id	sid	tid	token	lemma	pos
iajs-155	1	1	microsoft	microsoft	PROPN
iajs-155	1	2	word	word	NOUN
iajs-155	1	3	223	223	NUM
iajs-155	1	4	-	-	SYM
iajs-155	1	5	234	234	NUM
iajs-155	1	6	223	223	NUM
iajs-155	1	7	|	|	NOUN
iajs-155	1	8	mathematics	mathematic	NOUN
iajs-155	1	9	2015	2015	NUM
iajs-155	1	10	)	)	PUNCT
iajs-155	1	11	عام	عام	ADP
iajs-155	1	12	3العدد	3العدد	NUM
iajs-155	1	13	(	(	PUNCT
iajs-155	1	14	28مجلة	28مجلة	X
iajs-155	1	15	إبن	إبن	VERB
iajs-155	1	16	الھيثم	الھيثم	NOUN
iajs-155	1	17	للعلوم	للعلوم	NOUN
iajs-155	1	18	الصرفة	الصرفة	NOUN
iajs-155	2	1	و	و	PRON
iajs-155	2	2	التطبيقية	التطبيقية	ADV
iajs-155	2	3	المجلد	المجلد	VERB
iajs-155	2	4	ibn	ibn	PROPN
iajs-155	2	5	al	al	PROPN
iajs-155	2	6	-	-	PUNCT
iajs-155	2	7	haitham	haitham	PROPN
iajs-155	2	8	jour	jour	X
iajs-155	2	9	.	.	PROPN
iajs-155	3	1	for	for	ADP
iajs-155	3	2	pure	pure	ADJ
iajs-155	3	3	&	&	CCONJ
iajs-155	3	4	appl	appl	PROPN
iajs-155	3	5	.	.	PUNCT
iajs-155	4	1	sci	sci	PROPN
iajs-155	4	2	.	.	PUNCT
iajs-155	4	3	vol	vol	NOUN
iajs-155	4	4	.	.	PROPN
iajs-155	5	1	28	28	NUM
iajs-155	5	2	(	(	PUNCT
iajs-155	5	3	3	3	NUM
iajs-155	5	4	)	)	PUNCT
iajs-155	5	5	2015	2015	NUM
iajs-155	5	6	on	on	ADP
iajs-155	5	7	2	2	NUM
iajs-155	5	8	-	-	PUNCT
iajs-155	5	9	absorbing	absorb	VERB
iajs-155	5	10	submodules	submodule	NOUN
iajs-155	5	11	inaam	inaam	NOUN
iajs-155	5	12	m.ali	m.ali	PROPN
iajs-155	5	13	abdurehman	abdurehman	NOUN
iajs-155	5	14	a.	a.	PROPN
iajs-155	5	15	harfash	harfash	PROPN
iajs-155	5	16	dept	dept	PROPN
iajs-155	5	17	.	.	PROPN
iajs-155	6	1	of	of	ADP
iajs-155	6	2	mathematics/	mathematics/	NUM
iajs-155	6	3	college	college	NOUN
iajs-155	6	4	of	of	ADP
iajs-155	6	5	education	education	NOUN
iajs-155	6	6	for	for	ADP
iajs-155	6	7	pure	pure	ADJ
iajs-155	6	8	science	science	NOUN
iajs-155	6	9	(	(	PUNCT
iajs-155	6	10	ibn	ibn	PROPN
iajs-155	6	11	al	al	PROPN
iajs-155	6	12	haitham)/	haitham)/	PROPN
iajs-155	6	13	university	university	PROPN
iajs-155	6	14	of	of	ADP
iajs-155	6	15	baghdad	baghdad	PROPN
iajs-155	6	16	received	receive	VERB
iajs-155	6	17	in	in	ADP
iajs-155	6	18	:	:	PUNCT
iajs-155	6	19	26	26	NUM
iajs-155	6	20	/	/	SYM
iajs-155	6	21	may/205,accepted	may/205,accepte	VERB
iajs-155	6	22	in:1	in:1	PROPN
iajs-155	6	23	/	/	SYM
iajs-155	6	24	august/2015	august/2015	PROPN
iajs-155	6	25	abstract	abstract	ADV
iajs-155	6	26	let	let	VERB
iajs-155	6	27	r	r	PRON
iajs-155	6	28	be	be	AUX
iajs-155	6	29	a	a	DET
iajs-155	6	30	commutative	commutative	ADJ
iajs-155	6	31	ring	ring	NOUN
iajs-155	6	32	with	with	ADP
iajs-155	6	33	1	1	NUM
iajs-155	6	34	0	0	NUM
iajs-155	6	35	and	and	CCONJ
iajs-155	6	36	m	m	PROPN
iajs-155	6	37	is	be	AUX
iajs-155	6	38	a	a	DET
iajs-155	6	39	unitary	unitary	ADJ
iajs-155	6	40	r	r	NOUN
iajs-155	6	41	-	-	PUNCT
iajs-155	6	42	module	module	NOUN
iajs-155	6	43	.	.	PUNCT
iajs-155	7	1	in	in	ADP
iajs-155	7	2	this	this	DET
iajs-155	7	3	paper	paper	NOUN
iajs-155	7	4	,	,	PUNCT
iajs-155	7	5	our	our	PRON
iajs-155	7	6	aim	aim	NOUN
iajs-155	7	7	is	be	AUX
iajs-155	7	8	to	to	PART
iajs-155	7	9	continue	continue	VERB
iajs-155	7	10	studying	study	VERB
iajs-155	7	11	2	2	NUM
iajs-155	7	12	-	-	PUNCT
iajs-155	7	13	absorbing	absorb	VERB
iajs-155	7	14	submodules	submodule	NOUN
iajs-155	7	15	which	which	PRON
iajs-155	7	16	are	be	AUX
iajs-155	7	17	introduced	introduce	VERB
iajs-155	7	18	by	by	ADP
iajs-155	7	19	a.y	a.y	PROPN
iajs-155	7	20	.	.	PROPN
iajs-155	7	21	darani	darani	PROPN
iajs-155	7	22	and	and	CCONJ
iajs-155	7	23	f.	f.	PROPN
iajs-155	7	24	soheilina	soheilina	PROPN
iajs-155	7	25	.	.	PUNCT
iajs-155	8	1	many	many	ADJ
iajs-155	8	2	new	new	ADJ
iajs-155	8	3	properties	property	NOUN
iajs-155	8	4	and	and	CCONJ
iajs-155	8	5	characterizations	characterization	NOUN
iajs-155	8	6	are	be	AUX
iajs-155	8	7	given	give	VERB
iajs-155	8	8	.	.	PUNCT
iajs-155	9	1	key	key	ADJ
iajs-155	9	2	words	word	NOUN
iajs-155	9	3	:	:	PUNCT
iajs-155	9	4	prime	prime	ADJ
iajs-155	9	5	submodule	submodule	NOUN
iajs-155	9	6	,	,	PUNCT
iajs-155	9	7	2	2	NUM
iajs-155	9	8	-	-	PUNCT
iajs-155	9	9	absorbing	absorb	VERB
iajs-155	9	10	submodule	submodule	NOUN
iajs-155	9	11	,	,	PUNCT
iajs-155	9	12	quasiprime	quasiprime	PROPN
iajs-155	9	13	submodule	submodule	NOUN
iajs-155	9	14	,	,	PUNCT
iajs-155	9	15	multiplication	multiplication	NOUN
iajs-155	9	16	module	module	NOUN
iajs-155	9	17	,	,	PUNCT
iajs-155	9	18	p	p	NOUN
iajs-155	9	19	-	-	PUNCT
iajs-155	9	20	primary	primary	ADJ
iajs-155	9	21	submodule	submodule	NOUN
iajs-155	9	22	,	,	PUNCT
iajs-155	9	23	pure	pure	ADJ
iajs-155	9	24	submodule	submodule	NOUN
iajs-155	9	25	.	.	PUNCT
iajs-155	10	1	this	this	DET
iajs-155	10	2	paper	paper	NOUN
iajs-155	10	3	is	be	AUX
iajs-155	10	4	a	a	DET
iajs-155	10	5	part	part	NOUN
iajs-155	10	6	of	of	ADP
iajs-155	10	7	thesis	thesis	NOUN
iajs-155	10	8	submitted	submit	VERB
iajs-155	10	9	by	by	ADP
iajs-155	10	10	2nd	2nd	ADJ
iajs-155	10	11	author	author	NOUN
iajs-155	10	12	under	under	ADP
iajs-155	10	13	supervision	supervision	NOUN
iajs-155	10	14	by	by	ADP
iajs-155	10	15	1st	1st	ADJ
iajs-155	10	16	author	author	NOUN
iajs-155	10	17	224	224	NUM
iajs-155	10	18	|	|	NOUN
iajs-155	10	19	mathematics	mathematic	NOUN
iajs-155	10	20	2015	2015	NUM
iajs-155	10	21	)	)	PUNCT
iajs-155	10	22	عام	عام	ADP
iajs-155	10	23	3العدد	3العدد	NUM
iajs-155	10	24	(	(	PUNCT
iajs-155	10	25	28مجلة	28مجلة	X
iajs-155	10	26	إبن	إبن	VERB
iajs-155	10	27	الھيثم	الھيثم	NOUN
iajs-155	10	28	للعلوم	للعلوم	NOUN
iajs-155	10	29	الصرفة	الصرفة	NOUN
iajs-155	11	1	و	و	PRON
iajs-155	11	2	التطبيقية	التطبيقية	ADV
iajs-155	11	3	المجلد	المجلد	VERB
iajs-155	11	4	ibn	ibn	PROPN
iajs-155	11	5	al	al	PROPN
iajs-155	11	6	-	-	PUNCT
iajs-155	11	7	haitham	haitham	PROPN
iajs-155	11	8	jour	jour	X
iajs-155	11	9	.	.	PROPN
iajs-155	12	1	for	for	ADP
iajs-155	12	2	pure	pure	ADJ
iajs-155	12	3	&	&	CCONJ
iajs-155	12	4	appl	appl	PROPN
iajs-155	12	5	.	.	PUNCT
iajs-155	13	1	sci	sci	PROPN
iajs-155	13	2	.	.	PUNCT
iajs-155	13	3	vol	vol	NOUN
iajs-155	13	4	.	.	PROPN
iajs-155	14	1	28	28	NUM
iajs-155	14	2	(	(	PUNCT
iajs-155	14	3	3	3	NUM
iajs-155	14	4	)	)	PUNCT
iajs-155	14	5	2015	2015	NUM
iajs-155	14	6	introduction	introduction	NOUN
iajs-155	14	7	the	the	DET
iajs-155	14	8	concept	concept	NOUN
iajs-155	14	9	of	of	ADP
iajs-155	14	10	2	2	NUM
iajs-155	14	11	-	-	PUNCT
iajs-155	14	12	absorbing	absorbing	ADJ
iajs-155	14	13	ideal	ideal	NOUN
iajs-155	14	14	which	which	PRON
iajs-155	14	15	is	be	AUX
iajs-155	14	16	a	a	DET
iajs-155	14	17	generalization	generalization	NOUN
iajs-155	14	18	of	of	ADP
iajs-155	14	19	prime	prime	ADJ
iajs-155	14	20	ideal	ideal	NOUN
iajs-155	14	21	was	be	AUX
iajs-155	14	22	introduced	introduce	VERB
iajs-155	14	23	by	by	ADP
iajs-155	14	24	ayman	ayman	PROPN
iajs-155	14	25	badawi	badawi	PROPN
iajs-155	14	26	,	,	PUNCT
iajs-155	14	27	where	where	SCONJ
iajs-155	14	28	"	"	PUNCT
iajs-155	14	29	a	a	DET
iajs-155	14	30	nonzero	nonzero	ADJ
iajs-155	14	31	proper	proper	ADJ
iajs-155	14	32	ideal	ideal	NOUN
iajs-155	14	33	i	i	PRON
iajs-155	14	34	of	of	ADP
iajs-155	14	35	r	r	NOUN
iajs-155	14	36	is	be	AUX
iajs-155	14	37	called	call	VERB
iajs-155	14	38	a	a	DET
iajs-155	14	39	2	2	NUM
iajs-155	14	40	-	-	PUNCT
iajs-155	14	41	absorbing	absorbing	ADJ
iajs-155	14	42	ideal	ideal	NOUN
iajs-155	14	43	of	of	ADP
iajs-155	14	44	r	r	NOUN
iajs-155	14	45	if	if	SCONJ
iajs-155	14	46	whenever	whenever	SCONJ
iajs-155	14	47	a	a	DET
iajs-155	14	48	,	,	PUNCT
iajs-155	14	49	b	b	NOUN
iajs-155	14	50	,	,	PUNCT
iajs-155	14	51	c	c	X
iajs-155	14	52	∈r	∈r	ADJ
iajs-155	15	1	,	,	PUNCT
iajs-155	15	2	abc	abc	PROPN
iajs-155	15	3	∈	∈	PROPN
iajs-155	16	1	i	i	PRON
iajs-155	16	2	then	then	ADV
iajs-155	16	3	ab	ab	PROPN
iajs-155	16	4	∈	∈	PROPN
iajs-155	17	1	i	i	PRON
iajs-155	17	2	or	or	CCONJ
iajs-155	17	3	ac	ac	PROPN
iajs-155	17	4	∈i	∈i	ADP
iajs-155	17	5	or	or	CCONJ
iajs-155	17	6	bc	bc	PROPN
iajs-155	17	7	∈i	∈i	PROPN
iajs-155	17	8	"	"	PUNCT
iajs-155	17	9	,	,	PUNCT
iajs-155	17	10	[	[	X
iajs-155	17	11	1	1	NUM
iajs-155	17	12	]	]	PUNCT
iajs-155	17	13	.	.	PUNCT
iajs-155	18	1	this	this	DET
iajs-155	18	2	definition	definition	NOUN
iajs-155	18	3	can	can	AUX
iajs-155	18	4	obviously	obviously	ADV
iajs-155	18	5	be	be	AUX
iajs-155	18	6	made	make	VERB
iajs-155	18	7	for	for	ADP
iajs-155	18	8	any	any	DET
iajs-155	18	9	proper	proper	ADJ
iajs-155	18	10	ideal	ideal	NOUN
iajs-155	18	11	.	.	PUNCT
iajs-155	19	1	a.	a.	PROPN
iajs-155	19	2	y.	y.	PROPN
iajs-155	19	3	darani	darani	PROPN
iajs-155	19	4	and	and	CCONJ
iajs-155	19	5	f.	f.	PROPN
iajs-155	19	6	soheilnia	soheilnia	PROPN
iajs-155	19	7	in	in	ADP
iajs-155	19	8	[	[	X
iajs-155	19	9	2	2	NUM
iajs-155	19	10	]	]	PUNCT
iajs-155	19	11	introduced	introduce	VERB
iajs-155	19	12	the	the	DET
iajs-155	19	13	concept	concept	NOUN
iajs-155	19	14	of	of	ADP
iajs-155	19	15	2	2	NUM
iajs-155	19	16	-	-	PUNCT
iajs-155	19	17	absorbing	absorb	VERB
iajs-155	19	18	submodule	submodule	NOUN
iajs-155	19	19	where	where	SCONJ
iajs-155	19	20	"	"	PUNCT
iajs-155	19	21	a	a	DET
iajs-155	19	22	proper	proper	ADJ
iajs-155	19	23	submodule	submodule	NOUN
iajs-155	19	24	n	n	PROPN
iajs-155	19	25	of	of	ADP
iajs-155	19	26	m	m	PROPN
iajs-155	19	27	is	be	AUX
iajs-155	19	28	called	call	VERB
iajs-155	19	29	2	2	NUM
iajs-155	19	30	-	-	PUNCT
iajs-155	19	31	absorbing	absorb	VERB
iajs-155	19	32	submodule	submodule	NOUN
iajs-155	19	33	of	of	ADP
iajs-155	19	34	m	m	PRON
iajs-155	19	35	if	if	SCONJ
iajs-155	19	36	whenever	whenever	SCONJ
iajs-155	19	37	a	a	DET
iajs-155	19	38	,	,	PUNCT
iajs-155	19	39	b	b	X
iajs-155	19	40	∈	∈	PROPN
iajs-155	19	41	r	r	NOUN
iajs-155	19	42	,	,	PUNCT
iajs-155	19	43	m∈	m∈	PROPN
iajs-155	19	44	m	m	ADJ
iajs-155	19	45	and	and	CCONJ
iajs-155	19	46	abm	abm	PROPN
iajs-155	19	47	∈n	∈n	NOUN
iajs-155	19	48	,	,	PUNCT
iajs-155	19	49	then	then	ADV
iajs-155	19	50	am	be	AUX
iajs-155	19	51	∈n	∈n	ADJ
iajs-155	19	52	or	or	CCONJ
iajs-155	19	53	bm	bm	PRON
iajs-155	19	54	∈n	∈n	NOUN
iajs-155	19	55	or	or	CCONJ
iajs-155	19	56	ab	ab	PROPN
iajs-155	19	57	∈(n	∈(n	PROPN
iajs-155	19	58	:	:	PUNCT
iajs-155	19	59	m	m	NUM
iajs-155	19	60	)	)	PUNCT
iajs-155	19	61	"	"	PUNCT
iajs-155	19	62	our	our	PRON
iajs-155	19	63	concern	concern	NOUN
iajs-155	19	64	in	in	ADP
iajs-155	19	65	this	this	DET
iajs-155	19	66	paper	paper	NOUN
iajs-155	19	67	is	be	AUX
iajs-155	19	68	to	to	PART
iajs-155	19	69	give	give	VERB
iajs-155	19	70	a	a	DET
iajs-155	19	71	comprehensive	comprehensive	ADJ
iajs-155	19	72	study	study	NOUN
iajs-155	19	73	of	of	ADP
iajs-155	19	74	2	2	NUM
iajs-155	19	75	-	-	PUNCT
iajs-155	19	76	absorbing	absorb	VERB
iajs-155	19	77	submodule	submodule	NOUN
iajs-155	19	78	,	,	PUNCT
iajs-155	19	79	where	where	SCONJ
iajs-155	19	80	we	we	PRON
iajs-155	19	81	give	give	VERB
iajs-155	19	82	many	many	ADJ
iajs-155	19	83	new	new	ADJ
iajs-155	19	84	properties	property	NOUN
iajs-155	19	85	and	and	CCONJ
iajs-155	19	86	characterizations	characterization	NOUN
iajs-155	19	87	.	.	PUNCT
iajs-155	20	1	1	1	X
iajs-155	20	2	.	.	X
iajs-155	20	3	2	2	NUM
iajs-155	20	4	-	-	PUNCT
iajs-155	20	5	absorbing	absorbing	ADJ
iajs-155	20	6	submodules	submodule	NOUN
iajs-155	20	7	–	–	PUNCT
iajs-155	20	8	basic	basic	ADJ
iajs-155	20	9	properties	property	NOUN
iajs-155	20	10	"	"	PUNCT
iajs-155	20	11	definition	definition	NOUN
iajs-155	20	12	1.1	1.1	NUM
iajs-155	20	13	:	:	PUNCT
iajs-155	20	14	let	let	VERB
iajs-155	20	15	m	m	PRON
iajs-155	20	16	be	be	AUX
iajs-155	20	17	an	an	DET
iajs-155	20	18	r	r	NOUN
iajs-155	20	19	-	-	PUNCT
iajs-155	20	20	module	module	NOUN
iajs-155	20	21	.	.	PUNCT
iajs-155	21	1	n	n	CCONJ
iajs-155	21	2	a	a	DET
iajs-155	21	3	proper	proper	ADJ
iajs-155	21	4	submodule	submodule	NOUN
iajs-155	21	5	of	of	ADP
iajs-155	21	6	m	m	PROPN
iajs-155	21	7	is	be	AUX
iajs-155	21	8	called	call	VERB
iajs-155	21	9	2	2	NUM
iajs-155	21	10	-	-	PUNCT
iajs-155	21	11	absorbing	absorb	VERB
iajs-155	21	12	submodule	submodule	NOUN
iajs-155	21	13	if	if	SCONJ
iajs-155	21	14	whenever	whenever	SCONJ
iajs-155	21	15	a	a	DET
iajs-155	21	16	,	,	PUNCT
iajs-155	21	17	b	b	X
iajs-155	21	18	∈	∈	PROPN
iajs-155	21	19	r	r	NOUN
iajs-155	21	20	,	,	PUNCT
iajs-155	21	21	m	m	VERB
iajs-155	21	22	∈	∈	PROPN
iajs-155	21	23	m	m	ADJ
iajs-155	21	24	and	and	CCONJ
iajs-155	21	25	abm	abm	PROPN
iajs-155	21	26	∈	∈	PROPN
iajs-155	21	27	n	n	CCONJ
iajs-155	21	28	,	,	PUNCT
iajs-155	21	29	then	then	ADV
iajs-155	21	30	ab	ab	PROPN
iajs-155	21	31	∈	∈	PROPN
iajs-155	21	32	(	(	PUNCT
iajs-155	21	33	n	n	NOUN
iajs-155	21	34	:	:	PUNCT
iajs-155	21	35	m	m	X
iajs-155	21	36	)	)	PUNCT
iajs-155	21	37	or	or	CCONJ
iajs-155	21	38	am	be	AUX
iajs-155	21	39	∈	∈	PROPN
iajs-155	21	40	n	n	ADJ
iajs-155	21	41	or	or	CCONJ
iajs-155	21	42	bm	bm	PROPN
iajs-155	21	43	∈	∈	PROPN
iajs-155	21	44	n.	n.	PROPN
iajs-155	21	45	"[2	"[2	PROPN
iajs-155	21	46	]	]	PUNCT
iajs-155	21	47	note	note	VERB
iajs-155	21	48	that	that	SCONJ
iajs-155	21	49	an	an	DET
iajs-155	21	50	2	2	NUM
iajs-155	21	51	-	-	PUNCT
iajs-155	21	52	absorbing	absorbing	ADJ
iajs-155	21	53	ideal	ideal	NOUN
iajs-155	21	54	of	of	ADP
iajs-155	21	55	a	a	DET
iajs-155	21	56	ring	ring	NOUN
iajs-155	21	57	r	r	NOUN
iajs-155	21	58	is	be	AUX
iajs-155	21	59	a	a	DET
iajs-155	21	60	2	2	NUM
iajs-155	21	61	-	-	PUNCT
iajs-155	21	62	absorbing	absorb	VERB
iajs-155	21	63	submodule	submodule	NOUN
iajs-155	21	64	of	of	ADP
iajs-155	21	65	the	the	DET
iajs-155	21	66	r	r	NOUN
iajs-155	21	67	-	-	PUNCT
iajs-155	21	68	module	module	NOUN
iajs-155	21	69	r	r	NOUN
iajs-155	21	70	.	.	PUNCT
iajs-155	22	1	remarks	remark	NOUN
iajs-155	22	2	and	and	CCONJ
iajs-155	22	3	examples.1.2	examples.1.2	NOUN
iajs-155	22	4	:	:	PUNCT
iajs-155	22	5	(	(	PUNCT
iajs-155	22	6	1	1	X
iajs-155	22	7	)	)	PUNCT
iajs-155	22	8	"	"	PUNCT
iajs-155	22	9	the	the	DET
iajs-155	22	10	intersection	intersection	NOUN
iajs-155	22	11	of	of	ADP
iajs-155	22	12	each	each	DET
iajs-155	22	13	pair	pair	NOUN
iajs-155	22	14	of	of	ADP
iajs-155	22	15	distinct	distinct	ADJ
iajs-155	22	16	prime	prime	ADJ
iajs-155	22	17	submodules	submodule	NOUN
iajs-155	22	18	of	of	ADP
iajs-155	22	19	r	r	NOUN
iajs-155	22	20	-	-	PUNCT
iajs-155	22	21	module	module	NOUN
iajs-155	22	22	m	m	NOUN
iajs-155	22	23	is	be	AUX
iajs-155	22	24	2absorbing	2absorbe	VERB
iajs-155	22	25	"	"	PUNCT
iajs-155	22	26	[	[	X
iajs-155	22	27	2	2	NUM
iajs-155	22	28	]	]	PUNCT
iajs-155	22	29	(	(	PUNCT
iajs-155	22	30	2	2	X
iajs-155	22	31	)	)	PUNCT
iajs-155	22	32	it	it	PRON
iajs-155	22	33	is	be	AUX
iajs-155	22	34	clear	clear	ADJ
iajs-155	22	35	that	that	SCONJ
iajs-155	22	36	every	every	DET
iajs-155	22	37	prime	prime	ADJ
iajs-155	22	38	submodule	submodule	NOUN
iajs-155	22	39	is	be	AUX
iajs-155	22	40	2	2	NUM
iajs-155	22	41	-	-	PUNCT
iajs-155	22	42	absorbing	absorbing	ADJ
iajs-155	22	43	.	.	PUNCT
iajs-155	23	1	however	however	ADV
iajs-155	23	2	the	the	DET
iajs-155	23	3	converse	converse	NOUN
iajs-155	23	4	is	be	AUX
iajs-155	23	5	not	not	PART
iajs-155	23	6	true	true	ADJ
iajs-155	23	7	in	in	ADP
iajs-155	23	8	general	general	ADJ
iajs-155	23	9	,	,	PUNCT
iajs-155	23	10	for	for	ADP
iajs-155	23	11	example	example	NOUN
iajs-155	23	12	:	:	PUNCT
iajs-155	23	13	consider	consider	VERB
iajs-155	23	14	z6	z6	NOUN
iajs-155	23	15	as	as	ADP
iajs-155	23	16	z	z	NOUN
iajs-155	23	17	-	-	PUNCT
iajs-155	23	18	module	module	NOUN
iajs-155	23	19	,	,	PUNCT
iajs-155	23	20	0	0	NUM
iajs-155	23	21	not	not	PART
iajs-155	23	22	prime	prime	ADJ
iajs-155	23	23	submodule	submodule	NOUN
iajs-155	23	24	of	of	ADP
iajs-155	23	25	z6	z6	PROPN
iajs-155	23	26	since	since	SCONJ
iajs-155	23	27	2	2	NUM
iajs-155	23	28	.3	.3	NUM
iajs-155	23	29	0	0	NUM
iajs-155	23	30	∈	∈	NOUN
iajs-155	23	31	0	0	PUNCT
iajs-155	24	1	but	but	CCONJ
iajs-155	24	2	3	3	NUM
iajs-155	24	3	∉	∉	X
iajs-155	24	4	0	0	NUM
iajs-155	24	5	and	and	CCONJ
iajs-155	24	6	2	2	NUM
iajs-155	24	7	∉	∉	X
iajs-155	24	8	(	(	PUNCT
iajs-155	24	9	0	0	NUM
iajs-155	24	10	:	:	PUNCT
iajs-155	24	11	z	z	NOUN
iajs-155	24	12	z6)=6z	z6)=6z	PROPN
iajs-155	24	13	but	but	CCONJ
iajs-155	24	14	0	0	NUM
iajs-155	24	15	=	=	SYM
iajs-155	24	16	2	2	NUM
iajs-155	24	17	∩	∩	NOUN
iajs-155	24	18	3	3	NUM
iajs-155	24	19	is	be	AUX
iajs-155	24	20	2	2	NUM
iajs-155	24	21	-	-	PUNCT
iajs-155	24	22	absorbing	absorb	VERB
iajs-155	24	23	submodule	submodule	NOUN
iajs-155	24	24	of	of	ADP
iajs-155	24	25	z6	z6	PROPN
iajs-155	24	26	as	as	ADP
iajs-155	24	27	zmodule	zmodule	NOUN
iajs-155	24	28	by	by	ADP
iajs-155	24	29	part	part	NOUN
iajs-155	24	30	(	(	PUNCT
iajs-155	24	31	1	1	NUM
iajs-155	24	32	)	)	PUNCT
iajs-155	24	33	(	(	PUNCT
iajs-155	24	34	3	3	X
iajs-155	24	35	)	)	PUNCT
iajs-155	24	36	it	it	PRON
iajs-155	24	37	is	be	AUX
iajs-155	24	38	clear	clear	ADJ
iajs-155	24	39	that	that	SCONJ
iajs-155	24	40	every	every	DET
iajs-155	24	41	quasi	quasi	ADJ
iajs-155	24	42	-	-	ADJ
iajs-155	24	43	prime	prime	ADJ
iajs-155	24	44	submodule	submodule	NOUN
iajs-155	24	45	is	be	AUX
iajs-155	24	46	2	2	NUM
iajs-155	24	47	-	-	PUNCT
iajs-155	24	48	absorbing	absorbing	ADJ
iajs-155	24	49	,	,	PUNCT
iajs-155	24	50	where	where	SCONJ
iajs-155	24	51	"	"	PUNCT
iajs-155	24	52	a	a	DET
iajs-155	24	53	proper	proper	ADJ
iajs-155	24	54	submodule	submodule	NOUN
iajs-155	24	55	n	n	PROPN
iajs-155	24	56	of	of	ADP
iajs-155	24	57	m	m	PROPN
iajs-155	24	58	is	be	AUX
iajs-155	24	59	called	call	VERB
iajs-155	24	60	quasi	quasi	ADJ
iajs-155	24	61	-	-	ADJ
iajs-155	24	62	prime	prime	ADJ
iajs-155	24	63	submodule	submodule	NOUN
iajs-155	24	64	if	if	SCONJ
iajs-155	24	65	whenever	whenever	SCONJ
iajs-155	24	66	a	a	DET
iajs-155	24	67	,	,	PUNCT
iajs-155	24	68	b	b	X
iajs-155	24	69	∈	∈	PROPN
iajs-155	24	70	r	r	NOUN
iajs-155	24	71	,	,	PUNCT
iajs-155	24	72	m∈	m∈	NOUN
iajs-155	24	73	m	m	ADJ
iajs-155	24	74	and	and	CCONJ
iajs-155	24	75	a	a	DET
iajs-155	24	76	b	b	NOUN
iajs-155	24	77	m∈	m∈	ADJ
iajs-155	24	78	n	n	NOUN
iajs-155	24	79	,	,	PUNCT
iajs-155	24	80	then	then	ADV
iajs-155	24	81	a	a	DET
iajs-155	24	82	m∈	m∈	PROPN
iajs-155	24	83	n	n	NOUN
iajs-155	24	84	or	or	CCONJ
iajs-155	24	85	b	b	NOUN
iajs-155	24	86	m∈	m∈	PROPN
iajs-155	24	87	n"[3	n"[3	PROPN
iajs-155	24	88	]	]	PUNCT
iajs-155	24	89	however	however	ADV
iajs-155	24	90	a	a	DET
iajs-155	24	91	2	2	NUM
iajs-155	24	92	-	-	PUNCT
iajs-155	24	93	absorbing	absorb	VERB
iajs-155	24	94	submodule	submodule	NOUN
iajs-155	24	95	may	may	AUX
iajs-155	24	96	not	not	PART
iajs-155	24	97	be	be	AUX
iajs-155	24	98	quasi	quasi	ADJ
iajs-155	24	99	-	-	ADJ
iajs-155	24	100	prime	prime	ADJ
iajs-155	24	101	,	,	PUNCT
iajs-155	24	102	as	as	SCONJ
iajs-155	24	103	the	the	DET
iajs-155	24	104	following	follow	VERB
iajs-155	24	105	example	example	NOUN
iajs-155	24	106	shown	show	VERB
iajs-155	24	107	:	:	PUNCT
iajs-155	24	108	consider	consider	VERB
iajs-155	24	109	the	the	DET
iajs-155	24	110	z	z	NOUN
iajs-155	24	111	-	-	PUNCT
iajs-155	24	112	module	module	NOUN
iajs-155	24	113	z	z	NOUN
iajs-155	24	114	.	.	PUNCT
iajs-155	25	1	the	the	DET
iajs-155	25	2	submodule	submodule	NOUN
iajs-155	25	3	n	n	PROPN
iajs-155	25	4	=	=	NOUN
iajs-155	25	5	4z	4z	NOUN
iajs-155	25	6	is	be	AUX
iajs-155	25	7	a	a	DET
iajs-155	25	8	2	2	NUM
iajs-155	25	9	-	-	PUNCT
iajs-155	25	10	absorbing	absorb	VERB
iajs-155	25	11	submodule	submodule	NOUN
iajs-155	25	12	of	of	ADP
iajs-155	25	13	z	z	PROPN
iajs-155	25	14	since	since	SCONJ
iajs-155	25	15	,	,	PUNCT
iajs-155	25	16	if	if	SCONJ
iajs-155	25	17	a	a	DET
iajs-155	25	18	,	,	PUNCT
iajs-155	25	19	b	b	NOUN
iajs-155	25	20	,	,	PUNCT
iajs-155	25	21	c	c	PROPN
iajs-155	25	22	∈z	∈z	PROPN
iajs-155	25	23	with	with	ADP
iajs-155	25	24	abc	abc	PROPN
iajs-155	25	25	∈4z	∈4z	PROPN
iajs-155	25	26	=	=	PROPN
iajs-155	25	27	n	n	PROPN
iajs-155	25	28	,	,	PUNCT
iajs-155	25	29	then	then	ADV
iajs-155	25	30	at	at	ADV
iajs-155	25	31	least	least	ADV
iajs-155	25	32	two	two	NUM
iajs-155	25	33	of	of	ADP
iajs-155	25	34	a	a	DET
iajs-155	25	35	,	,	PUNCT
iajs-155	25	36	b	b	NOUN
iajs-155	25	37	,	,	PUNCT
iajs-155	25	38	c	c	PROPN
iajs-155	25	39	are	be	AUX
iajs-155	25	40	even	even	ADV
iajs-155	25	41	.	.	PUNCT
iajs-155	26	1	hence	hence	ADV
iajs-155	26	2	either	either	CCONJ
iajs-155	26	3	ab	ab	PROPN
iajs-155	26	4	∈	∈	PROPN
iajs-155	26	5	n	n	CCONJ
iajs-155	26	6	or	or	CCONJ
iajs-155	26	7	ac∈	ac∈	PROPN
iajs-155	26	8	n	n	PRON
iajs-155	26	9	or	or	CCONJ
iajs-155	26	10	bc	bc	PROPN
iajs-155	26	11	∈	∈	PROPN
iajs-155	26	12	n	n	PRON
iajs-155	26	13	but	but	CCONJ
iajs-155	26	14	4z	4z	NOUN
iajs-155	26	15	is	be	AUX
iajs-155	26	16	not	not	PART
iajs-155	26	17	quasi	quasi	ADJ
iajs-155	26	18	-	-	NOUN
iajs-155	26	19	prime	prime	ADJ
iajs-155	26	20	,	,	PUNCT
iajs-155	26	21	since	since	SCONJ
iajs-155	26	22	2.2.1	2.2.1	NUM
iajs-155	26	23	∈4z	∈4z	NOUN
iajs-155	26	24	but	but	CCONJ
iajs-155	26	25	2.1∉4z	2.1∉4z	NUM
iajs-155	26	26	.	.	PUNCT
iajs-155	27	1	(	(	PUNCT
iajs-155	27	2	4	4	X
iajs-155	27	3	)	)	PUNCT
iajs-155	27	4	let	let	VERB
iajs-155	27	5	n	n	PRON
iajs-155	27	6	,	,	PUNCT
iajs-155	27	7	w	w	PROPN
iajs-155	27	8	be	be	AUX
iajs-155	27	9	two	two	NUM
iajs-155	27	10	submodules	submodule	NOUN
iajs-155	27	11	of	of	ADP
iajs-155	27	12	an	an	DET
iajs-155	27	13	r	r	NOUN
iajs-155	27	14	-	-	PUNCT
iajs-155	27	15	module	module	NOUN
iajs-155	27	16	m	m	NOUN
iajs-155	27	17	and	and	CCONJ
iajs-155	27	18	w	w	NOUN
iajs-155	27	19	<	<	X
iajs-155	27	20	n	n	PROPN
iajs-155	27	21	.if	.if	PROPN
iajs-155	28	1	n	n	PRON
iajs-155	28	2	is	be	AUX
iajs-155	28	3	2	2	NUM
iajs-155	28	4	-	-	PUNCT
iajs-155	28	5	absorbing	absorb	VERB
iajs-155	28	6	submodule	submodule	NOUN
iajs-155	28	7	of	of	ADP
iajs-155	28	8	m	m	PRON
iajs-155	28	9	then	then	ADV
iajs-155	28	10	it	it	PRON
iajs-155	28	11	is	be	AUX
iajs-155	28	12	not	not	PART
iajs-155	28	13	necessary	necessary	ADJ
iajs-155	28	14	that	that	SCONJ
iajs-155	28	15	w	w	NOUN
iajs-155	28	16	is	be	AUX
iajs-155	28	17	2absorbing	2absorbing	NUM
iajs-155	28	18	submodule	submodule	NOUN
iajs-155	28	19	of	of	ADP
iajs-155	28	20	m	m	PRON
iajs-155	28	21	,	,	PUNCT
iajs-155	28	22	for	for	ADP
iajs-155	28	23	example	example	NOUN
iajs-155	28	24	in	in	ADP
iajs-155	28	25	z24	z24	PROPN
iajs-155	28	26	as	as	ADP
iajs-155	28	27	z	z	NOUN
iajs-155	28	28	-	-	PUNCT
iajs-155	28	29	module	module	NOUN
iajs-155	28	30	.	.	PUNCT
iajs-155	29	1	take	take	VERB
iajs-155	29	2	n=	n=	ADJ
iajs-155	29	3	6	6	NUM
iajs-155	29	4	,	,	PUNCT
iajs-155	29	5	w=	w=	NOUN
iajs-155	29	6	12	12	NUM
iajs-155	29	7	.	.	PUNCT
iajs-155	30	1	since	since	SCONJ
iajs-155	30	2	n	n	NOUN
iajs-155	30	3	=	=	SYM
iajs-155	30	4	2	2	NUM
iajs-155	30	5	∩	∩	X
iajs-155	30	6	3	3	NUM
iajs-155	30	7	and	and	CCONJ
iajs-155	30	8	both	both	PRON
iajs-155	30	9	of	of	ADP
iajs-155	30	10	them	they	PRON
iajs-155	30	11	are	be	AUX
iajs-155	30	12	prime	prime	ADJ
iajs-155	30	13	submodules	submodule	NOUN
iajs-155	30	14	then	then	ADV
iajs-155	30	15	n	n	AUX
iajs-155	30	16	is	be	AUX
iajs-155	30	17	2	2	NUM
iajs-155	30	18	-	-	PUNCT
iajs-155	30	19	absorbing	absorb	VERB
iajs-155	30	20	submodule	submodule	NOUN
iajs-155	30	21	by	by	ADP
iajs-155	30	22	part	part	NOUN
