-557	173	16	,	,	PUNCT
iajs-557	173	17	not	not	PART
iajs-557	173	18	qh	qh	NOUN
iajs-557	173	19	.	.	PUNCT
iajs-557	174	1	now	now	ADV
iajs-557	174	2	we	we	PRON
iajs-557	174	3	find	find	VERB
iajs-557	174	4	that	that	SCONJ
iajs-557	174	5	under	under	ADP
iajs-557	174	6	the	the	DET
iajs-557	174	7	class	class	NOUN
iajs-557	174	8	of	of	ADP
iajs-557	174	9	distributive	distributive	ADJ
iajs-557	174	10	of	of	ADP
iajs-557	174	11	modules	module	NOUN
iajs-557	174	12	,	,	PUNCT
iajs-557	174	13	the	the	DET
iajs-557	174	14	concepts	concept	NOUN
iajs-557	174	15	,	,	PUNCT
iajs-557	174	16	qhsubmodules	qhsubmodule	NOUN
iajs-557	174	17	and	and	CCONJ
iajs-557	174	18	sh	sh	NOUN
iajs-557	174	19	-	-	PUNCT
iajs-557	174	20	submodules	submodule	NOUN
iajs-557	174	21	are	be	AUX
iajs-557	174	22	equivalent	equivalent	ADJ
iajs-557	174	23	,	,	PUNCT
iajs-557	174	24	as	as	SCONJ
iajs-557	174	25	the	the	DET
iajs-557	174	26	following	follow	VERB
iajs-557	174	27	proposition	proposition	NOUN
iajs-557	174	28	shows	show	VERB
iajs-557	174	29	:	:	PUNCT
iajs-557	174	30	proposition	proposition	NOUN
iajs-557	174	31	:	:	PUNCT
iajs-557	174	32	let	let	VERB
iajs-557	174	33	m	m	PRON
iajs-557	174	34	be	be	AUX
iajs-557	174	35	distributive	distributive	ADJ
iajs-557	174	36	r	r	NOUN
iajs-557	174	37	-	-	PUNCT
iajs-557	174	38	module	module	NOUN
iajs-557	174	39	,	,	PUNCT
iajs-557	174	40	and	and	CCONJ
iajs-557	174	41	0	0	NUM
iajs-557	174	42	≠	≠	PROPN
iajs-557	174	43	n	n	PRON
iajs-557	174	44	≨	≨	PROPN
iajs-557	174	45	m.	m.	NOUN
iajs-557	175	1	then	then	ADV
iajs-557	175	2	n	n	PRON
iajs-557	175	3	is	be	AUX
iajs-557	175	4	sh	sh	PROPN
iajs-557	175	5	-	-	PUNCT
iajs-557	175	6	submodule	submodule	NOUN
iajs-557	175	7	if	if	SCONJ
iajs-557	175	8	and	and	CCONJ
iajs-557	175	9	only	only	ADV
iajs-557	175	10	if	if	SCONJ
iajs-557	175	11	n	n	NOUN
iajs-557	175	12	is	be	AUX
iajs-557	175	13	qh	qh	NOUN
iajs-557	175	14	-	-	NOUN
iajs-557	175	15	submodule	submodule	NOUN
iajs-557	175	16	.	.	PUNCT
iajs-557	176	1	mathematics	mathematic	NOUN
iajs-557	176	2	398	398	NUM
iajs-557	176	3	مجلة	مجلة	NOUN
iajs-557	176	4	إبن	إبن	VERB
iajs-557	176	5	الهيثم	الهيثم	ADJ
iajs-557	176	6	للعلوم	للعلوم	NOUN
iajs-557	176	7	الصرفة	الصرفة	NOUN
iajs-557	176	8	و	و	PRON
iajs-557	176	9	التطبيقية	التطبيقية	ADJ
iajs-557	176	10	2012	2012	NUM
iajs-557	176	11	السنة	السنة	NOUN
iajs-557	176	12	25	25	NUM
iajs-557	176	13	المجلد	المجلد	NOUN
iajs-557	176	14	3	3	NUM
iajs-557	176	15	العدد	العدد	PROPN
iajs-557	176	16	ibn	ibn	PROPN
iajs-557	176	17	al	al	PROPN
iajs-557	176	18	-	-	PUNCT
iajs-557	176	19	haitham	haitham	PROPN
iajs-557	176	20	journal	journal	PROPN
iajs-557	176	21	for	for	ADP
iajs-557	176	22	pure	pure	ADJ
iajs-557	176	23	and	and	CCONJ
iajs-557	176	24	applied	apply	VERB
iajs-557	176	25	science	science	NOUN
iajs-557	176	26	no	no	NOUN
iajs-557	176	27	.	.	NOUN
iajs-557	176	28	3	3	NUM
iajs-557	176	29	vol	vol	NOUN
iajs-557	176	30	.	.	PUNCT
iajs-557	177	1	25	25	NUM
iajs-557	177	2	year	year	NOUN
iajs-557	177	3	2012	2012	NUM
iajs-557	177	4	proof	proof	NOUN
iajs-557	177	5	:	:	PUNCT
iajs-557	177	6	(	(	PUNCT
iajs-557	177	7	⇒	⇒	NOUN
iajs-557	177	8	)	)	PUNCT
iajs-557	177	9	clear	clear	ADJ
iajs-557	177	10	by	by	ADP
iajs-557	177	11	remark	remark	NOUN
iajs-557	177	12	1.15(3	1.15(3	NUM
iajs-557	177	13	)	)	PUNCT
iajs-557	177	14	.	.	PUNCT
iajs-557	178	1	(	(	PUNCT
iajs-557	178	2	⇐	⇐	ADJ
iajs-557	178	3	)	)	PUNCT
iajs-557	178	4	assume	assume	VERB
iajs-557	178	5	n	n	PRON
iajs-557	178	6	is	be	AUX
iajs-557	178	7	qh	qh	NOUN
iajs-557	178	8	-	-	NOUN
iajs-557	178	9	submodule	submodule	NOUN
iajs-557	178	10	.	.	PUNCT
iajs-557	179	1	let	let	VERB
iajs-557	179	2	n	n	PRON
iajs-557	179	3	⊆	⊆	NUM
iajs-557	179	4	l1	l1	PROPN
iajs-557	179	5	+	+	CCONJ
iajs-557	179	6	l2	l2	PROPN
iajs-557	179	7	where	where	SCONJ
iajs-557	179	8	l1	l1	PROPN
iajs-557	179	9	,	,	PUNCT
iajs-557	179	10	l2	l2	NOUN
iajs-557	179	11	≤	≤	ADJ
iajs-557	179	12	m.	m.	NOUN
iajs-557	179	13	then	then	ADV
iajs-557	179	14	n	n	NOUN
iajs-557	179	15	=	=	SYM
iajs-557	179	16	n	n	NOUN
iajs-557	179	17	∩	∩	NOUN
iajs-557	179	18	(	(	PUNCT
iajs-557	179	19	l1	l1	PROPN
iajs-557	179	20	+	+	CCONJ
iajs-557	179	21	l2	l2	NOUN
iajs-557	179	22	)	)	PUNCT
iajs-557	179	23	,	,	PUNCT
iajs-557	180	1	so	so	ADV
iajs-557	180	2	n	n	ADV
iajs-557	180	3	=	=	SYM
iajs-557	180	4	(	(	PUNCT
iajs-557	180	5	n	n	X
iajs-557	180	6	∩	∩	ADJ
iajs-557	180	7	l1	l1	PROPN
iajs-557	180	8	)	)	PUNCT
iajs-557	181	1	+	+	CCONJ
iajs-557	181	2	(	(	PUNCT
iajs-557	181	3	n	n	CCONJ
iajs-557	181	4	∩	∩	ADJ
iajs-557	181	5	l2	l2	NOUN
iajs-557	181	6	)	)	PUNCT
iajs-557	181	7	,	,	PUNCT
iajs-557	181	8	since	since	SCONJ
iajs-557	181	9	m	m	PROPN
iajs-557	181	10	is	be	AUX
iajs-557	181	11	distributive	distributive	ADJ
iajs-557	181	12	.	.	PUNCT
iajs-557	182	1	then	then	ADV
iajs-557	182	2	n	n	PROPN
iajs-557	182	3	=	=	SYM
iajs-557	182	4	n	n	NOUN
iajs-557	182	5	∩	∩	X
iajs-557	182	6	l1	l1	PROPN
iajs-557	182	7	or	or	CCONJ
iajs-557	182	8	n	n	NOUN
iajs-557	182	9	=	=	SYM
iajs-557	182	10	n	n	NOUN
iajs-557	182	11	∩	∩	ADJ
iajs-557	182	12	l2	l2	NOUN
iajs-557	182	13	since	since	SCONJ
iajs-557	182	14	n	n	ADV
iajs-557	182	15	is	be	AUX
iajs-557	182	16	qh	qh	NOUN
iajs-557	182	17	.	.	PUNCT
iajs-557	183	1	it	it	PRON
iajs-557	183	2	follows	follow	VERB
iajs-557	183	3	that	that	SCONJ
iajs-557	183	4	either	either	CCONJ
iajs-557	183	5	n	n	PROPN
iajs-557	183	6	⊆	⊆	NUM
iajs-557	183	7	l1	l1	PROPN
iajs-557	183	8	or	or	CCONJ
iajs-557	183	9	n	n	CCONJ
iajs-557	183	10	⊆	⊆	NUM
iajs-557	183	11	l2	l2	NOUN
iajs-557	183	12	.	.	PUNCT
iajs-557	184	1	hence	hence	ADV
iajs-557	184	2	n	n	ADV
iajs-557	184	3	is	be	AUX
iajs-557	184	4	a	a	DET
iajs-557	184	5	sh	sh	NOUN
iajs-557	184	6	-	-	PUNCT
iajs-557	184	7	submodule	submodule	NOUN
iajs-557	184	8	.	.	PUNCT
iajs-557	185	1	corollary	corollary	NOUN
iajs-557	185	2	:	:	PUNCT
iajs-557	185	3	let	let	VERB
iajs-557	185	4	m	m	PRON
iajs-557	185	5	be	be	AUX
iajs-557	185	6	distributive	distributive	ADJ
iajs-557	185	7	r	r	NOUN
iajs-557	185	8	-	-	PUNCT
iajs-557	185	9	module	module	NOUN
iajs-557	185	10	,	,	PUNCT
iajs-557	185	11	and	and	CCONJ
iajs-557	185	12	<	<	X
iajs-557	185	13	0	0	NUM
iajs-557	185	14	>	>	X
iajs-557	185	15	≠	≠	PROPN
iajs-557	185	16	n	n	PRON
iajs-557	185	17	≨	≨	PROPN
iajs-557	185	18	m.	m.	NOUN
iajs-557	185	19	if	if	SCONJ
iajs-557	185	20	n	n	PRON
iajs-557	185	21	is	be	AUX
iajs-557	185	22	ch	ch	NOUN
iajs-557	185	23	-	-	PUNCT
iajs-557	185	24	submodule	submodule	NOUN
iajs-557	185	25	,	,	PUNCT
iajs-557	185	26	then	then	ADV
iajs-557	185	27	n	n	X
iajs-557	185	28	is	be	AUX
iajs-557	185	29	sh	sh	PROPN
iajs-557	185	30	.	.	PUNCT
iajs-557	185	31	proof	proof	NOUN
iajs-557	185	32	:	:	PUNCT
iajs-557	185	33	it	it	PRON
iajs-557	185	34	follows	follow	VERB
iajs-557	185	35	by	by	ADP
iajs-557	185	36	remark	remark	NOUN
iajs-557	185	37	1.15(1	1.15(1	NUM
iajs-557	185	38	)	)	PUNCT
iajs-557	185	39	and	and	CCONJ
iajs-557	185	40	previous	previous	ADJ
iajs-557	185	41	proposition	proposition	NOUN
iajs-557	185	42	.	.	PUNCT
iajs-557	186	1	remark	remark	NOUN
iajs-557	186	2	:	:	PUNCT
iajs-557	186	3	let	let	VERB
iajs-557	186	4	m	m	PRON
iajs-557	186	5	be	be	AUX
iajs-557	186	6	an	an	DET
iajs-557	186	7	r	r	NOUN
iajs-557	186	8	-	-	PUNCT
iajs-557	186	9	module	module	NOUN
iajs-557	186	10	,	,	PUNCT
iajs-557	186	11	n	n	PRON
iajs-557	187	1	⊆	⊆	NUM
iajs-557	187	2	k	k	PROPN
iajs-557	187	3	⊆	⊆	NUM
iajs-557	187	4	m.	m.	NOUN
iajs-557	187	5	if	if	SCONJ
iajs-557	187	6	n	n	PRON
iajs-557	187	7	is	be	AUX
iajs-557	187	8	sh(qh)-submodule	sh(qh)-submodule	NOUN
iajs-557	187	9	in	in	ADP
iajs-557	187	10	m	m	PROPN
iajs-557	187	11	,	,	PUNCT
iajs-557	187	12	then	then	ADV
iajs-557	187	13	n	n	PROPN
iajs-557	187	14	is	be	AUX
iajs-557	187	15	sh(qh	sh(qh	VERB
iajs-557	187	16	)	)	PUNCT
iajs-557	187	17	in	in	ADP
iajs-557	187	18	k.	k.	PROPN
iajs-557	187	19	proof	proof	PROPN
iajs-557	187	20	:	:	PUNCT
iajs-557	187	21	it	it	PRON
iajs-557	187	22	is	be	AUX
iajs-557	187	23	clear	clear	ADJ
iajs-557	187	24	the	the	DET
iajs-557	187	25	converse	converse	NOUN
iajs-557	187	26	of	of	ADP
iajs-557	187	27	this	this	DET
iajs-557	187	28	remark	remark	NOUN
iajs-557	187	29	is	be	AUX
iajs-557	187	30	true	true	ADJ
iajs-557	187	31	under	under	ADP
iajs-557	187	32	the	the	DET
iajs-557	187	33	class	class	NOUN
iajs-557	187	34	of	of	ADP
iajs-557	187	35	distributive	distributive	ADJ
iajs-557	187	36	module	module	NOUN
iajs-557	187	37	as	as	SCONJ
iajs-557	187	38	follows	follow	VERB
iajs-557	187	39	:	:	PUNCT
iajs-557	187	40	proposition	proposition	NOUN
iajs-557	187	41	:	:	PUNCT
iajs-557	187	42	let	let	VERB
iajs-557	187	43	m	m	PRON
iajs-557	187	44	be	be	AUX
iajs-557	187	45	a	a	DET
iajs-557	187	46	distributive	distributive	ADJ
iajs-557	187	47	r	r	NOUN
iajs-557	187	48	-	-	PUNCT
iajs-557	187	49	module	module	NOUN
iajs-557	187	50	.	.	PUNCT
iajs-557	188	1	let	let	VERB
iajs-557	188	2	n	n	PRON
iajs-557	188	3	⊆	⊆	NUM
iajs-557	188	4	k	k	PROPN
iajs-557	188	5	⊆	⊆	NUM
iajs-557	188	6	m.	m.	NOUN
iajs-557	188	7	then	then	ADV
iajs-557	188	8	n	n	PRON
iajs-557	188	9	is	be	AUX
iajs-557	188	10	sh(qh)-submodule	sh(qh)-submodule	ADJ
iajs-557	188	11	in	in	ADP
iajs-557	188	12	m	m	PROPN
iajs-557	188	13	if	if	SCONJ
iajs-557	189	1	and	and	CCONJ
iajs-557	189	2	only	only	ADV
iajs-557	189	3	if	if	SCONJ
iajs-557	189	4	n	n	PRON
iajs-557	189	5	is	be	AUX
iajs-557	189	6	sh(qh	sh(qh	VERB
iajs-557	189	7	)	)	PUNCT
iajs-557	189	8	in	in	ADP
iajs-557	189	9	k.	k.	PROPN
iajs-557	189	10	proof	proof	PROPN
iajs-557	189	11	:	:	PUNCT
iajs-557	189	12	(	(	PUNCT
iajs-557	189	13	⇒	⇒	NOUN
iajs-557	189	14	)	)	PUNCT
iajs-557	189	15	it	it	PRON
iajs-557	189	16	follows	follow	VERB
iajs-557	189	17	by	by	ADP
iajs-557	189	18	previous	previous	ADJ
iajs-557	189	19	remark	remark	NOUN
iajs-557	189	20	.	.	PUNCT
iajs-557	190	1	(	(	PUNCT
iajs-557	190	2	⇐	⇐	ADJ
iajs-557	190	3	)	)	PUNCT
iajs-557	190	4	assume	assume	VERB
iajs-557	190	5	n	n	PRON
iajs-557	190	6	is	be	AUX
iajs-557	190	7	sh	sh	PROPN
iajs-557	190	8	-	-	PUNCT
iajs-557	190	9	submodule	submodule	NOUN
iajs-557	190	10	in	in	ADP
iajs-557	190	11	k.	k.	PROPN
iajs-557	190	12	let	let	VERB
iajs-557	190	13	n	n	PRON
iajs-557	190	14	⊆	⊆	NUM
iajs-557	190	15	l1	l1	PROPN
iajs-557	190	16	+	+	CCONJ
iajs-557	190	17	l2	l2	PROPN
iajs-557	190	18	where	where	SCONJ
iajs-557	190	19	l1	l1	PROPN
iajs-557	190	20	,	,	PUNCT
iajs-557	190	21	l2	l2	NOUN
iajs-557	190	22	≤	≤	ADJ
iajs-557	190	23	m.	m.	NOUN
iajs-557	190	24	since	since	SCONJ
iajs-557	190	25	n	n	PROPN
iajs-557	190	26	⊆	⊆	NUM
iajs-557	190	27	k	k	NOUN
iajs-557	190	28	then	then	ADV
iajs-557	190	29	n	n	NOUN
iajs-557	190	30	=	=	SYM
iajs-557	190	31	n	n	NOUN
iajs-557	190	32	∩	∩	NOUN
iajs-557	190	33	k	k	PROPN
iajs-557	190	34	⊆	⊆	NUM
iajs-557	190	35	(	(	PUNCT
iajs-557	190	36	l1	l1	PROPN
iajs-557	190	37	+	+	CCONJ
iajs-557	190	38	l2	l2	NOUN
iajs-557	190	39	)	)	PUNCT
iajs-557	190	40	∩	∩	NOUN
iajs-557	190	41	k	k	PROPN
iajs-557	191	1	=	=	PRON
iajs-557	191	2	(	(	PUNCT
iajs-557	191	3	l1	l1	PROPN
iajs-557	191	4	∩	∩	PROPN
iajs-557	191	5	k	k	PROPN
iajs-557	191	6	)	)	PUNCT
iajs-557	191	7	+	+	CCONJ
iajs-557	191	8	(	(	PUNCT
iajs-557	191	9	l2	l2	NOUN
iajs-557	191	10	∩	∩	ADJ
iajs-557	191	11	k	k	NOUN
iajs-557	191	12	)	)	PUNCT
iajs-557	191	13	,	,	PUNCT
iajs-557	191	14	since	since	SCONJ
iajs-557	191	15	m	m	PROPN
iajs-557	191	16	is	be	AUX
iajs-557	191	17	distributive	distributive	ADJ
iajs-557	191	18	so	so	ADV
iajs-557	191	19	n	n	PRON
iajs-557	191	20	⊆	⊆	NUM
iajs-557	191	21	(	(	PUNCT
iajs-557	191	22	l1	l1	PROPN
iajs-557	191	23	∩	∩	PROPN
iajs-557	191	24	k	k	PROPN
iajs-557	191	25	)	)	PUNCT
iajs-557	191	26	+	+	CCONJ
iajs-557	191	27	(	(	PUNCT
iajs-557	191	28	l2	l2	NOUN
iajs-557	191	29	∩	∩	ADJ
iajs-557	191	30	k	k	NOUN
iajs-557	191	31	)	)	PUNCT
iajs-557	191	32	.	.	PUNCT
iajs-557	192	1	then	then	ADV
iajs-557	192	2	n	n	PROPN
iajs-557	192	3	⊆	⊆	NUM
iajs-557	192	4	l1	l1	PROPN
iajs-557	192	5	∩	∩	PROPN
iajs-557	192	6	k	k	PROPN
iajs-557	192	7	or	or	CCONJ
iajs-557	192	8	n	n	CCONJ
iajs-557	192	9	⊆	⊆	NUM
iajs-557	192	10	l2	l2	NOUN
iajs-557	192	11	∩	∩	ADJ
iajs-557	192	12	k	k	NOUN
iajs-557	192	13	,	,	PUNCT
iajs-557	192	14	since	since	SCONJ
iajs-557	192	15	n	n	ADV
iajs-557	192	16	is	be	AUX
iajs-557	192	17	sh	sh	PROPN
iajs-557	192	18	in	in	ADP
iajs-557	192	19	k.	k.	PROPN
iajs-557	193	1	then	then	ADV
iajs-557	193	2	n	n	PROPN
iajs-557	193	3	⊆	⊆	NUM
iajs-557	193	4	l1	l1	PROPN
iajs-557	193	5	or	or	CCONJ
iajs-557	193	6	n	n	CCONJ
iajs-557	193	7	⊆	⊆	NUM
iajs-557	193	8	l2	l2	NOUN
iajs-557	193	9	.	.	PUNCT
iajs-557	194	1	thus	thus	ADV
iajs-557	194	2	n	n	X
iajs-557	194	3	is	be	AUX
iajs-557	194	4	sh	sh	PROPN
iajs-557	194	5	in	in	ADP
iajs-557	194	6	m.	m.	NOUN
iajs-557	194	7	by	by	ADP
iajs-557	194	8	a	a	DET
iajs-557	194	9	similar	similar	ADJ
iajs-557	194	10	proof	proof	NOUN
iajs-557	194	11	,	,	PUNCT
iajs-557	194	12	if	if	SCONJ
iajs-557	194	13	n	n	ADV
iajs-557	194	14	is	be	AUX
iajs-557	194	15	qh	qh	NOUN
iajs-557	194	16	in	in	ADP
iajs-557	194	17	k	k	PROPN
iajs-557	194	18	,	,	PUNCT
iajs-557	194	19	then	then	ADV
iajs-557	194	20	n	n	PROPN
iajs-557	194	21	is	be	AUX
iajs-557	194	22	qh	qh	NOUN
iajs-557	194	23	in	in	ADP
iajs-557	194	24	m.	m.	NOUN
iajs-557	194	25	now	now	ADV
iajs-557	194	26	we	we	PRON
iajs-557	194	27	turn	turn	VERB
iajs-557	194	28	our	our	PRON
iajs-557	194	29	attention	attention	NOUN
iajs-557	194	30	to	to	ADP
iajs-557	194	31	image	image	NOUN
