id	sid	tid	token	lemma	pos
iajs-2001	1	1	microsoft	microsoft	PROPN
iajs-2001	1	2	word	word	NOUN
iajs-2001	1	3	118	118	NUM
iajs-2001	1	4	-	-	SYM
iajs-2001	1	5	125	125	NUM
iajs-2001	1	6	mathematics	mathematic	NOUN
iajs-2001	1	7	|	|	ADV
iajs-2001	1	8	118	118	NUM
iajs-2001	1	9	ibn	ibn	PROPN
iajs-2001	1	10	al	al	PROPN
iajs-2001	1	11	-	-	PUNCT
iajs-2001	1	12	haitham	haitham	PROPN
iajs-2001	1	13	jour	jour	X
iajs-2001	1	14	.	.	PROPN
iajs-2001	2	1	for	for	ADP
iajs-2001	2	2	pure	pure	ADJ
iajs-2001	2	3	&	&	CCONJ
iajs-2001	2	4	appl	appl	PROPN
iajs-2001	2	5	.	.	PUNCT
iajs-2001	3	1	sci	sci	PROPN
iajs-2001	3	2	.	.	PROPN
iajs-2001	3	3	ihjpas	ihjpa	VERB
iajs-2001	3	4	https://doi.org/10.30526/31.3.2001	https://doi.org/10.30526/31.3.2001	PROPN
iajs-2001	3	5	vol	vol	NOUN
iajs-2001	3	6	.	.	PROPN
iajs-2001	4	1	31	31	NUM
iajs-2001	5	1	(	(	PUNCT
iajs-2001	5	2	3	3	NUM
iajs-2001	5	3	)	)	PUNCT
iajs-2001	5	4	2018	2018	NUM
iajs-2001	5	5	wn-2	wn-2	ADP
iajs-2001	5	6	-	-	PUNCT
iajs-2001	5	7	absorbing	absorbing	ADJ
iajs-2001	5	8	submodules	submodule	NOUN
iajs-2001	5	9	and	and	CCONJ
iajs-2001	5	10	wns-2	wns-2	NOUN
iajs-2001	5	11	-	-	PUNCT
iajs-2001	5	12	absorbing	absorb	VERB
iajs-2001	5	13	submodules	submodule	NOUN
iajs-2001	6	1	wissam	wissam	PROPN
iajs-2001	6	2	a.	a.	PROPN
iajs-2001	6	3	hussain	hussain	PROPN
iajs-2001	6	4	departmentt	departmentt	PROPN
iajs-2001	6	5	of"mathematics	of"mathematics	PROPN
iajs-2001	6	6	,	,	PUNCT
iajs-2001	6	7	college	college	NOUN
iajs-2001	6	8	of	of	ADP
iajs-2001	6	9	education	education	NOUN
iajs-2001	6	10	for	for	ADP
iajs-2001	6	11	pure	pure	ADJ
iajs-2001	6	12	science	science	NOUN
iajs-2001	6	13	,	,	PUNCT
iajs-2001	6	14	tikrit"university	tikrit"university	NOUN
iajs-2001	6	15	,	,	PUNCT
iajs-2001	6	16	iraq	iraq	PROPN
iajs-2001	6	17	wissam.abbas1987@gmail.com	wissam.abbas1987@gmail.com	NUM
iajs-2001	6	18	'	'	PUNCT
iajs-2001	6	19	haibt	haibt	PROPN
iajs-2001	6	20	k.	k.	PROPN
iajs-2001	6	21	mohammd	mohammd	PROPN
iajs-2001	6	22	ali	ali	PROPN
iajs-2001	6	23	'	'	PROPN
iajs-2001	6	24	department	department	PROPN
iajs-2001	6	25	of"mathematics	of"mathematics	PROPN
iajs-2001	6	26	,	,	PUNCT
iajs-2001	6	27	college"of'"computer	college"of'"computer	PROPN
iajs-2001	6	28	'	'	PUNCT
iajs-2001	6	29	science"and"mathematics	science"and"mathematic	NOUN
iajs-2001	6	30	,	,	PUNCT
iajs-2001	6	31	tikrit"university	tikrit"university	NOUN
iajs-2001	6	32	,	,	PUNCT
iajs-2001	6	33	iraq'/	iraq'/	X
iajs-2001	6	34	///	///	NOUN
iajs-2001	6	35	'	'	PUNCT
iajs-2001	6	36	article	article	NOUN
iajs-2001	6	37	history	history	NOUN
iajs-2001	6	38	:	:	PUNCT
iajs-2001	6	39	received	receive	VERB
iajs-2001	6	40	30	30	NUM
iajs-2001	6	41	july	july	PROPN
iajs-2001	6	42	2018	2018	NUM
iajs-2001	6	43	,	,	PUNCT
iajs-2001	6	44	accepted	accept	VERB
iajs-2001	6	45	3	3	NUM
iajs-2001	6	46	september	september	PROPN
iajs-2001	6	47	2018	2018	NUM
iajs-2001	6	48	,	,	PUNCT
iajs-2001	6	49	published	publish	VERB
iajs-2001	6	50	december	december	PROPN
iajs-2001	6	51	2018	2018	NUM
iajs-2001	6	52	"	"	PUNCT
iajs-2001	6	53	"	"	PUNCT
iajs-2001	6	54	abstract	abstract	ADJ
iajs-2001	6	55	"	"	PUNCT
iajs-2001	6	56	"	"	PUNCT
iajs-2001	6	57	"	"	PUNCT
iajs-2001	6	58	in''this"article	in''this"article	PROPN
iajs-2001	6	59	,	,	PUNCT
iajs-2001	6	60	we"study	we"study	PROPN
iajs-2001	6	61	,	,	PUNCT
iajs-2001	6	62	the"concept"of	the"concept"of	PROPN
iajs-2001	6	63	wn"-"2"-''absorbing'''submodules	wn"-"2"-''absorbing'''submodule	NOUN
iajs-2001	6	64	and	and	CCONJ
iajs-2001	6	65	wns''''2''-''absorbing"submodules	wns''''2''-''absorbing"submodule	NOUN
iajs-2001	6	66	as	as	ADP
iajs-2001	6	67	generalization	generalization	NOUN
iajs-2001	6	68	of	of	ADP
iajs-2001	6	69	weakly	weakly	ADJ
iajs-2001	6	70	2	2	NUM
iajs-2001	6	71	-	-	PUNCT
iajs-2001	6	72	absorbing	absorbing	ADJ
iajs-2001	6	73	and	and	CCONJ
iajs-2001	6	74	weakly	weakly	ADJ
iajs-2001	6	75	semi	semi	ADJ
iajs-2001	6	76	2absorbing	2absorbing	NOUN
iajs-2001	6	77	submodules	submodule	NOUN
iajs-2001	6	78	respectively	respectively	ADV
iajs-2001	6	79	.	.	PUNCT
iajs-2001	7	1	we	we	PRON
iajs-2001	7	2	investigate	investigate	VERB
iajs-2001	7	3	some	some	PRON
iajs-2001	7	4	of	of	ADP
iajs-2001	7	5	basic	basic	ADJ
iajs-2001	7	6	properties	property	NOUN
iajs-2001	7	7	,	,	PUNCT
iajs-2001	7	8	examples	example	NOUN
iajs-2001	7	9	and	and	CCONJ
iajs-2001	7	10	characterizations	characterization	NOUN
iajs-2001	7	11	of	of	ADP
iajs-2001	7	12	them	they	PRON
iajs-2001	7	13	.	.	PUNCT
iajs-2001	8	1	also	also	ADV
iajs-2001	8	2	,	,	PUNCT
iajs-2001	8	3	prove	prove	VERB
iajs-2001	8	4	,	,	PUNCT
iajs-2001	8	5	the	the	DET
iajs-2001	8	6	class	class	NOUN
iajs-2001	8	7	of	of	ADP
iajs-2001	8	8	wn-2	wn-2	NOUN
iajs-2001	8	9	-	-	ADJ
iajs-2001	8	10	absorbing	absorbing	ADJ
iajs-2001	8	11	"	"	PUNCT
iajs-2001	8	12	submodules	submodule	NOUN
iajs-2001	8	13	is	be	AUX
iajs-2001	8	14	contained	contain	VERB
iajs-2001	8	15	in	in	ADP
iajs-2001	8	16	the	the	DET
iajs-2001	8	17	class	class	NOUN
iajs-2001	8	18	of	of	ADP
iajs-2001	8	19	wns-2	wns-2	NOUN
iajs-2001	8	20	-	-	PUNCT
iajs-2001	8	21	absorbing	absorbing	ADJ
iajs-2001	8	22	"	"	PUNCT
iajs-2001	8	23	submodules	submodule	NOUN
iajs-2001	8	24	.	.	PUNCT
iajs-2001	9	1	moreover	moreover	ADV
iajs-2001	9	2	,	,	PUNCT
iajs-2001	9	3	many	many	ADJ
iajs-2001	9	4	interesting	interesting	ADJ
iajs-2001	9	5	results	result	NOUN
iajs-2001	9	6	about	about	ADP
iajs-2001	9	7	these	these	DET
iajs-2001	9	8	concepts	concept	NOUN
iajs-2001	9	9	,	,	PUNCT
iajs-2001	9	10	were	be	AUX
iajs-2001	9	11	proven	prove	VERB
iajs-2001	9	12	.	.	PUNCT
iajs-2001	10	1	keywords:"wn-2	keywords:"wn-2	NOUN
