id	sid	tid	token	lemma	pos
iajs-2148	1	1	301	301	NUM
iajs-2148	1	2	ibn	ibn	PROPN
iajs-2148	1	3	al	al	PROPN
iajs-2148	1	4	-	-	PUNCT
iajs-2148	1	5	haitham	haitham	PROPN
iajs-2148	1	6	jour.for	jour.for	ADP
iajs-2148	1	7	pure	pure	ADJ
iajs-2148	1	8	&	&	CCONJ
iajs-2148	1	9	appl	appl	PROPN
iajs-2148	1	10	.sci	.sci	PROPN
iajs-2148	1	11	.	.	PUNCT
iajs-2148	2	1	32	32	NUM
iajs-2148	2	2	(	(	PUNCT
iajs-2148	2	3	2	2	NUM
iajs-2148	2	4	)	)	PUNCT
iajs-2148	2	5	2019	2019	NUM
iajs-2148	2	6	abstract	abstract	NOUN
iajs-2148	2	7	in	in	ADP
iajs-2148	2	8	this	this	DET
iajs-2148	2	9	research	research	NOUN
iajs-2148	2	10	note	note	NOUN
iajs-2148	2	11	approximately	approximately	ADV
iajs-2148	2	12	prime	prime	ADJ
iajs-2148	2	13	submodules	submodule	NOUN
iajs-2148	2	14	is	be	AUX
iajs-2148	2	15	defined	define	VERB
iajs-2148	2	16	as	as	ADP
iajs-2148	2	17	a	a	DET
iajs-2148	2	18	new	new	ADJ
iajs-2148	2	19	generalization	generalization	NOUN
iajs-2148	2	20	of	of	ADP
iajs-2148	2	21	prime	prime	ADJ
iajs-2148	2	22	submodules	submodule	NOUN
iajs-2148	2	23	of	of	ADP
iajs-2148	2	24	unitary	unitary	ADJ
iajs-2148	2	25	modules	module	NOUN
iajs-2148	2	26	over	over	ADP
iajs-2148	2	27	a	a	DET
iajs-2148	2	28	commutative	commutative	ADJ
iajs-2148	2	29	ring	ring	NOUN
iajs-2148	2	30	with	with	ADP
iajs-2148	2	31	identity	identity	NOUN
iajs-2148	2	32	.	.	PUNCT
iajs-2148	3	1	a	a	DET
iajs-2148	3	2	proper	proper	ADJ
iajs-2148	3	3	submodule	submodule	NOUN
iajs-2148	3	4	of	of	ADP
iajs-2148	3	5	an	an	DET
iajs-2148	3	6	-module	-module	NOUN
iajs-2148	3	7	is	be	AUX
iajs-2148	3	8	called	call	VERB
iajs-2148	3	9	an	an	DET
iajs-2148	3	10	approximaitly	approximaitly	ADV
iajs-2148	3	11	prime	prime	ADJ
iajs-2148	3	12	submodule	submodule	NOUN
iajs-2148	3	13	of	of	ADP
iajs-2148	3	14	(	(	PUNCT
iajs-2148	3	15	for	for	ADP
iajs-2148	3	16	short	short	ADJ
iajs-2148	3	17	app	app	ADJ
iajs-2148	3	18	-	-	PUNCT
iajs-2148	3	19	prime	prime	NOUN
iajs-2148	3	20	submodule	submodule	NOUN
iajs-2148	3	21	)	)	PUNCT
iajs-2148	3	22	,	,	PUNCT
iajs-2148	3	23	if	if	SCONJ
iajs-2148	3	24	when	when	SCONJ
iajs-2148	3	25	ever	ever	ADV
iajs-2148	3	26	,	,	PUNCT
iajs-2148	3	27	where	where	SCONJ
iajs-2148	3	28	,	,	PUNCT
iajs-2148	3	29	,	,	PUNCT
iajs-2148	3	30	implies	imply	VERB
iajs-2148	3	31	that	that	SCONJ
iajs-2148	3	32	either	either	ADV
iajs-2148	3	33	or	or	CCONJ
iajs-2148	3	34	.	.	PUNCT
iajs-2148	4	1	so	so	ADV
iajs-2148	4	2	,	,	PUNCT
iajs-2148	4	3	an	an	DET
iajs-2148	4	4	ideal	ideal	NOUN
iajs-2148	4	5	of	of	ADP
iajs-2148	4	6	a	a	DET
iajs-2148	4	7	ring	ring	NOUN
iajs-2148	4	8	is	be	AUX
iajs-2148	4	9	called	call	VERB
iajs-2148	4	10	app	app	ADJ
iajs-2148	4	11	-	-	PUNCT
iajs-2148	4	12	prime	prime	ADJ
iajs-2148	4	13	ideal	ideal	NOUN
iajs-2148	4	14	of	of	ADP
iajs-2148	4	15	if	if	SCONJ
iajs-2148	4	16	is	be	AUX
iajs-2148	4	17	an	an	DET
iajs-2148	4	18	app	app	ADJ
iajs-2148	4	19	-	-	PUNCT
iajs-2148	4	20	prime	prime	NOUN
iajs-2148	4	21	submodule	submodule	NOUN
iajs-2148	4	22	of	of	ADP
iajs-2148	4	23	-module	-module	PROPN
iajs-2148	4	24	.	.	PUNCT
iajs-2148	5	1	several	several	ADJ
iajs-2148	5	2	basic	basic	ADJ
iajs-2148	5	3	properties	property	NOUN
iajs-2148	5	4	,	,	PUNCT
iajs-2148	5	5	characterizations	characterization	NOUN
iajs-2148	5	6	and	and	CCONJ
iajs-2148	5	7	examples	example	NOUN
iajs-2148	5	8	of	of	ADP
iajs-2148	5	9	approximaitly	approximaitly	ADV
iajs-2148	5	10	prime	prime	ADJ
iajs-2148	5	11	submodules	submodule	NOUN
iajs-2148	5	12	were	be	AUX
iajs-2148	5	13	given	give	VERB
iajs-2148	5	14	.	.	PUNCT
iajs-2148	6	1	furthermore	furthermore	ADV
iajs-2148	6	2	,	,	PUNCT
iajs-2148	6	3	the	the	DET
iajs-2148	6	4	definition	definition	NOUN
iajs-2148	6	5	of	of	ADP
iajs-2148	6	6	approximaitly	approximaitly	ADV
iajs-2148	6	7	prime	prime	ADJ
iajs-2148	6	8	radical	radical	ADJ
iajs-2148	6	9	of	of	ADP
iajs-2148	6	10	submodules	submodule	NOUN
iajs-2148	6	11	of	of	ADP
iajs-2148	6	12	modules	module	NOUN
iajs-2148	6	13	were	be	AUX
iajs-2148	6	14	introduced	introduce	VERB
iajs-2148	6	15	,	,	PUNCT
iajs-2148	6	16	and	and	CCONJ
iajs-2148	6	17	some	some	PRON
iajs-2148	6	18	of	of	ADP
iajs-2148	6	19	it	it	PRON
iajs-2148	6	20	is	be	AUX
iajs-2148	6	21	properties	property	NOUN
iajs-2148	6	22	were	be	AUX
iajs-2148	6	23	established	establish	VERB
iajs-2148	6	24	.	.	PUNCT
iajs-2148	7	1	keywords	keyword	NOUN
iajs-2148	7	2	:	:	PUNCT
iajs-2148	7	3	prime	prime	ADJ
iajs-2148	7	4	submodules	submodule	NOUN
iajs-2148	7	5	,	,	PUNCT
iajs-2148	7	6	approximaitly	approximaitly	ADV
iajs-2148	7	7	prime	prime	ADJ
iajs-2148	7	8	submodules	submodule	NOUN
iajs-2148	7	9	,	,	PUNCT
iajs-2148	7	10	approximaitly	approximaitly	ADV
iajs-2148	7	11	prime	prime	ADJ
iajs-2148	7	12	radical	radical	ADJ
iajs-2148	7	13	,	,	PUNCT
iajs-2148	7	14	socle	socle	NOUN
iajs-2148	7	15	of	of	ADP
iajs-2148	7	16	submodules	submodule	NOUN
iajs-2148	7	17	.	.	PUNCT
iajs-2148	8	1	1	1	X
iajs-2148	8	2	.	.	X
iajs-2148	8	3	introduction	introduction	NOUN
iajs-2148	8	4	throughout	throughout	ADP
iajs-2148	8	5	this	this	DET
iajs-2148	8	6	article	article	NOUN
iajs-2148	8	7	,	,	PUNCT
iajs-2148	8	8	we	we	PRON
iajs-2148	8	9	consider	consider	VERB
iajs-2148	8	10	all	all	DET
iajs-2148	8	11	rings	ring	NOUN
iajs-2148	8	12	as	as	ADP
iajs-2148	8	13	commutative	commutative	ADJ
iajs-2148	8	14	rings	ring	NOUN
iajs-2148	8	15	with	with	ADP
iajs-2148	8	16	identity	identity	NOUN
iajs-2148	8	17	,	,	PUNCT
iajs-2148	8	18	and	and	CCONJ
iajs-2148	8	19	all	all	DET
iajs-2148	8	20	modules	module	NOUN
iajs-2148	8	21	as	as	ADP
iajs-2148	8	22	unital	unital	ADJ
iajs-2148	8	23	-modules	-module	NOUN
iajs-2148	8	24	.	.	PUNCT
iajs-2148	9	1	a	a	DET
iajs-2148	9	2	proper	proper	ADJ
iajs-2148	9	3	submodule	submodule	NOUN
iajs-2148	9	4	of	of	ADP
iajs-2148	9	5	an	an	DET
iajs-2148	9	6	module	module	NOUN
iajs-2148	9	7	is	be	AUX
iajs-2148	9	8	prime	prime	ADJ
iajs-2148	9	9	,	,	PUNCT
iajs-2148	9	10	if	if	SCONJ
iajs-2148	9	11	whenever	whenever	ADV
iajs-2148	9	12	,	,	PUNCT
iajs-2148	9	13	for	for	ADP
iajs-2148	9	14	,	,	PUNCT
iajs-2148	9	15	then	then	ADV
iajs-2148	9	16	either	either	CCONJ
iajs-2148	9	17	or	or	CCONJ
iajs-2148	9	18	[	[	PUNCT
iajs-2148	9	19	]	]	X
iajs-2148	9	20	where	where	SCONJ
iajs-2148	9	21	[	[	X
iajs-2148	9	22	]	]	X
iajs-2148	9	23	.	.	PUNCT
iajs-2148	10	1	the	the	DET
iajs-2148	10	2	class	class	NOUN
iajs-2148	10	3	of	of	ADP
iajs-2148	10	4	prime	prime	ADJ
iajs-2148	10	5	submodules	submodule	NOUN
iajs-2148	10	6	was	be	AUX
iajs-2148	10	7	introduced	introduce	VERB
iajs-2148	10	8	and	and	CCONJ
iajs-2148	10	9	systematically	systematically	ADV
iajs-2148	10	10	studied	study	VERB
iajs-2148	10	11	in	in	ADP
iajs-2148	10	12	1978	1978	NUM
iajs-2148	10	13	by	by	ADP
iajs-2148	10	14	dauns	daun	NOUN
iajs-2148	10	15	[	[	X
iajs-2148	10	16	1	1	NUM
iajs-2148	10	17	]	]	PUNCT
iajs-2148	10	18	.	.	PUNCT
iajs-2148	11	1	as	as	ADP
iajs-2148	11	2	a	a	DET
iajs-2148	11	3	generalization	generalization	NOUN
iajs-2148	11	4	of	of	ADP
iajs-2148	11	5	the	the	DET
iajs-2148	11	6	class	class	NOUN
iajs-2148	11	7	of	of	ADP
iajs-2148	11	8	prime	prime	ADJ
iajs-2148	11	9	ideals	ideal	NOUN
iajs-2148	11	10	of	of	ADP
iajs-2148	11	11	rings	ring	NOUN
iajs-2148	11	12	and	and	CCONJ
iajs-2148	11	13	recently	recently	ADV
iajs-2148	11	14	has	have	AUX
iajs-2148	11	15	received	receive	VERB
iajs-2148	11	16	a	a	DET
iajs-2148	11	17	good	good	NOUN
iajs-2148	11	18	of	of	ADP
iajs-2148	11	19	attention	attention	NOUN
iajs-2148	11	20	from	from	ADP
iajs-2148	11	21	several	several	ADJ
iajs-2148	11	22	authors	author	NOUN
iajs-2148	11	23	see	see	VERB
iajs-2148	11	24	[	[	X
iajs-2148	11	25	2	2	NUM
iajs-2148	11	26	-	-	SYM
iajs-2148	11	27	8	8	NUM
iajs-2148	11	28	]	]	PUNCT
iajs-2148	11	29	.	.	PUNCT
iajs-2148	12	1	in	in	ADP
iajs-2148	12	2	this	this	DET
iajs-2148	12	3	paper	paper	NOUN
iajs-2148	12	4	,	,	PUNCT
iajs-2148	12	5	we	we	PRON
iajs-2148	12	6	will	will	AUX
iajs-2148	12	7	recall	recall	VERB
iajs-2148	12	8	some	some	DET
iajs-2148	12	9	basic	basic	ADJ
iajs-2148	12	10	definitions	definition	NOUN
iajs-2148	12	11	.	.	PUNCT
iajs-2148	13	1	the	the	DET
iajs-2148	13	2	socle	socle	NOUN
iajs-2148	13	3	of	of	ADP
iajs-2148	13	4	a	a	DET
iajs-2148	13	5	module	module	NOUN
iajs-2148	13	6	denoted	denote	VERB
iajs-2148	13	7	by	by	ADP
iajs-2148	13	8	is	be	AUX
iajs-2148	13	9	the	the	DET
iajs-2148	13	10	intersection	intersection	NOUN
iajs-2148	13	11	of	of	ADP
iajs-2148	13	12	all	all	DET
iajs-2148	13	13	essential	essential	ADJ
iajs-2148	13	14	submodules	submodule	NOUN
iajs-2148	13	15	of	of	ADP
iajs-2148	13	16	[	[	X
iajs-2148	13	17	9	9	NUM
iajs-2148	13	18	]	]	PUNCT
iajs-2148	13	19	.	.	PUNCT
iajs-2148	14	1	where	where	SCONJ
iajs-2148	14	2	a	a	DET
iajs-2148	14	3	non	non	ADJ
iajs-2148	14	4	-	-	ADJ
iajs-2148	14	5	zero	zero	NUM
iajs-2148	14	6	submodule	submodule	NOUN
iajs-2148	14	7	of	of	ADP
iajs-2148	14	8	an	an	DET
iajs-2148	14	9	-module	-module	NOUN
iajs-2148	14	10	is	be	AUX
iajs-2148	14	11	called	call	VERB
iajs-2148	14	12	essential	essential	ADJ
iajs-2148	14	13	if	if	SCONJ
iajs-2148	14	14	for	for	ADP
iajs-2148	14	15	each	each	DET
iajs-2148	14	16	non	non	ADJ
iajs-2148	14	17	-	-	ADJ
iajs-2148	14	18	zero	zero	NUM
iajs-2148	14	19	submodule	submodule	NOUN
iajs-2148	14	20	of	of	ADP
iajs-2148	14	21	[	[	X
iajs-2148	14	22	9	9	NUM
iajs-2148	14	23	]	]	PUNCT
iajs-2148	14	24	.	.	PUNCT
iajs-2148	15	1	an	an	DET
iajs-2148	15	2	element	element	NOUN
iajs-2148	15	3	in	in	ADP
iajs-2148	15	4	a	a	DET
iajs-2148	15	5	module	module	NOUN
iajs-2148	15	6	over	over	ADP
iajs-2148	15	7	integral	integral	ADJ
iajs-2148	15	8	domain	domain	NOUN
iajs-2148	15	9	is	be	AUX
iajs-2148	15	10	torsion	torsion	NOUN
iajs-2148	15	11	element	element	NOUN
iajs-2148	15	12	if	if	SCONJ
iajs-2148	15	13	for	for	ADP
iajs-2148	15	14	all	all	PRON
iajs-2148	15	15	[	[	X
iajs-2148	15	16	9	9	NUM
iajs-2148	15	17	]	]	PUNCT
iajs-2148	15	18	.	.	PUNCT
iajs-2148	16	1	the	the	DET
iajs-2148	16	2	set	set	NOUN
iajs-2148	16	3	of	of	ADP
iajs-2148	16	4	all	all	DET
iajs-2148	16	5	torsion	torsion	NOUN
iajs-2148	16	6	elements	element	NOUN
iajs-2148	16	7	of	of	ADP
iajs-2148	16	8	denoted	denote	VERB
iajs-2148	16	9	by	by	ADP
iajs-2148	16	10	is	be	AUX
iajs-2148	16	11	a	a	DET
iajs-2148	16	12	submodule	submodule	NOUN
iajs-2148	16	13	of	of	ADP
iajs-2148	16	14	.	.	PUNCT
iajs-2148	17	1	if	if	SCONJ
iajs-2148	17	2	then	then	ADV
iajs-2148	17	3	is	be	AUX
iajs-2148	17	4	said	say	VERB
iajs-2148	17	5	to	to	PART
iajs-2148	17	6	be	be	AUX
iajs-2148	17	7	torsion	torsion	NOUN
iajs-2148	17	8	free	free	ADJ
iajs-2148	17	9	[	[	X
iajs-2148	17	10	9	9	NUM
iajs-2148	17	11	]	]	PUNCT
iajs-2148	17	12	.	.	PUNCT
iajs-2148	18	1	an	an	DET
iajs-2148	18	2	module	module	NOUN
iajs-2148	18	3	is	be	AUX
iajs-2148	18	4	multiplication	multiplication	NOUN
iajs-2148	18	5	if	if	SCONJ
iajs-2148	18	6	each	each	DET
iajs-2148	18	7	submodule	submodule	NOUN
iajs-2148	18	8	is	be	AUX
iajs-2148	18	9	the	the	DET
iajs-2148	18	10	form	form	NOUN
iajs-2148	18	11	for	for	ADP
iajs-2148	18	12	some	some	DET
iajs-2148	18	13	ideal	ideal	NOUN
iajs-2148	18	14	of	of	ADP
iajs-2148	18	15	or	or	CCONJ
iajs-2148	18	16	[	[	PUNCT
iajs-2148	18	17	]	]	X
iajs-2148	19	1	[	[	X
iajs-2148	19	2	10	10	NUM
iajs-2148	19	3	]	]	PUNCT
iajs-2148	19	4	.	.	PUNCT
iajs-2148	20	1	a	a	DET
iajs-2148	20	2	subset	subset	NOUN
iajs-2148	20	3	of	of	ADP
iajs-2148	20	4	a	a	DET
iajs-2148	20	5	ring	ring	NOUN
iajs-2148	20	6	is	be	AUX
iajs-2148	20	7	called	call	VERB
iajs-2148	20	8	multiplicatively	multiplicatively	ADV
iajs-2148	20	9	closed	close	VERB
iajs-2148	20	10	subset	subset	NOUN
iajs-2148	20	11	of	of	ADP
iajs-2148	20	12	if	if	SCONJ
iajs-2148	20	13	and	and	CCONJ
iajs-2148	20	14	for	for	ADP
iajs-2148	20	15	every	every	DET
iajs-2148	20	16	[	[	X
iajs-2148	20	17	11	11	NUM
iajs-2148	20	18	]	]	PUNCT
iajs-2148	20	19	.	.	PUNCT
iajs-2148	21	1	if	if	SCONJ
iajs-2148	21	2	is	be	AUX
iajs-2148	21	3	a	a	DET
iajs-2148	21	4	submodule	submodule	NOUN
iajs-2148	21	5	of	of	ADP
iajs-2148	21	6	an	an	DET
iajs-2148	21	7	-module	-module	NOUN
iajs-2148	21	8	,	,	PUNCT
iajs-2148	21	9	and	and	CCONJ
iajs-2148	21	10	is	be	AUX
iajs-2148	21	11	a	a	DET
iajs-2148	21	12	multiplicatively	multiplicatively	ADV
iajs-2148	21	13	closed	close	VERB
iajs-2148	21	14	subset	subset	NOUN
iajs-2148	21	15	of	of	ADP
iajs-2148	21	16	,	,	PUNCT
iajs-2148	21	17	then	then	ADV
iajs-2148	21	18	is	be	AUX
iajs-2148	21	19	a	a	DET
iajs-2148	21	20	submodule	submodule	NOUN
iajs-2148	21	21	of	of	ADP
iajs-2148	21	22	and	and	CCONJ
iajs-2148	21	23	ali	ali	PROPN
iajs-2148	21	24	sh	sh	PROPN
iajs-2148	21	25	.	.	PROPN
iajs-2148	21	26	ajeel	ajeel	PROPN
iajs-2148	21	27	haibat	haibat	PROPN
iajs-2148	21	28	k.	k.	PROPN
iajs-2148	21	29	mohammadali	mohammadali	PROPN
iajs-2148	21	30	approximaitly	approximaitly	ADV
iajs-2148	21	31	prime	prime	ADJ
iajs-2148	21	32	submodules	submodule	NOUN
iajs-2148	21	33	and	and	CCONJ
iajs-2148	21	34	some	some	DET
iajs-2148	21	35	related	relate	VERB
iajs-2148	21	36	concepts	concept	NOUN
iajs-2148	21	37	ibn	ibn	PROPN
iajs-2148	21	38	al	al	PROPN
iajs-2148	21	39	haitham	haitham	PROPN
iajs-2148	21	40	journal	journal	PROPN
iajs-2148	21	41	for	for	ADP
iajs-2148	21	42	pure	pure	ADJ
iajs-2148	21	43	and	and	CCONJ
iajs-2148	21	44	applied	apply	VERB
iajs-2148	21	45	science	science	NOUN
iajs-2148	21	46	journal	journal	PROPN
iajs-2148	21	47	homepage	homepage	NOUN
iajs-2148	21	48	:	:	PUNCT
iajs-2148	21	49	http://jih.uobaghdad.edu.iq/index.php/j/index	http://jih.uobaghdad.edu.iq/index.php/j/index	NOUN
iajs-2148	21	50	ali.shebl@st.tu.edu.iq	ali.shebl@st.tu.edu.iq	ADJ
iajs-2148	21	51	article	article	NOUN
iajs-2148	21	52	history	history	NOUN
iajs-2148	21	53	:	:	PUNCT
iajs-2148	21	54	received	receive	VERB
iajs-2148	21	55	27	27	NUM
iajs-2148	21	56	december	december	PROPN
iajs-2148	21	57	2018	2018	NUM
iajs-2148	21	58	,	,	PUNCT
iajs-2148	21	59	accepted	accept	VERB
iajs-2148	21	60	20	20	NUM
iajs-2148	21	61	january	january	PROPN
iajs-2148	21	62	2019	2019	NUM
iajs-2148	21	63	,	,	PUNCT
iajs-2148	21	64	publish	publish	VERB
iajs-2148	21	65	may	may	AUX
iajs-2148	21	66	2019	2019	NUM
iajs-2148	21	67	10.30526/32.2.2148	10.30526/32.2.2148	NUM
iajs-2148	21	68	doi	doi	NOUN
iajs-2148	21	69	:	:	PUNCT
iajs-2148	21	70	department	department	NOUN
iajs-2148	21	71	of	of	ADP
iajs-2148	21	72	mathematics	mathematics	PROPN
iajs-2148	21	73	,	,	PUNCT
iajs-2148	21	74	college	college	NOUN
iajs-2148	21	75	of	of	ADP
iajs-2148	21	76	computer	computer	NOUN
iajs-2148	21	77	science	science	NOUN
iajs-2148	21	78	and	and	CCONJ
iajs-2148	21	79	mathematics	mathematic	NOUN
iajs-2148	21	80	,	,	PUNCT
iajs-2148	21	81	university	university	NOUN
iajs-2148	21	82	of	of	ADP
iajs-2148	21	83	tikrit	tikrit	NOUN
iajs-2148	21	84	,	,	PUNCT
iajs-2148	21	85	tikrit	tikrit	NOUN
iajs-2148	21	86	,	,	PUNCT
iajs-2148	21	87	iraq	iraq	PROPN
iajs-2148	21	88	h.mohammadali@tu.edu.iq	h.mohammadali@tu.edu.iq	PROPN
iajs-2148	21	89	mailto:ali.shebl@st.tu.edu.iq	mailto:ali.shebl@st.tu.edu.iq	ADJ
iajs-2148	21	90	mailto:ali.shebl@st.tu.edu.iq	mailto:ali.shebl@st.tu.edu.iq	ADJ
iajs-2148	21	91	mailto:h.mohammadali@tu.edu.iq	mailto:h.mohammadali@tu.edu.iq	ADJ
iajs-2148	21	92	mailto:ali.shebl@st.tu.edu.iq	mailto:ali.shebl@st.tu.edu.iq	ADJ
iajs-2148	21	93	mailto:ali.shebl@st.tu.edu.iq	mailto:ali.shebl@st.tu.edu.iq	ADJ
iajs-2148	21	94	mailto:ali.shebl@st.tu.edu.iq	mailto:ali.shebl@st.tu.edu.iq	PROPN
iajs-2148	21	95	301	301	NUM
iajs-2148	21	96	ibn	ibn	PROPN
iajs-2148	21	97	al	al	PROPN
iajs-2148	21	98	-	-	PUNCT
iajs-2148	21	99	haitham	haitham	PROPN
iajs-2148	21	100	jour.for	jour.for	ADP
iajs-2148	21	101	pure	pure	ADJ
iajs-2148	21	102	&	&	CCONJ
iajs-2148	21	103	appl	appl	PROPN
iajs-2148	21	104	.sci	.sci	PROPN
iajs-2148	21	105	.	.	PUNCT
iajs-2148	22	1	32	32	NUM
iajs-2148	22	2	(	(	PUNCT
iajs-2148	22	3	2	2	NUM
iajs-2148	22	4	)	)	PUNCT
iajs-2148	22	5	2019	2019	NUM
iajs-2148	23	1	[	[	X
iajs-2148	23	2	11	11	NUM
iajs-2148	23	3	]	]	PUNCT
iajs-2148	23	4	.	.	PUNCT
iajs-2148	24	1	a	a	DET
iajs-2148	24	2	non	non	ADJ
iajs-2148	24	3	-	-	ADJ
iajs-2148	24	4	zero	zero	NUM
iajs-2148	24	5	-module	-module	NOUN
iajs-2148	24	6	is	be	AUX
iajs-2148	24	7	compressible	compressible	ADJ
iajs-2148	24	8	,	,	PUNCT
iajs-2148	24	9	if	if	SCONJ
iajs-2148	24	10	it	it	PRON
iajs-2148	24	11	is	be	AUX
iajs-2148	24	12	passable	passable	ADJ
iajs-2148	24	13	to	to	PART
iajs-2148	24	14	embed	embed	VERB
iajs-2148	24	15	in	in	ADP
iajs-2148	24	16	every	every	DET
iajs-2148	24	17	non	non	ADJ
iajs-2148	24	18	-	-	ADJ
iajs-2148	24	19	zero	zero	NUM
iajs-2148	24	20	submodule	submodule	NOUN
iajs-2148	24	21	of	of	ADP
iajs-2148	24	22	[	[	X
iajs-2148	24	23	12	12	NUM
iajs-2148	24	24	]	]	PUNCT
iajs-2148	24	25	.	.	PUNCT
iajs-2148	25	1	2	2	X
iajs-2148	25	2	.	.	X
iajs-2148	25	3	approximaitly	approximaitly	ADV
iajs-2148	25	4	prime	prime	ADJ
iajs-2148	25	5	submodules	submodule	NOUN
iajs-2148	25	6	in	in	ADP
iajs-2148	25	7	this	this	DET
iajs-2148	25	8	section	section	NOUN
iajs-2148	25	9	,	,	PUNCT
iajs-2148	25	10	we	we	PRON
iajs-2148	25	11	introduce	introduce	VERB
iajs-2148	25	12	the	the	DET
iajs-2148	25	13	definition	definition	NOUN
iajs-2148	25	14	of	of	ADP
iajs-2148	25	15	approximaitly	approximaitly	ADV
iajs-2148	25	16	prime	prime	ADJ
iajs-2148	25	17	submodule	submodule	NOUN
iajs-2148	25	18	as	as	ADP
iajs-2148	25	19	a	a	DET
iajs-2148	25	20	generalization	generalization	NOUN
iajs-2148	25	21	of	of	ADP
iajs-2148	25	22	a	a	DET
iajs-2148	25	23	prime	prime	ADJ
iajs-2148	25	24	submodule	submodule	NOUN
iajs-2148	25	25	,	,	PUNCT
iajs-2148	25	26	and	and	CCONJ
iajs-2148	25	27	give	give	VERB
iajs-2148	25	28	some	some	DET
iajs-2148	25	29	basic	basic	ADJ
iajs-2148	25	30	properties	property	NOUN
iajs-2148	25	31	,	,	PUNCT
iajs-2148	25	32	examples	example	NOUN
iajs-2148	25	33	and	and	CCONJ
iajs-2148	25	34	characterizations	characterization	NOUN
iajs-2148	25	35	of	of	ADP
iajs-2148	25	36	this	this	DET
iajs-2148	25	37	concept	concept	NOUN
iajs-2148	25	38	.	.	PUNCT
iajs-2148	26	1	definition	definition	NOUN
iajs-2148	26	2	(	(	PUNCT
iajs-2148	26	3	1	1	X
iajs-2148	26	4	)	)	PUNCT
iajs-2148	26	5	a	a	DET
iajs-2148	26	6	proper	proper	ADJ
iajs-2148	26	7	submodule	submodule	NOUN
iajs-2148	26	8	of	of	ADP
iajs-2148	26	9	an	an	DET
iajs-2148	26	10	-module	-module	NOUN
iajs-2148	26	11	is	be	AUX
iajs-2148	26	12	called	call	VERB
iajs-2148	26	13	an	an	DET
iajs-2148	26	14	approximaitly	approximaitly	ADV
iajs-2148	26	15	prime	prime	ADJ
iajs-2148	26	16	submodule	submodule	NOUN
iajs-2148	26	17	of	of	ADP
iajs-2148	26	18	(	(	PUNCT
iajs-2148	26	19	for	for	ADP
iajs-2148	26	20	short	short	ADJ
iajs-2148	26	21	app	app	ADJ
iajs-2148	26	22	-	-	PUNCT
iajs-2148	26	23	prime	prime	NOUN
iajs-2148	26	24	submodule	submodule	NOUN
iajs-2148	26	25	)	)	PUNCT
iajs-2148	26	26	,	,	PUNCT
iajs-2148	26	27	if	if	SCONJ
iajs-2148	26	28	whenever	whenever	ADV
iajs-2148	26	29	,	,	PUNCT
iajs-2148	26	30	where	where	SCONJ
iajs-2148	26	31	,	,	PUNCT
iajs-2148	26	32	,	,	PUNCT