iajs-557	194	32	and	and	CCONJ
iajs-557	194	33	inverse	inverse	NOUN
iajs-557	194	34	image	image	NOUN
iajs-557	194	35	of	of	ADP
iajs-557	194	36	sh	sh	PROPN
iajs-557	194	37	,	,	PUNCT
iajs-557	194	38	qh	qh	NOUN
iajs-557	194	39	and	and	CCONJ
iajs-557	194	40	ch	ch	NOUN
iajs-557	194	41	-	-	PUNCT
iajs-557	194	42	submodules	submodules	NOUN
iajs-557	194	43	.	.	PUNCT
iajs-557	195	1	proposition	proposition	NOUN
iajs-557	195	2	:	:	PUNCT
iajs-557	195	3	let	let	VERB
iajs-557	195	4	m	m	PRON
iajs-557	195	5	and	and	CCONJ
iajs-557	195	6	m	m	VERB
iajs-557	195	7	'	'	PUNCT
iajs-557	195	8	be	be	VERB
iajs-557	195	9	r	r	NOUN
iajs-557	195	10	-	-	PUNCT
iajs-557	195	11	modules	module	NOUN
iajs-557	195	12	and	and	CCONJ
iajs-557	195	13	n	n	CCONJ
iajs-557	195	14	be	be	AUX
iajs-557	195	15	a	a	DET
iajs-557	195	16	sh	sh	NOUN
iajs-557	195	17	-	-	PUNCT
iajs-557	195	18	submodule	submodule	NOUN
iajs-557	195	19	of	of	ADP
iajs-557	195	20	m.	m.	NOUN
iajs-557	195	21	if	if	SCONJ
iajs-557	195	22	f	f	PROPN
iajs-557	195	23	:	:	PUNCT
iajs-557	195	24	m	m	VERB
iajs-557	195	25	→	→	VERB
iajs-557	195	26	m	m	PRON
iajs-557	195	27	'	'	PUNCT
iajs-557	195	28	be	be	VERB
iajs-557	195	29	an	an	DET
iajs-557	195	30	repimorphism	repimorphism	NOUN
iajs-557	195	31	,	,	PUNCT
iajs-557	195	32	then	then	ADV
iajs-557	195	33	f	f	PROPN
iajs-557	195	34	(	(	PUNCT
iajs-557	195	35	n	n	CCONJ
iajs-557	195	36	)	)	PUNCT
iajs-557	195	37	is	be	AUX
iajs-557	195	38	sh	sh	PROPN
iajs-557	195	39	-	-	PUNCT
iajs-557	195	40	submodule	submodule	NOUN
iajs-557	195	41	of	of	ADP
iajs-557	195	42	m	m	NOUN
iajs-557	195	43	'	'	PUNCT
iajs-557	195	44	.	.	PUNCT
iajs-557	196	1	proof	proof	NOUN
iajs-557	196	2	:	:	PUNCT
iajs-557	196	3	let	let	VERB
iajs-557	196	4	f	f	PROPN
iajs-557	196	5	(	(	PUNCT
iajs-557	196	6	n	n	CCONJ
iajs-557	196	7	)	)	PUNCT
iajs-557	196	8	⊆	⊆	NUM
iajs-557	196	9	l1	l1	NOUN
iajs-557	196	10	+	+	CCONJ
iajs-557	196	11	l2	l2	PROPN
iajs-557	196	12	where	where	SCONJ
iajs-557	196	13	l1	l1	PROPN
iajs-557	196	14	,	,	PUNCT
iajs-557	196	15	l2	l2	VERB
iajs-557	196	16	≤	≤	NUM
iajs-557	196	17	m	m	NOUN
iajs-557	196	18	'	'	NUM
iajs-557	196	19	.	.	PUNCT
iajs-557	197	1	then	then	ADV
iajs-557	197	2	n	n	PROPN
iajs-557	197	3	⊆	⊆	NUM
iajs-557	197	4	f	f	PROPN
iajs-557	197	5	1f	1f	X
iajs-557	197	6	(	(	PUNCT
iajs-557	197	7	n	n	CCONJ
iajs-557	197	8	)	)	PUNCT
iajs-557	197	9	⊆	⊆	NUM
iajs-557	197	10	f	f	PROPN
iajs-557	197	11	1(l1	1(l1	NUM
iajs-557	197	12	+	+	NUM
iajs-557	197	13	l2	l2	NOUN
iajs-557	197	14	)	)	PUNCT
iajs-557	197	15	.	.	PUNCT
iajs-557	198	1	but	but	CCONJ
iajs-557	198	2	f	f	PROPN
iajs-557	198	3	1(l1	1(l1	X
iajs-557	199	1	+	+	CCONJ
iajs-557	199	2	l2	l2	NOUN
iajs-557	199	3	)	)	PUNCT
iajs-557	200	1	=	=	PUNCT
iajs-557	201	1	f	f	PROPN
iajs-557	201	2	1(l1	1(l1	NUM
iajs-557	201	3	)	)	PUNCT
iajs-557	202	1	+	+	NUM
iajs-557	202	2	f	f	PROPN
iajs-557	202	3	1(l2	1(l2	NUM
iajs-557	202	4	)	)	PUNCT
iajs-557	202	5	see	see	VERB
iajs-557	202	6	[	[	X
iajs-557	202	7	2,3.1.10(c	2,3.1.10(c	NUM
iajs-557	202	8	)	)	PUNCT
iajs-557	202	9	]	]	PUNCT
iajs-557	202	10	,	,	PUNCT
iajs-557	202	11	so	so	ADV
iajs-557	202	12	n	n	CCONJ
iajs-557	203	1	⊆	⊆	NUM
iajs-557	203	2	f	f	PROPN
iajs-557	203	3	1(l1	1(l1	NUM
iajs-557	203	4	)	)	PUNCT
iajs-557	204	1	+	+	NUM
iajs-557	204	2	f	f	PROPN
iajs-557	204	3	1(l2	1(l2	NUM
iajs-557	204	4	)	)	PUNCT
iajs-557	204	5	,	,	PUNCT
iajs-557	204	6	then	then	ADV
iajs-557	204	7	n	n	PROPN
iajs-557	204	8	⊆	⊆	NUM
iajs-557	204	9	f	f	PROPN
iajs-557	204	10	1(l1	1(l1	NUM
iajs-557	204	11	)	)	PUNCT
iajs-557	204	12	or	or	CCONJ
iajs-557	204	13	n	n	PRON
iajs-557	204	14	⊆	⊆	NUM
iajs-557	204	15	f	f	PROPN
iajs-557	204	16	1(l2	1(l2	NUM
iajs-557	204	17	)	)	PUNCT
iajs-557	204	18	.	.	PUNCT
iajs-557	205	1	it	it	PRON
iajs-557	205	2	follows	follow	VERB
iajs-557	205	3	f	f	PROPN
iajs-557	205	4	(	(	PUNCT
iajs-557	205	5	n	n	CCONJ
iajs-557	205	6	)	)	PUNCT
iajs-557	205	7	⊆	⊆	NUM
iajs-557	205	8	f	f	X
iajs-557	205	9	f	f	PROPN
iajs-557	205	10	–	–	PUNCT
iajs-557	205	11	1(l1	1(l1	NUM
iajs-557	205	12	)	)	PUNCT
iajs-557	205	13	=	=	SYM
iajs-557	205	14	l1	l1	PROPN
iajs-557	205	15	or	or	CCONJ
iajs-557	205	16	f	f	PROPN
iajs-557	205	17	(	(	PUNCT
iajs-557	205	18	n	n	CCONJ
iajs-557	205	19	)	)	PUNCT
iajs-557	206	1	⊆	⊆	NUM
iajs-557	206	2	f	f	PROPN
iajs-557	206	3	f	f	PROPN
iajs-557	206	4	–	–	PUNCT
iajs-557	206	5	1(l2	1(l2	NUM
iajs-557	206	6	)	)	PUNCT
iajs-557	206	7	=	=	SYM
iajs-557	206	8	l2	l2	NOUN
iajs-557	206	9	.	.	PUNCT
iajs-557	207	1	hence	hence	ADV
iajs-557	207	2	f	f	PROPN
iajs-557	207	3	(	(	PUNCT
iajs-557	207	4	n	n	CCONJ
iajs-557	207	5	)	)	PUNCT
iajs-557	207	6	⊆	⊆	NUM
iajs-557	207	7	l1	l1	PROPN
iajs-557	207	8	or	or	CCONJ
iajs-557	207	9	f	f	PROPN
iajs-557	207	10	(	(	PUNCT
iajs-557	207	11	n	n	CCONJ
iajs-557	207	12	)	)	PUNCT
iajs-557	207	13	⊆	⊆	NUM
iajs-557	207	14	l2	l2	NOUN
iajs-557	207	15	.	.	PUNCT
iajs-557	208	1	the	the	DET
iajs-557	208	2	condition	condition	NOUN
iajs-557	208	3	f	f	PROPN
iajs-557	208	4	is	be	AUX
iajs-557	208	5	an	an	DET
iajs-557	208	6	epimorphism	epimorphism	NOUN
iajs-557	208	7	is	be	AUX
iajs-557	208	8	necessary	necessary	ADJ
iajs-557	208	9	in	in	ADP
iajs-557	208	10	proposition	proposition	NOUN
iajs-557	208	11	1.20	1.20	NUM
iajs-557	208	12	,	,	PUNCT
iajs-557	208	13	for	for	ADP
iajs-557	208	14	example	example	NOUN
iajs-557	208	15	,	,	PUNCT
iajs-557	208	16	let	let	VERB
iajs-557	208	17	f	f	PRON
iajs-557	208	18	:	:	PUNCT
iajs-557	208	19	z12	z12	PROPN
iajs-557	208	20	→	→	PROPN
iajs-557	208	21	z12	z12	NUM
iajs-557	208	22	,	,	PUNCT
iajs-557	208	23	f	f	PROPN
iajs-557	208	24	(	(	PUNCT
iajs-557	208	25	x	x	X
iajs-557	208	26	)	)	PUNCT
iajs-557	209	1	=	=	SYM
iajs-557	209	2	4x	4x	NOUN
iajs-557	209	3	fo	fo	INTJ
iajs-557	209	4	each	each	DET
iajs-557	209	5	x	x	PROPN
iajs-557	209	6	∈	∈	PROPN
iajs-557	209	7	z12	z12	PROPN
iajs-557	209	8	,	,	PUNCT
iajs-557	209	9	where	where	SCONJ
iajs-557	209	10	z12	z12	PROPN
iajs-557	209	11	considered	consider	VERB
iajs-557	209	12	as	as	ADP
iajs-557	209	13	z	z	NOUN
iajs-557	209	14	-	-	NOUN
iajs-557	209	15	module	module	NOUN
iajs-557	209	16	.	.	PUNCT
iajs-557	210	1	it	it	PRON
iajs-557	210	2	is	be	AUX
iajs-557	210	3	clear	clear	ADJ
iajs-557	210	4	that	that	SCONJ
iajs-557	210	5	f	f	PROPN
iajs-557	210	6	is	be	AUX
iajs-557	210	7	not	not	PART
iajs-557	210	8	epimorphism	epimorphism	NOUN
iajs-557	210	9	.	.	PUNCT
iajs-557	211	1	let	let	VERB
iajs-557	211	2	n	n	NOUN
iajs-557	211	3	=	=	SYM
iajs-557	211	4	3	3	NUM
iajs-557	211	5	<	<	X
iajs-557	211	6	>	>	X
iajs-557	211	7	,	,	PUNCT
iajs-557	211	8	n	n	PROPN
iajs-557	211	9	is	be	AUX
iajs-557	211	10	a	a	DET
iajs-557	211	11	sh	sh	PROPN
iajs-557	211	12	submodule	submodule	NOUN
iajs-557	211	13	of	of	ADP
iajs-557	211	14	z12	z12	PROPN
iajs-557	211	15	.	.	PUNCT
iajs-557	212	1	but	but	CCONJ
iajs-557	212	2	f	f	PROPN
iajs-557	212	3	(	(	PUNCT
iajs-557	212	4	n	n	CCONJ
iajs-557	212	5	)	)	PUNCT
iajs-557	212	6	=	=	PUNCT
iajs-557	213	1	0	0	X
iajs-557	213	2	<	<	X
iajs-557	213	3	>	>	X
iajs-557	213	4	is	be	AUX
iajs-557	213	5	not	not	PART
iajs-557	213	6	sh	sh	PROPN
iajs-557	213	7	.	.	PROPN
iajs-557	214	1	1.21	1.21	NUM
iajs-557	214	2	corollary	corollary	NOUN
iajs-557	214	3	:	:	PUNCT
iajs-557	214	4	let	let	VERB
iajs-557	214	5	n	n	PRON
iajs-557	214	6	be	be	AUX
iajs-557	214	7	a	a	DET
iajs-557	214	8	sh	sh	NOUN
iajs-557	214	9	-	-	PUNCT
iajs-557	214	10	submodule	submodule	NOUN
iajs-557	214	11	of	of	ADP
iajs-557	214	12	an	an	DET
iajs-557	214	13	r	r	NOUN
iajs-557	214	14	-	-	PUNCT
iajs-557	214	15	module	module	NOUN
iajs-557	214	16	m.	m.	NOUN
iajs-557	214	17	let	let	VERB
iajs-557	214	18	l	l	PROPN
iajs-557	214	19	≨	≨	VERB
iajs-557	214	20	n	n	CCONJ
iajs-557	214	21	,	,	PUNCT
iajs-557	214	22	then	then	ADV
iajs-557	214	23	n	n	CCONJ
iajs-557	214	24	/	/	SYM
iajs-557	214	25	l	l	NOUN
iajs-557	214	26	is	be	AUX
iajs-557	214	27	sh	sh	PROPN
iajs-557	214	28	-	-	PUNCT
iajs-557	214	29	submodule	submodule	NOUN
iajs-557	214	30	of	of	ADP
iajs-557	214	31	m	m	PROPN
iajs-557	214	32	/	/	SYM
iajs-557	214	33	l.	l.	PROPN
iajs-557	214	34	corollary	corollary	PROPN
iajs-557	214	35	:	:	PUNCT
iajs-557	214	36	let	let	VERB
iajs-557	214	37	m	m	PROPN
iajs-557	214	38	≅	≅	PROPN
iajs-557	214	39	m	m	PROPN
iajs-557	214	40	'	'	PUNCT
iajs-557	214	41	be	be	VERB
iajs-557	214	42	r	r	NOUN
iajs-557	214	43	-	-	PUNCT
iajs-557	214	44	module	module	NOUN
iajs-557	214	45	,	,	PUNCT
iajs-557	214	46	if	if	SCONJ
iajs-557	214	47	n	n	PRON
iajs-557	214	48	≤	≤	X
iajs-557	214	49	m.	m.	NOUN
iajs-557	214	50	then	then	ADV
iajs-557	214	51	n	n	PRON
iajs-557	214	52	is	be	AUX
iajs-557	214	53	a	a	DET
iajs-557	214	54	sh	sh	NOUN
iajs-557	214	55	-	-	PUNCT
iajs-557	214	56	submodule	submodule	NOUN
iajs-557	214	57	of	of	ADP
iajs-557	214	58	m	m	PROPN
iajs-557	214	59	iff	iff	PROPN
iajs-557	214	60	f	f	PROPN
iajs-557	214	61	(	(	PUNCT
iajs-557	214	62	n	n	CCONJ
iajs-557	214	63	)	)	PUNCT
iajs-557	214	64	is	be	AUX
iajs-557	214	65	a	a	DET
iajs-557	214	66	shsubmodule	shsubmodule	NOUN
iajs-557	214	67	of	of	ADP
iajs-557	214	68	m	m	NOUN
iajs-557	214	69	'	'	PUNCT
iajs-557	214	70	.	.	PUNCT
iajs-557	215	1	proposition	proposition	NOUN
iajs-557	215	2	:	:	PUNCT
iajs-557	215	3	let	let	VERB
iajs-557	215	4	m	m	PRON
iajs-557	215	5	and	and	CCONJ
iajs-557	215	6	m	m	AUX
iajs-557	215	7	'	'	PUNCT
iajs-557	215	8	be	be	VERB
iajs-557	215	9	r	r	NOUN
iajs-557	215	10	-	-	PUNCT
iajs-557	215	11	modules	module	NOUN
iajs-557	215	12	and	and	CCONJ
iajs-557	215	13	f	f	NOUN
iajs-557	215	14	:	:	PUNCT
iajs-557	215	15	m	m	VERB
iajs-557	215	16	→	→	VERB
iajs-557	215	17	m	m	PRON
iajs-557	215	18	'	'	PUNCT
iajs-557	215	19	be	be	VERB
iajs-557	215	20	an	an	DET
iajs-557	215	21	isomorphism	isomorphism	NOUN
iajs-557	215	22	,	,	PUNCT
iajs-557	215	23	let	let	VERB
iajs-557	215	24	<	<	X
iajs-557	215	25	0	0	NUM
iajs-557	215	26	>	>	X
iajs-557	215	27	≠	≠	PROPN
iajs-557	215	28	n	n	PRON
iajs-557	215	29	≤	≤	NOUN
iajs-557	215	30	m.	m.	NOUN
iajs-557	215	31	if	if	SCONJ
iajs-557	215	32	n	n	PRON
iajs-557	215	33	is	be	AUX
iajs-557	215	34	qh(ch)-sbmodule	qh(ch)-sbmodule	NOUN
iajs-557	215	35	of	of	ADP
iajs-557	215	36	m	m	PROPN
iajs-557	215	37	,	,	PUNCT
iajs-557	215	38	then	then	ADV
iajs-557	215	39	f	f	PROPN
iajs-557	215	40	(	(	PUNCT
iajs-557	215	41	n	n	CCONJ
iajs-557	215	42	)	)	PUNCT
iajs-557	215	43	is	be	AUX
iajs-557	215	44	qh	qh	NOUN
iajs-557	215	45	(	(	PUNCT
iajs-557	215	46	ch)-submodule	ch)-submodule	NOUN
iajs-557	215	47	of	of	ADP
iajs-557	215	48	m	m	NOUN
iajs-557	215	49	'	'	PUNCT
iajs-557	215	50	.	.	PUNCT
iajs-557	216	1	mathematics	mathematic	NOUN
iajs-557	216	2	399	399	NUM
iajs-557	216	3	مجلة	مجلة	NOUN
iajs-557	216	4	إبن	إبن	VERB
iajs-557	216	5	الهيثم	الهيثم	ADJ
iajs-557	216	6	للعلوم	للعلوم	NOUN
iajs-557	216	7	الصرفة	الصرفة	NOUN
iajs-557	216	8	و	و	PRON
iajs-557	216	9	التطبيقية	التطبيقية	ADJ
iajs-557	216	10	2012	2012	NUM
iajs-557	216	11	السنة	السنة	NOUN
iajs-557	217	1	25	25	NUM
iajs-557	217	2	المجلد	المجلد	NOUN
iajs-557	217	3	3	3	NUM
iajs-557	217	4	العدد	العدد	PROPN
iajs-557	217	5	ibn	ibn	PROPN
iajs-557	217	6	al	al	PROPN
iajs-557	217	7	-	-	PUNCT
iajs-557	217	8	haitham	haitham	PROPN
iajs-557	217	9	journal	journal	PROPN
iajs-557	217	10	for	for	ADP
iajs-557	217	11	pure	pure	ADJ
iajs-557	217	12	and	and	CCONJ
iajs-557	217	13	applied	apply	VERB
iajs-557	217	14	science	science	NOUN
iajs-557	217	15	no	no	NOUN
iajs-557	217	16	.	.	NOUN
iajs-557	217	17	3	3	NUM
iajs-557	217	18	vol	vol	NOUN
iajs-557	217	19	.	.	PUNCT
iajs-557	217	20	25	25	NUM
iajs-557	217	21	year	year	NOUN
iajs-557	217	22	2012	2012	NUM
iajs-557	217	23	proof	proof	NOUN
iajs-557	217	24	:	:	PUNCT
iajs-557	217	25	if	if	SCONJ
iajs-557	217	26	n	n	PRON
iajs-557	217	27	is	be	AUX
iajs-557	217	28	qh	qh	NOUN
iajs-557	217	29	-	-	NOUN
iajs-557	217	30	submodule	submodule	NOUN
iajs-557	217	31	of	of	ADP
iajs-557	217	32	m.	m.	NOUN
iajs-557	217	33	assume	assume	VERB
iajs-557	217	34	f	f	PROPN
iajs-557	217	35	(	(	PUNCT
iajs-557	217	36	n	n	CCONJ
iajs-557	217	37	)	)	PUNCT
iajs-557	218	1	=	=	SYM
iajs-557	218	2	w1	w1	NOUN
iajs-557	218	3	+	+	NOUN
iajs-557	218	4	w2	w2	NOUN
iajs-557	218	5	for	for	ADP
iajs-557	218	6	some	some	DET
iajs-557	218	7	w1	w1	NOUN
iajs-557	218	8	,	,	PUNCT
iajs-557	218	9	w2	w2	NOUN
iajs-557	218	10	≤	≤	PUNCT
iajs-557	218	11	m	m	PROPN
iajs-557	218	12	'	'	PUNCT
iajs-557	218	13	.	.	PUNCT
iajs-557	219	1	since	since	SCONJ
iajs-557	219	2	f	f	PROPN
iajs-557	219	3	is	be	AUX
iajs-557	219	4	isomorphism	isomorphism	NOUN
iajs-557	219	5	,	,	PUNCT
iajs-557	219	6	so	so	ADV
iajs-557	219	7	w1	w1	NOUN
iajs-557	219	8	=	=	SYM
iajs-557	219	9	f	f	PROPN
iajs-557	219	10	(	(	PUNCT
iajs-557	219	11	l1	l1	PROPN
iajs-557	219	12	)	)	PUNCT
iajs-557	219	13	,	,	PUNCT
iajs-557	219	14	w2	w2	NOUN
iajs-557	219	15	=	=	SYM
iajs-557	219	16	f	f	PROPN
iajs-557	219	17	(	(	PUNCT
iajs-557	219	18	l2	l2	PROPN
iajs-557	219	19	)	)	PUNCT
iajs-557	219	20	for	for	ADP
iajs-557	219	21	some	some	DET
iajs-557	219	22	l1	l1	NOUN
iajs-557	219	23	,	,	PUNCT
iajs-557	219	24	l2	l2	NOUN
iajs-557	219	25	≤	≤	ADJ
iajs-557	219	26	m.	m.	NOUN
iajs-557	219	27	thus	thus	ADV
iajs-557	219	28	f	f	PROPN
iajs-557	219	29	(	(	PUNCT
iajs-557	219	30	n	n	CCONJ
iajs-557	219	31	)	)	PUNCT
iajs-557	220	1	=	=	SYM
iajs-557	220	2	f	f	PROPN
iajs-557	220	3	(	(	PUNCT
iajs-557	220	4	l1	l1	PROPN
iajs-557	220	5	)	)	PUNCT
iajs-557	221	1	+	+	NUM
iajs-557	221	2	f	f	X
iajs-557	221	3	(	(	PUNCT
iajs-557	221	4	l2	l2	NOUN
iajs-557	221	5	)	)	PUNCT
iajs-557	221	6	.	.	PUNCT