iajs-2001	10	2	-	-	PUNCT
iajs-2001	10	3	absorbing	absorb	VERB
iajs-2001	10	4	submodules	submodule	NOUN
iajs-2001	10	5	,	,	PUNCT
iajs-2001	10	6	wns-2	wns-2	NOUN
iajs-2001	10	7	-	-	PUNCT
iajs-2001	10	8	absorbing	absorb	VERB
iajs-2001	10	9	submodules	submodule	NOUN
iajs-2001	10	10	,	,	PUNCT
iajs-2001	10	11	weakly	weakly	ADJ
iajs-2001	10	12	2absorbing	2absorbing	NUM
iajs-2001	10	13	submodules	submodule	NOUN
iajs-2001	10	14	,	,	PUNCT
iajs-2001	10	15	weakly	weakly	ADJ
iajs-2001	10	16	semi-2	semi-2	NOUN
iajs-2001	10	17	-	-	PUNCT
iajs-2001	10	18	absorbing	absorbing	ADJ
iajs-2001	10	19	submodules	submodule	NOUN
iajs-2001	10	20	.	.	PUNCT
iajs-2001	11	1	1	1	X
iajs-2001	11	2	.	.	X
iajs-2001	11	3	introduction	introduction	NOUN
iajs-2001	11	4	"	"	PUNCT
iajs-2001	11	5	weakly''2''-''absorbing'''submodules'''was"introduced	weakly''2''-''absorbing'''submodules'''was"introduce	VERB
iajs-2001	11	6	by"darani	by"darani	PROPN
iajs-2001	11	7	and''"soheilinia	and''"soheilinia	PROPN
iajs-2001	11	8	,	,	PUNCT
iajs-2001	11	9	in	in	ADP
iajs-2001	11	10	2011	2011	NUM
iajs-2001	11	11	,	,	PUNCT
iajs-2001	11	12	where	where	SCONJ
iajs-2001	11	13	a''proper''submodule''b	a''proper''submodule''b	NOUN
iajs-2001	11	14	of	of	ADP
iajs-2001	11	15	an"r''-''module"y	an"r''-''module"y	PROPN
iajs-2001	11	16	is''called'''weakly'''"2	is''called'''weakly'''"2	NOUN
iajs-2001	11	17	absorbing''submodule,''if''whenever'''0	absorbing''submodule,''if''whenever'''0	PROPN
iajs-2001	11	18	≠	≠	PROPN
iajs-2001	11	19	aby	aby	PROPN
iajs-2001	11	20	∊	∊	PROPN
iajs-2001	11	21	b	b	PROPN
iajs-2001	11	22	,	,	PUNCT
iajs-2001	11	23	with	with	ADP
iajs-2001	11	24	a	a	DET
iajs-2001	11	25	,	,	PUNCT
iajs-2001	11	26	b	b	PROPN
iajs-2001	11	27	∊	∊	PROPN
iajs-2001	11	28	r	r	NOUN
iajs-2001	11	29	,	,	PUNCT
iajs-2001	11	30	y	y	PROPN
iajs-2001	11	31	∊	∊	PROPN
iajs-2001	11	32	y	y	PROPN
iajs-2001	11	33	,	,	PUNCT
iajs-2001	11	34	implies	imply	VERB
iajs-2001	11	35	that	that	SCONJ
iajs-2001	11	36	either	either	CCONJ
iajs-2001	11	37	ay	ay	PROPN
iajs-2001	11	38	∊	∊	PROPN
iajs-2001	11	39	b	b	PROPN
iajs-2001	11	40	or	or	CCONJ
iajs-2001	11	41	by	by	ADP
iajs-2001	11	42	∊	∊	PROPN
iajs-2001	11	43	b	b	PROPN
iajs-2001	11	44	or	or	CCONJ
iajs-2001	11	45	ab	ab	ADJ
iajs-2001	11	46	∊	∊	PROPN
iajs-2001	12	1	[	[	X
iajs-2001	12	2	b	b	X
iajs-2001	12	3	:	:	PUNCT
iajs-2001	12	4	y	y	NOUN
iajs-2001	12	5	]	]	X
iajs-2001	13	1	[	[	X
iajs-2001	13	2	1	1	NUM
iajs-2001	13	3	]	]	X
iajs-2001	13	4	"	"	PUNCT
iajs-2001	13	5	.	.	PUNCT
iajs-2001	14	1	and	and	CCONJ
iajs-2001	14	2	the	the	DET
iajs-2001	14	3	concept	concept	NOUN
iajs-2001	14	4	of	of	ADP
iajs-2001	14	5	a	a	DET
iajs-2001	14	6	weakly	weakly	ADJ
iajs-2001	14	7	semi	semi	ADJ
iajs-2001	14	8	2	2	NUM
iajs-2001	14	9	-	-	PUNCT
iajs-2001	14	10	absorbing	absorb	VERB
iajs-2001	14	11	submodule	submodule	NOUN
iajs-2001	14	12	was	be	AUX
iajs-2001	14	13	introduce	introduce	VERB
iajs-2001	14	14	by"haibt	by"haibt	NOUN
iajs-2001	14	15	and	and	CCONJ
iajs-2001	14	16	khalaf	khalaf	PROPN
iajs-2001	14	17	in	in	ADP
iajs-2001	14	18	2018	2018	NUM
iajs-2001	14	19	,	,	PUNCT
iajs-2001	14	20	where	where	SCONJ
iajs-2001	14	21	a''proper	a''proper	NOUN
iajs-2001	14	22	''	''	PUNCT
iajs-2001	14	23	submodule''b	submodule''b	PROPN
iajs-2001	14	24	of	of	ADP
iajs-2001	14	25	an"r"'''module"'''y'''is'''called	an"r"'''module"'''y'''is'''calle	VERB
iajs-2001	14	26	a"weakly"semi'''2-"absorbing"submodule	a"weakly"semi'''2-"absorbing"submodule	NOUN
iajs-2001	14	27	,	,	PUNCT
iajs-2001	14	28	if"whenever	if"whenever	NOUN
iajs-2001	14	29	0	0	NUM
iajs-2001	14	30	≠	≠	PROPN
iajs-2001	14	31	a	a	DET
iajs-2001	14	32	y	y	PROPN
iajs-2001	14	33	∊	∊	PROPN
iajs-2001	14	34	b	b	PROPN
iajs-2001	14	35	,	,	PUNCT
iajs-2001	14	36	with	with	ADP
iajs-2001	14	37	a	a	DET
iajs-2001	14	38	∊	∊	PROPN
iajs-2001	14	39	r	r	NOUN
iajs-2001	14	40	,	,	PUNCT
iajs-2001	14	41	y	y	PROPN
iajs-2001	14	42	∊	∊	PROPN
iajs-2001	14	43	y	y	PROPN
iajs-2001	14	44	,	,	PUNCT
iajs-2001	14	45	implies	imply	VERB
iajs-2001	14	46	that	that	SCONJ
iajs-2001	14	47	either	either	CCONJ
iajs-2001	14	48	ay	ay	PROPN
iajs-2001	14	49	∊	∊	PROPN
iajs-2001	14	50	b	b	PROPN
iajs-2001	14	51	or'''a	or'''a	NOUN
iajs-2001	15	1	∊	∊	PROPN
iajs-2001	15	2	[	[	PUNCT
iajs-2001	15	3	b	b	NOUN
iajs-2001	15	4	:	:	PUNCT
iajs-2001	15	5	y	y	X
iajs-2001	15	6	]	]	X
iajs-2001	16	1	[	[	X
iajs-2001	16	2	2	2	NUM
iajs-2001	16	3	]	]	PUNCT
iajs-2001	16	4	.	.	PUNCT
iajs-2001	17	1	"	"	PUNCT
iajs-2001	17	2	these	these	DET
iajs-2001	17	3	two	two	NUM
iajs-2001	17	4	concepts	concept	NOUN
iajs-2001	17	5	are	be	AUX
iajs-2001	17	6	generalized	generalize	VERB
iajs-2001	17	7	in	in	ADP
iajs-2001	17	8	this	this	DET
iajs-2001	17	9	article	article	NOUN
iajs-2001	17	10	,	,	PUNCT
iajs-2001	17	11	to	to	ADP
iajs-2001	17	12	wn-2	wn-2	ADP
iajs-2001	17	13	-	-	PUNCT
iajs-2001	17	14	absorbing	absorbing	ADJ
iajs-2001	17	15	submodules	submodule	NOUN
iajs-2001	17	16	and	and	CCONJ
iajs-2001	17	17	wns-2	wns-2	NOUN
iajs-2001	17	18	-	-	PUNCT
iajs-2001	17	19	absorbing	absorb	VERB
iajs-2001	17	20	submodules	submodule	NOUN
iajs-2001	17	21	,	,	PUNCT
iajs-2001	17	22	we	we	PRON
iajs-2001	17	23	prove	prove	VERB
iajs-2001	17	24	that	that	SCONJ
iajs-2001	17	25	the	the	DET
iajs-2001	17	26	class	class	NOUN
iajs-2001	17	27	of	of	ADP
iajs-2001	17	28	wn-2	wn-2	ADP
iajs-2001	17	29	-	-	ADJ
iajs-2001	17	30	absorbing	absorbing	ADJ
iajs-2001	17	31	submodules	submodule	NOUN
iajs-2001	17	32	is	be	AUX
iajs-2001	17	33	contained	contain	VERB
iajs-2001	17	34	in	in	ADP
iajs-2001	17	35	the	the	DET
iajs-2001	17	36	class	class	NOUN
iajs-2001	17	37	of	of	ADP
iajs-2001	17	38	wns-2	wns-2	NOUN