iajs-155	30	23	(	(	PUNCT
iajs-155	30	24	1	1	NUM
iajs-155	30	25	)	)	PUNCT
iajs-155	30	26	.	.	PUNCT
iajs-155	31	1	but	but	CCONJ
iajs-155	31	2	2.2.3	2.2.3	NUM
iajs-155	31	3	∈	∈	NOUN
iajs-155	31	4	w	w	NOUN
iajs-155	31	5	,	,	PUNCT
iajs-155	31	6	2.3	2.3	NUM
iajs-155	31	7	∉	∉	X
iajs-155	31	8	w	w	NOUN
iajs-155	31	9	and	and	CCONJ
iajs-155	31	10	2.2=4	2.2=4	NUM
iajs-155	31	11	∉	∉	PROPN
iajs-155	31	12	w	w	X
iajs-155	31	13	:	:	PUNCT
iajs-155	31	14	z	z	PROPN
iajs-155	31	15	=	=	SYM
iajs-155	31	16	12z	12z	X
iajs-155	31	17	.	.	PUNCT
iajs-155	32	1	thus	thus	ADV
iajs-155	32	2	w	w	NOUN
iajs-155	32	3	is	be	AUX
iajs-155	32	4	not	not	PART
iajs-155	32	5	2	2	NUM
iajs-155	32	6	-	-	PUNCT
iajs-155	32	7	absorbing	absorb	VERB
iajs-155	32	8	submodule	submodule	NOUN
iajs-155	32	9	in	in	ADP
iajs-155	32	10	m	m	PROPN
iajs-155	32	11	.	.	PUNCT
iajs-155	33	1	225	225	NUM
iajs-155	33	2	|	|	ADV
iajs-155	33	3	mathematics	mathematic	NOUN
iajs-155	33	4	2015	2015	NUM
iajs-155	33	5	)	)	PUNCT
iajs-155	33	6	عام	عام	ADP
iajs-155	33	7	3العدد	3العدد	NUM
iajs-155	33	8	(	(	PUNCT
iajs-155	33	9	28مجلة	28مجلة	X
iajs-155	33	10	إبن	إبن	VERB
iajs-155	33	11	الھيثم	الھيثم	NOUN
iajs-155	33	12	للعلوم	للعلوم	NOUN
iajs-155	33	13	الصرفة	الصرفة	NOUN
iajs-155	34	1	و	و	PRON
iajs-155	34	2	التطبيقية	التطبيقية	ADV
iajs-155	34	3	المجلد	المجلد	VERB
iajs-155	34	4	ibn	ibn	PROPN
iajs-155	34	5	al	al	PROPN
iajs-155	34	6	-	-	PUNCT
iajs-155	34	7	haitham	haitham	PROPN
iajs-155	34	8	jour	jour	X
iajs-155	34	9	.	.	PROPN
iajs-155	35	1	for	for	ADP
iajs-155	35	2	pure	pure	ADJ
iajs-155	35	3	&	&	CCONJ
iajs-155	35	4	appl	appl	PROPN
iajs-155	35	5	.	.	PUNCT
iajs-155	36	1	sci	sci	PROPN
iajs-155	36	2	.	.	PUNCT
iajs-155	36	3	vol	vol	NOUN
iajs-155	36	4	.	.	PROPN
iajs-155	37	1	28	28	NUM
iajs-155	37	2	(	(	PUNCT
iajs-155	37	3	3	3	NUM
iajs-155	37	4	)	)	PUNCT
iajs-155	37	5	2015	2015	NUM
iajs-155	37	6	(	(	PUNCT
iajs-155	37	7	5	5	X
iajs-155	37	8	)	)	PUNCT
iajs-155	37	9	let	let	VERB
iajs-155	37	10	n	n	PRON
iajs-155	37	11	,	,	PUNCT
iajs-155	37	12	w	w	PROPN
iajs-155	37	13	be	be	AUX
iajs-155	37	14	two	two	NUM
iajs-155	37	15	submodules	submodule	NOUN
iajs-155	37	16	of	of	ADP
iajs-155	37	17	an	an	DET
iajs-155	37	18	r	r	NOUN
iajs-155	37	19	-	-	PUNCT
iajs-155	37	20	module	module	NOUN
iajs-155	37	21	m	m	NOUN
iajs-155	37	22	and	and	CCONJ
iajs-155	37	23	n	n	CCONJ
iajs-155	37	24	<	<	X
iajs-155	37	25	w	w	NOUN
iajs-155	37	26	.	.	PUNCT
iajs-155	38	1	if	if	SCONJ
iajs-155	38	2	n	n	PRON
iajs-155	38	3	is	be	AUX
iajs-155	38	4	2	2	NUM
iajs-155	38	5	-	-	PUNCT
iajs-155	38	6	absorbing	absorb	VERB
iajs-155	38	7	submodule	submodule	NOUN
iajs-155	38	8	of	of	ADP
iajs-155	38	9	m	m	PROPN
iajs-155	38	10	,	,	PUNCT
iajs-155	38	11	then	then	ADV
iajs-155	38	12	n	n	PRON
iajs-155	38	13	is	be	AUX
iajs-155	38	14	a	a	DET
iajs-155	38	15	2	2	NUM
iajs-155	38	16	-	-	PUNCT
iajs-155	38	17	absorbing	absorb	VERB
iajs-155	38	18	submodule	submodule	NOUN
iajs-155	38	19	of	of	ADP
iajs-155	38	20	w	w	NOUN
iajs-155	38	21	proof	proof	NOUN
iajs-155	38	22	:	:	PUNCT
iajs-155	38	23	if	if	SCONJ
iajs-155	38	24	w	w	PROPN
iajs-155	38	25	=	=	NOUN
iajs-155	38	26	m	m	ADJ
iajs-155	38	27	,	,	PUNCT
iajs-155	38	28	then	then	ADV
iajs-155	38	29	nothing	nothing	PRON
iajs-155	38	30	to	to	PART
iajs-155	38	31	prove	prove	VERB
iajs-155	38	32	let	let	VERB
iajs-155	38	33	a	a	DET
iajs-155	38	34	b	b	NOUN
iajs-155	38	35	x	x	X
iajs-155	38	36	∈n	∈n	NOUN
iajs-155	38	37	,	,	PUNCT
iajs-155	38	38	where	where	SCONJ
iajs-155	38	39	a	a	DET
iajs-155	38	40	,	,	PUNCT
iajs-155	38	41	b	b	NOUN
iajs-155	38	42	∈r	∈r	NOUN
iajs-155	38	43	,	,	PUNCT
iajs-155	38	44	x	x	PRON
iajs-155	38	45	∈w	∈w	NOUN
iajs-155	38	46	.	.	PUNCT
iajs-155	39	1	since	since	SCONJ
iajs-155	39	2	x	x	X
iajs-155	39	3	∈w	∈w	VERB
iajs-155	39	4	then	then	ADV
iajs-155	39	5	x	x	SYM
iajs-155	39	6	∈	∈	PROPN
iajs-155	39	7	m	m	VERB
iajs-155	39	8	.	.	PUNCT
iajs-155	40	1	but	but	CCONJ
iajs-155	40	2	n	n	PRON
iajs-155	40	3	is	be	AUX
iajs-155	40	4	2	2	NUM
iajs-155	40	5	-	-	PUNCT
iajs-155	40	6	absorbing	absorb	VERB
iajs-155	40	7	submodule	submodule	NOUN
iajs-155	40	8	of	of	ADP
iajs-155	40	9	m	m	PROPN
iajs-155	40	10	,	,	PUNCT
iajs-155	40	11	so	so	ADV
iajs-155	40	12	either	either	ADV
iajs-155	40	13	:	:	PUNCT
iajs-155	40	14	a	a	DET
iajs-155	40	15	x	x	X
iajs-155	40	16	∈	∈	PROPN
iajs-155	40	17	n	n	NOUN
iajs-155	40	18	or	or	CCONJ
iajs-155	40	19	b	b	NOUN
iajs-155	40	20	x	x	SYM
iajs-155	40	21	∈	∈	PROPN
iajs-155	40	22	n	n	PRON
iajs-155	40	23	or	or	CCONJ
iajs-155	40	24	a	a	DET
iajs-155	40	25	b	b	NOUN
iajs-155	40	26	∈	∈	PROPN
iajs-155	40	27	(	(	PUNCT
iajs-155	40	28	n	n	NUM
iajs-155	40	29	:	:	PUNCT
iajs-155	40	30	m	m	NUM
iajs-155	40	31	)	)	PUNCT
iajs-155	40	32	and	and	CCONJ
iajs-155	40	33	since	since	SCONJ
iajs-155	40	34	n	n	CCONJ
iajs-155	40	35	<	<	X
iajs-155	40	36	w	w	NOUN
iajs-155	40	37	implies	imply	VERB
iajs-155	40	38	(	(	PUNCT
iajs-155	40	39	n	n	NUM
iajs-155	40	40	:	:	PUNCT
iajs-155	40	41	m	m	NUM
iajs-155	40	42	)	)	PUNCT
iajs-155	40	43	(	(	PUNCT
iajs-155	40	44	n	n	CCONJ
iajs-155	40	45	:	:	PUNCT
iajs-155	40	46	w	w	X
iajs-155	40	47	)	)	PUNCT
iajs-155	40	48	,	,	PUNCT
iajs-155	40	49	then	then	ADV
iajs-155	40	50	either	either	CCONJ
iajs-155	40	51	a	a	DET
iajs-155	40	52	x	x	SYM
iajs-155	40	53	∈	∈	PROPN
iajs-155	40	54	w	w	NOUN
iajs-155	40	55	or	or	CCONJ
iajs-155	40	56	b	b	NOUN
iajs-155	40	57	x	x	SYM
iajs-155	40	58	∈	∈	PROPN
iajs-155	40	59	w	w	NOUN
iajs-155	40	60	or	or	CCONJ
iajs-155	40	61	a	a	DET
iajs-155	40	62	b	b	NOUN
iajs-155	40	63	∈	∈	PROPN
iajs-155	40	64	(	(	PUNCT
iajs-155	40	65	n	n	NUM
iajs-155	40	66	:	:	PUNCT
iajs-155	40	67	w	w	X
iajs-155	40	68	)	)	PUNCT
iajs-155	40	69	hence	hence	ADV
iajs-155	40	70	n	n	PRON
iajs-155	40	71	is	be	AUX
iajs-155	40	72	2	2	NUM
iajs-155	40	73	-	-	ADJ
iajs-155	40	74	absorbing	absorbing	ADJ
iajs-155	40	75	in	in	ADP
iajs-155	40	76	w	w	ADP
iajs-155	40	77	∎	∎	PROPN
iajs-155	40	78	(	(	PUNCT
iajs-155	40	79	6	6	NUM
iajs-155	40	80	)	)	PUNCT
iajs-155	40	81	the	the	DET
iajs-155	40	82	sum	sum	NOUN
iajs-155	40	83	of	of	ADP
iajs-155	40	84	2	2	NUM
iajs-155	40	85	-	-	PUNCT
iajs-155	40	86	absorbing	absorb	VERB
iajs-155	40	87	submodules	submodule	NOUN
iajs-155	40	88	is	be	AUX
iajs-155	40	89	not	not	PART
iajs-155	40	90	necessary	necessary	ADJ
iajs-155	40	91	2	2	NUM
iajs-155	40	92	-	-	ADJ
iajs-155	40	93	absorbing	absorbing	ADJ
iajs-155	40	94	.	.	PUNCT
iajs-155	41	1	for	for	ADP
iajs-155	41	2	example	example	NOUN
iajs-155	41	3	:	:	PUNCT
iajs-155	41	4	let	let	VERB
iajs-155	41	5	n1	n1	PROPN
iajs-155	41	6	2z	2z	NUM
iajs-155	41	7	,	,	PUNCT
iajs-155	41	8	n2	n2	ADJ
iajs-155	41	9	3z	3z	NUM
iajs-155	41	10	,	,	PUNCT
iajs-155	41	11	each	each	PRON
iajs-155	41	12	of	of	ADP
iajs-155	41	13	n1	n1	PROPN
iajs-155	41	14	and	and	CCONJ
iajs-155	41	15	n2	n2	NOUN
iajs-155	41	16	is	be	AUX
iajs-155	41	17	2	2	NUM
iajs-155	41	18	-	-	PUNCT
iajs-155	41	19	absorbing	absorb	VERB
iajs-155	41	20	submodule	submodule	NOUN
iajs-155	41	21	in	in	ADP
iajs-155	41	22	the	the	DET
iajs-155	41	23	z	z	NOUN
iajs-155	41	24	-	-	PUNCT
iajs-155	41	25	module	module	NOUN
iajs-155	41	26	z	z	NOUN
iajs-155	41	27	,	,	PUNCT
iajs-155	41	28	but	but	CCONJ
iajs-155	41	29	n1	n1	PROPN
iajs-155	41	30	+	+	CCONJ
iajs-155	41	31	n2	n2	NOUN
iajs-155	41	32	=	=	NOUN
iajs-155	41	33	z	z	NOUN
iajs-155	41	34	which	which	PRON
iajs-155	41	35	is	be	AUX
iajs-155	41	36	not	not	PART
iajs-155	41	37	2	2	NUM
iajs-155	41	38	-	-	PUNCT
iajs-155	41	39	absorbing	absorbing	ADJ
iajs-155	41	40	.	.	PUNCT
iajs-155	42	1	(	(	PUNCT
iajs-155	42	2	7	7	X
iajs-155	42	3	)	)	PUNCT
iajs-155	42	4	let	let	VERB
iajs-155	42	5	n	n	PRON
iajs-155	42	6	and	and	CCONJ
iajs-155	42	7	w	w	PROPN
iajs-155	42	8	be	be	AUX
iajs-155	42	9	two	two	NUM
iajs-155	42	10	submodules	submodule	NOUN
iajs-155	42	11	of	of	ADP
iajs-155	42	12	an	an	DET
iajs-155	42	13	r	r	NOUN
iajs-155	42	14	-	-	PUNCT
iajs-155	42	15	module	module	NOUN
iajs-155	42	16	m	m	NOUN
iajs-155	42	17	such	such	ADJ
iajs-155	42	18	that	that	SCONJ
iajs-155	42	19	n	n	PROPN
iajs-155	42	20	≅	≅	PROPN
iajs-155	42	21	w	w	PROPN
iajs-155	42	22	if	if	SCONJ
iajs-155	42	23	n	n	PRON
iajs-155	42	24	is	be	AUX
iajs-155	42	25	2	2	NUM
iajs-155	42	26	-	-	PUNCT
iajs-155	42	27	absorbing	absorb	VERB
iajs-155	42	28	submodule	submodule	NOUN
iajs-155	42	29	,	,	PUNCT
iajs-155	42	30	it	it	PRON
iajs-155	42	31	is	be	AUX
iajs-155	42	32	not	not	PART
iajs-155	42	33	necessary	necessary	ADJ
iajs-155	42	34	that	that	SCONJ
iajs-155	42	35	w	w	NOUN
iajs-155	42	36	is	be	AUX
iajs-155	42	37	2	2	NUM
iajs-155	42	38	-	-	PUNCT
iajs-155	42	39	absorbing	absorb	VERB
iajs-155	42	40	submodule	submodule	NOUN
iajs-155	42	41	as	as	ADP
iajs-155	42	42	the	the	DET
iajs-155	42	43	following	following	ADJ
iajs-155	42	44	example	example	NOUN
iajs-155	42	45	explains	explain	VERB
iajs-155	42	46	this	this	PRON
iajs-155	42	47	:	:	PUNCT
iajs-155	42	48	consider	consider	VERB
iajs-155	42	49	the	the	DET
iajs-155	42	50	z	z	NOUN
iajs-155	42	51	-	-	PUNCT
iajs-155	42	52	module	module	NOUN
iajs-155	42	53	z	z	NOUN
iajs-155	42	54	,	,	PUNCT
iajs-155	42	55	the	the	DET
iajs-155	42	56	submodule	submodule	NOUN
iajs-155	42	57	2z	2z	NUM
iajs-155	42	58	is	be	AUX
iajs-155	42	59	2	2	NUM
iajs-155	42	60	-	-	PUNCT
iajs-155	42	61	absorbing	absorb	VERB
iajs-155	42	62	submodule	submodule	NOUN
iajs-155	42	63	but	but	CCONJ
iajs-155	42	64	2z	2z	NUM
iajs-155	42	65	≅	≅	PROPN
iajs-155	42	66	30z	30z	PROPN
iajs-155	42	67	and	and	CCONJ
iajs-155	42	68	30z	30z	NOUN
iajs-155	42	69	is	be	AUX
iajs-155	42	70	not	not	PART
iajs-155	42	71	2	2	NUM
iajs-155	42	72	-	-	ADJ
iajs-155	42	73	absorbing	absorbing	ADJ
iajs-155	42	74	since	since	SCONJ
iajs-155	42	75	2.3.5	2.3.5	NUM
iajs-155	42	76	30	30	NUM
iajs-155	42	77	∈	∈	NOUN
iajs-155	42	78	30	30	NUM
iajs-155	42	79	but	but	CCONJ
iajs-155	42	80	2.5	2.5	NUM
iajs-155	42	81	∉	∉	PROPN
iajs-155	42	82	30z	30z	PROPN
iajs-155	42	83	and	and	CCONJ
iajs-155	42	84	3.5	3.5	NUM
iajs-155	42	85	∉	∉	PROPN
iajs-155	42	86	30z	30z	PROPN
iajs-155	42	87	and	and	CCONJ
iajs-155	42	88	2.3	2.3	NUM
iajs-155	42	89	6	6	NUM
iajs-155	42	90	∉	∉	PROPN
iajs-155	42	91	30z	30z	NOUN
iajs-155	42	92	(	(	PUNCT
iajs-155	42	93	8)	8)	NUM
iajs-155	42	94	the	the	DET
iajs-155	42	95	intersection	intersection	NOUN
iajs-155	42	96	of	of	ADP
iajs-155	42	97	two	two	NUM
iajs-155	42	98	2	2	NUM
iajs-155	42	99	-	-	PUNCT
iajs-155	42	100	absorbing	absorb	VERB
iajs-155	42	101	submodule	submodule	NOUN
iajs-155	42	102	need	need	AUX
iajs-155	42	103	not	not	PART
iajs-155	42	104	be	be	AUX
iajs-155	42	105	2	2	NUM
iajs-155	42	106	-	-	PUNCT
iajs-155	42	107	absorbing	absorb	VERB
iajs-155	42	108	submodule	submodule	NOUN
iajs-155	42	109	for	for	ADP
iajs-155	42	110	example	example	NOUN
iajs-155	42	111	:	:	PUNCT
iajs-155	42	112	6z	6z	NOUN
iajs-155	42	113	and	and	CCONJ
iajs-155	42	114	5z	5z	NOUN
iajs-155	42	115	are	be	AUX
iajs-155	42	116	2	2	NUM
iajs-155	42	117	-	-	PUNCT
iajs-155	42	118	absorbing	absorb	VERB
iajs-155	42	119	submodule	submodule	NOUN
iajs-155	42	120	in	in	ADP
iajs-155	42	121	the	the	DET
iajs-155	42	122	z	z	NOUN
iajs-155	42	123	-	-	PUNCT
iajs-155	42	124	module	module	NOUN
iajs-155	42	125	z	z	NOUN
iajs-155	42	126	,	,	PUNCT
iajs-155	42	127	but	but	CCONJ
iajs-155	42	128	6z	6z	NUM
iajs-155	42	129	∩	∩	ADJ
iajs-155	42	130	5z	5z	NOUN
iajs-155	42	131	=	=	SYM
iajs-155	42	132	30z	30z	NOUN
iajs-155	42	133	which	which	PRON
iajs-155	42	134	is	be	AUX
iajs-155	42	135	not	not	PART
iajs-155	42	136	2	2	NUM
iajs-155	42	137	-	-	PUNCT
iajs-155	42	138	absorbing	absorbing	ADJ
iajs-155	42	139	(	(	PUNCT
iajs-155	42	140	9	9	NUM
iajs-155	42	141	)	)	PUNCT
iajs-155	42	142	let	let	VERB
iajs-155	42	143	n	n	PRON
iajs-155	42	144	be	be	AUX
iajs-155	42	145	a	a	DET
iajs-155	42	146	2	2	NUM
iajs-155	42	147	-	-	PUNCT
iajs-155	42	148	absorbing	absorb	VERB
iajs-155	42	149	submodule	submodule	NOUN
iajs-155	42	150	of	of	ADP
iajs-155	42	151	m	m	PROPN
iajs-155	42	152	,	,	PUNCT
iajs-155	42	153	then	then	ADV
iajs-155	42	154	for	for	ADP
iajs-155	42	155	each	each	DET
iajs-155	42	156	a⊆m	a⊆m	PROPN
iajs-155	42	157	,	,	PUNCT
iajs-155	42	158	either	either	CCONJ
iajs-155	42	159	a⊆n	a⊆n	PROPN
iajs-155	42	160	or	or	CCONJ
iajs-155	42	161	a∩n	a∩n	NOUN
iajs-155	42	162	is	be	AUX
iajs-155	42	163	a	a	DET
iajs-155	42	164	2	2	NUM
iajs-155	42	165	-	-	PUNCT
iajs-155	42	166	absorbing	absorb	VERB
iajs-155	42	167	submodule	submodule	NOUN
iajs-155	42	168	of	of	ADP
iajs-155	42	169	a	a	DET
iajs-155	42	170	proof	proof	NOUN
iajs-155	42	171	:	:	PUNCT
iajs-155	42	172	suppose	suppose	VERB
iajs-155	42	173	a⊈n	a⊈n	VERB
iajs-155	42	174	.	.	PUNCT
iajs-155	43	1	then	then	ADV
iajs-155	43	2	a∩n	a∩n	PROPN
iajs-155	43	3	≨	≨	VERB
iajs-155	43	4	a	a	PRON
iajs-155	43	5	.	.	PUNCT
iajs-155	44	1	let	let	VERB
iajs-155	44	2	abx	abx	NOUN
iajs-155	44	3	∈	∈	VERB
iajs-155	44	4	a∩n	a∩n	NOUN
iajs-155	44	5	,	,	PUNCT
iajs-155	44	6	and	and	CCONJ
iajs-155	44	7	x∈a	x∈a	NOUN
iajs-155	44	8	,	,	PUNCT
iajs-155	44	9	a	a	DET
iajs-155	44	10	,	,	PUNCT
iajs-155	44	11	b∈r	b∈r	NOUN
iajs-155	44	12	.	.	PUNCT
iajs-155	45	1	so	so	ADV
iajs-155	45	2	abx	abx	NOUN
iajs-155	45	3	∈	∈	PROPN
iajs-155	45	4	n	n	NOUN
iajs-155	45	5	.	.	PUNCT
iajs-155	46	1	since	since	SCONJ
iajs-155	46	2	n	n	NUM
iajs-155	46	3	is	be	AUX
iajs-155	46	4	2	2	NUM
iajs-155	46	5	-	-	PUNCT
iajs-155	46	6	absorbing	absorbing	ADJ
iajs-155	46	7	,	,	PUNCT
iajs-155	46	8	either	either	CCONJ
iajs-155	46	9	ax	ax	NOUN
iajs-155	46	10	∈	∈	PROPN
iajs-155	46	11	n	n	CCONJ
iajs-155	46	12	or	or	CCONJ
iajs-155	46	13	bx	bx	NOUN
iajs-155	46	14	∈	∈	PROPN
iajs-155	46	15	n	n	PRON
iajs-155	46	16	or	or	CCONJ
iajs-155	46	17	ab	ab	PROPN
iajs-155	46	18	∈	∈	PROPN
iajs-155	46	19	(	(	PUNCT
iajs-155	46	20	n	n	NUM
iajs-155	46	21	:	:	PUNCT
iajs-155	46	22	m	m	NUM
iajs-155	46	23	)	)	PUNCT
iajs-155	46	24	,	,	PUNCT
iajs-155	46	25	then	then	ADV
iajs-155	46	26	ax	ax	NOUN
iajs-155	46	27	∈	∈	PROPN
iajs-155	46	28	a∩n	a∩n	NOUN
iajs-155	46	29	or	or	CCONJ
iajs-155	46	30	bx	bx	PROPN
iajs-155	46	31	∈	∈	PROPN
iajs-155	46	32	a∩n	a∩n	NOUN
iajs-155	46	33	or	or	CCONJ
iajs-155	46	34	ab	ab	PROPN
iajs-155	46	35	∈	∈	PROPN
iajs-155	46	36	(	(	PUNCT
iajs-155	46	37	a∩n	a∩n	NOUN
iajs-155	46	38	:	:	PUNCT
iajs-155	46	39	a	a	X
iajs-155	46	40	)	)	PUNCT
iajs-155	46	41	.	.	PUNCT
iajs-155	47	1	proposition	proposition	NOUN
iajs-155	47	2	.1.3	.1.3	PROPN
iajs-155	47	3	:	:	PUNCT
iajs-155	47	4	let	let	VERB
iajs-155	47	5	φ	φ	PROPN
iajs-155	47	6	:	:	PUNCT
iajs-155	47	7	m	m	NOUN
iajs-155	47	8	⟶	⟶	ADJ
iajs-155	47	9	m′	m′	NOUN
iajs-155	47	10	be	be	AUX
iajs-155	47	11	an	an	DET
iajs-155	47	12	r	r	NOUN
iajs-155	47	13	-	-	PUNCT
iajs-155	47	14	epimorphism	epimorphism	NOUN
iajs-155	47	15	.	.	PUNCT
iajs-155	48	1	if	if	SCONJ
iajs-155	48	2	w	w	NOUN
iajs-155	48	3	is	be	AUX
iajs-155	48	4	2	2	NUM
iajs-155	48	5	-	-	PUNCT
iajs-155	48	6	absorbing	absorb	VERB
iajs-155	48	7	submodule	submodule	NOUN
iajs-155	48	8	of	of	ADP
iajs-155	48	9	m′	m′	PROPN
iajs-155	48	10	,	,	PUNCT
iajs-155	48	11	then	then	ADV
iajs-155	48	12	φ	φ	PROPN
iajs-155	48	13	(	(	PUNCT
iajs-155	48	14	w	w	NOUN
iajs-155	48	15	)	)	PUNCT
iajs-155	48	16	is	be	AUX
iajs-155	48	17	2	2	NUM
iajs-155	48	18	-	-	PUNCT
iajs-155	48	19	absorbing	absorb	VERB
iajs-155	48	20	submodule	submodule	NOUN
iajs-155	48	21	of	of	ADP
iajs-155	48	22	m.	m.	NOUN
iajs-155	48	23	proof	proof	NOUN
iajs-155	48	24	:	:	PUNCT
iajs-155	48	25	it	it	PRON
iajs-155	48	26	is	be	AUX
iajs-155	48	27	straight	straight	ADJ
iajs-155	48	28	for	for	ADP
iajs-155	48	29	word	word	NOUN
iajs-155	48	30	so	so	SCONJ
iajs-155	48	31	it	it	PRON
iajs-155	48	32	is	be	AUX
iajs-155	48	33	omitted	omit	VERB
iajs-155	49	1	∎	∎	PROPN
iajs-155	49	2	proposition	proposition	NOUN
iajs-155	49	3	1.4	1.4	NUM
iajs-155	49	4	:	:	PUNCT
iajs-155	49	5	let	let	VERB
iajs-155	49	6	f	f	PRON
iajs-155	49	7	:	:	PUNCT
iajs-155	50	1	m	m	VERB
iajs-155	50	2	⟶	⟶	ADJ
iajs-155	50	3	m′	m′	NOUN
iajs-155	50	4	be	be	AUX
iajs-155	50	5	an	an	DET
iajs-155	50	6	epimorphism	epimorphism	NOUN
iajs-155	50	7	,	,	PUNCT
iajs-155	50	8	n	n	CCONJ
iajs-155	50	9	<	<	X
iajs-155	50	10	m	m	VERB
iajs-155	50	11	such	such	ADJ
iajs-155	50	12	that	that	DET
iajs-155	50	13	kerf	kerf	NOUN
iajs-155	50	14	⊆	⊆	NUM
iajs-155	50	15	n	n	NOUN
iajs-155	50	16	,	,	PUNCT
iajs-155	50	17	then	then	ADV
iajs-155	50	18	n	n	PRON
iajs-155	50	19	is	be	AUX
iajs-155	50	20	2	2	NUM
iajs-155	50	21	-	-	PUNCT
iajs-155	50	22	absorbing	absorb	VERB
iajs-155	50	23	submodule	submodule	NOUN
iajs-155	50	24	of	of	ADP
iajs-155	50	25	m	m	PROPN
iajs-155	50	26	if	if	SCONJ
iajs-155	51	1	and	and	CCONJ
iajs-155	51	2	only	only	ADV
iajs-155	51	3	if	if	SCONJ
iajs-155	51	4	f(n	f(n	PROPN
iajs-155	51	5	)	)	PUNCT
iajs-155	51	6	is	be	AUX
iajs-155	51	7	2	2	NUM
iajs-155	51	8	-	-	PUNCT
iajs-155	51	9	absorbing	absorb	VERB
iajs-155	51	10	submodule	submodule	NOUN
iajs-155	51	11	of	of	ADP
iajs-155	51	12	m′	m′	PROPN
iajs-155	51	13	226	226	NUM
iajs-155	51	14	|	|	NOUN
iajs-155	51	15	mathematics	mathematic	NOUN
iajs-155	51	16	2015	2015	NUM
iajs-155	51	17	)	)	PUNCT
iajs-155	51	18	عام	عام	ADP
iajs-155	51	19	3العدد	3العدد	NUM
iajs-155	51	20	(	(	PUNCT
iajs-155	51	21	28مجلة	28مجلة	X
iajs-155	51	22	إبن	إبن	VERB
iajs-155	51	23	الھيثم	الھيثم	NOUN
iajs-155	51	24	للعلوم	للعلوم	NOUN
iajs-155	51	25	الصرفة	الصرفة	NOUN
iajs-155	52	1	و	و	PRON
iajs-155	52	2	التطبيقية	التطبيقية	ADV
iajs-155	52	3	المجلد	المجلد	VERB
iajs-155	52	4	ibn	ibn	PROPN
iajs-155	52	5	al	al	PROPN
iajs-155	52	6	-	-	PUNCT
iajs-155	52	7	haitham	haitham	PROPN
iajs-155	52	8	jour	jour	X
iajs-155	52	9	.	.	PROPN
iajs-155	53	1	for	for	ADP
iajs-155	53	2	pure	pure	ADJ
iajs-155	53	3	&	&	CCONJ
iajs-155	53	4	appl	appl	PROPN
iajs-155	53	5	.	.	PUNCT
iajs-155	54	1	sci	sci	PROPN
iajs-155	54	2	.	.	PUNCT
iajs-155	54	3	vol	vol	NOUN
iajs-155	54	4	.	.	PROPN
iajs-155	55	1	28	28	NUM
iajs-155	55	2	(	(	PUNCT
iajs-155	55	3	3	3	NUM
iajs-155	55	4	)	)	PUNCT
iajs-155	55	5	2015	2015	NUM
iajs-155	55	6	proof	proof	NOUN
iajs-155	55	7	:	:	PUNCT
iajs-155	55	8	⟹	⟹	NUM
iajs-155	55	9	let	let	VERB
iajs-155	55	10	abḿ	abḿ	INTJ
iajs-155	55	11	∈	∈	PROPN
iajs-155	55	12	f	f	PROPN
iajs-155	55	13	n	n	INTJ
iajs-155	55	14	,	,	PUNCT
iajs-155	55	15	where	where	SCONJ
iajs-155	55	16	ḿ	ḿ	PRON
iajs-155	55	17	∈	∈	PROPN
iajs-155	55	18	m′	m′	NOUN
iajs-155	55	19	a	a	NOUN
iajs-155	55	20	,	,	PUNCT
iajs-155	55	21	b	b	X
iajs-155	55	22	∈	∈	PROPN
iajs-155	55	23	r	r	NOUN
iajs-155	55	24	.	.	PUNCT
iajs-155	56	1	ḿ	ḿ	PRON
iajs-155	56	2	f	f	X
iajs-155	56	3	m	m	VERB
iajs-155	56	4	for	for	ADP
iajs-155	56	5	some	some	DET
iajs-155	56	6	m	m	NOUN
iajs-155	56	7	∈	∈	NOUN
iajs-155	56	8	m	m	NOUN
iajs-155	56	9	,	,	PUNCT
iajs-155	56	10	since	since	SCONJ
iajs-155	56	11	f	f	PROPN
iajs-155	56	12	is	be	AUX
iajs-155	56	13	onto	onto	ADP
iajs-155	56	14	.	.	PUNCT
iajs-155	57	1	then	then	ADV
iajs-155	57	2	abf(m	abf(m	PROPN
iajs-155	57	3	)	)	PUNCT
iajs-155	57	4	∈	∈	PROPN
iajs-155	57	5	f	f	PROPN
iajs-155	57	6	n	n	NOUN
iajs-155	57	7	,	,	PUNCT
iajs-155	57	8	so	so	ADV
iajs-155	57	9	abf(m)=f(n	abf(m)=f(n	PROPN
iajs-155	57	10	)	)	PUNCT
iajs-155	57	11	for	for	ADP
iajs-155	57	12	some	some	DET
iajs-155	57	13	n	n	PRON
iajs-155	57	14	∈	∈	NOUN
iajs-155	57	15	n	n	NOUN
iajs-155	57	16	and	and	CCONJ
iajs-155	57	17	hence	hence	ADV
iajs-155	57	18	f(abm)f(n	f(abm)f(n	PROPN
iajs-155	57	19	)	)	PUNCT
iajs-155	57	20	=	=	SYM
iajs-155	57	21	0	0	NUM
iajs-155	57	22	.	.	PUNCT
iajs-155	58	1	thus	thus	ADV
iajs-155	58	2	we	we	PRON
iajs-155	58	3	get	get	VERB
iajs-155	58	4	that	that	DET
iajs-155	58	5	abm	abm	PROPN
iajs-155	58	6	–	–	PUNCT
iajs-155	58	7	n	n	CCONJ
iajs-155	58	8	∈	∈	NOUN
iajs-155	58	9	kerf	kerf	NOUN
iajs-155	58	10	⊆	⊆	NUM
iajs-155	58	11	n	n	NUM
iajs-155	58	12	which	which	PRON
iajs-155	58	13	implies	imply	VERB
iajs-155	58	14	that	that	SCONJ
iajs-155	58	15	abm	abm	PROPN
iajs-155	58	16	∈	∈	PROPN
iajs-155	58	17	n	n	NOUN
iajs-155	58	18	.	.	PUNCT
iajs-155	59	1	but	but	CCONJ
iajs-155	59	2	n	n	PRON
iajs-155	59	3	is	be	AUX
iajs-155	59	4	2	2	NUM
iajs-155	59	5	-	-	PUNCT
iajs-155	59	6	absorbing	absorbing	ADJ
iajs-155	59	7	so	so	CCONJ
iajs-155	59	8	either	either	PRON
iajs-155	59	9	am	be	AUX
iajs-155	59	10	∈	∈	PROPN
iajs-155	59	11	n	n	ADJ
iajs-155	59	12	or	or	CCONJ
iajs-155	59	13	bm	bm	PROPN
iajs-155	59	14	∈	∈	PROPN
iajs-155	59	15	n	n	PRON
iajs-155	59	16	or	or	CCONJ
iajs-155	59	17	ab	ab	PROPN
iajs-155	59	18	∈	∈	PROPN
iajs-155	59	19	(	(	PUNCT
iajs-155	59	20	n	n	NOUN
iajs-155	59	21	:	:	PUNCT
iajs-155	59	22	m	m	X
iajs-155	59	23	)	)	PUNCT
iajs-155	59	24	.	.	PUNCT
iajs-155	60	1	if	if	SCONJ
iajs-155	60	2	am	be	AUX
iajs-155	60	3	∈	∈	PROPN
iajs-155	60	4	n	n	NOUN
iajs-155	60	5	,	,	PUNCT
iajs-155	60	6	then	then	ADV
iajs-155	60	7	f	f	X
iajs-155	60	8	(	(	PUNCT
iajs-155	60	9	am	am	PROPN
iajs-155	60	10	)	)	PUNCT
iajs-155	60	11	∈	∈	PROPN
iajs-155	60	12	f	f	PROPN
iajs-155	60	13	n	n	PROPN
iajs-155	60	14	,	,	PUNCT
iajs-155	60	15	that	that	PRON
iajs-155	60	16	is	be	AUX
iajs-155	60	17	a	a	DET
iajs-155	60	18	f(m	f(m	PROPN
iajs-155	60	19	)	)	PUNCT
iajs-155	60	20	∈	∈	PROPN
iajs-155	60	21	f	f	PROPN
iajs-155	61	1	n	n	ADV
iajs-155	61	2	so	so	ADV
iajs-155	61	3	a	a	DET
iajs-155	61	4	ḿ	ḿ	X
iajs-155	61	5	∈	∈	PROPN
iajs-155	61	6	f	f	PROPN
iajs-155	61	7	n	n	ADV
iajs-155	61	8	.	.	PUNCT
iajs-155	62	1	similarly	similarly	ADV
iajs-155	62	2	,	,	PUNCT
iajs-155	62	3	bm	bm	PROPN
iajs-155	62	4	∈	∈	PROPN
iajs-155	62	5	n	n	PRON
iajs-155	62	6	implies	imply	VERB
iajs-155	62	7	that	that	SCONJ
iajs-155	62	8	bḿ	bḿ	NOUN
iajs-155	62	9	∈	∈	PROPN
iajs-155	62	10	f	f	PROPN
iajs-155	62	11	n	n	ADV
iajs-155	62	12	.	.	PUNCT
iajs-155	63	1	if	if	SCONJ
iajs-155	63	2	ab	ab	PROPN
iajs-155	63	3	∈	∈	PROPN
iajs-155	63	4	(	(	PUNCT
iajs-155	63	5	n	n	NOUN
iajs-155	63	6	:	:	PUNCT
iajs-155	63	7	m	m	X
iajs-155	63	8	)	)	PUNCT
iajs-155	63	9	,	,	PUNCT
iajs-155	63	10	then	then	ADV
iajs-155	63	11	abm	abm	PROPN
iajs-155	63	12	⊆	⊆	NUM
iajs-155	63	13	n	n	PROPN
iajs-155	63	14	and	and	CCONJ
iajs-155	63	15	so	so	ADV
iajs-155	63	16	f(abm	f(abm	ADJ
iajs-155	63	17	)	)	PUNCT
iajs-155	63	18	⊆	⊆	NUM
iajs-155	63	19	f	f	PROPN
iajs-155	63	20	n	n	PRON
iajs-155	63	21	which	which	PRON
iajs-155	63	22	implies	imply	VERB
iajs-155	63	23	that	that	SCONJ
iajs-155	63	24	abm′	abm′	VERB
iajs-155	63	25	⊆	⊆	NUM
iajs-155	63	26	f	f	PROPN
iajs-155	63	27	n	n	NOUN
iajs-155	63	28	and	and	CCONJ
iajs-155	63	29	we	we	PRON
iajs-155	63	30	get	get	VERB
iajs-155	63	31	that	that	SCONJ
iajs-155	63	32	ab	ab	PROPN
iajs-155	63	33	∈	∈	PROPN
iajs-155	63	34	(	(	PUNCT
iajs-155	63	35	f	f	NOUN
iajs-155	63	36	n	n	CCONJ
iajs-155	63	37	:	:	PUNCT
iajs-155	63	38	m′	m′	NUM
iajs-155	63	39	)	)	PUNCT
iajs-155	63	40	.	.	PUNCT
iajs-155	64	1	⟸	⟸	PROPN
iajs-155	64	2	let	let	VERB
iajs-155	64	3	abm	abm	PROPN
iajs-155	64	4	∈	∈	PROPN
iajs-155	64	5	n	n	CCONJ
iajs-155	64	6	then	then	ADV
iajs-155	64	7	f(abm	f(abm	PROPN
iajs-155	64	8	)	)	PUNCT
iajs-155	64	9	∈	∈	PROPN
iajs-155	65	1	f	f	PROPN
iajs-155	65	2	n	n	CCONJ