iajs-557	222	1	but	but	CCONJ
iajs-557	222	2	f	f	PROPN
iajs-557	222	3	(	(	PUNCT
iajs-557	222	4	l1	l1	PROPN
iajs-557	222	5	+	+	CCONJ
iajs-557	222	6	l2	l2	NOUN
iajs-557	222	7	)	)	PUNCT
iajs-557	223	1	=	=	SYM
iajs-557	223	2	f	f	PROPN
iajs-557	223	3	(	(	PUNCT
iajs-557	223	4	l1	l1	PROPN
iajs-557	223	5	)	)	PUNCT
iajs-557	224	1	+	+	NUM
iajs-557	224	2	f	f	X
iajs-557	224	3	(	(	PUNCT
iajs-557	224	4	l2	l2	PROPN
iajs-557	224	5	)	)	PUNCT
iajs-557	224	6	,	,	PUNCT
iajs-557	224	7	see	see	VERB
iajs-557	224	8	[	[	X
iajs-557	224	9	2	2	NUM
iajs-557	224	10	,	,	PUNCT
iajs-557	224	11	3.1.10(a	3.1.10(a	NUM
iajs-557	224	12	)	)	PUNCT
iajs-557	224	13	]	]	PUNCT
iajs-557	224	14	.	.	PUNCT
iajs-557	225	1	then	then	ADV
iajs-557	225	2	f	f	PROPN
iajs-557	225	3	(	(	PUNCT
iajs-557	225	4	n	n	CCONJ
iajs-557	225	5	)	)	PUNCT
iajs-557	226	1	=	=	SYM
iajs-557	226	2	f	f	PROPN
iajs-557	226	3	(	(	PUNCT
iajs-557	226	4	l1	l1	PROPN
iajs-557	226	5	+	+	CCONJ
iajs-557	226	6	l2	l2	NOUN
iajs-557	226	7	)	)	PUNCT
iajs-557	226	8	.	.	PUNCT
iajs-557	227	1	since	since	SCONJ
iajs-557	227	2	f	f	PROPN
iajs-557	227	3	is	be	AUX
iajs-557	227	4	monomorphism	monomorphism	NOUN
iajs-557	227	5	,	,	PUNCT
iajs-557	227	6	we	we	PRON
iajs-557	227	7	get	get	VERB
iajs-557	227	8	n	n	PRON
iajs-557	227	9	=	=	PROPN
iajs-557	227	10	l1	l1	PROPN
iajs-557	227	11	+	+	CCONJ
iajs-557	227	12	l2	l2	NOUN
iajs-557	227	13	.	.	PUNCT
iajs-557	228	1	it	it	PRON
iajs-557	228	2	follows	follow	VERB
iajs-557	228	3	that	that	SCONJ
iajs-557	228	4	n	n	PROPN
iajs-557	228	5	=	=	PROPN
iajs-557	228	6	l1	l1	PROPN
iajs-557	228	7	or	or	CCONJ
iajs-557	228	8	n	n	CCONJ
iajs-557	228	9	=	=	NOUN
iajs-557	228	10	l2	l2	NOUN
iajs-557	228	11	.	.	PUNCT
iajs-557	229	1	hence	hence	ADV
iajs-557	229	2	f	f	PROPN
iajs-557	229	3	(	(	PUNCT
iajs-557	229	4	n	n	CCONJ
iajs-557	229	5	)	)	PUNCT
iajs-557	229	6	=	=	SYM
iajs-557	229	7	f	f	PROPN
iajs-557	229	8	(	(	PUNCT
iajs-557	229	9	l1	l1	PROPN
iajs-557	229	10	)	)	PUNCT
iajs-557	230	1	=	=	SYM
iajs-557	230	2	w1	w1	NOUN
iajs-557	230	3	or	or	CCONJ
iajs-557	230	4	f	f	PROPN
iajs-557	230	5	(	(	PUNCT
iajs-557	230	6	n	n	CCONJ
iajs-557	230	7	)	)	PUNCT
iajs-557	231	1	=	=	SYM
iajs-557	231	2	f	f	PROPN
iajs-557	231	3	(	(	PUNCT
iajs-557	231	4	l2	l2	NOUN
iajs-557	231	5	)	)	PUNCT
iajs-557	231	6	=	=	SYM
iajs-557	231	7	w2	w2	NOUN
iajs-557	231	8	.	.	PUNCT
iajs-557	232	1	by	by	ADP
iajs-557	232	2	a	a	DET
iajs-557	232	3	similar	similar	ADJ
iajs-557	232	4	proof	proof	NOUN
iajs-557	232	5	,	,	PUNCT
iajs-557	232	6	n	n	PRON
iajs-557	232	7	is	be	AUX
iajs-557	232	8	ch	ch	NOUN
iajs-557	232	9	-	-	PUNCT
iajs-557	232	10	submodule	submodule	NOUN
iajs-557	232	11	of	of	ADP
iajs-557	232	12	m	m	PROPN
iajs-557	232	13	implies	imply	VERB
iajs-557	232	14	f	f	PROPN
iajs-557	232	15	(	(	PUNCT
iajs-557	232	16	n	n	CCONJ
iajs-557	232	17	)	)	PUNCT
iajs-557	232	18	is	be	AUX
iajs-557	232	19	ch	ch	NOUN
iajs-557	232	20	of	of	ADP
iajs-557	232	21	m	m	NOUN
iajs-557	232	22	'	'	PUNCT
iajs-557	232	23	.	.	PUNCT
iajs-557	233	1	proposition	proposition	NOUN
iajs-557	233	2	:	:	PUNCT
iajs-557	233	3	let	let	VERB
iajs-557	233	4	f	f	PRON
iajs-557	233	5	:	:	PUNCT
iajs-557	233	6	m	m	VERB
iajs-557	233	7	→	→	VERB
iajs-557	233	8	m	m	PRON
iajs-557	233	9	'	'	PUNCT
iajs-557	233	10	be	be	VERB
iajs-557	233	11	an	an	DET
iajs-557	233	12	isomorphism	isomorphism	NOUN
iajs-557	233	13	r	r	NOUN
iajs-557	233	14	-	-	PUNCT
iajs-557	233	15	module	module	NOUN
iajs-557	233	16	.	.	PUNCT
iajs-557	234	1	if	if	SCONJ
iajs-557	234	2	k	k	PROPN
iajs-557	234	3	is	be	AUX
iajs-557	234	4	sh(qh	sh(qh	ADJ
iajs-557	234	5	or	or	CCONJ
iajs-557	234	6	ch)-submodule	ch)-submodule	NOUN
iajs-557	234	7	of	of	ADP
iajs-557	234	8	m	m	NOUN
iajs-557	234	9	'	'	PUNCT
iajs-557	234	10	,	,	PUNCT
iajs-557	234	11	then	then	ADV
iajs-557	234	12	f	f	PROPN
iajs-557	234	13	-1(k	-1(k	PROPN
iajs-557	234	14	)	)	PUNCT
iajs-557	234	15	is	be	AUX
iajs-557	234	16	sh(qh	sh(qh	VERB
iajs-557	234	17	or	or	CCONJ
iajs-557	234	18	ch)-submodule	ch)-submodule	NOUN
iajs-557	234	19	of	of	ADP
iajs-557	234	20	m.	m.	NOUN
iajs-557	234	21	proof	proof	NOUN
iajs-557	234	22	:	:	PUNCT
iajs-557	234	23	assume	assume	VERB
iajs-557	234	24	k	k	PROPN
iajs-557	234	25	is	be	AUX
iajs-557	234	26	sh	sh	PROPN
iajs-557	234	27	in	in	ADP
iajs-557	234	28	m	m	NOUN
iajs-557	234	29	'	'	PUNCT
iajs-557	234	30	.	.	PUNCT
iajs-557	235	1	let	let	VERB
iajs-557	235	2	f	f	PROPN
iajs-557	235	3	-1(k	-1(k	PROPN
iajs-557	235	4	)	)	PUNCT
iajs-557	235	5	⊆	⊆	NUM
iajs-557	235	6	l1	l1	NOUN
iajs-557	235	7	+	+	CCONJ
iajs-557	235	8	l2	l2	PROPN
iajs-557	235	9	where	where	SCONJ
iajs-557	235	10	l1	l1	PROPN
iajs-557	235	11	,	,	PUNCT
iajs-557	235	12	l2	l2	NOUN
iajs-557	235	13	≤	≤	ADJ
iajs-557	235	14	m.	m.	NOUN
iajs-557	235	15	then	then	ADV
iajs-557	235	16	f	f	PROPN
iajs-557	235	17	f	f	PROPN
iajs-557	235	18	-1(k	-1(k	PROPN
iajs-557	235	19	)	)	PUNCT
iajs-557	235	20	⊆	⊆	NUM
iajs-557	235	21	f	f	X
iajs-557	235	22	(	(	PUNCT
iajs-557	235	23	l1	l1	PROPN
iajs-557	235	24	+	+	CCONJ
iajs-557	235	25	l2	l2	NOUN
iajs-557	235	26	)	)	PUNCT
iajs-557	235	27	=	=	SYM
iajs-557	236	1	f	f	PROPN
iajs-557	236	2	(	(	PUNCT
iajs-557	236	3	l1	l1	PROPN
iajs-557	236	4	)	)	PUNCT
iajs-557	237	1	+	+	NUM
iajs-557	237	2	f	f	X
iajs-557	237	3	(	(	PUNCT
iajs-557	237	4	l2	l2	PROPN
iajs-557	237	5	)	)	PUNCT
iajs-557	237	6	,	,	PUNCT
iajs-557	237	7	see	see	VERB
iajs-557	237	8	[	[	X
iajs-557	237	9	2,3.1.10(a	2,3.1.10(a	NUM
iajs-557	237	10	)	)	PUNCT
iajs-557	237	11	]	]	PUNCT
iajs-557	237	12	.	.	PUNCT
iajs-557	238	1	since	since	SCONJ
iajs-557	238	2	f	f	PROPN
iajs-557	238	3	is	be	AUX
iajs-557	238	4	epimorphism	epimorphism	NOUN
iajs-557	238	5	k	k	PROPN
iajs-557	238	6	=	=	PUNCT
iajs-557	238	7	f	f	PROPN
iajs-557	238	8	f	f	PROPN
iajs-557	238	9	-1(k	-1(k	PROPN
iajs-557	238	10	)	)	PUNCT
iajs-557	238	11	⊆	⊆	NUM
iajs-557	238	12	f	f	X
iajs-557	238	13	(	(	PUNCT
iajs-557	238	14	l1	l1	PROPN
iajs-557	238	15	)	)	PUNCT
iajs-557	239	1	+	+	NUM
iajs-557	239	2	f	f	X
iajs-557	239	3	(	(	PUNCT
iajs-557	239	4	l2	l2	NOUN
iajs-557	239	5	)	)	PUNCT
iajs-557	239	6	.	.	PUNCT
iajs-557	240	1	so	so	ADV
iajs-557	240	2	k	k	PROPN
iajs-557	240	3	⊆	⊆	NUM
iajs-557	240	4	f	f	X
iajs-557	240	5	(	(	PUNCT
iajs-557	240	6	l1	l1	PROPN
iajs-557	240	7	)	)	PUNCT
iajs-557	240	8	or	or	CCONJ
iajs-557	240	9	k	k	X
iajs-557	240	10	⊆	⊆	NUM
iajs-557	240	11	f	f	PROPN
iajs-557	240	12	(	(	PUNCT
iajs-557	240	13	l2	l2	PROPN
iajs-557	240	14	)	)	PUNCT
iajs-557	240	15	.	.	PUNCT
iajs-557	241	1	thus	thus	ADV
iajs-557	241	2	f	f	PROPN
iajs-557	241	3	-1(k	-1(k	PROPN
iajs-557	241	4	)	)	PUNCT
iajs-557	241	5	⊆	⊆	NUM
iajs-557	241	6	f	f	PROPN
iajs-557	241	7	f	f	PROPN
iajs-557	241	8	-1(l1	-1(l1	PUNCT
iajs-557	241	9	)	)	PUNCT
iajs-557	241	10	=	=	SYM
iajs-557	241	11	l1	l1	PROPN
iajs-557	241	12	or	or	CCONJ
iajs-557	241	13	f	f	PROPN
iajs-557	241	14	-1(k	-1(k	PROPN
iajs-557	241	15	)	)	PUNCT
iajs-557	241	16	⊆	⊆	NUM
iajs-557	241	17	f	f	PROPN
iajs-557	241	18	f	f	PROPN
iajs-557	241	19	-1(l2	-1(l2	PUNCT
iajs-557	241	20	)	)	PUNCT
iajs-557	241	21	=	=	SYM
iajs-557	241	22	l2	l2	NOUN
iajs-557	241	23	.	.	PUNCT
iajs-557	242	1	since	since	SCONJ
iajs-557	242	2	f	f	PROPN
iajs-557	242	3	is	be	AUX
iajs-557	242	4	monomorphism	monomorphism	NOUN
iajs-557	242	5	.	.	PUNCT
iajs-557	243	1	hence	hence	ADV
iajs-557	243	2	f	f	PROPN
iajs-557	243	3	-1(k	-1(k	PROPN
iajs-557	243	4	)	)	PUNCT
iajs-557	243	5	⊆	⊆	NUM
iajs-557	243	6	l1	l1	PROPN
iajs-557	243	7	or	or	CCONJ
iajs-557	243	8	f	f	PROPN
iajs-557	243	9	-1(k	-1(k	PROPN
iajs-557	243	10	)	)	PUNCT
iajs-557	243	11	⊆	⊆	NUM
iajs-557	243	12	l2	l2	NOUN
iajs-557	243	13	.	.	PUNCT
iajs-557	244	1	by	by	ADP
iajs-557	244	2	a	a	DET
iajs-557	244	3	similar	similar	ADJ
iajs-557	244	4	proof	proof	NOUN
iajs-557	244	5	,	,	PUNCT
iajs-557	244	6	k	k	PROPN
iajs-557	244	7	is	be	AUX
iajs-557	244	8	qh(ch	qh(ch	PROPN
iajs-557	244	9	)	)	PUNCT
iajs-557	244	10	of	of	ADP
iajs-557	244	11	m	m	PROPN
iajs-557	244	12	'	'	PUNCT
iajs-557	244	13	,	,	PUNCT
iajs-557	244	14	then	then	ADV
iajs-557	244	15	f	f	PROPN
iajs-557	244	16	-1(k	-1(k	PROPN
iajs-557	244	17	)	)	PUNCT
iajs-557	244	18	is	be	AUX
iajs-557	244	19	qh(ch	qh(ch	PROPN
iajs-557	244	20	)	)	PUNCT
iajs-557	244	21	.	.	PUNCT
iajs-557	245	1	the	the	DET
iajs-557	245	2	condition	condition	NOUN
iajs-557	245	3	that	that	SCONJ
iajs-557	245	4	f	f	PROPN
iajs-557	245	5	is	be	AUX
iajs-557	245	6	an	an	DET
iajs-557	245	7	isomorphism	isomorphism	NOUN
iajs-557	245	8	is	be	AUX
iajs-557	245	9	necessary	necessary	ADJ
iajs-557	245	10	in	in	ADP
iajs-557	245	11	proposition	proposition	NOUN
iajs-557	245	12	1.24	1.24	NUM
iajs-557	245	13	.	.	PUNCT
iajs-557	246	1	for	for	ADP
iajs-557	246	2	example	example	NOUN
iajs-557	246	3	,	,	PUNCT
iajs-557	246	4	consider	consider	VERB
iajs-557	246	5	the	the	DET
iajs-557	246	6	z	z	NOUN
iajs-557	246	7	-	-	PUNCT
iajs-557	246	8	module	module	NOUN
iajs-557	246	9	z	z	NOUN
iajs-557	246	10	and	and	CCONJ
iajs-557	246	11	let	let	VERB
iajs-557	246	12	π	π	NOUN
iajs-557	246	13	:	:	PUNCT
iajs-557	246	14	z	z	X
iajs-557	246	15	→	→	VERB
iajs-557	246	16	z/<4	z/<4	NOUN
iajs-557	246	17	>	>	X
iajs-557	246	18	≃	≃	NOUN
iajs-557	246	19	z4	z4	PROPN
iajs-557	246	20	be	be	AUX
iajs-557	246	21	the	the	DET
iajs-557	246	22	natural	natural	ADJ
iajs-557	246	23	projection	projection	NOUN
iajs-557	246	24	.	.	PUNCT
iajs-557	247	1	let	let	VERB
iajs-557	248	1	k	k	NOUN
iajs-557	248	2	=	=	PUNCT
iajs-557	248	3	2	2	NUM
iajs-557	248	4	<	<	X
iajs-557	248	5	>	>	X
iajs-557	248	6	⊆	⊆	NUM
iajs-557	248	7	z4	z4	X
iajs-557	248	8	,	,	PUNCT
iajs-557	248	9	k	k	X
iajs-557	248	10	is	be	AUX
iajs-557	248	11	sh(qh	sh(qh	PROPN
iajs-557	248	12	or	or	CCONJ
iajs-557	248	13	ch	ch	NOUN
iajs-557	248	14	)	)	PUNCT
iajs-557	248	15	of	of	ADP
iajs-557	248	16	z4	z4	PROPN
iajs-557	248	17	.	.	PUNCT
iajs-557	249	1	but	but	CCONJ
iajs-557	249	2	π	π	NOUN
iajs-557	249	3	1(k	1(k	NUM
iajs-557	249	4	)	)	PUNCT
iajs-557	249	5	=	=	NOUN
iajs-557	249	6	2z	2z	NUM
iajs-557	249	7	is	be	AUX
iajs-557	249	8	not	not	PART
iajs-557	249	9	sh	sh	INTJ
iajs-557	249	10	(	(	PUNCT
iajs-557	249	11	not	not	PART
iajs-557	249	12	qh	qh	NOUN
iajs-557	249	13	,	,	PUNCT
iajs-557	249	14	not	not	PART
iajs-557	249	15	ch	ch	NOUN
iajs-557	249	16	)	)	PUNCT
iajs-557	249	17	in	in	ADP
iajs-557	249	18	z.	z.	PROPN
iajs-557	250	1	now	now	ADV
iajs-557	250	2	we	we	PRON
iajs-557	250	3	give	give	VERB
iajs-557	250	4	the	the	DET
iajs-557	250	5	next	next	ADJ
iajs-557	250	6	result	result	NOUN
iajs-557	250	7	of	of	ADP
iajs-557	250	8	this	this	DET
iajs-557	250	9	section	section	NOUN
iajs-557	250	10	.	.	PUNCT
iajs-557	251	1	proposition	proposition	NOUN
iajs-557	251	2	:	:	PUNCT
iajs-557	251	3	let	let	VERB
iajs-557	251	4	m1	m1	PROPN
iajs-557	251	5	,	,	PUNCT
iajs-557	251	6	m2	m2	PROPN
iajs-557	251	7	be	be	VERB
iajs-557	251	8	r	r	NOUN
iajs-557	251	9	-	-	PUNCT
iajs-557	251	10	modules	module	NOUN
iajs-557	251	11	.	.	PUNCT
iajs-557	252	1	let	let	VERB
iajs-557	252	2	m	m	NOUN
iajs-557	252	3	=	=	VERB
iajs-557	252	4	m1⊕m2	m1⊕m2	PROPN
iajs-557	252	5	,	,	PUNCT
iajs-557	252	6	and	and	CCONJ
iajs-557	252	7	let	let	VERB
iajs-557	252	8	<	<	X
iajs-557	252	9	0	0	NUM
iajs-557	252	10	>	>	X
iajs-557	252	11	≠	≠	PROPN
iajs-557	252	12	k	k	PROPN
iajs-557	252	13	⊆	⊆	NUM
iajs-557	252	14	m1⊕m2	m1⊕m2	NOUN
iajs-557	252	15	.	.	PUNCT
iajs-557	253	1	if	if	SCONJ
iajs-557	253	2	k	k	PROPN
iajs-557	253	3	=	=	SYM
iajs-557	253	4	n	n	PROPN
iajs-557	253	5	⊕	⊕	PROPN
iajs-557	253	6	w	w	PROPN
iajs-557	253	7	for	for	ADP
iajs-557	253	8	some	some	DET
iajs-557	253	9	n	n	PRON
iajs-557	253	10	≤	≤	NOUN
iajs-557	253	11	m1	m1	NOUN
iajs-557	253	12	,	,	PUNCT
iajs-557	253	13	w	w	PROPN
iajs-557	253	14	≤	≤	NUM
iajs-557	253	15	m2	m2	NOUN
iajs-557	253	16	such	such	ADJ
iajs-557	253	17	that	that	SCONJ
iajs-557	253	18	k	k	PROPN
iajs-557	253	19	is	be	AUX
iajs-557	253	20	sh(qh)-submodule	sh(qh)-submodule	NOUN
iajs-557	253	21	.	.	PUNCT
iajs-557	254	1	then	then	ADV
iajs-557	254	2	n	n	PROPN
iajs-557	254	3	is	be	AUX
iajs-557	254	4	sh(qh	sh(qh	PROPN
iajs-557	254	5	)	)	PUNCT
iajs-557	254	6	of	of	ADP
iajs-557	254	7	m1	m1	PROPN
iajs-557	254	8	and	and	CCONJ
iajs-557	254	9	w	w	PROPN
iajs-557	254	10	is	be	AUX
iajs-557	254	11	sh(qh	sh(qh	PROPN
iajs-557	254	12	)	)	PUNCT
iajs-557	254	13	of	of	ADP
iajs-557	254	14	m2	m2	PROPN
iajs-557	254	15	.	.	PUNCT
iajs-557	255	1	proof	proof	NOUN
iajs-557	255	2	:	:	PUNCT
iajs-557	255	3	assume	assume	VERB
iajs-557	255	4	k	k	X
iajs-557	255	5	=	=	PUNCT
iajs-557	255	6	n	n	PROPN
iajs-557	255	7	⊕	⊕	PROPN
iajs-557	255	8	w	w	PROPN
iajs-557	255	9	,	,	PUNCT
iajs-557	255	10	k	k	PROPN
iajs-557	255	11	is	be	AUX
iajs-557	255	12	a	a	DET
iajs-557	255	13	sh	sh	PROPN
iajs-557	255	14	submodule	submodule	NOUN
iajs-557	255	15	in	in	ADP
iajs-557	255	16	m	m	PROPN
iajs-557	255	17	,	,	PUNCT
iajs-557	255	18	k	k	X
iajs-557	255	19	=	=	PUNCT
iajs-557	255	20	(	(	PUNCT
iajs-557	255	21	n	n	PROPN
iajs-557	255	22	⊕	⊕	PROPN
iajs-557	255	23	<	<	X
iajs-557	255	24	0	0	NUM
iajs-557	255	25	>	>	X
iajs-557	255	26	)	)	PUNCT
iajs-557	256	1	+	+	CCONJ
iajs-557	256	2	(	(	PUNCT
iajs-557	256	3	<	<	X
iajs-557	256	4	0	0	NUM
iajs-557	256	5	>	>	X
iajs-557	256	6	⊕	⊕	PROPN
iajs-557	256	7	w	w	PROPN
iajs-557	256	8	)	)	PUNCT
iajs-557	256	9	,	,	PUNCT
iajs-557	256	10	so	so	ADV
iajs-557	256	11	k	k	PROPN
iajs-557	256	12	=	=	PUNCT
iajs-557	256	13	n	n	PROPN
iajs-557	256	14	⊕	⊕	PROPN
iajs-557	256	15	<	<	X
iajs-557	256	16	0	0	NUM
iajs-557	256	17	>	>	X