iajs-2001	17	39	-	-	PUNCT
iajs-2001	17	40	absorbing	absorb	VERB
iajs-2001	17	41	submodules	submodule	NOUN
iajs-2001	17	42	while	while	SCONJ
iajs-2001	17	43	the	the	DET
iajs-2001	17	44	converse	converse	NOUN
iajs-2001	17	45	is	be	AUX
iajs-2001	17	46	not	not	PART
iajs-2001	17	47	true	true	ADJ
iajs-2001	17	48	see	see	VERB
iajs-2001	17	49	example	example	NOUN
iajs-2001	17	50	(	(	PUNCT
iajs-2001	17	51	3.14).''''recall'''that''a''submodule	3.14).''''recall'''that''a''submodule	NUM
iajs-2001	17	52	a	a	PRON
iajs-2001	17	53	of	of	ADP
iajs-2001	17	54	an	an	DET
iajs-2001	17	55	r"-''module	r"-''module	PROPN
iajs-2001	17	56	y	y	PROPN
iajs-2001	17	57	is	be	AUX
iajs-2001	17	58	''	''	PUNCT
iajs-2001	17	59	called'''small	called'''small	PROPN
iajs-2001	17	60	if"for	if"for	PROPN
iajs-2001	17	61	any"submodule	any"submodule	NOUN
iajs-2001	17	62	b	b	PROPN
iajs-2001	17	63	of	of	ADP
iajs-2001	17	64	y	y	PROPN
iajs-2001	17	65	,	,	PUNCT
iajs-2001	17	66	y	y	PROPN
iajs-2001	17	67	=	=	PUNCT
iajs-2001	17	68	a	a	DET
iajs-2001	17	69	+	+	NUM
iajs-2001	17	70	b	b	NOUN
iajs-2001	17	71	,	,	PUNCT
iajs-2001	17	72	implies	imply	VERB
iajs-2001	17	73	that	that	SCONJ
iajs-2001	17	74	a	a	DET
iajs-2001	17	75	=	=	SYM
iajs-2001	17	76	y	y	PROPN
iajs-2001	18	1	[	[	X
iajs-2001	18	2	3	3	NUM
iajs-2001	18	3	]	]	PUNCT
iajs-2001	18	4	.	.	PUNCT
iajs-2001	19	1	recall	recall	VERB
iajs-2001	19	2	that	that	SCONJ
iajs-2001	19	3	an	an	DET
iajs-2001	19	4	r	r	NOUN
iajs-2001	19	5	-	-	PUNCT
iajs-2001	19	6	epimorphism	epimorphism	NOUN
iajs-2001	19	7	f	f	PROPN
iajs-2001	19	8	∶	∶	PROPN
iajs-2001	19	9	y	y	PROPN
iajs-2001	19	10	→	→	PUNCT
iajs-2001	19	11	y	y	PROPN
iajs-2001	19	12	is	be	AUX
iajs-2001	19	13	called	call	VERB
iajs-2001	19	14	small	small	ADJ
iajs-2001	19	15	if	if	SCONJ
iajs-2001	19	16	kerf	kerf	NOUN
iajs-2001	19	17	is	be	AUX
iajs-2001	19	18	a	a	DET
iajs-2001	19	19	small	small	ADJ
iajs-2001	19	20	submodule	submodule	NOUN
iajs-2001	19	21	of	of	ADP
iajs-2001	19	22	y	y	PROPN
iajs-2001	19	23	,	,	PUNCT
iajs-2001	19	24	and	and	CCONJ
iajs-2001	19	25	f	f	AUX
iajs-2001	19	26	ȷ	ȷ	NOUN
iajs-2001	19	27	m	m	VERB
iajs-2001	19	28	ȷ	ȷ	NOUN
iajs-2001	19	29	m	m	NOUN
iajs-2001	19	30	`	`	PUNCT
iajs-2001	19	31	ȷ	ȷ	X
iajs-2001	19	32	f	f	NOUN
iajs-2001	19	33	m	m	VERB
iajs-2001	19	34	and	and	CCONJ
iajs-2001	19	35	ȷ	ȷ	ADP
iajs-2001	19	36	m	m	VERB
iajs-2001	19	37	f	f	NOUN
iajs-2001	20	1	ȷ	ȷ	NOUN
iajs-2001	20	2	m	m	PRON
iajs-2001	20	3	[	[	X
iajs-2001	20	4	3	3	NUM
iajs-2001	20	5	]	]	PUNCT
iajs-2001	20	6	.	.	PUNCT
iajs-2001	21	1	a	a	DET
iajs-2001	21	2	ring	ring	NOUN
iajs-2001	21	3	r	r	NOUN
iajs-2001	21	4	is	be	AUX
iajs-2001	21	5	a	a	DET
iajs-2001	21	6	good	good	ADJ
iajs-2001	21	7	ring	ring	NOUN
iajs-2001	21	8	if	if	SCONJ
iajs-2001	21	9	𝚥(r	𝚥(r	NOUN
iajs-2001	21	10	)	)	PUNCT
iajs-2001	21	11	y	y	PROPN
iajs-2001	21	12	=	=	SYM
iajs-2001	21	13	𝚥(y	𝚥(y	PROPN
iajs-2001	21	14	)	)	PUNCT
iajs-2001	21	15	,	,	PUNCT
iajs-2001	21	16	where	where	SCONJ
iajs-2001	21	17	y	y	PROPN
iajs-2001	21	18	is	be	AUX
iajs-2001	21	19	an	an	DET
iajs-2001	21	20	rmodule	rmodule	NOUN
iajs-2001	21	21	equivalently	equivalently	ADV
iajs-2001	21	22	r	r	NOUN
iajs-2001	21	23	is	be	AUX
iajs-2001	21	24	a	a	DET
iajs-2001	21	25	good	good	ADJ
iajs-2001	21	26	ring	ring	NOUN
iajs-2001	21	27	if	if	SCONJ
iajs-2001	21	28	𝚥(y	𝚥(y	NOUN
iajs-2001	21	29	)	)	PUNCT
iajs-2001	21	30	∩	∩	NOUN
iajs-2001	21	31	a	a	DET
iajs-2001	21	32	=	=	X
iajs-2001	21	33	(	(	PUNCT
iajs-2001	21	34	a	a	NOUN
iajs-2001	21	35	)	)	PUNCT
iajs-2001	21	36	for	for	ADP
iajs-2001	21	37	every	every	DET
iajs-2001	21	38	submodule	submodule	NOUN
iajs-2001	21	39	a	a	PRON
iajs-2001	21	40	of	of	ADP
iajs-2001	21	41	y	y	PROPN
iajs-2001	22	1	[	[	X
iajs-2001	22	2	3	3	NUM
iajs-2001	22	3	]	]	PUNCT
iajs-2001	22	4	.	.	PUNCT
iajs-2001	23	1	if	if	SCONJ
iajs-2001	23	2	y	y	PROPN
iajs-2001	23	3	is	be	AUX
iajs-2001	23	4	an	an	DET
iajs-2001	23	5	r	r	NOUN
iajs-2001	23	6	-	-	PUNCT
iajs-2001	23	7	module	module	NOUN
iajs-2001	23	8	and	and	CCONJ
iajs-2001	23	9	a	a	DET
iajs-2001	23	10	,	,	PUNCT
iajs-2001	23	11	b	b	NOUN
iajs-2001	23	12	,	,	PUNCT
iajs-2001	23	13	c	c	PROPN
iajs-2001	23	14	are	be	AUX
iajs-2001	23	15	submodules	submodule	NOUN
iajs-2001	23	16	of	of	ADP
iajs-2001	23	17	y	y	PROPN
iajs-2001	23	18	with	with	ADP
iajs-2001	23	19	b	b	PROPN
iajs-2001	23	20	⊆	⊆	NUM
iajs-2001	23	21	c.	c.	NOUN
iajs-2001	23	22	then	then	ADV
iajs-2001	23	23	(	(	PUNCT
iajs-2001	23	24	a	a	DET
iajs-2001	23	25	+	+	NOUN
iajs-2001	23	26	b	b	NOUN
iajs-2001	23	27	)	)	PUNCT
iajs-2001	23	28	∩	∩	NOUN
iajs-2001	23	29	c	c	NOUN
iajs-2001	24	1	=	=	SYM
iajs-2001	24	2	(	(	PUNCT
iajs-2001	24	3	a	a	DET
iajs-2001	24	4	∩	∩	ADJ
iajs-2001	24	5	c	c	NOUN
iajs-2001	24	6	)	)	PUNCT
iajs-2001	25	1	+	+	CCONJ
iajs-2001	25	2	(	(	PUNCT
iajs-2001	25	3	b	b	NOUN
iajs-2001	25	4	∩	∩	X
iajs-2001	25	5	c	c	NOUN
iajs-2001	25	6	)	)	PUNCT
iajs-2001	26	1	=	=	SYM
iajs-2001	26	2	(	(	PUNCT
iajs-2001	26	3	a	a	DET
iajs-2001	26	4	∩	∩	ADJ
iajs-2001	26	5	c	c	NOUN
iajs-2001	26	6	)	)	PUNCT
iajs-2001	27	1	+	+	PUNCT
iajs-2001	27	2	b	b	X
iajs-2001	28	1	[	[	X
iajs-2001	28	2	3	3	NUM
iajs-2001	28	3	]	]	PUNCT
iajs-2001	28	4	.	.	PUNCT
iajs-2001	29	1	recall	recall	VERB
iajs-2001	29	2	that	that	SCONJ
iajs-2001	29	3	an	an	DET
iajs-2001	29	4	"	"	PUNCT
iajs-2001	29	5	r-"module	r-"module	PROPN
iajs-2001	29	6	y	y	PROPN
iajs-2001	29	7	is	be	AUX
iajs-2001	29	8	''	''	PUNCT
iajs-2001	29	9	'	'	PUNCT
iajs-2001	29	10	regular	regular	ADJ
iajs-2001	29	11	''	''	PUNCT
iajs-2001	29	12	if'''r'/'ann'(x)''is''regular''ring'''[4].'''recall''that''a''subset''s'''of'''a''ring'r	if'''r'/'ann'(x)''is''regular''ring'''[4].'''recall''that''a''subset''s'''of'''a''ring'r	NOUN