iajs-155	65	3	so	so	ADV
iajs-155	65	4	abf(m	abf(m	PROPN
iajs-155	65	5	)	)	PUNCT
iajs-155	65	6	∈	∈	PROPN
iajs-155	65	7	f	f	PROPN
iajs-155	65	8	n	n	PROPN
iajs-155	65	9	.	.	PUNCT
iajs-155	66	1	since	since	SCONJ
iajs-155	66	2	f(n	f(n	PROPN
iajs-155	66	3	)	)	PUNCT
iajs-155	66	4	is	be	AUX
iajs-155	66	5	2	2	NUM
iajs-155	66	6	-	-	PUNCT
iajs-155	66	7	absorbing	absorbing	ADJ
iajs-155	66	8	either	either	CCONJ
iajs-155	66	9	af(m	af(m	NOUN
iajs-155	66	10	)	)	PUNCT
iajs-155	67	1	∈	∈	PROPN
iajs-155	67	2	f	f	PROPN
iajs-155	67	3	n	n	NOUN
iajs-155	67	4	or	or	CCONJ
iajs-155	67	5	bf(m	bf(m	NUM
iajs-155	67	6	)	)	PUNCT
iajs-155	67	7	∈	∈	PROPN
iajs-155	68	1	f	f	PROPN
iajs-155	68	2	n	n	PROPN
iajs-155	68	3	or	or	CCONJ
iajs-155	68	4	ab	ab	PROPN
iajs-155	68	5	∈	∈	PROPN
iajs-155	68	6	(	(	PUNCT
iajs-155	68	7	f	f	NOUN
iajs-155	68	8	n	n	PROPN
iajs-155	68	9	:	:	PUNCT
iajs-155	68	10	m′	m′	NUM
iajs-155	68	11	)	)	PUNCT
iajs-155	68	12	.	.	PUNCT
iajs-155	69	1	1	1	X
iajs-155	69	2	)	)	PUNCT
iajs-155	69	3	if	if	SCONJ
iajs-155	69	4	af(m	af(m	NOUN
iajs-155	69	5	)	)	PUNCT
iajs-155	69	6	∈	∈	PROPN
iajs-155	69	7	f	f	PROPN
iajs-155	69	8	n	n	CCONJ
iajs-155	69	9	then	then	ADV
iajs-155	69	10	f(am	f(am	VERB
iajs-155	69	11	)	)	PUNCT
iajs-155	69	12	f	f	PROPN
iajs-155	69	13	n	n	PROPN
iajs-155	69	14	for	for	ADP
iajs-155	69	15	some	some	DET
iajs-155	69	16	n	n	NOUN
iajs-155	69	17	∈	∈	PROPN
iajs-155	69	18	f	f	PROPN
iajs-155	69	19	n	n	PROPN
iajs-155	69	20	,	,	PUNCT
iajs-155	69	21	henc	henc	PROPN
iajs-155	69	22	am	am	PROPN
iajs-155	69	23	-	-	PUNCT
iajs-155	69	24	n	n	CCONJ
iajs-155	69	25	∈	∈	NOUN
iajs-155	69	26	kerf	kerf	NOUN
iajs-155	69	27	⊆	⊆	NUM
iajs-155	69	28	n	n	NOUN
iajs-155	69	29	so	so	ADV
iajs-155	69	30	am∈	am∈	ADJ
iajs-155	69	31	n	n	PRON
iajs-155	69	32	2)if	2)if	NOUN
iajs-155	69	33	bf(m	bf(m	NOUN
iajs-155	69	34	)	)	PUNCT
iajs-155	70	1	∈	∈	PROPN
iajs-155	70	2	f	f	PROPN
iajs-155	70	3	n	n	CCONJ
iajs-155	70	4	then	then	ADV
iajs-155	70	5	similarly	similarly	ADV
iajs-155	70	6	that	that	SCONJ
iajs-155	70	7	bm	bm	PROPN
iajs-155	70	8	∈	∈	PROPN
iajs-155	70	9	n	n	PRON
iajs-155	70	10	3	3	NUM
iajs-155	70	11	)	)	PUNCT
iajs-155	70	12	if	if	SCONJ
iajs-155	70	13	ab	ab	PROPN
iajs-155	70	14	∈	∈	PROPN
iajs-155	70	15	(	(	PUNCT
iajs-155	70	16	f	f	NOUN
iajs-155	70	17	n	n	CCONJ
iajs-155	70	18	:	:	PUNCT
iajs-155	70	19	m′	m′	NUM
iajs-155	70	20	)	)	PUNCT
iajs-155	70	21	then	then	ADV
iajs-155	70	22	abm′	abm′	VERB
iajs-155	70	23	⊆	⊆	NUM
iajs-155	70	24	f	f	PROPN
iajs-155	70	25	n	n	PRON
iajs-155	70	26	so	so	ADV
iajs-155	70	27	abf(x	abf(x	NOUN
iajs-155	70	28	)	)	PUNCT
iajs-155	70	29	∈	∈	PROPN
iajs-155	70	30	f	f	PROPN
iajs-155	70	31	n	n	PROPN
iajs-155	70	32	for	for	ADP
iajs-155	70	33	each	each	DET
iajs-155	70	34	x	x	SYM
iajs-155	70	35	∈	∈	PROPN
iajs-155	70	36	m	m	VERB
iajs-155	70	37	so	so	SCONJ
iajs-155	70	38	that	that	SCONJ
iajs-155	70	39	f(abx)=f(n	f(abx)=f(n	NUM
iajs-155	70	40	)	)	PUNCT
iajs-155	70	41	for	for	ADP
iajs-155	70	42	some	some	DET
iajs-155	70	43	n	n	PRON
iajs-155	70	44	∈	∈	NOUN
iajs-155	70	45	n	n	ADV
iajs-155	70	46	and	and	CCONJ
iajs-155	70	47	hence	hence	ADV
iajs-155	70	48	abx	abx	VERB
iajs-155	70	49	∈	∈	PROPN
iajs-155	70	50	n	n	NOUN
iajs-155	70	51	for	for	ADP
iajs-155	70	52	each	each	DET
iajs-155	70	53	x	x	SYM
iajs-155	70	54	∈	∈	PROPN
iajs-155	70	55	m.	m.	NOUN
iajs-155	70	56	thus	thus	ADV
iajs-155	70	57	ab	ab	PROPN
iajs-155	70	58	∈	∈	PROPN
iajs-155	70	59	(	(	PUNCT
iajs-155	70	60	n	n	NOUN
iajs-155	70	61	:	:	PUNCT
iajs-155	70	62	m	m	X
iajs-155	70	63	)	)	PUNCT
iajs-155	70	64	.	.	PUNCT
iajs-155	71	1	∎	∎	VERB
iajs-155	71	2	by	by	ADP
iajs-155	71	3	using	use	VERB
iajs-155	71	4	proposition	proposition	NOUN
iajs-155	71	5	1.4	1.4	NUM
iajs-155	71	6	we	we	PRON
iajs-155	71	7	can	can	AUX
iajs-155	71	8	get	get	VERB
iajs-155	71	9	the	the	DET
iajs-155	71	10	following	following	ADJ
iajs-155	71	11	result	result	NOUN
iajs-155	71	12	which	which	PRON
iajs-155	71	13	is	be	AUX
iajs-155	71	14	given	give	VERB
iajs-155	71	15	in	in	ADP
iajs-155	71	16	[	[	PUNCT
iajs-155	71	17	2	2	NUM
iajs-155	71	18	]	]	PUNCT
iajs-155	71	19	as	as	ADP
iajs-155	71	20	a	a	DET
iajs-155	71	21	direct	direct	ADJ
iajs-155	71	22	consequence	consequence	NOUN
iajs-155	71	23	"	"	PUNCT
iajs-155	71	24	corollary	corollary	ADJ
iajs-155	71	25	1.5	1.5	NUM
iajs-155	71	26	:	:	PUNCT
iajs-155	71	27	let	let	VERB
iajs-155	71	28	r	r	PRON
iajs-155	71	29	be	be	AUX
iajs-155	71	30	a	a	DET
iajs-155	71	31	ring	ring	NOUN
iajs-155	71	32	,	,	PUNCT
iajs-155	71	33	m	m	VERB
iajs-155	71	34	an	an	DET
iajs-155	71	35	rmodule	rmodule	NOUN
iajs-155	71	36	and	and	CCONJ
iajs-155	71	37	n	n	NOUN
iajs-155	71	38	,	,	PUNCT
iajs-155	71	39	k	k	PROPN
iajs-155	71	40	submodules	submodule	NOUN
iajs-155	71	41	of	of	ADP
iajs-155	71	42	m	m	PROPN
iajs-155	71	43	with	with	ADP
iajs-155	71	44	k	k	PROPN
iajs-155	71	45	⊆n	⊆n	ADJ
iajs-155	71	46	.	.	PUNCT
iajs-155	72	1	then	then	ADV
iajs-155	72	2	n	n	PRON
iajs-155	72	3	is	be	AUX
iajs-155	72	4	a	a	DET
iajs-155	72	5	2	2	NUM
iajs-155	72	6	-	-	PUNCT
iajs-155	72	7	absorbing	absorb	VERB
iajs-155	72	8	submodule	submodule	NOUN
iajs-155	72	9	of	of	ADP
iajs-155	72	10	m	m	PROPN
iajs-155	72	11	if	if	SCONJ
iajs-155	73	1	and	and	CCONJ
iajs-155	73	2	only	only	ADV
iajs-155	73	3	if	if	SCONJ
iajs-155	73	4	is	be	AUX
iajs-155	73	5	a	a	DET
iajs-155	73	6	2	2	NUM
iajs-155	73	7	-	-	PUNCT
iajs-155	73	8	absorbing	absorb	VERB
iajs-155	73	9	submodule	submodule	NOUN
iajs-155	73	10	of	of	ADP
iajs-155	73	11	"	"	PUNCT
iajs-155	73	12	a.y	a.y	PROPN
iajs-155	73	13	.	.	PROPN
iajs-155	73	14	darani	darani	PROPN
iajs-155	73	15	and	and	CCONJ
iajs-155	73	16	f.	f.	PROPN
iajs-155	73	17	soheilina	soheilina	PROPN
iajs-155	73	18	in	in	ADP
iajs-155	73	19	[	[	X
iajs-155	73	20	2	2	NUM
iajs-155	73	21	]	]	PUNCT
iajs-155	73	22	introduced	introduce	VERB
iajs-155	73	23	the	the	DET
iajs-155	73	24	following	following	NOUN
iajs-155	73	25	:	:	PUNCT
iajs-155	73	26	"	"	PUNCT
iajs-155	73	27	proposition	proposition	NOUN
iajs-155	73	28	1.6	1.6	NUM
iajs-155	73	29	:	:	PUNCT
iajs-155	73	30	let	let	VERB
iajs-155	73	31	r	r	PRON
iajs-155	73	32	be	be	AUX
iajs-155	73	33	a	a	DET
iajs-155	73	34	commutative	commutative	ADJ
iajs-155	73	35	ring	ring	NOUN
iajs-155	73	36	,	,	PUNCT
iajs-155	73	37	m	m	VERB
iajs-155	73	38	is	be	AUX
iajs-155	73	39	a	a	DET
iajs-155	73	40	cyclic	cyclic	ADJ
iajs-155	73	41	r	r	NOUN
iajs-155	73	42	-	-	PUNCT
iajs-155	73	43	module	module	NOUN
iajs-155	73	44	and	and	CCONJ
iajs-155	73	45	n	n	NOUN
iajs-155	73	46	is	be	AUX
iajs-155	73	47	a	a	DET
iajs-155	73	48	submodule	submodule	NOUN
iajs-155	73	49	of	of	ADP
iajs-155	73	50	m	m	PROPN
iajs-155	73	51	.	.	PUNCT
iajs-155	74	1	then	then	ADV
iajs-155	74	2	n	n	PRON
iajs-155	74	3	is	be	AUX
iajs-155	74	4	a	a	DET
iajs-155	74	5	2	2	NUM
iajs-155	74	6	-	-	PUNCT
iajs-155	74	7	absorbing	absorb	VERB
iajs-155	74	8	submodule	submodule	NOUN
iajs-155	74	9	of	of	ADP
iajs-155	74	10	m	m	PROPN
iajs-155	74	11	if	if	SCONJ
iajs-155	75	1	and	and	CCONJ
iajs-155	75	2	only	only	ADV
iajs-155	75	3	if	if	SCONJ
iajs-155	75	4	(	(	PUNCT
iajs-155	75	5	n	n	X
iajs-155	75	6	:	:	PUNCT
iajs-155	75	7	m	m	X
iajs-155	75	8	)	)	PUNCT
iajs-155	75	9	is	be	AUX
iajs-155	75	10	a	a	DET
iajs-155	75	11	2	2	NUM
iajs-155	75	12	-	-	PUNCT
iajs-155	75	13	absorbing	absorbing	ADJ
iajs-155	75	14	ideal	ideal	NOUN
iajs-155	75	15	of	of	ADP
iajs-155	75	16	r	r	NOUN
iajs-155	75	17	"	"	PUNCT
iajs-155	75	18	.	.	PUNCT
iajs-155	76	1	sh	sh	PROPN
iajs-155	76	2	.	.	PROPN
iajs-155	76	3	payrovi	payrovi	PROPN
iajs-155	76	4	and	and	CCONJ
iajs-155	76	5	s.	s.	PROPN
iajs-155	76	6	babaei	babaei	NOUN
iajs-155	76	7	in	in	ADP
iajs-155	76	8	[	[	X
iajs-155	76	9	4	4	NUM
iajs-155	76	10	]	]	PUNCT
iajs-155	76	11	and	and	CCONJ
iajs-155	76	12	s.	s.	PROPN
iajs-155	76	13	moradi	moradi	PROPN
iajs-155	76	14	and	and	CCONJ
iajs-155	76	15	a.	a.	NOUN
iajs-155	76	16	azizi	azizi	PROPN
iajs-155	76	17	in	in	ADP
iajs-155	76	18	[	[	X
iajs-155	76	19	5	5	NUM
iajs-155	76	20	]	]	PUNCT
iajs-155	76	21	introduced	introduce	VERB
iajs-155	76	22	the	the	DET
iajs-155	76	23	following	following	NOUN
iajs-155	76	24	:	:	PUNCT
iajs-155	76	25	theorem	theorem	VERB
iajs-155	76	26	1.7	1.7	NUM
iajs-155	76	27	:	:	PUNCT
iajs-155	76	28	"	"	PUNCT
iajs-155	76	29	if	if	SCONJ
iajs-155	76	30	n	n	PRON
iajs-155	76	31	is	be	AUX
iajs-155	76	32	a	a	DET
iajs-155	76	33	2	2	NUM
iajs-155	76	34	-	-	PUNCT
iajs-155	76	35	absorbing	absorb	VERB
iajs-155	76	36	submodule	submodule	NOUN
iajs-155	76	37	of	of	ADP
iajs-155	76	38	m	m	PROPN
iajs-155	76	39	,	,	PUNCT
iajs-155	76	40	then	then	ADV
iajs-155	76	41	(	(	PUNCT
iajs-155	76	42	n	n	X
iajs-155	76	43	:	:	PUNCT
iajs-155	76	44	m	m	X
iajs-155	76	45	)	)	PUNCT
iajs-155	76	46	is	be	AUX
iajs-155	76	47	a	a	DET
iajs-155	76	48	2	2	NUM
iajs-155	76	49	-	-	PUNCT
iajs-155	76	50	absorbing	absorbing	ADJ
iajs-155	76	51	ideal	ideal	NOUN
iajs-155	76	52	of	of	ADP
iajs-155	76	53	r	r	NOUN
iajs-155	76	54	"	"	PUNCT
iajs-155	76	55	.	.	PUNCT
iajs-155	77	1	sh	sh	INTJ
iajs-155	77	2	.	.	PUNCT
iajs-155	78	1	payrovi	payrovi	PROPN
iajs-155	78	2	,	,	PUNCT
iajs-155	78	3	babaei	babaei	VERB
iajs-155	78	4	in	in	ADP
iajs-155	78	5	[	[	X
iajs-155	78	6	4	4	NUM
iajs-155	78	7	]	]	PUNCT
iajs-155	78	8	proved	prove	VERB
iajs-155	78	9	the	the	DET
iajs-155	78	10	following	follow	VERB
iajs-155	78	11	"	"	PUNCT
iajs-155	78	12	let	let	VERB
iajs-155	78	13	r	r	PRON
iajs-155	78	14	be	be	AUX
iajs-155	78	15	a	a	DET
iajs-155	78	16	noetherian	noetherian	ADJ
iajs-155	78	17	ring	ring	NOUN
iajs-155	78	18	,	,	PUNCT
iajs-155	78	19	m	m	VERB
iajs-155	78	20	a	a	DET
iajs-155	78	21	finitely	finitely	ADV
iajs-155	78	22	generated	generate	VERB
iajs-155	78	23	multiplication	multiplication	NOUN
iajs-155	78	24	r	r	NOUN
iajs-155	78	25	-	-	PUNCT
iajs-155	78	26	module	module	NOUN
iajs-155	78	27	and	and	CCONJ
iajs-155	78	28	n	n	DET
iajs-155	78	29	a	a	DET
iajs-155	78	30	proper	proper	ADJ
iajs-155	78	31	submodule	submodule	NOUN
iajs-155	78	32	of	of	ADP
iajs-155	78	33	m	m	PRON
iajs-155	78	34	such	such	ADJ
iajs-155	78	35	that	that	PRON
iajs-155	78	36	assr(m	assr(m	PROPN
iajs-155	78	37	/	/	SYM
iajs-155	78	38	n	n	CCONJ
iajs-155	78	39	)	)	PUNCT
iajs-155	78	40	is	be	AUX
iajs-155	78	41	a	a	DET
iajs-155	78	42	totally	totally	ADV
iajs-155	78	43	ordered	order	VERB
iajs-155	78	44	set	set	NOUN
iajs-155	78	45	.	.	PUNCT
iajs-155	79	1	if	if	SCONJ
iajs-155	79	2	(	(	PUNCT
iajs-155	79	3	n	n	X
iajs-155	79	4	:	:	PUNCT
iajs-155	79	5	r	r	NOUN
iajs-155	79	6	m	m	VERB
iajs-155	79	7	)	)	PUNCT
iajs-155	79	8	is	be	AUX
iajs-155	79	9	a	a	DET
iajs-155	79	10	2	2	NUM
iajs-155	79	11	-	-	PUNCT
iajs-155	79	12	absorbing	absorbing	ADJ
iajs-155	79	13	ideal	ideal	NOUN
iajs-155	79	14	of	of	ADP
iajs-155	79	15	r	r	NOUN
iajs-155	79	16	,	,	PUNCT
iajs-155	79	17	then	then	ADV
iajs-155	79	18	n	n	PRON
iajs-155	79	19	is	be	AUX
iajs-155	79	20	a	a	DET
iajs-155	79	21	2absorbing	2absorbing	NUM
iajs-155	79	22	submodule	submodule	NOUN
iajs-155	79	23	of	of	ADP
iajs-155	79	24	m	m	PROPN
iajs-155	79	25	"	"	PUNCT
iajs-155	79	26	,	,	PUNCT
iajs-155	79	27	where	where	SCONJ
iajs-155	79	28	r	r	NOUN
iajs-155	79	29	is	be	AUX
iajs-155	79	30	a	a	DET
iajs-155	79	31	commutative	commutative	ADJ
iajs-155	79	32	ring	ring	NOUN
iajs-155	79	33	with	with	ADP
iajs-155	79	34	non	non	ADJ
iajs-155	79	35	zero	zero	NUM
iajs-155	79	36	identity	identity	NOUN
iajs-155	79	37	.	.	PUNCT
iajs-155	80	1	227	227	NUM
iajs-155	80	2	|	|	NOUN
iajs-155	80	3	mathematics	mathematic	NOUN
iajs-155	80	4	2015	2015	NUM
iajs-155	80	5	)	)	PUNCT
iajs-155	80	6	عام	عام	ADP
iajs-155	80	7	3العدد	3العدد	NUM
iajs-155	80	8	(	(	PUNCT
iajs-155	80	9	28مجلة	28مجلة	X
iajs-155	80	10	إبن	إبن	VERB
iajs-155	80	11	الھيثم	الھيثم	NOUN
iajs-155	80	12	للعلوم	للعلوم	NOUN
iajs-155	80	13	الصرفة	الصرفة	NOUN
iajs-155	81	1	و	و	PRON
iajs-155	81	2	التطبيقية	التطبيقية	ADV
iajs-155	81	3	المجلد	المجلد	VERB
iajs-155	81	4	ibn	ibn	PROPN
iajs-155	81	5	al	al	PROPN
iajs-155	81	6	-	-	PUNCT
iajs-155	81	7	haitham	haitham	PROPN
iajs-155	81	8	jour	jour	X
iajs-155	81	9	.	.	PROPN
iajs-155	82	1	for	for	ADP
iajs-155	82	2	pure	pure	ADJ
iajs-155	82	3	&	&	CCONJ
iajs-155	82	4	appl	appl	PROPN
iajs-155	82	5	.	.	PUNCT
iajs-155	83	1	sci	sci	PROPN
iajs-155	83	2	.	.	PUNCT
iajs-155	83	3	vol	vol	NOUN
iajs-155	83	4	.	.	PROPN
iajs-155	84	1	28	28	NUM
iajs-155	84	2	(	(	PUNCT
iajs-155	84	3	3	3	NUM
iajs-155	84	4	)	)	PUNCT
iajs-155	84	5	2015	2015	NUM
iajs-155	84	6	however	however	SCONJ
iajs-155	84	7	we	we	PRON
iajs-155	84	8	get	get	VERB
iajs-155	84	9	the	the	DET
iajs-155	84	10	same	same	ADJ
iajs-155	84	11	conclusion	conclusion	NOUN
iajs-155	84	12	under	under	ADP
iajs-155	84	13	the	the	DET
iajs-155	84	14	class	class	NOUN
iajs-155	84	15	of	of	ADP
iajs-155	84	16	multiplication	multiplication	NOUN
iajs-155	84	17	modules	module	NOUN
iajs-155	84	18	.	.	PUNCT
iajs-155	85	1	before	before	ADP
iajs-155	85	2	giving	give	VERB
iajs-155	85	3	our	our	PRON
iajs-155	85	4	result	result	NOUN
iajs-155	85	5	,	,	PUNCT
iajs-155	85	6	recall	recall	VERB
iajs-155	85	7	that	that	SCONJ
iajs-155	85	8	"	"	PUNCT
iajs-155	85	9	an	an	DET
iajs-155	85	10	r	r	NOUN
iajs-155	85	11	-	-	PUNCT
iajs-155	85	12	module	module	NOUN
iajs-155	85	13	is	be	AUX
iajs-155	85	14	called	call	VERB
iajs-155	85	15	a	a	DET
iajs-155	85	16	multiplication	multiplication	NOUN
iajs-155	85	17	module	module	NOUN
iajs-155	85	18	if	if	SCONJ
iajs-155	85	19	for	for	ADP
iajs-155	85	20	every	every	DET
iajs-155	85	21	submodule	submodule	NOUN
iajs-155	85	22	n	n	PROPN
iajs-155	85	23	of	of	ADP
iajs-155	85	24	m	m	PRON
iajs-155	85	25	,	,	PUNCT
iajs-155	85	26	there	there	PRON
iajs-155	85	27	exists	exist	VERB
iajs-155	85	28	an	an	DET
iajs-155	85	29	ideal	ideal	NOUN
iajs-155	85	30	i	i	PRON
iajs-155	85	31	of	of	ADP
iajs-155	85	32	r	r	NOUN
iajs-155	86	1	such	such	ADJ
iajs-155	86	2	that	that	SCONJ
iajs-155	86	3	i	i	PRON
iajs-155	86	4	m	m	VERB
iajs-155	86	5	=	=	ADJ
iajs-155	86	6	n	n	X
iajs-155	86	7	.	.	PUNCT
iajs-155	87	1	equivalently	equivalently	ADV
iajs-155	87	2	,	,	PUNCT
iajs-155	87	3	m	m	VERB
iajs-155	87	4	is	be	AUX
iajs-155	87	5	a	a	DET
iajs-155	87	6	multiplication	multiplication	NOUN
iajs-155	87	7	module	module	NOUN
iajs-155	87	8	if	if	SCONJ
iajs-155	87	9	for	for	ADP
iajs-155	87	10	every	every	DET
iajs-155	87	11	submodule	submodule	NOUN
iajs-155	87	12	n	n	PROPN
iajs-155	87	13	of	of	ADP
iajs-155	87	14	m	m	PROPN
iajs-155	87	15	,	,	PUNCT
iajs-155	87	16	n=(n	n=(n	PROPN
iajs-155	87	17	:	:	PUNCT
iajs-155	87	18	rm)m	rm)m	VERB
iajs-155	87	19	"	"	PUNCT
iajs-155	87	20	.	.	PUNCT
iajs-155	88	1	[	[	X
iajs-155	88	2	6	6	NUM
iajs-155	88	3	]	]	PUNCT
iajs-155	88	4	theorem	theorem	VERB
iajs-155	88	5	1.8	1.8	NUM
iajs-155	88	6	:	:	PUNCT
iajs-155	88	7	let	let	VERB
iajs-155	88	8	m	m	PRON
iajs-155	88	9	be	be	AUX
iajs-155	88	10	a	a	DET
iajs-155	88	11	multiplication	multiplication	NOUN
iajs-155	88	12	r	r	NOUN
iajs-155	88	13	-	-	PUNCT
iajs-155	88	14	module	module	NOUN
iajs-155	88	15	and	and	CCONJ
iajs-155	88	16	n	n	NOUN
iajs-155	88	17	is	be	AUX
iajs-155	88	18	a	a	DET
iajs-155	88	19	proper	proper	ADJ
iajs-155	88	20	submodule	submodule	NOUN
iajs-155	88	21	of	of	ADP
iajs-155	88	22	m	m	PROPN
iajs-155	88	23	,	,	PUNCT
iajs-155	88	24	if	if	SCONJ
iajs-155	88	25	n	n	X
iajs-155	88	26	:	:	PUNCT
iajs-155	88	27	m	m	VERB
iajs-155	88	28	is	be	AUX
iajs-155	88	29	a	a	DET
iajs-155	88	30	2absorbing	2absorbing	NUM
iajs-155	88	31	ideal	ideal	NOUN
iajs-155	88	32	of	of	ADP
iajs-155	88	33	r	r	NOUN
iajs-155	88	34	,	,	PUNCT
iajs-155	88	35	then	then	ADV
iajs-155	88	36	n	n	PROPN
iajs-155	88	37	is	be	AUX
iajs-155	88	38	2	2	NUM
iajs-155	88	39	-	-	PUNCT
iajs-155	88	40	absorbing	absorb	VERB
iajs-155	88	41	submodule	submodule	NOUN
iajs-155	88	42	of	of	ADP
iajs-155	88	43	m	m	PROPN
iajs-155	88	44	proof	proof	NOUN
iajs-155	88	45	:	:	PUNCT
iajs-155	88	46	let	let	VERB
iajs-155	88	47	abm	abm	PROPN
iajs-155	88	48	∈	∈	PROPN
iajs-155	88	49	n	n	CCONJ
iajs-155	88	50	where	where	SCONJ
iajs-155	88	51	a	a	DET
iajs-155	88	52	,	,	PUNCT
iajs-155	88	53	b	b	X
iajs-155	88	54	∈	∈	PROPN
iajs-155	88	55	r	r	NOUN
iajs-155	88	56	,	,	PUNCT
iajs-155	88	57	m	m	VERB
iajs-155	88	58	∈	∈	ADJ
iajs-155	88	59	m	m	VERB
iajs-155	88	60	then	then	ADV
iajs-155	88	61	ab(m)⊆	ab(m)⊆	VERB
iajs-155	88	62	n	n	PRON
iajs-155	88	63	.	.	PUNCT
iajs-155	89	1	but	but	CCONJ
iajs-155	89	2	(	(	PUNCT
iajs-155	89	3	m)=	m)=	NOUN
iajs-155	89	4	i	i	X
iajs-155	89	5	m	m	VERB
iajs-155	89	6	for	for	ADP
iajs-155	89	7	some	some	PRON
iajs-155	90	1	i	i	PRON
iajs-155	90	2	r	r	NOUN
iajs-155	90	3	since	since	SCONJ
iajs-155	90	4	m	m	PROPN
iajs-155	90	5	is	be	AUX
iajs-155	90	6	a	a	DET
iajs-155	90	7	multiplication	multiplication	NOUN
iajs-155	90	8	r	r	NOUN
iajs-155	90	9	-	-	PUNCT
iajs-155	90	10	module	module	NOUN
iajs-155	90	11	,	,	PUNCT
iajs-155	90	12	so	so	ADV
iajs-155	90	13	abim	abim	NOUN
iajs-155	90	14	⊆	⊆	NUM
iajs-155	90	15	n	n	NOUN
iajs-155	90	16	.	.	PUNCT
iajs-155	91	1	hence	hence	ADV
iajs-155	91	2	abi	abi	PROPN
iajs-155	91	3	⊆	⊆	NUM
iajs-155	91	4	n	n	NUM
iajs-155	91	5	:	:	PUNCT
iajs-155	91	6	m	m	X
iajs-155	91	7	,	,	PUNCT
iajs-155	91	8	so	so	ADV
iajs-155	91	9	we	we	PRON
iajs-155	91	10	get	get	VERB
iajs-155	91	11	that	that	PRON
iajs-155	91	12	(	(	PUNCT
iajs-155	91	13	a)(b)i	a)(b)i	NUM
iajs-155	91	14	⊆	⊆	NUM
iajs-155	91	15	n	n	NUM
iajs-155	91	16	:	:	PUNCT
iajs-155	91	17	m	m	VERB
iajs-155	91	18	.	.	PUNCT
iajs-155	92	1	since	since	SCONJ
iajs-155	92	2	n	n	PROPN
iajs-155	92	3	:	:	PUNCT
iajs-155	92	4	m	m	VERB
iajs-155	92	5	is	be	AUX
iajs-155	92	6	a	a	DET
iajs-155	92	7	2	2	NUM
iajs-155	92	8	-	-	PUNCT
iajs-155	92	9	absorbing	absorbing	ADJ
iajs-155	92	10	ideal	ideal	NOUN
iajs-155	92	11	,	,	PUNCT
iajs-155	92	12	therefore	therefore	ADV
iajs-155	92	13	(	(	PUNCT
iajs-155	92	14	a)i	a)i	X
iajs-155	92	15	⊆	⊆	NUM
iajs-155	92	16	n	n	NUM
iajs-155	92	17	:	:	PUNCT
iajs-155	92	18	m	m	VERB
iajs-155	92	19	or	or	CCONJ
iajs-155	92	20	(	(	PUNCT
iajs-155	92	21	b)i	b)i	NOUN
iajs-155	92	22	⊆	⊆	NUM
iajs-155	92	23	n	n	NUM
iajs-155	92	24	:	:	PUNCT
iajs-155	92	25	m	m	VERB
iajs-155	92	26	or	or	CCONJ
iajs-155	92	27	(	(	PUNCT
iajs-155	92	28	a)(b	a)(b	ADJ
iajs-155	92	29	)	)	PUNCT
iajs-155	93	1	⊆	⊆	NUM
iajs-155	93	2	n	n	NUM
iajs-155	93	3	:	:	PUNCT
iajs-155	93	4	m	m	AUX
iajs-155	93	5	by	by	ADP
iajs-155	93	6	[	[	X
iajs-155	93	7	1	1	NUM
iajs-155	93	8	]	]	SYM
iajs-155	93	9	1	1	NUM
iajs-155	93	10	)	)	PUNCT
iajs-155	93	11	if	if	SCONJ
iajs-155	93	12	(	(	PUNCT
iajs-155	93	13	a)i	a)i	X
iajs-155	93	14	⊆	⊆	NUM
iajs-155	93	15	n	n	NUM
iajs-155	93	16	:	:	PUNCT
iajs-155	93	17	m	m	X
iajs-155	93	18	,	,	PUNCT
iajs-155	93	19	then	then	ADV
iajs-155	93	20	(	(	PUNCT
iajs-155	93	21	a)im	a)im	PROPN
iajs-155	93	22	⊆	⊆	NUM
iajs-155	93	23	n	n	NOUN
iajs-155	93	24	so	so	ADV
iajs-155	93	25	(	(	PUNCT
iajs-155	93	26	a)(m)⊆	a)(m)⊆	NOUN
iajs-155	93	27	n	n	CCONJ
iajs-155	93	28	thus	thus	ADV
iajs-155	93	29	am	be	AUX
iajs-155	93	30	∈	∈	PROPN
iajs-155	93	31	n	n	PRON
iajs-155	93	32	2	2	NUM
iajs-155	93	33	)	)	PUNCT
iajs-155	93	34	if	if	SCONJ
iajs-155	93	35	(	(	PUNCT
iajs-155	93	36	b)i	b)i	NOUN
iajs-155	93	37	⊆	⊆	NUM
iajs-155	93	38	n	n	NUM
iajs-155	93	39	:	:	PUNCT
iajs-155	93	40	m	m	X
iajs-155	93	41	,	,	PUNCT
iajs-155	93	42	then	then	ADV
iajs-155	93	43	similarly	similarly	ADV
iajs-155	93	44	bm	bm	PROPN
iajs-155	93	45	∈	∈	PROPN
iajs-155	93	46	n.	n.	NOUN
iajs-155	93	47	3	3	NUM
iajs-155	93	48	)	)	PUNCT
iajs-155	93	49	if	if	SCONJ
iajs-155	93	50	(	(	PUNCT
iajs-155	93	51	a)(b	a)(b	ADJ
iajs-155	93	52	)	)	PUNCT
iajs-155	93	53	⊆	⊆	NUM
iajs-155	93	54	n	n	NUM
iajs-155	93	55	:	:	PUNCT
iajs-155	93	56	m	m	X
iajs-155	93	57	,	,	PUNCT
iajs-155	93	58	then	then	ADV
iajs-155	93	59	ab	ab	PROPN
iajs-155	93	60	∈	∈	PROPN
iajs-155	94	1	n	n	CCONJ
iajs-155	94	2	:	:	PUNCT
iajs-155	94	3	m	m	VERB
iajs-155	94	4	∎	∎	ADJ
iajs-155	94	5	corollary	corollary	ADJ
iajs-155	94	6	1.9	1.9	NUM
iajs-155	94	7	:	:	PUNCT
iajs-155	94	8	let	let	VERB
iajs-155	94	9	m	m	PRON
iajs-155	94	10	be	be	AUX
iajs-155	94	11	a	a	DET
iajs-155	94	12	multiplication	multiplication	NOUN
iajs-155	94	13	r	r	NOUN
iajs-155	94	14	-	-	PUNCT
iajs-155	94	15	module	module	NOUN
iajs-155	94	16	and	and	CCONJ
iajs-155	94	17	n	n	NOUN
iajs-155	94	18	is	be	AUX
iajs-155	94	19	a	a	DET
iajs-155	94	20	proper	proper	ADJ
iajs-155	94	21	submodule	submodule	NOUN
iajs-155	94	22	of	of	ADP
iajs-155	94	23	m	m	PROPN
iajs-155	94	24	,	,	PUNCT
iajs-155	94	25	then	then	ADV
iajs-155	94	26	n	n	PRON
iajs-155	94	27	is	be	AUX
iajs-155	94	28	2absorbing	2absorbing	NUM
iajs-155	94	29	submodule	submodule	NOUN
iajs-155	94	30	of	of	ADP
iajs-155	94	31	m	m	PROPN
iajs-155	94	32	if	if	SCONJ
iajs-155	95	1	and	and	CCONJ
iajs-155	95	2	only	only	ADV
iajs-155	95	3	if	if	SCONJ
iajs-155	95	4	n	n	X
iajs-155	95	5	:	:	PUNCT
iajs-155	95	6	m	m	VERB
iajs-155	95	7	is	be	AUX
iajs-155	95	8	2	2	NUM
iajs-155	95	9	-	-	PUNCT
iajs-155	95	10	absorbing	absorbing	ADJ
iajs-155	95	11	ideal	ideal	NOUN
iajs-155	95	12	.	.	PUNCT
iajs-155	96	1	remark	remark	VERB
iajs-155	96	2	1.10	1.10	NUM
iajs-155	96	3	:	:	PUNCT
iajs-155	96	4	the	the	DET
iajs-155	96	5	condition	condition	NOUN
iajs-155	96	6	m	m	VERB
iajs-155	96	7	is	be	AUX
iajs-155	96	8	a	a	DET
iajs-155	96	9	multiplication	multiplication	NOUN
iajs-155	96	10	r	r	NOUN
iajs-155	96	11	-	-	PUNCT
iajs-155	96	12	module	module	NOUN
iajs-155	96	13	ca	can	AUX
iajs-155	96	14	n't	not	PART
iajs-155	96	15	be	be	AUX
iajs-155	96	16	dropped	drop	VERB
iajs-155	96	17	from	from	ADP
iajs-155	96	18	theorem	theorem	ADJ
iajs-155	96	19	1.8	1.8	NUM
iajs-155	96	20	.	.	PUNCT
iajs-155	97	1	consider	consider	VERB
iajs-155	97	2	the	the	DET
iajs-155	97	3	following	follow	VERB
iajs-155	97	4	example	example	NOUN
iajs-155	97	5	:	:	PUNCT
iajs-155	97	6	let	let	VERB
iajs-155	97	7	m	m	PRON
iajs-155	97	8	be	be	AUX
iajs-155	97	9	the	the	DET
iajs-155	97	10	z	z	NOUN
iajs-155	97	11	-	-	PUNCT
iajs-155	97	12	module	module	NOUN
iajs-155	97	13	z	z	NOUN
iajs-155	97	14	and	and	CCONJ
iajs-155	97	15	n	n	PROPN
iajs-155	97	16	=	=	NOUN
iajs-155	97	17	0	0	NUM
iajs-155	97	18	.	.	PUNCT
iajs-155	98	1	n	n	PRON
iajs-155	98	2	is	be	AUX
iajs-155	98	3	not	not	PART
iajs-155	98	4	2	2	NUM
iajs-155	98	5	-	-	PUNCT
iajs-155	98	6	absorbing	absorbing	ADJ
iajs-155	98	7	since	since	SCONJ
iajs-155	98	8	:	:	PUNCT
iajs-155	98	9	p.p	p.p	INTJ
iajs-155	98	10	.	.	NOUN
iajs-155	98	11	z	z	NOUN
iajs-155	98	12	=	=	SYM
iajs-155	98	13	0	0	NUM
iajs-155	99	1	but	but	CCONJ
iajs-155	99	2	p.	p.	NOUN
iajs-155	99	3	z	z	NOUN
iajs-155	99	4	0	0	PUNCT
iajs-155	100	1	and	and	CCONJ
iajs-155	100	2	p	p	PROPN
iajs-155	100	3	∉	∉	PROPN
iajs-155	100	4	z	z	PROPN
iajs-155	100	5	∶	∶	PROPN
iajs-155	100	6	z	z	NOUN
iajs-155	100	7	0	0	NUM
iajs-155	100	8	,	,	PUNCT
iajs-155	100	9	also	also	ADV
iajs-155	100	10	notice	notice	VERB
iajs-155	100	11	that	that	SCONJ
iajs-155	100	12	(	(	PUNCT
iajs-155	100	13	n	n	NUM
iajs-155	100	14	:	:	PUNCT
iajs-155	100	15	z	z	NOUN