iajs-557	256	18	or	or	CCONJ
iajs-557	256	19	k	k	NOUN
iajs-557	257	1	=	=	PUNCT
iajs-557	257	2	<	<	X
iajs-557	257	3	0	0	NUM
iajs-557	257	4	>	>	X
iajs-557	257	5	⊕	⊕	PROPN
iajs-557	257	6	w	w	PROPN
iajs-557	257	7	since	since	SCONJ
iajs-557	257	8	k	k	PROPN
iajs-557	257	9	is	be	AUX
iajs-557	257	10	sh	sh	INTJ
iajs-557	257	11	of	of	ADP
iajs-557	257	12	m.	m.	NOUN
iajs-557	257	13	if	if	SCONJ
iajs-557	257	14	k	k	PROPN
iajs-557	257	15	=	=	SYM
iajs-557	257	16	n	n	PROPN
iajs-557	257	17	⊕	⊕	PROPN
iajs-557	257	18	<	<	X
iajs-557	257	19	0	0	PROPN
iajs-557	257	20	>	>	X
iajs-557	257	21	.	.	PUNCT
iajs-557	258	1	we	we	PRON
iajs-557	258	2	claim	claim	VERB
iajs-557	258	3	that	that	SCONJ
iajs-557	258	4	n	n	PRON
iajs-557	258	5	is	be	AUX
iajs-557	258	6	sh	sh	PROPN
iajs-557	258	7	of	of	ADP
iajs-557	258	8	m1	m1	PROPN
iajs-557	258	9	.	.	PUNCT
iajs-557	259	1	assume	assume	VERB
iajs-557	259	2	n	n	PRON
iajs-557	259	3	⊆	⊆	NUM
iajs-557	259	4	l1	l1	PROPN
iajs-557	259	5	+	+	CCONJ
iajs-557	259	6	l2	l2	PROPN
iajs-557	259	7	where	where	SCONJ
iajs-557	259	8	l1	l1	PROPN
iajs-557	259	9	,	,	PUNCT
iajs-557	259	10	l2	l2	VERB
iajs-557	259	11	≤	≤	ADJ
iajs-557	259	12	m1	m1	NOUN
iajs-557	259	13	.	.	PUNCT
iajs-557	260	1	so	so	ADV
iajs-557	260	2	k	k	PROPN
iajs-557	260	3	⊆	⊆	NUM
iajs-557	260	4	(	(	PUNCT
iajs-557	260	5	l1	l1	PROPN
iajs-557	260	6	+	+	CCONJ
iajs-557	260	7	l2	l2	NOUN
iajs-557	260	8	)	)	PUNCT
iajs-557	260	9	⊕	⊕	PROPN
iajs-557	260	10	<	<	X
iajs-557	260	11	0	0	PROPN
iajs-557	260	12	>	>	X
iajs-557	260	13	.	.	PUNCT
iajs-557	261	1	then	then	ADV
iajs-557	261	2	k	k	PROPN
iajs-557	261	3	⊆	⊆	X
iajs-557	261	4	(	(	PUNCT
iajs-557	261	5	l1⊕	l1⊕	NOUN
iajs-557	261	6	<	<	X
iajs-557	261	7	0	0	NUM
iajs-557	261	8	>	>	X
iajs-557	261	9	)	)	PUNCT
iajs-557	262	1	+	+	CCONJ
iajs-557	262	2	(	(	PUNCT
iajs-557	262	3	l2⊕	l2⊕	VERB
iajs-557	262	4	<	<	X
iajs-557	262	5	0	0	NUM
iajs-557	262	6	>	>	NUM
iajs-557	262	7	)	)	PUNCT
iajs-557	262	8	.	.	PUNCT
iajs-557	263	1	then	then	ADV
iajs-557	263	2	k	k	PROPN
iajs-557	263	3	⊆	⊆	NUM
iajs-557	263	4	l1⊕	l1⊕	PROPN
iajs-557	263	5	<	<	X
iajs-557	263	6	0	0	NUM
iajs-557	263	7	>	>	X
iajs-557	263	8	or	or	CCONJ
iajs-557	263	9	k	k	PROPN
iajs-557	263	10	⊆	⊆	NUM
iajs-557	263	11	l2⊕	l2⊕	NOUN
iajs-557	263	12	<	<	X
iajs-557	263	13	0	0	NUM
iajs-557	263	14	>	>	X
iajs-557	263	15	.	.	PUNCT
iajs-557	264	1	thus	thus	ADV
iajs-557	264	2	n	n	PROPN
iajs-557	264	3	⊕	⊕	PROPN
iajs-557	264	4	<	<	X
iajs-557	264	5	0	0	NUM
iajs-557	264	6	>	>	X
iajs-557	264	7	⊆	⊆	NUM
iajs-557	264	8	l1⊕	l1⊕	NOUN
iajs-557	264	9	<	<	X
iajs-557	264	10	0	0	NUM
iajs-557	264	11	>	>	X
iajs-557	264	12	or	or	CCONJ
iajs-557	264	13	n	n	PRON
iajs-557	264	14	⊕	⊕	PROPN
iajs-557	264	15	<	<	X
iajs-557	264	16	0	0	NUM
iajs-557	264	17	>	>	X
iajs-557	264	18	⊆	⊆	NUM
iajs-557	264	19	l2⊕	l2⊕	NOUN
iajs-557	264	20	<	<	X
iajs-557	264	21	0	0	NUM
iajs-557	264	22	>	>	X
iajs-557	264	23	.	.	PUNCT
iajs-557	265	1	hence	hence	ADV
iajs-557	265	2	n	n	NOUN
iajs-557	265	3	≤	≤	NOUN
iajs-557	265	4	l1	l1	PROPN
iajs-557	265	5	or	or	CCONJ
iajs-557	265	6	n	n	PRON
iajs-557	265	7	≤	≤	NOUN
iajs-557	265	8	l2	l2	NOUN
iajs-557	265	9	.	.	PUNCT
iajs-557	266	1	then	then	ADV
iajs-557	266	2	n	n	X
iajs-557	266	3	is	be	AUX
iajs-557	266	4	sh	sh	PROPN
iajs-557	266	5	of	of	ADP
iajs-557	266	6	m1	m1	PROPN
iajs-557	266	7	.	.	PUNCT
iajs-557	267	1	similarly	similarly	ADV
iajs-557	267	2	,	,	PUNCT
iajs-557	267	3	if	if	SCONJ
iajs-557	267	4	k	k	PROPN
iajs-557	267	5	=	=	PUNCT
iajs-557	268	1	<	<	X
iajs-557	268	2	0	0	NUM
iajs-557	268	3	>	>	X
iajs-557	268	4	⊕	⊕	PROPN
iajs-557	268	5	w	w	PROPN
iajs-557	268	6	,	,	PUNCT
iajs-557	268	7	then	then	ADV
iajs-557	268	8	w	w	PROPN
iajs-557	268	9	is	be	AUX
iajs-557	268	10	sh	sh	PROPN
iajs-557	268	11	of	of	ADP
iajs-557	268	12	m2	m2	PROPN
iajs-557	268	13	.	.	PUNCT
iajs-557	269	1	by	by	ADP
iajs-557	269	2	a	a	DET
iajs-557	269	3	similar	similar	ADJ
iajs-557	269	4	proof	proof	NOUN
iajs-557	269	5	,	,	PUNCT
iajs-557	269	6	if	if	SCONJ
iajs-557	269	7	k	k	PROPN
iajs-557	269	8	is	be	AUX
iajs-557	269	9	qh	qh	NOUN
iajs-557	269	10	,	,	PUNCT
iajs-557	269	11	then	then	ADV
iajs-557	269	12	n	n	CCONJ
iajs-557	269	13	,	,	PUNCT
iajs-557	269	14	w	w	PROPN
iajs-557	269	15	are	be	AUX
iajs-557	269	16	qh	qh	NOUN
iajs-557	269	17	in	in	ADP
iajs-557	269	18	m1	m1	PROPN
iajs-557	269	19	,	,	PUNCT
iajs-557	269	20	m2	m2	PROPN
iajs-557	269	21	respectively	respectively	ADV
iajs-557	269	22	.	.	PUNCT
iajs-557	270	1	the	the	DET
iajs-557	270	2	converse	converse	NOUN
iajs-557	270	3	of	of	ADP
iajs-557	270	4	proposition	proposition	NOUN
iajs-557	270	5	1.25	1.25	NUM
iajs-557	270	6	is	be	AUX
iajs-557	270	7	not	not	PART
iajs-557	270	8	true	true	ADJ
iajs-557	270	9	in	in	ADP
iajs-557	270	10	general	general	ADJ
iajs-557	270	11	.	.	PUNCT
iajs-557	271	1	for	for	ADP
iajs-557	271	2	example	example	NOUN
iajs-557	271	3	in	in	ADP
iajs-557	271	4	z	z	NOUN
iajs-557	271	5	-	-	PUNCT
iajs-557	271	6	module	module	NOUN
iajs-557	271	7	m	m	NOUN
iajs-557	271	8	=	=	SYM
iajs-557	271	9	z4⊕z6	z4⊕z6	PROPN
iajs-557	271	10	.	.	PUNCT
iajs-557	272	1	if	if	SCONJ
iajs-557	272	2	k	k	PROPN
iajs-557	272	3	=	=	SYM
iajs-557	272	4	2	2	NUM
iajs-557	272	5	3	3	NUM
iajs-557	272	6	<	<	X
iajs-557	272	7	>	>	X
iajs-557	272	8	⊕	⊕	PROPN
iajs-557	272	9	<	<	X
iajs-557	272	10	>	>	X
iajs-557	272	11	,	,	PUNCT
iajs-557	272	12	then	then	ADV
iajs-557	272	13	k	k	PROPN
iajs-557	272	14	is	be	AUX
iajs-557	272	15	not	not	PART
iajs-557	272	16	sh	sh	INTJ
iajs-557	272	17	(	(	PUNCT
iajs-557	272	18	not	not	PART
iajs-557	272	19	qh	qh	NOUN
iajs-557	272	20	)	)	PUNCT
iajs-557	272	21	.	.	PUNCT
iajs-557	273	1	but	but	CCONJ
iajs-557	273	2	2	2	NUM
iajs-557	273	3	<	<	X
iajs-557	273	4	>	>	X
iajs-557	273	5	is	be	AUX
iajs-557	273	6	sh(qh	sh(qh	NOUN
iajs-557	273	7	)	)	PUNCT
iajs-557	273	8	in	in	ADP
iajs-557	273	9	z4	z4	PROPN
iajs-557	273	10	and	and	CCONJ
iajs-557	273	11	3	3	NUM
iajs-557	273	12	<	<	X
iajs-557	273	13	>	>	X
iajs-557	273	14	is	be	AUX
iajs-557	273	15	sh(qh	sh(qh	NOUN
iajs-557	273	16	)	)	PUNCT
iajs-557	273	17	in	in	ADP
iajs-557	273	18	z6	z6	PROPN
iajs-557	273	19	.	.	PUNCT
iajs-557	274	1	2sh	2sh	ADJ
iajs-557	274	2	and	and	CCONJ
iajs-557	274	3	qh(ch)-submodules	qh(ch)-submodule	NOUN
iajs-557	274	4	and	and	CCONJ
iajs-557	274	5	multiplication	multiplication	NOUN
iajs-557	274	6	modules	module	NOUN
iajs-557	274	7	in	in	ADP
iajs-557	274	8	this	this	DET
iajs-557	274	9	section	section	NOUN
iajs-557	274	10	,	,	PUNCT
iajs-557	274	11	we	we	PRON
iajs-557	274	12	introduce	introduce	VERB
iajs-557	274	13	some	some	DET
iajs-557	274	14	properties	property	NOUN
iajs-557	274	15	of	of	ADP
iajs-557	274	16	sh	sh	PROPN
iajs-557	274	17	and	and	CCONJ
iajs-557	274	18	qh(ch	qh(ch	PROPN
iajs-557	274	19	)	)	PUNCT
iajs-557	274	20	submodles	submodle	NOUN
iajs-557	274	21	in	in	ADP
iajs-557	274	22	the	the	DET
iajs-557	274	23	class	class	NOUN
iajs-557	274	24	of	of	ADP
iajs-557	274	25	multiplication	multiplication	NOUN
iajs-557	274	26	modules	module	NOUN
iajs-557	274	27	.	.	PUNCT
iajs-557	275	1	recall	recall	VERB
iajs-557	275	2	that	that	SCONJ
iajs-557	275	3	an	an	DET
iajs-557	275	4	r	r	NOUN
iajs-557	275	5	-	-	PUNCT
iajs-557	275	6	module	module	NOUN
iajs-557	275	7	m	m	NOUN
iajs-557	275	8	is	be	AUX
iajs-557	275	9	called	call	VERB
iajs-557	275	10	multiplication	multiplication	NOUN
iajs-557	275	11	if	if	SCONJ
iajs-557	275	12	every	every	DET
iajs-557	275	13	n	n	ADV
iajs-557	275	14	≤	≤	NUM
iajs-557	275	15	m	m	NOUN
iajs-557	275	16	,	,	PUNCT
iajs-557	275	17	n	n	X
iajs-557	275	18	is	be	AUX
iajs-557	275	19	of	of	ADP
iajs-557	275	20	the	the	DET
iajs-557	275	21	form	form	NOUN
iajs-557	276	1	n	n	NOUN
iajs-557	276	2	=	=	X
iajs-557	277	1	i	i	PRON
iajs-557	277	2	m	m	VERB
iajs-557	277	3	for	for	ADP
iajs-557	277	4	some	some	DET
iajs-557	277	5	ideal	ideal	NOUN
iajs-557	277	6	i	i	NOUN
iajs-557	277	7	≤	≤	PUNCT
iajs-557	277	8	r.	r.	PROPN
iajs-557	277	9	equivalently	equivalently	PROPN
iajs-557	277	10	,	,	PUNCT
iajs-557	277	11	n	n	PROPN
iajs-557	277	12	=	=	SYM
iajs-557	277	13	(	(	PUNCT
iajs-557	277	14	n	n	NOUN
iajs-557	277	15	r	r	NOUN
iajs-557	277	16	:	:	PUNCT
iajs-557	277	17	m)⋅m	m)⋅m	NOUN
iajs-557	277	18	,	,	PUNCT
iajs-557	277	19	where	where	SCONJ
iajs-557	277	20	(	(	PUNCT
iajs-557	277	21	n	n	NOUN
iajs-557	277	22	r	r	NOUN
iajs-557	277	23	:	:	PUNCT
iajs-557	277	24	m	m	X
iajs-557	277	25	)	)	PUNCT
iajs-557	277	26	=	=	PRON
iajs-557	278	1	{	{	PUNCT
iajs-557	278	2	r	r	NOUN
iajs-557	278	3	∈	∈	PROPN
iajs-557	278	4	r	r	NOUN
iajs-557	278	5	,	,	PUNCT
iajs-557	278	6	rm	rm	NOUN
iajs-557	278	7	⊆	⊆	NUM
iajs-557	278	8	n	n	CCONJ
iajs-557	278	9	}	}	PUNCT
iajs-557	278	10	,	,	PUNCT
iajs-557	278	11	see	see	VERB
iajs-557	278	12	[	[	X
iajs-557	278	13	4	4	NUM
iajs-557	278	14	]	]	PUNCT
iajs-557	278	15	.	.	PUNCT
iajs-557	279	1	proposition	proposition	NOUN
iajs-557	279	2	:	:	PUNCT
iajs-557	279	3	let	let	VERB
iajs-557	279	4	m	m	PRON
iajs-557	279	5	be	be	AUX
iajs-557	279	6	a	a	DET
iajs-557	279	7	faithful	faithful	ADJ
iajs-557	279	8	finitely	finitely	ADV
iajs-557	279	9	generated	generate	VERB
iajs-557	279	10	multiplication	multiplication	NOUN
iajs-557	279	11	r	r	NOUN
iajs-557	279	12	-	-	PUNCT
iajs-557	279	13	module	module	NOUN
iajs-557	279	14	,	,	PUNCT
iajs-557	279	15	n	n	CCONJ
iajs-557	279	16	≤	≤	NOUN
iajs-557	279	17	m.	m.	NOUN
iajs-557	279	18	then	then	ADV
iajs-557	279	19	the	the	DET
iajs-557	279	20	following	follow	VERB
iajs-557	279	21	statements	statement	NOUN
iajs-557	279	22	are	be	AUX
iajs-557	279	23	equivalent	equivalent	ADJ
iajs-557	279	24	:	:	PUNCT
iajs-557	279	25	(	(	PUNCT
iajs-557	279	26	1	1	X
iajs-557	279	27	)	)	PUNCT
iajs-557	279	28	n	n	PRON
iajs-557	279	29	is	be	AUX
iajs-557	279	30	sh(qh)-submodule	sh(qh)-submodule	NOUN
iajs-557	279	31	.	.	PUNCT
iajs-557	280	1	mathematics	mathematic	NOUN
iajs-557	280	2	400	400	NUM
iajs-557	280	3	مجلة	مجلة	NOUN
iajs-557	280	4	إبن	إبن	VERB
iajs-557	280	5	الهيثم	الهيثم	ADJ
iajs-557	280	6	للعلوم	للعلوم	NOUN
iajs-557	280	7	الصرفة	الصرفة	NOUN
iajs-557	281	1	و	و	PRON
iajs-557	281	2	التطبيقية	التطبيقية	ADJ
iajs-557	281	3	2012	2012	NUM
iajs-557	281	4	السنة	السنة	NOUN
iajs-557	281	5	25	25	NUM
iajs-557	281	6	المجلد	المجلد	NOUN
iajs-557	281	7	3	3	NUM
iajs-557	281	8	العدد	العدد	PROPN
iajs-557	281	9	ibn	ibn	PROPN
iajs-557	281	10	al	al	PROPN
iajs-557	281	11	-	-	PUNCT
iajs-557	281	12	haitham	haitham	PROPN
iajs-557	281	13	journal	journal	PROPN
iajs-557	281	14	for	for	ADP
iajs-557	281	15	pure	pure	ADJ
iajs-557	281	16	and	and	CCONJ
iajs-557	281	17	applied	apply	VERB
iajs-557	281	18	science	science	NOUN
iajs-557	281	19	no	no	NOUN
iajs-557	281	20	.	.	NOUN
iajs-557	281	21	3	3	NUM
iajs-557	281	22	vol	vol	NOUN
iajs-557	281	23	.	.	PUNCT
iajs-557	282	1	25	25	NUM
iajs-557	282	2	year	year	NOUN
iajs-557	282	3	2012	2012	NUM
iajs-557	282	4	(	(	PUNCT
iajs-557	282	5	2	2	NUM
iajs-557	282	6	)	)	PUNCT
iajs-557	282	7	(	(	PUNCT
iajs-557	282	8	n	n	CCONJ
iajs-557	282	9	r	r	NOUN
iajs-557	282	10	:	:	PUNCT
iajs-557	282	11	m	m	X
iajs-557	282	12	)	)	PUNCT
iajs-557	282	13	is	be	AUX
iajs-557	282	14	sh(qh)-ideal	sh(qh)-ideal	NOUN
iajs-557	282	15	.	.	PUNCT
iajs-557	283	1	(	(	PUNCT
iajs-557	283	2	3	3	X
iajs-557	283	3	)	)	PUNCT
iajs-557	283	4	n	n	NOUN
iajs-557	283	5	=	=	PUNCT
iajs-557	284	1	i	i	NOUN
iajs-557	284	2	m	m	PROPN
iajs-557	284	3	,	,	PUNCT
iajs-557	284	4	i	i	PRON
iajs-557	284	5	is	be	AUX
iajs-557	284	6	sh(qh)-ideal	sh(qh)-ideal	ADJ
iajs-557	284	7	for	for	ADP
iajs-557	284	8	some	some	DET
iajs-557	284	9	<	<	NOUN
iajs-557	284	10	0	0	NUM
iajs-557	284	11	>	>	X
iajs-557	284	12	≠	≠	PROPN
iajs-557	284	13	i	i	PROPN
iajs-557	284	14	≤	≤	X
iajs-557	284	15	r.	r.	NOUN
iajs-557	284	16	proof	proof	NOUN
iajs-557	284	17	:	:	PUNCT
iajs-557	284	18	(	(	PUNCT
iajs-557	284	19	1	1	X
iajs-557	284	20	)	)	PUNCT
iajs-557	284	21	⇒	⇒	NOUN
iajs-557	284	22	(	(	PUNCT
iajs-557	284	23	2	2	X
iajs-557	284	24	)	)	PUNCT
iajs-557	284	25	assume	assume	VERB
iajs-557	284	26	n	n	PRON
iajs-557	284	27	is	be	AUX
iajs-557	284	28	a	a	DET
iajs-557	284	29	sh	sh	NOUN
iajs-557	284	30	-	-	PUNCT
iajs-557	284	31	submodule	submodule	NOUN
iajs-557	284	32	of	of	ADP
iajs-557	284	33	m	m	PROPN
iajs-557	284	34	,	,	PUNCT
iajs-557	284	35	and	and	CCONJ
iajs-557	284	36	let	let	VERB
iajs-557	284	37	(	(	PUNCT
iajs-557	284	38	n	n	NOUN
iajs-557	284	39	r	r	NOUN
iajs-557	284	40	:	:	PUNCT
iajs-557	284	41	m	m	X
iajs-557	284	42	)	)	PUNCT
iajs-557	284	43	⊆	⊆	NUM
iajs-557	284	44	i1	i1	PROPN
iajs-557	284	45	+	+	CCONJ
iajs-557	284	46	i2	i2	PROPN
iajs-557	284	47	where	where	SCONJ
iajs-557	284	48	i1	i1	PROPN
iajs-557	284	49	,	,	PUNCT
iajs-557	284	50	i2	i2	PROPN
iajs-557	284	51	are	be	AUX
iajs-557	284	52	ideals	ideal	NOUN
iajs-557	284	53	of	of	ADP
iajs-557	284	54	r.	r.	PROPN
iajs-557	284	55	then	then	ADV
iajs-557	284	56	(	(	PUNCT
iajs-557	284	57	n	n	CCONJ
iajs-557	284	58	r	r	NOUN
iajs-557	284	59	:	:	PUNCT
iajs-557	284	60	m)⋅m	m)⋅m	NOUN
iajs-557	284	61	⊆	⊆	NUM
iajs-557	284	62	(	(	PUNCT
iajs-557	284	63	i1	i1	PROPN
iajs-557	284	64	+	+	CCONJ
iajs-557	284	65	i2)⋅m	i2)⋅m	PROPN
iajs-557	284	66	.	.	PUNCT
iajs-557	285	1	but	but	CCONJ
iajs-557	285	2	(	(	PUNCT
iajs-557	285	3	i1	i1	PROPN
iajs-557	285	4	+	+	CCONJ
iajs-557	285	5	i2)⋅m	i2)⋅m	PROPN
iajs-557	285	6	=	=	SYM
iajs-557	285	7	i1	i1	PROPN
iajs-557	285	8	m	m	PROPN
iajs-557	285	9	+	+	PROPN
iajs-557	285	10	i2	i2	PROPN
iajs-557	285	11	m	m	VERB
iajs-557	285	12	since	since	SCONJ
iajs-557	285	13	m	m	PROPN