iajs-2001	29	13	is	be	AUX
iajs-2001	29	14	called	call	VERB
iajs-2001	29	15	multiplicatively'''''closed'''subset''''of	multiplicatively'''''closed'''subset''''of	PROPN
iajs-2001	29	16	r	r	NOUN
iajs-2001	29	17	if	if	SCONJ
iajs-2001	29	18	''	''	PUNCT
iajs-2001	29	19	'	'	NUM
iajs-2001	29	20	1'∈'s''and'''ab	1'∈'s''and'''ab	NUM
iajs-2001	29	21	∈	∈	NOUN
iajs-2001	29	22	s	s	PART
iajs-2001	29	23	"	"	PUNCT
iajs-2001	29	24	for	for	ADP
iajs-2001	29	25	all'''a'',''b'∈	all'''a'',''b'∈	PROPN
iajs-2001	29	26	s	s	PART
iajs-2001	29	27	[	[	X
iajs-2001	29	28	5	5	NUM
iajs-2001	29	29	]	]	PUNCT
iajs-2001	29	30	"	"	PUNCT
iajs-2001	29	31	.	.	PUNCT
iajs-2001	30	1	this	this	DET
iajs-2001	30	2	note	note	NOUN
iajs-2001	30	3	consists	consist	VERB
iajs-2001	30	4	of	of	ADP
iajs-2001	30	5	two	two	NUM
iajs-2001	30	6	parts	part	NOUN
iajs-2001	30	7	in	in	ADP
iajs-2001	30	8	the	the	DET
iajs-2001	30	9	first	first	ADJ
iajs-2001	30	10	part	part	NOUN
iajs-2001	30	11	,	,	PUNCT
iajs-2001	30	12	we	we	PRON
iajs-2001	30	13	introduced'''the''concept''of	introduced'''the''concept''of	VERB
iajs-2001	30	14	"	"	PUNCT
iajs-2001	30	15	wn''-2"-''absorbing''submodule	wn''-2"-''absorbing''submodule	PROPN
iajs-2001	30	16	''	''	PUNCT
iajs-2001	30	17	,	,	PUNCT
iajs-2001	30	18	and	and	CCONJ
iajs-2001	30	19	in	in	ADP
iajs-2001	30	20	the	the	DET
iajs-2001	30	21	second	second	ADJ
iajs-2001	30	22	part	part	NOUN
iajs-2001	30	23	we	we	PRON
iajs-2001	30	24	introduced"the''concept''of'''wns'-'2'-'absorbing''''submodule	introduced"the''concept''of'''wns'-'2'-'absorbing''''submodule	VERB
iajs-2001	30	25	'	'	PUNCT
iajs-2001	30	26	.	.	PUNCT
iajs-2001	31	1	mathematics	mathematic	NOUN
iajs-2001	31	2	|	|	ADV
iajs-2001	31	3	119	119	NUM
iajs-2001	31	4	ibn	ibn	PROPN
iajs-2001	31	5	al	al	PROPN
iajs-2001	31	6	-	-	PUNCT
iajs-2001	31	7	haitham	haitham	PROPN
iajs-2001	31	8	jour	jour	X
iajs-2001	31	9	.	.	PROPN
iajs-2001	32	1	for	for	ADP
iajs-2001	32	2	pure	pure	ADJ
iajs-2001	32	3	&	&	CCONJ
iajs-2001	32	4	appl	appl	PROPN
iajs-2001	32	5	.	.	PUNCT
iajs-2001	33	1	sci	sci	PROPN
iajs-2001	33	2	.	.	PROPN
iajs-2001	33	3	ihjpas	ihjpa	VERB
iajs-2001	33	4	https://doi.org/10.30526/31.3.2001	https://doi.org/10.30526/31.3.2001	PROPN
iajs-2001	33	5	vol	vol	NOUN
iajs-2001	33	6	.	.	PROPN
iajs-2001	34	1	31	31	NUM
iajs-2001	35	1	(	(	PUNCT
iajs-2001	35	2	3	3	NUM
iajs-2001	35	3	)	)	PUNCT
iajs-2001	35	4	2018	2018	NUM
iajs-2001	35	5	2	2	NUM
iajs-2001	35	6	.	.	NOUN
iajs-2001	35	7	wn-2	wn-2	NOUN
iajs-2001	35	8	-	-	PUNCT
iajs-2001	35	9	absorbing"submodules"and	absorbing"submodules"and	NOUN
iajs-2001	35	10	related	relate	VERB
iajs-2001	35	11	concept	concept	NOUN
iajs-2001	35	12	in	in	ADP
iajs-2001	35	13	this	this	DET
iajs-2001	35	14	part	part	NOUN
iajs-2001	35	15	of	of	ADP
iajs-2001	35	16	the	the	DET
iajs-2001	35	17	research,'we''''introduce''''and'''studied''''the''''concept'''of	research,'we''''introduce''''and'''studied''''the''''concept'''of	PROPN
iajs-2001	35	18	wn-2"'absorbing'''submodules'''as''a''generalization'''of	wn-2"'absorbing'''submodules'''as''a''generalization'''of	PROPN
iajs-2001	35	19	weakly"2'-'absorbing''submodules	weakly"2'-'absorbing''submodule	NOUN
iajs-2001	35	20	''	''	PUNCT
iajs-2001	35	21	.	.	PUNCT
iajs-2001	36	1	'	'	PUNCT
iajs-2001	36	2	definition'1	definition'1	NOUN
iajs-2001	36	3	a"proper'''submodule''b	a"proper'''submodule''b	NOUN
iajs-2001	36	4	of	of	ADP
iajs-2001	36	5	an"r'-'module'''y	an"r'-'module'''y	NOUN
iajs-2001	36	6	is''said''to"be	is''said''to"be	PROPN
iajs-2001	36	7	wn-'2'-'absorbing	wn-'2'-'absorbing	PROPN
iajs-2001	36	8	submodules'"if''whenever	submodules'"if''whenever	NOUN
iajs-2001	36	9	'	'	PART
iajs-2001	36	10	0	0	NUM
iajs-2001	36	11	aby	aby	PROPN
iajs-2001	36	12	∊	∊	PROPN
iajs-2001	36	13	"	"	PUNCT
iajs-2001	36	14	b	b	NOUN
iajs-2001	36	15	,	,	PUNCT
iajs-2001	36	16	where	where	SCONJ
iajs-2001	36	17	a	a	DET
iajs-2001	36	18	,	,	PUNCT
iajs-2001	36	19	b	b	PROPN
iajs-2001	36	20	∊"r	∊"r	PROPN
iajs-2001	36	21	,	,	PUNCT
iajs-2001	36	22	y	y	PROPN
iajs-2001	36	23	∊	∊	PROPN
iajs-2001	36	24	y	y	PROPN
iajs-2001	36	25	,	,	PUNCT
iajs-2001	36	26	implies	imply	VERB
iajs-2001	36	27	that	that	SCONJ
iajs-2001	36	28	either	either	CCONJ
iajs-2001	36	29	ay	ay	PROPN
iajs-2001	36	30	∊	∊	PROPN
iajs-2001	36	31	"	"	PUNCT
iajs-2001	36	32	b	b	NOUN
iajs-2001	36	33	′	′	NUM
iajs-2001	36	34	"	"	PUNCT
iajs-2001	36	35	ȷ	ȷ	PROPN
iajs-2001	36	36	y	y	NOUN
iajs-2001	36	37	or	or	CCONJ
iajs-2001	36	38	by	by	ADP
iajs-2001	36	39	∊	∊	PROPN
iajs-2001	36	40	"	"	PUNCT
iajs-2001	36	41	b	b	NOUN
iajs-2001	36	42	′	′	NUM
iajs-2001	36	43	"	"	PUNCT
iajs-2001	36	44	ȷ	ȷ	PROPN
iajs-2001	36	45	y	y	PROPN
iajs-2001	36	46	or	or	CCONJ
iajs-2001	36	47	ab	ab	PROPN
iajs-2001	36	48	∊	∊	PROPN
iajs-2001	36	49	"	"	PUNCT
iajs-2001	36	50	b	b	X
iajs-2001	36	51	′	′	NUM
iajs-2001	36	52	"	"	PUNCT
iajs-2001	36	53	ȷ	ȷ	PROPN
iajs-2001	36	54	y	y	NOUN
iajs-2001	36	55	:	:	PUNCT
iajs-2001	36	56	y	y	PROPN
iajs-2001	36	57	,	,	PUNCT
iajs-2001	36	58	where	where	SCONJ
iajs-2001	36	59	𝚥(y	𝚥(y	NOUN
iajs-2001	36	60	)	)	PUNCT
iajs-2001	36	61	is	be	AUX
iajs-2001	36	62	the	the	DET
iajs-2001	36	63	jacobsen	jacobsen	PROPN
iajs-2001	36	64	radical	radical	PROPN
iajs-2001	36	65	of	of	ADP
iajs-2001	36	66	y.	y.	PROPN
iajs-2001	36	67	an''ideal	an''ideal	PROPN
iajs-2001	36	68	'	'	PART
iajs-2001	36	69	i"of	i"of	NOUN
iajs-2001	36	70	a	a	DET
iajs-2001	36	71	ring	ring	NOUN
iajs-2001	36	72	r	r	NOUN
iajs-2001	36	73	'	'	PUNCT
iajs-2001	36	74	is"'said	is"'said	PROPN
iajs-2001	36	75	'	'	PUNCT
iajs-2001	36	76	to'"be	to'"be	NUM