iajs-2148	26	33	implies	imply	VERB
iajs-2148	26	34	that	that	SCONJ
iajs-2148	26	35	either	either	ADV
iajs-2148	26	36	or	or	CCONJ
iajs-2148	26	37	.	.	PUNCT
iajs-2148	27	1	so	so	ADV
iajs-2148	27	2	,	,	PUNCT
iajs-2148	27	3	an	an	DET
iajs-2148	27	4	ideal	ideal	NOUN
iajs-2148	27	5	of	of	ADP
iajs-2148	27	6	a	a	DET
iajs-2148	27	7	ring	ring	NOUN
iajs-2148	27	8	is	be	AUX
iajs-2148	27	9	called	call	VERB
iajs-2148	27	10	app	app	ADJ
iajs-2148	27	11	-	-	PUNCT
iajs-2148	27	12	prime	prime	ADJ
iajs-2148	27	13	ideal	ideal	NOUN
iajs-2148	27	14	of	of	ADP
iajs-2148	27	15	if	if	SCONJ
iajs-2148	27	16	is	be	AUX
iajs-2148	27	17	an	an	DET
iajs-2148	27	18	app	app	ADJ
iajs-2148	27	19	-	-	PUNCT
iajs-2148	27	20	prime	prime	NOUN
iajs-2148	27	21	submodule	submodule	NOUN
iajs-2148	27	22	of	of	ADP
iajs-2148	27	23	-module	-module	PROPN
iajs-2148	27	24	.	.	PUNCT
iajs-2148	28	1	the	the	DET
iajs-2148	28	2	following	follow	VERB
iajs-2148	28	3	results	result	NOUN
iajs-2148	28	4	are	be	AUX
iajs-2148	28	5	characterizations	characterization	NOUN
iajs-2148	28	6	of	of	ADP
iajs-2148	28	7	app	app	ADJ
iajs-2148	28	8	-	-	PUNCT
iajs-2148	28	9	prime	prime	NOUN
iajs-2148	28	10	submodules	submodule	NOUN
iajs-2148	28	11	.	.	PUNCT
iajs-2148	29	1	proposition	proposition	NOUN
iajs-2148	29	2	(	(	PUNCT
iajs-2148	29	3	2	2	X
iajs-2148	29	4	)	)	PUNCT
iajs-2148	29	5	let	let	AUX
iajs-2148	29	6	be	be	AUX
iajs-2148	29	7	an	an	DET
iajs-2148	29	8	-module	-module	NOUN
iajs-2148	29	9	,	,	PUNCT
iajs-2148	29	10	and	and	CCONJ
iajs-2148	29	11	be	be	AUX
iajs-2148	29	12	a	a	DET
iajs-2148	29	13	submodule	submodule	NOUN
iajs-2148	29	14	of	of	ADP
iajs-2148	29	15	.	.	PUNCT
iajs-2148	30	1	then	then	ADV
iajs-2148	30	2	is	be	AUX
iajs-2148	30	3	an	an	DET
iajs-2148	30	4	app	app	ADJ
iajs-2148	30	5	-	-	PUNCT
iajs-2148	30	6	prime	prime	NOUN
iajs-2148	30	7	submodule	submodule	NOUN
iajs-2148	30	8	of	of	ADP
iajs-2148	30	9	if	if	SCONJ
iajs-2148	30	10	and	and	CCONJ
iajs-2148	30	11	only	only	ADV
iajs-2148	30	12	if	if	SCONJ
iajs-2148	30	13	for	for	ADP
iajs-2148	30	14	every	every	DET
iajs-2148	30	15	submodule	submodule	NOUN
iajs-2148	30	16	of	of	ADP
iajs-2148	30	17	and	and	CCONJ
iajs-2148	30	18	every	every	DET
iajs-2148	30	19	ideal	ideal	NOUN
iajs-2148	30	20	of	of	ADP
iajs-2148	30	21	such	such	ADJ
iajs-2148	30	22	that	that	PRON
iajs-2148	30	23	,	,	PUNCT
iajs-2148	30	24	implies	imply	VERB
iajs-2148	30	25	that	that	SCONJ
iajs-2148	30	26	either	either	CCONJ
iajs-2148	30	27	or	or	CCONJ
iajs-2148	30	28	[	[	PUNCT
iajs-2148	30	29	]	]	X
iajs-2148	30	30	.	.	PUNCT
iajs-2148	31	1	proof	proof	ADJ
iajs-2148	31	2	⇒	⇒	NOUN
iajs-2148	31	3	assume	assume	VERB
iajs-2148	31	4	that	that	SCONJ
iajs-2148	31	5	,	,	PUNCT
iajs-2148	31	6	where	where	SCONJ
iajs-2148	31	7	is	be	AUX
iajs-2148	31	8	an	an	DET
iajs-2148	31	9	ideal	ideal	NOUN
iajs-2148	31	10	of	of	ADP
iajs-2148	31	11	,	,	PUNCT
iajs-2148	31	12	and	and	CCONJ
iajs-2148	31	13	is	be	AUX
iajs-2148	31	14	a	a	DET
iajs-2148	31	15	submodule	submodule	NOUN
iajs-2148	31	16	of	of	ADP
iajs-2148	31	17	,	,	PUNCT
iajs-2148	31	18	and	and	CCONJ
iajs-2148	31	19	suppose	suppose	VERB
iajs-2148	31	20	that	that	SCONJ
iajs-2148	31	21	,	,	PUNCT
iajs-2148	31	22	then	then	ADV
iajs-2148	31	23	there	there	PRON
iajs-2148	31	24	exists	exist	VERB
iajs-2148	31	25	such	such	ADJ
iajs-2148	31	26	that	that	PRON
iajs-2148	31	27	.	.	PUNCT
iajs-2148	32	1	since	since	SCONJ
iajs-2148	32	2	,	,	PUNCT
iajs-2148	32	3	then	then	ADV
iajs-2148	32	4	for	for	ADP
iajs-2148	32	5	any	any	PRON
iajs-2148	32	6	,	,	PUNCT
iajs-2148	32	7	.	.	PUNCT
iajs-2148	33	1	but	but	CCONJ
iajs-2148	33	2	is	be	AUX
iajs-2148	33	3	an	an	DET
iajs-2148	33	4	app	app	ADJ
iajs-2148	33	5	-	-	PUNCT
iajs-2148	33	6	prime	prime	NOUN
iajs-2148	33	7	submodule	submodule	NOUN
iajs-2148	33	8	of	of	ADP
iajs-2148	33	9	,	,	PUNCT
iajs-2148	33	10	and	and	CCONJ
iajs-2148	33	11	,	,	PUNCT
iajs-2148	33	12	hence	hence	ADV
iajs-2148	33	13	[	[	X
iajs-2148	33	14	]	]	X
iajs-2148	33	15	.	.	PUNCT
iajs-2148	34	1	thus	thus	ADV
iajs-2148	34	2	[	[	X
iajs-2148	34	3	]	]	X
iajs-2148	34	4	.	.	PUNCT
iajs-2148	35	1	⇐	⇐	PROPN
iajs-2148	35	2	assume	assume	VERB
iajs-2148	35	3	that	that	SCONJ
iajs-2148	35	4	,	,	PUNCT
iajs-2148	35	5	where	where	SCONJ
iajs-2148	35	6	,	,	PUNCT
iajs-2148	35	7	,	,	PUNCT
iajs-2148	35	8	then	then	ADV
iajs-2148	35	9	,	,	PUNCT
iajs-2148	35	10	so	so	ADV
iajs-2148	35	11	by	by	ADP
iajs-2148	35	12	hypothesis	hypothesis	NOUN
iajs-2148	35	13	either	either	CCONJ
iajs-2148	35	14	[	[	PUNCT
iajs-2148	35	15	]	]	X
iajs-2148	35	16	or	or	CCONJ
iajs-2148	35	17	.	.	PUNCT
iajs-2148	36	1	that	that	PRON
iajs-2148	36	2	is	be	AUX
iajs-2148	36	3	either	either	PRON
iajs-2148	36	4	or	or	CCONJ
iajs-2148	36	5	[	[	PUNCT
iajs-2148	36	6	]	]	X
iajs-2148	36	7	.	.	PUNCT
iajs-2148	37	1	the	the	DET
iajs-2148	37	2	following	follow	VERB
iajs-2148	37	3	corollary	corollary	NOUN
iajs-2148	37	4	is	be	AUX
iajs-2148	37	5	a	a	DET
iajs-2148	37	6	consequence	consequence	NOUN
iajs-2148	37	7	immediately	immediately	ADV
iajs-2148	37	8	of	of	ADP
iajs-2148	37	9	a	a	DET
iajs-2148	37	10	proposition	proposition	NOUN
iajs-2148	37	11	(	(	PUNCT
iajs-2148	37	12	2	2	NUM
iajs-2148	37	13	)	)	PUNCT
iajs-2148	37	14	.	.	PUNCT
iajs-2148	38	1	corollary	corollary	ADJ
iajs-2148	38	2	(	(	PUNCT
iajs-2148	38	3	3	3	X
iajs-2148	38	4	)	)	PUNCT
iajs-2148	38	5	let	let	AUX
iajs-2148	38	6	be	be	AUX
iajs-2148	38	7	an	an	DET
iajs-2148	38	8	-module	-module	NOUN
iajs-2148	38	9	,	,	PUNCT
iajs-2148	38	10	and	and	CCONJ
iajs-2148	38	11	be	be	AUX
iajs-2148	38	12	a	a	DET
iajs-2148	38	13	submodule	submodule	NOUN
iajs-2148	38	14	of	of	ADP
iajs-2148	38	15	.	.	PUNCT
iajs-2148	39	1	then	then	ADV
iajs-2148	39	2	is	be	AUX
iajs-2148	39	3	an	an	DET
iajs-2148	39	4	app	app	ADJ
iajs-2148	39	5	-	-	PUNCT
iajs-2148	39	6	prime	prime	NOUN
iajs-2148	39	7	submodule	submodule	NOUN
iajs-2148	39	8	of	of	ADP
iajs-2148	39	9	if	if	SCONJ
iajs-2148	39	10	and	and	CCONJ
iajs-2148	39	11	only	only	ADV
iajs-2148	39	12	if	if	SCONJ
iajs-2148	39	13	for	for	ADP
iajs-2148	39	14	every	every	DET
iajs-2148	39	15	submodule	submodule	NOUN
iajs-2148	39	16	of	of	ADP
iajs-2148	39	17	and	and	CCONJ
iajs-2148	39	18	any	any	PRON
iajs-2148	39	19	with	with	ADP
iajs-2148	39	20	,	,	PUNCT
iajs-2148	39	21	implies	imply	VERB
iajs-2148	39	22	that	that	SCONJ
iajs-2148	39	23	either	either	CCONJ
iajs-2148	39	24	or	or	CCONJ
iajs-2148	39	25	[	[	PUNCT
iajs-2148	39	26	]	]	X
iajs-2148	39	27	.	.	PUNCT
iajs-2148	40	1	remark	remark	NOUN
iajs-2148	40	2	(	(	PUNCT
iajs-2148	40	3	4	4	X
iajs-2148	40	4	)	)	PUNCT
iajs-2148	40	5	it	it	PRON
iajs-2148	40	6	is	be	AUX
iajs-2148	40	7	clear	clear	ADJ
iajs-2148	40	8	that	that	SCONJ
iajs-2148	40	9	every	every	DET
iajs-2148	40	10	prime	prime	ADJ
iajs-2148	40	11	submodule	submodule	NOUN
iajs-2148	40	12	of	of	ADP
iajs-2148	40	13	an	an	DET
iajs-2148	40	14	-module	-module	NOUN
iajs-2148	40	15	is	be	AUX
iajs-2148	40	16	an	an	DET
iajs-2148	40	17	app	app	ADJ
iajs-2148	40	18	-	-	PUNCT
iajs-2148	40	19	prime	prime	NOUN
iajs-2148	40	20	submodule	submodule	NOUN
iajs-2148	40	21	of	of	ADP
iajs-2148	40	22	,	,	PUNCT
iajs-2148	40	23	while	while	SCONJ
iajs-2148	40	24	the	the	DET
iajs-2148	40	25	converse	converse	NOUN
iajs-2148	40	26	is	be	AUX
iajs-2148	40	27	not	not	PART
iajs-2148	40	28	true	true	ADJ
iajs-2148	40	29	as	as	SCONJ
iajs-2148	40	30	the	the	DET
iajs-2148	40	31	following	follow	VERB
iajs-2148	40	32	example	example	NOUN
iajs-2148	40	33	shows	show	VERB
iajs-2148	40	34	that	that	SCONJ
iajs-2148	40	35	:	:	PUNCT
iajs-2148	40	36	example	example	NOUN
iajs-2148	40	37	(	(	PUNCT
iajs-2148	40	38	5	5	X
iajs-2148	40	39	)	)	PUNCT
iajs-2148	40	40	consider	consider	VERB
iajs-2148	40	41	the	the	DET
iajs-2148	40	42	-module	-module	NOUN
iajs-2148	40	43	,	,	PUNCT
iajs-2148	40	44	and	and	CCONJ
iajs-2148	41	1	〈	〈	PROPN
iajs-2148	41	2	̅	̅	NOUN
iajs-2148	41	3	〉	〉	NOUN
iajs-2148	41	4	,	,	PUNCT
iajs-2148	41	5	〈	〈	NOUN
iajs-2148	41	6	̅〉.therefore	̅〉.therefore	NOUN
iajs-2148	41	7	each	each	PRON
iajs-2148	41	8	,	,	PUNCT
iajs-2148	41	9	,	,	PUNCT
iajs-2148	41	10	if	if	SCONJ
iajs-2148	41	11	,	,	PUNCT
iajs-2148	41	12	then	then	ADV
iajs-2148	41	13	either	either	CCONJ
iajs-2148	41	14	〈	〈	PROPN
iajs-2148	41	15	̅	̅	NOUN
iajs-2148	41	16	〉	〉	NOUN
iajs-2148	41	17	〈	〈	NOUN
iajs-2148	41	18	̅	̅	NOUN
iajs-2148	41	19	〉	〉	NOUN
iajs-2148	41	20	〈	〈	NOUN
iajs-2148	41	21	̅	̅	NOUN
iajs-2148	41	22	〉	〉	NOUN
iajs-2148	41	23	or	or	CCONJ
iajs-2148	41	24	[	[	X
iajs-2148	41	25	〈	〈	NOUN
iajs-2148	41	26	̅	̅	NOUN
iajs-2148	41	27	〉	〉	NOUN
iajs-2148	41	28	〈	〈	NOUN
iajs-2148	41	29	̅	̅	NOUN
iajs-2148	41	30	〉	〉	NOUN
iajs-2148	41	31	]	]	PUNCT
iajs-2148	41	32	.	.	PUNCT
iajs-2148	42	1	thus	thus	ADV
iajs-2148	42	2	is	be	AUX
iajs-2148	42	3	an	an	DET
iajs-2148	42	4	appprime	appprime	NOUN
iajs-2148	42	5	submodule	submodule	NOUN
iajs-2148	42	6	of	of	ADP
iajs-2148	42	7	,	,	PUNCT
iajs-2148	42	8	but	but	CCONJ
iajs-2148	42	9	is	be	AUX
iajs-2148	42	10	not	not	PART
iajs-2148	42	11	prime	prime	ADJ
iajs-2148	42	12	submodule	submodule	NOUN
iajs-2148	42	13	of	of	ADP
iajs-2148	42	14	,	,	PUNCT
iajs-2148	42	15	because	because	SCONJ
iajs-2148	42	16	̅	̅	NOUN
iajs-2148	42	17	,	,	PUNCT
iajs-2148	42	18	but	but	CCONJ
iajs-2148	42	19	neither	neither	PRON
iajs-2148	42	20	̅	̅	NOUN
iajs-2148	42	21	nor	nor	CCONJ
iajs-2148	42	22	[	[	PUNCT
iajs-2148	42	23	]	]	X
iajs-2148	42	24	.	.	PUNCT
iajs-2148	43	1	proposition	proposition	NOUN
iajs-2148	43	2	(	(	PUNCT
iajs-2148	43	3	6	6	X
iajs-2148	43	4	)	)	PUNCT
iajs-2148	43	5	let	let	AUX
iajs-2148	43	6	be	be	AUX
iajs-2148	43	7	an	an	DET
iajs-2148	43	8	-module	-module	NOUN
iajs-2148	43	9	,	,	PUNCT
iajs-2148	43	10	and	and	CCONJ
iajs-2148	43	11	be	be	AUX
iajs-2148	43	12	a	a	DET
iajs-2148	43	13	submodule	submodule	NOUN
iajs-2148	43	14	of	of	ADP
iajs-2148	43	15	,	,	PUNCT
iajs-2148	43	16	with	with	ADP
iajs-2148	43	17	.	.	PUNCT
iajs-2148	44	1	then	then	ADV
iajs-2148	44	2	is	be	AUX
iajs-2148	44	3	a	a	DET
iajs-2148	44	4	prime	prime	ADJ
iajs-2148	44	5	submodule	submodule	NOUN
iajs-2148	44	6	of	of	ADP
iajs-2148	44	7	if	if	SCONJ
iajs-2148	44	8	and	and	CCONJ
iajs-2148	44	9	only	only	ADV
iajs-2148	44	10	if	if	SCONJ
iajs-2148	44	11	is	be	AUX
iajs-2148	44	12	an	an	DET
iajs-2148	44	13	app	app	ADJ
iajs-2148	44	14	-	-	PUNCT
iajs-2148	44	15	prime	prime	NOUN
iajs-2148	44	16	submodule	submodule	NOUN
iajs-2148	44	17	of	of	ADP
iajs-2148	44	18	.	.	PUNCT
iajs-2148	45	1	301	301	NUM
iajs-2148	45	2	ibn	ibn	PROPN
iajs-2148	45	3	al	al	PROPN
iajs-2148	45	4	-	-	PUNCT
iajs-2148	45	5	haitham	haitham	PROPN
iajs-2148	45	6	jour.for	jour.for	ADP
iajs-2148	45	7	pure	pure	ADJ
iajs-2148	45	8	&	&	CCONJ
iajs-2148	45	9	appl	appl	PROPN
iajs-2148	45	10	.sci	.sci	PROPN
iajs-2148	45	11	.	.	PUNCT
iajs-2148	46	1	32	32	NUM
iajs-2148	46	2	(	(	PUNCT
iajs-2148	46	3	2	2	NUM
iajs-2148	46	4	)	)	PUNCT
iajs-2148	46	5	2019	2019	NUM
iajs-2148	46	6	proof	proof	NOUN
iajs-2148	46	7	it	it	PRON
iajs-2148	46	8	is	be	AUX
iajs-2148	46	9	clear	clear	ADJ
iajs-2148	46	10	the	the	DET
iajs-2148	46	11	following	follow	VERB
iajs-2148	46	12	corollaries	corollary	NOUN
iajs-2148	46	13	are	be	AUX
iajs-2148	46	14	direct	direct	ADJ
iajs-2148	46	15	consequence	consequence	NOUN
iajs-2148	46	16	of	of	ADP
iajs-2148	46	17	proposition	proposition	NOUN
iajs-2148	46	18	(	(	PUNCT
iajs-2148	46	19	2.6	2.6	NUM
iajs-2148	46	20	)	)	PUNCT
iajs-2148	46	21	.	.	PUNCT
iajs-2148	47	1	corollary	corollary	ADJ
iajs-2148	47	2	(	(	PUNCT
iajs-2148	47	3	7	7	X
iajs-2148	47	4	)	)	PUNCT
iajs-2148	47	5	let	let	AUX
iajs-2148	47	6	be	be	AUX
iajs-2148	47	7	an	an	DET
iajs-2148	47	8	-module	-module	NOUN
iajs-2148	47	9	,	,	PUNCT
iajs-2148	47	10	and	and	CCONJ
iajs-2148	47	11	be	be	AUX
iajs-2148	47	12	a	a	DET
iajs-2148	47	13	submodule	submodule	NOUN
iajs-2148	47	14	of	of	ADP
iajs-2148	47	15	,	,	PUNCT
iajs-2148	47	16	with	with	ADP
iajs-2148	47	17	.	.	PUNCT
iajs-2148	48	1	then	then	ADV
iajs-2148	48	2	is	be	AUX
iajs-2148	48	3	a	a	DET
iajs-2148	48	4	prime	prime	ADJ
iajs-2148	48	5	submodule	submodule	NOUN
iajs-2148	48	6	of	of	ADP
iajs-2148	48	7	if	if	SCONJ
iajs-2148	48	8	and	and	CCONJ
iajs-2148	48	9	only	only	ADV
iajs-2148	48	10	if	if	SCONJ
iajs-2148	48	11	is	be	AUX
iajs-2148	48	12	an	an	DET
iajs-2148	48	13	app	app	ADJ
iajs-2148	48	14	-	-	PUNCT
iajs-2148	48	15	prime	prime	NOUN
iajs-2148	48	16	submodule	submodule	NOUN
iajs-2148	48	17	of	of	ADP
iajs-2148	48	18	.	.	PUNCT
iajs-2148	49	1	it	it	PRON
iajs-2148	49	2	is	be	AUX
iajs-2148	49	3	well	well	ADV
iajs-2148	49	4	-	-	PUNCT
iajs-2148	49	5	known	know	VERB
iajs-2148	49	6	that	that	SCONJ
iajs-2148	49	7	a	a	DET
iajs-2148	49	8	torsion	torsion	NOUN
iajs-2148	49	9	free	free	ADJ
iajs-2148	49	10	-module	-module	NOUN
iajs-2148	49	11	has	have	VERB
iajs-2148	49	12	zero	zero	NUM
iajs-2148	49	13	socle	socle	NOUN
iajs-2148	49	14	[	[	X
iajs-2148	49	15	13	13	NUM
iajs-2148	49	16	]	]	PUNCT
iajs-2148	49	17	.	.	PUNCT
iajs-2148	50	1	so	so	ADV
iajs-2148	50	2	we	we	PRON
iajs-2148	50	3	set	set	VERB
iajs-2148	50	4	the	the	DET
iajs-2148	50	5	following	follow	VERB
iajs-2148	50	6	result	result	NOUN
iajs-2148	50	7	.	.	PUNCT
iajs-2148	51	1	corollary	corollary	ADJ
iajs-2148	51	2	(	(	PUNCT
iajs-2148	51	3	8)	8)	NUM
iajs-2148	51	4	let	let	AUX
iajs-2148	51	5	be	be	AUX
iajs-2148	51	6	a	a	DET
iajs-2148	51	7	torsion	torsion	NOUN
iajs-2148	51	8	free	free	ADJ
iajs-2148	51	9	-module	-module	NOUN
iajs-2148	51	10	,	,	PUNCT
iajs-2148	51	11	and	and	CCONJ
iajs-2148	51	12	be	be	AUX
iajs-2148	51	13	a	a	DET
iajs-2148	51	14	submodule	submodule	NOUN
iajs-2148	51	15	of	of	ADP
iajs-2148	51	16	.	.	PUNCT
iajs-2148	52	1	then	then	ADV
iajs-2148	52	2	is	be	AUX
iajs-2148	52	3	a	a	DET
iajs-2148	52	4	prime	prime	ADJ
iajs-2148	52	5	submodule	submodule	NOUN
iajs-2148	52	6	of	of	ADP
iajs-2148	52	7	if	if	SCONJ
iajs-2148	52	8	and	and	CCONJ
iajs-2148	52	9	only	only	ADV
iajs-2148	52	10	if	if	SCONJ
iajs-2148	52	11	is	be	AUX
iajs-2148	52	12	an	an	DET
iajs-2148	52	13	app	app	ADJ
iajs-2148	52	14	-	-	PUNCT
iajs-2148	52	15	prime	prime	NOUN
iajs-2148	52	16	submodule	submodule	NOUN
iajs-2148	52	17	of	of	ADP
iajs-2148	52	18	.	.	PUNCT
iajs-2148	53	1	proposition	proposition	NOUN
iajs-2148	53	2	(	(	PUNCT
iajs-2148	53	3	9	9	X
iajs-2148	53	4	)	)	PUNCT
iajs-2148	53	5	let	let	AUX
iajs-2148	53	6	be	be	AUX
iajs-2148	53	7	an	an	DET
iajs-2148	53	8	app	app	ADJ
iajs-2148	53	9	-	-	PUNCT
iajs-2148	53	10	prime	prime	NOUN
iajs-2148	53	11	submodule	submodule	NOUN
iajs-2148	53	12	of	of	ADP
iajs-2148	53	13	an	an	DET
iajs-2148	53	14	-module	-module	NOUN
iajs-2148	53	15	,	,	PUNCT
iajs-2148	53	16	with	with	ADP
iajs-2148	53	17	.	.	PUNCT
iajs-2148	54	1	then	then	ADV
iajs-2148	54	2	[	[	PUNCT
iajs-2148	54	3	]	]	X
iajs-2148	54	4	is	be	AUX
iajs-2148	54	5	an	an	DET
iajs-2148	54	6	app	app	ADJ
iajs-2148	54	7	-	-	PUNCT
iajs-2148	54	8	prime	prime	ADJ
iajs-2148	54	9	ideal	ideal	NOUN
iajs-2148	54	10	of	of	ADP
iajs-2148	54	11	.	.	PUNCT
iajs-2148	55	1	proof	proof	NOUN
iajs-2148	55	2	it	it	PRON
iajs-2148	55	3	is	be	AUX
iajs-2148	55	4	followed	follow	VERB
iajs-2148	55	5	by	by	ADP
iajs-2148	55	6	proposition	proposition	NOUN
iajs-2148	55	7	(	(	PUNCT
iajs-2148	55	8	6	6	NUM
iajs-2148	55	9	)	)	PUNCT
iajs-2148	55	10	and	and	CCONJ
iajs-2148	55	11	by	by	ADP
iajs-2148	55	12	[	[	X
iajs-2148	55	13	14	14	NUM
iajs-2148	55	14	,	,	PUNCT
iajs-2148	55	15	prop	prop	NOUN
iajs-2148	55	16	.	.	PUNCT
iajs-2148	56	1	2.8	2.8	NUM
iajs-2148	56	2	]	]	PUNCT
iajs-2148	56	3	.	.	PUNCT
iajs-2148	57	1	the	the	DET
iajs-2148	57	2	convers	conver	NOUN
iajs-2148	57	3	of	of	ADP
iajs-2148	57	4	proposition	proposition	NOUN
iajs-2148	57	5	(	(	PUNCT
iajs-2148	57	6	9	9	NUM
iajs-2148	57	7	)	)	PUNCT
iajs-2148	57	8	is	be	AUX
iajs-2148	57	9	not	not	PART
iajs-2148	57	10	true	true	ADJ
iajs-2148	57	11	in	in	ADP
iajs-2148	57	12	general	general	ADJ
iajs-2148	57	13	,	,	PUNCT
iajs-2148	57	14	as	as	SCONJ
iajs-2148	57	15	the	the	DET
iajs-2148	57	16	following	follow	VERB
iajs-2148	57	17	example	example	NOUN
iajs-2148	57	18	explain	explain	VERB
iajs-2148	57	19	that	that	PRON
iajs-2148	57	20	.	.	PUNCT
iajs-2148	58	1	example	example	NOUN
iajs-2148	58	2	(	(	PUNCT
iajs-2148	58	3	10	10	NUM
iajs-2148	58	4	)	)	PUNCT
iajs-2148	58	5	let	let	VERB
iajs-2148	58	6	,	,	PUNCT
iajs-2148	58	7	,	,	PUNCT
iajs-2148	58	8	and	and	CCONJ
iajs-2148	58	9	〈	〈	PROPN
iajs-2148	58	10	〉	〉	NOUN
iajs-2148	58	11	,	,	PUNCT
iajs-2148	58	12	then	then	ADV
iajs-2148	58	13	[	[	PUNCT
iajs-2148	58	14	]	]	X
iajs-2148	58	15	is	be	AUX
iajs-2148	58	16	an	an	DET
iajs-2148	58	17	app	app	ADJ
iajs-2148	58	18	-	-	PUNCT
iajs-2148	58	19	prime	prime	ADJ
iajs-2148	58	20	ideal	ideal	NOUN
iajs-2148	58	21	in	in	ADP
iajs-2148	58	22	a	a	DET
iajs-2148	58	23	ring	ring	NOUN
iajs-2148	58	24	.	.	PUNCT
iajs-2148	59	1	but	but	CCONJ
iajs-2148	59	2	is	be	AUX
iajs-2148	59	3	not	not	PART
iajs-2148	59	4	an	an	DET
iajs-2148	59	5	app	app	ADJ
iajs-2148	59	6	-	-	PUNCT
iajs-2148	59	7	prime	prime	NOUN
iajs-2148	59	8	submodule	submodule	NOUN
iajs-2148	59	9	of	of	ADP
iajs-2148	59	10	.	.	PUNCT
iajs-2148	60	1	recall	recall	VERB
iajs-2148	60	2	that	that	SCONJ
iajs-2148	60	3	an	an	DET
iajs-2148	60	4	-module	-module	NOUN
iajs-2148	60	5	is	be	AUX
iajs-2148	60	6	called	call	VERB
iajs-2148	60	7	singular	singular	ADJ
iajs-2148	60	8	module	module	NOUN
iajs-2148	60	9	provided	provide	VERB
iajs-2148	60	10	.	.	PUNCT
iajs-2148	61	1	at	at	ADP
iajs-2148	61	2	the	the	DET
iajs-2148	61	3	other	other	ADJ
iajs-2148	61	4	extreme	extreme	NOUN
iajs-2148	61	5	,	,	PUNCT
iajs-2148	61	6	we	we	PRON
iajs-2148	61	7	say	say	VERB
iajs-2148	61	8	that	that	PRON
iajs-2148	61	9	is	be	AUX
iajs-2148	61	10	non	non	ADJ
iajs-2148	61	11	-	-	ADJ
iajs-2148	61	12	singular	singular	ADJ
iajs-2148	61	13	module	module	NOUN
iajs-2148	61	14	provided	provide	VERB
iajs-2148	61	15	,	,	PUNCT
iajs-2148	61	16	where	where	SCONJ
iajs-2148	61	17	where	where	SCONJ
iajs-2148	61	18	the	the	DET
iajs-2148	61	19	set	set	NOUN
iajs-2148	61	20	of	of	ADP
iajs-2148	61	21	all	all	DET
iajs-2148	61	22	essential	essential	ADJ
iajs-2148	61	23	right	right	ADJ
iajs-2148	61	24	ideals	ideal	NOUN
iajs-2148	61	25	of	of	ADP
iajs-2148	61	26	the	the	DET
iajs-2148	61	27	ring	ring	NOUN
iajs-2148	61	28	,	,	PUNCT
iajs-2148	61	29	[	[	X
iajs-2148	61	30	9	9	NUM
iajs-2148	61	31	]	]	PUNCT
iajs-2148	61	32	.	.	PUNCT
iajs-2148	62	1	the	the	DET
iajs-2148	62	2	following	follow	VERB
iajs-2148	62	3	proposition	proposition	NOUN
iajs-2148	62	4	shows	show	VERB
iajs-2148	62	5	that	that	SCONJ
iajs-2148	62	6	the	the	DET
iajs-2148	62	7	converse	converse	NOUN
iajs-2148	62	8	of	of	ADP
iajs-2148	62	9	proposition	proposition	NOUN
iajs-2148	62	10	(	(	PUNCT
iajs-2148	62	11	9	9	NUM
iajs-2148	62	12	)	)	PUNCT
iajs-2148	62	13	is	be	AUX
iajs-2148	62	14	true	true	ADJ
iajs-2148	62	15	under	under	ADP
iajs-2148	62	16	certain	certain	ADJ
iajs-2148	62	17	conditions	condition	NOUN
iajs-2148	62	18	.	.	PUNCT
iajs-2148	63	1	proposition	proposition	NOUN
iajs-2148	63	2	(	(	PUNCT