iajs-557	285	14	is	be	AUX
iajs-557	285	15	finitely	finitely	ADV
iajs-557	285	16	generated	generate	VERB
iajs-557	285	17	.	.	PUNCT
iajs-557	286	1	it	it	PRON
iajs-557	286	2	follows	follow	VERB
iajs-557	286	3	(	(	PUNCT
iajs-557	286	4	n	n	CCONJ
iajs-557	286	5	r	r	NOUN
iajs-557	286	6	:	:	PUNCT
iajs-557	286	7	m)⋅m	m)⋅m	NOUN
iajs-557	286	8	⊆	⊆	NUM
iajs-557	286	9	i1	i1	PROPN
iajs-557	286	10	m	m	PROPN
iajs-557	286	11	+	+	PROPN
iajs-557	286	12	i2	i2	NOUN
iajs-557	286	13	m.	m.	NOUN
iajs-557	286	14	but	but	CCONJ
iajs-557	286	15	(	(	PUNCT
iajs-557	286	16	n	n	CCONJ
iajs-557	286	17	r	r	NOUN
iajs-557	286	18	:	:	PUNCT
iajs-557	286	19	m)⋅m	m)⋅m	NOUN
iajs-557	286	20	=	=	PUNCT
iajs-557	286	21	n	n	CCONJ
iajs-557	286	22	since	since	SCONJ
iajs-557	286	23	m	m	PROPN
iajs-557	286	24	is	be	AUX
iajs-557	286	25	multiplication	multiplication	NOUN
iajs-557	286	26	.	.	PUNCT
iajs-557	287	1	then	then	ADV
iajs-557	287	2	n	n	PROPN
iajs-557	287	3	⊆	⊆	NUM
iajs-557	287	4	i1	i1	PROPN
iajs-557	287	5	m	m	PROPN
iajs-557	287	6	+	+	PROPN
iajs-557	287	7	i2	i2	PROPN
iajs-557	287	8	m	m	PROPN
iajs-557	287	9	,	,	PUNCT
iajs-557	287	10	so	so	SCONJ
iajs-557	287	11	either	either	CCONJ
iajs-557	287	12	n	n	PROPN
iajs-557	287	13	⊆	⊆	NUM
iajs-557	287	14	i1	i1	PROPN
iajs-557	287	15	m	m	PROPN
iajs-557	287	16	or	or	CCONJ
iajs-557	287	17	n	n	CCONJ
iajs-557	287	18	⊆	⊆	NUM
iajs-557	287	19	i2	i2	NOUN
iajs-557	287	20	m	m	NOUN
iajs-557	287	21	,	,	PUNCT
iajs-557	287	22	that	that	ADV
iajs-557	287	23	is	is	ADV
iajs-557	287	24	(	(	PUNCT
iajs-557	287	25	n	n	NOUN
iajs-557	287	26	r	r	NOUN
iajs-557	287	27	:	:	PUNCT
iajs-557	287	28	m)⋅m	m)⋅m	NOUN
iajs-557	287	29	⊆	⊆	NUM
iajs-557	287	30	i1	i1	PROPN
iajs-557	287	31	m	m	PROPN
iajs-557	287	32	or	or	CCONJ
iajs-557	287	33	(	(	PUNCT
iajs-557	287	34	n	n	CCONJ
iajs-557	287	35	r	r	NOUN
iajs-557	287	36	:	:	PUNCT
iajs-557	287	37	m)⋅m	m)⋅m	NOUN
iajs-557	287	38	⊆	⊆	NUM
iajs-557	287	39	i2	i2	NOUN
iajs-557	287	40	m.	m.	NOUN
iajs-557	287	41	thus	thus	ADV
iajs-557	287	42	(	(	PUNCT
iajs-557	287	43	n	n	PRON
iajs-557	287	44	r	r	NOUN
iajs-557	287	45	:	:	PUNCT
iajs-557	287	46	m	m	X
iajs-557	287	47	)	)	PUNCT
iajs-557	288	1	⊆	⊆	NUM
iajs-557	288	2	i1	i1	PROPN
iajs-557	288	3	or	or	CCONJ
iajs-557	288	4	(	(	PUNCT
iajs-557	288	5	n	n	CCONJ
iajs-557	288	6	r	r	NOUN
iajs-557	288	7	:	:	PUNCT
iajs-557	288	8	m	m	X
iajs-557	288	9	)	)	PUNCT
iajs-557	288	10	⊆	⊆	NUM
iajs-557	288	11	i2	i2	NOUN
iajs-557	288	12	by	by	ADP
iajs-557	288	13	[	[	X
iajs-557	288	14	10	10	NUM
iajs-557	288	15	,	,	PUNCT
iajs-557	288	16	theorem	theorem	VERB
iajs-557	288	17	3.1	3.1	NUM
iajs-557	288	18	]	]	PUNCT
iajs-557	288	19	.	.	PUNCT
iajs-557	289	1	hence	hence	ADV
iajs-557	289	2	(	(	PUNCT
iajs-557	289	3	n	n	CCONJ
iajs-557	289	4	r	r	NOUN
iajs-557	289	5	:	:	PUNCT
iajs-557	289	6	m	m	X
iajs-557	289	7	)	)	PUNCT
iajs-557	289	8	is	be	AUX
iajs-557	289	9	a	a	DET
iajs-557	289	10	sh	sh	NOUN
iajs-557	289	11	-	-	PUNCT
iajs-557	289	12	ideal	ideal	NOUN
iajs-557	289	13	of	of	ADP
iajs-557	289	14	r.	r.	PROPN
iajs-557	289	15	by	by	ADP
iajs-557	289	16	a	a	DET
iajs-557	289	17	similar	similar	ADJ
iajs-557	289	18	proof	proof	NOUN
iajs-557	289	19	,	,	PUNCT
iajs-557	289	20	(	(	PUNCT
iajs-557	289	21	n	n	CCONJ
iajs-557	289	22	r	r	NOUN
iajs-557	289	23	:	:	PUNCT
iajs-557	289	24	m	m	X
iajs-557	289	25	)	)	PUNCT
iajs-557	289	26	is	be	AUX
iajs-557	289	27	qh	qh	NOUN
iajs-557	289	28	-	-	NOUN
iajs-557	289	29	ideal	ideal	NOUN
iajs-557	289	30	if	if	SCONJ
iajs-557	289	31	n	n	PRON
iajs-557	289	32	is	be	AUX
iajs-557	289	33	qh	qh	PROPN
iajs-557	289	34	.	.	PROPN
iajs-557	289	35	(	(	PUNCT
iajs-557	289	36	2	2	X
iajs-557	289	37	)	)	PUNCT
iajs-557	289	38	⇒	⇒	NOUN
iajs-557	289	39	(	(	PUNCT
iajs-557	289	40	3	3	X
iajs-557	289	41	)	)	PUNCT
iajs-557	289	42	assume	assume	NOUN
iajs-557	289	43	(	(	PUNCT
iajs-557	289	44	n	n	NOUN
iajs-557	289	45	r	r	NOUN
iajs-557	289	46	:	:	PUNCT
iajs-557	289	47	m	m	X
iajs-557	289	48	)	)	PUNCT
iajs-557	289	49	is	be	AUX
iajs-557	289	50	sh	sh	NOUN
iajs-557	289	51	-	-	PUNCT
iajs-557	289	52	ideal	ideal	NOUN
iajs-557	289	53	.	.	PUNCT
iajs-557	290	1	put	put	VERB
iajs-557	290	2	i	i	PRON
iajs-557	290	3	=	=	PUNCT
iajs-557	290	4	(	(	PUNCT
iajs-557	290	5	n	n	NOUN
iajs-557	290	6	r	r	NOUN
iajs-557	290	7	:	:	PUNCT
iajs-557	290	8	m	m	NOUN
iajs-557	290	9	)	)	PUNCT
iajs-557	290	10	.	.	PUNCT
iajs-557	291	1	since	since	SCONJ
iajs-557	291	2	m	m	PROPN
iajs-557	291	3	is	be	AUX
iajs-557	291	4	multiplication	multiplication	NOUN
iajs-557	291	5	,	,	PUNCT
iajs-557	291	6	then	then	ADV
iajs-557	291	7	n	n	NOUN
iajs-557	291	8	=	=	SYM
iajs-557	291	9	(	(	PUNCT
iajs-557	291	10	n	n	NOUN
iajs-557	291	11	r	r	NOUN
iajs-557	291	12	:	:	PUNCT
iajs-557	291	13	m)⋅m	m)⋅m	NOUN
iajs-557	291	14	.	.	PUNCT
iajs-557	292	1	hence	hence	ADV
iajs-557	292	2	n	n	NOUN
iajs-557	292	3	=	=	X
iajs-557	292	4	i	i	NOUN
iajs-557	292	5	m	m	PROPN
iajs-557	292	6	,	,	PUNCT
iajs-557	292	7	and	and	CCONJ
iajs-557	292	8	i	i	PRON
iajs-557	292	9	is	be	AUX
iajs-557	292	10	a	a	DET
iajs-557	292	11	sh	sh	NOUN
iajs-557	292	12	-	-	PUNCT
iajs-557	292	13	ideal	ideal	NOUN
iajs-557	292	14	.	.	PUNCT
iajs-557	293	1	similarly	similarly	ADV
iajs-557	293	2	i	i	PRON
iajs-557	293	3	is	be	AUX
iajs-557	293	4	a	a	DET
iajs-557	293	5	qh	qh	NOUN
iajs-557	293	6	-	-	NOUN
iajs-557	293	7	ideal	ideal	NOUN
iajs-557	293	8	.	.	PUNCT
iajs-557	294	1	(	(	PUNCT
iajs-557	294	2	3	3	X
iajs-557	294	3	)	)	PUNCT
iajs-557	294	4	⇒	⇒	NOUN
iajs-557	294	5	(	(	PUNCT
iajs-557	294	6	1	1	X
iajs-557	294	7	)	)	PUNCT
iajs-557	294	8	assume	assume	VERB
iajs-557	294	9	that	that	SCONJ
iajs-557	294	10	n	n	PRON
iajs-557	294	11	=	=	VERB
iajs-557	295	1	i	i	PRON
iajs-557	295	2	m	m	VERB
iajs-557	295	3	for	for	ADP
iajs-557	295	4	some	some	DET
iajs-557	295	5	sh	sh	NOUN
iajs-557	295	6	-	-	PUNCT
iajs-557	295	7	ideal	ideal	NOUN
iajs-557	295	8	i	i	PRON
iajs-557	295	9	of	of	ADP
iajs-557	295	10	r	r	NOUN
iajs-557	295	11	,	,	PUNCT
iajs-557	295	12	and	and	CCONJ
iajs-557	295	13	let	let	VERB
iajs-557	295	14	n	n	PRON
iajs-557	295	15	⊆	⊆	NUM
iajs-557	295	16	l1+l2	l1+l2	PROPN
iajs-557	295	17	where	where	SCONJ
iajs-557	295	18	l1	l1	PROPN
iajs-557	295	19	,	,	PUNCT
iajs-557	295	20	l2	l2	NOUN
iajs-557	295	21	≤	≤	NOUN
iajs-557	295	22	m.	m.	NOUN
iajs-557	295	23	but	but	CCONJ
iajs-557	295	24	l1	l1	PROPN
iajs-557	295	25	=	=	PROPN
iajs-557	295	26	i1	i1	PROPN
iajs-557	295	27	m	m	PROPN
iajs-557	295	28	,	,	PUNCT
iajs-557	295	29	l2	l2	NOUN
iajs-557	295	30	=	=	SYM
iajs-557	295	31	i2	i2	PROPN
iajs-557	295	32	m	m	VERB
iajs-557	295	33	since	since	SCONJ
iajs-557	295	34	m	m	PROPN
iajs-557	295	35	is	be	AUX
iajs-557	295	36	multiplication	multiplication	NOUN
iajs-557	295	37	for	for	ADP
iajs-557	295	38	some	some	DET
iajs-557	295	39	ideals	ideal	NOUN
iajs-557	295	40	i1	i1	PROPN
iajs-557	295	41	,	,	PUNCT
iajs-557	295	42	i2	i2	PROPN
iajs-557	295	43	of	of	ADP
iajs-557	295	44	r.	r.	PROPN
iajs-557	296	1	so	so	ADV
iajs-557	296	2	i	i	PRON
iajs-557	296	3	m	m	VERB
iajs-557	296	4	⊆	⊆	NUM
iajs-557	296	5	i1m+i2	i1m+i2	PROPN
iajs-557	296	6	m	m	VERB
iajs-557	296	7	⊆	⊆	NUM
iajs-557	296	8	(	(	PUNCT
iajs-557	296	9	i1	i1	PROPN
iajs-557	296	10	+	+	CCONJ
iajs-557	296	11	i2)⋅m	i2)⋅m	PROPN
iajs-557	296	12	,	,	PUNCT
iajs-557	296	13	since	since	SCONJ
iajs-557	296	14	m	m	PROPN
iajs-557	296	15	is	be	AUX
iajs-557	296	16	finitely	finitely	ADV
iajs-557	296	17	generated	generate	VERB
iajs-557	296	18	.	.	PUNCT
iajs-557	297	1	then	then	ADV
iajs-557	297	2	i	i	PRON
iajs-557	297	3	m	m	VERB
iajs-557	297	4	⊆	⊆	NUM
iajs-557	297	5	(	(	PUNCT
iajs-557	297	6	i1	i1	PROPN
iajs-557	297	7	+	+	CCONJ
iajs-557	297	8	i2)⋅m	i2)⋅m	PROPN
iajs-557	297	9	,	,	PUNCT
iajs-557	297	10	so	so	SCONJ
iajs-557	297	11	i	i	PROPN
iajs-557	297	12	⊆	⊆	NUM
iajs-557	297	13	i1	i1	PROPN
iajs-557	297	14	+	+	CCONJ
iajs-557	297	15	i2	i2	PROPN
iajs-557	297	16	by	by	ADP
iajs-557	297	17	[	[	X
iajs-557	297	18	10	10	NUM
iajs-557	297	19	,	,	PUNCT
iajs-557	297	20	theorem	theorem	VERB
iajs-557	297	21	3.1	3.1	NUM
iajs-557	297	22	]	]	PUNCT
iajs-557	297	23	.	.	PUNCT
iajs-557	298	1	so	so	ADV
iajs-557	298	2	that	that	SCONJ
iajs-557	298	3	either	either	CCONJ
iajs-557	298	4	i	i	PROPN
iajs-557	298	5	⊆	⊆	NUM
iajs-557	298	6	i1	i1	PROPN
iajs-557	298	7	or	or	CCONJ
iajs-557	298	8	i	i	PROPN
iajs-557	298	9	⊆	⊆	NUM
iajs-557	298	10	i2	i2	NOUN
iajs-557	298	11	,	,	PUNCT
iajs-557	298	12	which	which	PRON
iajs-557	298	13	implies	imply	VERB
iajs-557	298	14	i	i	PRON
iajs-557	298	15	m	m	PROPN
iajs-557	298	16	⊆	⊆	NUM
iajs-557	298	17	i1	i1	PROPN
iajs-557	298	18	m	m	PROPN
iajs-557	298	19	or	or	CCONJ
iajs-557	298	20	i	i	PRON
iajs-557	298	21	m	m	PROPN
iajs-557	298	22	⊆	⊆	NUM
iajs-557	298	23	i2	i2	NOUN
iajs-557	298	24	m.	m.	NOUN
iajs-557	298	25	hence	hence	ADV
iajs-557	298	26	n	n	PROPN
iajs-557	298	27	⊆	⊆	NUM
iajs-557	298	28	l1	l1	PROPN
iajs-557	298	29	or	or	CCONJ
iajs-557	298	30	n	n	CCONJ
iajs-557	298	31	⊆	⊆	NUM
iajs-557	298	32	l2	l2	NOUN
iajs-557	298	33	.	.	PUNCT
iajs-557	299	1	thus	thus	ADV
iajs-557	299	2	n	n	X
iajs-557	299	3	is	be	AUX
iajs-557	299	4	sh	sh	PROPN
iajs-557	299	5	.	.	PUNCT
iajs-557	300	1	similarly	similarly	ADV
iajs-557	300	2	n	n	PROPN
iajs-557	300	3	is	be	AUX
iajs-557	300	4	qh	qh	NOUN
iajs-557	300	5	.	.	PUNCT
iajs-557	301	1	the	the	DET
iajs-557	301	2	condition	condition	NOUN
iajs-557	301	3	m	m	VERB
iajs-557	301	4	is	be	AUX
iajs-557	301	5	faithful	faithful	ADJ
iajs-557	301	6	is	be	AUX
iajs-557	301	7	necessary	necessary	ADJ
iajs-557	301	8	in	in	ADP
iajs-557	301	9	proposition	proposition	NOUN
iajs-557	301	10	2.1	2.1	NUM
iajs-557	301	11	.	.	PUNCT
iajs-557	302	1	fot	fot	ADJ
iajs-557	302	2	example	example	NOUN
iajs-557	302	3	,	,	PUNCT
iajs-557	302	4	the	the	DET
iajs-557	302	5	z	z	NOUN
iajs-557	302	6	-	-	PUNCT
iajs-557	302	7	module	module	NOUN
iajs-557	302	8	z6	z6	NOUN
iajs-557	302	9	is	be	AUX
iajs-557	302	10	finitely	finitely	ADV
iajs-557	302	11	generated	generate	VERB
iajs-557	302	12	multiplication	multiplication	NOUN
iajs-557	302	13	,	,	PUNCT
iajs-557	302	14	but	but	CCONJ
iajs-557	302	15	not	not	PART
iajs-557	302	16	faithful	faithful	ADJ
iajs-557	302	17	,	,	PUNCT
iajs-557	302	18	let	let	VERB
iajs-557	302	19	n	n	NOUN
iajs-557	302	20	=	=	SYM
iajs-557	303	1	2	2	NUM
iajs-557	303	2	<	<	X
iajs-557	303	3	>	>	X
iajs-557	303	4	,	,	PUNCT
iajs-557	303	5	n	n	PROPN
iajs-557	303	6	is	be	AUX
iajs-557	303	7	sh	sh	INTJ
iajs-557	303	8	of	of	ADP
iajs-557	303	9	z6,but	z6,but	INTJ
iajs-557	303	10	(	(	PUNCT
iajs-557	303	11	n	n	X
iajs-557	303	12	z	z	NOUN
iajs-557	303	13	:	:	PUNCT
iajs-557	303	14	m	m	X
iajs-557	303	15	)	)	PUNCT
iajs-557	304	1	=	=	SYM
iajs-557	304	2	6z	6z	NOUN
iajs-557	304	3	(	(	PUNCT
iajs-557	304	4	2	2	NUM
iajs-557	304	5	:	:	PUNCT
iajs-557	304	6	z	z	NOUN
iajs-557	304	7	)	)	PUNCT
iajs-557	304	8	<	<	X
iajs-557	304	9	>	>	X
iajs-557	304	10	=	=	SYM
iajs-557	304	11	2z	2z	NUM
iajs-557	304	12	is	be	AUX
iajs-557	304	13	not	not	PART
iajs-557	304	14	sh	sh	PROPN
iajs-557	304	15	of	of	ADP
iajs-557	304	16	z.	z.	PROPN
iajs-557	304	17	corollary	corollary	PROPN
iajs-557	304	18	:	:	PUNCT
iajs-557	304	19	let	let	VERB
iajs-557	304	20	m	m	PRON
iajs-557	304	21	be	be	AUX
iajs-557	304	22	a	a	DET
iajs-557	304	23	finitely	finitely	ADV
iajs-557	304	24	generated	generate	VERB
iajs-557	304	25	faithful	faithful	ADJ
iajs-557	304	26	multiplication	multiplication	NOUN
iajs-557	304	27	r	r	NOUN
iajs-557	304	28	-	-	NOUN
iajs-557	304	29	module	module	NOUN
iajs-557	304	30	.	.	PUNCT
iajs-557	305	1	then	then	ADV
iajs-557	305	2	every	every	DET
iajs-557	305	3	non	non	ADJ
iajs-557	305	4	-	-	ADJ
iajs-557	305	5	zero	zero	NUM
iajs-557	305	6	submodule	submodule	NOUN
iajs-557	305	7	of	of	ADP
iajs-557	305	8	m	m	PROPN
iajs-557	305	9	is	be	AUX
iajs-557	305	10	sh(qh	sh(qh	VERB
iajs-557	305	11	)	)	PUNCT
iajs-557	306	1	if	if	SCONJ
iajs-557	306	2	and	and	CCONJ
iajs-557	306	3	only	only	ADV
iajs-557	306	4	if	if	SCONJ
iajs-557	306	5	every	every	DET
iajs-557	306	6	non	non	ADJ
iajs-557	306	7	-	-	ADJ
iajs-557	306	8	zero	zero	NUM
iajs-557	306	9	ideal	ideal	NOUN
iajs-557	306	10	of	of	ADP
iajs-557	306	11	r	r	NOUN
iajs-557	306	12	is	be	AUX
iajs-557	306	13	sh(qh	sh(qh	PROPN
iajs-557	306	14	)	)	PUNCT
iajs-557	306	15	.	.	PUNCT
iajs-557	307	1	proposition	proposition	NOUN
iajs-557	307	2	:	:	PUNCT
iajs-557	307	3	let	let	VERB
iajs-557	307	4	m	m	PRON
iajs-557	307	5	be	be	AUX
iajs-557	307	6	a	a	DET
iajs-557	307	7	faithful	faithful	ADJ
iajs-557	307	8	finitely	finitely	ADV
iajs-557	307	9	generated	generate	VERB
iajs-557	307	10	multiplication	multiplication	NOUN
iajs-557	307	11	r	r	NOUN
iajs-557	307	12	-	-	PUNCT
iajs-557	307	13	module	module	NOUN
iajs-557	307	14	.	.	PUNCT
iajs-557	308	1	then	then	ADV
iajs-557	308	2	r	r	NOUN
iajs-557	308	3	satisfies	satisfie	NOUN
iajs-557	308	4	acc(dcc	acc(dcc	NOUN
iajs-557	308	5	)	)	PUNCT
iajs-557	308	6	on	on	ADP
iajs-557	308	7	sh	sh	NOUN
iajs-557	308	8	-	-	PUNCT
iajs-557	308	9	ideals	ideal	NOUN
iajs-557	308	10	if	if	SCONJ
iajs-557	308	11	and	and	CCONJ
iajs-557	308	12	only	only	ADV
iajs-557	308	13	if	if	SCONJ
iajs-557	308	14	m	m	NOUN
iajs-557	308	15	satisfies	satisfy	VERB
iajs-557	308	16	acc(dcc	acc(dcc	NOUN
iajs-557	308	17	)	)	PUNCT
iajs-557	308	18	on	on	ADP
iajs-557	308	19	sh	sh	PROPN
iajs-557	308	20	-	-	PUNCT
iajs-557	308	21	submodules	submodules	NOUN
iajs-557	308	22	.	.	PUNCT
iajs-557	309	1	proof	proof	NOUN
iajs-557	309	2	:	:	PUNCT
iajs-557	309	3	⇒	⇒	NOUN
iajs-557	309	4	we	we	PRON