iajs-2001	36	77	wn-'2-'absorbing	wn-'2-'absorbing	NOUN
iajs-2001	36	78	'	'	PUNCT
iajs-2001	36	79	ideal	ideal	ADJ
iajs-2001	36	80	of"r	of"r	PROPN
iajs-2001	36	81	,	,	PUNCT
iajs-2001	36	82	if'''i	if'''i	VERB
iajs-2001	36	83	is'''a	is'''a	NOUN
iajs-2001	36	84	wn-"2'absorbing'''submodules'"of"an"r'-'module'r	wn-"2'absorbing'''submodules'"of"an"r'-'module'r	NOUN
iajs-2001	36	85	.	.	PUNCT
iajs-2001	37	1	remark"2	remark"2	PROPN
iajs-2001	38	1	'	'	PUNCT
iajs-2001	38	2	every"weakly	every"weakly	ADJ
iajs-2001	38	3	2	2	NUM
iajs-2001	38	4	-	-	PUNCT
iajs-2001	38	5	absorbing"submodule	absorbing"submodule	ADJ
iajs-2001	38	6	of	of	ADP
iajs-2001	38	7	an	an	DET
iajs-2001	38	8	r	r	NOUN
iajs-2001	38	9	-	-	PUNCT
iajs-2001	38	10	module	module	NOUN
iajs-2001	38	11	y	y	NOUN
iajs-2001	38	12	is	be	AUX
iajs-2001	38	13	wn-2	wn-2	ADP
iajs-2001	38	14	-	-	ADJ
iajs-2001	38	15	absorbing	absorbing	ADJ
iajs-2001	38	16	submodules	submodule	NOUN
iajs-2001	38	17	,	,	PUNCT
iajs-2001	38	18	while	while	SCONJ
iajs-2001	38	19	the	the	DET
iajs-2001	38	20	converse	converse	NOUN
iajs-2001	38	21	is	be	AUX
iajs-2001	38	22	not	not	PART
iajs-2001	38	23	true	true	ADJ
iajs-2001	38	24	.	.	PUNCT
iajs-2001	39	1	proof	proof	NOUN
iajs-2001	39	2	clear	clear	ADJ
iajs-2001	39	3	.	.	PUNCT
iajs-2001	40	1	for	for	ADP
iajs-2001	40	2	the	the	DET
iajs-2001	40	3	converse	converse	NOUN
iajs-2001	40	4	consider	consider	VERB
iajs-2001	40	5	the	the	DET
iajs-2001	40	6	following	follow	VERB
iajs-2001	40	7	example	example	NOUN
iajs-2001	40	8	:	:	PUNCT
iajs-2001	40	9	let	let	VERB
iajs-2001	40	10	y	y	PROPN
iajs-2001	40	11	=	=	PROPN
iajs-2001	40	12	z	z	PROPN
iajs-2001	40	13	,	,	PUNCT
iajs-2001	40	14	r	r	NOUN
iajs-2001	40	15	=	=	SYM
iajs-2001	40	16	z	z	PROPN
iajs-2001	40	17	and	and	CCONJ
iajs-2001	40	18	b	b	X
iajs-2001	40	19	=	=	PUNCT
iajs-2001	40	20	〈	〈	NOUN
iajs-2001	40	21	8	8	NUM
iajs-2001	40	22	〉	〉	NOUN
iajs-2001	40	23	it	it	PRON
iajs-2001	40	24	is	be	AUX
iajs-2001	40	25	clear	clear	ADJ
iajs-2001	40	26	that	that	SCONJ
iajs-2001	40	27	b	b	NOUN
iajs-2001	40	28	is	be	AUX
iajs-2001	40	29	a	a	DET
iajs-2001	40	30	wn-2	wn-2	NOUN
iajs-2001	40	31	-	-	ADJ
iajs-2001	40	32	absorbing	absorbing	ADJ
iajs-2001	40	33	submodules	submodule	NOUN
iajs-2001	40	34	of	of	ADP
iajs-2001	40	35	y	y	PROPN
iajs-2001	40	36	since	since	SCONJ
iajs-2001	40	37	b	b	PROPN
iajs-2001	40	38	+	+	CCONJ
iajs-2001	40	39	𝚥(y	𝚥(y	NOUN
iajs-2001	40	40	)	)	PUNCT
iajs-2001	41	1	=	=	PUNCT
iajs-2001	41	2	〈	〈	NOUN
iajs-2001	41	3	8	8	NUM
iajs-2001	41	4	〉	〉	NOUN
iajs-2001	41	5	+	+	CCONJ
iajs-2001	41	6	〈	〈	PROPN
iajs-2001	41	7	2	2	NUM
iajs-2001	41	8	〉	〉	NOUN
iajs-2001	41	9	=	=	PUNCT
iajs-2001	41	10	〈	〈	NOUN
iajs-2001	41	11	2	2	NUM
iajs-2001	41	12	〉	〉	NOUN
iajs-2001	41	13	.	.	PUNCT
iajs-2001	42	1	but	but	CCONJ
iajs-2001	42	2	b	b	NOUN
iajs-2001	42	3	is	be	AUX
iajs-2001	42	4	not	not	PART
iajs-2001	42	5	weakly	weakly	ADJ
iajs-2001	42	6	2	2	NUM
iajs-2001	42	7	-	-	PUNCT
iajs-2001	42	8	absorbing	absorb	VERB
iajs-2001	42	9	submodule	submodule	NOUN
iajs-2001	42	10	of	of	ADP
iajs-2001	42	11	y	y	PROPN
iajs-2001	42	12	since	since	ADV
iajs-2001	42	13	,	,	PUNCT
iajs-2001	42	14	0	0	NUM
iajs-2001	42	15	≠	≠	PROPN
iajs-2001	42	16	2.2.2	2.2.2	NUM
iajs-2001	42	17	∊	∊	PROPN
iajs-2001	42	18	b	b	NOUN
iajs-2001	42	19	,	,	PUNCT
iajs-2001	42	20	but	but	CCONJ
iajs-2001	42	21	2.2	2.2	NUM
iajs-2001	42	22	∉	∉	PROPN
iajs-2001	42	23	b	b	PROPN
iajs-2001	42	24	and	and	CCONJ
iajs-2001	42	25	2.2	2.2	NUM
iajs-2001	42	26	∉	∉	PROPN
iajs-2001	43	1	[	[	X
iajs-2001	43	2	b	b	X
iajs-2001	43	3	:	:	PUNCT
iajs-2001	43	4	y	y	NOUN
iajs-2001	43	5	]	]	X
iajs-2001	43	6	=	=	SYM
iajs-2001	43	7	8z	8z	NUM
iajs-2001	43	8	.	.	PUNCT
iajs-2001	44	1	'	'	PUNCT
iajs-2001	44	2	proposition	proposition	NOUN
iajs-2001	44	3	'	'	PUNCT
iajs-2001	44	4	3	3	NUM
iajs-2001	44	5	let''y'''be'''an"r''-''module',''and'''b''a'''proper'''submodule'''of"y''''with'''𝚥''(y)''⊆b	let''y'''be'''an"r''-''module',''and'''b''a'''proper'''submodule'''of"y''''with'''𝚥''(y)''⊆b	NOUN
iajs-2001	44	6	then'''b'''is'''a''''weakly"2''-'''absorbing''''submodule''of'''y'''if''and''only"if''''b'''is"a"wn-2"'absorbing'''submodule'''of'''y	then'''b'''is'''a''''weakly"2''-'''absorbing''''submodule''of'''y'''if''and''only"if''''b'''is"a"wn-2"'absorbing'''submodule'''of'''y	NOUN
iajs-2001	44	7	.	.	PUNCT
iajs-2001	45	1	proof	proof	NOUN
iajs-2001	45	2	"	"	PUNCT
iajs-2001	45	3	⟹	⟹	PUNCT
iajs-2001	45	4	by	by	ADP
iajs-2001	45	5	remark	remark	NOUN
iajs-2001	45	6	(	(	PUNCT
iajs-2001	45	7	2.2	2.2	NUM
iajs-2001	45	8	)	)	PUNCT
iajs-2001	45	9	.	.	PUNCT
iajs-2001	46	1	(	(	PUNCT
iajs-2001	46	2	⟸	⟸	ADJ
iajs-2001	46	3	since	since	SCONJ
iajs-2001	46	4	𝚥	𝚥	PROPN
iajs-2001	46	5	(	(	PUNCT
iajs-2001	46	6	y	y	NOUN
iajs-2001	46	7	)	)	PUNCT
iajs-2001	46	8	⊆	⊆	NUM
iajs-2001	46	9	b	b	NOUN
iajs-2001	46	10	then	then	ADV
iajs-2001	46	11	b	b	PROPN
iajs-2001	46	12	+	+	CCONJ
iajs-2001	46	13	𝚥	𝚥	PROPN
iajs-2001	46	14	(	(	PUNCT
iajs-2001	46	15	y	y	NOUN
iajs-2001	46	16	)	)	PUNCT
iajs-2001	46	17	=	=	SYM
iajs-2001	46	18	b	b	NOUN
iajs-2001	46	19	,	,	PUNCT
iajs-2001	46	20	hence	hence	ADV
iajs-2001	46	21	proof	proof	NOUN
iajs-2001	46	22	is	be	AUX
iajs-2001	46	23	direct	direct	ADJ
iajs-2001	46	24	.	.	PUNCT
iajs-2001	47	1	proposition	proposition	NOUN
iajs-2001	47	2	4	4	NUM
iajs-2001	47	3	let	let	VERB
iajs-2001	47	4	y	y	PRON
iajs-2001	47	5	be	be	AUX
iajs-2001	47	6	an"r	an"r	NOUN
iajs-2001	47	7	-	-	PUNCT
iajs-2001	47	8	module	module	NOUN
iajs-2001	47	9	,	,	PUNCT
iajs-2001	47	10	and	and	CCONJ
iajs-2001	47	11	b	b	X
iajs-2001	47	12	a"proper"submodule	a"proper"submodule	PROPN
iajs-2001	47	13	of	of	ADP