iajs-2148	63	3	11	11	NUM
iajs-2148	63	4	)	)	PUNCT
iajs-2148	63	5	let	let	AUX
iajs-2148	63	6	be	be	AUX
iajs-2148	63	7	a	a	DET
iajs-2148	63	8	multiplication	multiplication	NOUN
iajs-2148	63	9	non	non	ADJ
iajs-2148	63	10	-	-	ADJ
iajs-2148	63	11	singular	singular	ADJ
iajs-2148	63	12	-module	-module	NOUN
iajs-2148	63	13	,	,	PUNCT
iajs-2148	63	14	and	and	CCONJ
iajs-2148	63	15	be	be	AUX
iajs-2148	63	16	a	a	DET
iajs-2148	63	17	proper	proper	ADJ
iajs-2148	63	18	submodule	submodule	NOUN
iajs-2148	63	19	of	of	ADP
iajs-2148	63	20	,	,	PUNCT
iajs-2148	63	21	with	with	ADP
iajs-2148	63	22	.	.	PUNCT
iajs-2148	64	1	then	then	ADV
iajs-2148	64	2	is	be	AUX
iajs-2148	64	3	an	an	DET
iajs-2148	64	4	app	app	ADJ
iajs-2148	64	5	-	-	PUNCT
iajs-2148	64	6	prime	prime	NOUN
iajs-2148	64	7	submodule	submodule	NOUN
iajs-2148	64	8	of	of	ADP
iajs-2148	64	9	if	if	SCONJ
iajs-2148	64	10	and	and	CCONJ
iajs-2148	64	11	only	only	ADV
iajs-2148	64	12	if	if	SCONJ
iajs-2148	64	13	[	[	PUNCT
iajs-2148	64	14	]	]	X
iajs-2148	64	15	is	be	AUX
iajs-2148	64	16	an	an	DET
iajs-2148	64	17	appprime	appprime	NOUN
iajs-2148	64	18	ideal	ideal	NOUN
iajs-2148	64	19	of	of	ADP
iajs-2148	64	20	.	.	PUNCT
iajs-2148	65	1	proof	proof	ADJ
iajs-2148	65	2	⇒	⇒	PROPN
iajs-2148	65	3	follows	follow	VERB
iajs-2148	65	4	by	by	ADP
iajs-2148	65	5	proposition	proposition	NOUN
iajs-2148	65	6	(	(	PUNCT
iajs-2148	65	7	9	9	NUM
iajs-2148	65	8	)	)	PUNCT
iajs-2148	65	9	.	.	PUNCT
iajs-2148	66	1	⇐	⇐	PROPN
iajs-2148	66	2	let	let	VERB
iajs-2148	66	3	,	,	PUNCT
iajs-2148	66	4	where	where	SCONJ
iajs-2148	66	5	,	,	PUNCT
iajs-2148	66	6	,	,	PUNCT
iajs-2148	66	7	then	then	ADV
iajs-2148	66	8	.	.	PUNCT
iajs-2148	67	1	but	but	CCONJ
iajs-2148	67	2	is	be	AUX
iajs-2148	67	3	multiplication	multiplication	NOUN
iajs-2148	67	4	,	,	PUNCT
iajs-2148	67	5	then	then	ADV
iajs-2148	67	6	for	for	ADP
iajs-2148	67	7	some	some	DET
iajs-2148	67	8	ideal	ideal	NOUN
iajs-2148	67	9	of	of	ADP
iajs-2148	67	10	.	.	PUNCT
iajs-2148	68	1	thus	thus	ADV
iajs-2148	68	2	,	,	PUNCT
iajs-2148	68	3	it	it	PRON
iajs-2148	68	4	follows	follow	VERB
iajs-2148	68	5	that	that	SCONJ
iajs-2148	68	6	[	[	X
iajs-2148	68	7	]	]	X
iajs-2148	68	8	.	.	PUNCT
iajs-2148	69	1	since	since	SCONJ
iajs-2148	69	2	[	[	PUNCT
iajs-2148	69	3	]	]	X
iajs-2148	69	4	is	be	AUX
iajs-2148	69	5	an	an	DET
iajs-2148	69	6	app	app	ADJ
iajs-2148	69	7	-	-	PUNCT
iajs-2148	69	8	prime	prime	ADJ
iajs-2148	69	9	ideal	ideal	NOUN
iajs-2148	69	10	of	of	ADP
iajs-2148	69	11	,	,	PUNCT
iajs-2148	69	12	then	then	ADV
iajs-2148	69	13	either	either	CCONJ
iajs-2148	69	14	[	[	PUNCT
iajs-2148	69	15	]	]	X
iajs-2148	69	16	or	or	CCONJ
iajs-2148	69	17	[	[	X
iajs-2148	69	18	[	[	X
iajs-2148	69	19	]	]	X
iajs-2148	69	20	]	]	PUNCT
iajs-2148	69	21	[	[	PUNCT
iajs-2148	69	22	]	]	X
iajs-2148	69	23	.	.	PUNCT
iajs-2148	70	1	hence	hence	ADV
iajs-2148	70	2	either	either	CCONJ
iajs-2148	70	3	[	[	PUNCT
iajs-2148	70	4	]	]	X
iajs-2148	70	5	or	or	CCONJ
iajs-2148	70	6	[	[	PUNCT
iajs-2148	70	7	]	]	X
iajs-2148	70	8	.	.	PUNCT
iajs-2148	71	1	but	but	CCONJ
iajs-2148	71	2	is	be	AUX
iajs-2148	71	3	a	a	DET
iajs-2148	71	4	nonsingular	nonsingular	ADJ
iajs-2148	71	5	,	,	PUNCT
iajs-2148	71	6	then	then	ADV
iajs-2148	71	7	by	by	ADP
iajs-2148	71	8	[	[	PUNCT
iajs-2148	71	9	9	9	NUM
iajs-2148	71	10	]	]	PUNCT
iajs-2148	71	11	.	.	PUNCT
iajs-2148	72	1	we	we	PRON
iajs-2148	72	2	have	have	VERB
iajs-2148	72	3	.	.	PUNCT
iajs-2148	73	1	thus	thus	ADV
iajs-2148	73	2	,	,	PUNCT
iajs-2148	73	3	either	either	ADV
iajs-2148	73	4	or	or	CCONJ
iajs-2148	73	5	.	.	PUNCT
iajs-2148	74	1	hence	hence	ADV
iajs-2148	74	2	either	either	PRON
iajs-2148	74	3	or	or	CCONJ
iajs-2148	74	4	[	[	PUNCT
iajs-2148	74	5	]	]	X
iajs-2148	74	6	.	.	PUNCT
iajs-2148	75	1	hence	hence	ADV
iajs-2148	75	2	is	be	AUX
iajs-2148	75	3	an	an	DET
iajs-2148	75	4	app	app	ADJ
iajs-2148	75	5	-	-	PUNCT
iajs-2148	75	6	prime	prime	NOUN
iajs-2148	75	7	submodule	submodule	NOUN
iajs-2148	75	8	of	of	ADP
iajs-2148	75	9	.	.	PUNCT
iajs-2148	76	1	proposition	proposition	NOUN
iajs-2148	76	2	(	(	PUNCT
iajs-2148	76	3	12	12	NUM
iajs-2148	76	4	)	)	PUNCT
iajs-2148	76	5	let	let	AUX
iajs-2148	76	6	be	be	AUX
iajs-2148	76	7	a	a	DET
iajs-2148	76	8	faithful	faithful	ADJ
iajs-2148	76	9	multiplication	multiplication	NOUN
iajs-2148	76	10	-module	-module	NOUN
iajs-2148	76	11	,	,	PUNCT
iajs-2148	76	12	and	and	CCONJ
iajs-2148	76	13	be	be	AUX
iajs-2148	76	14	a	a	DET
iajs-2148	76	15	proper	proper	ADJ
iajs-2148	76	16	submodule	submodule	NOUN
iajs-2148	76	17	of	of	ADP
iajs-2148	76	18	,	,	PUNCT
iajs-2148	76	19	with	with	ADP
iajs-2148	76	20	.	.	PUNCT
iajs-2148	77	1	then	then	ADV
iajs-2148	77	2	is	be	AUX
iajs-2148	77	3	an	an	DET
iajs-2148	77	4	app	app	ADJ
iajs-2148	77	5	-	-	PUNCT
iajs-2148	77	6	prime	prime	NOUN
iajs-2148	77	7	submodule	submodule	NOUN
iajs-2148	77	8	of	of	ADP
iajs-2148	77	9	if	if	SCONJ
iajs-2148	77	10	and	and	CCONJ
iajs-2148	77	11	only	only	ADV
iajs-2148	77	12	if	if	SCONJ
iajs-2148	77	13	[	[	PUNCT
iajs-2148	77	14	]	]	X
iajs-2148	77	15	is	be	AUX
iajs-2148	77	16	an	an	DET
iajs-2148	77	17	app	app	ADJ
iajs-2148	77	18	-	-	PUNCT
iajs-2148	77	19	prime	prime	ADJ
iajs-2148	77	20	ideal	ideal	NOUN
iajs-2148	77	21	of	of	ADP
iajs-2148	77	22	.	.	PUNCT
iajs-2148	78	1	301	301	NUM
iajs-2148	78	2	ibn	ibn	PROPN
iajs-2148	78	3	al	al	PROPN
iajs-2148	78	4	-	-	PUNCT
iajs-2148	78	5	haitham	haitham	PROPN
iajs-2148	78	6	jour.for	jour.for	ADP
iajs-2148	78	7	pure	pure	ADJ
iajs-2148	78	8	&	&	CCONJ
iajs-2148	78	9	appl	appl	PROPN
iajs-2148	78	10	.sci	.sci	PROPN
iajs-2148	78	11	.	.	PUNCT
iajs-2148	79	1	32	32	NUM
iajs-2148	79	2	(	(	PUNCT
iajs-2148	79	3	2	2	NUM
iajs-2148	79	4	)	)	PUNCT
iajs-2148	79	5	2019	2019	NUM
iajs-2148	79	6	proof	proof	NOUN
iajs-2148	79	7	⇒	⇒	NOUN
iajs-2148	79	8	follows	follow	VERB
iajs-2148	79	9	by	by	ADP
iajs-2148	79	10	proposition	proposition	NOUN
iajs-2148	79	11	(	(	PUNCT
iajs-2148	79	12	9	9	NUM
iajs-2148	79	13	)	)	PUNCT
iajs-2148	79	14	.	.	PUNCT
iajs-2148	80	1	⇐	⇐	PROPN
iajs-2148	80	2	let	let	VERB
iajs-2148	80	3	,	,	PUNCT
iajs-2148	80	4	where	where	SCONJ
iajs-2148	80	5	,	,	PUNCT
iajs-2148	80	6	,	,	PUNCT
iajs-2148	80	7	then	then	ADV
iajs-2148	80	8	.	.	PUNCT
iajs-2148	81	1	but	but	CCONJ
iajs-2148	81	2	is	be	AUX
iajs-2148	81	3	multiplication	multiplication	NOUN
iajs-2148	81	4	,	,	PUNCT
iajs-2148	81	5	then	then	ADV
iajs-2148	81	6	for	for	ADP
iajs-2148	81	7	some	some	DET
iajs-2148	81	8	ideal	ideal	NOUN
iajs-2148	81	9	of	of	ADP
iajs-2148	81	10	.	.	PUNCT
iajs-2148	82	1	thus	thus	ADV
iajs-2148	82	2	,	,	PUNCT
iajs-2148	82	3	it	it	PRON
iajs-2148	82	4	follows	follow	VERB
iajs-2148	82	5	that	that	SCONJ
iajs-2148	82	6	[	[	X
iajs-2148	82	7	]	]	X
iajs-2148	82	8	.	.	PUNCT
iajs-2148	83	1	since	since	SCONJ
iajs-2148	83	2	[	[	PUNCT
iajs-2148	83	3	]	]	X
iajs-2148	83	4	is	be	AUX
iajs-2148	83	5	an	an	DET
iajs-2148	83	6	app	app	ADJ
iajs-2148	83	7	-	-	PUNCT
iajs-2148	83	8	prime	prime	ADJ
iajs-2148	83	9	ideal	ideal	NOUN
iajs-2148	83	10	of	of	ADP
iajs-2148	83	11	,	,	PUNCT
iajs-2148	83	12	then	then	ADV
iajs-2148	83	13	by	by	ADP
iajs-2148	83	14	corollary	corollary	ADJ
iajs-2148	83	15	(	(	PUNCT
iajs-2148	83	16	3	3	NUM
iajs-2148	83	17	)	)	PUNCT
iajs-2148	83	18	either	either	CCONJ
iajs-2148	83	19	[	[	PUNCT
iajs-2148	83	20	]	]	X
iajs-2148	83	21	or	or	CCONJ
iajs-2148	83	22	[	[	X
iajs-2148	83	23	[	[	X
iajs-2148	83	24	]	]	X
iajs-2148	83	25	]	]	PUNCT
iajs-2148	83	26	[	[	PUNCT
iajs-2148	83	27	]	]	X
iajs-2148	83	28	.	.	PUNCT
iajs-2148	84	1	thus	thus	ADV
iajs-2148	84	2	either	either	CCONJ
iajs-2148	84	3	[	[	PUNCT
iajs-2148	84	4	]	]	X
iajs-2148	84	5	or	or	CCONJ
iajs-2148	84	6	[	[	PUNCT
iajs-2148	84	7	]	]	X
iajs-2148	84	8	.	.	PUNCT
iajs-2148	85	1	since	since	SCONJ
iajs-2148	85	2	is	be	AUX
iajs-2148	85	3	a	a	DET
iajs-2148	85	4	faithful	faithful	ADJ
iajs-2148	85	5	multiplication	multiplication	NOUN
iajs-2148	85	6	,	,	PUNCT
iajs-2148	85	7	then	then	ADV
iajs-2148	85	8	by	by	ADP
iajs-2148	85	9	[	[	X
iajs-2148	85	10	10,cor.2.14	10,cor.2.14	NOUN
iajs-2148	85	11	]	]	PUNCT
iajs-2148	85	12	.	.	PUNCT
iajs-2148	86	1	we	we	PRON
iajs-2148	86	2	have	have	VERB
iajs-2148	86	3	.	.	PUNCT
iajs-2148	87	1	it	it	PRON
iajs-2148	87	2	follows	follow	VERB
iajs-2148	87	3	that	that	SCONJ
iajs-2148	87	4	either	either	CCONJ
iajs-2148	87	5	or	or	CCONJ
iajs-2148	87	6	.	.	PUNCT
iajs-2148	88	1	hence	hence	ADV
iajs-2148	88	2	either	either	PRON
iajs-2148	88	3	or	or	CCONJ
iajs-2148	88	4	[	[	PUNCT
iajs-2148	88	5	]	]	X
iajs-2148	88	6	.	.	PUNCT
iajs-2148	89	1	hence	hence	ADV
iajs-2148	89	2	is	be	AUX
iajs-2148	89	3	an	an	DET
iajs-2148	89	4	app	app	ADJ
iajs-2148	89	5	-	-	PUNCT
iajs-2148	89	6	prime	prime	NOUN
iajs-2148	89	7	submodule	submodule	NOUN
iajs-2148	89	8	of	of	ADP
iajs-2148	89	9	.	.	PUNCT
iajs-2148	90	1	proposition	proposition	NOUN
iajs-2148	90	2	(	(	PUNCT
iajs-2148	90	3	13	13	NUM
iajs-2148	90	4	)	)	PUNCT
iajs-2148	90	5	let	let	AUX
iajs-2148	90	6	be	be	AUX
iajs-2148	90	7	an	an	DET
iajs-2148	90	8	-module	-module	NOUN
iajs-2148	90	9	,	,	PUNCT
iajs-2148	90	10	and	and	CCONJ
iajs-2148	90	11	be	be	AUX
iajs-2148	90	12	a	a	DET
iajs-2148	90	13	submodule	submodule	NOUN
iajs-2148	90	14	of	of	ADP
iajs-2148	90	15	such	such	ADJ
iajs-2148	90	16	that	that	PRON
iajs-2148	90	17	[	[	PUNCT
iajs-2148	90	18	]	]	X
iajs-2148	90	19	is	be	AUX
iajs-2148	90	20	a	a	DET
iajs-2148	90	21	maximal	maximal	ADJ
iajs-2148	90	22	ideal	ideal	NOUN
iajs-2148	90	23	of	of	ADP
iajs-2148	90	24	.	.	PUNCT
iajs-2148	91	1	then	then	ADV
iajs-2148	91	2	is	be	AUX
iajs-2148	91	3	an	an	DET
iajs-2148	91	4	app	app	ADJ
iajs-2148	91	5	-	-	PUNCT
iajs-2148	91	6	prime	prime	NOUN
iajs-2148	91	7	submodule	submodule	NOUN
iajs-2148	91	8	of	of	ADP
iajs-2148	91	9	.	.	PUNCT
iajs-2148	92	1	proof	proof	NOUN
iajs-2148	92	2	let	let	VERB
iajs-2148	92	3	,	,	PUNCT
iajs-2148	92	4	where	where	SCONJ
iajs-2148	92	5	,	,	PUNCT
iajs-2148	92	6	,	,	PUNCT
iajs-2148	92	7	with	with	ADP
iajs-2148	92	8	[	[	PUNCT
iajs-2148	92	9	]	]	X
iajs-2148	92	10	.	.	PUNCT
iajs-2148	93	1	since	since	SCONJ
iajs-2148	93	2	[	[	PUNCT
iajs-2148	93	3	]	]	X
iajs-2148	93	4	is	be	AUX
iajs-2148	93	5	a	a	DET
iajs-2148	93	6	maximal	maximal	ADJ
iajs-2148	93	7	ideal	ideal	NOUN
iajs-2148	93	8	of	of	ADP
iajs-2148	93	9	,	,	PUNCT
iajs-2148	93	10	then	then	ADV
iajs-2148	93	11	〈	〈	PROPN
iajs-2148	93	12	〉	〉	NOUN
iajs-2148	93	13	[	[	PUNCT
iajs-2148	93	14	]	]	X
iajs-2148	93	15	,	,	PUNCT
iajs-2148	93	16	where	where	SCONJ
iajs-2148	93	17	〈	〈	PROPN
iajs-2148	93	18	〉	〉	NOUN
iajs-2148	93	19	is	be	AUX
iajs-2148	93	20	an	an	DET
iajs-2148	93	21	ideal	ideal	NOUN
iajs-2148	93	22	of	of	ADP
iajs-2148	93	23	generated	generate	VERB
iajs-2148	93	24	by	by	ADP
iajs-2148	93	25	.	.	PUNCT
iajs-2148	94	1	thus	thus	ADV
iajs-2148	94	2	,	,	PUNCT
iajs-2148	94	3	there	there	PRON
iajs-2148	94	4	exists	exist	VERB
iajs-2148	94	5	and	and	CCONJ
iajs-2148	94	6	[	[	PUNCT
iajs-2148	94	7	]	]	X
iajs-2148	94	8	such	such	ADJ
iajs-2148	94	9	that	that	PRON
iajs-2148	94	10	,	,	PUNCT
iajs-2148	94	11	it	it	PRON
iajs-2148	94	12	follows	follow	VERB
iajs-2148	94	13	that	that	PRON
iajs-2148	94	14	.	.	PUNCT
iajs-2148	95	1	hence	hence	ADV
iajs-2148	95	2	is	be	AUX
iajs-2148	95	3	an	an	DET
iajs-2148	95	4	app	app	ADJ
iajs-2148	95	5	-	-	PUNCT
iajs-2148	95	6	prime	prime	NOUN
iajs-2148	95	7	submodule	submodule	NOUN
iajs-2148	95	8	of	of	ADP
iajs-2148	95	9	.	.	PUNCT
iajs-2148	96	1	proposition	proposition	NOUN
iajs-2148	96	2	(	(	PUNCT
iajs-2148	96	3	14	14	NUM
iajs-2148	96	4	)	)	PUNCT
iajs-2148	96	5	let	let	AUX
iajs-2148	96	6	be	be	AUX
iajs-2148	96	7	an	an	DET
iajs-2148	96	8	-module	-module	NOUN
iajs-2148	96	9	,	,	PUNCT
iajs-2148	96	10	and	and	CCONJ
iajs-2148	96	11	be	be	AUX
iajs-2148	96	12	a	a	DET
iajs-2148	96	13	proper	proper	ADJ
iajs-2148	96	14	submodule	submodule	NOUN
iajs-2148	96	15	of	of	ADP
iajs-2148	96	16	,	,	PUNCT
iajs-2148	96	17	with	with	ADP
iajs-2148	96	18	[	[	PUNCT
iajs-2148	96	19	]	]	X
iajs-2148	96	20	[	[	PUNCT
iajs-2148	96	21	]	]	X
iajs-2148	96	22	,	,	PUNCT
iajs-2148	96	23	and	and	CCONJ
iajs-2148	96	24	proper	proper	ADJ
iajs-2148	96	25	submodule	submodule	NOUN
iajs-2148	96	26	of	of	ADP
iajs-2148	96	27	for	for	ADP
iajs-2148	96	28	each	each	DET
iajs-2148	96	29	submodule	submodule	NOUN
iajs-2148	96	30	of	of	ADP
iajs-2148	96	31	such	such	ADJ
iajs-2148	96	32	that	that	PRON
iajs-2148	96	33	[	[	PUNCT
iajs-2148	96	34	]	]	X
iajs-2148	96	35	is	be	AUX
iajs-2148	96	36	a	a	DET
iajs-2148	96	37	prime	prime	ADJ
iajs-2148	96	38	ideal	ideal	NOUN
iajs-2148	96	39	of	of	ADP
iajs-2148	96	40	.	.	PUNCT
iajs-2148	97	1	then	then	ADV
iajs-2148	97	2	is	be	AUX
iajs-2148	97	3	an	an	DET
iajs-2148	97	4	app	app	ADJ
iajs-2148	97	5	-	-	PUNCT
iajs-2148	97	6	prime	prime	NOUN
iajs-2148	97	7	submodule	submodule	NOUN
iajs-2148	97	8	of	of	ADP
iajs-2148	97	9	.	.	PUNCT
iajs-2148	98	1	proof	proof	NOUN
iajs-2148	98	2	assume	assume	VERB
iajs-2148	98	3	that	that	SCONJ
iajs-2148	98	4	,	,	PUNCT
iajs-2148	98	5	where	where	SCONJ
iajs-2148	98	6	,	,	PUNCT
iajs-2148	98	7	,	,	PUNCT
iajs-2148	98	8	with	with	ADP
iajs-2148	98	9	.	.	PUNCT
iajs-2148	99	1	then	then	ADV
iajs-2148	99	2	〈	〈	PROPN
iajs-2148	99	3	〉	〉	NOUN
iajs-2148	99	4	and	and	CCONJ
iajs-2148	99	5	so	so	ADV
iajs-2148	99	6	[	[	X
iajs-2148	99	7	]	]	X
iajs-2148	99	8	[	[	PUNCT
iajs-2148	99	9	]	]	X
iajs-2148	99	10	,	,	PUNCT
iajs-2148	99	11	then	then	ADV
iajs-2148	99	12	there	there	PRON
iajs-2148	99	13	exists	exist	VERB
iajs-2148	99	14	[	[	PUNCT
iajs-2148	99	15	]	]	PUNCT
iajs-2148	99	16	and	and	CCONJ
iajs-2148	99	17	[	[	PUNCT
iajs-2148	99	18	]	]	X
iajs-2148	99	19	.	.	PUNCT
iajs-2148	100	1	that	that	PRON
iajs-2148	100	2	is	be	AUX
iajs-2148	100	3	and	and	CCONJ
iajs-2148	100	4	.	.	PUNCT
iajs-2148	101	1	that	that	PRON
iajs-2148	101	2	is	be	AUX
iajs-2148	101	3	implies	imply	VERB
iajs-2148	101	4	that	that	SCONJ
iajs-2148	101	5	〈	〈	PROPN
iajs-2148	101	6	〉	〉	NOUN
iajs-2148	101	7	.	.	PUNCT
iajs-2148	102	1	it	it	PRON
iajs-2148	102	2	follows	follow	VERB
iajs-2148	102	3	that	that	SCONJ
iajs-2148	102	4	[	[	X
iajs-2148	102	5	]	]	X
iajs-2148	102	6	.	.	PUNCT
iajs-2148	103	1	but	but	CCONJ
iajs-2148	103	2	[	[	PUNCT
iajs-2148	103	3	]	]	X
iajs-2148	103	4	is	be	AUX
iajs-2148	103	5	a	a	DET
iajs-2148	103	6	prime	prime	ADJ
iajs-2148	103	7	ideal	ideal	NOUN
iajs-2148	103	8	of	of	ADP
iajs-2148	103	9	,	,	PUNCT
iajs-2148	103	10	then	then	ADV
iajs-2148	103	11	[	[	X
iajs-2148	103	12	]	]	X
iajs-2148	103	13	.	.	PUNCT
iajs-2148	104	1	hence	hence	ADV
iajs-2148	104	2	is	be	AUX
iajs-2148	104	3	an	an	DET
iajs-2148	104	4	app	app	ADJ
iajs-2148	104	5	-	-	PUNCT
iajs-2148	104	6	prime	prime	NOUN
iajs-2148	104	7	submodule	submodule	NOUN
iajs-2148	104	8	of	of	ADP
iajs-2148	104	9	.	.	PUNCT
iajs-2148	105	1	note	note	VERB
iajs-2148	105	2	it	it	PRON
iajs-2148	105	3	is	be	AUX
iajs-2148	105	4	well	well	ADV
iajs-2148	105	5	-	-	PUNCT
iajs-2148	105	6	know	know	VERB
iajs-2148	105	7	that	that	SCONJ
iajs-2148	105	8	if	if	SCONJ
iajs-2148	105	9	is	be	AUX
iajs-2148	105	10	a	a	DET
iajs-2148	105	11	non	non	ADJ
iajs-2148	105	12	-	-	ADJ
iajs-2148	105	13	zero	zero	NUM
iajs-2148	105	14	multiplication	multiplication	NOUN
iajs-2148	105	15	module	module	NOUN
iajs-2148	105	16	,	,	PUNCT
iajs-2148	105	17	then	then	ADV
iajs-2148	105	18	[	[	PUNCT
iajs-2148	105	19	]	]	X
iajs-2148	105	20	[	[	PUNCT
iajs-2148	105	21	]	]	X
iajs-2148	105	22	for	for	ADP
iajs-2148	105	23	each	each	DET
iajs-2148	105	24	submodule	submodule	NOUN
iajs-2148	105	25	of	of	ADP
iajs-2148	105	26	with	with	ADP
iajs-2148	105	27	,	,	PUNCT
iajs-2148	105	28	where	where	SCONJ
iajs-2148	105	29	is	be	AUX
iajs-2148	105	30	a	a	DET
iajs-2148	105	31	proper	proper	ADJ
iajs-2148	105	32	submodule	submodule	NOUN
iajs-2148	105	33	of	of	ADP
iajs-2148	105	34	[	[	X
iajs-2148	105	35	14	14	NUM
iajs-2148	105	36	,	,	PUNCT
iajs-2148	105	37	rem.2.15	rem.2.15	NOUN
iajs-2148	105	38	]	]	PUNCT
iajs-2148	105	39	.	.	PUNCT
iajs-2148	106	1	now	now	ADV
iajs-2148	106	2	,	,	PUNCT
iajs-2148	106	3	we	we	PRON
iajs-2148	106	4	get	get	VERB
iajs-2148	106	5	the	the	DET
iajs-2148	106	6	following	following	NOUN
iajs-2148	106	7	corollary	corollary	NOUN
iajs-2148	106	8	as	as	ADP
iajs-2148	106	9	a	a	DET
iajs-2148	106	10	direct	direct	ADJ
iajs-2148	106	11	consequence	consequence	NOUN
iajs-2148	106	12	of	of	ADP
iajs-2148	106	13	proposition	proposition	NOUN
iajs-2148	106	14	(	(	PUNCT
iajs-2148	106	15	14	14	NUM
iajs-2148	106	16	)	)	PUNCT
iajs-2148	106	17	.	.	PUNCT
iajs-2148	107	1	corollary	corollary	ADJ
iajs-2148	107	2	(	(	PUNCT
iajs-2148	107	3	15	15	NUM
iajs-2148	107	4	)	)	PUNCT
iajs-2148	107	5	let	let	AUX
iajs-2148	107	6	be	be	AUX
iajs-2148	107	7	a	a	DET
iajs-2148	107	8	multiplication	multiplication	NOUN
iajs-2148	107	9	-module	-module	NOUN
iajs-2148	107	10	,	,	PUNCT
iajs-2148	107	11	and	and	CCONJ
iajs-2148	107	12	be	be	AUX
iajs-2148	107	13	a	a	DET
iajs-2148	107	14	proper	proper	ADJ
iajs-2148	107	15	submodule	submodule	NOUN
iajs-2148	107	16	of	of	ADP
iajs-2148	107	17	,	,	PUNCT
iajs-2148	107	18	with	with	SCONJ
iajs-2148	107	19	[	[	PUNCT
iajs-2148	107	20	]	]	X
iajs-2148	107	21	is	be	AUX
iajs-2148	107	22	prime	prime	ADJ
iajs-2148	107	23	ideal	ideal	NOUN
iajs-2148	107	24	of	of	ADP
iajs-2148	107	25	,	,	PUNCT
iajs-2148	107	26	and	and	CCONJ
iajs-2148	107	27	for	for	ADP
iajs-2148	107	28	each	each	DET
iajs-2148	107	29	submodule	submodule	NOUN
iajs-2148	107	30	of	of	ADP
iajs-2148	107	31	.	.	PUNCT
iajs-2148	108	1	then	then	ADV
iajs-2148	108	2	is	be	AUX
iajs-2148	108	3	an	an	DET
iajs-2148	108	4	app	app	ADJ
iajs-2148	108	5	-	-	PUNCT
iajs-2148	108	6	prime	prime	NOUN
iajs-2148	108	7	submodule	submodule	NOUN
iajs-2148	108	8	of	of	ADP
iajs-2148	108	9	.	.	PUNCT
iajs-2148	109	1	proposition	proposition	NOUN
iajs-2148	109	2	(	(	PUNCT
iajs-2148	109	3	16	16	NUM
iajs-2148	109	4	)	)	PUNCT
iajs-2148	109	5	let	let	AUX
iajs-2148	109	6	be	be	AUX
iajs-2148	109	7	an	an	DET
iajs-2148	109	8	-module	-module	NOUN
iajs-2148	109	9	,	,	PUNCT
iajs-2148	109	10	and	and	CCONJ
iajs-2148	109	11	are	be	AUX
iajs-2148	109	12	submodules	submodule	NOUN
iajs-2148	109	13	of	of	ADP
iajs-2148	109	14	,	,	PUNCT
iajs-2148	109	15	with	with	ADP
iajs-2148	109	16	.	.	PUNCT
iajs-2148	110	1	if	if	SCONJ
iajs-2148	110	2	is	be	AUX
iajs-2148	110	3	an	an	DET
iajs-2148	110	4	app	app	ADJ
iajs-2148	110	5	-	-	PUNCT
iajs-2148	110	6	prime	prime	NOUN
iajs-2148	110	7	submodule	submodule	NOUN
iajs-2148	110	8	of	of	ADP
iajs-2148	110	9	and	and	CCONJ
iajs-2148	110	10	,	,	PUNCT
iajs-2148	110	11	then	then	ADV
iajs-2148	110	12	is	be	AUX
iajs-2148	110	13	an	an	DET
iajs-2148	110	14	app	app	ADJ
iajs-2148	110	15	-	-	PUNCT
iajs-2148	110	16	prime	prime	NOUN
iajs-2148	110	17	submodule	submodule	NOUN
iajs-2148	110	18	of	of	ADP
iajs-2148	110	19	.	.	PUNCT
iajs-2148	111	1	proof	proof	NOUN
iajs-2148	111	2	suppose	suppose	VERB
iajs-2148	111	3	that	that	SCONJ
iajs-2148	111	4	,	,	PUNCT
iajs-2148	111	5	where	where	SCONJ
iajs-2148	111	6	,	,	PUNCT
iajs-2148	111	7	.	.	PUNCT
iajs-2148	112	1	since	since	SCONJ