iajs-557	309	5	take	take	VERB
iajs-557	309	6	the	the	DET
iajs-557	309	7	case	case	NOUN
iajs-557	309	8	of	of	ADP
iajs-557	309	9	acc	acc	PROPN
iajs-557	309	10	.	.	PROPN
iajs-557	310	1	let	let	VERB
iajs-557	310	2	l1	l1	PROPN
iajs-557	310	3	⊆	⊆	NUM
iajs-557	310	4	l2	l2	NOUN
iajs-557	310	5	…	…	PUNCT
iajs-557	310	6	be	be	AUX
iajs-557	310	7	an	an	DET
iajs-557	310	8	ascending	ascend	VERB
iajs-557	310	9	chain	chain	NOUN
iajs-557	310	10	of	of	ADP
iajs-557	310	11	sh	sh	PROPN
iajs-557	310	12	-	-	PUNCT
iajs-557	310	13	submodules	submodule	NOUN
iajs-557	310	14	of	of	ADP
iajs-557	310	15	m.	m.	NOUN
iajs-557	310	16	since	since	SCONJ
iajs-557	310	17	li	li	PROPN
iajs-557	310	18	is	be	AUX
iajs-557	310	19	sh	sh	PROPN
iajs-557	310	20	-	-	PUNCT
iajs-557	310	21	submodule	submodule	NOUN
iajs-557	310	22	,	,	PUNCT
iajs-557	310	23	then	then	ADV
iajs-557	310	24	(	(	PUNCT
iajs-557	310	25	li	li	PROPN
iajs-557	310	26	r	r	NOUN
iajs-557	310	27	:	:	PUNCT
iajs-557	310	28	m	m	X
iajs-557	310	29	)	)	PUNCT
iajs-557	310	30	is	be	AUX
iajs-557	310	31	sh	sh	NOUN
iajs-557	310	32	-	-	PUNCT
iajs-557	310	33	ideal	ideal	NOUN
iajs-557	310	34	for	for	ADP
iajs-557	310	35	each	each	PRON
iajs-557	310	36	i	i	NOUN
iajs-557	310	37	=	=	NOUN
iajs-557	310	38	1,2	1,2	NUM
iajs-557	310	39	,	,	PUNCT
iajs-557	310	40	…	…	PUNCT
iajs-557	310	41	by	by	ADP
iajs-557	310	42	proposition	proposition	NOUN
iajs-557	310	43	2.1	2.1	NUM
iajs-557	310	44	,	,	PUNCT
iajs-557	310	45	and	and	CCONJ
iajs-557	310	46	(	(	PUNCT
iajs-557	310	47	l1	l1	PROPN
iajs-557	310	48	r	r	NOUN
iajs-557	310	49	:	:	PUNCT
iajs-557	310	50	m	m	X
iajs-557	310	51	)	)	PUNCT
iajs-557	310	52	⊆	⊆	NUM
iajs-557	310	53	(	(	PUNCT
iajs-557	310	54	l2	l2	NOUN
iajs-557	310	55	r	r	NOUN
iajs-557	310	56	:	:	PUNCT
iajs-557	310	57	m	m	X
iajs-557	310	58	)	)	PUNCT
iajs-557	310	59	⊆	⊆	NUM
iajs-557	310	60	…	…	PUNCT
iajs-557	310	61	by	by	ADP
iajs-557	310	62	[	[	X
iajs-557	310	63	10,theorem	10,theorem	NUM
iajs-557	310	64	3.1	3.1	NUM
iajs-557	310	65	]	]	PUNCT
iajs-557	310	66	.	.	PUNCT
iajs-557	311	1	but	but	CCONJ
iajs-557	311	2	r	r	NOUN
iajs-557	311	3	satisfies	satisfy	VERB
iajs-557	311	4	acc	acc	PROPN
iajs-557	311	5	on	on	ADP
iajs-557	311	6	any	any	DET
iajs-557	311	7	ascending	ascend	VERB
iajs-557	311	8	chain	chain	NOUN
iajs-557	311	9	of	of	ADP
iajs-557	311	10	sh	sh	NOUN
iajs-557	311	11	-	-	PUNCT
iajs-557	311	12	ideals	ideal	NOUN
iajs-557	311	13	.	.	PUNCT
iajs-557	312	1	so	so	ADV
iajs-557	312	2	ther	ther	ADV
iajs-557	312	3	exists	exist	VERB
iajs-557	312	4	n	n	PRON
iajs-557	312	5	∈	∈	PROPN
iajs-557	312	6	z+	z+	NUM
iajs-557	312	7	such	such	ADJ
iajs-557	312	8	that	that	SCONJ
iajs-557	312	9	(	(	PUNCT
iajs-557	312	10	ln	ln	NOUN
iajs-557	312	11	r	r	NOUN
iajs-557	312	12	:	:	PUNCT
iajs-557	312	13	m	m	X
iajs-557	312	14	)	)	PUNCT
iajs-557	312	15	=	=	SYM
iajs-557	312	16	(	(	PUNCT
iajs-557	313	1	ln	ln	ADJ
iajs-557	313	2	+	+	CCONJ
iajs-557	313	3	1	1	NUM
iajs-557	313	4	r	r	NOUN
iajs-557	313	5	:	:	PUNCT
iajs-557	313	6	m	m	X
iajs-557	313	7	)	)	PUNCT
iajs-557	313	8	=	=	SYM
iajs-557	313	9	…	…	PUNCT
iajs-557	313	10	.	.	PUNCT
iajs-557	314	1	then	then	ADV
iajs-557	314	2	(	(	PUNCT
iajs-557	314	3	ln	ln	NOUN
iajs-557	314	4	r	r	NOUN
iajs-557	314	5	:	:	PUNCT
iajs-557	314	6	m)⋅m	m)⋅m	NOUN
iajs-557	314	7	=	=	SYM
iajs-557	314	8	(	(	PUNCT
iajs-557	314	9	ln	ln	ADJ
iajs-557	314	10	+	+	CCONJ
iajs-557	314	11	1	1	NUM
iajs-557	314	12	r	r	NOUN
iajs-557	314	13	:	:	PUNCT
iajs-557	314	14	m)⋅m	m)⋅m	NOUN
iajs-557	314	15	=	=	SYM
iajs-557	314	16	…	…	PUNCT
iajs-557	314	17	..	..	PUNCT
iajs-557	314	18	thus	thus	ADV
iajs-557	314	19	ln	ln	ADV
iajs-557	314	20	=	=	PUNCT
iajs-557	314	21	ln	ln	NOUN
iajs-557	315	1	+	+	NOUN
iajs-557	315	2	1	1	NUM
iajs-557	315	3	=	=	SYM
iajs-557	315	4	…	…	PUNCT
iajs-557	315	5	for	for	ADP
iajs-557	315	6	some	some	DET
iajs-557	315	7	n	n	PRON
iajs-557	315	8	∈	∈	NOUN
iajs-557	315	9	z+	z+	NOUN
iajs-557	315	10	.	.	PUNCT
iajs-557	316	1	hence	hence	ADV
iajs-557	316	2	m	m	VERB
iajs-557	316	3	satisfies	satisfy	VERB
iajs-557	316	4	acc	acc	PROPN
iajs-557	316	5	on	on	ADP
iajs-557	316	6	sh	sh	PROPN
iajs-557	316	7	-	-	PUNCT
iajs-557	316	8	submodules	submodules	NOUN
iajs-557	316	9	.	.	PUNCT
iajs-557	317	1	⇐	⇐	ADP
iajs-557	317	2	the	the	DET
iajs-557	317	3	proof	proof	NOUN
iajs-557	317	4	is	be	AUX
iajs-557	317	5	similar	similar	ADJ
iajs-557	317	6	.	.	PUNCT
iajs-557	318	1	sh	sh	PROPN
iajs-557	318	2	,	,	PUNCT
iajs-557	318	3	qh(ch)-submodules	qh(ch)-submodule	NOUN
iajs-557	318	4	and	and	CCONJ
iajs-557	318	5	other	other	ADJ
iajs-557	318	6	related	related	ADJ
iajs-557	318	7	concepts	concept	NOUN
iajs-557	318	8	recall	recall	VERB
iajs-557	318	9	that	that	SCONJ
iajs-557	318	10	an	an	DET
iajs-557	318	11	r	r	NOUN
iajs-557	318	12	-	-	PUNCT
iajs-557	318	13	module	module	NOUN
iajs-557	318	14	m	m	NOUN
iajs-557	318	15	is	be	AUX
iajs-557	318	16	called	call	VERB
iajs-557	318	17	scalar	scalar	ADJ
iajs-557	318	18	if	if	SCONJ
iajs-557	318	19	for	for	ADP
iajs-557	318	20	each	each	DET
iajs-557	318	21	f	f	PROPN
iajs-557	318	22	∈	∈	PROPN
iajs-557	318	23	endr(m	endr(m	PROPN
iajs-557	318	24	)	)	PUNCT
iajs-557	318	25	,	,	PUNCT
iajs-557	318	26	there	there	PRON
iajs-557	318	27	exists	exist	VERB
iajs-557	318	28	r	r	NOUN
iajs-557	318	29	∈	∈	PROPN
iajs-557	318	30	r	r	NOUN
iajs-557	318	31	such	such	ADJ
iajs-557	318	32	that	that	SCONJ
iajs-557	318	33	f	f	PROPN
iajs-557	318	34	(	(	PUNCT
iajs-557	318	35	x	x	X
iajs-557	318	36	)	)	PUNCT
iajs-557	318	37	=	=	PUNCT
iajs-557	318	38	rx	rx	VERB
iajs-557	318	39	for	for	ADP
iajs-557	318	40	all	all	PRON
iajs-557	318	41	x	x	SYM
iajs-557	318	42	∈	∈	PROPN
iajs-557	318	43	m	m	PRON
iajs-557	318	44	,	,	PUNCT
iajs-557	318	45	see	see	VERB
iajs-557	318	46	[	[	X
iajs-557	318	47	11	11	NUM
iajs-557	318	48	]	]	PUNCT
iajs-557	318	49	.	.	PUNCT
iajs-557	319	1	proposition	proposition	NOUN
iajs-557	319	2	:	:	PUNCT
iajs-557	319	3	let	let	VERB
iajs-557	319	4	m	m	PRON
iajs-557	319	5	be	be	AUX
iajs-557	319	6	a	a	DET
iajs-557	319	7	scalar	scalar	ADJ
iajs-557	319	8	r	r	NOUN
iajs-557	319	9	-	-	PUNCT
iajs-557	319	10	module	module	NOUN
iajs-557	319	11	and	and	CCONJ
iajs-557	319	12	r	r	NOUN
iajs-557	319	13	is	be	AUX
iajs-557	319	14	sh	sh	NOUN
iajs-557	319	15	-	-	PUNCT
iajs-557	319	16	ring	ring	NOUN
iajs-557	319	17	,	,	PUNCT
iajs-557	319	18	then	then	ADV
iajs-557	319	19	endr(m	endr(m	VERB
iajs-557	319	20	)	)	PUNCT
iajs-557	319	21	is	be	AUX
iajs-557	319	22	sh	sh	NOUN
iajs-557	319	23	-	-	PUNCT
iajs-557	319	24	ring	ring	NOUN
iajs-557	319	25	.	.	PUNCT
iajs-557	320	1	mathematics	mathematic	NOUN
iajs-557	320	2	401	401	NUM
iajs-557	320	3	مجلة	مجلة	NOUN
iajs-557	320	4	إبن	إبن	VERB
iajs-557	320	5	الهيثم	الهيثم	ADJ
iajs-557	320	6	للعلوم	للعلوم	NOUN
iajs-557	320	7	الصرفة	الصرفة	NOUN
iajs-557	321	1	و	و	PRON
iajs-557	321	2	التطبيقية	التطبيقية	ADJ
iajs-557	321	3	2012	2012	NUM
iajs-557	321	4	السنة	السنة	NOUN
iajs-557	321	5	25	25	NUM
iajs-557	321	6	المجلد	المجلد	NOUN
iajs-557	321	7	3	3	NUM
iajs-557	321	8	العدد	العدد	PROPN
iajs-557	321	9	ibn	ibn	PROPN
iajs-557	321	10	al	al	PROPN
iajs-557	321	11	-	-	PUNCT
iajs-557	321	12	haitham	haitham	PROPN
iajs-557	321	13	journal	journal	PROPN
iajs-557	321	14	for	for	ADP
iajs-557	321	15	pure	pure	ADJ
iajs-557	321	16	and	and	CCONJ
iajs-557	321	17	applied	apply	VERB
iajs-557	321	18	science	science	NOUN
iajs-557	321	19	no	no	NOUN
iajs-557	321	20	.	.	NOUN
iajs-557	321	21	3	3	NUM
iajs-557	321	22	vol	vol	NOUN
iajs-557	321	23	.	.	PUNCT
iajs-557	322	1	25	25	NUM
iajs-557	322	2	year	year	NOUN
iajs-557	322	3	2012	2012	NUM
iajs-557	322	4	proof	proof	NOUN
iajs-557	322	5	:	:	PUNCT
iajs-557	322	6	since	since	SCONJ
iajs-557	322	7	m	m	PROPN
iajs-557	322	8	is	be	AUX
iajs-557	322	9	a	a	DET
iajs-557	322	10	scalar	scalar	ADJ
iajs-557	322	11	r	r	NOUN
iajs-557	322	12	-	-	PUNCT
iajs-557	322	13	module	module	NOUN
iajs-557	322	14	,	,	PUNCT
iajs-557	322	15	then	then	ADV
iajs-557	322	16	endr(m	endr(m	PROPN
iajs-557	322	17	)	)	PUNCT
iajs-557	322	18	≃	≃	VERB
iajs-557	322	19	r/	r/	ADV
iajs-557	322	20	r	r	PROPN
iajs-557	322	21	ann	ann	PROPN
iajs-557	322	22	m	m	PROPN
iajs-557	322	23	,	,	PUNCT
iajs-557	322	24	see	see	VERB
iajs-557	322	25	[	[	X
iajs-557	322	26	12	12	NUM
iajs-557	322	27	,	,	PUNCT
iajs-557	322	28	lemma	lemma	PROPN
iajs-557	322	29	3.6.1	3.6.1	NUM
iajs-557	322	30	]	]	PUNCT
iajs-557	322	31	.	.	PUNCT
iajs-557	323	1	since	since	SCONJ
iajs-557	323	2	r	r	NOUN
iajs-557	323	3	is	be	AUX
iajs-557	323	4	sh	sh	NOUN
iajs-557	323	5	-	-	PUNCT
iajs-557	323	6	ring	ring	NOUN
iajs-557	323	7	,	,	PUNCT
iajs-557	323	8	then	then	ADV
iajs-557	323	9	r/	r/	ADV
iajs-557	323	10	r	r	PROPN
iajs-557	323	11	ann	ann	PROPN
iajs-557	323	12	m	m	PROPN
iajs-557	323	13	is	be	AUX
iajs-557	323	14	sh	sh	NOUN
iajs-557	323	15	-	-	PUNCT
iajs-557	323	16	ring	ring	NOUN
iajs-557	323	17	by	by	ADP
iajs-557	323	18	corollary	corollary	ADJ
iajs-557	323	19	1.21	1.21	NUM
iajs-557	323	20	.	.	PUNCT
iajs-557	324	1	thus	thus	ADV
iajs-557	324	2	endr(m	endr(m	VERB
iajs-557	324	3	)	)	PUNCT
iajs-557	324	4	is	be	AUX
iajs-557	324	5	sh	sh	NOUN
iajs-557	324	6	-	-	PUNCT
iajs-557	324	7	ring	ring	NOUN
iajs-557	324	8	by	by	ADP
iajs-557	324	9	corollary	corollary	ADJ
iajs-557	324	10	1.22	1.22	NUM
iajs-557	324	11	.	.	PUNCT
iajs-557	325	1	corollary	corollary	NOUN
iajs-557	325	2	:	:	PUNCT
iajs-557	325	3	let	let	VERB
iajs-557	325	4	m	m	PRON
iajs-557	325	5	be	be	AUX
iajs-557	325	6	a	a	DET
iajs-557	325	7	finitely	finitely	ADV
iajs-557	325	8	generated	generate	VERB
iajs-557	325	9	multiplication	multiplication	NOUN
iajs-557	325	10	module	module	NOUN
iajs-557	325	11	over	over	ADP
iajs-557	325	12	sh	sh	NOUN
iajs-557	325	13	-	-	PUNCT
iajs-557	325	14	ring	ring	NOUN
iajs-557	325	15	.	.	PUNCT
iajs-557	326	1	then	then	ADV
iajs-557	326	2	endr(m	endr(m	VERB
iajs-557	326	3	)	)	PUNCT
iajs-557	326	4	is	be	AUX
iajs-557	326	5	shring	shre	VERB
iajs-557	326	6	.	.	PUNCT
iajs-557	327	1	proof	proof	NOUN
iajs-557	327	2	:	:	PUNCT
iajs-557	327	3	since	since	SCONJ
iajs-557	327	4	m	m	PROPN
iajs-557	327	5	is	be	AUX
iajs-557	327	6	finitely	finitely	ADV
iajs-557	327	7	generated	generate	VERB
iajs-557	327	8	multiplication	multiplication	NOUN
iajs-557	327	9	,	,	PUNCT
iajs-557	327	10	then	then	ADV
iajs-557	327	11	m	m	VERB
iajs-557	327	12	is	be	AUX
iajs-557	327	13	scalar	scalar	ADJ
iajs-557	327	14	,	,	PUNCT
iajs-557	327	15	see	see	VERB
iajs-557	327	16	[	[	X
iajs-557	327	17	11	11	NUM
iajs-557	327	18	]	]	PUNCT
iajs-557	327	19	.	.	PUNCT
iajs-557	328	1	hence	hence	ADV
iajs-557	328	2	the	the	DET
iajs-557	328	3	result	result	NOUN
iajs-557	328	4	is	be	AUX
iajs-557	328	5	obtained	obtain	VERB
iajs-557	328	6	by	by	ADP
iajs-557	328	7	proposition	proposition	NOUN
iajs-557	328	8	3.1	3.1	NUM
iajs-557	328	9	.	.	PUNCT
iajs-557	329	1	next	next	ADV
iajs-557	329	2	we	we	PRON
iajs-557	329	3	shall	shall	AUX
iajs-557	329	4	prove	prove	VERB
iajs-557	329	5	in	in	ADP
iajs-557	329	6	the	the	DET
iajs-557	329	7	class	class	NOUN
iajs-557	329	8	of	of	ADP
iajs-557	329	9	comultiplication	comultiplication	NOUN
iajs-557	329	10	prime	prime	ADJ
iajs-557	329	11	modules	module	NOUN
iajs-557	329	12	,	,	PUNCT
iajs-557	329	13	every	every	DET
iajs-557	329	14	submodule	submodule	NOUN
iajs-557	329	15	of	of	ADP
iajs-557	329	16	m	m	PROPN
iajs-557	329	17	is	be	AUX
iajs-557	329	18	sh	sh	INTJ
iajs-557	329	19	(	(	PUNCT
iajs-557	329	20	qh)-module	qh)-module	ADV
iajs-557	329	21	.	.	PUNCT
iajs-557	330	1	but	but	CCONJ
iajs-557	330	2	first	first	ADV
iajs-557	330	3	we	we	PRON
iajs-557	330	4	prove	prove	VERB
iajs-557	330	5	the	the	DET
iajs-557	330	6	following	follow	VERB
iajs-557	330	7	proposition	proposition	NOUN
iajs-557	330	8	and	and	CCONJ
iajs-557	330	9	lemma	lemma	PROPN
iajs-557	330	10	.	.	PUNCT
iajs-557	331	1	proposition	proposition	NOUN
iajs-557	331	2	:	:	PUNCT
iajs-557	331	3	let	let	VERB
iajs-557	331	4	m	m	PRON
iajs-557	331	5	be	be	AUX
iajs-557	331	6	a	a	DET
iajs-557	331	7	comultiplication	comultiplication	NOUN
iajs-557	331	8	r	r	NOUN
iajs-557	331	9	-	-	NOUN
iajs-557	331	10	module	module	NOUN
iajs-557	331	11	.	.	PUNCT
iajs-557	332	1	if	if	SCONJ
iajs-557	332	2	m	m	NOUN
iajs-557	332	3	is	be	AUX
iajs-557	332	4	prime	prime	ADJ
iajs-557	332	5	(	(	PUNCT
iajs-557	332	6	quasi	quasi	ADJ
iajs-557	332	7	-	-	ADJ
iajs-557	332	8	prime	prime	ADJ
iajs-557	332	9	or	or	CCONJ
iajs-557	332	10	second	second	ADJ
iajs-557	332	11	)	)	PUNCT
iajs-557	332	12	.	.	PUNCT
iajs-557	333	1	then	then	ADV
iajs-557	333	2	m	m	VERB
iajs-557	333	3	is	be	AUX
iajs-557	333	4	hollow	hollow	ADJ
iajs-557	333	5	.	.	PUNCT
iajs-557	334	1	proof	proof	NOUN
iajs-557	334	2	:	:	PUNCT
iajs-557	334	3	since	since	SCONJ
iajs-557	334	4	m	m	PROPN
iajs-557	334	5	is	be	AUX
iajs-557	334	6	prime	prime	ADJ
iajs-557	334	7	(	(	PUNCT
iajs-557	334	8	quasi	quasi	ADJ
iajs-557	334	9	-	-	ADJ
iajs-557	334	10	prime	prime	ADJ
iajs-557	334	11	or	or	CCONJ
iajs-557	334	12	second	second	ADJ
iajs-557	334	13	)	)	PUNCT
iajs-557	334	14	,	,	PUNCT
iajs-557	334	15	then	then	ADV
iajs-557	334	16	r	r	PROPN
iajs-557	334	17	ann	ann	PROPN
iajs-557	334	18	m	m	NOUN
iajs-557	334	19	is	be	AUX
iajs-557	334	20	prime	prime	ADJ
iajs-557	334	21	,	,	PUNCT
iajs-557	334	22	see	see	VERB
iajs-557	334	23	[	[	X
iajs-557	334	24	8	8	NUM
iajs-557	334	25	]	]	PUNCT
iajs-557	334	26	,	,	PUNCT
iajs-557	335	1	[	[	X
iajs-557	335	2	9	9	NUM
iajs-557	335	3	]	]	PUNCT
iajs-557	335	4	,	,	PUNCT
iajs-557	335	5	[	[	X