iajs-155	100	16	)	)	PUNCT
iajs-155	101	1	=	=	SYM
iajs-155	101	2	0	0	NUM
iajs-155	101	3	,	,	PUNCT
iajs-155	101	4	and	and	CCONJ
iajs-155	101	5	0	0	NUM
iajs-155	101	6	is	be	AUX
iajs-155	101	7	a	a	DET
iajs-155	101	8	prime	prime	ADJ
iajs-155	101	9	ideal	ideal	NOUN
iajs-155	101	10	in	in	ADP
iajs-155	101	11	z	z	PROPN
iajs-155	101	12	,	,	PUNCT
iajs-155	102	1	so	so	ADV
iajs-155	102	2	0	0	NUM
iajs-155	102	3	is	be	AUX
iajs-155	102	4	2	2	NUM
iajs-155	102	5	-	-	PUNCT
iajs-155	102	6	absorbing	absorbing	ADJ
iajs-155	102	7	ideal	ideal	NOUN
iajs-155	102	8	in	in	ADP
iajs-155	102	9	z	z	PROPN
iajs-155	102	10	.	.	PUNCT
iajs-155	103	1	recall	recall	VERB
iajs-155	103	2	that	that	SCONJ
iajs-155	103	3	"	"	PUNCT
iajs-155	103	4	a	a	DET
iajs-155	103	5	proper	proper	ADJ
iajs-155	103	6	submodule	submodule	NOUN
iajs-155	103	7	n	n	PROPN
iajs-155	103	8	of	of	ADP
iajs-155	103	9	an	an	DET
iajs-155	103	10	r	r	NOUN
iajs-155	103	11	-	-	PUNCT
iajs-155	103	12	module	module	NOUN
iajs-155	103	13	m	m	NOUN
iajs-155	103	14	is	be	AUX
iajs-155	103	15	called	call	VERB
iajs-155	103	16	a	a	DET
iajs-155	103	17	p	p	NOUN
iajs-155	103	18	-	-	PUNCT
iajs-155	103	19	primary	primary	ADJ
iajs-155	103	20	submodule	submodule	NOUN
iajs-155	103	21	of	of	ADP
iajs-155	103	22	m	m	PROPN
iajs-155	103	23	if	if	SCONJ
iajs-155	103	24	whenever	whenever	SCONJ
iajs-155	103	25	a	a	DET
iajs-155	103	26	∈	∈	NOUN
iajs-155	103	27	r	r	NOUN
iajs-155	103	28	and	and	CCONJ
iajs-155	103	29	m	m	NOUN
iajs-155	103	30	∈m	∈m	NOUN
iajs-155	103	31	and	and	CCONJ
iajs-155	103	32	am	be	AUX
iajs-155	103	33	∈	∈	PROPN
iajs-155	103	34	n	n	NOUN
iajs-155	103	35	,	,	PUNCT
iajs-155	103	36	then	then	ADV
iajs-155	103	37	m	m	VERB
iajs-155	103	38	∈	∈	PROPN
iajs-155	103	39	n	n	NOUN
iajs-155	103	40	or	or	CCONJ
iajs-155	103	41	a	a	DET
iajs-155	103	42	∈	∈	NOUN
iajs-155	103	43	√n	√n	NOUN
iajs-155	103	44	∶	∶	NOUN
iajs-155	103	45	m	m	NOUN
iajs-155	103	46	=	=	NOUN
iajs-155	103	47	p	p	X
iajs-155	103	48	"	"	PUNCT
iajs-155	103	49	[	[	X
iajs-155	103	50	7	7	NUM
iajs-155	103	51	]	]	X
iajs-155	103	52	darani	darani	X
iajs-155	103	53	in	in	ADP
iajs-155	103	54	[	[	X
iajs-155	103	55	7	7	NUM
iajs-155	103	56	]	]	PUNCT
iajs-155	103	57	proved	prove	VERB
iajs-155	103	58	the	the	DET
iajs-155	103	59	following	follow	VERB
iajs-155	103	60	:	:	PUNCT
iajs-155	103	61	"	"	PUNCT
iajs-155	103	62	let	let	VERB
iajs-155	103	63	n	n	PRON
iajs-155	103	64	be	be	AUX
iajs-155	103	65	a	a	DET
iajs-155	103	66	p	p	NOUN
iajs-155	103	67	-	-	PUNCT
iajs-155	103	68	primary	primary	ADJ
iajs-155	103	69	submodule	submodule	NOUN
iajs-155	103	70	of	of	ADP
iajs-155	103	71	a	a	DET
iajs-155	103	72	cyclic	cyclic	ADJ
iajs-155	103	73	r	r	NOUN
iajs-155	103	74	-	-	PUNCT
iajs-155	103	75	module	module	NOUN
iajs-155	103	76	.	.	PUNCT
iajs-155	104	1	then	then	ADV
iajs-155	104	2	n	n	PROPN
iajs-155	104	3	is	be	AUX
iajs-155	104	4	2	2	NUM
iajs-155	104	5	-	-	PUNCT
iajs-155	104	6	absorbing	absorbing	ADJ
iajs-155	104	7	if	if	SCONJ
iajs-155	104	8	and	and	CCONJ
iajs-155	104	9	only	only	ADV
iajs-155	104	10	if	if	SCONJ
iajs-155	104	11	(	(	PUNCT
iajs-155	104	12	pm)2	pm)2	PROPN
iajs-155	104	13	⊆	⊆	NUM
iajs-155	104	14	n	n	CCONJ
iajs-155	104	15	"	"	PUNCT
iajs-155	104	16	however	however	ADV
iajs-155	104	17	we	we	PRON
iajs-155	104	18	improve	improve	VERB
iajs-155	104	19	this	this	DET
iajs-155	104	20	theorem	theorem	NOUN
iajs-155	104	21	as	as	SCONJ
iajs-155	104	22	follows	follow	VERB
iajs-155	104	23	.	.	PUNCT
iajs-155	105	1	proposition	proposition	NOUN
iajs-155	105	2	1.11	1.11	NUM
iajs-155	105	3	:	:	PUNCT
iajs-155	105	4	let	let	VERB
iajs-155	105	5	n	n	PRON
iajs-155	105	6	be	be	AUX
iajs-155	105	7	a	a	DET
iajs-155	105	8	p	p	NOUN
iajs-155	105	9	-	-	PUNCT
iajs-155	105	10	primary	primary	ADJ
iajs-155	105	11	submodule	submodule	NOUN
iajs-155	105	12	of	of	ADP
iajs-155	105	13	a	a	DET
iajs-155	105	14	multiplication	multiplication	NOUN
iajs-155	105	15	r	r	NOUN
iajs-155	105	16	-	-	PUNCT
iajs-155	105	17	module	module	NOUN
iajs-155	105	18	m	m	NOUN
iajs-155	105	19	.	.	PUNCT
iajs-155	106	1	then	then	ADV
iajs-155	106	2	n	n	PROPN
iajs-155	106	3	is	be	AUX
iajs-155	106	4	2	2	NUM
iajs-155	106	5	-	-	PUNCT
iajs-155	106	6	absorbing	absorbing	ADJ
iajs-155	106	7	if	if	SCONJ
iajs-155	106	8	and	and	CCONJ
iajs-155	106	9	only	only	ADV
iajs-155	106	10	if	if	SCONJ
iajs-155	106	11	(	(	PUNCT
iajs-155	106	12	pm)2	pm)2	PROPN
iajs-155	106	13	⊆	⊆	NUM
iajs-155	106	14	n	n	CCONJ
iajs-155	106	15	(	(	PUNCT
iajs-155	106	16	where	where	SCONJ
iajs-155	106	17	(	(	PUNCT
iajs-155	106	18	pm)2	pm)2	NOUN
iajs-155	106	19	=	=	PUNCT
iajs-155	106	20	p2	p2	PROPN
iajs-155	106	21	m	m	NOUN
iajs-155	106	22	)	)	PUNCT
iajs-155	106	23	.	.	PUNCT
iajs-155	107	1	228	228	NUM
iajs-155	107	2	|	|	ADV
iajs-155	107	3	mathematics	mathematic	NOUN
iajs-155	107	4	2015	2015	NUM
iajs-155	107	5	)	)	PUNCT
iajs-155	107	6	عام	عام	ADP
iajs-155	107	7	3العدد	3العدد	NUM
iajs-155	107	8	(	(	PUNCT
iajs-155	107	9	28مجلة	28مجلة	X
iajs-155	107	10	إبن	إبن	VERB
iajs-155	107	11	الھيثم	الھيثم	NOUN
iajs-155	107	12	للعلوم	للعلوم	NOUN
iajs-155	107	13	الصرفة	الصرفة	NOUN
iajs-155	108	1	و	و	PRON
iajs-155	108	2	التطبيقية	التطبيقية	ADV
iajs-155	108	3	المجلد	المجلد	VERB
iajs-155	108	4	ibn	ibn	PROPN
iajs-155	108	5	al	al	PROPN
iajs-155	108	6	-	-	PUNCT
iajs-155	108	7	haitham	haitham	PROPN
iajs-155	108	8	jour	jour	X
iajs-155	108	9	.	.	PROPN
iajs-155	109	1	for	for	ADP
iajs-155	109	2	pure	pure	ADJ
iajs-155	109	3	&	&	CCONJ
iajs-155	109	4	appl	appl	PROPN
iajs-155	109	5	.	.	PUNCT
iajs-155	110	1	sci	sci	PROPN
iajs-155	110	2	.	.	PUNCT
iajs-155	110	3	vol	vol	NOUN
iajs-155	110	4	.	.	PROPN
iajs-155	111	1	28	28	NUM
iajs-155	111	2	(	(	PUNCT
iajs-155	111	3	3	3	NUM
iajs-155	111	4	)	)	PUNCT
iajs-155	111	5	2015	2015	NUM
iajs-155	111	6	proof	proof	NOUN
iajs-155	111	7	:	:	PUNCT
iajs-155	111	8	⟹	⟹	NUM
iajs-155	111	9	since	since	SCONJ
iajs-155	111	10	m	m	PROPN
iajs-155	111	11	is	be	AUX
iajs-155	111	12	a	a	DET
iajs-155	111	13	multiplication	multiplication	NOUN
iajs-155	111	14	r	r	NOUN
iajs-155	111	15	-	-	PUNCT
iajs-155	111	16	module	module	NOUN
iajs-155	111	17	,	,	PUNCT
iajs-155	111	18	m	m	NOUN
iajs-155	111	19	-	-	PUNCT
iajs-155	111	20	radn=√n	radn=√n	VERB
iajs-155	111	21	:	:	PUNCT
iajs-155	111	22	m	m	VERB
iajs-155	111	23	m	m	VERB
iajs-155	111	24	by[7,th	by[7,th	PUNCT
iajs-155	111	25	2.10	2.10	NUM
iajs-155	111	26	]	]	PUNCT
iajs-155	111	27	but	but	CCONJ
iajs-155	111	28	n	n	PRON
iajs-155	111	29	is	be	AUX
iajs-155	111	30	p	p	NOUN
iajs-155	111	31	-	-	PUNCT
iajs-155	111	32	primary	primary	ADJ
iajs-155	111	33	,	,	PUNCT
iajs-155	111	34	so	so	SCONJ
iajs-155	111	35	p	p	X
iajs-155	111	36	=	=	NOUN
iajs-155	111	37	√n	√n	ADJ
iajs-155	111	38	:	:	PUNCT
iajs-155	111	39	m	m	NOUN
iajs-155	111	40	is	be	AUX
iajs-155	111	41	a	a	DET
iajs-155	111	42	prime	prime	ADJ
iajs-155	111	43	ideal	ideal	NOUN
iajs-155	111	44	.	.	PUNCT
iajs-155	112	1	;	;	PUNCT
iajs-155	112	2	that	that	PRON
iajs-155	112	3	is	be	AUX
iajs-155	112	4	m	m	NOUN
iajs-155	112	5	-	-	NOUN
iajs-155	112	6	radn	radn	NOUN
iajs-155	112	7	=	=	NOUN
iajs-155	112	8	pm	pm	NOUN
iajs-155	112	9	.	.	PUNCT
iajs-155	113	1	it	it	PRON
iajs-155	113	2	follows	follow	VERB
iajs-155	113	3	that	that	SCONJ
iajs-155	113	4	p2	p2	PROPN
iajs-155	113	5	⊆	⊆	NUM
iajs-155	113	6	(	(	PUNCT
iajs-155	113	7	n	n	NUM
iajs-155	113	8	:	:	PUNCT
iajs-155	113	9	m	m	VERB
iajs-155	113	10	)	)	PUNCT
iajs-155	113	11	by	by	ADP
iajs-155	113	12	[	[	X
iajs-155	113	13	1	1	NUM
iajs-155	113	14	]	]	X
iajs-155	113	15	th	th	NOUN
iajs-155	113	16	.	.	PROPN
iajs-155	113	17	2.4	2.4	NUM
iajs-155	113	18	.	.	PUNCT
iajs-155	114	1	thus	thus	ADV
iajs-155	114	2	p2	p2	NOUN
iajs-155	114	3	m	m	NOUN
iajs-155	114	4	⊆n	⊆n	ADJ
iajs-155	114	5	.	.	PUNCT
iajs-155	115	1	hence	hence	ADV
iajs-155	115	2	(	(	PUNCT
iajs-155	115	3	pm)2	pm)2	PROPN
iajs-155	115	4	⊆n	⊆n	ADJ
iajs-155	115	5	⟸	⟸	ADJ
iajs-155	115	6	let	let	VERB
iajs-155	115	7	abm∈n	abm∈n	ADV
iajs-155	115	8	where	where	SCONJ
iajs-155	115	9	a	a	DET
iajs-155	115	10	,	,	PUNCT
iajs-155	115	11	b	b	NOUN
iajs-155	115	12	∈r	∈r	NOUN
iajs-155	115	13	,	,	PUNCT
iajs-155	115	14	m∈m	m∈m	NOUN
iajs-155	115	15	assume	assume	VERB
iajs-155	115	16	that	that	SCONJ
iajs-155	115	17	am∉n	am∉n	PROPN
iajs-155	115	18	and	and	CCONJ
iajs-155	115	19	bm∉n	bm∉n	PROPN
iajs-155	115	20	.	.	PUNCT
iajs-155	116	1	as	as	SCONJ
iajs-155	116	2	n	n	PRON
iajs-155	116	3	is	be	AUX
iajs-155	116	4	primary	primary	ADJ
iajs-155	116	5	with	with	ADP
iajs-155	116	6	abm∈n	abm∈n	ADJ
iajs-155	116	7	and	and	CCONJ
iajs-155	116	8	bm∉n	bm∉n	PROPN
iajs-155	116	9	,	,	PUNCT
iajs-155	116	10	we	we	PRON
iajs-155	116	11	get	get	VERB
iajs-155	116	12	a	a	DET
iajs-155	116	13	∈	∈	NOUN
iajs-155	116	14	√n	√n	NOUN
iajs-155	116	15	:	:	PUNCT
iajs-155	116	16	m	m	NOUN
iajs-155	116	17	=	=	NOUN
iajs-155	116	18	p	p	X
iajs-155	116	19	also	also	ADV
iajs-155	116	20	abm∈n	abm∈n	ADJ
iajs-155	116	21	and	and	CCONJ
iajs-155	116	22	am∉n	am∉n	PROPN
iajs-155	116	23	,	,	PUNCT
iajs-155	116	24	we	we	PRON
iajs-155	116	25	get	get	VERB
iajs-155	116	26	b	b	NOUN
iajs-155	116	27	∈	∈	PROPN
iajs-155	116	28	√n	√n	NOUN
iajs-155	116	29	:	:	PUNCT
iajs-155	116	30	m	m	NOUN
iajs-155	116	31	=	=	NOUN
iajs-155	116	32	p	p	X
iajs-155	116	33	thus	thus	ADV
iajs-155	116	34	ab	ab	PROPN
iajs-155	116	35	∈	∈	PROPN
iajs-155	116	36	p	p	PROPN
iajs-155	116	37	⊆	⊆	NUM
iajs-155	116	38	(	(	PUNCT
iajs-155	116	39	n	n	NUM
iajs-155	116	40	:	:	PUNCT
iajs-155	116	41	m	m	NUM
iajs-155	116	42	)	)	PUNCT
iajs-155	116	43	and	and	CCONJ
iajs-155	116	44	consequently	consequently	ADV
iajs-155	116	45	n	n	PRON
iajs-155	116	46	is	be	AUX
iajs-155	116	47	2	2	NUM
iajs-155	116	48	-	-	PUNCT
iajs-155	116	49	absorbing	absorb	VERB
iajs-155	116	50	submodule	submodule	NOUN
iajs-155	116	51	of	of	ADP
iajs-155	116	52	m	m	PROPN
iajs-155	116	53	∎	∎	PROPN
iajs-155	116	54	recall	recall	VERB
iajs-155	116	55	that	that	SCONJ
iajs-155	116	56	"	"	PUNCT
iajs-155	116	57	a	a	DET
iajs-155	116	58	proper	proper	ADJ
iajs-155	116	59	submodule	submodule	NOUN
iajs-155	116	60	n	n	PROPN
iajs-155	116	61	of	of	ADP
iajs-155	116	62	an	an	DET
iajs-155	116	63	r	r	NOUN
iajs-155	116	64	-	-	PUNCT
iajs-155	116	65	module	module	NOUN
iajs-155	116	66	m	m	NOUN
iajs-155	116	67	is	be	AUX
iajs-155	116	68	called	call	VERB
iajs-155	116	69	a	a	DET
iajs-155	116	70	2	2	NUM
iajs-155	116	71	-	-	PUNCT
iajs-155	116	72	absorbing	absorbing	ADJ
iajs-155	116	73	primary	primary	ADJ
iajs-155	116	74	submodule	submodule	NOUN
iajs-155	116	75	of	of	ADP
iajs-155	116	76	m	m	PROPN
iajs-155	116	77	if	if	SCONJ
iajs-155	116	78	whenever	whenever	SCONJ
iajs-155	116	79	a	a	DET
iajs-155	116	80	;	;	PUNCT
iajs-155	116	81	b	b	X
iajs-155	116	82	∈	∈	PROPN
iajs-155	116	83	r	r	NOUN
iajs-155	116	84	and	and	CCONJ
iajs-155	116	85	m	m	NOUN
iajs-155	116	86	∈m	∈m	NOUN
iajs-155	116	87	and	and	CCONJ
iajs-155	116	88	abm	abm	PROPN
iajs-155	116	89	∈	∈	PROPN
iajs-155	116	90	n	n	CCONJ
iajs-155	116	91	,	,	PUNCT
iajs-155	116	92	then	then	ADV
iajs-155	116	93	am	be	AUX
iajs-155	116	94	∈	∈	PROPN
iajs-155	116	95	m	m	PROPN
iajs-155	116	96	-	-	PUNCT
iajs-155	116	97	rad(n	rad(n	NOUN
iajs-155	116	98	)	)	PUNCT
iajs-155	116	99	or	or	CCONJ
iajs-155	116	100	bm	bm	PROPN
iajs-155	116	101	∈	∈	PROPN
iajs-155	116	102	m	m	PROPN
iajs-155	116	103	-	-	PUNCT
iajs-155	116	104	rad(n	rad(n	NOUN
iajs-155	116	105	)	)	PUNCT
iajs-155	116	106	or	or	CCONJ
iajs-155	116	107	ab	ab	PROPN
iajs-155	116	108	∈	∈	PROPN
iajs-155	116	109	(	(	PUNCT
iajs-155	116	110	n	n	NUM
iajs-155	116	111	:	:	PUNCT
iajs-155	116	112	r	r	NOUN
iajs-155	116	113	m	m	NOUN
iajs-155	116	114	)	)	PUNCT
iajs-155	116	115	"	"	PUNCT
iajs-155	117	1	[	[	X
iajs-155	117	2	8	8	NUM
iajs-155	117	3	]	]	X
iajs-155	117	4	it	it	PRON
iajs-155	117	5	is	be	AUX
iajs-155	117	6	clear	clear	ADJ
iajs-155	117	7	that	that	SCONJ
iajs-155	117	8	2	2	NUM
iajs-155	117	9	-	-	PUNCT
iajs-155	117	10	absorbing	absorb	VERB
iajs-155	117	11	submodule	submodule	NOUN
iajs-155	117	12	implies	imply	VERB
iajs-155	117	13	2	2	NUM
iajs-155	117	14	-	-	PUNCT
iajs-155	117	15	absorbing	absorb	VERB
iajs-155	117	16	primary	primary	NOUN
iajs-155	117	17	,	,	PUNCT
iajs-155	117	18	but	but	CCONJ
iajs-155	117	19	the	the	DET
iajs-155	117	20	converse	converse	NOUN
iajs-155	117	21	may	may	AUX
iajs-155	117	22	be	be	AUX
iajs-155	117	23	not	not	PART
iajs-155	117	24	hold	hold	VERB
iajs-155	117	25	,	,	PUNCT
iajs-155	117	26	as	as	SCONJ
iajs-155	117	27	the	the	DET
iajs-155	117	28	example	example	NOUN
iajs-155	117	29	shows	show	VERB
iajs-155	117	30	let	let	VERB
iajs-155	117	31	m	m	PRON
iajs-155	117	32	be	be	AUX
iajs-155	117	33	the	the	DET
iajs-155	117	34	z	z	NOUN
iajs-155	117	35	-	-	PUNCT
iajs-155	117	36	module	module	NOUN
iajs-155	117	37	z	z	NOUN
iajs-155	117	38	,	,	PUNCT
iajs-155	117	39	let	let	VERB
iajs-155	117	40	n=8z	n=8z	NOUN
iajs-155	117	41	,	,	PUNCT
iajs-155	117	42	so	so	CCONJ
iajs-155	117	43	n	n	PRON
iajs-155	117	44	is	be	AUX
iajs-155	117	45	2	2	NUM
iajs-155	117	46	-	-	PUNCT
iajs-155	117	47	absorbing	absorb	VERB
iajs-155	117	48	primary	primary	NOUN
iajs-155	117	49	,	,	PUNCT
iajs-155	117	50	but	but	CCONJ
iajs-155	117	51	it	it	PRON
iajs-155	117	52	is	be	AUX
iajs-155	117	53	not	not	PART
iajs-155	117	54	2	2	NUM
iajs-155	117	55	-	-	PUNCT
iajs-155	117	56	absorbing	absorb	VERB
iajs-155	117	57	submodule	submodule	NOUN
iajs-155	117	58	.	.	PUNCT
iajs-155	118	1	babaei	babaei	NOUN
iajs-155	118	2	in	in	ADP
iajs-155	118	3	[	[	X
iajs-155	118	4	4	4	NUM
iajs-155	118	5	]	]	PUNCT
iajs-155	118	6	proved	prove	VERB
iajs-155	118	7	the	the	DET
iajs-155	118	8	following	follow	VERB
iajs-155	118	9	:	:	PUNCT
iajs-155	118	10	"	"	PUNCT
iajs-155	118	11	let	let	VERB
iajs-155	118	12	r	r	PRON
iajs-155	118	13	be	be	AUX
iajs-155	118	14	a	a	DET
iajs-155	118	15	noetherian	noetherian	ADJ
iajs-155	118	16	ring	ring	NOUN
iajs-155	118	17	,	,	PUNCT
iajs-155	118	18	i	i	PRON
iajs-155	118	19	a	a	DET
iajs-155	118	20	2	2	NUM
iajs-155	118	21	-	-	PUNCT
iajs-155	118	22	absorbing	absorbing	ADJ
iajs-155	118	23	ideal	ideal	NOUN
iajs-155	118	24	of	of	ADP
iajs-155	118	25	r	r	NOUN
iajs-155	118	26	,	,	PUNCT
iajs-155	118	27	and	and	CCONJ
iajs-155	118	28	m	m	VERB
iajs-155	118	29	a	a	DET
iajs-155	118	30	faithful	faithful	ADJ
iajs-155	118	31	multiplication	multiplication	NOUN
iajs-155	118	32	rmodule	rmodule	NOUN
iajs-155	118	33	such	such	ADJ
iajs-155	118	34	that	that	SCONJ
iajs-155	118	35	assr(m	assr(m	PROPN
iajs-155	118	36	/im	/im	PUNCT
iajs-155	118	37	)	)	PUNCT
iajs-155	118	38	is	be	AUX
iajs-155	118	39	a	a	DET
iajs-155	118	40	non	non	ADJ
iajs-155	118	41	-	-	ADJ
iajs-155	118	42	empty	empty	ADJ
iajs-155	118	43	totally	totally	ADV
iajs-155	118	44	ordered	order	VERB
iajs-155	118	45	set	set	NOUN
iajs-155	118	46	.	.	PUNCT
iajs-155	119	1	then	then	ADV
iajs-155	119	2	abm	abm	PROPN
iajs-155	119	3	∈	∈	PROPN
iajs-155	119	4	i	i	PRON
iajs-155	119	5	m	m	VERB
iajs-155	119	6	implies	imply	VERB
iajs-155	119	7	that	that	PRON
iajs-155	119	8	am	be	AUX
iajs-155	119	9	∈im	∈im	ADJ
iajs-155	119	10	or	or	CCONJ
iajs-155	119	11	bm	bm	PROPN
iajs-155	119	12	∈	∈	PROPN
iajs-155	120	1	i	i	PRON
iajs-155	120	2	m	m	VERB
iajs-155	120	3	or	or	CCONJ
iajs-155	120	4	ab	ab	ADJ
iajs-155	120	5	∈i	∈i	NOUN
iajs-155	120	6	whenever	whenever	SCONJ
iajs-155	120	7	a	a	PRON
iajs-155	120	8	;	;	PUNCT
iajs-155	120	9	b	b	X
iajs-155	120	10	∈	∈	PROPN
iajs-155	120	11	r	r	NOUN
iajs-155	120	12	and	and	CCONJ
iajs-155	120	13	m	m	PROPN
iajs-155	120	14	∈	∈	NOUN
iajs-155	120	15	m	m	NOUN
iajs-155	120	16	"	"	PUNCT
iajs-155	121	1	[	[	X
iajs-155	121	2	4	4	NUM
iajs-155	121	3	]	]	PUNCT
iajs-155	121	4	,	,	PUNCT
iajs-155	121	5	that	that	SCONJ
iajs-155	121	6	i	i	PRON
iajs-155	121	7	m	m	VERB
iajs-155	121	8	is	be	AUX
iajs-155	121	9	2	2	NUM
iajs-155	121	10	-	-	PUNCT
iajs-155	121	11	absorbing	absorbing	ADJ
iajs-155	121	12	.	.	PUNCT
iajs-155	122	1	where	where	SCONJ
iajs-155	122	2	r	r	NOUN
iajs-155	122	3	is	be	AUX
iajs-155	122	4	a	a	DET
iajs-155	122	5	commutative	commutative	ADJ
iajs-155	122	6	ring	ring	NOUN
iajs-155	122	7	with	with	ADP
iajs-155	122	8	nonzero	nonzero	PROPN
iajs-155	122	9	identity	identity	NOUN
iajs-155	122	10	.	.	PUNCT
iajs-155	123	1	however	however	ADV
iajs-155	123	2	we	we	PRON
iajs-155	123	3	get	get	VERB
iajs-155	123	4	the	the	DET
iajs-155	123	5	same	same	ADJ
iajs-155	123	6	conclusion	conclusion	NOUN
iajs-155	123	7	of	of	ADP
iajs-155	123	8	this	this	DET
iajs-155	123	9	theorem	theorem	NOUN
iajs-155	123	10	but	but	CCONJ
iajs-155	123	11	with	with	ADP
iajs-155	123	12	less	less	ADJ
iajs-155	123	13	conditions	condition	NOUN
iajs-155	123	14	and	and	CCONJ
iajs-155	123	15	also	also	ADV
iajs-155	123	16	we	we	PRON
iajs-155	123	17	give	give	VERB
iajs-155	123	18	a	a	DET
iajs-155	123	19	simple	simple	ADJ
iajs-155	123	20	proof	proof	NOUN
iajs-155	123	21	.	.	PUNCT
iajs-155	124	1	proposition	proposition	NOUN
iajs-155	124	2	1.12	1.12	NUM
iajs-155	124	3	:	:	PUNCT
iajs-155	124	4	suppose	suppose	VERB
iajs-155	124	5	m	m	PRON
iajs-155	124	6	is	be	AUX
iajs-155	124	7	a	a	DET
iajs-155	124	8	finitely	finitely	ADV
iajs-155	124	9	generated	generate	VERB
iajs-155	124	10	multiplication	multiplication	NOUN
iajs-155	124	11	r	r	NOUN
iajs-155	124	12	-	-	NOUN
iajs-155	124	13	module	module	NOUN
iajs-155	124	14	.	.	PUNCT
iajs-155	125	1	if	if	SCONJ
iajs-155	125	2	i	i	PRON
iajs-155	125	3	is	be	AUX
iajs-155	125	4	2	2	NUM
iajs-155	125	5	-	-	PUNCT
iajs-155	125	6	absorbing	absorbing	ADJ
iajs-155	125	7	ideal	ideal	NOUN
iajs-155	125	8	of	of	ADP
iajs-155	125	9	r	r	NOUN
iajs-155	125	10	such	such	ADJ
iajs-155	125	11	that	that	DET
iajs-155	125	12	annm	annm	NOUN
iajs-155	125	13	⊆i	⊆i	NOUN
iajs-155	125	14	,	,	PUNCT
iajs-155	125	15	then	then	ADV
iajs-155	125	16	i	i	PRON
iajs-155	125	17	m	m	VERB
iajs-155	125	18	is	be	AUX
iajs-155	125	19	2	2	NUM
iajs-155	125	20	-	-	PUNCT
iajs-155	125	21	absorbing	absorb	VERB
iajs-155	125	22	submodule	submodule	NOUN
iajs-155	125	23	of	of	ADP
iajs-155	125	24	m.	m.	NOUN
iajs-155	125	25	proof	proof	NOUN
iajs-155	125	26	:	:	PUNCT
iajs-155	125	27	let	let	VERB
iajs-155	125	28	abm	abm	PROPN
iajs-155	125	29	∈	∈	PROPN
iajs-155	126	1	i	i	PRON
iajs-155	126	2	m	m	VERB
iajs-155	126	3	where	where	SCONJ
iajs-155	126	4	a	a	DET
iajs-155	126	5	,	,	PUNCT
iajs-155	126	6	b	b	X
iajs-155	126	7	∈	∈	PROPN
iajs-155	126	8	r	r	NOUN
iajs-155	126	9	,	,	PUNCT
iajs-155	126	10	m	m	VERB
iajs-155	126	11	∈	∈	ADJ
iajs-155	126	12	m	m	NOUN
iajs-155	126	13	,	,	PUNCT
iajs-155	126	14	hence	hence	ADV
iajs-155	126	15	ab(m	ab(m	ADV
iajs-155	126	16	)	)	PUNCT
iajs-155	126	17	∈	∈	PROPN
iajs-155	127	1	i	i	PRON
iajs-155	127	2	m	m	VERB
iajs-155	127	3	.	.	PUNCT
iajs-155	128	1	since	since	SCONJ
iajs-155	128	2	m	m	PROPN
iajs-155	128	3	is	be	AUX
iajs-155	128	4	multiplication	multiplication	NOUN
iajs-155	128	5	,	,	PUNCT
iajs-155	128	6	then	then	ADV
iajs-155	128	7	(	(	PUNCT
iajs-155	128	8	m)=jm	m)=jm	NOUN
iajs-155	128	9	for	for	ADP
iajs-155	128	10	some	some	DET
iajs-155	128	11	ideal	ideal	ADJ
iajs-155	128	12	j	j	PROPN
iajs-155	128	13	of	of	ADP
iajs-155	128	14	r	r	NOUN
iajs-155	128	15	.	.	PUNCT
iajs-155	129	1	thus	thus	ADV
iajs-155	129	2	abjm	abjm	VERB
iajs-155	129	3	⊆	⊆	NUM
iajs-155	129	4	i	i	PRON
iajs-155	129	5	m	m	PROPN
iajs-155	129	6	and	and	CCONJ
iajs-155	129	7	so	so	ADV
iajs-155	129	8	abj	abj	PROPN
iajs-155	129	9	⊆	⊆	NUM
iajs-155	129	10	i+annm	i+annm	PROPN
iajs-155	129	11	=	=	SYM
iajs-155	130	1	i	i	PROPN
iajs-155	130	2	.	.	PUNCT
iajs-155	131	1	but	but	CCONJ
iajs-155	131	2	i	i	PRON
iajs-155	131	3	is	be	AUX
iajs-155	131	4	a	a	DET
iajs-155	131	5	2absorbing	2absorbing	NUM
iajs-155	131	6	ideal	ideal	NOUN
iajs-155	131	7	of	of	ADP
iajs-155	131	8	r	r	NOUN
iajs-155	131	9	,	,	PUNCT
iajs-155	131	10	so	so	CCONJ
iajs-155	131	11	either	either	CCONJ
iajs-155	131	12	ab	ab	PROPN
iajs-155	131	13	∈	∈	PROPN
iajs-155	131	14	i	i	PRON
iajs-155	131	15	or	or	CCONJ
iajs-155	131	16	aj	aj	PROPN
iajs-155	131	17	⊆	⊆	NUM
iajs-155	131	18	i	i	PRON
iajs-155	131	19	or	or	CCONJ
iajs-155	131	20	bj	bj	VERB
iajs-155	131	21	⊆	⊆	NUM
iajs-155	131	22	i	i	PRON
iajs-155	131	23	,	,	PUNCT
iajs-155	131	24	it	it	PRON
iajs-155	131	25	follows	follow	VERB
iajs-155	131	26	that	that	SCONJ
iajs-155	131	27	ab	ab	PROPN
iajs-155	131	28	∈	∈	PROPN
iajs-155	132	1	i	i	PRON
iajs-155	132	2	m	m	VERB
iajs-155	132	3	:	:	PUNCT
iajs-155	132	4	m	m	VERB
iajs-155	132	5	or	or	CCONJ
iajs-155	132	6	ajm	ajm	VERB
iajs-155	132	7	⊆	⊆	NUM
iajs-155	132	8	i	i	PRON
iajs-155	132	9	m	m	VERB
iajs-155	132	10	or	or	CCONJ
iajs-155	132	11	bjm	bjm	VERB
iajs-155	132	12	⊆	⊆	NUM
iajs-155	132	13	i	i	NOUN
iajs-155	132	14	m	m	VERB
iajs-155	132	15	;	;	PUNCT
iajs-155	132	16	that	that	PRON
iajs-155	132	17	is	be	AUX
iajs-155	132	18	either	either	CCONJ
iajs-155	132	19	ab	ab	PROPN
iajs-155	132	20	∈	∈	PROPN
iajs-155	133	1	i	i	PRON
iajs-155	133	2	m	m	VERB
iajs-155	133	3	:	:	PUNCT
iajs-155	133	4	m	m	VERB
iajs-155	133	5	or	or	CCONJ
iajs-155	133	6	a(m	a(m	ADJ
iajs-155	133	7	)	)	PUNCT
iajs-155	134	1	⊆	⊆	NUM
iajs-155	134	2	i	i	PRON
iajs-155	134	3	m	m	VERB
iajs-155	134	4	or	or	CCONJ
iajs-155	134	5	b(m	b(m	PROPN
iajs-155	134	6	)	)	PUNCT
iajs-155	134	7	⊆	⊆	NUM
iajs-155	135	1	i	i	PRON
iajs-155	135	2	m	m	VERB
iajs-155	135	3	thus	thus	ADV
iajs-155	135	4	ab	ab	PROPN
iajs-155	135	5	∈	∈	PROPN
iajs-155	136	1	i	i	PRON
iajs-155	136	2	m	m	VERB
iajs-155	136	3	:	:	PUNCT
iajs-155	136	4	m	m	VERB
iajs-155	136	5	or	or	CCONJ
iajs-155	136	6	am	be	AUX
iajs-155	136	7	∈	∈	PROPN
iajs-155	137	1	i	i	PRON
iajs-155	137	2	m	m	VERB
iajs-155	137	3	or	or	CCONJ
iajs-155	137	4	bm	bm	PROPN
iajs-155	137	5	∈	∈	PROPN
iajs-155	138	1	i	i	PRON
iajs-155	138	2	m	m	VERB
iajs-155	139	1	and	and	CCONJ
iajs-155	139	2	so	so	ADV
iajs-155	139	3	i	i	PRON
iajs-155	139	4	m	m	VERB
iajs-155	139	5	is	be	AUX
iajs-155	139	6	a	a	DET
iajs-155	139	7	2	2	NUM
iajs-155	139	8	-	-	PUNCT
iajs-155	139	9	absorbing	absorb	VERB
iajs-155	139	10	submodule	submodule	NOUN
iajs-155	139	11	of	of	ADP
iajs-155	139	12	m	m	PROPN
iajs-155	139	13	∎	∎	PROPN
iajs-155	139	14	229	229	NUM
iajs-155	140	1	|	|	ADV
iajs-155	140	2	mathematics	mathematic	NOUN
iajs-155	140	3	2015	2015	NUM
iajs-155	140	4	)	)	PUNCT
iajs-155	140	5	عام	عام	ADP
iajs-155	140	6	3العدد	3العدد	NUM
iajs-155	140	7	(	(	PUNCT
iajs-155	140	8	28مجلة	28مجلة	X
iajs-155	140	9	إبن	إبن	VERB
iajs-155	140	10	الھيثم	الھيثم	NOUN
iajs-155	140	11	للعلوم	للعلوم	NOUN
iajs-155	140	12	الصرفة	الصرفة	NOUN
iajs-155	141	1	و	و	PRON
iajs-155	141	2	التطبيقية	التطبيقية	ADV
iajs-155	141	3	المجلد	المجلد	VERB
iajs-155	141	4	ibn	ibn	PROPN
iajs-155	141	5	al	al	PROPN
iajs-155	141	6	-	-	PUNCT
iajs-155	141	7	haitham	haitham	PROPN
iajs-155	141	8	jour	jour	X
iajs-155	141	9	.	.	PROPN
iajs-155	142	1	for	for	ADP
iajs-155	142	2	pure	pure	ADJ
iajs-155	142	3	&	&	CCONJ
iajs-155	142	4	appl	appl	PROPN
iajs-155	142	5	.	.	PUNCT
iajs-155	143	1	sci	sci	PROPN
iajs-155	143	2	.	.	PUNCT
iajs-155	143	3	vol	vol	NOUN
iajs-155	143	4	.	.	PROPN