iajs-2148	112	2	is	be	AUX
iajs-2148	112	3	an	an	DET
iajs-2148	112	4	app	app	ADJ
iajs-2148	112	5	-	-	PUNCT
iajs-2148	112	6	prime	prime	NOUN
iajs-2148	112	7	submodule	submodule	NOUN
iajs-2148	112	8	of	of	ADP
iajs-2148	112	9	,	,	PUNCT
iajs-2148	112	10	then	then	ADV
iajs-2148	112	11	either	either	ADV
iajs-2148	112	12	or	or	CCONJ
iajs-2148	112	13	.	.	PUNCT
iajs-2148	113	1	but	but	CCONJ
iajs-2148	113	2	,	,	PUNCT
iajs-2148	113	3	implies	imply	VERB
iajs-2148	113	4	that	that	SCONJ
iajs-2148	113	5	either	either	CCONJ
iajs-2148	113	6	or	or	CCONJ
iajs-2148	113	7	.	.	PUNCT
iajs-2148	114	1	thus	thus	ADV
iajs-2148	114	2	is	be	AUX
iajs-2148	114	3	an	an	DET
iajs-2148	114	4	app	app	ADJ
iajs-2148	114	5	-	-	PUNCT
iajs-2148	114	6	prime	prime	NOUN
iajs-2148	114	7	submodule	submodule	NOUN
iajs-2148	114	8	of	of	ADP
iajs-2148	114	9	.	.	PUNCT
iajs-2148	115	1	301	301	NUM
iajs-2148	115	2	ibn	ibn	PROPN
iajs-2148	115	3	al	al	PROPN
iajs-2148	115	4	-	-	PUNCT
iajs-2148	115	5	haitham	haitham	PROPN
iajs-2148	115	6	jour.for	jour.for	ADP
iajs-2148	115	7	pure	pure	ADJ
iajs-2148	115	8	&	&	CCONJ
iajs-2148	115	9	appl	appl	PROPN
iajs-2148	115	10	.sci	.sci	PROPN
iajs-2148	115	11	.	.	PUNCT
iajs-2148	116	1	32	32	NUM
iajs-2148	116	2	(	(	PUNCT
iajs-2148	116	3	2	2	NUM
iajs-2148	116	4	)	)	SYM
iajs-2148	116	5	2019	2019	NUM
iajs-2148	116	6	proposition	proposition	NOUN
iajs-2148	116	7	(	(	PUNCT
iajs-2148	116	8	17	17	NUM
iajs-2148	116	9	)	)	PUNCT
iajs-2148	116	10	let	let	AUX
iajs-2148	116	11	be	be	AUX
iajs-2148	116	12	an	an	DET
iajs-2148	116	13	-module	-module	NOUN
iajs-2148	116	14	,	,	PUNCT
iajs-2148	116	15	and	and	CCONJ
iajs-2148	116	16	be	be	AUX
iajs-2148	116	17	a	a	DET
iajs-2148	116	18	submodule	submodule	NOUN
iajs-2148	116	19	of	of	ADP
iajs-2148	116	20	,	,	PUNCT
iajs-2148	116	21	with	with	SCONJ
iajs-2148	116	22	is	be	AUX
iajs-2148	116	23	an	an	DET
iajs-2148	116	24	app	app	ADJ
iajs-2148	116	25	-	-	PUNCT
iajs-2148	116	26	prime	prime	NOUN
iajs-2148	116	27	submodule	submodule	NOUN
iajs-2148	116	28	of	of	ADP
iajs-2148	116	29	.	.	PUNCT
iajs-2148	117	1	then	then	ADV
iajs-2148	117	2	is	be	AUX
iajs-2148	117	3	an	an	DET
iajs-2148	117	4	app	app	ADJ
iajs-2148	117	5	-	-	PUNCT
iajs-2148	117	6	prime	prime	NOUN
iajs-2148	117	7	submodule	submodule	NOUN
iajs-2148	117	8	of	of	ADP
iajs-2148	117	9	.	.	PUNCT
iajs-2148	118	1	proof	proof	NOUN
iajs-2148	118	2	suppose	suppose	VERB
iajs-2148	118	3	that	that	SCONJ
iajs-2148	118	4	,	,	PUNCT
iajs-2148	118	5	where	where	SCONJ
iajs-2148	118	6	,	,	PUNCT
iajs-2148	118	7	.	.	PUNCT
iajs-2148	119	1	hence	hence	ADV
iajs-2148	119	2	,	,	PUNCT
iajs-2148	119	3	and	and	CCONJ
iajs-2148	119	4	so	so	ADV
iajs-2148	119	5	.	.	PUNCT
iajs-2148	120	1	but	but	CCONJ
iajs-2148	120	2	is	be	AUX
iajs-2148	120	3	an	an	DET
iajs-2148	120	4	app	app	ADJ
iajs-2148	120	5	-	-	PUNCT
iajs-2148	120	6	prime	prime	NOUN
iajs-2148	120	7	submodule	submodule	NOUN
iajs-2148	120	8	of	of	ADP
iajs-2148	120	9	,	,	PUNCT
iajs-2148	120	10	then	then	ADV
iajs-2148	120	11	either	either	ADV
iajs-2148	120	12	or	or	CCONJ
iajs-2148	120	13	.	.	PUNCT
iajs-2148	121	1	thus	thus	ADV
iajs-2148	121	2	is	be	AUX
iajs-2148	121	3	an	an	DET
iajs-2148	121	4	app	app	ADJ
iajs-2148	121	5	-	-	PUNCT
iajs-2148	121	6	prime	prime	NOUN
iajs-2148	121	7	submodule	submodule	NOUN
iajs-2148	121	8	of	of	ADP
iajs-2148	121	9	.	.	PUNCT
iajs-2148	122	1	proposition	proposition	NOUN
iajs-2148	122	2	(	(	PUNCT
iajs-2148	122	3	18	18	NUM
iajs-2148	122	4	)	)	PUNCT
iajs-2148	122	5	let	let	AUX
iajs-2148	122	6	be	be	AUX
iajs-2148	122	7	an	an	DET
iajs-2148	122	8	-module	-module	NOUN
iajs-2148	122	9	,	,	PUNCT
iajs-2148	122	10	and	and	CCONJ
iajs-2148	122	11	be	be	AUX
iajs-2148	122	12	a	a	DET
iajs-2148	122	13	submodule	submodule	NOUN
iajs-2148	122	14	of	of	ADP
iajs-2148	122	15	,	,	PUNCT
iajs-2148	122	16	with	with	SCONJ
iajs-2148	122	17	[	[	PUNCT
iajs-2148	122	18	]	]	X
iajs-2148	122	19	is	be	AUX
iajs-2148	122	20	a	a	DET
iajs-2148	122	21	prime	prime	ADJ
iajs-2148	122	22	ideal	ideal	NOUN
iajs-2148	122	23	of	of	ADP
iajs-2148	122	24	.	.	PUNCT
iajs-2148	123	1	then	then	ADV
iajs-2148	123	2	for	for	ADP
iajs-2148	123	3	each	each	DET
iajs-2148	123	4	multiplicatively	multiplicatively	ADV
iajs-2148	123	5	closed	close	VERB
iajs-2148	123	6	subset	subset	NOUN
iajs-2148	123	7	of	of	ADP
iajs-2148	123	8	with	with	ADP
iajs-2148	123	9	[	[	PUNCT
iajs-2148	123	10	]	]	X
iajs-2148	123	11	if	if	SCONJ
iajs-2148	123	12	and	and	CCONJ
iajs-2148	123	13	only	only	ADV
iajs-2148	123	14	if	if	SCONJ
iajs-2148	123	15	is	be	AUX
iajs-2148	123	16	an	an	DET
iajs-2148	123	17	app	app	ADJ
iajs-2148	123	18	-	-	PUNCT
iajs-2148	123	19	prime	prime	NOUN
iajs-2148	123	20	submodule	submodule	NOUN
iajs-2148	123	21	of	of	ADP
iajs-2148	123	22	.	.	PUNCT
iajs-2148	124	1	proof	proof	ADJ
iajs-2148	124	2	⇒	⇒	PROPN
iajs-2148	124	3	assume	assume	VERB
iajs-2148	124	4	that	that	SCONJ
iajs-2148	124	5	,	,	PUNCT
iajs-2148	124	6	where	where	SCONJ
iajs-2148	124	7	,	,	PUNCT
iajs-2148	124	8	,	,	PUNCT
iajs-2148	124	9	and	and	CCONJ
iajs-2148	124	10	suppose	suppose	VERB
iajs-2148	124	11	that	that	SCONJ
iajs-2148	124	12	and	and	CCONJ
iajs-2148	124	13	[	[	PUNCT
iajs-2148	124	14	]	]	X
iajs-2148	124	15	.	.	PUNCT
iajs-2148	125	1	since	since	SCONJ
iajs-2148	125	2	is	be	AUX
iajs-2148	125	3	a	a	DET
iajs-2148	125	4	multiplicatively	multiplicatively	ADV
iajs-2148	125	5	closed	close	VERB
iajs-2148	125	6	subset	subset	NOUN
iajs-2148	125	7	of	of	ADP
iajs-2148	125	8	,	,	PUNCT
iajs-2148	125	9	then	then	ADV
iajs-2148	125	10	,	,	PUNCT
iajs-2148	125	11	and	and	CCONJ
iajs-2148	125	12	since	since	SCONJ
iajs-2148	125	13	[	[	PUNCT
iajs-2148	125	14	]	]	X
iajs-2148	125	15	is	be	AUX
iajs-2148	125	16	a	a	DET
iajs-2148	125	17	prime	prime	ADJ
iajs-2148	125	18	ideal	ideal	NOUN
iajs-2148	125	19	of	of	ADP
iajs-2148	125	20	,	,	PUNCT
iajs-2148	125	21	then	then	ADV
iajs-2148	125	22	it	it	PRON
iajs-2148	125	23	is	be	AUX
iajs-2148	125	24	clear	clear	ADJ
iajs-2148	125	25	that	that	SCONJ
iajs-2148	125	26	[	[	X
iajs-2148	125	27	]	]	X
iajs-2148	125	28	.	.	PUNCT
iajs-2148	126	1	but	but	CCONJ
iajs-2148	126	2	,	,	PUNCT
iajs-2148	126	3	implies	imply	VERB
iajs-2148	126	4	that	that	PRON
iajs-2148	126	5	and	and	CCONJ
iajs-2148	126	6	hence	hence	ADV
iajs-2148	126	7	which	which	DET
iajs-2148	126	8	contradiction	contradiction	NOUN
iajs-2148	126	9	.	.	PUNCT
iajs-2148	127	1	thus	thus	ADV
iajs-2148	127	2	,	,	PUNCT
iajs-2148	127	3	either	either	CCONJ
iajs-2148	127	4	or	or	CCONJ
iajs-2148	127	5	[	[	PUNCT
iajs-2148	127	6	]	]	X
iajs-2148	127	7	,	,	PUNCT
iajs-2148	127	8	therefore	therefore	ADV
iajs-2148	127	9	is	be	AUX
iajs-2148	127	10	an	an	DET
iajs-2148	127	11	app	app	ADJ
iajs-2148	127	12	-	-	PUNCT
iajs-2148	127	13	prime	prime	NOUN
iajs-2148	127	14	submodule	submodule	NOUN
iajs-2148	127	15	of	of	ADP
iajs-2148	127	16	.	.	PUNCT
iajs-2148	128	1	⇐	⇐	PROPN
iajs-2148	128	2	let	let	VERB
iajs-2148	128	3	,	,	PUNCT
iajs-2148	128	4	then	then	ADV
iajs-2148	128	5	there	there	PRON
iajs-2148	128	6	exists	exist	VERB
iajs-2148	128	7	such	such	ADJ
iajs-2148	128	8	that	that	PRON
iajs-2148	128	9	.	.	PUNCT
iajs-2148	129	1	but	but	CCONJ
iajs-2148	129	2	is	be	AUX
iajs-2148	129	3	an	an	DET
iajs-2148	129	4	app	app	ADJ
iajs-2148	129	5	-	-	PUNCT
iajs-2148	129	6	prime	prime	NOUN
iajs-2148	129	7	submodule	submodule	NOUN
iajs-2148	129	8	of	of	ADP
iajs-2148	129	9	,	,	PUNCT
iajs-2148	129	10	so	so	ADV
iajs-2148	129	11	either	either	PRON
iajs-2148	129	12	or	or	CCONJ
iajs-2148	129	13	[	[	PUNCT
iajs-2148	129	14	]	]	X
iajs-2148	129	15	.	.	PUNCT
iajs-2148	130	1	but	but	CCONJ
iajs-2148	130	2	[	[	PUNCT
iajs-2148	130	3	]	]	X
iajs-2148	130	4	,	,	PUNCT
iajs-2148	130	5	implies	imply	VERB
iajs-2148	130	6	that	that	SCONJ
iajs-2148	130	7	[	[	X
iajs-2148	130	8	]	]	X
iajs-2148	130	9	,	,	PUNCT
iajs-2148	130	10	which	which	PRON
iajs-2148	130	11	is	be	AUX
iajs-2148	130	12	a	a	DET
iajs-2148	130	13	contradiction	contradiction	NOUN
iajs-2148	130	14	.	.	PUNCT
iajs-2148	131	1	thus	thus	ADV
iajs-2148	131	2	and	and	CCONJ
iajs-2148	131	3	hence	hence	ADV
iajs-2148	131	4	.	.	PUNCT
iajs-2148	132	1	proposition	proposition	NOUN
iajs-2148	132	2	(	(	PUNCT
iajs-2148	132	3	19	19	NUM
iajs-2148	132	4	)	)	PUNCT
iajs-2148	132	5	let	let	AUX
iajs-2148	132	6	be	be	AUX
iajs-2148	132	7	an	an	DET
iajs-2148	132	8	-module	-module	NOUN
iajs-2148	132	9	,	,	PUNCT
iajs-2148	132	10	and	and	CCONJ
iajs-2148	132	11	be	be	AUX
iajs-2148	132	12	a	a	DET
iajs-2148	132	13	maximal	maximal	ADJ
iajs-2148	132	14	ideal	ideal	NOUN
iajs-2148	132	15	of	of	ADP
iajs-2148	132	16	,	,	PUNCT
iajs-2148	132	17	with	with	ADP
iajs-2148	132	18	.	.	PUNCT
iajs-2148	133	1	then	then	ADV
iajs-2148	133	2	is	be	AUX
iajs-2148	133	3	an	an	DET
iajs-2148	133	4	app	app	ADJ
iajs-2148	133	5	-	-	PUNCT
iajs-2148	133	6	prime	prime	NOUN
iajs-2148	133	7	submodule	submodule	NOUN
iajs-2148	133	8	of	of	ADP
iajs-2148	133	9	.	.	PUNCT
iajs-2148	134	1	proof	proof	NOUN
iajs-2148	134	2	clearly	clearly	ADV
iajs-2148	134	3	,	,	PUNCT
iajs-2148	134	4	[	[	PUNCT
iajs-2148	134	5	]	]	X
iajs-2148	134	6	.that	.that	PRON
iajs-2148	134	7	is	be	AUX
iajs-2148	134	8	there	there	PRON
iajs-2148	134	9	exists	exist	VERB
iajs-2148	134	10	[	[	PUNCT
iajs-2148	134	11	]	]	PUNCT
iajs-2148	134	12	and	and	CCONJ
iajs-2148	134	13	,	,	PUNCT
iajs-2148	134	14	then	then	ADV
iajs-2148	134	15	〈	〈	PROPN
iajs-2148	134	16	〉	〉	NOUN
iajs-2148	134	17	,	,	PUNCT
iajs-2148	134	18	where	where	SCONJ
iajs-2148	134	19	〈	〈	PROPN
iajs-2148	134	20	〉	〉	NOUN
iajs-2148	134	21	is	be	AUX
iajs-2148	134	22	an	an	DET
iajs-2148	134	23	ideal	ideal	NOUN
iajs-2148	134	24	of	of	ADP
iajs-2148	134	25	generated	generate	VERB
iajs-2148	134	26	by	by	ADP
iajs-2148	134	27	,	,	PUNCT
iajs-2148	134	28	thus	thus	ADV
iajs-2148	134	29	there	there	PRON
iajs-2148	134	30	exist	exist	VERB
iajs-2148	134	31	and	and	CCONJ
iajs-2148	134	32	such	such	ADJ
iajs-2148	134	33	that	that	PRON
iajs-2148	134	34	.	.	PUNCT
iajs-2148	135	1	hence	hence	ADV
iajs-2148	135	2	for	for	ADP
iajs-2148	135	3	each	each	PRON
iajs-2148	135	4	.	.	PUNCT
iajs-2148	136	1	it	it	PRON
iajs-2148	136	2	follows	follow	VERB
iajs-2148	136	3	that	that	SCONJ
iajs-2148	136	4	for	for	ADP
iajs-2148	136	5	each	each	PRON
iajs-2148	136	6	,	,	PUNCT
iajs-2148	136	7	hence	hence	ADV
iajs-2148	136	8	,	,	PUNCT
iajs-2148	136	9	it	it	PRON
iajs-2148	136	10	follows	follow	VERB
iajs-2148	136	11	that	that	SCONJ
iajs-2148	136	12	which	which	PRON
iajs-2148	136	13	is	be	AUX
iajs-2148	136	14	a	a	DET
iajs-2148	136	15	contradiction	contradiction	NOUN
iajs-2148	136	16	.	.	PUNCT
iajs-2148	137	1	then	then	ADV
iajs-2148	137	2	and	and	CCONJ
iajs-2148	137	3	hence	hence	ADV
iajs-2148	137	4	[	[	X
iajs-2148	137	5	]	]	X
iajs-2148	137	6	,	,	PUNCT
iajs-2148	137	7	it	it	PRON
iajs-2148	137	8	follows	follow	VERB
iajs-2148	137	9	that	that	SCONJ
iajs-2148	137	10	[	[	PUNCT
iajs-2148	137	11	]	]	X
iajs-2148	137	12	is	be	AUX
iajs-2148	137	13	a	a	DET
iajs-2148	137	14	maximal	maximal	ADJ
iajs-2148	137	15	ideal	ideal	NOUN
iajs-2148	137	16	of	of	ADP
iajs-2148	137	17	,	,	PUNCT
iajs-2148	137	18	hence	hence	ADV
iajs-2148	137	19	by	by	ADP
iajs-2148	137	20	proposition	proposition	NOUN
iajs-2148	137	21	(	(	PUNCT
iajs-2148	137	22	13	13	NUM
iajs-2148	137	23	)	)	PUNCT
iajs-2148	137	24	is	be	AUX
iajs-2148	137	25	an	an	DET
iajs-2148	137	26	app	app	ADJ
iajs-2148	137	27	-	-	PUNCT
iajs-2148	137	28	prime	prime	NOUN
iajs-2148	137	29	submodule	submodule	NOUN
iajs-2148	137	30	of	of	ADP
iajs-2148	137	31	.	.	PUNCT
iajs-2148	138	1	proposition	proposition	NOUN
iajs-2148	138	2	(	(	PUNCT
iajs-2148	138	3	20	20	NUM
iajs-2148	138	4	)	)	PUNCT
iajs-2148	138	5	let	let	AUX
iajs-2148	138	6	be	be	AUX
iajs-2148	138	7	a	a	DET
iajs-2148	138	8	faithful	faithful	ADJ
iajs-2148	138	9	multiplication	multiplication	NOUN
iajs-2148	138	10	-module	-module	NOUN
iajs-2148	138	11	,	,	PUNCT
iajs-2148	138	12	and	and	CCONJ
iajs-2148	138	13	is	be	AUX
iajs-2148	138	14	an	an	DET
iajs-2148	138	15	app	app	ADJ
iajs-2148	138	16	-	-	PUNCT
iajs-2148	138	17	prime	prime	ADJ
iajs-2148	138	18	ideal	ideal	NOUN
iajs-2148	138	19	of	of	ADP
iajs-2148	138	20	.	.	PUNCT
iajs-2148	139	1	then	then	ADV
iajs-2148	139	2	is	be	AUX
iajs-2148	139	3	an	an	DET
iajs-2148	139	4	app	app	ADJ
iajs-2148	139	5	-	-	PUNCT
iajs-2148	139	6	prime	prime	NOUN
iajs-2148	139	7	submodule	submodule	NOUN
iajs-2148	139	8	of	of	ADP
iajs-2148	139	9	.	.	PUNCT
iajs-2148	140	1	proof	proof	NOUN
iajs-2148	140	2	let	let	VERB
iajs-2148	140	3	,	,	PUNCT
iajs-2148	140	4	where	where	SCONJ
iajs-2148	140	5	,	,	PUNCT
iajs-2148	140	6	,	,	PUNCT
iajs-2148	140	7	then	then	ADV
iajs-2148	140	8	.	.	PUNCT
iajs-2148	141	1	but	but	CCONJ
iajs-2148	141	2	is	be	AUX
iajs-2148	141	3	multiplication	multiplication	NOUN
iajs-2148	141	4	,	,	PUNCT
iajs-2148	141	5	then	then	ADV
iajs-2148	141	6	for	for	ADP
iajs-2148	141	7	some	some	DET
iajs-2148	141	8	ideal	ideal	NOUN
iajs-2148	141	9	of	of	ADP
iajs-2148	141	10	.	.	PUNCT
iajs-2148	142	1	it	it	PRON
iajs-2148	142	2	follows	follow	VERB
iajs-2148	142	3	that	that	PRON
iajs-2148	142	4	,	,	PUNCT
iajs-2148	142	5	and	and	CCONJ
iajs-2148	142	6	so	so	ADV
iajs-2148	142	7	.	.	PUNCT
iajs-2148	143	1	but	but	CCONJ
iajs-2148	143	2	is	be	AUX
iajs-2148	143	3	an	an	DET
iajs-2148	143	4	appprime	appprime	NOUN
iajs-2148	143	5	ideal	ideal	NOUN
iajs-2148	143	6	of	of	ADP
iajs-2148	143	7	,	,	PUNCT
iajs-2148	143	8	then	then	ADV
iajs-2148	143	9	by	by	ADP
iajs-2148	143	10	corollary	corollary	ADJ
iajs-2148	143	11	(	(	PUNCT
iajs-2148	143	12	3	3	NUM
iajs-2148	143	13	)	)	PUNCT
iajs-2148	143	14	either	either	CCONJ
iajs-2148	143	15	or	or	CCONJ
iajs-2148	143	16	[	[	PUNCT
iajs-2148	143	17	]	]	X
iajs-2148	143	18	.	.	PUNCT
iajs-2148	144	1	thus	thus	ADV
iajs-2148	144	2	either	either	CCONJ
iajs-2148	144	3	or	or	CCONJ
iajs-2148	144	4	.	.	PUNCT
iajs-2148	145	1	but	but	CCONJ
iajs-2148	145	2	be	be	AUX
iajs-2148	145	3	a	a	DET
iajs-2148	145	4	faithful	faithful	ADJ
iajs-2148	145	5	multiplication	multiplication	NOUN
iajs-2148	145	6	module	module	NOUN
iajs-2148	145	7	then	then	ADV
iajs-2148	145	8	by	by	ADP
iajs-2148	145	9	[	[	X
iajs-2148	145	10	10	10	NUM
iajs-2148	145	11	,	,	PUNCT
iajs-2148	145	12	cor	cor	NOUN
iajs-2148	145	13	.	.	PROPN
iajs-2148	145	14	2.14	2.14	NUM
iajs-2148	145	15	]	]	PUNCT
iajs-2148	145	16	.	.	PUNCT
iajs-2148	146	1	we	we	PRON
iajs-2148	146	2	have	have	VERB
iajs-2148	146	3	.	.	PUNCT
iajs-2148	147	1	thus	thus	ADV
iajs-2148	147	2	either	either	CCONJ
iajs-2148	147	3	301	301	NUM
iajs-2148	147	4	ibn	ibn	PROPN
iajs-2148	147	5	al	al	PROPN
iajs-2148	147	6	-	-	PUNCT
iajs-2148	147	7	haitham	haitham	PROPN
iajs-2148	147	8	jour.for	jour.for	ADP
iajs-2148	147	9	pure	pure	ADJ
iajs-2148	147	10	&	&	CCONJ
iajs-2148	147	11	appl	appl	PROPN
iajs-2148	147	12	.sci	.sci	PROPN
iajs-2148	147	13	.	.	PUNCT
iajs-2148	148	1	32	32	NUM
iajs-2148	148	2	(	(	PUNCT
iajs-2148	148	3	2	2	NUM
iajs-2148	148	4	)	)	PUNCT
iajs-2148	148	5	2019	2019	NUM
iajs-2148	148	6	or	or	CCONJ
iajs-2148	148	7	.	.	PUNCT
iajs-2148	149	1	that	that	PRON
iajs-2148	149	2	is	be	AUX
iajs-2148	149	3	either	either	CCONJ
iajs-2148	149	4	[	[	PUNCT
iajs-2148	149	5	]	]	X
iajs-2148	149	6	or	or	CCONJ
iajs-2148	149	7	.	.	PUNCT
iajs-2148	150	1	hence	hence	ADV
iajs-2148	150	2	is	be	AUX
iajs-2148	150	3	an	an	DET
iajs-2148	150	4	app	app	ADJ
iajs-2148	150	5	-	-	PUNCT
iajs-2148	150	6	prime	prime	NOUN
iajs-2148	150	7	submodule	submodule	NOUN
iajs-2148	150	8	of	of	ADP
iajs-2148	150	9	.	.	PUNCT
iajs-2148	151	1	proposition	proposition	NOUN
iajs-2148	151	2	(	(	PUNCT
iajs-2148	151	3	21	21	NUM
iajs-2148	151	4	)	)	PUNCT
iajs-2148	151	5	let	let	AUX
iajs-2148	151	6	be	be	AUX
iajs-2148	151	7	a	a	DET
iajs-2148	151	8	finitely	finitely	ADV
iajs-2148	151	9	generated	generate	VERB
iajs-2148	151	10	multiplication	multiplication	NOUN
iajs-2148	151	11	non	non	ADJ
iajs-2148	151	12	-	-	ADJ
iajs-2148	151	13	singular	singular	ADJ
iajs-2148	151	14	-module	-module	NOUN
iajs-2148	151	15	,	,	PUNCT
iajs-2148	151	16	and	and	CCONJ
iajs-2148	151	17	is	be	AUX
iajs-2148	151	18	an	an	DET
iajs-2148	151	19	app	app	ADJ
iajs-2148	151	20	-	-	PUNCT
iajs-2148	151	21	prime	prime	ADJ
iajs-2148	151	22	ideal	ideal	NOUN
iajs-2148	151	23	of	of	ADP
iajs-2148	151	24	,	,	PUNCT
iajs-2148	151	25	with	with	ADP
iajs-2148	151	26	.	.	PUNCT
iajs-2148	152	1	then	then	ADV
iajs-2148	152	2	is	be	AUX
iajs-2148	152	3	an	an	DET
iajs-2148	152	4	app	app	ADJ
iajs-2148	152	5	-	-	PUNCT
iajs-2148	152	6	prime	prime	NOUN
iajs-2148	152	7	submodule	submodule	NOUN
iajs-2148	152	8	of	of	ADP
iajs-2148	152	9	.	.	PUNCT
iajs-2148	153	1	proof	proof	NOUN
iajs-2148	153	2	let	let	VERB
iajs-2148	153	3	,	,	PUNCT
iajs-2148	153	4	where	where	SCONJ
iajs-2148	153	5	,	,	PUNCT
iajs-2148	153	6	,	,	PUNCT
iajs-2148	153	7	then	then	ADV
iajs-2148	153	8	.	.	PUNCT
iajs-2148	154	1	but	but	CCONJ
iajs-2148	154	2	is	be	AUX
iajs-2148	154	3	multiplication	multiplication	NOUN
iajs-2148	154	4	,	,	PUNCT
iajs-2148	154	5	then	then	ADV
iajs-2148	154	6	for	for	ADP
iajs-2148	154	7	some	some	DET
iajs-2148	154	8	ideal	ideal	NOUN
iajs-2148	154	9	of	of	ADP
iajs-2148	154	10	.	.	PUNCT
iajs-2148	155	1	it	it	PRON
iajs-2148	155	2	follows	follow	VERB
iajs-2148	155	3	that	that	PRON
iajs-2148	155	4	,	,	PUNCT
iajs-2148	155	5	and	and	CCONJ
iajs-2148	155	6	so	so	ADV
iajs-2148	155	7	.	.	PUNCT
iajs-2148	156	1	but	but	CCONJ
iajs-2148	156	2	is	be	AUX
iajs-2148	156	3	an	an	DET
iajs-2148	156	4	appprime	appprime	NOUN
iajs-2148	156	5	ideal	ideal	NOUN
iajs-2148	156	6	of	of	ADP
iajs-2148	156	7	,	,	PUNCT
iajs-2148	156	8	then	then	ADV
iajs-2148	156	9	by	by	ADP
iajs-2148	156	10	corollary	corollary	ADJ
iajs-2148	156	11	(	(	PUNCT
iajs-2148	156	12	3	3	NUM
iajs-2148	156	13	)	)	PUNCT
iajs-2148	156	14	either	either	CCONJ
iajs-2148	156	15	or	or	CCONJ
iajs-2148	156	16	[	[	PUNCT
iajs-2148	156	17	]	]	X
iajs-2148	156	18	.	.	PUNCT
iajs-2148	157	1	thus	thus	ADV
iajs-2148	157	2	either	either	CCONJ
iajs-2148	157	3	or	or	CCONJ
iajs-2148	157	4	.	.	PUNCT
iajs-2148	158	1	but	but	CCONJ
iajs-2148	158	2	is	be	AUX
iajs-2148	158	3	non	non	ADJ
iajs-2148	158	4	-	-	ADJ
iajs-2148	158	5	singular	singular	ADJ
iajs-2148	158	6	,	,	PUNCT
iajs-2148	158	7	then	then	ADV
iajs-2148	158	8	by	by	ADP
iajs-2148	158	9	[	[	PUNCT
iajs-2148	158	10	9	9	NUM
iajs-2148	158	11	]	]	PUNCT
iajs-2148	158	12	.	.	PUNCT
iajs-2148	159	1	we	we	PRON
iajs-2148	159	2	have	have	VERB
iajs-2148	159	3	.	.	PUNCT
iajs-2148	160	1	thus	thus	ADV
iajs-2148	160	2	either	either	ADV
iajs-2148	160	3	or	or	CCONJ
iajs-2148	160	4	.	.	PUNCT
iajs-2148	161	1	that	that	PRON
iajs-2148	161	2	is	be	AUX
iajs-2148	161	3	either	either	CCONJ
iajs-2148	161	4	[	[	PUNCT
iajs-2148	161	5	]	]	X
iajs-2148	161	6	or	or	CCONJ
iajs-2148	161	7	.	.	PUNCT
iajs-2148	162	1	hence	hence	ADV
iajs-2148	162	2	is	be	AUX
iajs-2148	162	3	an	an	DET
iajs-2148	162	4	app	app	ADJ
iajs-2148	162	5	-	-	PUNCT
iajs-2148	162	6	prime	prime	NOUN
iajs-2148	162	7	submodule	submodule	NOUN
iajs-2148	162	8	of	of	ADP