iajs-2001	47	14	y	y	PROPN
iajs-2001	47	15	with	with	ADP
iajs-2001	47	16	a	a	DET
iajs-2001	47	17	⊂	⊂	PROPN
iajs-2001	47	18	b.	b.	PROPN
iajs-2001	48	1	if	if	SCONJ
iajs-2001	48	2	a	a	PRON
iajs-2001	48	3	is	be	AUX
iajs-2001	48	4	a	a	DET
iajs-2001	48	5	wn-2absorbing"submodule	wn-2absorbing"submodule	NOUN
iajs-2001	48	6	of	of	ADP
iajs-2001	48	7	y	y	PROPN
iajs-2001	48	8	and	and	CCONJ
iajs-2001	48	9	𝚥	𝚥	PROPN
iajs-2001	48	10	(	(	PUNCT
iajs-2001	48	11	y	y	NOUN
iajs-2001	48	12	)	)	PUNCT
iajs-2001	48	13	⊆𝚥(b	⊆𝚥(b	NOUN
iajs-2001	48	14	)	)	PUNCT
iajs-2001	48	15	,	,	PUNCT
iajs-2001	48	16	then	then	ADV
iajs-2001	48	17	a"is'''a"wn-2-''absorbing	a"is'''a"wn-2-''absorbe	VERB
iajs-2001	48	18	'	'	PUNCT
iajs-2001	48	19	'	'	PUNCT
iajs-2001	48	20	submodule"of	submodule"of	PRON
iajs-2001	48	21	b.	b.	NOUN
iajs-2001	48	22	'	'	PUNCT
iajs-2001	48	23	proof	proof	NOUN
iajs-2001	48	24	"	"	PUNCT
iajs-2001	48	25	'	'	PUNCT
iajs-2001	48	26	let	let	VERB
iajs-2001	48	27	0	0	NUM
iajs-2001	48	28	"	"	PUNCT
iajs-2001	48	29	aby	aby	X
iajs-2001	48	30	∊	∊	PROPN
iajs-2001	48	31	′a	′a	PROPN
iajs-2001	48	32	,	,	PUNCT
iajs-2001	48	33	where′a	where′a	PROPN
iajs-2001	48	34	,	,	PUNCT
iajs-2001	48	35	b	b	PROPN
iajs-2001	48	36	∊	∊	NUM
iajs-2001	48	37	r	r	NOUN
iajs-2001	48	38	,	,	PUNCT
iajs-2001	48	39	y	y	PROPN
iajs-2001	48	40	∊	∊	PROPN
iajs-2001	48	41	′b	′b	PROPN
iajs-2001	48	42	,	,	PUNCT
iajs-2001	48	43	since	since	SCONJ
iajs-2001	48	44	a	a	DET
iajs-2001	48	45	is	be	AUX
iajs-2001	48	46	a"wn-2	a"wn-2	NOUN
iajs-2001	48	47	-	-	PUNCT
iajs-2001	48	48	absorbing	absorb	VERB
iajs-2001	48	49	"	"	PUNCT
iajs-2001	48	50	submodule	submodule	NOUN
iajs-2001	48	51	of	of	ADP
iajs-2001	48	52	y	y	PROPN
iajs-2001	48	53	then	then	ADV
iajs-2001	48	54	either	either	CCONJ
iajs-2001	48	55	ay	ay	PROPN
iajs-2001	48	56	∊	∊	PROPN
iajs-2001	48	57	a	a	DET
iajs-2001	48	58	ȷ	ȷ	NOUN
iajs-2001	48	59	y	y	NOUN
iajs-2001	48	60	or	or	CCONJ
iajs-2001	48	61	by	by	ADP
iajs-2001	48	62	∊	∊	PROPN
iajs-2001	48	63	a	a	DET
iajs-2001	48	64	ȷ	ȷ	NOUN
iajs-2001	49	1	y	y	PROPN
iajs-2001	50	1	or	or	CCONJ
iajs-2001	50	2	ab	ab	PROPN
iajs-2001	50	3	∊	∊	PROPN
iajs-2001	50	4	a	a	DET
iajs-2001	50	5	ȷ	ȷ	X
iajs-2001	50	6	y	y	NOUN
iajs-2001	50	7	:	:	PUNCT
iajs-2001	50	8	y	y	PROPN
iajs-2001	50	9	,	,	PUNCT
iajs-2001	50	10	but	but	CCONJ
iajs-2001	50	11	𝚥	𝚥	PROPN
iajs-2001	50	12	(	(	PUNCT
iajs-2001	50	13	y	y	NOUN
iajs-2001	50	14	)	)	PUNCT
iajs-2001	50	15	⊆𝚥	⊆𝚥	NOUN
iajs-2001	50	16	(	(	PUNCT
iajs-2001	50	17	b	b	NOUN
iajs-2001	50	18	)	)	PUNCT
iajs-2001	50	19	,	,	PUNCT
iajs-2001	50	20	so	so	CCONJ
iajs-2001	51	1	either	either	CCONJ
iajs-2001	51	2	ay	ay	PROPN
iajs-2001	51	3	∊	∊	PROPN
iajs-2001	51	4	a	a	DET
iajs-2001	51	5	ȷ	ȷ	NOUN
iajs-2001	51	6	b	b	NOUN
iajs-2001	51	7	or	or	CCONJ
iajs-2001	51	8	by	by	ADP
iajs-2001	51	9	∊	∊	PROPN
iajs-2001	51	10	a	a	DET
iajs-2001	51	11	ȷ	ȷ	NOUN
iajs-2001	51	12	b	b	NOUN
iajs-2001	51	13	or	or	CCONJ
iajs-2001	51	14	ab	ab	PROPN
iajs-2001	51	15	∊	∊	PROPN
iajs-2001	51	16	a	a	DET
iajs-2001	51	17	ȷ	ȷ	X
iajs-2001	51	18	y	y	NOUN
iajs-2001	51	19	:	:	PUNCT
iajs-2001	51	20	y	y	PROPN
iajs-2001	51	21	⊆	⊆	NUM
iajs-2001	51	22	a	a	DET
iajs-2001	51	23	ȷ	ȷ	NOUN
iajs-2001	51	24	b	b	NOUN
iajs-2001	51	25	:	:	PUNCT
iajs-2001	51	26	y	y	PROPN
iajs-2001	51	27	⊆	⊆	PROPN
iajs-2001	51	28	a	a	DET
iajs-2001	51	29	ȷ	ȷ	NOUN
iajs-2001	51	30	b	b	NOUN
iajs-2001	51	31	:	:	PUNCT
iajs-2001	51	32	b	b	X
iajs-2001	51	33	since	since	SCONJ
iajs-2001	51	34	b	b	PROPN
iajs-2001	51	35	is	be	AUX
iajs-2001	51	36	a"submodule	a"submodule	ADP
iajs-2001	51	37	of	of	ADP
iajs-2001	51	38	y.	y.	NOUN
iajs-2001	51	39	hence	hence	ADV
iajs-2001	51	40	a	a	DET
iajs-2001	51	41	a	a	DET
iajs-2001	51	42	wn-2	wn-2	NOUN
iajs-2001	51	43	-	-	PUNCT
iajs-2001	51	44	absorbing"submodule	absorbing"submodule	ADJ
iajs-2001	51	45	of	of	ADP
iajs-2001	51	46	b.	b.	PROPN
iajs-2001	51	47	proposition	proposition	NOUN
iajs-2001	51	48	5	5	NUM
iajs-2001	51	49	let"y	let"y	NOUN
iajs-2001	51	50	be	be	VERB
iajs-2001	51	51	an"r"-'module,''and'''b	an"r"-'module,''and'''b	NOUN
iajs-2001	51	52	a'proper'''submodule''of	a'proper'''submodule''of	VERB
iajs-2001	51	53	y,"'if	y,"'if	PROPN
iajs-2001	51	54	b	b	PROPN
iajs-2001	51	55	+	+	CCONJ
iajs-2001	51	56	𝚥(y	𝚥(y	PROPN
iajs-2001	51	57	)	)	PUNCT
iajs-2001	51	58	is''a	is''a	NOUN
iajs-2001	51	59	wn-2"absorbing'''submodule'''of	wn-2"absorbing'''submodule'''of	PROPN
iajs-2001	51	60	y,''then'''b	y,''then'''b	PROPN
iajs-2001	51	61	is'''a	is'''a	PROPN
iajs-2001	51	62	wn-2"-'absorbing"'submodule''of'''y	wn-2"-'absorbing"'submodule''of'''y	PROPN
iajs-2001	51	63	.	.	PUNCT
iajs-2001	52	1	'	'	PUNCT
iajs-2001	52	2	proof	proof	NOUN
iajs-2001	52	3	"	"	PUNCT
iajs-2001	52	4	since'''b	since'''b	VERB
iajs-2001	52	5	⊆	⊆	NUM
iajs-2001	52	6	b	b	NOUN
iajs-2001	52	7	+	+	CCONJ
iajs-2001	52	8	𝚥(y	𝚥(y	PROPN
iajs-2001	52	9	)	)	PUNCT
iajs-2001	52	10	,	,	PUNCT
iajs-2001	52	11	hence	hence	ADV
iajs-2001	52	12	proof	proof	NOUN
iajs-2001	52	13	is	be	AUX
iajs-2001	52	14	clearly	clearly	ADV
iajs-2001	52	15	.	.	PUNCT
iajs-2001	53	1	mathematics	mathematic	NOUN
iajs-2001	53	2	|	|	ADV
iajs-2001	53	3	120	120	NUM
iajs-2001	53	4	ibn	ibn	PROPN
iajs-2001	53	5	al	al	PROPN
iajs-2001	53	6	-	-	PUNCT
iajs-2001	53	7	haitham	haitham	PROPN
iajs-2001	53	8	jour	jour	X
iajs-2001	53	9	.	.	PROPN
iajs-2001	54	1	for	for	ADP
iajs-2001	54	2	pure	pure	ADJ
iajs-2001	54	3	&	&	CCONJ
iajs-2001	54	4	appl	appl	PROPN
iajs-2001	54	5	.	.	PUNCT
iajs-2001	55	1	sci	sci	PROPN
iajs-2001	55	2	.	.	PROPN
iajs-2001	55	3	ihjpas	ihjpa	VERB
iajs-2001	55	4	https://doi.org/10.30526/31.3.2001	https://doi.org/10.30526/31.3.2001	PROPN
iajs-2001	55	5	vol	vol	NOUN