iajs-155	144	1	28	28	NUM
iajs-155	144	2	(	(	PUNCT
iajs-155	144	3	3	3	NUM
iajs-155	144	4	)	)	PUNCT
iajs-155	144	5	2015	2015	NUM
iajs-155	144	6	corollary	corollary	NOUN
iajs-155	144	7	1.13	1.13	NUM
iajs-155	144	8	:	:	PUNCT
iajs-155	144	9	suppose	suppose	VERB
iajs-155	144	10	m	m	PRON
iajs-155	144	11	is	be	AUX
iajs-155	144	12	a	a	DET
iajs-155	144	13	faithful	faithful	ADJ
iajs-155	144	14	finitely	finitely	ADV
iajs-155	144	15	generated	generate	VERB
iajs-155	144	16	multiplication	multiplication	NOUN
iajs-155	144	17	r	r	NOUN
iajs-155	144	18	-	-	NOUN
iajs-155	144	19	module	module	NOUN
iajs-155	144	20	.	.	PUNCT
iajs-155	145	1	if	if	SCONJ
iajs-155	145	2	i	i	PRON
iajs-155	145	3	is	be	AUX
iajs-155	145	4	a	a	DET
iajs-155	145	5	2	2	NUM
iajs-155	145	6	-	-	PUNCT
iajs-155	145	7	absorbing	absorbing	ADJ
iajs-155	145	8	ideal	ideal	NOUN
iajs-155	145	9	of	of	ADP
iajs-155	145	10	r	r	NOUN
iajs-155	145	11	,	,	PUNCT
iajs-155	145	12	then	then	ADV
iajs-155	145	13	i	i	PRON
iajs-155	145	14	m	m	VERB
iajs-155	145	15	is	be	AUX
iajs-155	145	16	a	a	DET
iajs-155	145	17	2	2	NUM
iajs-155	145	18	-	-	PUNCT
iajs-155	145	19	absorbing	absorb	VERB
iajs-155	145	20	submodule	submodule	NOUN
iajs-155	145	21	of	of	ADP
iajs-155	145	22	m.	m.	NOUN
iajs-155	145	23	proof	proof	NOUN
iajs-155	145	24	:	:	PUNCT
iajs-155	145	25	it	it	PRON
iajs-155	145	26	follows	follow	VERB
iajs-155	145	27	directly	directly	ADV
iajs-155	145	28	by	by	ADP
iajs-155	145	29	proposition	proposition	NOUN
iajs-155	145	30	1.12	1.12	NUM
iajs-155	145	31	∎	∎	PROPN
iajs-155	145	32	corollary	corollary	ADJ
iajs-155	145	33	1.14	1.14	NUM
iajs-155	145	34	:	:	PUNCT
iajs-155	145	35	let	let	VERB
iajs-155	145	36	m	m	PRON
iajs-155	145	37	be	be	AUX
iajs-155	145	38	a	a	DET
iajs-155	145	39	faithful	faithful	ADJ
iajs-155	145	40	finitely	finitely	ADV
iajs-155	145	41	generated	generate	VERB
iajs-155	145	42	multiplication	multiplication	NOUN
iajs-155	145	43	r	r	NOUN
iajs-155	145	44	-	-	NOUN
iajs-155	145	45	module	module	NOUN
iajs-155	145	46	.	.	PUNCT
iajs-155	146	1	then	then	ADV
iajs-155	146	2	every	every	DET
iajs-155	146	3	proper	proper	ADJ
iajs-155	146	4	submodule	submodule	NOUN
iajs-155	146	5	of	of	ADP
iajs-155	146	6	m	m	PROPN
iajs-155	146	7	is	be	AUX
iajs-155	146	8	2	2	NUM
iajs-155	146	9	-	-	PUNCT
iajs-155	146	10	absorbing	absorbing	ADJ
iajs-155	146	11	if	if	SCONJ
iajs-155	146	12	and	and	CCONJ
iajs-155	146	13	only	only	ADV
iajs-155	146	14	if	if	SCONJ
iajs-155	146	15	every	every	DET
iajs-155	146	16	proper	proper	ADJ
iajs-155	146	17	ideal	ideal	NOUN
iajs-155	146	18	of	of	ADP
iajs-155	146	19	r	r	NOUN
iajs-155	146	20	is	be	AUX
iajs-155	146	21	2	2	NUM
iajs-155	146	22	-	-	ADJ
iajs-155	146	23	absorbing	absorbing	ADJ
iajs-155	146	24	.	.	PUNCT
iajs-155	147	1	proof	proof	NOUN
iajs-155	147	2	:	:	PUNCT
iajs-155	147	3	⟸	⟸	ADJ
iajs-155	147	4	it	it	PRON
iajs-155	147	5	follows	follow	VERB
iajs-155	147	6	directly	directly	ADV
iajs-155	147	7	by	by	ADP
iajs-155	147	8	corollary	corollary	ADJ
iajs-155	147	9	(	(	PUNCT
iajs-155	147	10	1.13	1.13	NUM
iajs-155	147	11	)	)	PUNCT
iajs-155	147	12	.	.	PUNCT
iajs-155	148	1	⟹	⟹	PROPN
iajs-155	149	1	let	let	VERB
iajs-155	149	2	i	i	PRON
iajs-155	149	3	be	be	AUX
iajs-155	149	4	a	a	DET
iajs-155	149	5	proper	proper	ADJ
iajs-155	149	6	ideal	ideal	NOUN
iajs-155	149	7	of	of	ADP
iajs-155	149	8	r	r	NOUN
iajs-155	149	9	.	.	PUNCT
iajs-155	150	1	then	then	ADV
iajs-155	150	2	n=	n=	ADV
iajs-155	150	3	i	i	PRON
iajs-155	150	4	m	m	VERB
iajs-155	150	5	is	be	AUX
iajs-155	150	6	a	a	DET
iajs-155	150	7	proper	proper	ADJ
iajs-155	150	8	submodule	submodule	NOUN
iajs-155	150	9	of	of	ADP
iajs-155	150	10	m	m	PRON
iajs-155	150	11	so	so	ADV
iajs-155	150	12	it	it	PRON
iajs-155	150	13	is	be	AUX
iajs-155	150	14	2	2	NUM
iajs-155	150	15	-	-	PUNCT
iajs-155	150	16	absorbing	absorbing	ADJ
iajs-155	150	17	and	and	CCONJ
iajs-155	150	18	hence	hence	ADV
iajs-155	150	19	by	by	ADP
iajs-155	150	20	theorem	theorem	NOUN
iajs-155	150	21	(	(	PUNCT
iajs-155	150	22	1.7	1.7	NUM
iajs-155	150	23	)	)	PUNCT
iajs-155	150	24	,	,	PUNCT
iajs-155	150	25	(	(	PUNCT
iajs-155	150	26	n	n	CCONJ
iajs-155	150	27	:	:	PUNCT
iajs-155	150	28	m	m	X
iajs-155	150	29	)	)	PUNCT
iajs-155	150	30	is	be	AUX
iajs-155	150	31	2	2	NUM
iajs-155	150	32	-	-	PUNCT
iajs-155	150	33	absorbing	absorb	VERB
iajs-155	150	34	ideal	ideal	NOUN
iajs-155	150	35	.	.	PUNCT
iajs-155	151	1	but	but	CCONJ
iajs-155	151	2	m	m	PROPN
iajs-155	151	3	is	be	AUX
iajs-155	151	4	faithful	faithful	ADJ
iajs-155	151	5	finitely	finitely	ADV
iajs-155	151	6	generated	generate	VERB
iajs-155	151	7	multiplication	multiplication	NOUN
iajs-155	151	8	r	r	NOUN
iajs-155	151	9	-	-	NOUN
iajs-155	151	10	module	module	NOUN
iajs-155	151	11	,	,	PUNCT
iajs-155	151	12	so	so	CCONJ
iajs-155	151	13	(	(	PUNCT
iajs-155	151	14	n	n	CCONJ
iajs-155	151	15	:	:	PUNCT
iajs-155	151	16	m)=i	m)=i	VERB
iajs-155	151	17	by	by	ADP
iajs-155	151	18	(	(	PUNCT
iajs-155	151	19	[	[	X
iajs-155	151	20	6],th.3.1	6],th.3.1	ADJ
iajs-155	151	21	)	)	PUNCT
iajs-155	151	22	∎	∎	ADJ
iajs-155	151	23	"	"	PUNCT
iajs-155	151	24	proposition	proposition	NOUN
iajs-155	151	25	1.15	1.15	NUM
iajs-155	151	26	:	:	PUNCT
iajs-155	152	1	[	[	X
iajs-155	152	2	4,th	4,th	NUM
iajs-155	152	3	2.4	2.4	NUM
iajs-155	152	4	]	]	PUNCT
iajs-155	152	5	let	let	AUX
iajs-155	152	6	m	m	PRON
iajs-155	152	7	be	be	AUX
iajs-155	152	8	an	an	DET
iajs-155	152	9	r	r	NOUN
iajs-155	152	10	-	-	PUNCT
iajs-155	152	11	module	module	NOUN
iajs-155	152	12	,	,	PUNCT
iajs-155	152	13	n	n	CCONJ
iajs-155	152	14	a	a	DET
iajs-155	152	15	proper	proper	ADJ
iajs-155	152	16	submodule	submodule	NOUN
iajs-155	152	17	of	of	ADP
iajs-155	152	18	m	m	PRON
iajs-155	152	19	,	,	PUNCT
iajs-155	152	20	if	if	SCONJ
iajs-155	152	21	n	n	PRON
iajs-155	152	22	is	be	AUX
iajs-155	152	23	2	2	NUM
iajs-155	152	24	-	-	PUNCT
iajs-155	152	25	absorbing	absorbing	ADJ
iajs-155	152	26	then	then	ADV
iajs-155	152	27	n	n	CCONJ
iajs-155	152	28	:	:	PUNCT
iajs-155	152	29	m	m	VERB
iajs-155	152	30	is	be	AUX
iajs-155	152	31	2	2	NUM
iajs-155	152	32	-	-	PUNCT
iajs-155	152	33	absorbing	absorbing	ADJ
iajs-155	152	34	ideal	ideal	NOUN
iajs-155	152	35	for	for	ADP
iajs-155	152	36	each	each	DET
iajs-155	152	37	m	m	NOUN
iajs-155	152	38	∈m	∈m	NOUN
iajs-155	152	39	-	-	PUNCT
iajs-155	152	40	n	n	CCONJ
iajs-155	152	41	"	"	PUNCT
iajs-155	152	42	proof	proof	NOUN
iajs-155	152	43	:	:	PUNCT
iajs-155	152	44	first	first	ADJ
iajs-155	152	45	n	n	X
iajs-155	152	46	:	:	PUNCT
iajs-155	152	47	m	m	AUX
iajs-155	152	48	r	r	NOUN
iajs-155	152	49	for	for	ADP
iajs-155	152	50	any	any	DET
iajs-155	152	51	m	m	NOUN
iajs-155	152	52	∉	∉	PROPN
iajs-155	152	53	n	n	ADV
iajs-155	152	54	let	let	VERB
iajs-155	152	55	abc	abc	PROPN
iajs-155	152	56	∈	∈	PROPN
iajs-155	152	57	n	n	CCONJ
iajs-155	152	58	:	:	PUNCT
iajs-155	152	59	m	m	VERB
iajs-155	152	60	then	then	ADV
iajs-155	152	61	ab(cm	ab(cm	PROPN
iajs-155	152	62	)	)	PUNCT
iajs-155	153	1	∈	∈	PROPN
iajs-155	153	2	n	n	NOUN
iajs-155	153	3	,	,	PUNCT
iajs-155	153	4	but	but	CCONJ
iajs-155	153	5	n	n	PRON
iajs-155	153	6	is	be	AUX
iajs-155	153	7	2	2	NUM
iajs-155	153	8	-	-	PUNCT
iajs-155	153	9	absorbing	absorb	VERB
iajs-155	153	10	submodule	submodule	NOUN
iajs-155	153	11	then	then	ADV
iajs-155	153	12	a(cm	a(cm	PROPN
iajs-155	153	13	)	)	PUNCT
iajs-155	153	14	∈	∈	PROPN
iajs-155	153	15	n	n	PRON
iajs-155	153	16	or	or	CCONJ
iajs-155	153	17	b(cm	b(cm	NOUN
iajs-155	153	18	)	)	PUNCT
iajs-155	153	19	∈	∈	PROPN
iajs-155	153	20	n	n	NOUN
iajs-155	153	21	or	or	CCONJ
iajs-155	153	22	ab	ab	PROPN
iajs-155	153	23	∈	∈	PROPN
iajs-155	153	24	n	n	CCONJ
iajs-155	153	25	:	:	PUNCT
iajs-155	153	26	m	m	INTJ
iajs-155	153	27	,	,	PUNCT
iajs-155	153	28	so	so	SCONJ
iajs-155	153	29	that	that	SCONJ
iajs-155	153	30	acm	acm	PROPN
iajs-155	153	31	∈	∈	PROPN
iajs-155	153	32	n	n	NOUN
iajs-155	153	33	or	or	CCONJ
iajs-155	153	34	bcm	bcm	NOUN
iajs-155	153	35	∈	∈	PROPN
iajs-155	153	36	n	n	PRON
iajs-155	153	37	or	or	CCONJ
iajs-155	153	38	abm	abm	PROPN
iajs-155	153	39	⊆	⊆	NUM
iajs-155	153	40	n	n	PROPN
iajs-155	153	41	that	that	PRON
iajs-155	153	42	is	be	AUX
iajs-155	153	43	ac	ac	PROPN
iajs-155	153	44	∈	∈	PROPN
iajs-155	153	45	n	n	CCONJ
iajs-155	153	46	:	:	PUNCT
iajs-155	153	47	m	m	PROPN
iajs-155	153	48	or	or	CCONJ
iajs-155	153	49	bc	bc	PROPN
iajs-155	153	50	∈	∈	PROPN
iajs-155	153	51	n	n	CCONJ
iajs-155	153	52	:	:	PUNCT
iajs-155	153	53	m	m	VERB
iajs-155	153	54	or	or	CCONJ
iajs-155	153	55	ab	ab	PROPN
iajs-155	153	56	∈	∈	PROPN
iajs-155	154	1	n	n	CCONJ
iajs-155	154	2	:	:	PUNCT
iajs-155	154	3	m	m	VERB
iajs-155	154	4	hence	hence	ADV
iajs-155	154	5	n	n	ADJ
iajs-155	154	6	:	:	PUNCT
iajs-155	154	7	m	m	VERB
iajs-155	154	8	is	be	AUX
iajs-155	154	9	2	2	NUM
iajs-155	154	10	-	-	PUNCT
iajs-155	154	11	absorbing	absorbing	ADJ
iajs-155	154	12	ideal	ideal	NOUN
iajs-155	154	13	∎	∎	PROPN
iajs-155	154	14	recall	recall	VERB
iajs-155	154	15	that	that	SCONJ
iajs-155	154	16	"	"	PUNCT
iajs-155	154	17	a	a	DET
iajs-155	154	18	submodule	submodule	NOUN
iajs-155	154	19	n	n	PROPN
iajs-155	154	20	of	of	ADP
iajs-155	154	21	m	m	PROPN
iajs-155	154	22	is	be	AUX
iajs-155	154	23	called	call	VERB
iajs-155	154	24	a	a	DET
iajs-155	154	25	pure	pure	ADJ
iajs-155	154	26	submodule	submodule	NOUN
iajs-155	154	27	of	of	ADP
iajs-155	154	28	an	an	DET
iajs-155	154	29	r	r	NOUN
iajs-155	154	30	-	-	PUNCT
iajs-155	154	31	module	module	NOUN
iajs-155	154	32	m	m	NOUN
iajs-155	154	33	if	if	SCONJ
iajs-155	154	34	i	i	PRON
iajs-155	154	35	m	m	VERB
iajs-155	154	36	∩	∩	ADJ
iajs-155	154	37	n	n	NOUN
iajs-155	154	38	=	=	PUNCT
iajs-155	154	39	in	in	ADP
iajs-155	154	40	for	for	ADP
iajs-155	154	41	any	any	DET
iajs-155	154	42	ideal	ideal	NOUN
iajs-155	154	43	i	i	PRON
iajs-155	154	44	of	of	ADP
iajs-155	154	45	r	r	NOUN
iajs-155	154	46	"	"	PUNCT
iajs-155	154	47	[	[	X
iajs-155	154	48	9	9	NUM
iajs-155	154	49	]	]	PUNCT
iajs-155	154	50	proposition	proposition	NOUN
iajs-155	154	51	1.16	1.16	NUM
iajs-155	154	52	:	:	PUNCT
iajs-155	154	53	let	let	VERB
iajs-155	154	54	n	n	PRON
iajs-155	154	55	be	be	AUX
iajs-155	154	56	a	a	DET
iajs-155	154	57	proper	proper	ADJ
iajs-155	154	58	pure	pure	ADJ
iajs-155	154	59	submodule	submodule	NOUN
iajs-155	154	60	of	of	ADP
iajs-155	154	61	an	an	DET
iajs-155	154	62	r	r	NOUN
iajs-155	154	63	-	-	PUNCT
iajs-155	154	64	module	module	NOUN
iajs-155	154	65	m	m	NOUN
iajs-155	154	66	,	,	PUNCT
iajs-155	154	67	if	if	SCONJ
iajs-155	154	68	0	0	NUM
iajs-155	154	69	is	be	AUX
iajs-155	154	70	a	a	DET
iajs-155	154	71	2	2	NUM
iajs-155	154	72	-	-	PUNCT
iajs-155	154	73	absorbing	absorb	VERB
iajs-155	154	74	submodule	submodule	NOUN
iajs-155	154	75	of	of	ADP
iajs-155	154	76	m	m	PROPN
iajs-155	154	77	,	,	PUNCT
iajs-155	154	78	then	then	ADV
iajs-155	154	79	n	n	PROPN
iajs-155	154	80	is	be	AUX
iajs-155	154	81	2	2	NUM
iajs-155	154	82	-	-	PUNCT
iajs-155	154	83	absorbing	absorbing	ADJ
iajs-155	154	84	.	.	PUNCT
iajs-155	155	1	proof	proof	NOUN
iajs-155	155	2	:	:	PUNCT
iajs-155	155	3	let	let	VERB
iajs-155	155	4	abm	abm	PROPN
iajs-155	155	5	∈	∈	PROPN
iajs-155	155	6	n	n	CCONJ
iajs-155	155	7	where	where	SCONJ
iajs-155	155	8	a	a	DET
iajs-155	155	9	,	,	PUNCT
iajs-155	155	10	b	b	X
iajs-155	155	11	∈	∈	PROPN
iajs-155	155	12	r	r	NOUN
iajs-155	155	13	,	,	PUNCT
iajs-155	155	14	m	m	VERB
iajs-155	155	15	∈	∈	NOUN
iajs-155	155	16	m	m	AUX
iajs-155	155	17	put	put	VERB
iajs-155	155	18	i	i	PRON
iajs-155	155	19	=(	=(	PROPN
iajs-155	155	20	ab	ab	PROPN
iajs-155	155	21	)	)	PUNCT
iajs-155	156	1	then	then	ADV
iajs-155	156	2	abm	abm	PROPN
iajs-155	156	3	∈	∈	PROPN
iajs-155	157	1	i	i	PRON
iajs-155	157	2	m	m	VERB
iajs-155	157	3	∩	∩	ADJ
iajs-155	157	4	n	n	CCONJ
iajs-155	157	5	,	,	PUNCT
iajs-155	157	6	but	but	CCONJ
iajs-155	157	7	i	i	PRON
iajs-155	157	8	m	m	VERB
iajs-155	157	9	∩	∩	ADJ
iajs-155	157	10	n	n	NOUN
iajs-155	157	11	=	=	PUNCT
iajs-155	157	12	in	in	ADV
iajs-155	157	13	,	,	PUNCT
iajs-155	157	14	so	so	ADV
iajs-155	157	15	abm	abm	PROPN
iajs-155	157	16	=	=	NOUN
iajs-155	157	17	abn	abn	NOUN
iajs-155	157	18	for	for	ADP
iajs-155	157	19	some	some	DET
iajs-155	157	20	n	n	PRON
iajs-155	157	21	∈	∈	PROPN
iajs-155	157	22	n	n	NOUN
iajs-155	157	23	,	,	PUNCT
iajs-155	157	24	then	then	ADV
iajs-155	157	25	ab	ab	PROPN
iajs-155	157	26	(	(	PUNCT
iajs-155	157	27	m	m	PROPN
iajs-155	157	28	–	–	PUNCT
iajs-155	157	29	n	n	NOUN
iajs-155	157	30	)	)	PUNCT
iajs-155	158	1	=	=	SYM
iajs-155	158	2	0	0	NUM
iajs-155	158	3	,	,	PUNCT
iajs-155	158	4	but	but	CCONJ
iajs-155	158	5	(	(	PUNCT
iajs-155	158	6	o	o	NOUN
iajs-155	158	7	)	)	PUNCT
iajs-155	158	8	is	be	AUX
iajs-155	158	9	2	2	NUM
iajs-155	158	10	-	-	PUNCT
iajs-155	158	11	absorbing	absorbing	ADJ
iajs-155	158	12	then	then	ADV
iajs-155	158	13	a	a	DET
iajs-155	158	14	(	(	PUNCT
iajs-155	158	15	m	m	PROPN
iajs-155	158	16	–	–	PUNCT
iajs-155	158	17	n	n	NOUN
iajs-155	158	18	)	)	PUNCT
iajs-155	159	1	=	=	SYM
iajs-155	159	2	0	0	NUM
iajs-155	159	3	or	or	CCONJ
iajs-155	159	4	b	b	PROPN
iajs-155	159	5	(	(	PUNCT
iajs-155	159	6	m	m	PROPN
iajs-155	159	7	–	–	PUNCT
iajs-155	159	8	n	n	NOUN
iajs-155	159	9	)	)	PUNCT
iajs-155	159	10	=	=	SYM
iajs-155	159	11	0	0	NUM
iajs-155	159	12	or	or	CCONJ
iajs-155	159	13	ab	ab	PROPN
iajs-155	159	14	∈	∈	PROPN
iajs-155	159	15	annm	annm	NOUN
iajs-155	159	16	⊆	⊆	NUM
iajs-155	159	17	(	(	PUNCT
iajs-155	159	18	n	n	NUM
iajs-155	159	19	:	:	PUNCT
iajs-155	159	20	m	m	X
iajs-155	159	21	)	)	PUNCT
iajs-155	159	22	.	.	PUNCT
iajs-155	160	1	so	so	ADV
iajs-155	160	2	we	we	PRON
iajs-155	160	3	get	get	VERB
iajs-155	160	4	am	be	AUX
iajs-155	160	5	=	=	NOUN
iajs-155	160	6	an	an	DET
iajs-155	160	7	∈	∈	PROPN
iajs-155	160	8	n	n	NOUN
iajs-155	160	9	or	or	CCONJ
iajs-155	160	10	bm	bm	VERB
iajs-155	160	11	=	=	NOUN
iajs-155	160	12	bn	bn	NOUN
iajs-155	160	13	∈n	∈n	NOUN
iajs-155	160	14	or	or	CCONJ
iajs-155	160	15	ab	ab	PRON
iajs-155	160	16	∈	∈	PROPN
iajs-155	160	17	(	(	PUNCT
iajs-155	160	18	n	n	NOUN
iajs-155	160	19	:	:	PUNCT
iajs-155	160	20	m	m	X
iajs-155	160	21	)	)	PUNCT
iajs-155	160	22	,	,	PUNCT
iajs-155	160	23	thus	thus	ADV
iajs-155	160	24	n	n	PRON
iajs-155	160	25	is	be	AUX
iajs-155	160	26	a	a	DET
iajs-155	160	27	2	2	NUM
iajs-155	160	28	-	-	PUNCT
iajs-155	160	29	absorbing	absorb	VERB
iajs-155	160	30	submodule	submodule	NOUN
iajs-155	160	31	∎	∎	PROPN
iajs-155	160	32	2	2	NUM
iajs-155	160	33	.	.	NOUN
iajs-155	160	34	2	2	NUM
iajs-155	160	35	-	-	PUNCT
iajs-155	160	36	absorbing	absorb	VERB
iajs-155	160	37	submodules	submodule	NOUN
iajs-155	160	38	characterizations	characterization	NOUN
iajs-155	160	39	.	.	PUNCT
iajs-155	161	1	in	in	ADP
iajs-155	161	2	this	this	DET
iajs-155	161	3	section	section	NOUN
iajs-155	161	4	we	we	PRON
iajs-155	161	5	give	give	VERB
iajs-155	161	6	some	some	DET
iajs-155	161	7	characterizations	characterization	NOUN
iajs-155	161	8	of	of	ADP
iajs-155	161	9	2	2	NUM
iajs-155	161	10	-	-	PUNCT
iajs-155	161	11	absorbing	absorb	VERB
iajs-155	161	12	submodules	submodule	NOUN
iajs-155	161	13	.	.	PUNCT
iajs-155	162	1	we	we	PRON
iajs-155	162	2	start	start	VERB
iajs-155	162	3	with	with	ADP
iajs-155	162	4	the	the	DET
iajs-155	162	5	following	follow	VERB
iajs-155	162	6	proposition	proposition	NOUN
iajs-155	162	7	:	:	PUNCT
iajs-155	162	8	230	230	NUM
iajs-155	162	9	|	|	NOUN
iajs-155	162	10	mathematics	mathematic	NOUN
iajs-155	162	11	2015	2015	NUM
iajs-155	162	12	)	)	PUNCT
iajs-155	162	13	عام	عام	ADP
iajs-155	162	14	3العدد	3العدد	NUM
iajs-155	162	15	(	(	PUNCT
iajs-155	162	16	28مجلة	28مجلة	X
iajs-155	162	17	إبن	إبن	VERB
iajs-155	163	1	الھيثم	الھيثم	NOUN
iajs-155	163	2	للعلوم	للعلوم	NOUN
iajs-155	163	3	الصرفة	الصرفة	NOUN
iajs-155	164	1	و	و	PRON
iajs-155	164	2	التطبيقية	التطبيقية	ADV
iajs-155	164	3	المجلد	المجلد	VERB
iajs-155	164	4	ibn	ibn	PROPN
iajs-155	164	5	al	al	PROPN
iajs-155	164	6	-	-	PUNCT
iajs-155	164	7	haitham	haitham	PROPN
iajs-155	164	8	jour	jour	X
iajs-155	164	9	.	.	PROPN
iajs-155	165	1	for	for	ADP
iajs-155	165	2	pure	pure	ADJ
iajs-155	165	3	&	&	CCONJ
iajs-155	165	4	appl	appl	PROPN
iajs-155	165	5	.	.	PUNCT
iajs-155	166	1	sci	sci	PROPN
iajs-155	166	2	.	.	PUNCT
iajs-155	166	3	vol	vol	NOUN
iajs-155	166	4	.	.	PROPN
iajs-155	167	1	28	28	NUM
iajs-155	167	2	(	(	PUNCT
iajs-155	167	3	3	3	NUM
iajs-155	167	4	)	)	PUNCT
iajs-155	167	5	2015	2015	NUM
iajs-155	167	6	proposition	proposition	NOUN
iajs-155	167	7	2.1	2.1	NUM
iajs-155	167	8	:	:	PUNCT
iajs-155	167	9	let	let	VERB
iajs-155	167	10	n	n	PRON
iajs-155	167	11	a	a	DET
iajs-155	167	12	proper	proper	ADJ
iajs-155	167	13	submodule	submodule	NOUN
iajs-155	167	14	of	of	ADP
iajs-155	167	15	an	an	DET
iajs-155	167	16	r	r	NOUN
iajs-155	167	17	-	-	PUNCT
iajs-155	167	18	module	module	NOUN
iajs-155	167	19	m	m	NOUN
iajs-155	167	20	.	.	PUNCT
iajs-155	168	1	then	then	ADV
iajs-155	168	2	n	n	PROPN
iajs-155	168	3	is	be	AUX
iajs-155	168	4	2	2	NUM
iajs-155	168	5	-	-	PUNCT
iajs-155	168	6	absorbing	absorb	VERB
iajs-155	168	7	submodule	submodule	NOUN
iajs-155	168	8	of	of	ADP
iajs-155	168	9	m	m	PROPN
iajs-155	168	10	if	if	SCONJ
iajs-155	169	1	and	and	CCONJ
iajs-155	169	2	only	only	ADV
iajs-155	169	3	if	if	SCONJ
iajs-155	169	4	abk	abk	PROPN
iajs-155	169	5	⊆n	⊆n	ADJ
iajs-155	169	6	for	for	ADP
iajs-155	169	7	some	some	PRON
iajs-155	169	8	a	a	DET
iajs-155	169	9	,	,	PUNCT
iajs-155	169	10	b	b	X
iajs-155	169	11	∈r	∈r	NOUN
iajs-155	169	12	,	,	PUNCT
iajs-155	169	13	k	k	PROPN
iajs-155	169	14	m	m	PROPN
iajs-155	169	15	.	.	PUNCT
iajs-155	170	1	implies	imply	VERB
iajs-155	170	2	ab∈(n	ab∈(n	PROPN
iajs-155	170	3	:	:	PUNCT
iajs-155	170	4	m	m	NUM
iajs-155	170	5	)	)	PUNCT
iajs-155	170	6	,	,	PUNCT
iajs-155	170	7	or	or	CCONJ
iajs-155	170	8	ak	ak	PROPN
iajs-155	170	9	⊆n	⊆n	ADJ
iajs-155	170	10	or	or	CCONJ
iajs-155	170	11	bk	bk	VERB
iajs-155	170	12	⊆n	⊆n	ADJ
iajs-155	170	13	proof	proof	NOUN
iajs-155	170	14	:	:	PUNCT
iajs-155	170	15	suppose	suppose	VERB
iajs-155	170	16	that	that	SCONJ
iajs-155	170	17	ab	ab	PROPN
iajs-155	170	18	∉(n	∉(n	PROPN
iajs-155	170	19	:	:	PUNCT
iajs-155	170	20	m	m	X
iajs-155	170	21	)	)	PUNCT
iajs-155	170	22	and	and	CCONJ
iajs-155	170	23	ak	ak	PROPN
iajs-155	170	24	⊈	⊈	PROPN
iajs-155	170	25	n	n	CCONJ
iajs-155	170	26	and	and	CCONJ
iajs-155	170	27	bk	bk	VERB
iajs-155	170	28	⊈	⊈	PROPN
iajs-155	170	29	n	n	NOUN
iajs-155	170	30	.	.	PUNCT
iajs-155	171	1	then	then	ADV
iajs-155	171	2	there	there	PRON
iajs-155	171	3	exist	exist	VERB
iajs-155	171	4	m1	m1	PROPN
iajs-155	171	5	,	,	PUNCT
iajs-155	171	6	m2	m2	PROPN
iajs-155	171	7	in	in	ADP
iajs-155	171	8	k	k	PROPN
iajs-155	171	9	such	such	ADJ
iajs-155	171	10	that	that	SCONJ
iajs-155	171	11	am1	am1	PROPN
iajs-155	171	12	∉	∉	PROPN
iajs-155	171	13	n	n	PROPN
iajs-155	171	14	and	and	CCONJ
iajs-155	171	15	bm2	bm2	PROPN
iajs-155	171	16	∉	∉	PROPN
iajs-155	171	17	n	n	PROPN
iajs-155	171	18	.	.	PUNCT
iajs-155	172	1	since	since	SCONJ
iajs-155	172	2	abm1	abm1	PROPN
iajs-155	172	3	∈	∈	PROPN
iajs-155	172	4	n	n	PROPN
iajs-155	172	5	and	and	CCONJ
iajs-155	172	6	ab	ab	PROPN
iajs-155	172	7	∉(n	∉(n	PROPN
iajs-155	172	8	:	:	PUNCT
iajs-155	172	9	m	m	PROPN
iajs-155	172	10	)	)	PUNCT
iajs-155	172	11	,	,	PUNCT
iajs-155	172	12	am1	am1	PROPN
iajs-155	172	13	∉	∉	PROPN
iajs-155	172	14	n	n	PROPN
iajs-155	172	15	,	,	PUNCT
iajs-155	172	16	we	we	PRON
iajs-155	172	17	get	get	VERB
iajs-155	172	18	bm1	bm1	PROPN
iajs-155	172	19	∈	∈	PROPN
iajs-155	172	20	n.	n.	NOUN
iajs-155	172	21	also	also	ADV
iajs-155	172	22	since	since	SCONJ
iajs-155	172	23	abm2	abm2	ADJ
iajs-155	172	24	∈n	∈n	NOUN
iajs-155	172	25	and	and	CCONJ
iajs-155	172	26	ab	ab	PROPN
iajs-155	172	27	∉(n	∉(n	PROPN
iajs-155	172	28	:	:	PUNCT
iajs-155	172	29	m	m	PROPN
iajs-155	172	30	)	)	PUNCT
iajs-155	172	31	,	,	PUNCT
iajs-155	172	32	bm2	bm2	NOUN
iajs-155	172	33	∉	∉	PROPN
iajs-155	172	34	n	n	PROPN
iajs-155	172	35	,	,	PUNCT
iajs-155	172	36	we	we	PRON
iajs-155	172	37	get	get	VERB
iajs-155	172	38	am2	am2	DET
iajs-155	172	39	∈	∈	NOUN
iajs-155	172	40	n	n	CCONJ
iajs-155	172	41	now	now	ADV
iajs-155	172	42	,	,	PUNCT
iajs-155	172	43	since	since	SCONJ
iajs-155	172	44	ab(m1	ab(m1	PROPN
iajs-155	172	45	+	+	CCONJ
iajs-155	172	46	m2	m2	PROPN
iajs-155	172	47	)	)	PUNCT
iajs-155	172	48	∈	∈	PROPN
iajs-155	172	49	n	n	NOUN
iajs-155	172	50	and	and	CCONJ
iajs-155	172	51	ab	ab	PROPN
iajs-155	172	52	∉(n	∉(n	PROPN
iajs-155	172	53	:	:	PUNCT
iajs-155	172	54	m	m	X
iajs-155	172	55	)	)	PUNCT
iajs-155	172	56	we	we	PRON
iajs-155	172	57	have	have	VERB
iajs-155	172	58	a(m1	a(m1	NOUN
iajs-155	172	59	+	+	CCONJ
iajs-155	172	60	m2	m2	PROPN
iajs-155	172	61	)	)	PUNCT
iajs-155	172	62	∈	∈	PROPN
iajs-155	172	63	n	n	NOUN
iajs-155	172	64	or	or	CCONJ
iajs-155	172	65	b(m1	b(m1	NOUN
iajs-155	172	66	+	+	CCONJ
iajs-155	172	67	m2	m2	PROPN
iajs-155	172	68	)	)	PUNCT
iajs-155	172	69	∈	∈	PROPN
iajs-155	173	1	n	n	CCONJ
iajs-155	173	2	if	if	SCONJ
iajs-155	173	3	a(m1	a(m1	NOUN
iajs-155	173	4	+	+	CCONJ
iajs-155	173	5	m2	m2	PROPN
iajs-155	173	6	)	)	PUNCT
iajs-155	173	7	∈	∈	PROPN
iajs-155	173	8	n	n	NOUN
iajs-155	173	9	;	;	PUNCT
iajs-155	173	10	i.e.	i.e.	X
iajs-155	173	11	am1	am1	X
iajs-155	174	1	+	+	CCONJ
iajs-155	174	2	am2	am2	X
iajs-155	174	3	∈	∈	NOUN
iajs-155	174	4	n	n	NOUN
iajs-155	174	5	and	and	CCONJ
iajs-155	174	6	since	since	SCONJ
iajs-155	174	7	am2	am2	DET
iajs-155	174	8	∈	∈	PROPN
iajs-155	174	9	n	n	CCONJ
iajs-155	174	10	we	we	PRON
iajs-155	174	11	get	get	VERB
iajs-155	174	12	am1	am1	PROPN
iajs-155	174	13	∈	∈	PROPN
iajs-155	174	14	n	n	PRON
iajs-155	174	15	which	which	PRON
iajs-155	174	16	is	be	AUX
iajs-155	174	17	contradiction	contradiction	NOUN
iajs-155	174	18	!	!	PUNCT
iajs-155	175	1	if	if	SCONJ
iajs-155	175	2	b(m1	b(m1	NOUN
iajs-155	175	3	+	+	CCONJ
iajs-155	175	4	m2	m2	PROPN
iajs-155	175	5	)	)	PUNCT
iajs-155	175	6	∈	∈	PROPN
iajs-155	175	7	n	n	NOUN
iajs-155	175	8	;	;	PUNCT
iajs-155	175	9	i.e.	i.e.	X
iajs-155	175	10	bm1	bm1	X
iajs-155	175	11	+	+	NOUN
iajs-155	175	12	bm2	bm2	NOUN
iajs-155	175	13	∈	∈	NOUN
iajs-155	175	14	n	n	NOUN
iajs-155	175	15	and	and	CCONJ
iajs-155	175	16	since	since	SCONJ
iajs-155	175	17	bm2	bm2	NOUN
iajs-155	175	18	∈	∈	PROPN
iajs-155	175	19	n	n	CCONJ
iajs-155	175	20	we	we	PRON
iajs-155	175	21	get	get	VERB
iajs-155	175	22	bm1	bm1	ADJ
iajs-155	175	23	∈	∈	PROPN
iajs-155	175	24	n	n	PRON
iajs-155	175	25	which	which	PRON
iajs-155	175	26	is	be	AUX
iajs-155	175	27	contradiction	contradiction	NOUN
iajs-155	175	28	!	!	PUNCT
iajs-155	176	1	then	then	ADV
iajs-155	176	2	either	either	CCONJ
iajs-155	176	3	ab	ab	PROPN
iajs-155	176	4	∈(n	∈(n	PROPN
iajs-155	176	5	:	:	PUNCT
iajs-155	176	6	m	m	NUM
iajs-155	176	7	)	)	PUNCT
iajs-155	176	8	or	or	CCONJ
iajs-155	176	9	ak	ak	PROPN
iajs-155	176	10	⊆	⊆	NUM
iajs-155	176	11	n	n	PROPN
iajs-155	176	12	and	and	CCONJ
iajs-155	176	13	bk	bk	VERB
iajs-155	176	14	⊆	⊆	NUM
iajs-155	176	15	n	n	PRON
iajs-155	176	16	the	the	DET
iajs-155	176	17	converse	converse	NOUN
iajs-155	176	18	is	be	AUX
iajs-155	176	19	clear	clear	ADJ
iajs-155	176	20	∎	∎	PROPN