iajs-2148	162	9	.	.	PUNCT
iajs-2148	163	1	proposition	proposition	NOUN
iajs-2148	163	2	(	(	PUNCT
iajs-2148	163	3	22	22	NUM
iajs-2148	163	4	)	)	PUNCT
iajs-2148	163	5	let	let	AUX
iajs-2148	163	6	be	be	AUX
iajs-2148	163	7	an	an	DET
iajs-2148	163	8	-module	-module	NOUN
iajs-2148	163	9	,	,	PUNCT
iajs-2148	163	10	and	and	CCONJ
iajs-2148	163	11	be	be	AUX
iajs-2148	163	12	a	a	DET
iajs-2148	163	13	proper	proper	ADJ
iajs-2148	163	14	submodule	submodule	NOUN
iajs-2148	163	15	of	of	ADP
iajs-2148	163	16	with	with	ADP
iajs-2148	163	17	[	[	PUNCT
iajs-2148	163	18	]	]	X
iajs-2148	163	19	[	[	PUNCT
iajs-2148	163	20	]	]	X
iajs-2148	163	21	for	for	ADP
iajs-2148	163	22	each	each	DET
iajs-2148	163	23	submodule	submodule	NOUN
iajs-2148	163	24	of	of	ADP
iajs-2148	163	25	such	such	ADJ
iajs-2148	163	26	that	that	PRON
iajs-2148	163	27	and	and	CCONJ
iajs-2148	163	28	.	.	PUNCT
iajs-2148	164	1	then	then	ADV
iajs-2148	164	2	is	be	AUX
iajs-2148	164	3	an	an	DET
iajs-2148	164	4	app	app	ADJ
iajs-2148	164	5	-	-	PUNCT
iajs-2148	164	6	prime	prime	NOUN
iajs-2148	164	7	submodule	submodule	NOUN
iajs-2148	164	8	of	of	ADP
iajs-2148	164	9	if	if	SCONJ
iajs-2148	164	10	and	and	CCONJ
iajs-2148	164	11	only	only	ADV
iajs-2148	164	12	if	if	SCONJ
iajs-2148	164	13	is	be	AUX
iajs-2148	164	14	a	a	DET
iajs-2148	164	15	compressible	compressible	ADJ
iajs-2148	164	16	-module	-module	NOUN
iajs-2148	164	17	.	.	PUNCT
iajs-2148	165	1	proof	proof	ADJ
iajs-2148	165	2	⇒	⇒	PROPN
iajs-2148	165	3	assume	assume	VERB
iajs-2148	165	4	that	that	PRON
iajs-2148	165	5	is	be	AUX
iajs-2148	165	6	an	an	DET
iajs-2148	165	7	app	app	ADJ
iajs-2148	165	8	-	-	PUNCT
iajs-2148	165	9	prime	prime	NOUN
iajs-2148	165	10	submodule	submodule	NOUN
iajs-2148	165	11	of	of	ADP
iajs-2148	165	12	and	and	CCONJ
iajs-2148	165	13	be	be	AUX
iajs-2148	165	14	a	a	DET
iajs-2148	165	15	submodule	submodule	NOUN
iajs-2148	165	16	of	of	ADP
iajs-2148	165	17	with	with	ADP
iajs-2148	165	18	,	,	PUNCT
iajs-2148	165	19	therefore	therefore	ADV
iajs-2148	165	20	is	be	AUX
iajs-2148	165	21	a	a	DET
iajs-2148	165	22	non	non	ADJ
iajs-2148	165	23	-	-	ADJ
iajs-2148	165	24	zero	zero	NUM
iajs-2148	165	25	submodule	submodule	NOUN
iajs-2148	165	26	of	of	ADP
iajs-2148	165	27	,	,	PUNCT
iajs-2148	165	28	we	we	PRON
iajs-2148	165	29	are	be	AUX
iajs-2148	165	30	going	go	VERB
iajs-2148	165	31	to	to	PART
iajs-2148	165	32	emmbed	emmbed	VERB
iajs-2148	165	33	inside	inside	ADV
iajs-2148	165	34	.	.	PUNCT
iajs-2148	166	1	since	since	SCONJ
iajs-2148	166	2	[	[	PUNCT
iajs-2148	166	3	]	]	X
iajs-2148	166	4	[	[	PUNCT
iajs-2148	166	5	]	]	X
iajs-2148	166	6	,	,	PUNCT
iajs-2148	166	7	then	then	ADV
iajs-2148	166	8	there	there	PRON
iajs-2148	166	9	exists	exist	VERB
iajs-2148	166	10	[	[	PUNCT
iajs-2148	166	11	]	]	PUNCT
iajs-2148	166	12	and	and	CCONJ
iajs-2148	166	13	[	[	PUNCT
iajs-2148	166	14	]	]	X
iajs-2148	166	15	.	.	PUNCT
iajs-2148	167	1	that	that	PRON
iajs-2148	167	2	is	be	AUX
iajs-2148	167	3	.	.	PUNCT
iajs-2148	168	1	define	define	VERB
iajs-2148	168	2	→	→	SYM
iajs-2148	168	3	by	by	ADP
iajs-2148	168	4	for	for	ADP
iajs-2148	168	5	each	each	PRON
iajs-2148	168	6	.	.	PUNCT
iajs-2148	169	1	it	it	PRON
iajs-2148	169	2	is	be	AUX
iajs-2148	169	3	clear	clear	ADJ
iajs-2148	169	4	that	that	PRON
iajs-2148	169	5	is	be	AUX
iajs-2148	169	6	an	an	DET
iajs-2148	169	7	-homomorphism	-homomorphism	NOUN
iajs-2148	169	8	.	.	PUNCT
iajs-2148	170	1	to	to	PART
iajs-2148	170	2	prove	prove	VERB
iajs-2148	170	3	that	that	PRON
iajs-2148	170	4	is	be	AUX
iajs-2148	170	5	one	one	NUM
iajs-2148	170	6	to	to	ADP
iajs-2148	170	7	one	one	NUM
iajs-2148	170	8	.	.	PUNCT
iajs-2148	171	1	suppose	suppose	VERB
iajs-2148	171	2	that	that	SCONJ
iajs-2148	171	3	,	,	PUNCT
iajs-2148	171	4	then	then	ADV
iajs-2148	171	5	,	,	PUNCT
iajs-2148	171	6	so	so	CCONJ
iajs-2148	171	7	,	,	PUNCT
iajs-2148	171	8	that	that	PRON
iajs-2148	171	9	is	be	AUX
iajs-2148	171	10	.	.	PUNCT
iajs-2148	172	1	but	but	CCONJ
iajs-2148	172	2	is	be	AUX
iajs-2148	172	3	an	an	DET
iajs-2148	172	4	app	app	ADJ
iajs-2148	172	5	-	-	PUNCT
iajs-2148	172	6	prime	prime	NOUN
iajs-2148	172	7	submodule	submodule	NOUN
iajs-2148	172	8	of	of	ADP
iajs-2148	172	9	,	,	PUNCT
iajs-2148	172	10	so	so	ADV
iajs-2148	172	11	either	either	ADV
iajs-2148	172	12	or	or	CCONJ
iajs-2148	172	13	.	.	PUNCT
iajs-2148	173	1	since	since	ADV
iajs-2148	173	2	,	,	PUNCT
iajs-2148	173	3	it	it	PRON
iajs-2148	173	4	follows	follow	VERB
iajs-2148	173	5	that	that	SCONJ
iajs-2148	173	6	either	either	CCONJ
iajs-2148	173	7	or	or	CCONJ
iajs-2148	173	8	.	.	PUNCT
iajs-2148	174	1	but	but	CCONJ
iajs-2148	174	2	,	,	PUNCT
iajs-2148	174	3	hence	hence	ADV
iajs-2148	174	4	,	,	PUNCT
iajs-2148	174	5	so	so	ADV
iajs-2148	174	6	.	.	PUNCT
iajs-2148	175	1	thus	thus	ADV
iajs-2148	175	2	is	be	AUX
iajs-2148	175	3	monomorphism	monomorphism	NOUN
iajs-2148	175	4	and	and	CCONJ
iajs-2148	175	5	is	be	AUX
iajs-2148	175	6	a	a	DET
iajs-2148	175	7	compressible	compressible	NOUN
iajs-2148	175	8	.	.	PUNCT
iajs-2148	176	1	⇐	⇐	PROPN
iajs-2148	176	2	suppose	suppose	VERB
iajs-2148	176	3	that	that	PRON
iajs-2148	176	4	is	be	AUX
iajs-2148	176	5	a	a	DET
iajs-2148	176	6	compressible	compressible	ADJ
iajs-2148	176	7	-module	-module	NOUN
iajs-2148	176	8	,	,	PUNCT
iajs-2148	176	9	and	and	CCONJ
iajs-2148	176	10	,	,	PUNCT
iajs-2148	176	11	where	where	SCONJ
iajs-2148	176	12	,	,	PUNCT
iajs-2148	176	13	,	,	PUNCT
iajs-2148	176	14	with	with	ADP
iajs-2148	176	15	that	that	PRON
iajs-2148	176	16	is	be	AUX
iajs-2148	176	17	since	since	ADV
iajs-2148	176	18	.	.	PUNCT
iajs-2148	177	1	then	then	ADV
iajs-2148	177	2	〈	〈	PROPN
iajs-2148	177	3	〉	〉	NOUN
iajs-2148	177	4	is	be	AUX
iajs-2148	177	5	a	a	DET
iajs-2148	177	6	submodule	submodule	NOUN
iajs-2148	177	7	of	of	ADP
iajs-2148	177	8	,	,	PUNCT
iajs-2148	177	9	hence	hence	ADV
iajs-2148	177	10	there	there	PRON
iajs-2148	177	11	is	be	VERB
iajs-2148	177	12	a	a	DET
iajs-2148	177	13	monomorphism	monomorphism	NOUN
iajs-2148	177	14	→	→	SYM
iajs-2148	177	15	〈	〈	NOUN
iajs-2148	177	16	〉	〉	NOUN
iajs-2148	177	17	,	,	PUNCT
iajs-2148	177	18	that	that	ADV
iajs-2148	177	19	is	is	ADV
iajs-2148	177	20	.	.	PUNCT
iajs-2148	178	1	let	let	VERB
iajs-2148	178	2	,	,	PUNCT
iajs-2148	178	3	then	then	ADV
iajs-2148	178	4	〈	〈	PROPN
iajs-2148	178	5	〉	〉	NOUN
iajs-2148	178	6	,	,	PUNCT
iajs-2148	178	7	then	then	ADV
iajs-2148	178	8	such	such	ADJ
iajs-2148	178	9	that	that	SCONJ
iajs-2148	178	10	that	that	PRON
iajs-2148	178	11	is	is	ADV
iajs-2148	178	12	,	,	PUNCT
iajs-2148	178	13	implies	imply	VERB
iajs-2148	178	14	that	that	SCONJ
iajs-2148	178	15	that	that	PRON
iajs-2148	178	16	is	is	ADV
iajs-2148	178	17	,	,	PUNCT
iajs-2148	178	18	it	it	PRON
iajs-2148	178	19	follows	follow	VERB
iajs-2148	178	20	that	that	SCONJ
iajs-2148	178	21	for	for	ADP
iajs-2148	178	22	each	each	PRON
iajs-2148	178	23	,	,	PUNCT
iajs-2148	178	24	hence	hence	ADV
iajs-2148	178	25	,	,	PUNCT
iajs-2148	178	26	that	that	ADV
iajs-2148	178	27	is	is	ADV
iajs-2148	178	28	.	.	PUNCT
iajs-2148	179	1	thus	thus	ADV
iajs-2148	179	2	is	be	AUX
iajs-2148	179	3	an	an	DET
iajs-2148	179	4	app	app	ADJ
iajs-2148	179	5	-	-	PUNCT
iajs-2148	179	6	prime	prime	NOUN
iajs-2148	179	7	submodule	submodule	NOUN
iajs-2148	179	8	of	of	ADP
iajs-2148	179	9	.	.	PUNCT
iajs-2148	180	1	so	so	ADV
iajs-2148	180	2	,	,	PUNCT
iajs-2148	180	3	we	we	PRON
iajs-2148	180	4	get	get	VERB
iajs-2148	180	5	the	the	DET
iajs-2148	180	6	following	following	NOUN
iajs-2148	180	7	corollary	corollary	NOUN
iajs-2148	180	8	as	as	ADP
iajs-2148	180	9	a	a	DET
iajs-2148	180	10	direct	direct	ADJ
iajs-2148	180	11	consequence	consequence	NOUN
iajs-2148	180	12	of	of	ADP
iajs-2148	180	13	proposition	proposition	NOUN
iajs-2148	180	14	(	(	PUNCT
iajs-2148	180	15	22	22	NUM
iajs-2148	180	16	)	)	PUNCT
iajs-2148	180	17	.	.	PUNCT
iajs-2148	181	1	corollary	corollary	ADJ
iajs-2148	181	2	(	(	PUNCT
iajs-2148	181	3	23	23	NUM
iajs-2148	181	4	)	)	PUNCT
iajs-2148	181	5	let	let	AUX
iajs-2148	181	6	be	be	AUX
iajs-2148	181	7	a	a	DET
iajs-2148	181	8	multiplication	multiplication	NOUN
iajs-2148	181	9	-module	-module	NOUN
iajs-2148	181	10	,	,	PUNCT
iajs-2148	181	11	and	and	CCONJ
iajs-2148	181	12	be	be	AUX
iajs-2148	181	13	a	a	DET
iajs-2148	181	14	proper	proper	ADJ
iajs-2148	181	15	submodule	submodule	NOUN
iajs-2148	181	16	of	of	ADP
iajs-2148	181	17	with	with	ADP
iajs-2148	181	18	.	.	PUNCT
iajs-2148	182	1	then	then	ADV
iajs-2148	182	2	is	be	AUX
iajs-2148	182	3	an	an	DET
iajs-2148	182	4	app	app	ADJ
iajs-2148	182	5	-	-	PUNCT
iajs-2148	182	6	prime	prime	NOUN
iajs-2148	182	7	submodule	submodule	NOUN
iajs-2148	182	8	of	of	ADP
iajs-2148	182	9	if	if	SCONJ
iajs-2148	182	10	and	and	CCONJ
iajs-2148	182	11	only	only	ADV
iajs-2148	182	12	if	if	SCONJ
iajs-2148	182	13	is	be	AUX
iajs-2148	182	14	a	a	DET
iajs-2148	182	15	compressible	compressible	ADJ
iajs-2148	182	16	-module	-module	NOUN
iajs-2148	182	17	.	.	PUNCT
iajs-2148	183	1	301	301	NUM
iajs-2148	183	2	ibn	ibn	PROPN
iajs-2148	183	3	al	al	PROPN
iajs-2148	183	4	-	-	PUNCT
iajs-2148	183	5	haitham	haitham	PROPN
iajs-2148	183	6	jour.for	jour.for	ADP
iajs-2148	183	7	pure	pure	ADJ
iajs-2148	183	8	&	&	CCONJ
iajs-2148	183	9	appl	appl	PROPN
iajs-2148	183	10	.sci	.sci	PROPN
iajs-2148	183	11	.	.	PUNCT
iajs-2148	184	1	32	32	NUM
iajs-2148	184	2	(	(	PUNCT
iajs-2148	184	3	2	2	NUM
iajs-2148	184	4	)	)	SYM
iajs-2148	184	5	2019	2019	NUM
iajs-2148	184	6	proposition	proposition	NOUN
iajs-2148	184	7	(	(	PUNCT
iajs-2148	184	8	24	24	NUM
iajs-2148	184	9	)	)	PUNCT
iajs-2148	184	10	let	let	AUX
iajs-2148	184	11	be	be	AUX
iajs-2148	184	12	an	an	DET
iajs-2148	184	13	-module	-module	NOUN
iajs-2148	184	14	,	,	PUNCT
iajs-2148	184	15	and	and	CCONJ
iajs-2148	184	16	be	be	AUX
iajs-2148	184	17	a	a	DET
iajs-2148	184	18	proper	proper	ADJ
iajs-2148	184	19	submodule	submodule	NOUN
iajs-2148	184	20	of	of	ADP
iajs-2148	184	21	such	such	ADJ
iajs-2148	184	22	that	that	PRON
iajs-2148	184	23	[	[	X
iajs-2148	184	24	]	]	X
iajs-2148	184	25	[	[	PUNCT
iajs-2148	184	26	]	]	X
iajs-2148	184	27	for	for	ADP
iajs-2148	184	28	each	each	DET
iajs-2148	184	29	submodule	submodule	NOUN
iajs-2148	184	30	of	of	ADP
iajs-2148	184	31	with	with	ADP
iajs-2148	184	32	and	and	CCONJ
iajs-2148	184	33	.then	.then	X
iajs-2148	184	34	is	be	AUX
iajs-2148	184	35	an	an	DET
iajs-2148	184	36	app	app	ADJ
iajs-2148	184	37	-	-	PUNCT
iajs-2148	184	38	prime	prime	NOUN
iajs-2148	184	39	submodule	submodule	NOUN
iajs-2148	184	40	of	of	ADP
iajs-2148	184	41	.	.	PUNCT
iajs-2148	185	1	proof	proof	NOUN
iajs-2148	185	2	suppose	suppose	VERB
iajs-2148	185	3	that	that	SCONJ
iajs-2148	185	4	,	,	PUNCT
iajs-2148	185	5	where	where	SCONJ
iajs-2148	185	6	,	,	PUNCT
iajs-2148	185	7	,	,	PUNCT
iajs-2148	185	8	with	with	ADP
iajs-2148	185	9	.	.	PUNCT
iajs-2148	186	1	let	let	VERB
iajs-2148	186	2	〈	〈	PROPN
iajs-2148	186	3	〉	〉	NOUN
iajs-2148	186	4	,	,	PUNCT
iajs-2148	186	5	then	then	ADV
iajs-2148	186	6	〈	〈	PROPN
iajs-2148	186	7	〉	〉	NOUN
iajs-2148	186	8	,	,	PUNCT
iajs-2148	186	9	it	it	PRON
iajs-2148	186	10	follows	follow	VERB
iajs-2148	186	11	that	that	PRON
iajs-2148	186	12	.	.	PUNCT
iajs-2148	187	1	and	and	CCONJ
iajs-2148	187	2	so	so	ADV
iajs-2148	187	3	[	[	PUNCT
iajs-2148	187	4	〈	〈	NOUN
iajs-2148	187	5	〉	〉	NOUN
iajs-2148	187	6	]	]	PUNCT
iajs-2148	187	7	[	[	PUNCT
iajs-2148	187	8	〈	〈	NOUN
iajs-2148	187	9	〉	〉	NOUN
iajs-2148	187	10	]	]	PUNCT
iajs-2148	187	11	[	[	PUNCT
iajs-2148	187	12	]	]	X
iajs-2148	187	13	.	.	PUNCT
iajs-2148	188	1	hence	hence	ADV
iajs-2148	188	2	[	[	X
iajs-2148	188	3	]	]	X
iajs-2148	188	4	.	.	PUNCT
iajs-2148	189	1	it	it	PRON
iajs-2148	189	2	follows	follow	VERB
iajs-2148	189	3	that	that	PRON
iajs-2148	189	4	is	be	AUX
iajs-2148	189	5	an	an	DET
iajs-2148	189	6	app	app	ADJ
iajs-2148	189	7	-	-	PUNCT
iajs-2148	189	8	prime	prime	NOUN
iajs-2148	189	9	submodule	submodule	NOUN
iajs-2148	189	10	of	of	ADP
iajs-2148	189	11	.	.	PUNCT
iajs-2148	190	1	remark	remark	NOUN
iajs-2148	190	2	(	(	PUNCT
iajs-2148	190	3	25	25	NUM
iajs-2148	190	4	)	)	PUNCT
iajs-2148	190	5	the	the	DET
iajs-2148	190	6	intersection	intersection	NOUN
iajs-2148	190	7	of	of	ADP
iajs-2148	190	8	two	two	NUM
iajs-2148	190	9	app	app	ADJ
iajs-2148	190	10	-	-	PUNCT
iajs-2148	190	11	prime	prime	NOUN
iajs-2148	190	12	submodules	submodule	NOUN
iajs-2148	190	13	of	of	ADP
iajs-2148	190	14	an	an	DET
iajs-2148	190	15	-module	-module	NOUN
iajs-2148	190	16	need	need	AUX
iajs-2148	190	17	not	not	PART
iajs-2148	190	18	be	be	AUX
iajs-2148	190	19	an	an	DET
iajs-2148	190	20	appprime	appprime	NOUN
iajs-2148	190	21	submodule	submodule	NOUN
iajs-2148	190	22	of	of	ADP
iajs-2148	190	23	,	,	PUNCT
iajs-2148	190	24	as	as	SCONJ
iajs-2148	190	25	the	the	DET
iajs-2148	190	26	following	follow	VERB
iajs-2148	190	27	example	example	NOUN
iajs-2148	190	28	shows	show	VERB
iajs-2148	190	29	that	that	SCONJ
iajs-2148	190	30	:	:	PUNCT
iajs-2148	190	31	example	example	NOUN
iajs-2148	190	32	(	(	PUNCT
iajs-2148	190	33	26	26	NUM
iajs-2148	190	34	)	)	PUNCT
iajs-2148	190	35	let	let	VERB
iajs-2148	190	36	,	,	PUNCT
iajs-2148	190	37	,	,	PUNCT
iajs-2148	190	38	and	and	CCONJ
iajs-2148	190	39	.	.	PUNCT
iajs-2148	191	1	and	and	CCONJ
iajs-2148	191	2	are	be	AUX
iajs-2148	191	3	app	app	ADJ
iajs-2148	191	4	-	-	PUNCT
iajs-2148	191	5	prime	prime	NOUN
iajs-2148	191	6	submodules	submodule	NOUN
iajs-2148	191	7	of	of	ADP
iajs-2148	191	8	,	,	PUNCT
iajs-2148	191	9	but	but	CCONJ
iajs-2148	191	10	is	be	AUX
iajs-2148	191	11	not	not	PART
iajs-2148	191	12	app	app	ADJ
iajs-2148	191	13	-	-	PUNCT
iajs-2148	191	14	prime	prime	NOUN
iajs-2148	191	15	submodule	submodule	NOUN
iajs-2148	191	16	of	of	ADP
iajs-2148	191	17	since	since	SCONJ
iajs-2148	192	1	but	but	CCONJ
iajs-2148	192	2	and	and	CCONJ
iajs-2148	192	3	[	[	PUNCT
iajs-2148	192	4	]	]	X
iajs-2148	192	5	proposition	proposition	NOUN
iajs-2148	192	6	(	(	PUNCT
iajs-2148	192	7	27	27	NUM
iajs-2148	192	8	)	)	PUNCT
iajs-2148	192	9	let	let	AUX
iajs-2148	192	10	be	be	AUX
iajs-2148	192	11	an	an	DET
iajs-2148	192	12	-module	-module	NOUN
iajs-2148	192	13	,	,	PUNCT
iajs-2148	192	14	are	be	AUX
iajs-2148	192	15	two	two	NUM
iajs-2148	192	16	app	app	ADJ
iajs-2148	192	17	-	-	PUNCT
iajs-2148	192	18	prime	prime	NOUN
iajs-2148	192	19	submodules	submodule	NOUN
iajs-2148	192	20	of	of	ADP
iajs-2148	192	21	with	with	ADP
iajs-2148	192	22	and	and	CCONJ
iajs-2148	192	23	.	.	PUNCT
iajs-2148	193	1	then	then	ADV
iajs-2148	193	2	is	be	AUX
iajs-2148	193	3	an	an	DET
iajs-2148	193	4	app	app	ADJ
iajs-2148	193	5	-	-	PUNCT
iajs-2148	193	6	prime	prime	NOUN
iajs-2148	193	7	submodule	submodule	NOUN
iajs-2148	193	8	of	of	ADP
iajs-2148	193	9	.	.	PUNCT
iajs-2148	194	1	proof	proof	NOUN
iajs-2148	194	2	suppose	suppose	VERB
iajs-2148	194	3	that	that	SCONJ
iajs-2148	194	4	,	,	PUNCT
iajs-2148	194	5	where	where	SCONJ
iajs-2148	194	6	,	,	PUNCT
iajs-2148	194	7	,	,	PUNCT
iajs-2148	194	8	then	then	ADV
iajs-2148	194	9	and	and	CCONJ
iajs-2148	194	10	.	.	PUNCT
iajs-2148	195	1	since	since	SCONJ
iajs-2148	195	2	and	and	CCONJ
iajs-2148	195	3	are	be	AUX
iajs-2148	195	4	app	app	ADJ
iajs-2148	195	5	-	-	PUNCT
iajs-2148	195	6	prime	prime	NOUN
iajs-2148	195	7	submodules	submodule	NOUN
iajs-2148	195	8	of	of	ADP
iajs-2148	195	9	,	,	PUNCT
iajs-2148	195	10	so	so	ADV
iajs-2148	195	11	either	either	ADV
iajs-2148	195	12	or	or	CCONJ
iajs-2148	195	13	and	and	CCONJ
iajs-2148	195	14	either	either	ADV
iajs-2148	195	15	or	or	CCONJ
iajs-2148	195	16	.	.	PUNCT
iajs-2148	196	1	but	but	CCONJ
iajs-2148	196	2	and	and	CCONJ
iajs-2148	196	3	,	,	PUNCT
iajs-2148	196	4	then	then	ADV
iajs-2148	196	5	either	either	CCONJ
iajs-2148	196	6	or	or	CCONJ
iajs-2148	196	7	and	and	CCONJ
iajs-2148	196	8	either	either	ADV
iajs-2148	196	9	or	or	CCONJ
iajs-2148	196	10	,	,	PUNCT
iajs-2148	196	11	it	it	PRON
iajs-2148	196	12	follows	follow	VERB
iajs-2148	196	13	that	that	SCONJ
iajs-2148	196	14	either	either	CCONJ
iajs-2148	196	15	or	or	CCONJ
iajs-2148	196	16	.	.	PUNCT
iajs-2148	197	1	hence	hence	ADV
iajs-2148	197	2	is	be	AUX
iajs-2148	197	3	an	an	DET
iajs-2148	197	4	app	app	ADJ
iajs-2148	197	5	-	-	PUNCT
iajs-2148	197	6	prime	prime	NOUN
iajs-2148	197	7	submodule	submodule	NOUN
iajs-2148	197	8	of	of	ADP
iajs-2148	197	9	.	.	PUNCT
iajs-2148	198	1	proposition	proposition	NOUN
iajs-2148	198	2	(	(	PUNCT
iajs-2148	198	3	28	28	NUM
iajs-2148	198	4	)	)	PUNCT
iajs-2148	198	5	let	let	AUX
iajs-2148	198	6	be	be	AUX
iajs-2148	198	7	an	an	DET
iajs-2148	198	8	-module	-module	NOUN
iajs-2148	198	9	,	,	PUNCT
iajs-2148	198	10	are	be	AUX
iajs-2148	198	11	two	two	NUM
iajs-2148	198	12	submodules	submodule	NOUN
iajs-2148	198	13	of	of	ADP
iajs-2148	198	14	with	with	ADP
iajs-2148	198	15	is	be	AUX
iajs-2148	198	16	not	not	PART
iajs-2148	198	17	contained	contain	VERB
iajs-2148	198	18	in	in	ADP
iajs-2148	198	19	and	and	CCONJ
iajs-2148	198	20	.	.	PUNCT
iajs-2148	199	1	if	if	SCONJ
iajs-2148	199	2	is	be	AUX
iajs-2148	199	3	an	an	DET
iajs-2148	199	4	app	app	ADJ
iajs-2148	199	5	-	-	PUNCT
iajs-2148	199	6	prime	prime	NOUN
iajs-2148	199	7	submodule	submodule	NOUN
iajs-2148	199	8	of	of	ADP
iajs-2148	199	9	then	then	ADV
iajs-2148	199	10	is	be	AUX
iajs-2148	199	11	an	an	DET
iajs-2148	199	12	app	app	ADJ
iajs-2148	199	13	-	-	PUNCT
iajs-2148	199	14	prime	prime	NOUN
iajs-2148	199	15	submodule	submodule	NOUN
iajs-2148	199	16	of	of	ADP
iajs-2148	199	17	.	.	PUNCT
iajs-2148	200	1	proof	proof	NOUN
iajs-2148	200	2	since	since	SCONJ
iajs-2148	200	3	is	be	AUX
iajs-2148	200	4	not	not	PART
iajs-2148	200	5	contained	contain	VERB
iajs-2148	200	6	in	in	ADP
iajs-2148	200	7	,	,	PUNCT
iajs-2148	200	8	then	then	ADV
iajs-2148	200	9	is	be	AUX
iajs-2148	200	10	a	a	DET
iajs-2148	200	11	proper	proper	ADJ
iajs-2148	200	12	submodule	submodule	NOUN
iajs-2148	200	13	of	of	ADP
iajs-2148	200	14	.	.	PUNCT
iajs-2148	201	1	now	now	ADV
iajs-2148	201	2	,	,	PUNCT
iajs-2148	201	3	let	let	VERB
iajs-2148	201	4	,	,	PUNCT
iajs-2148	201	5	where	where	SCONJ
iajs-2148	201	6	,	,	PUNCT
iajs-2148	201	7	,	,	PUNCT
iajs-2148	201	8	then	then	ADV
iajs-2148	201	9	and	and	CCONJ
iajs-2148	201	10	.	.	PUNCT
iajs-2148	202	1	but	but	CCONJ
iajs-2148	202	2	is	be	AUX
iajs-2148	202	3	app	app	ADJ
iajs-2148	202	4	-	-	PUNCT
iajs-2148	202	5	prime	prime	NOUN
iajs-2148	202	6	submodule	submodule	NOUN
iajs-2148	202	7	of	of	ADP
iajs-2148	202	8	,	,	PUNCT
iajs-2148	202	9	then	then	ADV
iajs-2148	202	10	either	either	ADV
iajs-2148	202	11	or	or	CCONJ
iajs-2148	202	12	.	.	PUNCT
iajs-2148	203	1	but	but	CCONJ
iajs-2148	203	2	and	and	CCONJ
iajs-2148	203	3	,	,	PUNCT
iajs-2148	203	4	then	then	ADV
iajs-2148	203	5	we	we	PRON
iajs-2148	203	6	have	have	VERB
iajs-2148	203	7	either	either	CCONJ
iajs-2148	203	8	[	[	PUNCT
iajs-2148	203	9	]	]	X
iajs-2148	204	1	or	or	CCONJ
iajs-2148	204	2	[	[	PUNCT
iajs-2148	204	3	]	]	X
iajs-2148	204	4	.	.	PUNCT
iajs-2148	205	1	but	but	CCONJ
iajs-2148	205	2	,	,	PUNCT
iajs-2148	205	3	so	so	CCONJ
iajs-2148	205	4	by	by	ADP
iajs-2148	205	5	modular	modular	ADJ
iajs-2148	205	6	law	law	NOUN