iajs-2001	55	6	.	.	PROPN
iajs-2001	56	1	31	31	NUM
iajs-2001	57	1	(	(	PUNCT
iajs-2001	57	2	3	3	NUM
iajs-2001	57	3	)	)	PUNCT
iajs-2001	57	4	2018	2018	NUM
iajs-2001	57	5	remark	remark	NOUN
iajs-2001	57	6	6	6	NUM
iajs-2001	57	7	the"intersection	the"intersection	NOUN
iajs-2001	57	8	of	of	ADP
iajs-2001	57	9	two	two	NUM
iajs-2001	57	10	is	be	AUX
iajs-2001	57	11	a	a	DET
iajs-2001	57	12	"	"	PUNCT
iajs-2001	57	13	wn-2"-'absorbing''submodules'"of"an"r'-"module	wn-2"-'absorbing''submodules'"of"an"r'-"module	X
iajs-2001	57	14	y	y	PRON
iajs-2001	57	15	need	need	VERB
iajs-2001	57	16	not	not	PART
iajs-2001	57	17	to	to	PART
iajs-2001	57	18	be	be	AUX
iajs-2001	57	19	is	be	AUX
iajs-2001	57	20	a	a	DET
iajs-2001	57	21	wn-2	wn-2	NOUN
iajs-2001	57	22	-	-	PUNCT
iajs-2001	57	23	absorbing"submodule	absorbing"submodule	ADJ
iajs-2001	57	24	.	.	PUNCT
iajs-2001	58	1	the	the	DET
iajs-2001	58	2	following	follow	VERB
iajs-2001	58	3	example	example	NOUN
iajs-2001	58	4	explain	explain	VERB
iajs-2001	58	5	that	that	SCONJ
iajs-2001	58	6	:	:	PUNCT
iajs-2001	58	7	let	let	VERB
iajs-2001	58	8	y	y	PROPN
iajs-2001	58	9	=	=	PROPN
iajs-2001	58	10	z	z	PROPN
iajs-2001	58	11	,	,	PUNCT
iajs-2001	58	12	r	r	NOUN
iajs-2001	58	13	=	=	SYM
iajs-2001	58	14	z	z	PROPN
iajs-2001	58	15	,	,	PUNCT
iajs-2001	58	16	a	a	DET
iajs-2001	58	17	=	=	NOUN
iajs-2001	58	18	6z	6z	NOUN
iajs-2001	58	19	,	,	PUNCT
iajs-2001	59	1	b	b	X
iajs-2001	59	2	=	=	SYM
iajs-2001	59	3	7z	7z	PROPN
iajs-2001	59	4	.	.	PUNCT
iajs-2001	60	1	clearly	clearly	ADV
iajs-2001	60	2	a	a	PRON
iajs-2001	60	3	,	,	PUNCT
iajs-2001	60	4	b	b	NOUN
iajs-2001	60	5	is	be	AUX
iajs-2001	60	6	a	a	DET
iajs-2001	60	7	wn-2	wn-2	NOUN
iajs-2001	60	8	-	-	ADJ
iajs-2001	60	9	absorbing	absorbing	ADJ
iajs-2001	60	10	submodules	submodule	NOUN
iajs-2001	60	11	since	since	SCONJ
iajs-2001	60	12	they	they	PRON
iajs-2001	60	13	are	be	AUX
iajs-2001	60	14	weakly	weakly	ADJ
iajs-2001	60	15	2	2	NUM
iajs-2001	60	16	-	-	PUNCT
iajs-2001	60	17	absorbing	absorb	VERB
iajs-2001	60	18	submodules	submodule	NOUN
iajs-2001	60	19	of	of	ADP
iajs-2001	60	20	y	y	PROPN
iajs-2001	60	21	but	but	CCONJ
iajs-2001	60	22	a	a	DET
iajs-2001	60	23	∩	∩	ADJ
iajs-2001	60	24	b	b	NOUN
iajs-2001	60	25	=	=	SYM
iajs-2001	60	26	42z	42z	PROPN
iajs-2001	60	27	is	be	AUX
iajs-2001	60	28	not	not	PART
iajs-2001	60	29	wn-2	wn-2	ADP
iajs-2001	60	30	-	-	ADJ
iajs-2001	60	31	absorbing	absorbing	ADJ
iajs-2001	60	32	submodule	submodule	NOUN
iajs-2001	60	33	of	of	ADP
iajs-2001	60	34	y	y	PROPN
iajs-2001	60	35	since	since	SCONJ
iajs-2001	60	36	,	,	PUNCT
iajs-2001	60	37	if	if	SCONJ
iajs-2001	60	38	0	0	NUM
iajs-2001	60	39	≠	≠	PROPN
iajs-2001	60	40	2.3.7	2.3.7	NUM
iajs-2001	60	41	∈	∈	PROPN
iajs-2001	60	42	a	a	DET
iajs-2001	60	43	∩	∩	ADJ
iajs-2001	60	44	b	b	NOUN
iajs-2001	60	45	,	,	PUNCT
iajs-2001	60	46	but	but	CCONJ
iajs-2001	60	47	2.7	2.7	NUM
iajs-2001	60	48	∉	∉	PROPN
iajs-2001	60	49	a	a	DET
iajs-2001	60	50	∩	∩	ADJ
iajs-2001	60	51	𝚥(y	𝚥(y	NOUN
iajs-2001	60	52	)	)	PUNCT
iajs-2001	60	53	and	and	CCONJ
iajs-2001	61	1	3.7	3.7	NUM
iajs-2001	61	2	∉	∉	PROPN
iajs-2001	61	3	a	a	DET
iajs-2001	61	4	∩	∩	ADJ
iajs-2001	61	5	𝚥(y	𝚥(y	NOUN
iajs-2001	61	6	)	)	PUNCT
iajs-2001	61	7	and	and	CCONJ
iajs-2001	61	8	2.3	2.3	NUM
iajs-2001	61	9	∉	∉	X
iajs-2001	62	1	[	[	X
iajs-2001	62	2	a	a	DET
iajs-2001	62	3	∩	∩	ADJ
iajs-2001	62	4	𝚥(y	𝚥(y	PROPN
iajs-2001	62	5	):	):	PUNCT
iajs-2001	62	6	y	y	NOUN
iajs-2001	62	7	]	]	X
iajs-2001	62	8	=	=	X
iajs-2001	62	9	42z	42z	NUM
iajs-2001	62	10	.	.	PUNCT
iajs-2001	62	11	'	'	PUNCT
iajs-2001	62	12	proposition'7	proposition'7	NOUN
iajs-2001	62	13	'	'	PUNCT
iajs-2001	62	14	let	let	VERB
iajs-2001	62	15	y"be	y"be	PROPN
iajs-2001	62	16	an"r'-"module	an"r'-"module	PROPN
iajs-2001	62	17	,	,	PUNCT
iajs-2001	62	18	"	"	PUNCT
iajs-2001	62	19	and''a	and''a	NOUN
iajs-2001	62	20	,	,	PUNCT
iajs-2001	62	21	b	b	NOUN
iajs-2001	62	22	are	be	AUX
iajs-2001	62	23	wn-'2'-'absorbing''submodules"of''y	wn-'2'-'absorbing''submodules"of''y	NOUN
iajs-2001	62	24	with	with	ADP
iajs-2001	62	25	a	a	DET
iajs-2001	62	26	⊆	⊆	NUM
iajs-2001	62	27	𝚥	𝚥	PROPN
iajs-2001	62	28	(	(	PUNCT
iajs-2001	62	29	y	y	NOUN
iajs-2001	62	30	)	)	PUNCT
iajs-2001	62	31	and	and	CCONJ
iajs-2001	62	32	b	b	NOUN
iajs-2001	62	33	⊆	⊆	NUM
iajs-2001	62	34	𝚥(y	𝚥(y	NOUN
iajs-2001	62	35	)	)	PUNCT
iajs-2001	62	36	,	,	PUNCT
iajs-2001	62	37	then	then	ADV
iajs-2001	62	38	a	a	DET
iajs-2001	62	39	∩	∩	ADJ
iajs-2001	62	40	b	b	NOUN
iajs-2001	62	41	is	be	AUX
iajs-2001	62	42	wn-2	wn-2	ADP
iajs-2001	62	43	-	-	ADJ
iajs-2001	62	44	absorbing	absorbing	ADJ
iajs-2001	62	45	submodules	submodule	NOUN
iajs-2001	62	46	of	of	ADP
iajs-2001	62	47	y.	y.	PROPN
iajs-2001	62	48	proof	proof	NOUN
iajs-2001	62	49	"	"	PUNCT
iajs-2001	62	50	let	let	VERB
iajs-2001	62	51	0	0	NUM
iajs-2001	62	52	≠	≠	PROPN
iajs-2001	62	53	aby	aby	NOUN
iajs-2001	62	54	∈	∈	PROPN
iajs-2001	62	55	a	a	DET
iajs-2001	62	56	∩	∩	ADJ
iajs-2001	62	57	b	b	NOUN
iajs-2001	62	58	,	,	PUNCT
iajs-2001	62	59	with	with	ADP
iajs-2001	62	60	a	a	DET
iajs-2001	62	61	,	,	PUNCT
iajs-2001	62	62	b	b	X
iajs-2001	62	63	∈	∈	PROPN
iajs-2001	62	64	r	r	NOUN
iajs-2001	62	65	,	,	PUNCT
iajs-2001	62	66	y	y	PROPN
iajs-2001	62	67	∈	∈	PROPN
iajs-2001	62	68	y	y	PROPN
iajs-2001	62	69	,	,	PUNCT
iajs-2001	62	70	implies	imply	VERB
iajs-2001	62	71	that	that	SCONJ
iajs-2001	62	72	0	0	NUM
iajs-2001	62	73	≠	≠	PROPN
iajs-2001	62	74	aby	aby	NOUN
iajs-2001	62	75	∈	∈	PROPN
iajs-2001	62	76	a	a	DET
iajs-2001	62	77	and	and	CCONJ
iajs-2001	62	78	0	0	NUM
iajs-2001	62	79	≠	≠	PROPN
iajs-2001	62	80	aby	aby	PROPN
iajs-2001	62	81	∈	∈	PROPN
iajs-2001	62	82	b.	b.	PROPN
iajs-2001	63	1	it	it	PRON