iajs-155	176	21	the	the	DET
iajs-155	176	22	following	follow	VERB
iajs-155	176	23	theorem	theorem	NOUN
iajs-155	176	24	give	give	VERB
iajs-155	176	25	a	a	DET
iajs-155	176	26	useful	useful	ADJ
iajs-155	176	27	characterization	characterization	NOUN
iajs-155	176	28	of	of	ADP
iajs-155	176	29	2	2	NUM
iajs-155	176	30	-	-	PUNCT
iajs-155	176	31	absorbing	absorb	VERB
iajs-155	176	32	submodule	submodule	NOUN
iajs-155	176	33	.	.	PUNCT
iajs-155	177	1	theorem	theorem	VERB
iajs-155	177	2	2.2	2.2	NUM
iajs-155	177	3	:	:	PUNCT
iajs-155	177	4	let	let	VERB
iajs-155	177	5	n	n	PRON
iajs-155	177	6	a	a	DET
iajs-155	177	7	proper	proper	ADJ
iajs-155	177	8	submodule	submodule	NOUN
iajs-155	177	9	of	of	ADP
iajs-155	177	10	an	an	DET
iajs-155	177	11	r	r	NOUN
iajs-155	177	12	-	-	PUNCT
iajs-155	177	13	module	module	NOUN
iajs-155	177	14	m	m	NOUN
iajs-155	177	15	,	,	PUNCT
iajs-155	177	16	then	then	ADV
iajs-155	177	17	the	the	DET
iajs-155	177	18	following	follow	VERB
iajs-155	177	19	statement	statement	NOUN
iajs-155	177	20	are	be	AUX
iajs-155	177	21	equivalent	equivalent	ADJ
iajs-155	177	22	:	:	PUNCT
iajs-155	177	23	(	(	PUNCT
iajs-155	177	24	1	1	X
iajs-155	177	25	)	)	PUNCT
iajs-155	177	26	n	n	PRON
iajs-155	177	27	is	be	AUX
iajs-155	177	28	a	a	DET
iajs-155	177	29	2	2	NUM
iajs-155	177	30	-	-	PUNCT
iajs-155	177	31	absorbing	absorb	VERB
iajs-155	177	32	submodule	submodule	NOUN
iajs-155	177	33	of	of	ADP
iajs-155	177	34	m	m	PROPN
iajs-155	177	35	(	(	PUNCT
iajs-155	177	36	2	2	NUM
iajs-155	177	37	)	)	PUNCT
iajs-155	178	1	if	if	SCONJ
iajs-155	178	2	i	i	PRON
iajs-155	178	3	j	j	PROPN
iajs-155	178	4	k	k	PROPN
iajs-155	178	5	⊆	⊆	NUM
iajs-155	178	6	n	n	CCONJ
iajs-155	178	7	,	,	PUNCT
iajs-155	178	8	for	for	ADP
iajs-155	178	9	some	some	DET
iajs-155	178	10	ideal	ideal	NOUN
iajs-155	178	11	i	i	PRON
iajs-155	178	12	and	and	CCONJ
iajs-155	178	13	j	j	PROPN
iajs-155	178	14	of	of	ADP
iajs-155	178	15	r	r	NOUN
iajs-155	178	16	and	and	CCONJ
iajs-155	178	17	some	some	DET
iajs-155	178	18	submodule	submodule	NOUN
iajs-155	178	19	k	k	PROPN
iajs-155	178	20	of	of	ADP
iajs-155	178	21	m	m	PRON
iajs-155	178	22	then	then	ADV
iajs-155	178	23	either	either	CCONJ
iajs-155	178	24	i	i	PRON
iajs-155	178	25	k	k	PROPN
iajs-155	178	26	⊆	⊆	NUM
iajs-155	178	27	n	n	PROPN
iajs-155	178	28	or	or	CCONJ
iajs-155	178	29	j	j	PROPN
iajs-155	178	30	k⊆	k⊆	PROPN
iajs-155	178	31	n	n	CCONJ
iajs-155	178	32	or	or	CCONJ
iajs-155	178	33	i	i	PRON
iajs-155	178	34	j	j	PROPN
iajs-155	178	35	⊆	⊆	X
iajs-155	178	36	(	(	PUNCT
iajs-155	178	37	n	n	CCONJ
iajs-155	178	38	:	:	PUNCT
iajs-155	178	39	m	m	NUM
iajs-155	178	40	)	)	PUNCT
iajs-155	178	41	.	.	PUNCT
iajs-155	179	1	proof	proof	NOUN
iajs-155	179	2	:	:	PUNCT
iajs-155	179	3	(	(	PUNCT
iajs-155	179	4	1	1	X
iajs-155	179	5	)	)	PUNCT
iajs-155	179	6	⟹	⟹	NOUN
iajs-155	179	7	(	(	PUNCT
iajs-155	179	8	2	2	X
iajs-155	179	9	)	)	PUNCT
iajs-155	179	10	suppose	suppose	VERB
iajs-155	179	11	n	n	PRON
iajs-155	179	12	is	be	AUX
iajs-155	179	13	a	a	DET
iajs-155	179	14	2	2	NUM
iajs-155	179	15	-	-	PUNCT
iajs-155	179	16	absorbing	absorb	VERB
iajs-155	179	17	submodule	submodule	NOUN
iajs-155	179	18	of	of	ADP
iajs-155	179	19	m	m	PROPN
iajs-155	180	1	and	and	CCONJ
iajs-155	180	2	i	i	PRON
iajs-155	180	3	j	j	PROPN
iajs-155	181	1	k	k	PROPN
iajs-155	181	2	⊆	⊆	NUM
iajs-155	181	3	n	n	PROPN
iajs-155	181	4	for	for	ADP
iajs-155	181	5	some	some	DET
iajs-155	181	6	ideals	ideal	NOUN
iajs-155	181	7	i	i	PRON
iajs-155	181	8	and	and	CCONJ
iajs-155	181	9	j	j	PROPN
iajs-155	181	10	of	of	ADP
iajs-155	181	11	r	r	NOUN
iajs-155	181	12	and	and	CCONJ
iajs-155	181	13	some	some	DET
iajs-155	181	14	submodule	submodule	NOUN
iajs-155	181	15	k	k	PROPN
iajs-155	181	16	of	of	ADP
iajs-155	181	17	m	m	PROPN
iajs-155	182	1	and	and	CCONJ
iajs-155	182	2	i	i	PRON
iajs-155	182	3	j	j	PROPN
iajs-155	182	4	⊈	⊈	PROPN
iajs-155	182	5	(	(	PUNCT
iajs-155	182	6	n	n	CCONJ
iajs-155	182	7	:	:	PUNCT
iajs-155	182	8	m	m	NUM
iajs-155	182	9	)	)	PUNCT
iajs-155	182	10	.	.	PUNCT
iajs-155	183	1	to	to	PART
iajs-155	183	2	show	show	VERB
iajs-155	183	3	that	that	SCONJ
iajs-155	183	4	i	i	PRON
iajs-155	183	5	k	k	PROPN
iajs-155	183	6	⊆	⊆	NUM
iajs-155	183	7	n	n	PROPN
iajs-155	183	8	or	or	CCONJ
iajs-155	183	9	j	j	PROPN
iajs-155	183	10	k	k	PROPN
iajs-155	183	11	⊆	⊆	NUM
iajs-155	183	12	n.	n.	NOUN
iajs-155	183	13	suppose	suppose	VERB
iajs-155	183	14	i	i	PRON
iajs-155	183	15	k	k	PROPN
iajs-155	183	16	⊈	⊈	PROPN
iajs-155	183	17	n	n	PROPN
iajs-155	183	18	and	and	CCONJ
iajs-155	183	19	j	j	PROPN
iajs-155	184	1	k	k	PROPN
iajs-155	184	2	⊈	⊈	PROPN
iajs-155	185	1	n	n	CCONJ
iajs-155	185	2	.	.	PUNCT
iajs-155	186	1	then	then	ADV
iajs-155	186	2	there	there	PRON
iajs-155	186	3	exist	exist	VERB
iajs-155	186	4	a1	a1	NOUN
iajs-155	186	5	∈	∈	PROPN
iajs-155	187	1	i	i	PRON
iajs-155	187	2	and	and	CCONJ
iajs-155	187	3	a2	a2	PROPN
iajs-155	187	4	∈	∈	PROPN
iajs-155	187	5	j	j	PROPN
iajs-155	187	6	such	such	ADJ
iajs-155	187	7	that	that	DET
iajs-155	187	8	a1	a1	NOUN
iajs-155	187	9	k	k	X
iajs-155	187	10	⊈	⊈	PROPN
iajs-155	187	11	n	n	PROPN
iajs-155	187	12	and	and	CCONJ
iajs-155	187	13	a2	a2	PROPN
iajs-155	187	14	k	k	X
iajs-155	187	15	⊈	⊈	PROPN
iajs-155	187	16	n	n	CCONJ
iajs-155	187	17	.	.	PUNCT
iajs-155	188	1	but	but	CCONJ
iajs-155	188	2	a1	a1	PROPN
iajs-155	188	3	a2	a2	PROPN
iajs-155	188	4	k	k	PROPN
iajs-155	188	5	⊆	⊆	NUM
iajs-155	188	6	n	n	NUM
iajs-155	188	7	and	and	CCONJ
iajs-155	188	8	neither	neither	CCONJ
iajs-155	188	9	a1	a1	NOUN
iajs-155	188	10	k	k	PROPN
iajs-155	188	11	⊈	⊈	PROPN
iajs-155	188	12	n	n	CCONJ
iajs-155	188	13	nor	nor	CCONJ
iajs-155	188	14	a2	a2	PROPN
iajs-155	188	15	k	k	X
iajs-155	188	16	⊈	⊈	PROPN
iajs-155	188	17	n	n	NOUN
iajs-155	188	18	and	and	CCONJ
iajs-155	188	19	n	n	PROPN
iajs-155	188	20	is	be	AUX
iajs-155	188	21	2	2	NUM
iajs-155	188	22	-	-	ADJ
iajs-155	188	23	absorbing	absorbing	ADJ
iajs-155	188	24	,	,	PUNCT
iajs-155	188	25	so	so	ADV
iajs-155	188	26	we	we	PRON
iajs-155	188	27	have	have	VERB
iajs-155	188	28	a1	a1	NOUN
iajs-155	188	29	a2	a2	PROPN
iajs-155	188	30	∈	∈	PROPN
iajs-155	188	31	(	(	PUNCT
iajs-155	188	32	n	n	CCONJ
iajs-155	188	33	:	:	PUNCT
iajs-155	188	34	m	m	VERB
iajs-155	188	35	)	)	PUNCT
iajs-155	188	36	by	by	ADP
iajs-155	188	37	proposition	proposition	NOUN
iajs-155	188	38	(	(	PUNCT
iajs-155	188	39	2.1	2.1	NUM
iajs-155	188	40	)	)	PUNCT
iajs-155	188	41	since	since	SCONJ
iajs-155	188	42	i	i	PRON
iajs-155	188	43	j	j	PROPN
iajs-155	188	44	⊈	⊈	PROPN
iajs-155	188	45	(	(	PUNCT
iajs-155	188	46	n	n	CCONJ
iajs-155	188	47	:	:	PUNCT
iajs-155	188	48	m	m	NUM
iajs-155	188	49	)	)	PUNCT
iajs-155	188	50	,	,	PUNCT
iajs-155	188	51	then	then	ADV
iajs-155	188	52	there	there	PRON
iajs-155	188	53	exist	exist	VERB
iajs-155	188	54	b1	b1	NOUN
iajs-155	188	55	∈	∈	PROPN
iajs-155	188	56	i	i	PRON
iajs-155	188	57	and	and	CCONJ
iajs-155	188	58	b2	b2	PROPN
iajs-155	188	59	∈	∈	PROPN
iajs-155	188	60	j	j	NOUN
iajs-155	188	61	such	such	ADJ
iajs-155	188	62	that	that	DET
iajs-155	188	63	b1	b1	NOUN
iajs-155	188	64	b2	b2	NOUN
iajs-155	188	65	∉(n	∉(n	PROPN
iajs-155	188	66	:	:	PUNCT
iajs-155	188	67	m	m	NOUN
iajs-155	188	68	)	)	PUNCT
iajs-155	188	69	.	.	PUNCT
iajs-155	189	1	but	but	CCONJ
iajs-155	189	2	b1	b1	VERB
iajs-155	189	3	b2k	b2k	PROPN
iajs-155	189	4	⊆	⊆	NUM
iajs-155	189	5	n	n	NOUN
iajs-155	189	6	,	,	PUNCT
iajs-155	189	7	so	so	ADV
iajs-155	189	8	we	we	PRON
iajs-155	189	9	have	have	VERB
iajs-155	189	10	b1k	b1k	PROPN
iajs-155	189	11	⊆	⊆	NUM
iajs-155	189	12	n	n	ADJ
iajs-155	189	13	or	or	CCONJ
iajs-155	189	14	b2k	b2k	PROPN
iajs-155	189	15	⊆	⊆	NUM
iajs-155	189	16	n	n	X
iajs-155	189	17	by	by	ADP
iajs-155	189	18	proposition	proposition	NOUN
iajs-155	189	19	(	(	PUNCT
iajs-155	189	20	2.1	2.1	NUM
iajs-155	189	21	)	)	PUNCT
iajs-155	189	22	now	now	ADV
iajs-155	189	23	we	we	PRON
iajs-155	189	24	have	have	VERB
iajs-155	189	25	the	the	DET
iajs-155	189	26	following	follow	VERB
iajs-155	189	27	cases	case	NOUN
iajs-155	189	28	:	:	PUNCT
iajs-155	189	29	case	case	NOUN
iajs-155	189	30	(	(	PUNCT
iajs-155	189	31	1	1	NUM
iajs-155	189	32	)	)	PUNCT
iajs-155	189	33	b1k	b1k	NOUN
iajs-155	190	1	⊆	⊆	NUM
iajs-155	190	2	n	n	NOUN
iajs-155	190	3	and	and	CCONJ
iajs-155	190	4	b2k	b2k	PROPN
iajs-155	190	5	⊈	⊈	PROPN
iajs-155	190	6	n	n	PRON
iajs-155	190	7	231	231	NUM
iajs-155	190	8	|	|	NOUN
iajs-155	190	9	mathematics	mathematic	NOUN
iajs-155	190	10	2015	2015	NUM
iajs-155	190	11	)	)	PUNCT
iajs-155	190	12	عام	عام	ADP
iajs-155	190	13	3العدد	3العدد	NUM
iajs-155	190	14	(	(	PUNCT
iajs-155	190	15	28مجلة	28مجلة	X
iajs-155	190	16	إبن	إبن	VERB
iajs-155	190	17	الھيثم	الھيثم	NOUN
iajs-155	190	18	للعلوم	للعلوم	NOUN
iajs-155	190	19	الصرفة	الصرفة	NOUN
iajs-155	191	1	و	و	PRON
iajs-155	191	2	التطبيقية	التطبيقية	ADV
iajs-155	191	3	المجلد	المجلد	VERB
iajs-155	191	4	ibn	ibn	PROPN
iajs-155	191	5	al	al	PROPN
iajs-155	191	6	-	-	PUNCT
iajs-155	191	7	haitham	haitham	PROPN
iajs-155	191	8	jour	jour	X
iajs-155	191	9	.	.	PROPN
iajs-155	192	1	for	for	ADP
iajs-155	192	2	pure	pure	ADJ
iajs-155	192	3	&	&	CCONJ
iajs-155	192	4	appl	appl	PROPN
iajs-155	192	5	.	.	PUNCT
iajs-155	193	1	sci	sci	PROPN
iajs-155	193	2	.	.	PUNCT
iajs-155	193	3	vol	vol	NOUN
iajs-155	193	4	.	.	PROPN
iajs-155	194	1	28	28	NUM
iajs-155	194	2	(	(	PUNCT
iajs-155	194	3	3	3	NUM
iajs-155	194	4	)	)	PUNCT
iajs-155	194	5	2015	2015	NUM
iajs-155	194	6	since	since	SCONJ
iajs-155	194	7	a1b2k	a1b2k	NUM
iajs-155	194	8	⊆	⊆	NUM
iajs-155	194	9	n	n	NUM
iajs-155	194	10	and	and	CCONJ
iajs-155	194	11	b2k	b2k	PROPN
iajs-155	194	12	⊈	⊈	PROPN
iajs-155	194	13	n	n	CCONJ
iajs-155	194	14	and	and	CCONJ
iajs-155	194	15	a1	a1	PROPN
iajs-155	194	16	k	k	PROPN
iajs-155	194	17	⊈	⊈	PROPN
iajs-155	195	1	n	n	NOUN
iajs-155	195	2	so	so	SCONJ
iajs-155	195	3	that	that	SCONJ
iajs-155	195	4	a1	a1	NOUN
iajs-155	195	5	b2	b2	NOUN
iajs-155	195	6	∈	∈	PROPN
iajs-155	195	7	(	(	PUNCT
iajs-155	195	8	n	n	NUM
iajs-155	195	9	:	:	PUNCT
iajs-155	195	10	m	m	VERB
iajs-155	195	11	)	)	PUNCT
iajs-155	195	12	by	by	ADP
iajs-155	195	13	proposition	proposition	NOUN
iajs-155	195	14	(	(	PUNCT
iajs-155	195	15	2.1	2.1	NUM
iajs-155	195	16	)	)	PUNCT
iajs-155	195	17	since	since	SCONJ
iajs-155	195	18	b1k	b1k	PROPN
iajs-155	195	19	⊆	⊆	PROPN
iajs-155	195	20	n	n	NOUN
iajs-155	195	21	and	and	CCONJ
iajs-155	195	22	a1k	a1k	VERB
iajs-155	195	23	⊈	⊈	NUM
iajs-155	195	24	n	n	X
iajs-155	195	25	,	,	PUNCT
iajs-155	195	26	we	we	PRON
iajs-155	195	27	conclude	conclude	VERB
iajs-155	195	28	(	(	PUNCT
iajs-155	195	29	a1+b1	a1+b1	PROPN
iajs-155	195	30	)	)	PUNCT
iajs-155	195	31	k	k	NOUN
iajs-155	196	1	⊈	⊈	PROPN
iajs-155	196	2	n	n	CCONJ
iajs-155	196	3	.	.	PUNCT
iajs-155	197	1	on	on	ADP
iajs-155	197	2	the	the	DET
iajs-155	197	3	other	other	ADJ
iajs-155	197	4	hand	hand	NOUN
iajs-155	197	5	,	,	PUNCT
iajs-155	197	6	(	(	PUNCT
iajs-155	197	7	a1+b1	a1+b1	PROPN
iajs-155	197	8	)	)	PUNCT
iajs-155	197	9	b2	b2	NOUN
iajs-155	197	10	k	k	NOUN
iajs-155	197	11	⊆	⊆	NUM
iajs-155	197	12	n	n	PROPN
iajs-155	197	13	and	and	CCONJ
iajs-155	197	14	neither	neither	CCONJ
iajs-155	197	15	(	(	PUNCT
iajs-155	197	16	a1+b1	a1+b1	PROPN
iajs-155	197	17	)	)	PUNCT
iajs-155	197	18	k	k	PROPN
iajs-155	198	1	⊆	⊆	NUM
iajs-155	198	2	n	n	NUM
iajs-155	198	3	nor	nor	CCONJ
iajs-155	198	4	b2k	b2k	PROPN
iajs-155	198	5	⊆	⊆	NUM
iajs-155	198	6	n	n	NOUN
iajs-155	198	7	,	,	PUNCT
iajs-155	198	8	we	we	PRON
iajs-155	198	9	get	get	VERB
iajs-155	198	10	that	that	DET
iajs-155	198	11	(	(	PUNCT
iajs-155	198	12	a1+b1	a1+b1	PROPN
iajs-155	198	13	)	)	PUNCT
iajs-155	198	14	b2	b2	NOUN
iajs-155	198	15	∈(n	∈(n	NOUN
iajs-155	198	16	:	:	PUNCT
iajs-155	198	17	m	m	VERB
iajs-155	198	18	)	)	PUNCT
iajs-155	198	19	by	by	ADP
iajs-155	198	20	proposition	proposition	NOUN
iajs-155	198	21	(	(	PUNCT
iajs-155	198	22	2.1	2.1	NUM
iajs-155	198	23	)	)	PUNCT
iajs-155	198	24	but	but	CCONJ
iajs-155	198	25	(	(	PUNCT
iajs-155	198	26	(	(	PUNCT
iajs-155	198	27	a1+b1	a1+b1	PROPN
iajs-155	198	28	)	)	PUNCT
iajs-155	198	29	b2	b2	NOUN
iajs-155	198	30	=	=	SYM
iajs-155	198	31	a1b2+b1b2	a1b2+b1b2	PROPN
iajs-155	198	32	∈(n	∈(n	PROPN
iajs-155	198	33	:	:	PUNCT
iajs-155	198	34	m	m	NUM
iajs-155	198	35	)	)	PUNCT
iajs-155	198	36	and	and	CCONJ
iajs-155	198	37	a1b2	a1b2	PROPN
iajs-155	198	38	∈(n	∈(n	PROPN
iajs-155	198	39	:	:	PUNCT
iajs-155	198	40	m	m	NUM
iajs-155	198	41	)	)	PUNCT
iajs-155	198	42	,	,	PUNCT
iajs-155	198	43	we	we	PRON
iajs-155	198	44	get	get	VERB
iajs-155	198	45	b1b2	b1b2	ADP
iajs-155	198	46	∈(n	∈(n	NOUN
iajs-155	198	47	:	:	PUNCT
iajs-155	198	48	m	m	NOUN
iajs-155	198	49	)	)	PUNCT
iajs-155	198	50	which	which	PRON
iajs-155	198	51	is	be	AUX
iajs-155	198	52	a	a	DET
iajs-155	198	53	contradiction	contradiction	NOUN
iajs-155	198	54	!	!	PUNCT
iajs-155	199	1	case(2	case(2	NOUN
iajs-155	199	2	)	)	PUNCT
iajs-155	200	1	if	if	SCONJ
iajs-155	200	2	b2k	b2k	PROPN
iajs-155	200	3	⊆	⊆	NUM
iajs-155	200	4	n	n	NOUN
iajs-155	200	5	and	and	CCONJ
iajs-155	200	6	b1k	b1k	ADP
iajs-155	200	7	⊈	⊈	PROPN
iajs-155	200	8	n	n	CCONJ
iajs-155	200	9	.	.	PUNCT
iajs-155	201	1	by	by	ADP
iajs-155	201	2	a	a	DET
iajs-155	201	3	similar	similar	ADJ
iajs-155	201	4	argument	argument	NOUN
iajs-155	201	5	of	of	ADP
iajs-155	201	6	case	case	NOUN
iajs-155	201	7	(	(	PUNCT
iajs-155	201	8	1	1	NUM
iajs-155	201	9	)	)	PUNCT
iajs-155	201	10	,	,	PUNCT
iajs-155	201	11	we	we	PRON
iajs-155	201	12	reach	reach	VERB
iajs-155	201	13	to	to	ADP
iajs-155	201	14	a	a	DET
iajs-155	201	15	contradiction	contradiction	NOUN
iajs-155	201	16	!	!	PUNCT
iajs-155	202	1	case(3	case(3	NOUN
iajs-155	202	2	)	)	PUNCT
iajs-155	203	1	b1k	b1k	PROPN
iajs-155	204	1	⊆	⊆	NUM
iajs-155	204	2	n	n	NOUN
iajs-155	204	3	and	and	CCONJ
iajs-155	204	4	b2k	b2k	PROPN
iajs-155	204	5	⊆	⊆	NUM
iajs-155	204	6	n	n	NOUN
iajs-155	204	7	since	since	SCONJ
iajs-155	204	8	b2k	b2k	PROPN
iajs-155	204	9	⊆	⊆	NUM
iajs-155	204	10	n	n	NOUN
iajs-155	204	11	and	and	CCONJ
iajs-155	204	12	a2k	a2k	VERB
iajs-155	204	13	⊈	⊈	NUM
iajs-155	204	14	n	n	CCONJ
iajs-155	204	15	,	,	PUNCT
iajs-155	204	16	we	we	PRON
iajs-155	204	17	conclude	conclude	VERB
iajs-155	204	18	(	(	PUNCT
iajs-155	204	19	a2+b2	a2+b2	NOUN
iajs-155	204	20	)	)	PUNCT
iajs-155	204	21	k	k	NOUN
iajs-155	205	1	⊈	⊈	PROPN
iajs-155	205	2	n	n	NOUN
iajs-155	205	3	.	.	PUNCT
iajs-155	206	1	but	but	CCONJ
iajs-155	206	2	a1	a1	PROPN
iajs-155	206	3	(	(	PUNCT
iajs-155	206	4	a2+b2	a2+b2	NOUN
iajs-155	206	5	)	)	PUNCT
iajs-155	206	6	k	k	PROPN
iajs-155	207	1	⊆	⊆	NUM
iajs-155	207	2	n	n	NUM
iajs-155	207	3	and	and	CCONJ
iajs-155	207	4	neither	neither	CCONJ
iajs-155	207	5	a1k⊆n	a1k⊆n	PROPN
iajs-155	207	6	nor	nor	CCONJ
iajs-155	207	7	(	(	PUNCT
iajs-155	207	8	a2+b2)k	a2+b2)k	NOUN
iajs-155	207	9	⊆	⊆	NUM
iajs-155	207	10	n	n	CCONJ
iajs-155	207	11	,	,	PUNCT
iajs-155	207	12	hence	hence	ADV
iajs-155	207	13	a1	a1	NOUN
iajs-155	207	14	(	(	PUNCT
iajs-155	207	15	a2+b2)∈(n	a2+b2)∈(n	PROPN
iajs-155	207	16	:	:	PUNCT
iajs-155	207	17	m	m	VERB
iajs-155	207	18	)	)	PUNCT
iajs-155	207	19	by	by	ADP
iajs-155	207	20	proposition	proposition	NOUN
iajs-155	207	21	(	(	PUNCT
iajs-155	207	22	2.1	2.1	NUM
iajs-155	207	23	)	)	PUNCT
iajs-155	207	24	since	since	SCONJ
iajs-155	207	25	a1	a1	NOUN
iajs-155	207	26	a2∈(n	a2∈(n	NOUN
iajs-155	207	27	:	:	PUNCT
iajs-155	207	28	m	m	NUM
iajs-155	207	29	)	)	PUNCT
iajs-155	207	30	and	and	CCONJ
iajs-155	207	31	a1	a1	PROPN
iajs-155	207	32	a2	a2	PROPN
iajs-155	207	33	+	+	CCONJ
iajs-155	207	34	a1	a1	NOUN
iajs-155	207	35	b2	b2	NOUN
iajs-155	207	36	∈(n	∈(n	NOUN
iajs-155	207	37	:	:	PUNCT
iajs-155	207	38	m	m	NUM
iajs-155	207	39	)	)	PUNCT
iajs-155	207	40	,	,	PUNCT
iajs-155	207	41	we	we	PRON
iajs-155	207	42	have	have	VERB
iajs-155	207	43	a1	a1	NOUN
iajs-155	207	44	b2	b2	NOUN
iajs-155	207	45	∈(n	∈(n	NOUN
iajs-155	207	46	:	:	PUNCT
iajs-155	207	47	m	m	NUM
iajs-155	207	48	)	)	PUNCT
iajs-155	207	49	.	.	PUNCT
iajs-155	208	1	since	since	SCONJ
iajs-155	208	2	(	(	PUNCT
iajs-155	208	3	a1+b1	a1+b1	PROPN
iajs-155	208	4	)	)	PUNCT
iajs-155	208	5	a2	a2	PROPN
iajs-155	208	6	k	k	NOUN
iajs-155	208	7	⊆	⊆	NUM
iajs-155	208	8	n	n	NUM
iajs-155	208	9	and	and	CCONJ
iajs-155	208	10	neither	neither	CCONJ
iajs-155	208	11	a2	a2	PROPN
iajs-155	208	12	k	k	PROPN
iajs-155	208	13	⊆	⊆	NUM
iajs-155	208	14	n	n	NOUN
iajs-155	208	15	nor	nor	CCONJ
iajs-155	208	16	(	(	PUNCT
iajs-155	208	17	a1+b1)k	a1+b1)k	PROPN
iajs-155	208	18	⊆	⊆	NUM
iajs-155	208	19	n	n	NOUN
iajs-155	208	20	,	,	PUNCT
iajs-155	208	21	we	we	PRON
iajs-155	208	22	conclude	conclude	VERB
iajs-155	208	23	(	(	PUNCT
iajs-155	208	24	a1+b1	a1+b1	NOUN
iajs-155	208	25	)	)	PUNCT
iajs-155	208	26	a2∈(n	a2∈(n	NOUN
iajs-155	208	27	:	:	PUNCT
iajs-155	208	28	m	m	VERB
iajs-155	208	29	)	)	PUNCT
iajs-155	208	30	by	by	ADP
iajs-155	208	31	proposition	proposition	NOUN
iajs-155	208	32	(	(	PUNCT
iajs-155	208	33	2.1	2.1	NUM
iajs-155	208	34	)	)	PUNCT
iajs-155	208	35	but	but	CCONJ
iajs-155	208	36	(	(	PUNCT
iajs-155	208	37	a1+b1	a1+b1	PROPN
iajs-155	208	38	)	)	PUNCT
iajs-155	208	39	a2	a2	PROPN
iajs-155	208	40	=	=	PUNCT
iajs-155	208	41	a1a2+b1	a1a2+b1	NOUN
iajs-155	208	42	a2	a2	PROPN
iajs-155	208	43	,	,	PUNCT
iajs-155	208	44	so	so	ADV
iajs-155	208	45	a1a2+b1	a1a2+b1	NOUN
iajs-155	208	46	a2	a2	PROPN
iajs-155	208	47	∈(n	∈(n	PROPN
iajs-155	208	48	:	:	PUNCT
iajs-155	208	49	m	m	NUM
iajs-155	208	50	)	)	PUNCT
iajs-155	208	51	and	and	CCONJ
iajs-155	208	52	since	since	SCONJ
iajs-155	208	53	a1	a1	NOUN
iajs-155	208	54	a2∈(n	a2∈(n	NOUN
iajs-155	208	55	:	:	PUNCT
iajs-155	208	56	m	m	NUM
iajs-155	208	57	)	)	PUNCT
iajs-155	208	58	,	,	PUNCT
iajs-155	208	59	we	we	PRON
iajs-155	208	60	get	get	VERB
iajs-155	208	61	b1a2	b1a2	PUNCT
iajs-155	208	62	∈(n	∈(n	NOUN
iajs-155	208	63	:	:	PUNCT
iajs-155	208	64	m	m	NOUN
iajs-155	208	65	)	)	PUNCT
iajs-155	208	66	.	.	PUNCT
iajs-155	209	1	now	now	ADV
iajs-155	209	2	,	,	PUNCT
iajs-155	209	3	since	since	SCONJ
iajs-155	209	4	(	(	PUNCT
iajs-155	209	5	a1+b1	a1+b1	PROPN
iajs-155	209	6	)	)	PUNCT
iajs-155	209	7	(	(	PUNCT
iajs-155	209	8	a2+b2)k	a2+b2)k	NOUN
iajs-155	209	9	⊆n	⊆n	ADJ
iajs-155	209	10	and	and	CCONJ
iajs-155	209	11	neither	neither	PRON
iajs-155	209	12	(	(	PUNCT
iajs-155	209	13	a1+b1)k	a1+b1)k	ADV
iajs-155	209	14	⊆n	⊆n	ADJ
iajs-155	209	15	nor	nor	CCONJ
iajs-155	209	16	(	(	PUNCT
iajs-155	209	17	a2+b2)k	a2+b2)k	NOUN
iajs-155	209	18	⊆n	⊆n	ADJ
iajs-155	209	19	,	,	PUNCT
iajs-155	209	20	we	we	PRON
iajs-155	209	21	have	have	VERB
iajs-155	209	22	(	(	PUNCT
iajs-155	209	23	a1+b1	a1+b1	PROPN
iajs-155	209	24	)	)	PUNCT
iajs-155	209	25	(	(	PUNCT
iajs-155	209	26	a2+b2	a2+b2	NOUN
iajs-155	209	27	)	)	PUNCT
iajs-155	209	28	=	=	PUNCT
iajs-155	209	29	a1a2	a1a2	PROPN
iajs-155	209	30	+	+	X
iajs-155	209	31	a1b2+b1a2+b1b2	a1b2+b1a2+b1b2	PROPN
iajs-155	209	32	∈(n	∈(n	PROPN
iajs-155	209	33	:	:	PUNCT
iajs-155	209	34	m	m	VERB
iajs-155	209	35	)	)	PUNCT
iajs-155	209	36	by	by	ADP
iajs-155	209	37	proposition	proposition	NOUN
iajs-155	209	38	(	(	PUNCT
iajs-155	209	39	2.1	2.1	NUM
iajs-155	209	40	)	)	PUNCT
iajs-155	209	41	but	but	CCONJ
iajs-155	209	42	a1a2	a1a2	INTJ
iajs-155	209	43	,	,	PUNCT
iajs-155	209	44	a1b2	a1b2	PROPN
iajs-155	209	45	,	,	PUNCT
iajs-155	209	46	b1a2	b1a2	X
iajs-155	209	47	∈(n	∈(n	NOUN
iajs-155	209	48	:	:	PUNCT
iajs-155	209	49	m	m	NUM
iajs-155	209	50	)	)	PUNCT
iajs-155	209	51	,	,	PUNCT
iajs-155	209	52	so	so	ADV
iajs-155	209	53	b1b2	b1b2	VERB
iajs-155	209	54	∈(n	∈(n	PRON
iajs-155	209	55	:	:	PUNCT
iajs-155	209	56	m	m	NOUN
iajs-155	209	57	)	)	PUNCT
iajs-155	209	58	which	which	PRON
iajs-155	209	59	is	be	AUX
iajs-155	209	60	a	a	DET
iajs-155	209	61	contradiction	contradiction	NOUN
iajs-155	209	62	!	!	PUNCT
iajs-155	209	63	.	.	PUNCT
iajs-155	210	1	consequently	consequently	ADV
iajs-155	210	2	i1k	i1k	ADJ
iajs-155	210	3	⊆n	⊆n	ADJ
iajs-155	210	4	or	or	CCONJ
iajs-155	210	5	i2k	i2k	PRON
iajs-155	210	6	⊆n	⊆n	ADJ
iajs-155	210	7	(	(	PUNCT
iajs-155	210	8	2	2	NUM
iajs-155	210	9	)	)	PUNCT
iajs-155	210	10	⟹	⟹	NOUN
iajs-155	210	11	(	(	PUNCT
iajs-155	210	12	1	1	X
iajs-155	210	13	)	)	PUNCT
iajs-155	210	14	it	it	PRON
iajs-155	210	15	is	be	AUX
iajs-155	210	16	clear	clear	ADJ
iajs-155	210	17	∎	∎	PROPN
iajs-155	210	18	recall	recall	VERB
iajs-155	210	19	that	that	SCONJ
iajs-155	210	20	"	"	PUNCT
iajs-155	210	21	.	.	PUNCT
iajs-155	211	1	for	for	ADP
iajs-155	211	2	any	any	DET
iajs-155	211	3	two	two	NUM
iajs-155	211	4	submodules	submodule	NOUN
iajs-155	211	5	n	n	NOUN
iajs-155	211	6	,	,	PUNCT
iajs-155	211	7	k	k	PROPN
iajs-155	211	8	of	of	ADP
iajs-155	211	9	a	a	DET
iajs-155	211	10	multiplication	multiplication	NOUN
iajs-155	211	11	r	r	NOUN
iajs-155	211	12	-	-	PUNCT
iajs-155	211	13	module	module	NOUN
iajs-155	211	14	m	m	NOUN
iajs-155	211	15	with	with	ADP
iajs-155	211	16	n	n	PROPN
iajs-155	211	17	=	=	PROPN
iajs-155	211	18	i1	i1	PROPN
iajs-155	211	19	m	m	PROPN
iajs-155	211	20	and	and	CCONJ
iajs-155	211	21	k	k	PROPN
iajs-155	211	22	=	=	PROPN
iajs-155	211	23	i2	i2	PROPN
iajs-155	211	24	m	m	PROPN
iajs-155	211	25	for	for	ADP
iajs-155	211	26	some	some	DET
iajs-155	211	27	ideals	ideal	NOUN
iajs-155	211	28	i1	i1	PROPN
iajs-155	211	29	and	and	CCONJ
iajs-155	211	30	i2	i2	PROPN
iajs-155	211	31	of	of	ADP
iajs-155	211	32	r.	r.	PROPN
iajs-155	211	33	the	the	DET
iajs-155	211	34	product	product	NOUN
iajs-155	211	35	n	n	PROPN
iajs-155	211	36	and	and	CCONJ
iajs-155	211	37	k	k	PROPN
iajs-155	211	38	denoted	denote	VERB
iajs-155	211	39	by	by	ADP
iajs-155	211	40	nk	nk	PROPN
iajs-155	211	41	is	be	AUX
iajs-155	211	42	defined	define	VERB
iajs-155	211	43	by	by	ADP
iajs-155	211	44	nk	nk	PROPN
iajs-155	211	45	=	=	PUNCT
iajs-155	211	46	i1i2m	i1i2m	NOUN
iajs-155	211	47	.	.	PUNCT
iajs-155	212	1	"[2	"[2	PUNCT
iajs-155	212	2	]	]	PUNCT
iajs-155	212	3	by	by	ADP
iajs-155	212	4	using	use	VERB
iajs-155	212	5	this	this	DET
iajs-155	212	6	definition	definition	NOUN
iajs-155	212	7	of	of	ADP
iajs-155	212	8	product	product	NOUN
iajs-155	212	9	of	of	ADP
iajs-155	212	10	submodules	submodule	NOUN
iajs-155	212	11	,	,	PUNCT
iajs-155	212	12	we	we	PRON
iajs-155	212	13	give	give	VERB
iajs-155	212	14	the	the	DET
iajs-155	212	15	following	follow	VERB
iajs-155	212	16	characterization	characterization	NOUN
iajs-155	212	17	of	of	ADP
iajs-155	212	18	2	2	NUM
iajs-155	212	19	-	-	PUNCT
iajs-155	212	20	absorbing	absorb	VERB
iajs-155	212	21	submodules	submodule	NOUN
iajs-155	212	22	.	.	PUNCT
iajs-155	213	1	theorem	theorem	VERB
iajs-155	213	2	2.3	2.3	NUM
iajs-155	213	3	:	:	PUNCT
iajs-155	213	4	let	let	VERB
iajs-155	213	5	n	n	PRON
iajs-155	213	6	be	be	AUX
iajs-155	213	7	a	a	DET
iajs-155	213	8	proper	proper	ADJ
iajs-155	213	9	submodule	submodule	NOUN
iajs-155	213	10	of	of	ADP
iajs-155	213	11	a	a	DET
iajs-155	213	12	multiplication	multiplication	NOUN
iajs-155	213	13	r	r	NOUN
iajs-155	213	14	-	-	PUNCT
iajs-155	213	15	module	module	NOUN
iajs-155	213	16	m	m	NOUN
iajs-155	213	17	,	,	PUNCT
iajs-155	213	18	then	then	ADV
iajs-155	213	19	n	n	PRON
iajs-155	213	20	is	be	AUX
iajs-155	213	21	a	a	DET