iajs-2148	205	7	we	we	PRON
iajs-2148	205	8	have	have	VERB
iajs-2148	205	9	either	either	PRON
iajs-2148	205	10	or	or	CCONJ
iajs-2148	205	11	.	.	PUNCT
iajs-2148	206	1	but	but	CCONJ
iajs-2148	206	2	by	by	ADP
iajs-2148	206	3	[	[	PUNCT
iajs-2148	206	4	13	13	NUM
iajs-2148	206	5	]	]	PUNCT
iajs-2148	206	6	.	.	PUNCT
iajs-2148	207	1	coro	coro	PROPN
iajs-2148	207	2	.	.	PUNCT
iajs-2148	208	1	9.9	9.9	NUM
iajs-2148	208	2	]	]	PUNCT
iajs-2148	208	3	we	we	PRON
iajs-2148	208	4	have	have	VERB
iajs-2148	208	5	,	,	PUNCT
iajs-2148	208	6	hence	hence	ADV
iajs-2148	208	7	either	either	ADV
iajs-2148	208	8	or	or	CCONJ
iajs-2148	208	9	.	.	PUNCT
iajs-2148	209	1	thus	thus	ADV
iajs-2148	209	2	is	be	AUX
iajs-2148	209	3	an	an	DET
iajs-2148	209	4	app	app	ADJ
iajs-2148	209	5	-	-	PUNCT
iajs-2148	209	6	prime	prime	NOUN
iajs-2148	209	7	submodule	submodule	NOUN
iajs-2148	209	8	of	of	ADP
iajs-2148	209	9	.	.	PUNCT
iajs-2148	210	1	proposition	proposition	NOUN
iajs-2148	210	2	(	(	PUNCT
iajs-2148	210	3	29	29	NUM
iajs-2148	210	4	)	)	PUNCT
iajs-2148	210	5	let	let	AUX
iajs-2148	210	6	be	be	AUX
iajs-2148	210	7	an	an	DET
iajs-2148	210	8	-module	-module	NOUN
iajs-2148	210	9	,	,	PUNCT
iajs-2148	210	10	and	and	CCONJ
iajs-2148	210	11	be	be	AUX
iajs-2148	210	12	a	a	DET
iajs-2148	210	13	submodule	submodule	NOUN
iajs-2148	210	14	of	of	ADP
iajs-2148	210	15	such	such	ADJ
iajs-2148	210	16	that	that	PRON
iajs-2148	210	17	,	,	PUNCT
iajs-2148	210	18	where	where	SCONJ
iajs-2148	210	19	is	be	AUX
iajs-2148	210	20	a	a	DET
iajs-2148	210	21	prime	prime	ADJ
iajs-2148	210	22	submodule	submodule	NOUN
iajs-2148	210	23	of	of	ADP
iajs-2148	210	24	for	for	ADP
iajs-2148	210	25	each	each	PRON
iajs-2148	210	26	.	.	PUNCT
iajs-2148	211	1	then	then	ADV
iajs-2148	211	2	is	be	AUX
iajs-2148	211	3	an	an	DET
iajs-2148	211	4	app	app	ADJ
iajs-2148	211	5	-	-	PUNCT
iajs-2148	211	6	prime	prime	NOUN
iajs-2148	211	7	submodule	submodule	NOUN
iajs-2148	211	8	of	of	ADP
iajs-2148	211	9	.	.	PUNCT
iajs-2148	212	1	proof	proof	NOUN
iajs-2148	212	2	let	let	VERB
iajs-2148	212	3	,	,	PUNCT
iajs-2148	212	4	where	where	SCONJ
iajs-2148	212	5	,	,	PUNCT
iajs-2148	212	6	,	,	PUNCT
iajs-2148	212	7	then	then	ADV
iajs-2148	212	8	for	for	ADP
iajs-2148	212	9	each	each	PRON
iajs-2148	212	10	.	.	PUNCT
iajs-2148	213	1	since	since	SCONJ
iajs-2148	213	2	is	be	AUX
iajs-2148	213	3	a	a	DET
iajs-2148	213	4	prime	prime	ADJ
iajs-2148	213	5	submodule	submodule	NOUN
iajs-2148	213	6	of	of	ADP
iajs-2148	213	7	for	for	ADP
iajs-2148	213	8	each	each	PRON
iajs-2148	213	9	,	,	PUNCT
iajs-2148	213	10	so	so	ADV
iajs-2148	213	11	either	either	PRON
iajs-2148	213	12	or	or	CCONJ
iajs-2148	213	13	[	[	PUNCT
iajs-2148	213	14	]	]	X
iajs-2148	213	15	.	.	PUNCT
iajs-2148	214	1	that	that	PRON
iajs-2148	214	2	is	be	AUX
iajs-2148	214	3	either	either	CCONJ
iajs-2148	214	4	330	330	NUM
iajs-2148	214	5	ibn	ibn	PROPN
iajs-2148	214	6	al	al	PROPN
iajs-2148	214	7	-	-	PUNCT
iajs-2148	214	8	haitham	haitham	PROPN
iajs-2148	214	9	jour.for	jour.for	ADP
iajs-2148	214	10	pure	pure	ADJ
iajs-2148	214	11	&	&	CCONJ
iajs-2148	214	12	appl	appl	PROPN
iajs-2148	214	13	.sci	.sci	PROPN
iajs-2148	214	14	.	.	PUNCT
iajs-2148	215	1	32	32	NUM
iajs-2148	215	2	(	(	PUNCT
iajs-2148	215	3	2	2	NUM
iajs-2148	215	4	)	)	PUNCT
iajs-2148	215	5	2019	2019	NUM
iajs-2148	215	6	or	or	CCONJ
iajs-2148	216	1	,	,	PUNCT
iajs-2148	216	2	it	it	PRON
iajs-2148	216	3	follows	follow	VERB
iajs-2148	216	4	that	that	PRON
iajs-2148	216	5	.	.	PUNCT
iajs-2148	217	1	hence	hence	ADV
iajs-2148	217	2	either	either	PRON
iajs-2148	217	3	or	or	CCONJ
iajs-2148	217	4	[	[	PUNCT
iajs-2148	217	5	]	]	X
iajs-2148	217	6	.	.	PUNCT
iajs-2148	218	1	therefore	therefore	ADV
iajs-2148	218	2	is	be	AUX
iajs-2148	218	3	an	an	DET
iajs-2148	218	4	app	app	ADJ
iajs-2148	218	5	-	-	PUNCT
iajs-2148	218	6	prime	prime	NOUN
iajs-2148	218	7	submodule	submodule	NOUN
iajs-2148	218	8	of	of	ADP
iajs-2148	218	9	.	.	PUNCT
iajs-2148	219	1	the	the	DET
iajs-2148	219	2	following	follow	VERB
iajs-2148	219	3	proposition	proposition	NOUN
iajs-2148	219	4	shows	show	VERB
iajs-2148	219	5	that	that	SCONJ
iajs-2148	219	6	the	the	DET
iajs-2148	219	7	invers	inver	NOUN
iajs-2148	219	8	image	image	NOUN
iajs-2148	219	9	of	of	ADP
iajs-2148	219	10	app	app	ADJ
iajs-2148	219	11	-	-	PUNCT
iajs-2148	219	12	prime	prime	NOUN
iajs-2148	219	13	submodule	submodule	NOUN
iajs-2148	219	14	is	be	AUX
iajs-2148	219	15	appprime	appprime	ADJ
iajs-2148	219	16	.	.	PUNCT
iajs-2148	220	1	proposition	proposition	NOUN
iajs-2148	220	2	(	(	PUNCT
iajs-2148	220	3	30	30	NUM
iajs-2148	220	4	)	)	PUNCT
iajs-2148	220	5	let	let	AUX
iajs-2148	220	6	be	be	AUX
iajs-2148	220	7	an	an	DET
iajs-2148	220	8	-epimorphism	-epimorphism	NOUN
iajs-2148	220	9	,	,	PUNCT
iajs-2148	220	10	and	and	CCONJ
iajs-2148	220	11	be	be	AUX
iajs-2148	220	12	an	an	DET
iajs-2148	220	13	app	app	ADJ
iajs-2148	220	14	-	-	PUNCT
iajs-2148	220	15	prime	prime	NOUN
iajs-2148	220	16	submodule	submodule	NOUN
iajs-2148	220	17	of	of	ADP
iajs-2148	220	18	.	.	PUNCT
iajs-2148	221	1	then	then	ADV
iajs-2148	221	2	is	be	AUX
iajs-2148	221	3	an	an	DET
iajs-2148	221	4	app	app	ADJ
iajs-2148	221	5	-	-	PUNCT
iajs-2148	221	6	prime	prime	NOUN
iajs-2148	221	7	submodule	submodule	NOUN
iajs-2148	221	8	of	of	ADP
iajs-2148	221	9	.	.	PUNCT
iajs-2148	222	1	proof	proof	NOUN
iajs-2148	222	2	it	it	PRON
iajs-2148	222	3	is	be	AUX
iajs-2148	222	4	clear	clear	ADJ
iajs-2148	222	5	that	that	PRON
iajs-2148	222	6	is	be	AUX
iajs-2148	222	7	a	a	DET
iajs-2148	222	8	proper	proper	ADJ
iajs-2148	222	9	submodule	submodule	NOUN
iajs-2148	222	10	of	of	ADP
iajs-2148	222	11	.	.	PUNCT
iajs-2148	223	1	now	now	ADV
iajs-2148	223	2	,	,	PUNCT
iajs-2148	223	3	suppose	suppose	VERB
iajs-2148	223	4	that	that	SCONJ
iajs-2148	223	5	,	,	PUNCT
iajs-2148	223	6	where	where	SCONJ
iajs-2148	223	7	,	,	PUNCT
iajs-2148	223	8	,	,	PUNCT
iajs-2148	223	9	then	then	ADV
iajs-2148	223	10	.	.	PUNCT
iajs-2148	224	1	but	but	CCONJ
iajs-2148	224	2	is	be	AUX
iajs-2148	224	3	an	an	DET
iajs-2148	224	4	app	app	ADJ
iajs-2148	224	5	-	-	PUNCT
iajs-2148	224	6	prime	prime	NOUN
iajs-2148	224	7	submodule	submodule	NOUN
iajs-2148	224	8	of	of	ADP
iajs-2148	224	9	,	,	PUNCT
iajs-2148	224	10	implies	imply	VERB
iajs-2148	224	11	that	that	SCONJ
iajs-2148	224	12	either	either	ADV
iajs-2148	224	13	or	or	CCONJ
iajs-2148	224	14	.	.	PUNCT
iajs-2148	225	1	if	if	SCONJ
iajs-2148	225	2	,	,	PUNCT
iajs-2148	225	3	then	then	ADV
iajs-2148	225	4	(	(	PUNCT
iajs-2148	225	5	)	)	PUNCT
iajs-2148	226	1	[	[	X
iajs-2148	226	2	15,theo.(1.4)a	15,theo.(1.4)a	X
iajs-2148	226	3	]	]	X
iajs-2148	226	4	.	.	PUNCT
iajs-2148	227	1	that	that	PRON
iajs-2148	227	2	is	be	AUX
iajs-2148	227	3	.	.	PUNCT
iajs-2148	228	1	if	if	SCONJ
iajs-2148	228	2	and	and	CCONJ
iajs-2148	228	3	,	,	PUNCT
iajs-2148	228	4	then	then	ADV
iajs-2148	228	5	.	.	PUNCT
iajs-2148	229	1	that	that	PRON
iajs-2148	229	2	is	be	AUX
iajs-2148	229	3	(	(	PUNCT
iajs-2148	229	4	)	)	PUNCT
iajs-2148	230	1	[	[	X
iajs-2148	230	2	15,theo.(1.4)a	15,theo.(1.4)a	NUM
iajs-2148	230	3	]	]	PUNCT
iajs-2148	230	4	.	.	PUNCT
iajs-2148	231	1	hence	hence	ADV
iajs-2148	231	2	.	.	PUNCT
iajs-2148	232	1	thus	thus	ADV
iajs-2148	232	2	is	be	AUX
iajs-2148	232	3	an	an	DET
iajs-2148	232	4	app	app	ADJ
iajs-2148	232	5	-	-	PUNCT
iajs-2148	232	6	prime	prime	NOUN
iajs-2148	232	7	submodule	submodule	NOUN
iajs-2148	232	8	of	of	ADP
iajs-2148	232	9	.	.	PUNCT
iajs-2148	233	1	proposition	proposition	NOUN
iajs-2148	233	2	(	(	PUNCT
iajs-2148	233	3	31	31	NUM
iajs-2148	233	4	)	)	PUNCT
iajs-2148	233	5	let	let	AUX
iajs-2148	233	6	be	be	AUX
iajs-2148	233	7	an	an	DET
iajs-2148	233	8	-epimorphism	-epimorphism	NOUN
iajs-2148	233	9	,	,	PUNCT
iajs-2148	233	10	and	and	CCONJ
iajs-2148	233	11	be	be	AUX
iajs-2148	233	12	an	an	DET
iajs-2148	233	13	app	app	ADJ
iajs-2148	233	14	-	-	PUNCT
iajs-2148	233	15	prime	prime	NOUN
iajs-2148	233	16	submodule	submodule	NOUN
iajs-2148	233	17	of	of	ADP
iajs-2148	233	18	with	with	ADP
iajs-2148	233	19	.	.	PUNCT
iajs-2148	234	1	then	then	ADV
iajs-2148	234	2	is	be	AUX
iajs-2148	234	3	an	an	DET
iajs-2148	234	4	app	app	ADJ
iajs-2148	234	5	-	-	PUNCT
iajs-2148	234	6	prime	prime	NOUN
iajs-2148	234	7	submodule	submodule	NOUN
iajs-2148	234	8	of	of	ADP
iajs-2148	234	9	.	.	PUNCT
iajs-2148	235	1	proof	proof	NOUN
iajs-2148	235	2	is	be	AUX
iajs-2148	235	3	a	a	DET
iajs-2148	235	4	proper	proper	ADJ
iajs-2148	235	5	submodule	submodule	NOUN
iajs-2148	235	6	of	of	ADP
iajs-2148	235	7	.	.	PUNCT
iajs-2148	236	1	if	if	SCONJ
iajs-2148	236	2	not	not	PART
iajs-2148	236	3	,	,	PUNCT
iajs-2148	236	4	that	that	ADV
iajs-2148	236	5	is	is	ADV
iajs-2148	236	6	,	,	PUNCT
iajs-2148	236	7	let	let	VERB
iajs-2148	236	8	,	,	PUNCT
iajs-2148	236	9	then	then	ADV
iajs-2148	236	10	,	,	PUNCT
iajs-2148	236	11	so	so	ADV
iajs-2148	236	12	there	there	PRON
iajs-2148	236	13	exists	exist	VERB
iajs-2148	236	14	such	such	ADJ
iajs-2148	236	15	that	that	SCONJ
iajs-2148	236	16	,	,	PUNCT
iajs-2148	236	17	that	that	ADV
iajs-2148	236	18	is	is	ADV
iajs-2148	236	19	,	,	PUNCT
iajs-2148	236	20	implies	imply	VERB
iajs-2148	236	21	that	that	SCONJ
iajs-2148	236	22	,	,	PUNCT
iajs-2148	236	23	it	it	PRON
iajs-2148	236	24	follows	follow	VERB
iajs-2148	236	25	that	that	SCONJ
iajs-2148	236	26	,	,	PUNCT
iajs-2148	236	27	hence	hence	ADV
iajs-2148	236	28	contradiction	contradiction	NOUN
iajs-2148	236	29	.	.	PUNCT
iajs-2148	237	1	now	now	ADV
iajs-2148	237	2	suppose	suppose	VERB
iajs-2148	237	3	that	that	SCONJ
iajs-2148	237	4	,	,	PUNCT
iajs-2148	237	5	where	where	SCONJ
iajs-2148	237	6	,	,	PUNCT
iajs-2148	237	7	.	.	PUNCT
iajs-2148	238	1	since	since	SCONJ
iajs-2148	238	2	is	be	AUX
iajs-2148	238	3	an	an	DET
iajs-2148	238	4	epimorphism	epimorphism	NOUN
iajs-2148	238	5	,	,	PUNCT
iajs-2148	238	6	and	and	CCONJ
iajs-2148	238	7	,	,	PUNCT
iajs-2148	238	8	then	then	ADV
iajs-2148	238	9	there	there	PRON
iajs-2148	238	10	exists	exist	VERB
iajs-2148	238	11	such	such	ADJ
iajs-2148	238	12	that	that	SCONJ
iajs-2148	238	13	,	,	PUNCT
iajs-2148	238	14	that	that	ADV
iajs-2148	238	15	is	is	ADV
iajs-2148	238	16	,	,	PUNCT
iajs-2148	238	17	so	so	CCONJ
iajs-2148	238	18	there	there	PRON
iajs-2148	238	19	exists	exist	VERB
iajs-2148	238	20	such	such	ADJ
iajs-2148	238	21	that	that	PRON
iajs-2148	238	22	,	,	PUNCT
iajs-2148	238	23	it	it	PRON
iajs-2148	238	24	follows	follow	VERB
iajs-2148	238	25	that	that	SCONJ
iajs-2148	238	26	,	,	PUNCT
iajs-2148	238	27	so	so	ADV
iajs-2148	238	28	,	,	PUNCT
iajs-2148	238	29	then	then	ADV
iajs-2148	238	30	.	.	PUNCT
iajs-2148	239	1	but	but	CCONJ
iajs-2148	239	2	be	be	AUX
iajs-2148	239	3	an	an	DET
iajs-2148	239	4	app	app	ADJ
iajs-2148	239	5	-	-	PUNCT
iajs-2148	239	6	prime	prime	NOUN
iajs-2148	239	7	submodule	submodule	NOUN
iajs-2148	239	8	of	of	ADP
iajs-2148	239	9	,	,	PUNCT
iajs-2148	239	10	then	then	ADV
iajs-2148	239	11	either	either	CCONJ
iajs-2148	239	12	or	or	CCONJ
iajs-2148	239	13	,	,	PUNCT
iajs-2148	239	14	and	and	CCONJ
iajs-2148	239	15	hence	hence	ADV
iajs-2148	239	16	either	either	ADV
iajs-2148	239	17	or	or	CCONJ
iajs-2148	239	18	.	.	PUNCT
iajs-2148	240	1	but	but	CCONJ
iajs-2148	240	2	by	by	ADP
iajs-2148	240	3	[	[	X
iajs-2148	240	4	15,theo.(1.4)a	15,theo.(1.4)a	NUM
iajs-2148	240	5	]	]	PUNCT
iajs-2148	240	6	.	.	PUNCT
iajs-2148	241	1	we	we	PRON
iajs-2148	241	2	have	have	VERB
iajs-2148	241	3	.	.	PUNCT
iajs-2148	242	1	so	so	ADV
iajs-2148	242	2	,	,	PUNCT
iajs-2148	242	3	we	we	PRON
iajs-2148	242	4	have	have	VERB
iajs-2148	242	5	either	either	PRON
iajs-2148	242	6	or	or	CCONJ
iajs-2148	242	7	.	.	PUNCT
iajs-2148	243	1	hence	hence	ADV
iajs-2148	243	2	is	be	AUX
iajs-2148	243	3	an	an	DET
iajs-2148	243	4	app	app	ADJ
iajs-2148	243	5	-	-	PUNCT
iajs-2148	243	6	prime	prime	NOUN
iajs-2148	243	7	submodule	submodule	NOUN
iajs-2148	243	8	of	of	ADP
iajs-2148	243	9	.	.	PUNCT
iajs-2148	244	1	as	as	ADP
iajs-2148	244	2	a	a	DET
iajs-2148	244	3	direct	direct	ADJ
iajs-2148	244	4	consequence	consequence	NOUN
iajs-2148	244	5	of	of	ADP
iajs-2148	244	6	proposition	proposition	NOUN
iajs-2148	244	7	(	(	PUNCT
iajs-2148	244	8	30	30	NUM
iajs-2148	244	9	)	)	PUNCT
iajs-2148	244	10	and	and	CCONJ
iajs-2148	244	11	proposition	proposition	NOUN
iajs-2148	244	12	(	(	PUNCT
iajs-2148	244	13	31	31	NUM
iajs-2148	244	14	)	)	PUNCT
iajs-2148	244	15	,	,	PUNCT
iajs-2148	244	16	we	we	PRON
iajs-2148	244	17	set	set	VERB
iajs-2148	244	18	the	the	DET
iajs-2148	244	19	following	follow	VERB
iajs-2148	244	20	result	result	NOUN
iajs-2148	244	21	.	.	PUNCT
iajs-2148	245	1	corollary	corollary	ADJ
iajs-2148	245	2	(	(	PUNCT
iajs-2148	245	3	32	32	NUM
iajs-2148	245	4	)	)	PUNCT
iajs-2148	245	5	let	let	AUX
iajs-2148	245	6	be	be	AUX
iajs-2148	245	7	an	an	DET
iajs-2148	245	8	-module	-module	NOUN
iajs-2148	245	9	,	,	PUNCT
iajs-2148	245	10	are	be	AUX
iajs-2148	245	11	two	two	NUM
iajs-2148	245	12	submodules	submodule	NOUN
iajs-2148	245	13	of	of	ADP
iajs-2148	245	14	with	with	ADP
iajs-2148	245	15	.	.	PUNCT
iajs-2148	246	1	then	then	ADV
iajs-2148	246	2	is	be	AUX
iajs-2148	246	3	an	an	DET
iajs-2148	246	4	appprime	appprime	NOUN
iajs-2148	246	5	submodule	submodule	NOUN
iajs-2148	246	6	of	of	ADP
iajs-2148	246	7	if	if	SCONJ
iajs-2148	246	8	and	and	CCONJ
iajs-2148	246	9	only	only	ADV
iajs-2148	246	10	if	if	SCONJ
iajs-2148	246	11	is	be	AUX
iajs-2148	246	12	an	an	DET
iajs-2148	246	13	app	app	ADJ
iajs-2148	246	14	-	-	PUNCT
iajs-2148	246	15	prime	prime	NOUN
iajs-2148	246	16	submodule	submodule	NOUN
iajs-2148	246	17	of	of	ADP
iajs-2148	246	18	.	.	PUNCT
iajs-2148	247	1	proposition	proposition	NOUN
iajs-2148	247	2	(	(	PUNCT
iajs-2148	247	3	33	33	NUM
iajs-2148	247	4	)	)	PUNCT
iajs-2148	247	5	let	let	AUX
iajs-2148	247	6	be	be	AUX
iajs-2148	247	7	an	an	DET
iajs-2148	247	8	-module	-module	NOUN
iajs-2148	247	9	,	,	PUNCT
iajs-2148	247	10	are	be	AUX
iajs-2148	247	11	two	two	NUM
iajs-2148	247	12	submodules	submodule	NOUN
iajs-2148	247	13	of	of	ADP
iajs-2148	247	14	and	and	CCONJ
iajs-2148	247	15	is	be	AUX
iajs-2148	247	16	an	an	DET
iajs-2148	247	17	app	app	ADJ
iajs-2148	247	18	-	-	PUNCT
iajs-2148	247	19	prime	prime	NOUN
iajs-2148	247	20	submodule	submodule	NOUN
iajs-2148	247	21	of	of	ADP
iajs-2148	247	22	with	with	ADP
iajs-2148	247	23	and	and	CCONJ
iajs-2148	247	24	[	[	PUNCT
iajs-2148	247	25	]	]	X
iajs-2148	247	26	[	[	PUNCT
iajs-2148	247	27	]	]	X
iajs-2148	247	28	then	then	ADV
iajs-2148	247	29	.	.	PUNCT
iajs-2148	248	1	proof	proof	NOUN
iajs-2148	248	2	since	since	SCONJ
iajs-2148	248	3	[	[	PUNCT
iajs-2148	248	4	]	]	X
iajs-2148	248	5	[	[	PUNCT
iajs-2148	248	6	]	]	X
iajs-2148	248	7	,	,	PUNCT
iajs-2148	248	8	then	then	ADV
iajs-2148	248	9	there	there	PRON
iajs-2148	248	10	exists	exist	VERB
iajs-2148	248	11	[	[	PUNCT
iajs-2148	248	12	]	]	PUNCT
iajs-2148	248	13	but	but	CCONJ
iajs-2148	248	14	[	[	PUNCT
iajs-2148	248	15	]	]	X
iajs-2148	248	16	.	.	PUNCT
iajs-2148	249	1	let	let	VERB
iajs-2148	249	2	,	,	PUNCT
iajs-2148	250	1	so	so	ADV
iajs-2148	250	2	and	and	CCONJ
iajs-2148	250	3	,	,	PUNCT
iajs-2148	250	4	so	so	ADV
iajs-2148	250	5	,	,	PUNCT
iajs-2148	250	6	implies	imply	VERB
iajs-2148	250	7	that	that	PRON
iajs-2148	250	8	.	.	PUNCT
iajs-2148	251	1	but	but	CCONJ
iajs-2148	251	2	ia	ia	PROPN
iajs-2148	251	3	an	an	DET
iajs-2148	251	4	appprime	appprime	NOUN
iajs-2148	251	5	submodule	submodule	NOUN
iajs-2148	251	6	of	of	ADP
iajs-2148	251	7	and	and	CCONJ
iajs-2148	251	8	[	[	PUNCT
iajs-2148	251	9	]	]	X
iajs-2148	251	10	then	then	ADV
iajs-2148	251	11	.	.	PUNCT
iajs-2148	252	1	thus	thus	ADV
iajs-2148	252	2	.	.	PUNCT
iajs-2148	253	1	333	333	NUM
iajs-2148	253	2	ibn	ibn	PROPN
iajs-2148	253	3	al	al	PROPN
iajs-2148	253	4	-	-	PUNCT
iajs-2148	253	5	haitham	haitham	PROPN
iajs-2148	253	6	jour.for	jour.for	ADP
iajs-2148	253	7	pure	pure	ADJ
iajs-2148	253	8	&	&	CCONJ
iajs-2148	253	9	appl	appl	PROPN
iajs-2148	253	10	.sci	.sci	PROPN
iajs-2148	253	11	.	.	PUNCT
iajs-2148	254	1	32	32	NUM
iajs-2148	254	2	(	(	PUNCT
iajs-2148	254	3	2	2	NUM
iajs-2148	254	4	)	)	PUNCT
iajs-2148	254	5	2019	2019	NUM
iajs-2148	254	6	3	3	NUM
iajs-2148	254	7	.	.	PUNCT
iajs-2148	255	1	approximaitly	approximaitly	ADV
iajs-2148	255	2	prime	prime	ADJ
iajs-2148	255	3	radical	radical	ADJ
iajs-2148	255	4	of	of	ADP
iajs-2148	255	5	submodules	submodule	NOUN
iajs-2148	255	6	in	in	ADP
iajs-2148	255	7	this	this	DET
iajs-2148	255	8	section	section	NOUN
iajs-2148	255	9	we	we	PRON
iajs-2148	255	10	introduce	introduce	VERB
iajs-2148	255	11	the	the	DET
iajs-2148	255	12	notion	notion	NOUN
iajs-2148	255	13	of	of	ADP
iajs-2148	255	14	approximaitly	approximaitly	ADV
iajs-2148	255	15	prime	prime	ADJ
iajs-2148	255	16	radical	radical	ADJ
iajs-2148	255	17	of	of	ADP
iajs-2148	255	18	a	a	DET
iajs-2148	255	19	submodule	submodule	NOUN
iajs-2148	255	20	,	,	PUNCT
iajs-2148	255	21	and	and	CCONJ
iajs-2148	255	22	we	we	PRON
iajs-2148	255	23	establish	establish	VERB
iajs-2148	255	24	several	several	ADJ
iajs-2148	255	25	properties	property	NOUN
iajs-2148	255	26	of	of	ADP
iajs-2148	255	27	this	this	DET
iajs-2148	255	28	notion	notion	NOUN
iajs-2148	255	29	that	that	PRON
iajs-2148	255	30	are	be	AUX
iajs-2148	255	31	similarly	similarly	ADV
iajs-2148	255	32	to	to	ADP
iajs-2148	255	33	those	those	PRON
iajs-2148	255	34	of	of	ADP
iajs-2148	255	35	radical	radical	ADJ
iajs-2148	255	36	of	of	ADP
iajs-2148	255	37	submodules	submodule	NOUN
iajs-2148	255	38	.	.	PUNCT
iajs-2148	256	1	definition	definition	NOUN
iajs-2148	256	2	(	(	PUNCT
iajs-2148	256	3	34	34	NUM
iajs-2148	256	4	)	)	PUNCT
iajs-2148	256	5	let	let	VERB
iajs-2148	256	6	b	b	NOUN
iajs-2148	256	7	an	an	DET
iajs-2148	256	8	-module	-module	NOUN
iajs-2148	256	9	,	,	PUNCT
iajs-2148	256	10	and	and	CCONJ
iajs-2148	256	11	is	be	AUX
iajs-2148	256	12	a	a	DET
iajs-2148	256	13	submodule	submodule	NOUN
iajs-2148	256	14	of	of	ADP
iajs-2148	256	15	.an	.an	PUNCT
iajs-2148	256	16	app	app	ADJ
iajs-2148	256	17	-	-	PUNCT
iajs-2148	256	18	prime	prime	NOUN
iajs-2148	256	19	radical	radical	NOUN
iajs-2148	256	20	of	of	ADP
iajs-2148	256	21	a	a	DET
iajs-2148	256	22	submodule	submodule	NOUN
iajs-2148	256	23	denoted	denote	VERB
iajs-2148	256	24	by	by	ADP
iajs-2148	256	25	is	be	AUX
iajs-2148	256	26	defined	define	VERB
iajs-2148	256	27	as	as	ADP
iajs-2148	256	28	the	the	DET
iajs-2148	256	29	intersection	intersection	NOUN
iajs-2148	256	30	of	of	ADP
iajs-2148	256	31	all	all	DET
iajs-2148	256	32	approximaitly	approximaitly	ADV
iajs-2148	256	33	prime	prime	ADJ
iajs-2148	256	34	submodules	submodule	NOUN
iajs-2148	256	35	of	of	ADP
iajs-2148	256	36	which	which	PRON
iajs-2148	256	37	contain	contain	VERB
iajs-2148	256	38	,	,	PUNCT
iajs-2148	256	39	if	if	SCONJ
iajs-2148	256	40	there	there	PRON
iajs-2148	256	41	exists	exist	VERB
iajs-2148	256	42	no	no	DET
iajs-2148	256	43	approximaitly	approximaitly	ADV
iajs-2148	256	44	prime	prime	ADJ
iajs-2148	256	45	submodule	submodule	NOUN
iajs-2148	256	46	containing	contain	VERB
iajs-2148	256	47	,	,	PUNCT
iajs-2148	256	48	we	we	PRON