iajs-557	335	6	6	6	NUM
iajs-557	335	7	]	]	PUNCT
iajs-557	335	8	.	.	PUNCT
iajs-557	336	1	and	and	CCONJ
iajs-557	336	2	endr(m	endr(m	VERB
iajs-557	336	3	)	)	PUNCT
iajs-557	336	4	is	be	AUX
iajs-557	336	5	domain	domain	NOUN
iajs-557	336	6	,	,	PUNCT
iajs-557	336	7	see	see	VERB
iajs-557	336	8	[	[	X
iajs-557	336	9	5	5	NUM
iajs-557	336	10	,	,	PUNCT
iajs-557	336	11	corollary	corollary	ADJ
iajs-557	336	12	3.21	3.21	NUM
iajs-557	336	13	]	]	PUNCT
iajs-557	336	14	.	.	PUNCT
iajs-557	337	1	hence	hence	ADV
iajs-557	337	2	m	m	PROPN
iajs-557	337	3	is	be	AUX
iajs-557	337	4	hollow	hollow	ADJ
iajs-557	337	5	,	,	PUNCT
iajs-557	337	6	see	see	VERB
iajs-557	337	7	[	[	X
iajs-557	337	8	5	5	NUM
iajs-557	337	9	,	,	PUNCT
iajs-557	337	10	theorem	theorem	VERB
iajs-557	337	11	3.24	3.24	NUM
iajs-557	337	12	]	]	PUNCT
iajs-557	337	13	.	.	PUNCT
iajs-557	338	1	lemma	lemma	PROPN
iajs-557	338	2	:	:	PUNCT
iajs-557	338	3	let	let	VERB
iajs-557	338	4	m	m	PRON
iajs-557	338	5	be	be	AUX
iajs-557	338	6	a	a	DET
iajs-557	338	7	comultiplication	comultiplication	NOUN
iajs-557	338	8	r	r	NOUN
iajs-557	338	9	-	-	PUNCT
iajs-557	338	10	module	module	NOUN
iajs-557	338	11	and	and	CCONJ
iajs-557	338	12	n	n	PRON
iajs-557	338	13	≤	≤	NOUN
iajs-557	338	14	m.	m.	NOUN
iajs-557	338	15	then	then	ADV
iajs-557	338	16	n	n	PRON
iajs-557	338	17	is	be	AUX
iajs-557	338	18	a	a	DET
iajs-557	338	19	comultiplication	comultiplication	NOUN
iajs-557	338	20	rmodule	rmodule	NOUN
iajs-557	338	21	.	.	PUNCT
iajs-557	339	1	proof	proof	NOUN
iajs-557	339	2	:	:	PUNCT
iajs-557	339	3	let	let	VERB
iajs-557	339	4	w	w	NOUN
iajs-557	339	5	≤	≤	X
iajs-557	339	6	n.	n.	NOUN
iajs-557	340	1	so	so	ADV
iajs-557	340	2	w	w	PROPN
iajs-557	340	3	is	be	AUX
iajs-557	340	4	a	a	DET
iajs-557	340	5	submodule	submodule	NOUN
iajs-557	340	6	of	of	ADP
iajs-557	340	7	m.	m.	NOUN
iajs-557	340	8	then	then	ADV
iajs-557	340	9	there	there	PRON
iajs-557	340	10	exists	exist	VERB
iajs-557	340	11	i	i	PRON
iajs-557	340	12	≤	≤	NOUN
iajs-557	340	13	r	r	NOUN
iajs-557	340	14	such	such	ADJ
iajs-557	340	15	that	that	DET
iajs-557	340	16	w	w	PROPN
iajs-557	340	17	=	=	SYM
iajs-557	340	18	ann	ann	PROPN
iajs-557	340	19	μ	μ	PROPN
iajs-557	340	20	i.	i.	PROPN
iajs-557	340	21	we	we	PRON
iajs-557	340	22	claim	claim	VERB
iajs-557	340	23	that	that	SCONJ
iajs-557	340	24	w	w	PROPN
iajs-557	340	25	=	=	SYM
iajs-557	340	26	n	n	PROPN
iajs-557	340	27	ann	ann	PROPN
iajs-557	340	28	i.	i.	PROPN
iajs-557	340	29	to	to	PART
iajs-557	340	30	prove	prove	VERB
iajs-557	340	31	our	our	PRON
iajs-557	340	32	assertion	assertion	NOUN
iajs-557	340	33	.	.	PUNCT
iajs-557	341	1	let	let	VERB
iajs-557	341	2	m	m	PRON
iajs-557	341	3	∈	∈	VERB
iajs-557	341	4	w	w	NOUN
iajs-557	341	5	(	(	PUNCT
iajs-557	341	6	so	so	ADV
iajs-557	341	7	,	,	PUNCT
iajs-557	341	8	m	m	VERB
iajs-557	341	9	∈	∈	PROPN
iajs-557	341	10	n	n	CCONJ
iajs-557	341	11	)	)	PUNCT
iajs-557	341	12	.	.	PUNCT
iajs-557	342	1	hence	hence	ADV
iajs-557	342	2	mi	mi	PROPN
iajs-557	342	3	=	=	SYM
iajs-557	342	4	0	0	PROPN
iajs-557	342	5	,	,	PUNCT
iajs-557	342	6	so	so	SCONJ
iajs-557	342	7	m	m	VERB
iajs-557	342	8	∈	∈	PROPN
iajs-557	342	9	n	n	PRON
iajs-557	342	10	ann	ann	PROPN
iajs-557	342	11	i.	i.	PROPN
iajs-557	342	12	now	now	ADV
iajs-557	342	13	let	let	VERB
iajs-557	342	14	m	m	PRON
iajs-557	342	15	∈	∈	PROPN
iajs-557	342	16	n	n	PRON
iajs-557	342	17	ann	ann	PROPN
iajs-557	342	18	i	i	PROPN
iajs-557	342	19	,	,	PUNCT
iajs-557	342	20	so	so	ADV
iajs-557	342	21	m	m	VERB
iajs-557	342	22	∈	∈	PROPN
iajs-557	342	23	n	n	NOUN
iajs-557	342	24	and	and	CCONJ
iajs-557	342	25	mi	mi	PROPN
iajs-557	342	26	=	=	SYM
iajs-557	342	27	0	0	PROPN
iajs-557	342	28	,	,	PUNCT
iajs-557	342	29	m	m	PROPN
iajs-557	342	30	∈	∈	NOUN
iajs-557	342	31	m.	m.	NOUN
iajs-557	342	32	thus	thus	ADV
iajs-557	342	33	m	m	ADP
iajs-557	342	34	∈	∈	PROPN
iajs-557	342	35	ann	ann	PROPN
iajs-557	342	36	μ	μ	PROPN
iajs-557	342	37	i.	i.	PROPN
iajs-557	342	38	then	then	ADV
iajs-557	342	39	w	w	PROPN
iajs-557	342	40	=	=	SYM
iajs-557	342	41	n	n	PROPN
iajs-557	342	42	ann	ann	PROPN
iajs-557	342	43	i.	i.	PROPN
iajs-557	342	44	henc	henc	PROPN
iajs-557	342	45	n	n	PROPN
iajs-557	342	46	is	be	AUX
iajs-557	342	47	comultiplication	comultiplication	NOUN
iajs-557	342	48	.	.	PUNCT
iajs-557	343	1	theorem	theorem	VERB
iajs-557	343	2	:	:	PUNCT
iajs-557	343	3	let	let	VERB
iajs-557	343	4	m	m	PRON
iajs-557	343	5	be	be	AUX
iajs-557	343	6	a	a	DET
iajs-557	343	7	comultiplication	comultiplication	NOUN
iajs-557	343	8	prime	prime	ADJ
iajs-557	343	9	r	r	NOUN
iajs-557	343	10	-	-	PUNCT
iajs-557	343	11	module	module	NOUN
iajs-557	343	12	.	.	PUNCT
iajs-557	344	1	then	then	ADV
iajs-557	344	2	every	every	DET
iajs-557	344	3	non	non	ADJ
iajs-557	344	4	-	-	ADJ
iajs-557	344	5	zero	zero	NUM
iajs-557	344	6	submodule	submodule	NOUN
iajs-557	344	7	of	of	ADP
iajs-557	344	8	m	m	PROPN
iajs-557	344	9	is	be	AUX
iajs-557	344	10	a	a	DET
iajs-557	344	11	sh(qh	sh(qh	NOUN
iajs-557	344	12	)	)	PUNCT
iajs-557	344	13	r	r	NOUN
iajs-557	344	14	-	-	PUNCT
iajs-557	344	15	module	module	NOUN
iajs-557	344	16	.	.	PUNCT
iajs-557	345	1	proof	proof	NOUN
iajs-557	345	2	:	:	PUNCT
iajs-557	345	3	since	since	SCONJ
iajs-557	345	4	m	m	PROPN
iajs-557	345	5	is	be	AUX
iajs-557	345	6	comultiplication	comultiplication	NOUN
iajs-557	345	7	prime	prime	NOUN
iajs-557	345	8	,	,	PUNCT
iajs-557	345	9	then	then	ADV
iajs-557	345	10	by	by	ADP
iajs-557	345	11	proposition	proposition	NOUN
iajs-557	345	12	3.3	3.3	NUM
iajs-557	345	13	,	,	PUNCT
iajs-557	345	14	m	m	VERB
iajs-557	345	15	is	be	AUX
iajs-557	345	16	hollow	hollow	ADJ
iajs-557	345	17	.	.	PUNCT
iajs-557	346	1	let	let	VERB
iajs-557	346	2	n	n	PRON
iajs-557	346	3	be	be	AUX
iajs-557	346	4	a	a	DET
iajs-557	346	5	nonzero	nonzero	ADJ
iajs-557	346	6	submodule	submodule	NOUN
iajs-557	346	7	of	of	ADP
iajs-557	346	8	m	m	PROPN
iajs-557	346	9	,	,	PUNCT
iajs-557	346	10	then	then	ADV
iajs-557	346	11	n	n	PROPN
iajs-557	346	12	is	be	AUX
iajs-557	346	13	comultiplication	comultiplication	NOUN
iajs-557	346	14	by	by	ADP
iajs-557	346	15	lemma	lemma	PROPN
iajs-557	346	16	3.4	3.4	NUM
iajs-557	346	17	.	.	PUNCT
iajs-557	347	1	but	but	CCONJ
iajs-557	347	2	m	m	PROPN
iajs-557	347	3	is	be	AUX
iajs-557	347	4	prime	prime	ADJ
iajs-557	347	5	implies	implie	NOUN
iajs-557	347	6	n	n	AUX
iajs-557	347	7	is	be	AUX
iajs-557	347	8	a	a	DET
iajs-557	347	9	prime	prime	ADJ
iajs-557	347	10	r	r	NOUN
iajs-557	347	11	-	-	PUNCT
iajs-557	347	12	module	module	NOUN
iajs-557	347	13	.	.	PUNCT
iajs-557	348	1	thus	thus	ADV
iajs-557	348	2	n	n	PRON
iajs-557	348	3	is	be	AUX
iajs-557	348	4	a	a	DET
iajs-557	348	5	hollow	hollow	ADJ
iajs-557	348	6	r	r	NOUN
iajs-557	348	7	-	-	PUNCT
iajs-557	348	8	module	module	NOUN
iajs-557	348	9	by	by	ADP
iajs-557	348	10	proposition	proposition	NOUN
iajs-557	348	11	3.3	3.3	NUM
iajs-557	348	12	.	.	PUNCT
iajs-557	349	1	hence	hence	ADV
iajs-557	349	2	n	n	ADV
iajs-557	349	3	is	be	AUX
iajs-557	349	4	qh(sh)-rmodule	qh(sh)-rmodule	ADJ
iajs-557	349	5	,	,	PUNCT
iajs-557	349	6	see	see	VERB
iajs-557	349	7	remark	remark	NOUN
iajs-557	349	8	1.15(5	1.15(5	NUM
iajs-557	349	9	)	)	PUNCT
iajs-557	349	10	.	.	PUNCT
iajs-557	350	1	corollary	corollary	NOUN
iajs-557	350	2	:	:	PUNCT
iajs-557	350	3	let	let	VERB
iajs-557	350	4	m	m	PRON
iajs-557	350	5	be	be	AUX
iajs-557	350	6	a	a	DET
iajs-557	350	7	comultiplication	comultiplication	NOUN
iajs-557	350	8	prime	prime	ADJ
iajs-557	350	9	r	r	NOUN
iajs-557	350	10	-	-	PUNCT
iajs-557	350	11	module	module	NOUN
iajs-557	350	12	.	.	PUNCT
iajs-557	351	1	then	then	ADV
iajs-557	351	2	every	every	DET
iajs-557	351	3	non	non	ADJ
iajs-557	351	4	-	-	ADJ
iajs-557	351	5	zero	zero	NUM
iajs-557	351	6	submodule	submodule	NOUN
iajs-557	351	7	of	of	ADP
iajs-557	351	8	m	m	PROPN
iajs-557	351	9	is	be	AUX
iajs-557	351	10	qh	qh	NOUN
iajs-557	351	11	-	-	PUNCT
iajs-557	351	12	sumodule	sumodule	NOUN
iajs-557	351	13	of	of	ADP
iajs-557	351	14	m.	m.	NOUN
iajs-557	351	15	references	reference	NOUN
iajs-557	351	16	1	1	NUM
iajs-557	351	17	.	.	PUNCT
iajs-557	352	1	abuhlail	abuhlail	NOUN
iajs-557	352	2	,	,	PUNCT
iajs-557	352	3	j.y	j.y	PROPN
iajs-557	352	4	.	.	PROPN
iajs-557	352	5	(	(	PUNCT
iajs-557	352	6	2011	2011	NUM
iajs-557	352	7	)	)	PUNCT
iajs-557	352	8	zariski	zariski	NOUN
iajs-557	352	9	topologies	topology	NOUN
iajs-557	352	10	for	for	ADP
iajs-557	352	11	coprime	coprime	NOUN
iajs-557	352	12	and	and	CCONJ
iajs-557	352	13	second	second	ADJ
iajs-557	352	14	submodules	submodule	NOUN
iajs-557	352	15	,	,	PUNCT
iajs-557	352	16	deanshib	deanshib	NOUN
iajs-557	352	17	of	of	ADP
iajs-557	352	18	scientific	scientific	ADJ
iajs-557	352	19	research	research	NOUN
iajs-557	352	20	at	at	ADP
iajs-557	352	21	king	king	PROPN
iajs-557	352	22	fahad	fahad	PROPN
iajs-557	352	23	university	university	PROPN
iajs-557	352	24	of	of	ADP
iajs-557	352	25	peroleum	peroleum	NOUN
iajs-557	352	26	and	and	CCONJ
iajs-557	352	27	minerals	mineral	NOUN
iajs-557	352	28	february	february	NOUN
iajs-557	352	29	4	4	NUM
iajs-557	352	30	.	.	NOUN
iajs-557	352	31	2	2	NUM
iajs-557	352	32	.	.	X
iajs-557	352	33	kasch	kasch	PROPN
iajs-557	352	34	,	,	PUNCT
iajs-557	352	35	f.	f.	PROPN
iajs-557	352	36	(	(	PUNCT
iajs-557	352	37	1982	1982	NUM
iajs-557	352	38	)	)	PUNCT
iajs-557	352	39	modules	module	NOUN
iajs-557	352	40	and	and	CCONJ
iajs-557	352	41	rings	ring	NOUN
iajs-557	352	42	,	,	PUNCT
iajs-557	352	43	acadimic	acadimic	PROPN
iajs-557	352	44	press	press	PROPN
iajs-557	352	45	,	,	PUNCT
iajs-557	352	46	london	london	PROPN
iajs-557	352	47	.	.	PUNCT
iajs-557	353	1	3	3	X
iajs-557	353	2	.	.	X
iajs-557	353	3	osofsky	osofsky	ADJ
iajs-557	353	4	,	,	PUNCT
iajs-557	353	5	b.l	b.l	PROPN
iajs-557	353	6	.	.	PROPN
iajs-557	353	7	(	(	PUNCT
iajs-557	353	8	1991	1991	NUM
iajs-557	353	9	)	)	PUNCT
iajs-557	353	10	a	a	DET
iajs-557	353	11	contruction	contruction	NOUN
iajs-557	353	12	of	of	ADP
iajs-557	353	13	non	non	ADJ
iajs-557	353	14	standard	standard	ADJ
iajs-557	353	15	uniserial	uniserial	ADJ
iajs-557	353	16	modules	module	NOUN
iajs-557	353	17	over	over	ADP
iajs-557	353	18	valuation	valuation	NOUN
iajs-557	353	19	domains	domain	NOUN
iajs-557	353	20	,	,	PUNCT
iajs-557	353	21	bulletin	bulletin	NOUN
iajs-557	353	22	amer	amer	NOUN
iajs-557	353	23	.	.	PUNCT
iajs-557	354	1	math.soc	math.soc	X
iajs-557	354	2	.	.	PROPN
iajs-557	354	3	,	,	PUNCT
iajs-557	354	4	25:89	25:89	NUM
iajs-557	354	5	-	-	SYM
iajs-557	354	6	97	97	NUM
iajs-557	354	7	.	.	NOUN
iajs-557	355	1	4	4	NUM
iajs-557	355	2	.	.	X
iajs-557	355	3	barnard	barnard	PROPN
iajs-557	355	4	,	,	PUNCT
iajs-557	355	5	a.	a.	NOUN
iajs-557	355	6	(	(	PUNCT
iajs-557	355	7	1981	1981	NUM
iajs-557	355	8	)	)	PUNCT
iajs-557	355	9	multiplication	multiplication	NOUN
iajs-557	355	10	modules	module	NOUN
iajs-557	355	11	,	,	PUNCT
iajs-557	355	12	j.algebra	j.algebra	PROPN
iajs-557	355	13	,	,	PUNCT
iajs-557	355	14	71:174	71:174	NUM
iajs-557	355	15	-	-	SYM
iajs-557	355	16	178	178	NUM
iajs-557	355	17	.	.	PUNCT
iajs-557	356	1	mathematics	mathematic	NOUN
iajs-557	356	2	402	402	NUM
iajs-557	356	3	مجلة	مجلة	NOUN
iajs-557	356	4	إبن	إبن	VERB
iajs-557	356	5	الهيثم	الهيثم	ADJ
iajs-557	356	6	للعلوم	للعلوم	NOUN
iajs-557	356	7	الصرفة	الصرفة	NOUN
iajs-557	357	1	و	و	PRON
iajs-557	357	2	التطبيقية	التطبيقية	ADJ
iajs-557	357	3	2012	2012	NUM
iajs-557	357	4	السنة	السنة	NOUN
iajs-557	357	5	25	25	NUM
iajs-557	357	6	المجلد	المجلد	NOUN
iajs-557	357	7	3	3	NUM
iajs-557	357	8	العدد	العدد	PROPN
iajs-557	357	9	ibn	ibn	PROPN
iajs-557	357	10	al	al	PROPN
iajs-557	357	11	-	-	PUNCT
iajs-557	357	12	haitham	haitham	PROPN
iajs-557	357	13	journal	journal	PROPN
iajs-557	357	14	for	for	ADP
iajs-557	357	15	pure	pure	ADJ
iajs-557	357	16	and	and	CCONJ
iajs-557	357	17	applied	apply	VERB
iajs-557	357	18	science	science	NOUN
iajs-557	357	19	no	no	NOUN
iajs-557	357	20	.	.	NOUN
iajs-557	357	21	3	3	NUM
iajs-557	357	22	vol	vol	NOUN
iajs-557	357	23	.	.	PUNCT
iajs-557	358	1	25	25	NUM
iajs-557	358	2	year	year	NOUN
iajs-557	358	3	2012	2012	NUM
iajs-557	358	4	5	5	NUM
iajs-557	358	5	.	.	PUNCT
iajs-557	358	6	ansari	ansari	PROPN
iajs-557	358	7	toroghy	toroghy	PROPN
iajs-557	358	8	,	,	PUNCT
iajs-557	358	9	h.	h.	PROPN
iajs-557	358	10	and	and	CCONJ
iajs-557	358	11	farshadifar	farshadifar	PROPN
iajs-557	358	12	,	,	PUNCT
iajs-557	358	13	f.	f.	PROPN
iajs-557	358	14	(	(	PUNCT
iajs-557	358	15	2007	2007	NUM
iajs-557	358	16	)	)	PUNCT
iajs-557	358	17	the	the	DET
iajs-557	358	18	dual	dual	ADJ
iajs-557	358	19	notion	notion	NOUN
iajs-557	358	20	of	of	ADP
iajs-557	358	21	multiplication	multiplication	NOUN
iajs-557	358	22	modules	module	NOUN
iajs-557	358	23	,	,	PUNCT
iajs-557	358	24	taiwanese	taiwanese	ADJ
iajs-557	358	25	j.math	j.math	NOUN
iajs-557	358	26	.	.	PROPN
iajs-557	358	27	,	,	PUNCT
iajs-557	358	28	11(4):1189	11(4):1189	NUM
iajs-557	358	29	-	-	SYM
iajs-557	358	30	1201	1201	NUM
iajs-557	358	31	.	.	PUNCT
iajs-557	359	1	6	6	NUM
iajs-557	359	2	.	.	X
iajs-557	359	3	yassemi	yassemi	NOUN
iajs-557	359	4	,	,	PUNCT
iajs-557	359	5	s.	s.	PROPN
iajs-557	359	6	(	(	PUNCT
iajs-557	359	7	2001	2001	NUM
iajs-557	359	8	)	)	PUNCT
iajs-557	359	9	the	the	DET
iajs-557	359	10	dual	dual	ADJ
iajs-557	359	11	notion	notion	NOUN
iajs-557	359	12	of	of	ADP
iajs-557	359	13	prime	prime	ADJ
iajs-557	359	14	submodules	submodule	NOUN
iajs-557	359	15	,	,	PUNCT
iajs-557	359	16	arch.math	arch.math	NOUN
iajs-557	359	17	.	.	PUNCT
iajs-557	360	1	(	(	PUNCT
iajs-557	360	2	brno	brno	NOUN