iajs-155	213	22	2	2	NUM
iajs-155	213	23	-	-	PUNCT
iajs-155	213	24	asorbing	asorbe	VERB
iajs-155	213	25	submodule	submodule	NOUN
iajs-155	213	26	of	of	ADP
iajs-155	213	27	m	m	PROPN
iajs-155	213	28	if	if	SCONJ
iajs-155	214	1	and	and	CCONJ
iajs-155	214	2	only	only	ADV
iajs-155	214	3	if	if	SCONJ
iajs-155	214	4	n1	n1	PROPN
iajs-155	214	5	n2	n2	ADJ
iajs-155	214	6	n3	n3	PROPN
iajs-155	214	7	⊆n	⊆n	PROPN
iajs-155	214	8	implies	imply	VERB
iajs-155	214	9	that	that	SCONJ
iajs-155	214	10	n1	n1	PROPN
iajs-155	214	11	n2	n2	ADJ
iajs-155	214	12	⊆n	⊆n	ADJ
iajs-155	214	13	or	or	CCONJ
iajs-155	214	14	n1	n1	ADJ
iajs-155	214	15	n3	n3	NOUN
iajs-155	214	16	⊆n	⊆n	ADJ
iajs-155	214	17	or	or	CCONJ
iajs-155	214	18	n2	n2	ADJ
iajs-155	214	19	n3	n3	NOUN
iajs-155	214	20	⊆n	⊆n	ADJ
iajs-155	214	21	,	,	PUNCT
iajs-155	214	22	where	where	SCONJ
iajs-155	214	23	n1	n1	NOUN
iajs-155	214	24	,	,	PUNCT
iajs-155	214	25	n2	n2	NOUN
iajs-155	214	26	,	,	PUNCT
iajs-155	214	27	n3	n3	PROPN
iajs-155	214	28	are	be	AUX
iajs-155	214	29	submodules	submodule	NOUN
iajs-155	214	30	of	of	ADP
iajs-155	214	31	m.	m.	NOUN
iajs-155	214	32	proof	proof	NOUN
iajs-155	214	33	:	:	PUNCT
iajs-155	214	34	⟹	⟹	X
iajs-155	214	35	since	since	SCONJ
iajs-155	214	36	m	m	PROPN
iajs-155	214	37	is	be	AUX
iajs-155	214	38	multiplication	multiplication	NOUN
iajs-155	214	39	,	,	PUNCT
iajs-155	214	40	then	then	ADV
iajs-155	214	41	n1=	n1=	PROPN
iajs-155	214	42	i1	i1	PROPN
iajs-155	214	43	m	m	PROPN
iajs-155	214	44	,	,	PUNCT
iajs-155	214	45	n2=	n2=	PROPN
iajs-155	214	46	i2	i2	PROPN
iajs-155	214	47	m	m	PROPN
iajs-155	214	48	,	,	PUNCT
iajs-155	214	49	n3=	n3=	PROPN
iajs-155	214	50	i3	i3	VERB
iajs-155	214	51	m	m	PROPN
iajs-155	214	52	for	for	ADP
iajs-155	214	53	some	some	DET
iajs-155	214	54	ideals	ideal	NOUN
iajs-155	214	55	i1	i1	PROPN
iajs-155	214	56	,	,	PUNCT
iajs-155	214	57	i2	i2	PROPN
iajs-155	214	58	,	,	PUNCT
iajs-155	214	59	i3	i3	NOUN
iajs-155	214	60	of	of	ADP
iajs-155	214	61	r	r	NOUN
iajs-155	214	62	.	.	PUNCT
iajs-155	215	1	it	it	PRON
iajs-155	215	2	follows	follow	VERB
iajs-155	215	3	that	that	SCONJ
iajs-155	215	4	:	:	PUNCT
iajs-155	215	5	n1	n1	PROPN
iajs-155	215	6	n2	n2	NOUN
iajs-155	215	7	n3	n3	PROPN
iajs-155	215	8	=	=	PROPN
iajs-155	215	9	i1	i1	PROPN
iajs-155	215	10	i2	i2	PROPN
iajs-155	215	11	i3	i3	PROPN
iajs-155	215	12	m	m	PROPN
iajs-155	215	13	⊆	⊆	NUM
iajs-155	215	14	n	n	NOUN
iajs-155	215	15	.	.	PUNCT
iajs-155	216	1	hence	hence	ADV
iajs-155	216	2	i1	i1	PROPN
iajs-155	216	3	i2	i2	PROPN
iajs-155	216	4	i3	i3	PROPN
iajs-155	216	5	⊆	⊆	PROPN
iajs-155	216	6	(	(	PUNCT
iajs-155	216	7	n	n	NUM
iajs-155	216	8	:	:	PUNCT
iajs-155	216	9	m	m	NUM
iajs-155	216	10	)	)	PUNCT
iajs-155	216	11	.	.	PUNCT
iajs-155	217	1	but	but	CCONJ
iajs-155	217	2	n	n	PRON
iajs-155	217	3	is	be	AUX
iajs-155	217	4	2	2	NUM
iajs-155	217	5	-	-	PUNCT
iajs-155	217	6	absorbing	absorb	VERB
iajs-155	217	7	submodule	submodule	NOUN
iajs-155	217	8	of	of	ADP
iajs-155	217	9	m	m	PROPN
iajs-155	217	10	implies	imply	VERB
iajs-155	217	11	that	that	SCONJ
iajs-155	217	12	(	(	PUNCT
iajs-155	217	13	n	n	NUM
iajs-155	217	14	:	:	PUNCT
iajs-155	217	15	m	m	VERB
iajs-155	217	16	)	)	PUNCT
iajs-155	217	17	is	be	AUX
iajs-155	217	18	2	2	NUM
iajs-155	217	19	-	-	PUNCT
iajs-155	217	20	absorbing	absorb	VERB
iajs-155	217	21	ideal	ideal	NOUN
iajs-155	217	22	by	by	ADP
iajs-155	217	23	[	[	PUNCT
iajs-155	217	24	theorem	theorem	ADJ
iajs-155	217	25	1.7	1.7	NUM
iajs-155	217	26	]	]	PUNCT
iajs-155	217	27	.	.	PUNCT
iajs-155	218	1	so	so	ADV
iajs-155	218	2	by	by	ADP
iajs-155	218	3	[	[	X
iajs-155	218	4	1	1	NUM
iajs-155	218	5	,	,	PUNCT
iajs-155	218	6	th	th	X
iajs-155	218	7	2.13	2.13	NUM
iajs-155	218	8	]	]	PUNCT
iajs-155	218	9	either	either	CCONJ
iajs-155	218	10	i1	i1	PROPN
iajs-155	218	11	i2	i2	PROPN
iajs-155	218	12	⊆	⊆	NUM
iajs-155	218	13	(	(	PUNCT
iajs-155	218	14	n	n	NUM
iajs-155	218	15	:	:	PUNCT
iajs-155	218	16	m	m	NUM
iajs-155	218	17	)	)	PUNCT
iajs-155	218	18	or	or	CCONJ
iajs-155	218	19	i1	i1	PROPN
iajs-155	218	20	i3	i3	PROPN
iajs-155	218	21	⊆	⊆	NUM
iajs-155	218	22	(	(	PUNCT
iajs-155	218	23	n	n	NUM
iajs-155	218	24	:	:	PUNCT
iajs-155	218	25	m	m	NUM
iajs-155	218	26	)	)	PUNCT
iajs-155	218	27	or	or	CCONJ
iajs-155	218	28	i2	i2	PROPN
iajs-155	218	29	i3	i3	NOUN
iajs-155	218	30	⊆	⊆	NUM
iajs-155	218	31	(	(	PUNCT
iajs-155	218	32	n	n	NUM
iajs-155	218	33	:	:	PUNCT
iajs-155	218	34	m	m	NUM
iajs-155	218	35	)	)	PUNCT
iajs-155	218	36	.	.	PUNCT
iajs-155	219	1	then	then	ADV
iajs-155	219	2	i1	i1	PROPN
iajs-155	219	3	i2	i2	PROPN
iajs-155	219	4	m	m	PROPN
iajs-155	219	5	⊆	⊆	NUM
iajs-155	219	6	n	n	PROPN
iajs-155	219	7	or	or	CCONJ
iajs-155	219	8	i1	i1	PROPN
iajs-155	219	9	i3	i3	PROPN
iajs-155	219	10	m	m	PROPN
iajs-155	219	11	⊆	⊆	NUM
iajs-155	219	12	n	n	ADJ
iajs-155	219	13	or	or	CCONJ
iajs-155	219	14	i2	i2	PROPN
iajs-155	219	15	i3	i3	PROPN
iajs-155	219	16	m	m	PROPN
iajs-155	219	17	⊆	⊆	NUM
iajs-155	219	18	n	n	NOUN
iajs-155	219	19	.	.	PUNCT
iajs-155	220	1	thus	thus	ADV
iajs-155	220	2	n1	n1	ADJ
iajs-155	220	3	n2	n2	ADJ
iajs-155	220	4	⊆n	⊆n	ADJ
iajs-155	220	5	or	or	CCONJ
iajs-155	220	6	n1	n1	ADJ
iajs-155	220	7	n3	n3	NOUN
iajs-155	220	8	⊆n	⊆n	ADJ
iajs-155	220	9	or	or	CCONJ
iajs-155	220	10	n2	n2	ADJ
iajs-155	220	11	n3	n3	NOUN
iajs-155	220	12	⊆n	⊆n	VERB
iajs-155	220	13	232	232	NUM
iajs-155	220	14	|	|	NOUN
iajs-155	220	15	mathematics	mathematic	NOUN
iajs-155	220	16	2015	2015	NUM
iajs-155	220	17	)	)	PUNCT
iajs-155	220	18	عام	عام	ADP
iajs-155	220	19	3العدد	3العدد	NUM
iajs-155	220	20	(	(	PUNCT
iajs-155	220	21	28مجلة	28مجلة	X
iajs-155	220	22	إبن	إبن	VERB
iajs-155	220	23	الھيثم	الھيثم	NOUN
iajs-155	220	24	للعلوم	للعلوم	NOUN
iajs-155	220	25	الصرفة	الصرفة	NOUN
iajs-155	221	1	و	و	PRON
iajs-155	221	2	التطبيقية	التطبيقية	ADV
iajs-155	221	3	المجلد	المجلد	VERB
iajs-155	221	4	ibn	ibn	PROPN
iajs-155	221	5	al	al	PROPN
iajs-155	221	6	-	-	PUNCT
iajs-155	221	7	haitham	haitham	PROPN
iajs-155	221	8	jour	jour	X
iajs-155	221	9	.	.	PROPN
iajs-155	222	1	for	for	ADP
iajs-155	222	2	pure	pure	ADJ
iajs-155	222	3	&	&	CCONJ
iajs-155	222	4	appl	appl	PROPN
iajs-155	222	5	.	.	PUNCT
iajs-155	223	1	sci	sci	PROPN
iajs-155	223	2	.	.	PUNCT
iajs-155	223	3	vol	vol	NOUN
iajs-155	223	4	.	.	PROPN
iajs-155	224	1	28	28	NUM
iajs-155	224	2	(	(	PUNCT
iajs-155	224	3	3	3	NUM
iajs-155	224	4	)	)	PUNCT
iajs-155	224	5	2015	2015	NUM
iajs-155	224	6	⟸	⟸	NUM
iajs-155	224	7	let	let	VERB
iajs-155	224	8	i1	i1	PROPN
iajs-155	224	9	i2	i2	PROPN
iajs-155	224	10	k	k	PROPN
iajs-155	224	11	⊆	⊆	PROPN
iajs-155	224	12	n	n	PROPN
iajs-155	224	13	where	where	SCONJ
iajs-155	224	14	i1	i1	PROPN
iajs-155	224	15	,	,	PUNCT
iajs-155	224	16	i2	i2	PROPN
iajs-155	224	17	are	be	AUX
iajs-155	224	18	ideals	ideal	NOUN
iajs-155	224	19	of	of	ADP
iajs-155	224	20	r	r	NOUN
iajs-155	224	21	and	and	CCONJ
iajs-155	224	22	k	k	NOUN
iajs-155	224	23	m	m	PROPN
iajs-155	224	24	.	.	PUNCT
iajs-155	225	1	since	since	SCONJ
iajs-155	225	2	m	m	PROPN
iajs-155	225	3	is	be	AUX
iajs-155	225	4	multiplication	multiplication	NOUN
iajs-155	225	5	module	module	NOUN
iajs-155	225	6	,	,	PUNCT
iajs-155	225	7	k	k	PROPN
iajs-155	225	8	=	=	SYM
iajs-155	225	9	jm	jm	PROPN
iajs-155	225	10	for	for	ADP
iajs-155	225	11	some	some	DET
iajs-155	225	12	j	j	NOUN
iajs-155	225	13	of	of	ADP
iajs-155	225	14	r	r	NOUN
iajs-155	225	15	.	.	PUNCT
iajs-155	226	1	thus	thus	ADV
iajs-155	226	2	i1	i1	PROPN
iajs-155	226	3	i2j	i2j	ADV
iajs-155	226	4	m	m	VERB
iajs-155	226	5	⊆	⊆	NUM
iajs-155	226	6	n	n	NOUN
iajs-155	226	7	.	.	PUNCT
iajs-155	227	1	put	put	VERB
iajs-155	227	2	n1=	n1=	PROPN
iajs-155	227	3	i1	i1	PROPN
iajs-155	227	4	m	m	PROPN
iajs-155	227	5	n2=	n2=	PROPN
iajs-155	227	6	i2	i2	PROPN
iajs-155	227	7	m	m	VERB
iajs-155	227	8	.	.	PUNCT
iajs-155	228	1	it	it	PRON
iajs-155	228	2	follows	follow	VERB
iajs-155	228	3	that	that	SCONJ
iajs-155	228	4	n1	n1	PROPN
iajs-155	228	5	n2k=	n2k=	PROPN
iajs-155	228	6	i1	i1	PROPN
iajs-155	228	7	i2j	i2j	ADV
iajs-155	228	8	m	m	VERB
iajs-155	228	9	⊆	⊆	NUM
iajs-155	228	10	n	n	NOUN
iajs-155	228	11	.	.	PUNCT
iajs-155	229	1	so	so	ADV
iajs-155	229	2	by	by	ADP
iajs-155	229	3	hypotheses	hypothesis	NOUN
iajs-155	229	4	either	either	CCONJ
iajs-155	229	5	n1	n1	PROPN
iajs-155	229	6	k	k	PROPN
iajs-155	229	7	⊆	⊆	NUM
iajs-155	229	8	n	n	NOUN
iajs-155	229	9	or	or	CCONJ
iajs-155	229	10	n2	n2	PROPN
iajs-155	229	11	k	k	PROPN
iajs-155	229	12	⊆	⊆	NUM
iajs-155	229	13	n	n	ADJ
iajs-155	229	14	or	or	CCONJ
iajs-155	229	15	n1	n1	ADJ
iajs-155	229	16	n2	n2	NOUN
iajs-155	229	17	⊆	⊆	NUM
iajs-155	229	18	n	n	NOUN
iajs-155	229	19	then	then	ADV
iajs-155	229	20	i1j	i1j	PRON
iajs-155	229	21	m	m	VERB
iajs-155	229	22	⊆	⊆	NUM
iajs-155	229	23	n	n	ADP
iajs-155	229	24	or	or	CCONJ
iajs-155	229	25	i2j	i2j	ADV
iajs-155	229	26	m	m	PROPN
iajs-155	229	27	⊆	⊆	NUM
iajs-155	229	28	n	n	ADP
iajs-155	229	29	or	or	CCONJ
iajs-155	229	30	i1	i1	PROPN
iajs-155	229	31	i2	i2	PROPN
iajs-155	229	32	⊆	⊆	NUM
iajs-155	229	33	(	(	PUNCT
iajs-155	229	34	n	n	NUM
iajs-155	229	35	:	:	PUNCT
iajs-155	229	36	m	m	NUM
iajs-155	229	37	)	)	PUNCT
iajs-155	229	38	;	;	PUNCT
iajs-155	229	39	that	that	PRON
iajs-155	229	40	is	be	AUX
iajs-155	229	41	i1	i1	PROPN
iajs-155	229	42	k	k	PROPN
iajs-155	229	43	⊆	⊆	PROPN
iajs-155	229	44	n	n	PROPN
iajs-155	229	45	or	or	CCONJ
iajs-155	229	46	i2k	i2k	NOUN
iajs-155	229	47	⊆	⊆	NUM
iajs-155	229	48	n	n	NOUN
iajs-155	229	49	or	or	CCONJ
iajs-155	229	50	i1i2	i1i2	NUM
iajs-155	229	51	⊆	⊆	NUM
iajs-155	229	52	(	(	PUNCT
iajs-155	229	53	n	n	NUM
iajs-155	229	54	:	:	PUNCT
iajs-155	229	55	m	m	X
iajs-155	229	56	)	)	PUNCT
iajs-155	229	57	thus	thus	ADV
iajs-155	229	58	n	n	PRON
iajs-155	229	59	is	be	AUX
iajs-155	229	60	2	2	NUM
iajs-155	229	61	-	-	PUNCT
iajs-155	229	62	absorbing	absorb	VERB
iajs-155	229	63	submodule	submodule	NOUN
iajs-155	229	64	of	of	ADP
iajs-155	229	65	m	m	PROPN
iajs-155	229	66	∎	∎	PROPN
iajs-155	229	67	proposition	proposition	NOUN
iajs-155	229	68	2.4	2.4	NUM
iajs-155	229	69	:	:	PUNCT
iajs-155	229	70	let	let	VERB
iajs-155	229	71	n	n	PRON
iajs-155	229	72	be	be	AUX
iajs-155	229	73	a	a	DET
iajs-155	229	74	proper	proper	ADJ
iajs-155	229	75	submodule	submodule	NOUN
iajs-155	229	76	of	of	ADP
iajs-155	229	77	an	an	DET
iajs-155	229	78	r	r	NOUN
iajs-155	229	79	-	-	PUNCT
iajs-155	229	80	module	module	NOUN
iajs-155	229	81	m	m	NOUN
iajs-155	229	82	.	.	PUNCT
iajs-155	230	1	the	the	DET
iajs-155	230	2	following	follow	VERB
iajs-155	230	3	statements	statement	NOUN
iajs-155	230	4	are	be	AUX
iajs-155	230	5	equivalent	equivalent	ADJ
iajs-155	230	6	:	:	PUNCT
iajs-155	230	7	(	(	PUNCT
iajs-155	230	8	1	1	X
iajs-155	230	9	)	)	PUNCT
iajs-155	230	10	n	n	PRON
iajs-155	230	11	is	be	AUX
iajs-155	230	12	a	a	DET
iajs-155	230	13	2	2	NUM
iajs-155	230	14	-	-	PUNCT
iajs-155	230	15	absorrbing	absorrbe	VERB
iajs-155	230	16	submodule	submodule	NOUN
iajs-155	230	17	of	of	ADP
iajs-155	230	18	m	m	PROPN
iajs-155	230	19	(	(	PUNCT
iajs-155	230	20	2	2	NUM
iajs-155	230	21	)	)	PUNCT
iajs-155	230	22	(	(	PUNCT
iajs-155	230	23	n	n	X
iajs-155	230	24	:	:	PUNCT
iajs-155	230	25	i	i	NOUN
iajs-155	230	26	)	)	PUNCT
iajs-155	230	27	is	be	AUX
iajs-155	230	28	2	2	NUM
iajs-155	230	29	-	-	PUNCT
iajs-155	230	30	absorbing	absorbing	ADJ
iajs-155	230	31	,	,	PUNCT
iajs-155	230	32	for	for	ADP
iajs-155	230	33	each	each	DET
iajs-155	230	34	ideal	ideal	NOUN
iajs-155	230	35	i	i	PRON
iajs-155	230	36	of	of	ADP
iajs-155	230	37	r	r	NOUN
iajs-155	230	38	with	with	ADP
iajs-155	230	39	i	i	PRON
iajs-155	230	40	m	m	VERB
iajs-155	230	41	⊈n	⊈n	PROPN
iajs-155	230	42	(	(	PUNCT
iajs-155	230	43	3	3	NUM
iajs-155	230	44	)	)	PUNCT
iajs-155	230	45	(	(	PUNCT
iajs-155	230	46	n	n	X
iajs-155	230	47	:	:	PUNCT
iajs-155	230	48	(	(	PUNCT
iajs-155	230	49	r	r	NOUN
iajs-155	230	50	)	)	PUNCT
iajs-155	230	51	)	)	PUNCT
iajs-155	230	52	is	be	AUX
iajs-155	230	53	2	2	NUM
iajs-155	230	54	-	-	PUNCT
iajs-155	230	55	absorbing	absorb	VERB
iajs-155	230	56	submodule	submodule	NOUN
iajs-155	230	57	for	for	ADP
iajs-155	230	58	each	each	DET
iajs-155	230	59	r	r	NOUN
iajs-155	230	60	∈r	∈r	NOUN
iajs-155	230	61	with	with	ADP
iajs-155	230	62	rm	rm	NOUN
iajs-155	230	63	⊈n	⊈n	NUM
iajs-155	230	64	proof	proof	NOUN
iajs-155	230	65	:	:	PUNCT
iajs-155	230	66	(	(	PUNCT
iajs-155	230	67	1	1	X
iajs-155	230	68	)	)	PUNCT
iajs-155	230	69	⟹(2	⟹(2	PUNCT
iajs-155	230	70	)	)	PUNCT
iajs-155	230	71	let	let	VERB
iajs-155	230	72	i	i	PRON
iajs-155	230	73	be	be	AUX
iajs-155	230	74	an	an	DET
iajs-155	230	75	ideal	ideal	NOUN
iajs-155	230	76	of	of	ADP
iajs-155	230	77	r	r	NOUN
iajs-155	230	78	with	with	ADP
iajs-155	230	79	i	i	PRON
iajs-155	230	80	m	m	PROPN
iajs-155	230	81	⊈m	⊈m	NUM
iajs-155	230	82	,	,	PUNCT
iajs-155	230	83	then	then	ADV
iajs-155	230	84	(	(	PUNCT
iajs-155	230	85	n	n	X
iajs-155	230	86	:	:	PUNCT
iajs-155	231	1	i	i	NOUN
iajs-155	231	2	)	)	PUNCT
iajs-155	231	3	is	be	AUX
iajs-155	231	4	a	a	DET
iajs-155	231	5	proper	proper	ADJ
iajs-155	231	6	submodule	submodule	NOUN
iajs-155	231	7	of	of	ADP
iajs-155	231	8	m	m	PRON
iajs-155	231	9	let	let	VERB
iajs-155	231	10	a	a	DET
iajs-155	231	11	b	b	NOUN
iajs-155	231	12	m	m	VERB
iajs-155	231	13	∈	∈	NOUN
iajs-155	231	14	(	(	PUNCT
iajs-155	231	15	n	n	NOUN
iajs-155	231	16	:	:	PUNCT
iajs-155	231	17	i	i	NOUN
iajs-155	231	18	)	)	PUNCT
iajs-155	231	19	,	,	PUNCT
iajs-155	231	20	where	where	SCONJ
iajs-155	231	21	a	a	DET
iajs-155	231	22	,	,	PUNCT
iajs-155	231	23	b	b	X
iajs-155	231	24	∈	∈	PROPN
iajs-155	231	25	r	r	NOUN
iajs-155	231	26	,	,	PUNCT
iajs-155	231	27	m	m	VERB
iajs-155	231	28	∈	∈	ADJ
iajs-155	231	29	m	m	NOUN
iajs-155	231	30	.	.	PUNCT
iajs-155	232	1	then	then	ADV
iajs-155	232	2	ab(im	ab(im	PROPN
iajs-155	232	3	)	)	PUNCT
iajs-155	233	1	⊆	⊆	NUM
iajs-155	233	2	n	n	NOUN
iajs-155	233	3	.	.	PUNCT
iajs-155	234	1	but	but	CCONJ
iajs-155	234	2	n	n	PRON
iajs-155	234	3	is	be	AUX
iajs-155	234	4	2	2	NUM
iajs-155	234	5	-	-	PUNCT
iajs-155	234	6	absorbing	absorb	VERB
iajs-155	234	7	submodule	submodule	NOUN
iajs-155	234	8	of	of	ADP
iajs-155	234	9	m	m	PROPN
iajs-155	234	10	,	,	PUNCT
iajs-155	234	11	so	so	ADV
iajs-155	234	12	by	by	ADP
iajs-155	234	13	proposition(2.1	proposition(2.1	NOUN
iajs-155	234	14	)	)	PUNCT
iajs-155	234	15	,	,	PUNCT
iajs-155	234	16	either	either	CCONJ
iajs-155	234	17	a(im	a(im	NOUN
iajs-155	234	18	)	)	PUNCT
iajs-155	234	19	⊆	⊆	NUM
iajs-155	234	20	n	n	ADP
iajs-155	234	21	or	or	CCONJ
iajs-155	234	22	b(im	b(im	NUM
iajs-155	234	23	)	)	PUNCT
iajs-155	234	24	⊆	⊆	NUM
iajs-155	234	25	n	n	NUM
iajs-155	234	26	or	or	CCONJ
iajs-155	234	27	ab	ab	PROPN
iajs-155	234	28	∈	∈	PROPN
iajs-155	234	29	(	(	PUNCT
iajs-155	234	30	n	n	NOUN
iajs-155	234	31	:	:	PUNCT
iajs-155	234	32	m	m	X
iajs-155	234	33	)	)	PUNCT
iajs-155	235	1	⊆	⊆	X
iajs-155	235	2	(	(	PUNCT
iajs-155	235	3	(	(	PUNCT
iajs-155	235	4	n	n	X
iajs-155	235	5	:	:	PUNCT
iajs-155	235	6	i	i	X
iajs-155	235	7	):	):	PUNCT
iajs-155	235	8	m	m	PROPN
iajs-155	235	9	)	)	PUNCT
iajs-155	235	10	.	.	PUNCT
iajs-155	236	1	hence	hence	ADV
iajs-155	236	2	either	either	PRON
iajs-155	236	3	am	be	AUX
iajs-155	236	4	∈	∈	NOUN
iajs-155	236	5	(	(	PUNCT
iajs-155	236	6	n	n	NOUN
iajs-155	236	7	:	:	PUNCT
iajs-155	236	8	i	i	NOUN
iajs-155	236	9	)	)	PUNCT
iajs-155	236	10	or	or	CCONJ
iajs-155	236	11	bm	bm	X
iajs-155	236	12	∈(n	∈(n	NOUN
iajs-155	236	13	:	:	PUNCT
iajs-155	236	14	i	i	NOUN
iajs-155	236	15	)	)	PUNCT
iajs-155	236	16	or	or	CCONJ
iajs-155	236	17	ab	ab	PROPN
iajs-155	236	18	∈	∈	PROPN
iajs-155	236	19	(	(	PUNCT
iajs-155	236	20	(	(	PUNCT
iajs-155	236	21	n	n	X
iajs-155	236	22	:	:	PUNCT
iajs-155	236	23	i	i	PRON
iajs-155	236	24	):	):	PUNCT
iajs-155	236	25	m	m	VERB
iajs-155	236	26	.	.	PUNCT
iajs-155	237	1	thus	thus	ADV
iajs-155	237	2	(	(	PUNCT
iajs-155	237	3	n	n	X
iajs-155	237	4	:	:	PUNCT
iajs-155	237	5	i	i	NOUN
iajs-155	237	6	)	)	PUNCT
iajs-155	237	7	is	be	AUX
iajs-155	237	8	a	a	DET
iajs-155	237	9	2	2	NUM
iajs-155	237	10	-	-	PUNCT
iajs-155	237	11	absorbing	absorb	VERB
iajs-155	237	12	submodule	submodule	NOUN
iajs-155	237	13	.	.	PUNCT
iajs-155	238	1	(	(	PUNCT
iajs-155	238	2	2	2	NUM
iajs-155	238	3	)	)	PUNCT
iajs-155	238	4	⟹(3	⟹(3	PROPN
iajs-155	238	5	)	)	PUNCT
iajs-155	238	6	it	it	PRON
iajs-155	238	7	is	be	AUX
iajs-155	238	8	clear	clear	ADJ
iajs-155	238	9	.	.	PUNCT
iajs-155	239	1	(	(	PUNCT
iajs-155	239	2	3	3	NUM
iajs-155	239	3	)	)	PUNCT
iajs-155	239	4	⟹(1	⟹(1	NUM
iajs-155	239	5	)	)	PUNCT
iajs-155	239	6	take	take	VERB
iajs-155	239	7	r	r	NOUN
iajs-155	239	8	=	=	NOUN
iajs-155	239	9	1	1	NUM
iajs-155	239	10	then	then	ADV
iajs-155	239	11	(	(	PUNCT
iajs-155	239	12	n	n	X
iajs-155	239	13	:	:	SYM
iajs-155	239	14	1	1	X
iajs-155	239	15	)	)	PUNCT
iajs-155	239	16	=	=	SYM
iajs-155	240	1	n	n	X
iajs-155	240	2	,	,	PUNCT
iajs-155	240	3	so	so	CCONJ
iajs-155	240	4	n	n	PROPN
iajs-155	240	5	is	be	AUX
iajs-155	240	6	2	2	NUM
iajs-155	240	7	-	-	PUNCT
iajs-155	240	8	absorbing	absorbing	ADJ
iajs-155	240	9	.	.	PUNCT
iajs-155	241	1	∎	∎	PROPN
iajs-155	241	2	now	now	ADV
iajs-155	241	3	we	we	PRON
iajs-155	241	4	have	have	VERB
iajs-155	241	5	the	the	DET
iajs-155	241	6	following	following	NOUN
iajs-155	241	7	:	:	PUNCT
iajs-155	241	8	theorem	theorem	VERB
iajs-155	241	9	2.5	2.5	NUM
iajs-155	241	10	:	:	PUNCT
iajs-155	241	11	let	let	VERB
iajs-155	241	12	n	n	PRON
iajs-155	241	13	be	be	AUX
iajs-155	241	14	a	a	DET
iajs-155	241	15	proper	proper	ADJ
iajs-155	241	16	submodule	submodule	NOUN
iajs-155	241	17	of	of	ADP
iajs-155	241	18	an	an	DET
iajs-155	241	19	r	r	NOUN
iajs-155	241	20	-	-	PUNCT
iajs-155	241	21	module	module	NOUN
iajs-155	241	22	m	m	NOUN
iajs-155	241	23	.	.	PUNCT
iajs-155	242	1	consider	consider	VERB
iajs-155	242	2	the	the	DET
iajs-155	242	3	following	follow	VERB
iajs-155	242	4	statements	statement	NOUN
iajs-155	242	5	:	:	PUNCT
iajs-155	242	6	1	1	NUM
iajs-155	242	7	)	)	PUNCT
iajs-155	242	8	n	n	PRON
iajs-155	242	9	is	be	AUX
iajs-155	242	10	a	a	DET
iajs-155	242	11	2	2	NUM
iajs-155	242	12	-	-	PUNCT
iajs-155	242	13	absorbing	absorb	VERB
iajs-155	242	14	submodule	submodule	NOUN
iajs-155	242	15	of	of	ADP
iajs-155	242	16	m	m	PROPN
iajs-155	242	17	2	2	NUM
iajs-155	242	18	)	)	PUNCT
iajs-155	242	19	for	for	ADP
iajs-155	242	20	each	each	DET
iajs-155	242	21	a	a	NOUN
iajs-155	242	22	,	,	PUNCT
iajs-155	242	23	b	b	X
iajs-155	242	24	∈	∈	PROPN
iajs-155	242	25	r	r	NOUN
iajs-155	242	26	,	,	PUNCT
iajs-155	242	27	m	m	VERB
iajs-155	242	28	∈	∈	NOUN
iajs-155	242	29	m	m	VERB
iajs-155	242	30	.	.	PUNCT
iajs-155	243	1	if	if	SCONJ
iajs-155	243	2	abm	abm	PROPN
iajs-155	243	3	∉	∉	PROPN
iajs-155	243	4	n	n	PROPN
iajs-155	243	5	,	,	PUNCT
iajs-155	243	6	then	then	ADV
iajs-155	243	7	(	(	PUNCT
iajs-155	243	8	n	n	X
iajs-155	243	9	:	:	PUNCT
iajs-155	243	10	abm	abm	PROPN
iajs-155	243	11	)	)	PUNCT
iajs-155	243	12	=	=	PUNCT
iajs-155	243	13	(	(	PUNCT
iajs-155	243	14	n	n	CCONJ
iajs-155	243	15	:	:	PUNCT
iajs-155	243	16	am)∪(n	am)∪(n	PROPN
iajs-155	243	17	:	:	PUNCT
iajs-155	243	18	bm	bm	PROPN
iajs-155	243	19	)	)	PUNCT
iajs-155	243	20	3	3	NUM
iajs-155	243	21	)	)	PUNCT
iajs-155	243	22	for	for	ADP
iajs-155	243	23	each	each	PRON
iajs-155	243	24	a	a	NOUN
iajs-155	243	25	,	,	PUNCT
iajs-155	243	26	b	b	X
iajs-155	243	27	∈	∈	PROPN
iajs-155	243	28	r	r	NOUN
iajs-155	243	29	,	,	PUNCT
iajs-155	243	30	m∈m	m∈m	NOUN
iajs-155	243	31	if	if	SCONJ
iajs-155	243	32	abm∉n	abm∉n	PRON
iajs-155	243	33	then	then	ADV
iajs-155	243	34	(	(	PUNCT
iajs-155	243	35	n	n	CCONJ
iajs-155	243	36	:	:	PUNCT
iajs-155	243	37	abm)=(n	abm)=(n	PROPN
iajs-155	243	38	:	:	PUNCT
iajs-155	243	39	am	be	AUX
iajs-155	243	40	)	)	PUNCT
iajs-155	243	41	or	or	CCONJ
iajs-155	243	42	(	(	PUNCT
iajs-155	243	43	n	n	CCONJ
iajs-155	243	44	:	:	PUNCT
iajs-155	243	45	abm	abm	PROPN
iajs-155	243	46	)	)	PUNCT
iajs-155	243	47	=(	=(	PROPN
iajs-155	243	48	n	n	CCONJ
iajs-155	243	49	:	:	PUNCT
iajs-155	243	50	bm	bm	PROPN
iajs-155	243	51	)	)	PUNCT
iajs-155	243	52	then	then	ADV
iajs-155	243	53	(	(	PUNCT
iajs-155	243	54	1	1	X
iajs-155	243	55	)	)	PUNCT
iajs-155	243	56	⟹(2	⟹(2	NUM
iajs-155	243	57	)	)	PUNCT
iajs-155	244	1	⟹(3	⟹(3	X
iajs-155	244	2	)	)	PUNCT
iajs-155	244	3	and	and	CCONJ
iajs-155	244	4	if	if	SCONJ
iajs-155	244	5	m	m	NOUN
iajs-155	244	6	is	be	AUX
iajs-155	244	7	cyclic	cyclic	ADJ
iajs-155	244	8	,	,	PUNCT
iajs-155	244	9	then	then	ADV
iajs-155	244	10	(	(	PUNCT
iajs-155	244	11	3	3	NUM
iajs-155	244	12	)	)	PUNCT
iajs-155	244	13	⟹(1	⟹(1	NUM
iajs-155	244	14	)	)	PUNCT
iajs-155	244	15	.	.	PUNCT
iajs-155	245	1	proof	proof	NOUN
iajs-155	245	2	:	:	PUNCT
iajs-155	245	3	(	(	PUNCT
iajs-155	245	4	1	1	X
iajs-155	245	5	)	)	PUNCT
iajs-155	245	6	⟹(2	⟹(2	PUNCT
iajs-155	245	7	)	)	PUNCT
iajs-155	245	8	let	let	VERB
iajs-155	245	9	c	c	NOUN
iajs-155	245	10	∈	∈	PROPN
iajs-155	245	11	(	(	PUNCT
iajs-155	245	12	n	n	NOUN
iajs-155	245	13	:	:	PUNCT
iajs-155	245	14	abm	abm	PROPN
iajs-155	245	15	)	)	PUNCT
iajs-155	246	1	then	then	ADV
iajs-155	246	2	abcm	abcm	NOUN
iajs-155	246	3	∈	∈	PROPN
iajs-155	246	4	n	n	X
iajs-155	246	5	.	.	PUNCT
iajs-155	247	1	since	since	SCONJ
iajs-155	247	2	abm	abm	PROPN
iajs-155	247	3	∉	∉	PROPN
iajs-155	247	4	n	n	PROPN
iajs-155	247	5	and	and	CCONJ
iajs-155	247	6	n	n	PROPN
iajs-155	247	7	is	be	AUX
iajs-155	247	8	2	2	NUM
iajs-155	247	9	-	-	ADJ
iajs-155	247	10	absorbing	absorbing	ADJ
iajs-155	247	11	,	,	PUNCT
iajs-155	247	12	so	so	CCONJ
iajs-155	247	13	by	by	ADP
iajs-155	247	14	(	(	PUNCT
iajs-155	247	15	1.2	1.2	NUM
iajs-155	247	16	)	)	PUNCT
iajs-155	247	17	,	,	PUNCT
iajs-155	247	18	either	either	CCONJ
iajs-155	247	19	acm	acm	PROPN
iajs-155	247	20	∈	∈	PROPN
iajs-155	247	21	n	n	NOUN
iajs-155	247	22	or	or	CCONJ
iajs-155	247	23	bcm	bcm	NOUN
iajs-155	247	24	∈	∈	PROPN
iajs-155	247	25	n	n	NOUN
iajs-155	247	26	.	.	PUNCT
iajs-155	248	1	then	then	ADV
iajs-155	248	2	c	c	PROPN
iajs-155	248	3	∈	∈	PROPN
iajs-155	248	4	(	(	PUNCT
iajs-155	248	5	n	n	NOUN
iajs-155	248	6	:	:	PUNCT
iajs-155	248	7	am	be	AUX
iajs-155	248	8	)	)	PUNCT
iajs-155	248	9	or	or	CCONJ
iajs-155	248	10	c	c	NOUN
iajs-155	248	11	∈	∈	PROPN
iajs-155	248	12	(	(	PUNCT
iajs-155	248	13	n	n	NOUN
iajs-155	248	14	:	:	PUNCT
iajs-155	248	15	bm	bm	PROPN
iajs-155	248	16	)	)	PUNCT
iajs-155	248	17	thus	thus	ADV
iajs-155	248	18	(	(	PUNCT
iajs-155	248	19	n	n	X
iajs-155	248	20	:	:	PUNCT
iajs-155	248	21	abm	abm	PROPN
iajs-155	248	22	)	)	PUNCT
iajs-155	248	23	⊆	⊆	NUM
iajs-155	248	24	(	(	PUNCT
iajs-155	248	25	n	n	NUM
iajs-155	248	26	:	:	PUNCT
iajs-155	248	27	am	be	AUX
iajs-155	248	28	)	)	PUNCT
iajs-155	248	29	∪	∪	X
iajs-155	248	30	(	(	PUNCT
iajs-155	248	31	n	n	NOUN
iajs-155	248	32	:	:	PUNCT
iajs-155	248	33	bm	bm	PROPN
iajs-155	248	34	)	)	PUNCT
iajs-155	248	35	.	.	PUNCT
iajs-155	249	1	to	to	PART
iajs-155	249	2	prove	prove	VERB
iajs-155	249	3	reverse	reverse	ADJ
iajs-155	249	4	inclusion	inclusion	NOUN
iajs-155	249	5	:	:	PUNCT
iajs-155	249	6	let	let	VERB
iajs-155	249	7	c	c	PROPN
iajs-155	249	8	∈	∈	PROPN
iajs-155	249	9	(	(	PUNCT
iajs-155	249	10	n	n	NOUN
iajs-155	249	11	:	:	PUNCT