iajs-2148	256	49	put	put	VERB
iajs-2148	256	50	.	.	PUNCT
iajs-2148	257	1	in	in	ADP
iajs-2148	257	2	the	the	DET
iajs-2148	257	3	following	follow	VERB
iajs-2148	257	4	proposition	proposition	NOUN
iajs-2148	257	5	we	we	PRON
iajs-2148	257	6	introduce	introduce	VERB
iajs-2148	257	7	some	some	DET
iajs-2148	257	8	basic	basic	ADJ
iajs-2148	257	9	properties	property	NOUN
iajs-2148	257	10	of	of	ADP
iajs-2148	257	11	approximaitly	approximaitly	ADV
iajs-2148	257	12	prime	prime	ADJ
iajs-2148	257	13	radical	radical	ADJ
iajs-2148	257	14	.	.	PUNCT
iajs-2148	258	1	proposition	proposition	NOUN
iajs-2148	258	2	(	(	PUNCT
iajs-2148	258	3	35	35	NUM
iajs-2148	258	4	)	)	PUNCT
iajs-2148	258	5	let	let	AUX
iajs-2148	258	6	be	be	AUX
iajs-2148	258	7	an	an	DET
iajs-2148	258	8	-epimorphism	-epimorphism	NOUN
iajs-2148	258	9	,	,	PUNCT
iajs-2148	258	10	and	and	CCONJ
iajs-2148	258	11	is	be	AUX
iajs-2148	258	12	a	a	DET
iajs-2148	258	13	submodule	submodule	NOUN
iajs-2148	258	14	of	of	ADP
iajs-2148	258	15	wihe	wihe	NOUN
iajs-2148	258	16	.	.	PUNCT
iajs-2148	259	1	then	then	ADV
iajs-2148	259	2	(	(	PUNCT
iajs-2148	259	3	)	)	PUNCT
iajs-2148	259	4	.	.	PUNCT
iajs-2148	260	1	proof	proof	NOUN
iajs-2148	260	2	since	since	SCONJ
iajs-2148	260	3	where	where	SCONJ
iajs-2148	260	4	the	the	DET
iajs-2148	260	5	intersection	intersection	NOUN
iajs-2148	260	6	runs	run	VERB
iajs-2148	260	7	over	over	ADP
iajs-2148	260	8	all	all	DET
iajs-2148	260	9	app	app	ADJ
iajs-2148	260	10	-	-	PUNCT
iajs-2148	260	11	prime	prime	NOUN
iajs-2148	260	12	submodules	submodule	NOUN
iajs-2148	260	13	of	of	ADP
iajs-2148	260	14	with	with	ADP
iajs-2148	260	15	,	,	PUNCT
iajs-2148	260	16	so	so	CCONJ
iajs-2148	260	17	(	(	PUNCT
iajs-2148	260	18	)	)	PUNCT
iajs-2148	260	19	.	.	PUNCT
iajs-2148	261	1	since	since	SCONJ
iajs-2148	261	2	,	,	PUNCT
iajs-2148	261	3	then	then	ADV
iajs-2148	261	4	by	by	ADP
iajs-2148	261	5	[	[	PUNCT
iajs-2148	261	6	13	13	NUM
iajs-2148	261	7	,	,	PUNCT
iajs-2148	261	8	lemm.(3.1.10	lemm.(3.1.10	PROPN
iajs-2148	261	9	)	)	PUNCT
iajs-2148	261	10	c	c	NOUN
iajs-2148	261	11	]	]	PUNCT
iajs-2148	261	12	.	.	PUNCT
iajs-2148	262	1	(	(	PUNCT
iajs-2148	262	2	)	)	PUNCT
iajs-2148	262	3	where	where	SCONJ
iajs-2148	262	4	the	the	DET
iajs-2148	262	5	intersection	intersection	NOUN
iajs-2148	262	6	runs	run	VERB
iajs-2148	262	7	over	over	ADP
iajs-2148	262	8	all	all	DET
iajs-2148	262	9	appprime	appprime	ADJ
iajs-2148	262	10	submodules	submodule	NOUN
iajs-2148	262	11	of	of	ADP
iajs-2148	262	12	with	with	ADP
iajs-2148	262	13	.	.	PUNCT
iajs-2148	263	1	thus	thus	ADV
iajs-2148	263	2	(	(	PUNCT
iajs-2148	263	3	)	)	PUNCT
iajs-2148	263	4	.	.	PUNCT
iajs-2148	264	1	proposition	proposition	NOUN
iajs-2148	264	2	(	(	PUNCT
iajs-2148	264	3	36	36	NUM
iajs-2148	264	4	)	)	PUNCT
iajs-2148	264	5	let	let	AUX
iajs-2148	264	6	be	be	AUX
iajs-2148	264	7	an	an	DET
iajs-2148	264	8	-epimorphism	-epimorphism	NOUN
iajs-2148	264	9	,	,	PUNCT
iajs-2148	264	10	and	and	CCONJ
iajs-2148	264	11	is	be	AUX
iajs-2148	264	12	a	a	DET
iajs-2148	264	13	submodule	submodule	NOUN
iajs-2148	264	14	of	of	ADP
iajs-2148	264	15	.	.	PUNCT
iajs-2148	265	1	then	then	ADV
iajs-2148	265	2	(	(	PUNCT
iajs-2148	265	3	)	)	PUNCT
iajs-2148	265	4	.	.	PUNCT
iajs-2148	266	1	proof	proof	NOUN
iajs-2148	266	2	since	since	SCONJ
iajs-2148	266	3	where	where	SCONJ
iajs-2148	266	4	the	the	DET
iajs-2148	266	5	intersection	intersection	NOUN
iajs-2148	266	6	runs	run	VERB
iajs-2148	266	7	over	over	ADP
iajs-2148	266	8	all	all	DET
iajs-2148	266	9	app	app	ADJ
iajs-2148	266	10	-	-	PUNCT
iajs-2148	266	11	prime	prime	NOUN
iajs-2148	266	12	submodules	submodule	NOUN
iajs-2148	266	13	of	of	ADP
iajs-2148	266	14	with	with	ADP
iajs-2148	266	15	.	.	PUNCT
iajs-2148	267	1	hence	hence	ADV
iajs-2148	267	2	by	by	ADP
iajs-2148	267	3	[	[	X
iajs-2148	267	4	15,lemm.(3.1.10)a	15,lemm.(3.1.10)a	NUM
iajs-2148	267	5	]	]	PUNCT
iajs-2148	267	6	.	.	PUNCT
iajs-2148	268	1	(	(	PUNCT
iajs-2148	268	2	)	)	PUNCT
iajs-2148	268	3	where	where	SCONJ
iajs-2148	268	4	the	the	DET
iajs-2148	268	5	intersection	intersection	NOUN
iajs-2148	268	6	runs	run	VERB
iajs-2148	268	7	over	over	ADP
iajs-2148	268	8	all	all	DET
iajs-2148	268	9	app	app	ADJ
iajs-2148	268	10	-	-	PUNCT
iajs-2148	268	11	prime	prime	NOUN
iajs-2148	268	12	submodules	submodule	NOUN
iajs-2148	268	13	of	of	ADP
iajs-2148	268	14	with	with	ADP
iajs-2148	268	15	.	.	PUNCT
iajs-2148	269	1	it	it	PRON
iajs-2148	269	2	follows	follow	VERB
iajs-2148	269	3	that	that	PRON
iajs-2148	269	4	(	(	PUNCT
iajs-2148	269	5	)	)	PUNCT
iajs-2148	269	6	.	.	PUNCT
iajs-2148	270	1	proposition	proposition	NOUN
iajs-2148	270	2	(	(	PUNCT
iajs-2148	270	3	37	37	NUM
iajs-2148	270	4	)	)	PUNCT
iajs-2148	270	5	let	let	AUX
iajs-2148	270	6	be	be	AUX
iajs-2148	270	7	an	an	DET
iajs-2148	270	8	-module	-module	NOUN
iajs-2148	270	9	,	,	PUNCT
iajs-2148	270	10	and	and	CCONJ
iajs-2148	270	11	are	be	AUX
iajs-2148	270	12	two	two	NUM
iajs-2148	270	13	submodules	submodule	NOUN
iajs-2148	270	14	of	of	ADP
iajs-2148	270	15	.	.	PUNCT
iajs-2148	271	1	then	then	ADV
iajs-2148	271	2	:	:	PUNCT
iajs-2148	271	3	(	(	PUNCT
iajs-2148	271	4	1	1	X
iajs-2148	271	5	)	)	PUNCT
iajs-2148	271	6	.	.	PUNCT
iajs-2148	272	1	(	(	PUNCT
iajs-2148	272	2	2	2	X
iajs-2148	272	3	)	)	PUNCT
iajs-2148	272	4	if	if	SCONJ
iajs-2148	272	5	,	,	PUNCT
iajs-2148	272	6	then	then	ADV
iajs-2148	272	7	.	.	PUNCT
iajs-2148	273	1	(	(	PUNCT
iajs-2148	273	2	3	3	X
iajs-2148	273	3	)	)	PUNCT
iajs-2148	273	4	(	(	PUNCT
iajs-2148	273	5	)	)	PUNCT
iajs-2148	273	6	.	.	PUNCT
iajs-2148	274	1	(	(	PUNCT
iajs-2148	274	2	4	4	NUM
iajs-2148	274	3	)	)	PUNCT
iajs-2148	274	4	.	.	PUNCT
iajs-2148	275	1	(	(	PUNCT
iajs-2148	275	2	5	5	NUM
iajs-2148	275	3	)	)	PUNCT
iajs-2148	275	4	(	(	PUNCT
iajs-2148	275	5	)	)	PUNCT
iajs-2148	275	6	.	.	PUNCT
iajs-2148	276	1	proof	proof	NOUN
iajs-2148	276	2	(	(	PUNCT
iajs-2148	276	3	1	1	NUM
iajs-2148	276	4	)	)	PUNCT
iajs-2148	276	5	since	since	SCONJ
iajs-2148	276	6	where	where	SCONJ
iajs-2148	276	7	the	the	DET
iajs-2148	276	8	intersection	intersection	NOUN
iajs-2148	276	9	runs	run	VERB
iajs-2148	276	10	over	over	ADP
iajs-2148	276	11	all	all	DET
iajs-2148	276	12	app	app	ADJ
iajs-2148	276	13	-	-	PUNCT
iajs-2148	276	14	prime	prime	NOUN
iajs-2148	276	15	submodules	submodule	NOUN
iajs-2148	276	16	of	of	ADP
iajs-2148	276	17	with	with	ADP
iajs-2148	276	18	,	,	PUNCT
iajs-2148	276	19	so	so	ADV
iajs-2148	276	20	.	.	PUNCT
iajs-2148	277	1	(	(	PUNCT
iajs-2148	277	2	2	2	X
iajs-2148	277	3	)	)	PUNCT
iajs-2148	277	4	suppose	suppose	VERB
iajs-2148	277	5	that	that	SCONJ
iajs-2148	277	6	,	,	PUNCT
iajs-2148	277	7	and	and	CCONJ
iajs-2148	277	8	let	let	AUX
iajs-2148	277	9	be	be	AUX
iajs-2148	277	10	an	an	DET
iajs-2148	277	11	app	app	ADJ
iajs-2148	277	12	-	-	PUNCT
iajs-2148	277	13	prime	prime	NOUN
iajs-2148	277	14	submodule	submodule	NOUN
iajs-2148	277	15	of	of	ADP
iajs-2148	277	16	with	with	ADP
iajs-2148	277	17	,	,	PUNCT
iajs-2148	277	18	then	then	ADV
iajs-2148	277	19	,	,	PUNCT
iajs-2148	277	20	implies	imply	VERB
iajs-2148	277	21	that	that	PRON
iajs-2148	277	22	.	.	PUNCT
iajs-2148	278	1	thus	thus	ADV
iajs-2148	278	2	.	.	PUNCT
iajs-2148	279	1	(	(	PUNCT
iajs-2148	279	2	3	3	X
iajs-2148	279	3	)	)	PUNCT
iajs-2148	279	4	by	by	ADP
iajs-2148	279	5	part	part	NOUN
iajs-2148	279	6	(	(	PUNCT
iajs-2148	279	7	1	1	X
iajs-2148	279	8	)	)	PUNCT
iajs-2148	279	9	we	we	PRON
iajs-2148	279	10	have	have	VERB
iajs-2148	279	11	(	(	PUNCT
iajs-2148	279	12	)	)	PUNCT
iajs-2148	279	13	.	.	PUNCT
iajs-2148	280	1	but	but	CCONJ
iajs-2148	280	2	(	(	PUNCT
iajs-2148	280	3	)	)	PUNCT
iajs-2148	280	4	,	,	PUNCT
iajs-2148	280	5	where	where	SCONJ
iajs-2148	280	6	the	the	DET
iajs-2148	280	7	intersection	intersection	NOUN
iajs-2148	280	8	runs	run	VERB
iajs-2148	280	9	over	over	ADP
iajs-2148	280	10	all	all	DET
iajs-2148	280	11	app	app	ADJ
iajs-2148	280	12	-	-	PUNCT
iajs-2148	280	13	prime	prime	NOUN
iajs-2148	280	14	submodules	submodule	NOUN
iajs-2148	280	15	of	of	ADP
iajs-2148	280	16	with	with	ADP
iajs-2148	280	17	,	,	PUNCT
iajs-2148	280	18	again	again	ADV
iajs-2148	280	19	by	by	ADP
iajs-2148	280	20	(	(	PUNCT
iajs-2148	280	21	1	1	NUM
iajs-2148	280	22	)	)	PUNCT
iajs-2148	280	23	.	.	PUNCT
iajs-2148	281	1	thus	thus	ADV
iajs-2148	281	2	331	331	NUM
iajs-2148	281	3	ibn	ibn	PROPN
iajs-2148	281	4	al	al	PROPN
iajs-2148	281	5	-	-	PUNCT
iajs-2148	281	6	haitham	haitham	PROPN
iajs-2148	281	7	jour.for	jour.for	ADP
iajs-2148	281	8	pure	pure	ADJ
iajs-2148	281	9	&	&	CCONJ
iajs-2148	281	10	appl	appl	PROPN
iajs-2148	281	11	.sci	.sci	PROPN
iajs-2148	281	12	.	.	PUNCT
iajs-2148	282	1	32	32	NUM
iajs-2148	282	2	(	(	PUNCT
iajs-2148	282	3	2	2	NUM
iajs-2148	282	4	)	)	PUNCT
iajs-2148	282	5	2019	2019	NUM
iajs-2148	282	6	(	(	PUNCT
iajs-2148	282	7	)	)	PUNCT
iajs-2148	282	8	.	.	PUNCT
iajs-2148	283	1	hence	hence	ADV
iajs-2148	283	2	(	(	PUNCT
iajs-2148	283	3	)	)	PUNCT
iajs-2148	283	4	.	.	PUNCT
iajs-2148	284	1	(	(	PUNCT
iajs-2148	284	2	4	4	X
iajs-2148	284	3	)	)	PUNCT
iajs-2148	284	4	let	let	AUX
iajs-2148	284	5	be	be	AUX
iajs-2148	284	6	an	an	DET
iajs-2148	284	7	app	app	ADJ
iajs-2148	284	8	-	-	PUNCT
iajs-2148	284	9	prime	prime	NOUN
iajs-2148	284	10	submodule	submodule	NOUN
iajs-2148	284	11	of	of	ADP
iajs-2148	284	12	contining	contining	NOUN
iajs-2148	284	13	and	and	CCONJ
iajs-2148	284	14	.	.	PUNCT
iajs-2148	285	1	since	since	ADV
iajs-2148	285	2	,	,	PUNCT
iajs-2148	285	3	so	so	ADV
iajs-2148	285	4	:	:	PUNCT
iajs-2148	285	5	.	.	PUNCT
iajs-2148	286	1	thus	thus	ADV
iajs-2148	286	2	.	.	PUNCT
iajs-2148	287	1	by	by	ADP
iajs-2148	287	2	same	same	ADJ
iajs-2148	287	3	way	way	NOUN
iajs-2148	287	4	.	.	PUNCT
iajs-2148	288	1	hence	hence	ADV
iajs-2148	288	2	.	.	PUNCT
iajs-2148	289	1	(	(	PUNCT
iajs-2148	289	2	5	5	NUM
iajs-2148	289	3	)	)	PUNCT
iajs-2148	289	4	since	since	SCONJ
iajs-2148	289	5	,	,	PUNCT
iajs-2148	289	6	then	then	ADV
iajs-2148	289	7	by	by	ADP
iajs-2148	289	8	(	(	PUNCT
iajs-2148	289	9	2	2	X
iajs-2148	289	10	)	)	PUNCT
iajs-2148	289	11	we	we	PRON
iajs-2148	289	12	have	have	VERB
iajs-2148	289	13	(	(	PUNCT
iajs-2148	289	14	)	)	PUNCT
iajs-2148	289	15	.	.	PUNCT
iajs-2148	290	1	now	now	ADV
iajs-2148	290	2	,	,	PUNCT
iajs-2148	290	3	let	let	AUX
iajs-2148	290	4	be	be	AUX
iajs-2148	290	5	an	an	DET
iajs-2148	290	6	app	app	ADJ
iajs-2148	290	7	-	-	PUNCT
iajs-2148	290	8	prime	prime	NOUN
iajs-2148	290	9	submodule	submodule	NOUN
iajs-2148	290	10	of	of	ADP
iajs-2148	290	11	with	with	ADP
iajs-2148	290	12	,	,	PUNCT
iajs-2148	290	13	we	we	PRON
iajs-2148	290	14	prove	prove	VERB
iajs-2148	290	15	that	that	PRON
iajs-2148	290	16	.	.	PUNCT
iajs-2148	291	1	since	since	SCONJ
iajs-2148	291	2	and	and	CCONJ
iajs-2148	291	3	,	,	PUNCT
iajs-2148	291	4	,	,	PUNCT
iajs-2148	291	5	then	then	ADV
iajs-2148	291	6	and	and	CCONJ
iajs-2148	291	7	.	.	PUNCT
iajs-2148	292	1	hence	hence	ADV
iajs-2148	292	2	.	.	PUNCT
iajs-2148	293	1	therefore	therefore	ADV
iajs-2148	293	2	and	and	CCONJ
iajs-2148	293	3	we	we	PRON
iajs-2148	293	4	have	have	VERB
iajs-2148	293	5	(	(	PUNCT
iajs-2148	293	6	)	)	PUNCT
iajs-2148	293	7	.	.	PUNCT
iajs-2148	294	1	recall	recall	VERB
iajs-2148	294	2	that	that	SCONJ
iajs-2148	294	3	a	a	DET
iajs-2148	294	4	submodule	submodule	NOUN
iajs-2148	294	5	of	of	ADP
iajs-2148	294	6	an	an	DET
iajs-2148	294	7	-module	-module	NOUN
iajs-2148	294	8	is	be	AUX
iajs-2148	294	9	called	call	VERB
iajs-2148	294	10	completely	completely	ADV
iajs-2148	294	11	irreducible	irreducible	ADJ
iajs-2148	294	12	,	,	PUNCT
iajs-2148	294	13	if	if	SCONJ
iajs-2148	294	14	for	for	ADP
iajs-2148	294	15	any	any	DET
iajs-2148	294	16	submodules	submodule	NOUN
iajs-2148	294	17	of	of	ADP
iajs-2148	294	18	,	,	PUNCT
iajs-2148	294	19	,	,	PUNCT
iajs-2148	294	20	implies	imply	VERB
iajs-2148	294	21	that	that	SCONJ
iajs-2148	294	22	either	either	CCONJ
iajs-2148	294	23	or	or	CCONJ
iajs-2148	294	24	[	[	X
iajs-2148	294	25	11	11	NUM
iajs-2148	294	26	]	]	PUNCT
iajs-2148	294	27	.	.	PUNCT
iajs-2148	295	1	proposition	proposition	NOUN
iajs-2148	295	2	(	(	PUNCT
iajs-2148	295	3	38	38	NUM
iajs-2148	295	4	)	)	PUNCT
iajs-2148	295	5	let	let	VERB
iajs-2148	295	6	and	and	CCONJ
iajs-2148	295	7	are	be	AUX
iajs-2148	295	8	two	two	NUM
iajs-2148	295	9	submodules	submodule	NOUN
iajs-2148	295	10	of	of	ADP
iajs-2148	295	11	an	an	DET
iajs-2148	295	12	-module	-module	NOUN
iajs-2148	295	13	.	.	PUNCT
iajs-2148	296	1	if	if	SCONJ
iajs-2148	296	2	every	every	DET
iajs-2148	296	3	app	app	ADJ
iajs-2148	296	4	-	-	PUNCT
iajs-2148	296	5	prime	prime	NOUN
iajs-2148	296	6	submodule	submodule	NOUN
iajs-2148	296	7	of	of	ADP
iajs-2148	296	8	which	which	PRON
iajs-2148	296	9	contains	contain	VERB
iajs-2148	296	10	is	be	AUX
iajs-2148	296	11	a	a	DET
iajs-2148	296	12	completely	completely	ADV
iajs-2148	296	13	irreducible	irreducible	ADJ
iajs-2148	296	14	submodule	submodule	NOUN
iajs-2148	296	15	,	,	PUNCT
iajs-2148	296	16	then	then	ADV
iajs-2148	296	17	.	.	PUNCT
iajs-2148	297	1	proof	proof	NOUN
iajs-2148	297	2	holds	hold	VERB
iajs-2148	297	3	by	by	ADP
iajs-2148	297	4	proposition	proposition	NOUN
iajs-2148	297	5	(	(	PUNCT
iajs-2148	297	6	37)(4	37)(4	NOUN
iajs-2148	297	7	)	)	PUNCT
iajs-2148	297	8	.	.	PUNCT
iajs-2148	298	1	now	now	ADV
iajs-2148	298	2	,	,	PUNCT
iajs-2148	298	3	if	if	SCONJ
iajs-2148	298	4	,	,	PUNCT
iajs-2148	298	5	then	then	ADV
iajs-2148	298	6	.	.	PUNCT
iajs-2148	299	1	if	if	SCONJ
iajs-2148	299	2	,	,	PUNCT
iajs-2148	299	3	then	then	ADV
iajs-2148	299	4	there	there	PRON
iajs-2148	299	5	exists	exist	VERB
iajs-2148	299	6	an	an	DET
iajs-2148	299	7	app	app	ADJ
iajs-2148	299	8	-	-	PUNCT
iajs-2148	299	9	prime	prime	NOUN
iajs-2148	299	10	submodule	submodule	NOUN
iajs-2148	299	11	of	of	ADP
iajs-2148	299	12	such	such	ADJ
iajs-2148	299	13	that	that	PRON
iajs-2148	299	14	or	or	CCONJ
iajs-2148	299	15	,	,	PUNCT
iajs-2148	299	16	so	so	SCONJ
iajs-2148	299	17	that	that	SCONJ
iajs-2148	299	18	either	either	ADV
iajs-2148	299	19	or	or	CCONJ
iajs-2148	299	20	because	because	SCONJ
iajs-2148	299	21	every	every	DET
iajs-2148	299	22	app	app	ADJ
iajs-2148	299	23	-	-	PUNCT
iajs-2148	299	24	prime	prime	NOUN
iajs-2148	299	25	submodule	submodule	NOUN
iajs-2148	299	26	containing	contain	VERB
iajs-2148	299	27	is	be	AUX
iajs-2148	299	28	completely	completely	ADV
iajs-2148	299	29	irreducible	irreducible	ADJ
iajs-2148	299	30	,	,	PUNCT
iajs-2148	299	31	then	then	ADV
iajs-2148	299	32	we	we	PRON
iajs-2148	299	33	have	have	VERB
iajs-2148	299	34	either	either	PRON
iajs-2148	299	35	or	or	CCONJ
iajs-2148	299	36	.	.	PUNCT
iajs-2148	300	1	therefore	therefore	ADV
iajs-2148	300	2	,	,	PUNCT
iajs-2148	300	3	and	and	CCONJ
iajs-2148	300	4	hence	hence	ADV
iajs-2148	300	5	.	.	PUNCT
iajs-2148	301	1	4	4	X
iajs-2148	301	2	.	.	X
iajs-2148	301	3	conclusion	conclusion	NOUN
iajs-2148	301	4	in	in	ADP
iajs-2148	301	5	this	this	DET
iajs-2148	301	6	paper	paper	NOUN
iajs-2148	301	7	an	an	DET
iajs-2148	301	8	approximaitly	approximaitly	ADV
iajs-2148	301	9	prime	prime	ADJ
iajs-2148	301	10	submodules	submodule	NOUN
iajs-2148	301	11	are	be	AUX
iajs-2148	301	12	introduced	introduce	VERB
iajs-2148	301	13	and	and	CCONJ
iajs-2148	301	14	studied	study	VERB
iajs-2148	301	15	as	as	ADP
iajs-2148	301	16	a	a	DET
iajs-2148	301	17	new	new	ADJ
iajs-2148	301	18	generalization	generalization	NOUN
iajs-2148	301	19	of	of	ADP
iajs-2148	301	20	prime	prime	ADJ
iajs-2148	301	21	submodules	submodule	NOUN
iajs-2148	301	22	,	,	PUNCT
iajs-2148	301	23	also	also	ADV
iajs-2148	301	24	we	we	PRON
iajs-2148	301	25	introduced	introduce	VERB
iajs-2148	301	26	and	and	CCONJ
iajs-2148	301	27	studied	study	VERB
iajs-2148	301	28	the	the	DET
iajs-2148	301	29	approximaitly	approximaitly	ADV
iajs-2148	301	30	prime	prime	ADJ
iajs-2148	301	31	radical	radical	ADJ
iajs-2148	301	32	of	of	ADP
iajs-2148	301	33	modules	module	NOUN
iajs-2148	301	34	.	.	PUNCT
iajs-2148	302	1	the	the	DET
iajs-2148	302	2	main	main	ADJ
iajs-2148	302	3	results	result	NOUN
iajs-2148	302	4	of	of	ADP
iajs-2148	302	5	this	this	DET
iajs-2148	302	6	study	study	NOUN
iajs-2148	302	7	are	be	AUX
iajs-2148	302	8	the	the	DET
iajs-2148	302	9	following	following	NOUN
iajs-2148	302	10	.	.	PUNCT
iajs-2148	303	1	1	1	X
iajs-2148	303	2	)	)	PUNCT
iajs-2148	303	3	a	a	DET
iajs-2148	303	4	proper	proper	ADJ
iajs-2148	303	5	submodule	submodule	NOUN
iajs-2148	303	6	of	of	ADP
iajs-2148	303	7	an	an	DET
iajs-2148	303	8	-module	-module	NOUN
iajs-2148	303	9	is	be	AUX
iajs-2148	303	10	an	an	DET
iajs-2148	303	11	app	app	ADJ
iajs-2148	303	12	-	-	PUNCT
iajs-2148	303	13	prime	prime	NOUN
iajs-2148	303	14	submodule	submodule	NOUN
iajs-2148	303	15	if	if	SCONJ
iajs-2148	303	16	and	and	CCONJ
iajs-2148	303	17	only	only	ADV
iajs-2148	303	18	,	,	PUNCT
iajs-2148	303	19	with	with	SCONJ
iajs-2148	303	20	is	be	AUX
iajs-2148	303	21	an	an	DET
iajs-2148	303	22	ideal	ideal	NOUN
iajs-2148	303	23	of	of	ADP
iajs-2148	303	24	and	and	CCONJ
iajs-2148	303	25	is	be	AUX
iajs-2148	303	26	a	a	DET
iajs-2148	303	27	submodule	submodule	NOUN
iajs-2148	303	28	of	of	ADP
iajs-2148	303	29	,	,	PUNCT
iajs-2148	303	30	implies	imply	VERB
iajs-2148	303	31	that	that	SCONJ
iajs-2148	303	32	either	either	CCONJ
iajs-2148	303	33	or	or	CCONJ
iajs-2148	303	34	[	[	PUNCT
iajs-2148	303	35	]	]	X
iajs-2148	303	36	.	.	NOUN
iajs-2148	304	1	2	2	X
iajs-2148	304	2	)	)	PUNCT
iajs-2148	304	3	every	every	DET
iajs-2148	304	4	prime	prime	ADJ
iajs-2148	304	5	submodule	submodule	NOUN
iajs-2148	304	6	is	be	AUX
iajs-2148	304	7	an	an	DET
iajs-2148	304	8	app	app	ADJ
iajs-2148	304	9	-	-	PUNCT
iajs-2148	304	10	prime	prime	NOUN
iajs-2148	304	11	submodule	submodule	NOUN
iajs-2148	304	12	,	,	PUNCT
iajs-2148	304	13	while	while	SCONJ
iajs-2148	304	14	the	the	DET
iajs-2148	304	15	converse	converse	NOUN
iajs-2148	304	16	is	be	AUX
iajs-2148	304	17	not	not	PART
iajs-2148	304	18	true	true	ADJ
iajs-2148	304	19	see	see	NOUN
iajs-2148	304	20	remark	remark	NOUN
iajs-2148	304	21	(	(	PUNCT
iajs-2148	304	22	4	4	NUM
iajs-2148	304	23	)	)	PUNCT
iajs-2148	304	24	and	and	CCONJ
iajs-2148	304	25	example	example	NOUN
iajs-2148	304	26	(	(	PUNCT
iajs-2148	304	27	5	5	NUM
iajs-2148	304	28	)	)	PUNCT
iajs-2148	304	29	.	.	PUNCT
iajs-2148	305	1	3	3	X
iajs-2148	305	2	)	)	PUNCT
iajs-2148	305	3	in	in	ADP
iajs-2148	305	4	multiplication	multiplication	NOUN
iajs-2148	305	5	non	non	ADJ
iajs-2148	305	6	-	-	ADJ
iajs-2148	305	7	singular	singular	ADJ
iajs-2148	305	8	-module	-module	NOUN
iajs-2148	305	9	a	a	DET
iajs-2148	305	10	proper	proper	ADJ
iajs-2148	305	11	submodule	submodule	NOUN
iajs-2148	305	12	of	of	ADP
iajs-2148	305	13	with	with	ADP
iajs-2148	305	14	is	be	AUX
iajs-2148	305	15	an	an	DET
iajs-2148	305	16	app	app	ADJ
iajs-2148	305	17	-	-	PUNCT
iajs-2148	305	18	prime	prime	NOUN
iajs-2148	305	19	submodule	submodule	NOUN
iajs-2148	305	20	of	of	ADP
iajs-2148	305	21	if	if	SCONJ
iajs-2148	305	22	and	and	CCONJ
iajs-2148	305	23	only	only	ADV
iajs-2148	305	24	if	if	SCONJ
iajs-2148	305	25	[	[	PUNCT
iajs-2148	305	26	]	]	X
iajs-2148	305	27	is	be	AUX