iajs-557	360	3	)	)	PUNCT
iajs-557	360	4	37:273	37:273	NUM
iajs-557	360	5	-	-	SYM
iajs-557	360	6	278	278	NUM
iajs-557	360	7	.	.	PUNCT
iajs-557	361	1	7	7	X
iajs-557	361	2	.	.	X
iajs-557	361	3	bourbaki	bourbaki	NOUN
iajs-557	361	4	,	,	PUNCT
iajs-557	361	5	n.	n.	PROPN
iajs-557	361	6	(	(	PUNCT
iajs-557	361	7	1998	1998	NUM
iajs-557	361	8	)	)	PUNCT
iajs-557	361	9	commutative	commutative	ADJ
iajs-557	361	10	algebra	algebra	NOUN
iajs-557	361	11	,	,	PUNCT
iajs-557	361	12	springer	springer	NOUN
iajs-557	361	13	-	-	PUNCT
iajs-557	361	14	verlag	verlag	PROPN
iajs-557	361	15	.	.	PUNCT
iajs-557	362	1	8	8	NUM
iajs-557	362	2	.	.	PUNCT
iajs-557	362	3	desale	desale	NOUN
iajs-557	362	4	,	,	PUNCT
iajs-557	362	5	g.	g.	PROPN
iajs-557	362	6	and	and	CCONJ
iajs-557	362	7	nicholson	nicholson	PROPN
iajs-557	362	8	,	,	PUNCT
iajs-557	362	9	w.k	w.k	PROPN
iajs-557	362	10	.	.	PROPN
iajs-557	362	11	(	(	PUNCT
iajs-557	362	12	1981	1981	NUM
iajs-557	362	13	)	)	PUNCT
iajs-557	362	14	endoprimitive	endoprimitive	ADJ
iajs-557	362	15	rings	ring	NOUN
iajs-557	362	16	,	,	PUNCT
iajs-557	362	17	j.algebra	j.algebra	PROPN
iajs-557	362	18	,	,	PUNCT
iajs-557	362	19	7:546	7:546	NUM
iajs-557	362	20	-	-	SYM
iajs-557	362	21	560	560	NUM
iajs-557	362	22	.	.	NOUN
iajs-557	362	23	9	9	NUM
iajs-557	362	24	.	.	NUM
iajs-557	362	25	muntaha	muntaha	NOUN
iajs-557	362	26	.	.	PUNCT
iajs-557	363	1	a.al	a.al	PROPN
iajs-557	363	2	.	.	PROPN
iajs-557	363	3	(	(	PUNCT
iajs-557	363	4	1999	1999	NUM
iajs-557	363	5	)	)	PUNCT
iajs-557	363	6	quasi	quasi	ADJ
iajs-557	363	7	-	-	ADJ
iajs-557	363	8	prime	prime	ADJ
iajs-557	363	9	modules	module	NOUN
iajs-557	363	10	and	and	CCONJ
iajs-557	363	11	quasi	quasi	ADJ
iajs-557	363	12	-	-	ADJ
iajs-557	363	13	prime	prime	ADJ
iajs-557	363	14	submodules	submodule	NOUN
iajs-557	363	15	,	,	PUNCT
iajs-557	363	16	m.sc	m.sc	PROPN
iajs-557	364	1	.	.	PUNCT
iajs-557	364	2	thesis	thesis	NOUN
iajs-557	364	3	,	,	PUNCT
iajs-557	364	4	college	college	NOUN
iajs-557	364	5	of	of	ADP
iajs-557	364	6	education	education	PROPN
iajs-557	364	7	ibn	ibn	PROPN
iajs-557	364	8	-	-	PUNCT
iajs-557	364	9	al	al	PROPN
iajs-557	364	10	-	-	PUNCT
iajs-557	364	11	haitham	haitham	PROPN
iajs-557	364	12	,	,	PUNCT
iajs-557	364	13	university	university	NOUN
iajs-557	364	14	of	of	ADP
iajs-557	364	15	baghdad	baghdad	PROPN
iajs-557	364	16	.	.	PUNCT
iajs-557	365	1	10	10	NUM
iajs-557	365	2	.	.	X
iajs-557	366	1	abd	abd	PROPN
iajs-557	366	2	el	el	PROPN
iajs-557	366	3	-	-	PUNCT
iajs-557	366	4	bast	bast	NOUN
iajs-557	366	5	,	,	PUNCT
iajs-557	366	6	z.	z.	PROPN
iajs-557	366	7	and	and	CCONJ
iajs-557	366	8	simith	simith	PROPN
iajs-557	366	9	,	,	PUNCT
iajs-557	366	10	p.f	p.f	PROPN
iajs-557	366	11	.	.	PROPN
iajs-557	366	12	(	(	PUNCT
iajs-557	366	13	1988	1988	NUM
iajs-557	366	14	)	)	PUNCT
iajs-557	366	15	multiplication	multiplication	NOUN
iajs-557	366	16	modules	module	NOUN
iajs-557	366	17	,	,	PUNCT
iajs-557	366	18	commun.algebra	commun.algebra	PROPN
iajs-557	366	19	,	,	PUNCT
iajs-557	366	20	16	16	NUM
iajs-557	366	21	(	(	PUNCT
iajs-557	366	22	4):755	4):755	PROPN
iajs-557	366	23	-	-	PUNCT
iajs-557	366	24	779	779	NUM
iajs-557	366	25	.	.	PROPN
iajs-557	367	1	11	11	NUM
iajs-557	367	2	.	.	X
iajs-557	367	3	najad	najad	PROPN
iajs-557	367	4	,	,	PUNCT
iajs-557	367	5	b.	b.	PROPN
iajs-557	367	6	(	(	PUNCT
iajs-557	367	7	2004	2004	NUM
iajs-557	367	8	)	)	PUNCT
iajs-557	367	9	scalar	scalar	ADJ
iajs-557	367	10	reflexive	reflexive	ADJ
iajs-557	367	11	modules	module	NOUN
iajs-557	367	12	,	,	PUNCT
iajs-557	367	13	ph.d	ph.d	PROPN
iajs-557	367	14	.	.	PUNCT
iajs-557	368	1	thesis	thesis	PROPN
iajs-557	368	2	,	,	PUNCT
iajs-557	368	3	college	college	NOUN
iajs-557	368	4	of	of	ADP
iajs-557	368	5	science	science	NOUN
iajs-557	368	6	,	,	PUNCT
iajs-557	368	7	university	university	NOUN
iajs-557	368	8	of	of	ADP
iajs-557	368	9	baghdad	baghdad	PROPN
iajs-557	368	10	.	.	PUNCT
iajs-557	369	1	12	12	NUM
iajs-557	369	2	.	.	PUNCT
iajs-557	370	1	eman	eman	PROPN
iajs-557	370	2	,	,	PUNCT
iajs-557	370	3	a.	a.	PROPN
iajs-557	370	4	(	(	PUNCT
iajs-557	370	5	2006	2006	NUM
iajs-557	370	6	)	)	PUNCT
iajs-557	370	7	on	on	ADP
iajs-557	370	8	ikeda	ikeda	NOUN
iajs-557	370	9	-	-	PUNCT
iajs-557	370	10	nakayama	nakayama	PROPN
iajs-557	370	11	modules	module	NOUN
iajs-557	370	12	,	,	PUNCT
iajs-557	370	13	ph.d	ph.d	PROPN
iajs-557	370	14	.	.	PUNCT
iajs-557	371	1	thesis	thesis	PROPN
iajs-557	371	2	,	,	PUNCT
iajs-557	371	3	college	college	NOUN
iajs-557	371	4	of	of	ADP
iajs-557	371	5	education	education	NOUN
iajs-557	371	6	ibnal	ibnal	ADJ
iajs-557	371	7	-	-	PUNCT
iajs-557	371	8	haitham	haitham	PROPN
iajs-557	371	9	,	,	PUNCT
iajs-557	371	10	university	university	NOUN
iajs-557	371	11	of	of	ADP
iajs-557	371	12	baghdad	baghdad	PROPN
iajs-557	371	13	.	.	PUNCT
iajs-557	372	1	mathematics	mathematic	NOUN
iajs-557	372	2	403	403	NUM
iajs-557	372	3	مجلة	مجلة	VERB
iajs-557	372	4	إبن	إبن	NOUN
iajs-557	372	5	الهيثم	الهيثم	ADJ
iajs-557	372	6	للعلوم	للعلوم	NOUN
iajs-557	372	7	الصرفة	الصرفة	NOUN
iajs-557	373	1	و	و	PRON
iajs-557	373	2	التطبيقية	التطبيقية	ADJ
iajs-557	373	3	2012	2012	NUM
iajs-557	373	4	السنة	السنة	NOUN
iajs-557	373	5	25	25	NUM
iajs-557	373	6	المجلد	المجلد	NOUN
iajs-557	373	7	3	3	NUM
iajs-557	373	8	العدد	العدد	PROPN
iajs-557	373	9	ibn	ibn	PROPN
iajs-557	373	10	al	al	PROPN
iajs-557	373	11	-	-	PUNCT
iajs-557	373	12	haitham	haitham	PROPN
iajs-557	373	13	journal	journal	PROPN
iajs-557	373	14	for	for	ADP
iajs-557	373	15	pure	pure	ADJ
iajs-557	373	16	and	and	CCONJ
iajs-557	373	17	applied	apply	VERB
iajs-557	373	18	science	science	NOUN
iajs-557	373	19	no	no	NOUN
iajs-557	373	20	.	.	NOUN
iajs-557	373	21	3	3	NUM
iajs-557	373	22	vol	vol	NOUN
iajs-557	373	23	.	.	PUNCT
iajs-557	374	1	25	25	NUM
iajs-557	374	2	year	year	NOUN
iajs-557	374	3	2012	2012	NUM
iajs-557	374	4	iالمقاسات	iالمقاسات	PROPN
iajs-557	374	5	الجزئية	الجزئية	VERB
iajs-557	374	6	المجوفة	المجوفة	PROPN
iajs-557	374	7	(	(	PUNCT
iajs-557	374	8	التامة	التامة	NOUN
iajs-557	374	9	)	)	PUNCT
iajs-557	374	10	بقوة	بقوة	ADJ
iajs-557	374	11	غالب	غالب	ADJ
iajs-557	374	12	أحمد	أحمد	NOUN
iajs-557	374	13	حمود	حمود	VERB
iajs-557	374	14	هادي	هادي	NOUN
iajs-557	374	15	،	،	X
iajs-557	374	16	انعام	انعام	PROPN
iajs-557	374	17	محمد	محمد	PROPN
iajs-557	374	18	علي	علي	NOUN
iajs-557	374	19	،	،	PROPN
iajs-557	374	20	جامعة	جامعة	PROPN
iajs-557	374	21	بغداد	بغداد	PROPN
iajs-557	374	22	ابن	ابن	PROPN
iajs-557	374	23	الهيثم	الهيثم	PROPN
iajs-557	374	24	-قسم	-قسم	PUNCT
iajs-557	374	25	الرياضيات	الرياضيات	PROPN
iajs-557	374	26	،	،	PROPN
iajs-557	374	27	كلية	كلية	PROPN
iajs-557	374	28	التربية	التربية	PROPN
iajs-557	374	29	2012ايار	2012ايار	PROPN
iajs-557	374	30	21قبل	21قبل	NUM
iajs-557	374	31	البحث	البحث	NOUN
iajs-557	374	32	في	في	ADP
iajs-557	374	33	:	:	PUNCT
iajs-557	374	34	2012اذار	2012اذار	NUM
iajs-557	374	35	15في	15في	NOUN
iajs-557	374	36	:	:	PUNCT
iajs-557	374	37	استلم	استلم	PROPN
iajs-557	374	38	البحث	البحث	PROPN
iajs-557	374	39	الخالصة	الخالصة	PROPN
iajs-557	374	40	.	.	PUNCT
iajs-557	375	1	في	في	PRON
iajs-557	375	2	هذا	هذا	NOUN
iajs-557	375	3	البحث	البحث	PROPN
iajs-557	375	4	درسنا	درسنا	PROPN
iajs-557	375	5	المفاهيم	المفاهيم	PROPN
iajs-557	375	6	:	:	PUNCT
iajs-557	375	7	المقاسات	المقاسات	PROPN
iajs-557	375	8	الجزئية	الجزئية	VERB
iajs-557	375	9	rاسا	rاسا	NOUN
iajs-557	375	10	ً	ً	PROPN
iajs-557	375	11	على	على	NOUN
iajs-557	375	12	مق	مق	PRON
iajs-557	375	13	mحلقة	mحلقة	NOUN
iajs-557	375	14	ابدالية	ابدالية	NOUN
iajs-557	375	15	ذا	ذا	PRON
iajs-557	375	16	محايد	محايد	PROPN
iajs-557	375	17	.	.	PUNCT
iajs-557	376	1	وليكن	وليكن	PROPN
iajs-557	376	2	rلتكن	rلتكن	PROPN
iajs-557	376	3	المجوفة	المجوفة	PROPN
iajs-557	376	4	بقوة	بقوة	PROPN
iajs-557	376	5	(	(	PUNCT
iajs-557	376	6	التامة	التامة	NOUN
iajs-557	376	7	)	)	PUNCT
iajs-557	376	8	والمقاسات	والمقاسات	PROPN
iajs-557	376	9	الجزئية	الجزئية	PROPN
iajs-557	376	10	شبه	شبه	VERB
iajs-557	376	11	المجوفة	المجوفة	PROPN
iajs-557	376	12	وقدمنا	وقدمنا	PROPN
iajs-557	377	1	الخواص	الخواص	INTJ
iajs-557	377	2	المتعلقة	المتعلقة	VERB
iajs-557	377	3	بهم	بهم	PROPN
iajs-557	377	4	والعالقات	والعالقات	PROPN
iajs-557	377	5	فيما	فيما	PROPN
iajs-557	377	6	بينهم	بينهم	PROPN
iajs-557	377	7	.	.	PUNCT
iajs-557	378	1	كذلك	كذلك	PROPN
iajs-557	378	2	درسنا	درسنا	PROPN
iajs-557	378	3	لمقاسات	لمقاسات	PROPN
iajs-557	378	4	التوزيعية	التوزيعية	PROPN
iajs-557	378	5	،	،	PROPN
iajs-557	378	6	والمقاسات	والمقاسات	PROPN
iajs-557	378	7	الجدائية	الجدائية	PROPN
iajs-557	378	8	المضادة	المضادة	PROPN
iajs-557	378	9	،	،	PROPN
iajs-557	378	10	سلوك	سلوك	PROPN
iajs-557	378	11	هذه	هذه	PROPN
iajs-557	378	12	المقاسات	المقاسات	PROPN
iajs-557	378	13	الجزئية	الجزئية	VERB
iajs-557	378	14	في	في	DET
iajs-557	378	15	أصناف	أصناف	NOUN
iajs-557	378	16	معينة	معينة	NOUN
iajs-557	378	17	من	من	ADP
iajs-557	378	18	المقاسات	المقاسات	PROPN
iajs-557	378	19	،	،	PROPN
iajs-557	378	20	مثل	مثل	PROPN
iajs-557	378	21	ا	ا	PROPN
iajs-557	379	1	والمقاسات	والمقاسات	PROPN
iajs-557	379	2	الجدائية	الجدائية	PROPN
iajs-557	379	3	والمقاسات	والمقاسات	PROPN
iajs-557	379	4	القياسية	القياسية	PROPN
iajs-557	379	5	.	.	PUNCT
iajs-557	380	1	المقاس	المقاس	PROPN
iajs-557	380	2	�	�	PROPN
iajs-557	380	3	�	�	PROPN
iajs-557	380	4	�	�	PROPN
iajs-557	380	5	ات	ات	PROPN
iajs-557	380	6	الجزئي	الجزئي	PROPN
iajs-557	380	7	�	�	PROPN
iajs-557	380	8	�	�	PROPN
iajs-557	380	9	�	�	PROPN
iajs-557	381	1	ة	ة	ADP
iajs-557	381	2	المجوف	المجوف	PROPN
iajs-557	381	3	�	�	PROPN
iajs-557	381	4	�	�	PROPN
iajs-557	381	5	�	�	PROPN
iajs-557	381	6	ة	ة	PROPN
iajs-557	381	7	(	(	PUNCT
iajs-557	381	8	التام	التام	PROPN
iajs-557	381	9	�	�	PROPN
iajs-557	381	10	�	�	PROPN
iajs-557	381	11	�	�	PROPN
iajs-557	381	12	ة	ة	NOUN
iajs-557	381	13	)	)	PUNCT
iajs-557	381	14	بق	بق	PROPN
iajs-557	381	15	�	�	PROPN
iajs-557	381	16	�	�	PROPN
iajs-557	381	17	�	�	PROPN
iajs-557	381	18	وة	وة	PROPN
iajs-557	381	19	،	،	PROPN
iajs-557	381	20	المقاس	المقاس	PROPN
iajs-557	381	21	�	�	PROPN
iajs-557	381	22	�	�	PROPN
iajs-557	381	23	�	�	PROPN
iajs-557	381	24	ات	ات	PROPN
iajs-557	381	25	التوزيعي	التوزيعي	PROPN
iajs-557	381	26	�	�	PROPN
iajs-557	381	27	�	�	PROPN
iajs-557	381	28	�	�	PROPN
iajs-557	381	29	ة	ة	PROPN
iajs-557	381	30	،	،	PROPN
iajs-557	381	31	المقاس	المقاس	PROPN
iajs-557	381	32	�	�	PROPN
iajs-557	381	33	�	�	PROPN
iajs-557	381	34	�	�	PROPN
iajs-557	381	35	ات	ات	PROPN
iajs-557	381	36	الجدائي	الجدائي	PROPN
iajs-557	381	37	�	�	PROPN
iajs-557	381	38	�	�	PROPN
iajs-557	381	39	�	�	PROPN
iajs-557	381	40	ة	ة	PROPN
iajs-557	381	41	(	(	PUNCT
iajs-557	381	42	الجدائي	الجدائي	PROPN
iajs-557	381	43	�	�	PROPN
iajs-557	381	44	�	�	PROPN
iajs-557	381	45	�	�	PROPN
iajs-557	381	46	ة	ة	PROPN
iajs-557	381	47	الكلم	الكلم	PROPN
iajs-557	381	48	�	�	PROPN
iajs-557	381	49	�	�	PROPN
iajs-557	381	50	�	�	PROPN
iajs-557	381	51	ات	ات	ADP
iajs-557	381	52	المفتاحي	المفتاحي	ADJ
iajs-557	381	53	�	�	PROPN
iajs-557	381	54	�	�	PROPN
iajs-557	381	55	�	�	PROPN
iajs-557	381	56	ة	ة	NOUN
iajs-557	381	57	:	:	PUNCT
iajs-557	381	58	المضادة	المضادة	ADJ
iajs-557	381	59	)	)	PUNCT
iajs-557	381	60	،	،	PROPN
iajs-557	381	61	المقاسات	المقاسات	PROPN
iajs-557	381	62	القياسية	القياسية	PROPN
iajs-557	381	63	.	.	PUNCT
iajs-557	381	64	received	receive	VERB
iajs-557	381	65	in	in	ADP
iajs-557	381	66	:	:	PUNCT
iajs-557	381	67	15	15	NUM
iajs-557	381	68	march	march	NOUN
iajs-557	381	69	2012	2012	NUM
iajs-557	381	70	accepted	accept	VERB
iajs-557	381	71	in	in	ADP
iajs-557	381	72	:	:	PUNCT
iajs-557	381	73	21	21	NUM
iajs-557	381	74	may	may	PROPN
iajs-557	381	75	2012	2012	NUM
iajs-557	381	76	introduction	introduction	NOUN
iajs-557	381	77	قسم	قسم	PROPN
iajs-557	381	78	الرياضيات	الرياضيات	PROPN
iajs-557	381	79	،	،	PROPN
iajs-557	381	80	كلية	كلية	PROPN
iajs-557	381	81	التربيةابن	التربيةابن	PROPN
iajs-557	381	82	الهيثم	الهيثم	PROPN
iajs-557	381	83	،	،	PROPN
iajs-557	381	84	جامعة	جامعة	PROPN
iajs-557	381	85	بغداد	بغداد	PROPN
iajs-557	381	86	لتكن	لتكن	PROPN
iajs-557	381	87	r	r	NOUN
iajs-557	381	88	حلقة	حلقة	PROPN
iajs-557	381	89	ابدالية	ابدالية	NOUN
iajs-557	381	90	ذا	ذا	PRON
iajs-557	381	91	محايد	محايد	PROPN
iajs-557	381	92	.	.	PUNCT
iajs-557	382	1	وليكن	وليكن	NOUN
iajs-557	382	2	m	m	VERB
iajs-557	382	3	مقاسا	مقاسا	NOUN
iajs-557	382	4	ً	ً	NOUN
iajs-557	382	5	على	على	NOUN
iajs-557	382	6	r	r	NOUN
iajs-557	382	7	.	.	PUNCT
iajs-557	383	1	في	في	PRON
iajs-557	383	2	هذا	هذا	NOUN
iajs-557	383	3	البحث	البحث	PROPN
iajs-557	383	4	درسنا	درسنا	PROPN
iajs-557	383	5	المفاهيم	المفاهيم	PROPN
iajs-557	383	6	:	:	PUNCT
iajs-557	383	7	المقاسات	المقاسات	PROPN
iajs-557	383	8	الجزئية	الجزئية	PROPN
iajs-557	383	9	المجوفة	المجوفة	PROPN
iajs-557	383	10	بقوة	بقوة	PROPN
iajs-557	383	11	(	(	PUNCT
iajs-557	383	12	التامة	التامة	NOUN
iajs-557	383	13	)	)	PUNCT
iajs-557	383	14	والمقاسات	والمقاسات	PROPN
iajs-557	383	15	الجزئية	الجزئية	PROPN
iajs-557	383	16	شبه	شبه	VERB
iajs-557	383	17	المجوفة	المجوفة	PROPN
iajs-557	383	18	وقدمنا	وقدمنا	PROPN
iajs-557	384	1	الخواص	الخواص	INTJ
iajs-557	384	2	المتعلقة	المتعلقة	VERB
iajs-557	384	3	بهم	بهم	VERB
iajs-557	384	4	والعلاقات	والعلاقات	PROPN
iajs-557	384	5	فيما	فيما	PROPN
iajs-557	384	6	بينهم	بينهم	NOUN
iajs-557	384	7	.	.	PUNCT
iajs-557	385	1	كذلك	كذلك	PROPN
iajs-557	385	2	درسنا	درسنا	PROPN
iajs-557	385	3	سلوك	سلوك	PROPN
iajs-557	385	4	هذه	هذه	PROPN
iajs-557	385	5	المقاسات	المقاسات	PROPN
iajs-557	385	6	الجزئية	الجزئية	VERB
iajs-557	385	7	ف	ف	NOUN
iajs-557	385	8	...	...	PUNCT