iajs-155	249	12	am	be	AUX
iajs-155	249	13	)	)	PUNCT
iajs-155	249	14	∪	∪	X
iajs-155	249	15	(	(	PUNCT
iajs-155	249	16	n	n	NOUN
iajs-155	249	17	:	:	PUNCT
iajs-155	249	18	bm	bm	PROPN
iajs-155	249	19	)	)	PUNCT
iajs-155	249	20	,	,	PUNCT
iajs-155	249	21	then	then	ADV
iajs-155	249	22	c	c	PROPN
iajs-155	249	23	∈	∈	PROPN
iajs-155	249	24	(	(	PUNCT
iajs-155	249	25	n	n	NOUN
iajs-155	249	26	:	:	PUNCT
iajs-155	249	27	am	be	AUX
iajs-155	249	28	)	)	PUNCT
iajs-155	249	29	or	or	CCONJ
iajs-155	249	30	c	c	NOUN
iajs-155	249	31	∈	∈	PROPN
iajs-155	249	32	(	(	PUNCT
iajs-155	249	33	n	n	NOUN
iajs-155	249	34	:	:	PUNCT
iajs-155	249	35	bm	bm	PROPN
iajs-155	249	36	)	)	PUNCT
iajs-155	249	37	so	so	SCONJ
iajs-155	249	38	that	that	PRON
iajs-155	249	39	acm	acm	PROPN
iajs-155	249	40	∈	∈	PROPN
iajs-155	249	41	n	n	NOUN
iajs-155	249	42	or	or	CCONJ
iajs-155	249	43	bcm	bcm	NOUN
iajs-155	249	44	∈	∈	PROPN
iajs-155	249	45	n	n	CCONJ
iajs-155	249	46	it	it	PRON
iajs-155	249	47	follows	follow	VERB
iajs-155	249	48	that	that	SCONJ
iajs-155	249	49	abcm	abcm	NOUN
iajs-155	249	50	∈	∈	PROPN
iajs-155	249	51	bn	bn	ADP
iajs-155	249	52	⊆	⊆	NUM
iajs-155	249	53	n	n	ADP
iajs-155	249	54	or	or	CCONJ
iajs-155	249	55	abcm	abcm	NOUN
iajs-155	249	56	∈	∈	NOUN
iajs-155	249	57	an	an	DET
iajs-155	249	58	⊆	⊆	NUM
iajs-155	249	59	n	n	PRON
iajs-155	249	60	thus	thus	ADV
iajs-155	249	61	abcm	abcm	NOUN
iajs-155	249	62	∈	∈	NOUN
iajs-155	249	63	n	n	NOUN
iajs-155	249	64	and	and	CCONJ
iajs-155	249	65	hence	hence	ADV
iajs-155	249	66	c	c	NOUN
iajs-155	249	67	∈	∈	PROPN
iajs-155	249	68	n	n	CCONJ
iajs-155	249	69	:	:	PUNCT
iajs-155	249	70	abm	abm	PROPN
iajs-155	249	71	.	.	PUNCT
iajs-155	250	1	233	233	NUM
iajs-155	250	2	|	|	ADV
iajs-155	250	3	mathematics	mathematic	NOUN
iajs-155	250	4	2015	2015	NUM
iajs-155	250	5	)	)	PUNCT
iajs-155	250	6	عام	عام	ADP
iajs-155	250	7	3العدد	3العدد	NUM
iajs-155	250	8	(	(	PUNCT
iajs-155	250	9	28مجلة	28مجلة	X
iajs-155	250	10	إبن	إبن	VERB
iajs-155	250	11	الھيثم	الھيثم	NOUN
iajs-155	250	12	للعلوم	للعلوم	NOUN
iajs-155	250	13	الصرفة	الصرفة	NOUN
iajs-155	251	1	و	و	PRON
iajs-155	251	2	التطبيقية	التطبيقية	ADV
iajs-155	251	3	المجلد	المجلد	VERB
iajs-155	251	4	ibn	ibn	PROPN
iajs-155	251	5	al	al	PROPN
iajs-155	251	6	-	-	PUNCT
iajs-155	251	7	haitham	haitham	PROPN
iajs-155	251	8	jour	jour	X
iajs-155	251	9	.	.	PROPN
iajs-155	252	1	for	for	ADP
iajs-155	252	2	pure	pure	ADJ
iajs-155	252	3	&	&	CCONJ
iajs-155	252	4	appl	appl	PROPN
iajs-155	252	5	.	.	PUNCT
iajs-155	253	1	sci	sci	PROPN
iajs-155	253	2	.	.	PUNCT
iajs-155	253	3	vol	vol	NOUN
iajs-155	253	4	.	.	PROPN
iajs-155	254	1	28	28	NUM
iajs-155	254	2	(	(	PUNCT
iajs-155	254	3	3	3	NUM
iajs-155	254	4	)	)	PUNCT
iajs-155	254	5	2015	2015	NUM
iajs-155	254	6	then	then	ADV
iajs-155	254	7	(	(	PUNCT
iajs-155	254	8	n	n	CCONJ
iajs-155	254	9	:	:	PUNCT
iajs-155	254	10	am	be	AUX
iajs-155	254	11	)	)	PUNCT
iajs-155	254	12	∪	∪	NOUN
iajs-155	254	13	(	(	PUNCT
iajs-155	254	14	n	n	CCONJ
iajs-155	254	15	:	:	PUNCT
iajs-155	254	16	bm	bm	PROPN
iajs-155	254	17	)	)	PUNCT
iajs-155	254	18	⊆	⊆	NUM
iajs-155	254	19	(	(	PUNCT
iajs-155	254	20	n	n	CCONJ
iajs-155	254	21	:	:	PUNCT
iajs-155	254	22	abm	abm	PROPN
iajs-155	254	23	)	)	PUNCT
iajs-155	254	24	.	.	PUNCT
iajs-155	255	1	so	so	ADV
iajs-155	255	2	we	we	PRON
iajs-155	255	3	get	get	VERB
iajs-155	255	4	(	(	PUNCT
iajs-155	255	5	n	n	CCONJ
iajs-155	255	6	:	:	PUNCT
iajs-155	255	7	abm)=(n	abm)=(n	ADJ
iajs-155	255	8	:	:	PUNCT
iajs-155	255	9	am)∪(n	am)∪(n	PROPN
iajs-155	255	10	:	:	PUNCT
iajs-155	255	11	bm	bm	PROPN
iajs-155	255	12	)	)	PUNCT
iajs-155	255	13	.	.	PUNCT
iajs-155	256	1	(	(	PUNCT
iajs-155	256	2	2	2	NUM
iajs-155	256	3	)	)	PUNCT
iajs-155	256	4	⟹(3	⟹(3	NOUN
iajs-155	256	5	)	)	PUNCT
iajs-155	256	6	since	since	SCONJ
iajs-155	256	7	(	(	PUNCT
iajs-155	256	8	n	n	NUM
iajs-155	256	9	:	:	PUNCT
iajs-155	256	10	abm	abm	PROPN
iajs-155	256	11	)	)	PUNCT
iajs-155	256	12	is	be	AUX
iajs-155	256	13	an	an	DET
iajs-155	256	14	ideal	ideal	NOUN
iajs-155	256	15	of	of	ADP
iajs-155	256	16	r	r	NOUN
iajs-155	256	17	,	,	PUNCT
iajs-155	256	18	and	and	CCONJ
iajs-155	256	19	(	(	PUNCT
iajs-155	256	20	n	n	CCONJ
iajs-155	256	21	:	:	PUNCT
iajs-155	256	22	abm)=(n	abm)=(n	ADJ
iajs-155	256	23	:	:	PUNCT
iajs-155	256	24	am)∪(n	am)∪(n	PROPN
iajs-155	256	25	:	:	PUNCT
iajs-155	256	26	bm	bm	PROPN
iajs-155	256	27	)	)	PUNCT
iajs-155	256	28	.	.	PUNCT
iajs-155	257	1	so	so	ADV
iajs-155	257	2	either	either	ADV
iajs-155	257	3	(	(	PUNCT
iajs-155	257	4	n	n	X
iajs-155	257	5	:	:	PUNCT
iajs-155	257	6	bm	bm	PROPN
iajs-155	257	7	)	)	PUNCT
iajs-155	257	8	⊆	⊆	NUM
iajs-155	257	9	(	(	PUNCT
iajs-155	257	10	n	n	NUM
iajs-155	257	11	:	:	PUNCT
iajs-155	257	12	am	be	AUX
iajs-155	257	13	)	)	PUNCT
iajs-155	257	14	or	or	CCONJ
iajs-155	257	15	(	(	PUNCT
iajs-155	257	16	n	n	X
iajs-155	257	17	:	:	PUNCT
iajs-155	257	18	am	be	AUX
iajs-155	257	19	)	)	PUNCT
iajs-155	257	20	⊆	⊆	NUM
iajs-155	257	21	(	(	PUNCT
iajs-155	257	22	n	n	NOUN
iajs-155	257	23	:	:	PUNCT
iajs-155	257	24	bm	bm	PROPN
iajs-155	257	25	)	)	PUNCT
iajs-155	257	26	.	.	PUNCT
iajs-155	258	1	then	then	ADV
iajs-155	258	2	(	(	PUNCT
iajs-155	258	3	n	n	X
iajs-155	258	4	:	:	PUNCT
iajs-155	258	5	abm	abm	PROPN
iajs-155	258	6	)	)	PUNCT
iajs-155	258	7	=	=	PRON
iajs-155	258	8	(	(	PUNCT
iajs-155	258	9	n	n	CCONJ
iajs-155	258	10	:	:	PUNCT
iajs-155	258	11	am	be	AUX
iajs-155	258	12	)	)	PUNCT
iajs-155	258	13	or	or	CCONJ
iajs-155	258	14	(	(	PUNCT
iajs-155	258	15	n	n	X
iajs-155	258	16	:	:	PUNCT
iajs-155	258	17	abm	abm	PROPN
iajs-155	258	18	)	)	PUNCT
iajs-155	258	19	(	(	PUNCT
iajs-155	258	20	n	n	X
iajs-155	258	21	:	:	PUNCT
iajs-155	258	22	bm	bm	PROPN
iajs-155	258	23	)	)	PUNCT
iajs-155	258	24	(	(	PUNCT
iajs-155	258	25	3	3	NUM
iajs-155	258	26	)	)	PUNCT
iajs-155	258	27	⟹(1	⟹(1	NUM
iajs-155	258	28	)	)	PUNCT
iajs-155	258	29	now	now	ADV
iajs-155	258	30	suppose	suppose	VERB
iajs-155	258	31	that	that	SCONJ
iajs-155	258	32	m=	m=	AUX
iajs-155	258	33	m	m	VERB
iajs-155	258	34	for	for	ADP
iajs-155	258	35	some	some	DET
iajs-155	258	36	m1	m1	NOUN
iajs-155	258	37	∈m	∈m	NOUN
iajs-155	258	38	let	let	VERB
iajs-155	258	39	abm	abm	PROPN
iajs-155	258	40	∈	∈	PROPN
iajs-155	258	41	n	n	X
iajs-155	258	42	.	.	PUNCT
iajs-155	259	1	but	but	CCONJ
iajs-155	259	2	m	m	PROPN
iajs-155	259	3	=	=	PROPN
iajs-155	259	4	rm1	rm1	PROPN
iajs-155	259	5	for	for	ADP
iajs-155	259	6	some	some	DET
iajs-155	259	7	r	r	NOUN
iajs-155	259	8	∈	∈	NOUN
iajs-155	259	9	r	r	NOUN
iajs-155	259	10	then	then	ADV
iajs-155	259	11	abrm1	abrm1	PUNCT
iajs-155	260	1	∈	∈	PROPN
iajs-155	260	2	n	n	ADV
iajs-155	261	1	and	and	CCONJ
iajs-155	261	2	suppose	suppose	VERB
iajs-155	261	3	that	that	SCONJ
iajs-155	261	4	a	a	DET
iajs-155	261	5	b	b	NOUN
iajs-155	261	6	∉	∉	X
iajs-155	261	7	n	n	PROPN
iajs-155	261	8	:	:	PUNCT
iajs-155	261	9	m)=	m)=	NOUN
iajs-155	261	10	(	(	PUNCT
iajs-155	261	11	n	n	NUM
iajs-155	261	12	:	:	PUNCT
iajs-155	261	13	m	m	PROPN
iajs-155	261	14	)	)	PUNCT
iajs-155	261	15	,	,	PUNCT
iajs-155	261	16	abm1	abm1	PROPN
iajs-155	261	17	∉	∉	PROPN
iajs-155	261	18	n	n	PROPN
iajs-155	261	19	and	and	CCONJ
iajs-155	261	20	r∈	r∈	PROPN
iajs-155	261	21	(	(	PUNCT
iajs-155	261	22	n	n	PROPN
iajs-155	261	23	:	:	PUNCT
iajs-155	261	24	abm1	abm1	PROPN
iajs-155	261	25	)	)	PUNCT
iajs-155	261	26	.	.	PUNCT
iajs-155	262	1	by	by	ADP
iajs-155	262	2	(	(	PUNCT
iajs-155	262	3	3	3	X
iajs-155	262	4	)	)	PUNCT
iajs-155	262	5	(	(	PUNCT
iajs-155	262	6	n	n	X
iajs-155	262	7	:	:	PUNCT
iajs-155	262	8	abm1	abm1	PROPN
iajs-155	262	9	)	)	PUNCT
iajs-155	262	10	=	=	PRON
iajs-155	262	11	(	(	PUNCT
iajs-155	262	12	n	n	X
iajs-155	262	13	:	:	PUNCT
iajs-155	262	14	a	a	DET
iajs-155	262	15	m1	m1	NOUN
iajs-155	262	16	)	)	PUNCT
iajs-155	262	17	or	or	CCONJ
iajs-155	262	18	(	(	PUNCT
iajs-155	262	19	n	n	X
iajs-155	262	20	:	:	PUNCT
iajs-155	262	21	ab	ab	PROPN
iajs-155	262	22	m1	m1	PROPN
iajs-155	262	23	)	)	PUNCT
iajs-155	262	24	=	=	PUNCT
iajs-155	262	25	(	(	PUNCT
iajs-155	262	26	n	n	X
iajs-155	262	27	:	:	PUNCT
iajs-155	262	28	b	b	X
iajs-155	262	29	m1	m1	NOUN
iajs-155	262	30	)	)	PUNCT
iajs-155	262	31	.	.	PUNCT
iajs-155	263	1	since	since	SCONJ
iajs-155	263	2	r	r	NOUN
iajs-155	263	3	∈	∈	PROPN
iajs-155	263	4	(	(	PUNCT
iajs-155	263	5	n	n	NOUN
iajs-155	263	6	:	:	PUNCT
iajs-155	263	7	abm1	abm1	PROPN
iajs-155	263	8	)	)	PUNCT
iajs-155	263	9	,	,	PUNCT
iajs-155	263	10	so	so	CCONJ
iajs-155	263	11	either	either	CCONJ
iajs-155	263	12	r	r	NOUN
iajs-155	263	13	∈	∈	PROPN
iajs-155	263	14	(	(	PUNCT
iajs-155	263	15	n	n	NOUN
iajs-155	263	16	:	:	PUNCT
iajs-155	263	17	a	a	DET
iajs-155	263	18	m1	m1	NOUN
iajs-155	263	19	)	)	PUNCT
iajs-155	263	20	or	or	CCONJ
iajs-155	263	21	r	r	NOUN
iajs-155	263	22	∈	∈	PROPN
iajs-155	263	23	(	(	PUNCT
iajs-155	263	24	n	n	NOUN
iajs-155	263	25	:	:	PUNCT
iajs-155	263	26	b	b	X
iajs-155	263	27	m1	m1	NOUN
iajs-155	263	28	)	)	PUNCT
iajs-155	263	29	.	.	PUNCT
iajs-155	264	1	then	then	ADV
iajs-155	264	2	a	a	DET
iajs-155	264	3	rm1	rm1	PROPN
iajs-155	264	4	∈	∈	PROPN
iajs-155	264	5	n	n	PRON
iajs-155	264	6	or	or	CCONJ
iajs-155	264	7	b	b	PROPN
iajs-155	264	8	rm1	rm1	PROPN
iajs-155	264	9	∈	∈	PROPN
iajs-155	264	10	n	n	X
iajs-155	264	11	,	,	PUNCT
iajs-155	264	12	we	we	PRON
iajs-155	264	13	get	get	VERB
iajs-155	264	14	a	a	DET
iajs-155	264	15	m	m	NOUN
iajs-155	264	16	∈	∈	NOUN
iajs-155	264	17	n	n	NOUN
iajs-155	264	18	or	or	CCONJ
iajs-155	264	19	b	b	NOUN
iajs-155	264	20	m	m	VERB
iajs-155	264	21	∈	∈	NOUN
iajs-155	264	22	n	n	NOUN
iajs-155	264	23	.	.	PUNCT
iajs-155	265	1	hence	hence	ADV
iajs-155	265	2	n	n	NOUN
iajs-155	265	3	is	be	AUX
iajs-155	265	4	2	2	NUM
iajs-155	265	5	-	-	PUNCT
iajs-155	265	6	absorbing	absorbing	ADJ
iajs-155	265	7	∎	∎	PROPN
iajs-155	265	8	references	reference	NOUN
iajs-155	265	9	1	1	NUM
iajs-155	265	10	.	.	PUNCT
iajs-155	266	1	badawi	badawi	PROPN
iajs-155	266	2	,	,	PUNCT
iajs-155	266	3	a(2007	a(2007	PROPN
iajs-155	266	4	)	)	PUNCT
iajs-155	266	5	,	,	PUNCT
iajs-155	266	6	on	on	ADP
iajs-155	266	7	2	2	NUM
iajs-155	266	8	-	-	PUNCT
iajs-155	266	9	absorbing	absorbing	ADJ
iajs-155	266	10	ideals	ideal	NOUN
iajs-155	266	11	of	of	ADP
iajs-155	266	12	commutative	commutative	ADJ
iajs-155	266	13	rings	ring	NOUN
iajs-155	266	14	,	,	PUNCT
iajs-155	266	15	bull	bull	NOUN
iajs-155	266	16	.	.	PUNCT
iajs-155	267	1	austral	austral	PROPN
iajs-155	267	2	.	.	PUNCT
iajs-155	268	1	math	math	NOUN
iajs-155	268	2	.	.	PUNCT
iajs-155	269	1	soc	soc	PROPN
iajs-155	269	2	.	.	PUNCT
iajs-155	270	1	75	75	NUM
iajs-155	270	2	,	,	PUNCT
iajs-155	270	3	417–429	417–429	NUM
iajs-155	270	4	.	.	NOUN
iajs-155	271	1	2	2	NUM
iajs-155	271	2	.	.	X
iajs-155	271	3	yousefian	yousefian	ADJ
iajs-155	271	4	darani	darani	PROPN
iajs-155	271	5	,	,	PUNCT
iajs-155	271	6	a.	a.	NOUN
iajs-155	271	7	and	and	CCONJ
iajs-155	271	8	soheilnia	soheilnia	PROPN
iajs-155	271	9	,	,	PUNCT
iajs-155	271	10	f.(2011	f.(2011	ADJ
iajs-155	271	11	)	)	PUNCT
iajs-155	271	12	,	,	PUNCT
iajs-155	271	13	on	on	ADP
iajs-155	271	14	2	2	NUM
iajs-155	271	15	-	-	PUNCT
iajs-155	271	16	absorbing	absorbing	ADJ
iajs-155	271	17	and	and	CCONJ
iajs-155	271	18	weakly	weakly	ADJ
iajs-155	271	19	2	2	NUM
iajs-155	271	20	-	-	PUNCT
iajs-155	271	21	absorbing	absorb	VERB
iajs-155	271	22	submodules	submodule	NOUN
iajs-155	271	23	,	,	PUNCT
iajs-155	271	24	thai	thai	PROPN
iajs-155	271	25	j.	j.	PROPN
iajs-155	271	26	math	math	PROPN
iajs-155	271	27	.	.	PUNCT
iajs-155	271	28	,	,	PUNCT
iajs-155	271	29	9	9	NUM
iajs-155	271	30	,	,	PUNCT
iajs-155	271	31	577	577	NUM
iajs-155	271	32	-	-	SYM
iajs-155	271	33	584	584	NUM
iajs-155	271	34	3	3	NUM
iajs-155	271	35	.	.	PUNCT
iajs-155	271	36	abdul	abdul	PROPN
iajs-155	271	37	-	-	PUNCT
iajs-155	271	38	razak	razak	PROPN
iajs-155	271	39	,	,	PUNCT
iajs-155	271	40	h.m	h.m	PROPN
iajs-155	271	41	(	(	PUNCT
iajs-155	271	42	1999	1999	NUM
iajs-155	271	43	)	)	PUNCT
iajs-155	271	44	,	,	PUNCT
iajs-155	271	45	"	"	PUNCT
iajs-155	271	46	quasi	quasi	ADJ
iajs-155	271	47	-	-	ADJ
iajs-155	271	48	prime	prime	ADJ
iajs-155	271	49	modules	module	NOUN
iajs-155	271	50	and	and	CCONJ
iajs-155	271	51	quasi	quasi	ADJ
iajs-155	271	52	-	-	ADJ
iajs-155	271	53	prime	prime	ADJ
iajs-155	271	54	submodules	submodule	NOUN
iajs-155	271	55	"	"	PUNCT
iajs-155	271	56	,	,	PUNCT
iajs-155	271	57	m.sc.thesis	m.sc.thesis	NOUN
iajs-155	271	58	,	,	PUNCT
iajs-155	271	59	college	college	NOUN
iajs-155	271	60	of	of	ADP
iajs-155	271	61	education	education	PROPN
iajs-155	271	62	ibn	ibn	PROPN
iajs-155	271	63	al	al	PROPN
iajs-155	271	64	-	-	PUNCT
iajs-155	271	65	haitham	haitham	PROPN
iajs-155	271	66	,	,	PUNCT
iajs-155	271	67	university	university	PROPN
iajs-155	271	68	of	of	ADP
iajs-155	271	69	baghdad	baghdad	PROPN
iajs-155	271	70	,	,	PUNCT
iajs-155	271	71	4	4	NUM
iajs-155	271	72	.	.	PUNCT
iajs-155	272	1	payrovi	payrovi	PROPN
iajs-155	272	2	,	,	PUNCT
iajs-155	272	3	sh	sh	PROPN
iajs-155	272	4	and	and	CCONJ
iajs-155	272	5	babaei	babaei	PROPN
iajs-155	272	6	,	,	PUNCT
iajs-155	272	7	s.(2012	s.(2012	PROPN
iajs-155	272	8	)	)	PUNCT
iajs-155	272	9	,	,	PUNCT
iajs-155	272	10	on	on	ADP
iajs-155	272	11	2	2	NUM
iajs-155	272	12	-	-	PUNCT
iajs-155	272	13	absorbing	absorbing	ADJ
iajs-155	272	14	submodules	submodule	NOUN
iajs-155	272	15	,	,	PUNCT
iajs-155	272	16	algebra	algebra	NOUN
iajs-155	272	17	colloq	colloq	PROPN
iajs-155	272	18	.	.	PUNCT
iajs-155	272	19	,	,	PUNCT
iajs-155	272	20	19	19	NUM
iajs-155	272	21	,	,	PUNCT
iajs-155	272	22	913920	913920	NUM
iajs-155	272	23	.	.	PUNCT
iajs-155	273	1	5	5	X
iajs-155	273	2	.	.	X
iajs-155	273	3	moradi	moradi	NOUN
iajs-155	273	4	,	,	PUNCT
iajs-155	273	5	s.	s.	PROPN
iajs-155	273	6	and	and	CCONJ
iajs-155	273	7	azizi	azizi	PROPN
iajs-155	273	8	,	,	PUNCT
iajs-155	273	9	a.(2012	a.(2012	PROPN
iajs-155	273	10	)	)	PUNCT
iajs-155	273	11	,	,	PUNCT
iajs-155	273	12	2	2	NUM
iajs-155	273	13	-	-	PUNCT
iajs-155	273	14	absorbing	absorbing	ADJ
iajs-155	273	15	and	and	CCONJ
iajs-155	273	16	n	n	CCONJ
iajs-155	273	17	-	-	PUNCT
iajs-155	273	18	weakly	weakly	ADJ
iajs-155	273	19	prime	prime	ADJ
iajs-155	273	20	submodule	submodule	NOUN
iajs-155	273	21	,	,	PUNCT
iajs-155	273	22	miskolc	miskolc	ADJ
iajs-155	273	23	mathematical	mathematical	ADJ
iajs-155	273	24	notes	note	NOUN
iajs-155	273	25	,	,	PUNCT
iajs-155	273	26	13	13	NUM
iajs-155	273	27	,	,	PUNCT
iajs-155	273	28	1,pp	1,pp	NUM
iajs-155	273	29	.	.	PUNCT
iajs-155	274	1	75	75	NUM
iajs-155	274	2	-	-	SYM
iajs-155	274	3	86	86	NUM
iajs-155	274	4	.	.	PUNCT
iajs-155	275	1	6.elbast	6.elbast	NUM
iajs-155	275	2	,	,	PUNCT
iajs-155	275	3	z.a	z.a	PROPN
iajs-155	275	4	.	.	PROPN
iajs-155	275	5	and	and	CCONJ
iajs-155	275	6	smith	smith	PROPN
iajs-155	275	7	,	,	PUNCT
iajs-155	275	8	p.f.(1988	p.f.(1988	PROPN
iajs-155	275	9	)	)	PUNCT
iajs-155	275	10	,	,	PUNCT
iajs-155	275	11	multiplication	multiplication	NOUN
iajs-155	275	12	modules	module	NOUN
iajs-155	275	13	,	,	PUNCT
iajs-155	275	14	comm	comm	NOUN
iajs-155	275	15	.	.	PUNCT
iajs-155	276	1	in	in	ADP
iajs-155	276	2	algebra	algebra	PROPN
iajs-155	276	3	,	,	PUNCT
iajs-155	276	4	16(4	16(4	NUM
iajs-155	276	5	)	)	PUNCT
iajs-155	276	6	,	,	PUNCT
iajs-155	276	7	755	755	NUM
iajs-155	276	8	-	-	SYM
iajs-155	276	9	779	779	NUM
iajs-155	276	10	.	.	NOUN
iajs-155	276	11	7	7	X
iajs-155	276	12	.	.	X
iajs-155	276	13	moore	moore	PROPN
iajs-155	276	14	,	,	PUNCT
iajs-155	276	15	m.e	m.e	PROPN
iajs-155	276	16	.	.	PROPN
iajs-155	276	17	and	and	CCONJ
iajs-155	276	18	smith	smith	PROPN
iajs-155	276	19	,	,	PUNCT
iajs-155	276	20	s.j.(2002	s.j.(2002	PROPN
iajs-155	276	21	)	)	PUNCT
iajs-155	276	22	,	,	PUNCT
iajs-155	276	23	prime	prime	ADJ
iajs-155	276	24	and	and	CCONJ
iajs-155	276	25	radical	radical	ADJ
iajs-155	276	26	submodules	submodule	NOUN
iajs-155	276	27	of	of	ADP
iajs-155	276	28	modules	module	NOUN
iajs-155	276	29	over	over	ADP
iajs-155	276	30	commutative	commutative	ADJ
iajs-155	276	31	rings	ring	NOUN
iajs-155	276	32	,	,	PUNCT
iajs-155	276	33	comm	comm	NOUN
iajs-155	276	34	.	.	PUNCT
iajs-155	277	1	algebra	algebra	NOUN
iajs-155	277	2	,	,	PUNCT
iajs-155	277	3	30	30	NUM
iajs-155	277	4	,	,	PUNCT
iajs-155	277	5	5073	5073	NUM
iajs-155	277	6	-	-	SYM
iajs-155	277	7	5064	5064	NUM
iajs-155	277	8	.	.	PUNCT
iajs-155	278	1	8	8	X
iajs-155	278	2	.	.	X
iajs-155	279	1	hojjat	hojjat	PROPN
iajs-155	279	2	mostafanasab	mostafanasab	PROPN
iajs-155	279	3	,	,	PUNCT
iajs-155	279	4	ece	ece	PROPN
iajs-155	279	5	yetkin	yetkin	PROPN
iajs-155	279	6	,	,	PUNCT
iajs-155	279	7	unsal	unsal	PROPN
iajs-155	279	8	tekir	tekir	PROPN
iajs-155	279	9	and	and	CCONJ
iajs-155	279	10	a.	a.	PROPN
iajs-155	279	11	y.	y.	PROPN
iajs-155	279	12	darani(2015	darani(2015	PROPN
iajs-155	279	13	)	)	PUNCT
iajs-155	279	14	,	,	PUNCT
iajs-155	279	15	on	on	ADP
iajs-155	279	16	2	2	NUM
iajs-155	279	17	-	-	PUNCT
iajs-155	279	18	absorbing	absorbing	ADJ
iajs-155	279	19	primary	primary	ADJ
iajs-155	279	20	submodules	submodule	NOUN
iajs-155	279	21	of	of	ADP
iajs-155	279	22	modules	module	NOUN
iajs-155	279	23	over	over	ADP
iajs-155	279	24	commutative	commutative	ADJ
iajs-155	279	25	rings	ring	NOUN
iajs-155	279	26	,	,	PUNCT
iajs-155	279	27	arxiv	arxiv	NOUN
iajs-155	279	28	:	:	PUNCT
iajs-155	279	29	1503	1503	NUM
iajs-155	279	30	.	.	PUNCT
iajs-155	280	1	00308v1	00308v1	PUNCT
iajs-155	281	1	[	[	X
iajs-155	281	2	math.ac	math.ac	X
iajs-155	281	3	]	]	X
iajs-155	281	4	mar	mar	PROPN
iajs-155	281	5	2015	2015	NUM
iajs-155	281	6	.	.	PUNCT
iajs-155	282	1	9	9	X
iajs-155	282	2	.	.	X
iajs-155	282	3	anderson	anderson	PROPN
iajs-155	282	4	,	,	PUNCT
iajs-155	282	5	e.w	e.w	PROPN
iajs-155	282	6	.	.	PROPN
iajs-155	282	7	and	and	CCONJ
iajs-155	282	8	fuller	full	ADJ
iajs-155	282	9	,	,	PUNCT
iajs-155	282	10	k.r.(1992	k.r.(1992	PROPN
iajs-155	282	11	)	)	PUNCT
iajs-155	282	12	,	,	PUNCT
iajs-155	282	13	"	"	PUNCT
iajs-155	282	14	rings	ring	NOUN
iajs-155	282	15	and	and	CCONJ
iajs-155	282	16	categories	category	NOUN
iajs-155	282	17	of	of	ADP
iajs-155	282	18	modules	module	NOUN
iajs-155	282	19	"	"	PUNCT
iajs-155	282	20	,	,	PUNCT
iajs-155	282	21	springerverlage	springerverlage	NOUN
iajs-155	282	22	,	,	PUNCT
iajs-155	282	23	new	new	PROPN
iajs-155	282	24	york	york	PROPN
iajs-155	282	25	.	.	PUNCT
iajs-155	283	1	234	234	NUM
iajs-155	283	2	|	|	NOUN
iajs-155	283	3	mathematics	mathematic	NOUN
iajs-155	283	4	2015	2015	NUM
iajs-155	283	5	)	)	PUNCT
iajs-155	283	6	عام	عام	ADP
iajs-155	283	7	3العدد	3العدد	NUM
iajs-155	283	8	(	(	PUNCT
iajs-155	283	9	28مجلة	28مجلة	X
iajs-155	283	10	إبن	إبن	VERB
iajs-155	283	11	الھيثم	الھيثم	NOUN
iajs-155	283	12	للعلوم	للعلوم	NOUN
iajs-155	283	13	الصرفة	الصرفة	NOUN
iajs-155	284	1	و	و	PRON
iajs-155	284	2	التطبيقية	التطبيقية	ADV
iajs-155	284	3	المجلد	المجلد	VERB
iajs-155	284	4	ibn	ibn	PROPN
iajs-155	284	5	al	al	PROPN
iajs-155	284	6	-	-	PUNCT
iajs-155	284	7	haitham	haitham	PROPN
iajs-155	284	8	jour	jour	X
iajs-155	284	9	.	.	PROPN
iajs-155	285	1	for	for	ADP
iajs-155	285	2	pure	pure	ADJ
iajs-155	285	3	&	&	CCONJ
iajs-155	285	4	appl	appl	PROPN
iajs-155	285	5	.	.	PUNCT
iajs-155	286	1	sci	sci	PROPN
iajs-155	286	2	.	.	PUNCT
iajs-155	286	3	vol	vol	NOUN
iajs-155	286	4	.	.	PROPN
iajs-155	287	1	28	28	NUM
iajs-155	287	2	(	(	PUNCT
iajs-155	287	3	3	3	NUM
iajs-155	287	4	)	)	PUNCT
iajs-155	287	5	2015	2015	NUM
iajs-155	287	6	2حول	2حول	NUM
iajs-155	287	7	المقاسات	المقاسات	NOUN
iajs-155	287	8	الجزئية	الجزئية	VERB
iajs-155	287	9	من	من	PRON
iajs-155	287	10	النمط	النمط	PROPN
iajs-155	287	11	المستحوذه	المستحوذه	PROPN
iajs-155	287	12	على	على	PROPN
iajs-155	287	13	انعام	انعام	ADJ
iajs-155	287	14	محمد	محمد	PROPN
iajs-155	287	15	علي	علي	NOUN
iajs-155	287	16	ھادي	ھادي	NOUN
iajs-155	287	17	عبدالرحمن	عبدالرحمن	ADJ
iajs-155	287	18	عبدهللا	عبدهللا	VERB
iajs-155	287	19	حرفش	حرفش	PROPN
iajs-155	287	20	بغداد	بغداد	PROPN
iajs-155	287	21	/	/	SYM
iajs-155	287	22	جامعة	جامعة	NOUN
iajs-155	287	23	(	(	PUNCT
iajs-155	287	24	ابن	ابن	PROPN
iajs-155	287	25	الھيثم	الھيثم	PROPN
iajs-155	287	26	)	)	PUNCT
iajs-155	287	27	للعلوم	للعلوم	PROPN
iajs-155	287	28	الصرفة	الصرفة	NOUN
iajs-155	287	29	كلية	كلية	NOUN
iajs-155	287	30	التربية	التربية	PROPN
iajs-155	287	31	/قسم	/قسم	PUNCT
iajs-155	287	32	الرياضيات	الرياضيات	NOUN
iajs-155	287	33	2015	2015	NUM
iajs-155	287	34	/	/	SYM
iajs-155	287	35	اب/1،قبل	اب/1،قبل	PROPN
iajs-155	287	36	البحث	البحث	VERB
iajs-155	287	37	في:2015	في:2015	PROPN
iajs-155	287	38	/	/	SYM
iajs-155	287	39	أيار/26استلم	أيار/26استلم	ADJ
iajs-155	287	40	البحث	البحث	NOUN
iajs-155	287	41	في	في	NOUN
iajs-155	287	42	:	:	PUNCT
iajs-155	287	43	خالصة	خالصة	VERB
iajs-155	287	44	ال	ال	ADP
iajs-155	287	45	0	0	PUNCT
iajs-155	288	1	حلقة	حلقة	NOUN
iajs-155	288	2	ابدالية	ابدالية	PROPN
iajs-155	288	3	ذات	ذات	NOUN
iajs-155	288	4	محايد	محايد	NOUN
iajs-155	288	5	rلتكن	rلتكن	NOUN
iajs-155	288	6	.	.	PUNCT
iajs-155	289	1	rمقاس	rمقاس	PROPN
iajs-155	289	2	احادي	احادي	NOUN
iajs-155	289	3	على	على	NOUN
iajs-155	289	4	الحلقة	الحلقة	NOUN
iajs-155	289	5	mو	mو	ADP
iajs-155	289	6	1	1	NUM
iajs-155	289	7	والتي	والتي	NOUN
iajs-155	289	8	قدمت	قدمت	PROPN
iajs-155	290	1	من	من	PROPN
iajs-155	290	2	قبل	قبل	PROPN
iajs-155	290	3	الباحثين	الباحثين	PROPN
iajs-155	291	1	2ھدفنا	2ھدفنا	NUM
iajs-155	291	2	في	في	ADP
iajs-155	291	3	ھذا	ھذا	NOUN
iajs-155	291	4	البحث	البحث	VERB
iajs-155	291	5	االستمرار	االستمرار	PROPN
iajs-155	292	1	في	في	SCONJ
iajs-155	292	2	دراسة	دراسة	PROPN
iajs-155	292	3	المقاسات	المقاسات	PROPN
iajs-155	292	4	الجزئية	الجزئية	VERB
iajs-155	292	5	من	من	PRON
iajs-155	292	6	النمط	النمط	PROPN
iajs-155	292	7	المستحوذة	المستحوذة	PROPN
iajs-155	292	8	على	على	PROPN
iajs-155	292	9	دارياني	دارياني	PROPN
iajs-155	292	10	وسھيليني	وسھيليني	NOUN
iajs-155	292	11	.	.	PUNCT
iajs-155	293	1	العديد	العديد	INTJ
iajs-155	294	1	من	من	INTJ
iajs-155	294	2	الخواص	الخواص	PROPN
iajs-155	294	3	والتمييزات	والتمييزات	PROPN
iajs-155	294	4	لھذا	لھذا	PROPN
iajs-155	294	5	المفھوم	المفھوم	PROPN
iajs-155	294	6	قد	قد	INTJ
iajs-155	294	7	اعطيت	اعطيت	PROPN
iajs-155	294	8	،	،	PROPN
iajs-155	294	9	المقاسات	المقاسات	PROPN
iajs-155	294	10	الجزئية	الجزئية	PROPN
iajs-155	294	11	شبه	شبه	VERB
iajs-155	294	12	االولية	االولية	NOUN
iajs-155	294	13	،	،	X
iajs-155	294	14	2المقاسات	2المقاسات	NUM
iajs-155	294	15	الجزئية	الجزئية	NOUN
iajs-155	294	16	االولية	االولية	NOUN
iajs-155	294	17	،	،	PROPN
iajs-155	294	18	المقاسات	المقاسات	PROPN
iajs-155	294	19	الجزئية	الجزئية	VERB
iajs-155	294	20	المستحوذة	المستحوذة	ADJ
iajs-155	294	21	على	على	NOUN
iajs-155	294	22	:	:	PUNCT
iajs-155	294	23	الكلمات	الكلمات	VERB
iajs-155	294	24	المفتاحية	المفتاحية	NOUN
iajs-155	294	25	المقاسات	المقاسات	PROPN
iajs-155	294	26	الجدائية	الجدائية	PROPN
iajs-155	294	27	،	،	PROPN
iajs-155	294	28	المقاسات	المقاسات	PROPN
iajs-155	294	29	الجزئية	الجزئية	VERB
iajs-155	294	30	االبتدائية	االبتدائية	PROPN
iajs-155	294	31	،	،	PROPN
iajs-155	294	32	المقاسات	المقاسات	PROPN
iajs-155	294	33	الجزئية	الجزئية	VERB
iajs-155	294	34	النقية	النقية	PRON
iajs-155	294	35	.	.	PUNCT
iajs-155	294	36	  	  	SPACE