iajs-2148	305	28	an	an	DET
iajs-2148	305	29	app	app	ADJ
iajs-2148	305	30	-	-	PUNCT
iajs-2148	305	31	prime	prime	ADJ
iajs-2148	305	32	ideal	ideal	NOUN
iajs-2148	305	33	of	of	ADP
iajs-2148	305	34	.	.	PUNCT
iajs-2148	306	1	also	also	ADV
iajs-2148	306	2	,	,	PUNCT
iajs-2148	306	3	this	this	DET
iajs-2148	306	4	result	result	NOUN
iajs-2148	306	5	is	be	AUX
iajs-2148	306	6	satisfied	satisfied	ADJ
iajs-2148	306	7	if	if	SCONJ
iajs-2148	306	8	is	be	AUX
iajs-2148	306	9	faithful	faithful	ADJ
iajs-2148	306	10	multiplication	multiplication	NOUN
iajs-2148	306	11	with	with	ADP
iajs-2148	306	12	see	see	NOUN
iajs-2148	306	13	proposition	proposition	NOUN
iajs-2148	306	14	(	(	PUNCT
iajs-2148	306	15	12	12	NUM
iajs-2148	306	16	)	)	PUNCT
iajs-2148	306	17	.	.	PUNCT
iajs-2148	307	1	4	4	X
iajs-2148	307	2	)	)	PUNCT
iajs-2148	307	3	if	if	SCONJ
iajs-2148	307	4	is	be	AUX
iajs-2148	307	5	a	a	DET
iajs-2148	307	6	proper	proper	ADJ
iajs-2148	307	7	submodule	submodule	NOUN
iajs-2148	307	8	of	of	ADP
iajs-2148	307	9	,	,	PUNCT
iajs-2148	307	10	with	with	SCONJ
iajs-2148	307	11	[	[	PUNCT
iajs-2148	307	12	]	]	X
iajs-2148	307	13	is	be	AUX
iajs-2148	307	14	a	a	DET
iajs-2148	307	15	prime	prime	ADJ
iajs-2148	307	16	ideal	ideal	NOUN
iajs-2148	307	17	of	of	ADP
iajs-2148	307	18	.	.	PUNCT
iajs-2148	308	1	then	then	ADV
iajs-2148	308	2	with	with	ADP
iajs-2148	308	3	[	[	PUNCT
iajs-2148	308	4	]	]	X
iajs-2148	308	5	if	if	SCONJ
iajs-2148	308	6	and	and	CCONJ
iajs-2148	308	7	only	only	ADV
iajs-2148	308	8	if	if	SCONJ
iajs-2148	308	9	is	be	AUX
iajs-2148	308	10	an	an	DET
iajs-2148	308	11	app	app	ADJ
iajs-2148	308	12	-	-	PUNCT
iajs-2148	308	13	prime	prime	NOUN
iajs-2148	308	14	submodule	submodule	NOUN
iajs-2148	308	15	of	of	ADP
iajs-2148	308	16	.	.	PUNCT
iajs-2148	309	1	331	331	NUM
iajs-2148	309	2	ibn	ibn	PROPN
iajs-2148	309	3	al	al	PROPN
iajs-2148	309	4	-	-	PUNCT
iajs-2148	309	5	haitham	haitham	PROPN
iajs-2148	309	6	jour.for	jour.for	ADP
iajs-2148	309	7	pure	pure	ADJ
iajs-2148	309	8	&	&	CCONJ
iajs-2148	309	9	appl	appl	PROPN
iajs-2148	309	10	.sci	.sci	PROPN
iajs-2148	309	11	.	.	PUNCT
iajs-2148	310	1	32	32	NUM
iajs-2148	310	2	(	(	PUNCT
iajs-2148	310	3	2	2	NUM
iajs-2148	310	4	)	)	PUNCT
iajs-2148	310	5	2019	2019	NUM
iajs-2148	310	6	5	5	NUM
iajs-2148	310	7	)	)	PUNCT
iajs-2148	310	8	if	if	SCONJ
iajs-2148	310	9	is	be	AUX
iajs-2148	310	10	a	a	DET
iajs-2148	310	11	maximal	maximal	ADJ
iajs-2148	310	12	ideal	ideal	NOUN
iajs-2148	310	13	of	of	ADP
iajs-2148	310	14	,	,	PUNCT
iajs-2148	310	15	with	with	ADP
iajs-2148	310	16	.	.	PUNCT
iajs-2148	311	1	then	then	ADV
iajs-2148	311	2	is	be	AUX
iajs-2148	311	3	an	an	DET
iajs-2148	311	4	app	app	ADJ
iajs-2148	311	5	-	-	PUNCT
iajs-2148	311	6	prime	prime	NOUN
iajs-2148	311	7	submodule	submodule	NOUN
iajs-2148	311	8	this	this	DET
iajs-2148	311	9	result	result	NOUN
iajs-2148	311	10	is	be	AUX
iajs-2148	311	11	true	true	ADJ
iajs-2148	311	12	if	if	SCONJ
iajs-2148	311	13	faithful	faithful	ADJ
iajs-2148	311	14	multiplication	multiplication	NOUN
iajs-2148	311	15	(	(	PUNCT
iajs-2148	311	16	finitely	finitely	ADV
iajs-2148	311	17	generated	generate	VERB
iajs-2148	311	18	multiplication	multiplication	NOUN
iajs-2148	311	19	non	non	ADJ
iajs-2148	311	20	-	-	ADJ
iajs-2148	311	21	singular	singular	ADJ
iajs-2148	311	22	)	)	PUNCT
iajs-2148	311	23	see	see	VERB
iajs-2148	311	24	proposition	proposition	NOUN
iajs-2148	311	25	(	(	PUNCT
iajs-2148	311	26	20	20	NUM
iajs-2148	311	27	)	)	PUNCT
iajs-2148	311	28	,	,	PUNCT
iajs-2148	311	29	proposition	proposition	NOUN
iajs-2148	311	30	(	(	PUNCT
iajs-2148	311	31	21	21	NUM
iajs-2148	311	32	)	)	PUNCT
iajs-2148	311	33	.	.	PUNCT
iajs-2148	312	1	6	6	X
iajs-2148	312	2	)	)	PUNCT
iajs-2148	312	3	if	if	SCONJ
iajs-2148	312	4	[	[	X
iajs-2148	312	5	]	]	X
iajs-2148	312	6	[	[	PUNCT
iajs-2148	312	7	]	]	X
iajs-2148	312	8	for	for	ADP
iajs-2148	312	9	each	each	DET
iajs-2148	312	10	submodule	submodule	NOUN
iajs-2148	312	11	of	of	ADP
iajs-2148	312	12	with	with	ADP
iajs-2148	312	13	and	and	CCONJ
iajs-2148	312	14	.then	.then	X
iajs-2148	312	15	is	be	AUX
iajs-2148	312	16	an	an	DET
iajs-2148	312	17	app	app	ADJ
iajs-2148	312	18	-	-	PUNCT
iajs-2148	312	19	prime	prime	NOUN
iajs-2148	312	20	.	.	PUNCT
iajs-2148	313	1	7	7	X
iajs-2148	313	2	)	)	PUNCT
iajs-2148	313	3	the	the	DET
iajs-2148	313	4	invers	inver	NOUN
iajs-2148	313	5	image	image	NOUN
iajs-2148	313	6	and	and	CCONJ
iajs-2148	313	7	homomorphic	homomorphic	ADJ
iajs-2148	313	8	image	image	NOUN
iajs-2148	313	9	of	of	ADP
iajs-2148	313	10	an	an	DET
iajs-2148	313	11	app	app	ADJ
iajs-2148	313	12	-	-	PUNCT
iajs-2148	313	13	prime	prime	NOUN
iajs-2148	313	14	submodule	submodule	NOUN
iajs-2148	313	15	is	be	AUX
iajs-2148	313	16	an	an	DET
iajs-2148	313	17	app	app	ADJ
iajs-2148	313	18	-	-	PUNCT
iajs-2148	313	19	prime	prime	NOUN
iajs-2148	313	20	submodule	submodule	NOUN
iajs-2148	313	21	see	see	VERB
iajs-2148	313	22	proposition	proposition	NOUN
iajs-2148	313	23	(	(	PUNCT
iajs-2148	313	24	30	30	NUM
iajs-2148	313	25	)	)	PUNCT
iajs-2148	313	26	,	,	PUNCT
iajs-2148	313	27	proposition	proposition	NOUN
iajs-2148	313	28	(	(	PUNCT
iajs-2148	313	29	31	31	NUM
iajs-2148	313	30	)	)	PUNCT
iajs-2148	313	31	.	.	PUNCT
iajs-2148	314	1	8)	8)	NUM
iajs-2148	314	2	we	we	PRON
iajs-2148	314	3	introduced	introduce	VERB
iajs-2148	314	4	and	and	CCONJ
iajs-2148	314	5	studied	study	VERB
iajs-2148	314	6	and	and	CCONJ
iajs-2148	314	7	state	state	NOUN
iajs-2148	314	8	several	several	ADJ
iajs-2148	314	9	basic	basic	ADJ
iajs-2148	314	10	properties	property	NOUN
iajs-2148	314	11	of	of	ADP
iajs-2148	314	12	this	this	DET
iajs-2148	314	13	notion	notion	NOUN
iajs-2148	314	14	for	for	ADP
iajs-2148	314	15	example	example	NOUN
iajs-2148	314	16	see	see	VERB
iajs-2148	314	17	proposition	proposition	NOUN
iajs-2148	314	18	(	(	PUNCT
iajs-2148	314	19	36	36	NUM
iajs-2148	314	20	)	)	PUNCT
iajs-2148	314	21	,	,	PUNCT
iajs-2148	314	22	proposition	proposition	NOUN
iajs-2148	314	23	(	(	PUNCT
iajs-2148	314	24	37	37	NUM
iajs-2148	314	25	)	)	PUNCT
iajs-2148	314	26	and	and	CCONJ
iajs-2148	314	27	proposition	proposition	NOUN
iajs-2148	314	28	(	(	PUNCT
iajs-2148	314	29	38	38	NUM
iajs-2148	314	30	)	)	PUNCT
iajs-2148	314	31	.	.	PUNCT
iajs-2148	315	1	references	reference	NOUN
iajs-2148	315	2	1	1	NUM
iajs-2148	315	3	.	.	PUNCT
iajs-2148	315	4	dauns	daun	NOUN
iajs-2148	315	5	,	,	PUNCT
iajs-2148	315	6	j.	j.	PROPN
iajs-2148	315	7	prime	prime	PROPN
iajs-2148	315	8	modules	modules	PROPN
iajs-2148	315	9	.	.	PUNCT
iajs-2148	316	1	j.	j.	PROPN
iajs-2148	316	2	reine	reine	PROPN
iajs-2148	316	3	angew	angew	PROPN
iajs-2148	316	4	,	,	PUNCT
iajs-2148	316	5	math.1978	math.1978	PROPN
iajs-2148	316	6	,	,	PUNCT
iajs-2148	316	7	2	2	NUM
iajs-2148	316	8	,	,	PUNCT
iajs-2148	316	9	156	156	NUM
iajs-2148	316	10	-	-	SYM
iajs-2148	316	11	181	181	NUM
iajs-2148	316	12	.	.	PUNCT
iajs-2148	317	1	2	2	NUM
iajs-2148	317	2	.	.	X
iajs-2148	317	3	lu	lu	PROPN
iajs-2148	317	4	,	,	PUNCT
iajs-2148	317	5	c.p	c.p	PROPN
iajs-2148	317	6	.	.	PROPN
iajs-2148	317	7	prime	prime	ADJ
iajs-2148	317	8	submodules	submodule	NOUN
iajs-2148	317	9	of	of	ADP
iajs-2148	317	10	modules	module	NOUN
iajs-2148	317	11	.	.	PUNCT
iajs-2148	318	1	comm	comm	NOUN
iajs-2148	318	2	.	.	PUNCT
iajs-2148	319	1	math	math	NOUN
iajs-2148	319	2	.	.	PUNCT
iajs-2148	320	1	univ	univ	PROPN
iajs-2148	320	2	.	.	PUNCT
iajs-2148	321	1	sancti	sancti	PROPN
iajs-2148	321	2	pauli.1984	pauli.1984	PROPN
iajs-2148	321	3	,	,	PUNCT
iajs-2148	321	4	33	33	NUM
iajs-2148	321	5	,	,	PUNCT
iajs-2148	321	6	6169	6169	NUM
iajs-2148	321	7	.	.	PUNCT
iajs-2148	322	1	3	3	X
iajs-2148	322	2	.	.	X
iajs-2148	322	3	mccasland	mccasland	PROPN
iajs-2148	322	4	,	,	PUNCT
iajs-2148	322	5	r.l	r.l	PROPN
iajs-2148	322	6	.	.	PROPN
iajs-2148	322	7	;	;	PUNCT
iajs-2148	322	8	smith	smith	PROPN
iajs-2148	322	9	,	,	PUNCT
iajs-2148	322	10	p.f	p.f	PROPN
iajs-2148	322	11	.	.	PROPN
iajs-2148	322	12	prime	prime	ADJ
iajs-2148	322	13	submodules	submodule	NOUN
iajs-2148	322	14	of	of	ADP
iajs-2148	322	15	noetherian	noetherian	ADJ
iajs-2148	322	16	modules	module	NOUN
iajs-2148	322	17	,	,	PUNCT
iajs-2148	322	18	rocky	rocky	ADJ
iajs-2148	322	19	mountain	mountain	NOUN
iajs-2148	322	20	.	.	PUNCT
iajs-2148	323	1	j.	j.	PROPN
iajs-2148	323	2	of	of	ADP
iajs-2148	323	3	math	math	PROPN
iajs-2148	323	4	.	.	PUNCT
iajs-2148	324	1	1993	1993	NUM
iajs-2148	324	2	,	,	PUNCT
iajs-2148	324	3	23	23	NUM
iajs-2148	324	4	,	,	PUNCT
iajs-2148	324	5	3	3	NUM
iajs-2148	324	6	,	,	PUNCT
iajs-2148	324	7	1041	1041	NUM
iajs-2148	324	8	-	-	SYM
iajs-2148	324	9	1062	1062	NUM
iajs-2148	324	10	.	.	PUNCT
iajs-2148	325	1	4	4	X
iajs-2148	325	2	.	.	X
iajs-2148	325	3	mccasland	mccasland	PROPN
iajs-2148	325	4	,	,	PUNCT
iajs-2148	325	5	r.l	r.l	PROPN
iajs-2148	325	6	.	.	PROPN
iajs-2148	325	7	;	;	PUNCT
iajs-2148	325	8	moore	moore	PROPN
iajs-2148	325	9	,	,	PUNCT
iajs-2148	325	10	m.c	m.c	PROPN
iajs-2148	325	11	.	.	PROPN
iajs-2148	325	12	;	;	PUNCT
iajs-2148	325	13	smith	smith	PROPN
iajs-2148	325	14	,	,	PUNCT
iajs-2148	325	15	p.f	p.f	PROPN
iajs-2148	325	16	.	.	PROPN
iajs-2148	325	17	on	on	ADP
iajs-2148	325	18	spectrum	spectrum	NOUN
iajs-2148	325	19	of	of	ADP
iajs-2148	325	20	modules	module	NOUN
iajs-2148	325	21	over	over	ADP
iajs-2148	325	22	commutative	commutative	ADJ
iajs-2148	325	23	rings	ring	NOUN
iajs-2148	325	24	.	.	PUNCT
iajs-2148	326	1	comm	comm	NOUN
iajs-2148	326	2	.	.	PUNCT
iajs-2148	327	1	algebra.1997	algebra.1997	PROPN
iajs-2148	327	2	,	,	PUNCT
iajs-2148	327	3	25	25	NUM
iajs-2148	327	4	,	,	PUNCT
iajs-2148	327	5	1	1	NUM
iajs-2148	327	6	,	,	PUNCT
iajs-2148	327	7	79	79	NUM
iajs-2148	327	8	-	-	SYM
iajs-2148	327	9	103	103	NUM
iajs-2148	327	10	.	.	PUNCT
iajs-2148	328	1	5	5	NUM
iajs-2148	328	2	.	.	X
iajs-2148	328	3	moore	moore	PROPN
iajs-2148	328	4	,	,	PUNCT
iajs-2148	328	5	m.e	m.e	PROPN
iajs-2148	328	6	.	.	PROPN
iajs-2148	328	7	;	;	PUNCT
iajs-2148	329	1	smith	smith	PROPN
iajs-2148	329	2	,	,	PUNCT
iajs-2148	329	3	s.j	s.j	PROPN
iajs-2148	329	4	.	.	PROPN
iajs-2148	329	5	prime	prime	ADJ
iajs-2148	329	6	and	and	CCONJ
iajs-2148	329	7	radical	radical	ADJ
iajs-2148	329	8	submodules	submodule	NOUN
iajs-2148	329	9	over	over	ADP
iajs-2148	329	10	commutative	commutative	ADJ
iajs-2148	329	11	rings	ring	NOUN
iajs-2148	329	12	.	.	PUNCT
iajs-2148	330	1	comm	comm	NOUN
iajs-2148	330	2	.	.	PUNCT
iajs-2148	331	1	algebra	algebra	PROPN
iajs-2148	331	2	.	.	PUNCT
iajs-2148	332	1	2002	2002	NUM
iajs-2148	332	2	,	,	PUNCT
iajs-2148	332	3	30	30	NUM
iajs-2148	332	4	,	,	PUNCT
iajs-2148	332	5	10	10	NUM
iajs-2148	332	6	,	,	PUNCT
iajs-2148	332	7	5037	5037	NUM
iajs-2148	332	8	-	-	SYM
iajs-2148	332	9	5064	5064	NUM
iajs-2148	332	10	.	.	PUNCT
iajs-2148	333	1	6	6	NUM
iajs-2148	333	2	.	.	X
iajs-2148	333	3	ebrahimi	ebrahimi	PROPN
iajs-2148	333	4	,	,	PUNCT
iajs-2148	333	5	s.a	s.a	PROPN
iajs-2148	333	6	.	.	PROPN
iajs-2148	333	7	;	;	PUNCT
iajs-2148	333	8	esmaeili	esmaeili	NOUN
iajs-2148	333	9	,	,	PUNCT
iajs-2148	333	10	k.s	k.s	PROPN
iajs-2148	333	11	.	.	PROPN
iajs-2148	334	1	on	on	ADP
iajs-2148	334	2	mccasland	mccasland	PROPN
iajs-2148	334	3	submodules	submodule	NOUN
iajs-2148	334	4	.	.	PUNCT
iajs-2148	335	1	international	international	ADJ
iajs-2148	335	2	math	math	NOUN
iajs-2148	335	3	.	.	PUNCT
iajs-2148	336	1	forum	forum	PROPN
iajs-2148	336	2	.	.	PROPN
iajs-2148	337	1	2007	2007	NUM
iajs-2148	337	2	,	,	PUNCT
iajs-2148	337	3	2	2	NUM
iajs-2148	337	4	,	,	PUNCT
iajs-2148	337	5	46	46	NUM
iajs-2148	337	6	,	,	PUNCT
iajs-2148	337	7	2255	2255	NUM
iajs-2148	337	8	-	-	SYM
iajs-2148	337	9	2260	2260	NUM
iajs-2148	337	10	.	.	PUNCT
iajs-2148	338	1	7	7	X
iajs-2148	338	2	.	.	X
iajs-2148	338	3	hussin	hussin	PROPN
iajs-2148	338	4	,	,	PUNCT
iajs-2148	338	5	s.a.we	s.a.we	X
iajs-2148	338	6	-	-	PUNCT
iajs-2148	338	7	prime	prime	NOUN
iajs-2148	338	8	submodules	submodule	NOUN
iajs-2148	338	9	and	and	CCONJ
iajs-2148	338	10	we	we	PRON
iajs-2148	338	11	-	-	PUNCT
iajs-2148	338	12	semi	semi	ADJ
iajs-2148	338	13	-	-	ADJ
iajs-2148	338	14	prime	prime	ADJ
iajs-2148	338	15	submodules	submodule	NOUN
iajs-2148	338	16	.	.	PUNCT
iajs-2148	339	1	ibn	ibn	PROPN
iajs-2148	339	2	-	-	PUNCT
iajs-2148	339	3	al	al	PROPN
iajs-2148	339	4	-	-	PUNCT
iajs-2148	339	5	haitham	haitham	PROPN
iajs-2148	339	6	journal	journal	PROPN
iajs-2148	339	7	for	for	ADP
iajs-2148	339	8	pure	pure	ADJ
iajs-2148	339	9	and	and	CCONJ
iajs-2148	339	10	applied	applied	ADJ
iajs-2148	339	11	science	science	NOUN
iajs-2148	339	12	.	.	PUNCT
iajs-2148	340	1	2018	2018	NUM
iajs-2148	340	2	,	,	PUNCT
iajs-2148	340	3	3	3	NUM
iajs-2148	340	4	,	,	PUNCT
iajs-2148	340	5	3	3	NUM
iajs-2148	340	6	,	,	PUNCT
iajs-2148	340	7	109	109	NUM
iajs-2148	340	8	-	-	SYM
iajs-2148	340	9	117	117	NUM
iajs-2148	340	10	.	.	NOUN
iajs-2148	340	11	8	8	NUM
iajs-2148	340	12	.	.	X
iajs-2148	341	1	hussin	hussin	PROPN
iajs-2148	341	2	,	,	PUNCT
iajs-2148	341	3	w.a	w.a	PROPN
iajs-2148	341	4	.	.	PROPN
iajs-2148	341	5	wn-2	wn-2	NOUN
iajs-2148	341	6	-	-	ADJ
iajs-2148	341	7	absorbing	absorbing	ADJ
iajs-2148	341	8	and	and	CCONJ
iajs-2148	341	9	wes2	wes2	NOUN
iajs-2148	341	10	-	-	PUNCT
iajs-2148	341	11	absorbing	absorbing	NOUN
iajs-2148	341	12	submoduled	submodule	VERB
iajs-2148	341	13	.	.	PUNCT
iajs-2148	342	1	ibn	ibn	PROPN
iajs-2148	342	2	-	-	PUNCT
iajs-2148	342	3	al	al	PROPN
iajs-2148	342	4	-	-	PUNCT
iajs-2148	342	5	haitham	haitham	PROPN
iajs-2148	342	6	jornal	jornal	NOUN
iajs-2148	342	7	for	for	ADP
iajs-2148	342	8	pure	pure	ADJ
iajs-2148	342	9	and	and	CCONJ
iajs-2148	342	10	applied	apply	VERB
iajs-2148	342	11	science.2018	science.2018	PROPN
iajs-2148	342	12	,	,	PUNCT
iajs-2148	342	13	31	31	NUM
iajs-2148	342	14	,	,	PUNCT
iajs-2148	342	15	3	3	NUM
iajs-2148	342	16	,	,	PUNCT
iajs-2148	342	17	118	118	NUM
iajs-2148	342	18	-	-	SYM
iajs-2148	342	19	125	125	NUM
iajs-2148	342	20	.	.	NOUN
iajs-2148	343	1	9	9	NUM
iajs-2148	343	2	.	.	X
iajs-2148	343	3	gooderal	gooderal	ADJ
iajs-2148	343	4	,	,	PUNCT
iajs-2148	343	5	k.r	k.r	PROPN
iajs-2148	343	6	.	.	PROPN
iajs-2148	343	7	ring	ring	PROPN
iajs-2148	343	8	theory	theory	NOUN
iajs-2148	343	9	,	,	PUNCT
iajs-2148	343	10	nonsingular	nonsingular	ADJ
iajs-2148	343	11	ring	ring	NOUN
iajs-2148	343	12	and	and	CCONJ
iajs-2148	343	13	modules	module	NOUN
iajs-2148	343	14	.	.	PUNCT
iajs-2148	344	1	marcel	marcel	PROPN
iajs-2148	344	2	.	.	PUNCT
iajs-2148	344	3	dekker	dekker	PROPN
iajs-2148	344	4	.	.	PUNCT
iajs-2148	345	1	new	new	PROPN
iajs-2148	345	2	york	york	PROPN
iajs-2148	345	3	.	.	PUNCT
iajs-2148	345	4	1976	1976	NUM
iajs-2148	345	5	.	.	PUNCT
iajs-2148	346	1	10	10	NUM
iajs-2148	346	2	.	.	X
iajs-2148	347	1	abd	abd	PROPN
iajs-2148	347	2	el	el	PROPN
iajs-2148	347	3	-	-	PUNCT
iajs-2148	347	4	bast	bast	NOUN
iajs-2148	347	5	,	,	PUNCT
iajs-2148	347	6	z.	z.	PROPN
iajs-2148	347	7	;	;	PUNCT
iajs-2148	347	8	smith	smith	PROPN
iajs-2148	347	9	,	,	PUNCT
iajs-2148	347	10	p.f	p.f	PROPN
iajs-2148	347	11	.	.	PROPN
iajs-2148	347	12	multiplication	multiplication	NOUN
iajs-2148	347	13	modules	module	NOUN
iajs-2148	347	14	.	.	PUNCT
iajs-2148	348	1	comm	comm	NOUN
iajs-2148	348	2	.	.	PUNCT
iajs-2148	349	1	algebra.1988	algebra.1988	PROPN
iajs-2148	349	2	,	,	PUNCT
iajs-2148	349	3	16	16	NUM
iajs-2148	349	4	,	,	PUNCT
iajs-2148	349	5	4	4	NUM
iajs-2148	349	6	,	,	PUNCT
iajs-2148	349	7	755779	755779	NUM
iajs-2148	349	8	.	.	PUNCT
iajs-2148	350	1	11	11	NUM
iajs-2148	350	2	.	.	X
iajs-2148	351	1	larsen	larsen	PROPN
iajs-2148	351	2	,	,	PUNCT
iajs-2148	351	3	m.d	m.d	PROPN
iajs-2148	351	4	.	.	PROPN
iajs-2148	351	5	;	;	PUNCT
iajs-2148	351	6	mccarthy	mccarthy	PROPN
iajs-2148	351	7	,	,	PUNCT
iajs-2148	351	8	p.j	p.j	PROPN
iajs-2148	351	9	.	.	PROPN
iajs-2148	351	10	multiplicative	multiplicative	PROPN
iajs-2148	351	11	theory	theory	NOUN
iajs-2148	351	12	of	of	ADP
iajs-2148	351	13	ideals	ideal	NOUN
iajs-2148	351	14	.	.	PUNCT
iajs-2148	352	1	academic	academic	ADJ
iajs-2148	352	2	press	press	NOUN
iajs-2148	352	3	.	.	PUNCT
iajs-2148	353	1	new	new	PROPN
iajs-2148	353	2	york	york	PROPN
iajs-2148	353	3	and	and	CCONJ
iajs-2148	353	4	london	london	PROPN
iajs-2148	353	5	.	.	PUNCT
iajs-2148	354	1	1971	1971	NUM
iajs-2148	354	2	12	12	NUM
iajs-2148	354	3	.	.	PUNCT
iajs-2148	354	4	zalmanowiz	zalmanowiz	NOUN
iajs-2148	354	5	,	,	PUNCT
iajs-2148	354	6	j.	j.	PROPN
iajs-2148	354	7	dense	dense	ADJ
iajs-2148	354	8	rings	ring	NOUN
iajs-2148	354	9	of	of	ADP
iajs-2148	354	10	liner	liner	NOUN
iajs-2148	354	11	transformations	transformation	NOUN
iajs-2148	354	12	.	.	PUNCT
iajs-2148	355	1	ring	ring	NOUN
iajs-2148	355	2	theory	theory	PROPN
iajs-2148	355	3	ii	ii	PROPN
iajs-2148	355	4	.	.	PUNCT
iajs-2148	356	1	marcel	marcel	PROPN
iajs-2148	356	2	dekker	dekker	PROPN
iajs-2148	356	3	new	new	PROPN
iajs-2148	356	4	york	york	PROPN
iajs-2148	356	5	.	.	PUNCT
iajs-2148	357	1	1997	1997	NUM
iajs-2148	357	2	13	13	NUM
iajs-2148	357	3	.	.	PUNCT
iajs-2148	358	1	anderson	anderson	PROPN
iajs-2148	358	2	,	,	PUNCT
iajs-2148	358	3	f.w	f.w	PROPN
iajs-2148	358	4	.	.	PROPN
iajs-2148	358	5	;	;	PUNCT
iajs-2148	358	6	fuller	full	ADJ
iajs-2148	358	7	,	,	PUNCT
iajs-2148	358	8	k.r	k.r	PROPN
iajs-2148	358	9	.	.	PROPN
iajs-2148	358	10	rings	ring	NOUN
iajs-2148	358	11	and	and	CCONJ
iajs-2148	358	12	categories	category	NOUN
iajs-2148	358	13	of	of	ADP
iajs-2148	358	14	modules	module	NOUN
iajs-2148	358	15	.	.	PUNCT
iajs-2148	359	1	springer	springer	NOUN
iajs-2148	359	2	-	-	PUNCT
iajs-2148	359	3	verlag	verlag	PROPN
iajs-2148	359	4	.	.	PUNCT
iajs-2148	360	1	new	new	PROPN
iajs-2148	360	2	york	york	PROPN
iajs-2148	360	3	.	.	PUNCT
iajs-2148	361	1	1992	1992	NUM
iajs-2148	361	2	14	14	NUM
iajs-2148	361	3	.	.	PUNCT
iajs-2148	361	4	athab	athab	PROPN
iajs-2148	361	5	,	,	PUNCT
iajs-2148	361	6	e.a	e.a	PROPN
iajs-2148	361	7	.	.	PROPN
iajs-2148	361	8	prime	prime	PROPN
iajs-2148	361	9	and	and	CCONJ
iajs-2148	361	10	semi	semi	ADJ
iajs-2148	361	11	prime	prime	ADJ
iajs-2148	361	12	submodules	submodule	NOUN
iajs-2148	361	13	.	.	PUNCT
iajs-2148	362	1	m.sc	m.sc	NOUN
iajs-2148	362	2	.	.	PUNCT
iajs-2148	363	1	thesis	thesis	NOUN
iajs-2148	363	2	.	.	PUNCT
iajs-2148	364	1	college	college	NOUN
iajs-2148	364	2	of	of	ADP
iajs-2148	364	3	science	science	NOUN
iajs-2148	364	4	,	,	PUNCT
iajs-2148	364	5	university	university	NOUN
iajs-2148	364	6	of	of	ADP
iajs-2148	364	7	baghdad	baghdad	PROPN
iajs-2148	364	8	.	.	PUNCT
iajs-2148	365	1	1996	1996	NUM
iajs-2148	365	2	15	15	NUM
iajs-2148	365	3	.	.	PUNCT
iajs-2148	366	1	kasch	kasch	PROPN
iajs-2148	366	2	,	,	PUNCT
iajs-2148	366	3	f.	f.	PROPN
iajs-2148	366	4	modules	module	NOUN
iajs-2148	366	5	and	and	CCONJ
iajs-2148	366	6	rings	ring	NOUN
iajs-2148	366	7	.	.	PUNCT
iajs-2148	367	1	london	london	PROPN
iajs-2148	367	2	math	math	PROPN
iajs-2148	367	3	.	.	PUNCT
iajs-2148	368	1	soc	soc	PROPN
iajs-2148	368	2	.	.	PUNCT
iajs-2148	369	1	monographs	monograph	NOUN
iajs-2148	369	2	.	.	PUNCT
iajs-2148	370	1	new	new	PROPN
iajs-2148	370	2	york	york	PROPN
iajs-2148	370	3	.	.	PUNCT
iajs-2148	371	1	1982	1982	NUM