iajs-2001	63	2	follows	follow	VERB
iajs-2001	63	3	that	that	SCONJ
iajs-2001	63	4	either	either	CCONJ
iajs-2001	63	5	ay	ay	PROPN
iajs-2001	63	6	∊	∊	PROPN
iajs-2001	63	7	"	"	PUNCT
iajs-2001	63	8	′a	′a	ADP
iajs-2001	63	9	′"ȷ	′"ȷ	NOUN
iajs-2001	63	10	y	y	PROPN
iajs-2001	63	11	or	or	CCONJ
iajs-2001	63	12	by	by	ADP
iajs-2001	63	13	∊	∊	PROPN
iajs-2001	63	14	"	"	PUNCT
iajs-2001	63	15	′a′	′a′	ADJ
iajs-2001	63	16	"	"	PUNCT
iajs-2001	63	17	ȷ	ȷ	PROPN
iajs-2001	63	18	y	y	PROPN
iajs-2001	63	19	or	or	CCONJ
iajs-2001	63	20	ab	ab	PROPN
iajs-2001	63	21	∊	∊	PROPN
iajs-2001	63	22	a	a	DET
iajs-2001	63	23	ȷ	ȷ	X
iajs-2001	63	24	y	y	NOUN
iajs-2001	63	25	:	:	PUNCT
iajs-2001	63	26	y	y	PROPN
iajs-2001	63	27	,	,	PUNCT
iajs-2001	63	28	and	and	CCONJ
iajs-2001	63	29	either	either	ADV
iajs-2001	63	30	ay	ay	PROPN
iajs-2001	63	31	∊	∊	PROPN
iajs-2001	63	32	′b′	′b′	NOUN
iajs-2001	63	33	′ȷ	′ȷ	PROPN
iajs-2001	63	34	y	y	PROPN
iajs-2001	63	35	or	or	CCONJ
iajs-2001	63	36	by	by	ADP
iajs-2001	63	37	∊	∊	PROPN
iajs-2001	63	38	′b	′b	PROPN
iajs-2001	63	39	′ȷ	′ȷ	PROPN
iajs-2001	63	40	y	y	PROPN
iajs-2001	63	41	or	or	CCONJ
iajs-2001	63	42	ab	ab	PROPN
iajs-2001	63	43	∊	∊	PROPN
iajs-2001	63	44	"	"	PUNCT
iajs-2001	63	45	′b	′b	PROPN
iajs-2001	63	46	′ȷ	′ȷ	NOUN
iajs-2001	63	47	y	y	PROPN
iajs-2001	63	48	:	:	PUNCT
iajs-2001	64	1	y	y	PROPN
iajs-2001	64	2	.	.	PUNCT
iajs-2001	65	1	but"a	but"a	ADP
iajs-2001	65	2	⊆	⊆	NUM
iajs-2001	65	3	𝚥	𝚥	PROPN
iajs-2001	65	4	(	(	PUNCT
iajs-2001	65	5	y	y	NOUN
iajs-2001	65	6	)	)	PUNCT
iajs-2001	65	7	and	and	CCONJ
iajs-2001	65	8	b	b	NOUN
iajs-2001	65	9	⊆	⊆	NUM
iajs-2001	65	10	𝚥(y	𝚥(y	NOUN
iajs-2001	65	11	)	)	PUNCT
iajs-2001	65	12	,	,	PUNCT
iajs-2001	65	13	then	then	ADV
iajs-2001	65	14	a	a	DET
iajs-2001	65	15	+	+	ADJ
iajs-2001	65	16	𝚥(y	𝚥(y	NOUN
iajs-2001	65	17	)	)	PUNCT
iajs-2001	65	18	=	=	SYM
iajs-2001	65	19	𝚥(y	𝚥(y	PROPN
iajs-2001	65	20	)	)	PUNCT
iajs-2001	65	21	and	and	CCONJ
iajs-2001	65	22	b	b	NOUN
iajs-2001	65	23	+	+	CCONJ
iajs-2001	65	24	𝚥(y	𝚥(y	PROPN
iajs-2001	65	25	)	)	PUNCT
iajs-2001	65	26	=	=	SYM
iajs-2001	65	27	𝚥(y	𝚥(y	PROPN
iajs-2001	65	28	)	)	PUNCT
iajs-2001	65	29	.	.	PUNCT
iajs-2001	66	1	hence	hence	ADV
iajs-2001	66	2	ay	ay	NOUN
iajs-2001	66	3	∊	∊	PROPN
iajs-2001	66	4	ȷ	ȷ	PROPN
iajs-2001	66	5	y	y	PROPN
iajs-2001	66	6	or	or	CCONJ
iajs-2001	66	7	by	by	ADP
iajs-2001	66	8	∊	∊	PROPN
iajs-2001	66	9	′ȷ	′ȷ	PROPN
iajs-2001	66	10	y	y	PROPN
iajs-2001	66	11	or	or	CCONJ
iajs-2001	66	12	'	'	PUNCT
iajs-2001	66	13	ab′	ab′	NOUN
iajs-2001	66	14	∊	∊	NUM
iajs-2001	66	15	′	′	NUM
iajs-2001	67	1	ȷ	ȷ	NOUN
iajs-2001	67	2	y	y	NOUN
iajs-2001	67	3	:	:	PUNCT
iajs-2001	68	1	y	y	PROPN
iajs-2001	68	2	.	.	PUNCT
iajs-2001	69	1	thus'a	thus'a	X
iajs-2001	69	2	∩	∩	NOUN
iajs-2001	69	3	b	b	NOUN
iajs-2001	69	4	⊆	⊆	NUM
iajs-2001	69	5	𝚥(y	𝚥(y	NOUN
iajs-2001	69	6	)	)	PUNCT
iajs-2001	69	7	,	,	PUNCT
iajs-2001	69	8	implies	imply	VERB
iajs-2001	69	9	that	that	SCONJ
iajs-2001	69	10	a	a	DET
iajs-2001	69	11	∩	∩	ADJ
iajs-2001	69	12	b	b	X
iajs-2001	69	13	+	+	CCONJ
iajs-2001	69	14	𝚥(y	𝚥(y	PROPN
iajs-2001	69	15	)	)	PUNCT
iajs-2001	69	16	=	=	SYM
iajs-2001	69	17	𝚥(y	𝚥(y	PROPN
iajs-2001	69	18	)	)	PUNCT
iajs-2001	69	19	thus	thus	ADV
iajs-2001	69	20	,	,	PUNCT
iajs-2001	69	21	we	we	PRON
iajs-2001	69	22	have	have	VERB
iajs-2001	69	23	ay	ay	PROPN
iajs-2001	69	24	∊	∊	PROPN
iajs-2001	69	25	a	a	DET
iajs-2001	69	26	∩	∩	ADJ
iajs-2001	69	27	b	b	NOUN
iajs-2001	69	28	ȷ	ȷ	SYM
iajs-2001	69	29	y	y	PROPN
iajs-2001	69	30	or	or	CCONJ
iajs-2001	69	31	by	by	ADP
iajs-2001	69	32	∊	∊	PROPN
iajs-2001	69	33	a	a	DET
iajs-2001	69	34	∩	∩	ADJ
iajs-2001	69	35	b	b	NOUN
iajs-2001	69	36	ȷ	ȷ	SYM
iajs-2001	69	37	y	y	PROPN
iajs-2001	69	38	or	or	CCONJ
iajs-2001	69	39	ab	ab	PROPN
iajs-2001	69	40	∊	∊	PROPN
iajs-2001	69	41	a	a	DET
iajs-2001	69	42	∩	∩	ADJ
iajs-2001	69	43	b	b	NOUN
iajs-2001	69	44	ȷ	ȷ	X
iajs-2001	69	45	y	y	NOUN
iajs-2001	69	46	:	:	PUNCT
iajs-2001	69	47	y	y	PROPN
iajs-2001	69	48	.	.	PUNCT
iajs-2001	70	1	so	so	ADV
iajs-2001	70	2	,	,	PUNCT
iajs-2001	70	3	a	a	DET
iajs-2001	70	4	∩	∩	ADJ
iajs-2001	70	5	b	b	NOUN
iajs-2001	70	6	is	be	AUX
iajs-2001	70	7	a	a	DET
iajs-2001	70	8	wn-2'absorbing'''submodule	wn-2'absorbing'''submodule	PROPN
iajs-2001	70	9	of'"y	of'"y	NOUN
iajs-2001	70	10	.	.	PUNCT
iajs-2001	70	11	'	'	PUNCT
iajs-2001	71	1	proposition"8	proposition"8	PROPN
iajs-2001	71	2	'	'	PUNCT
iajs-2001	71	3	let"y	let"y	PROPN
iajs-2001	71	4	be	be	VERB
iajs-2001	71	5	an	an	DET
iajs-2001	71	6	r'-"module	r'-"module	NOUN
iajs-2001	71	7	,	,	PUNCT
iajs-2001	71	8	over"a	over"a	DET
iajs-2001	71	9	good	good	ADJ
iajs-2001	71	10	ring	ring	NOUN
iajs-2001	71	11	and	and	CCONJ
iajs-2001	71	12	a	a	DET
iajs-2001	71	13	,	,	PUNCT
iajs-2001	71	14	b	b	NOUN
iajs-2001	71	15	are	be	AUX
iajs-2001	71	16	submodules	submodule	NOUN
iajs-2001	71	17	of	of	ADP
iajs-2001	71	18	y	y	PROPN
iajs-2001	71	19	,	,	PUNCT
iajs-2001	71	20	a	a	DET
iajs-2001	71	21	⊈	⊈	PROPN
iajs-2001	71	22	b	b	NOUN
iajs-2001	71	23	and	and	CCONJ
iajs-2001	71	24	𝚥(y	𝚥(y	NOUN
iajs-2001	71	25	)	)	PUNCT
iajs-2001	71	26	⊆	⊆	NUM
iajs-2001	71	27	a	a	PRON
iajs-2001	71	28	,	,	PUNCT
iajs-2001	71	29	if	if	SCONJ
iajs-2001	71	30	b	b	PROPN
iajs-2001	71	31	is	be	AUX
iajs-2001	71	32	wn-2	wn-2	VERB
iajs-2001	71	33	-	-	ADJ
iajs-2001	71	34	absorbing	absorbing	ADJ
iajs-2001	71	35	submodules	submodule	NOUN
iajs-2001	71	36	of	of	ADP
iajs-2001	71	37	y	y	PROPN
iajs-2001	71	38	,	,	PUNCT
iajs-2001	71	39	then	then	ADV
iajs-2001	71	40	a	a	DET
iajs-2001	71	41	∩	∩	ADJ
iajs-2001	71	42	b	b	NOUN
iajs-2001	71	43	is	be	AUX
iajs-2001	71	44	wn-2	wn-2	ADP
iajs-2001	71	45	-	-	ADJ
iajs-2001	71	46	absorbing	absorbing	ADJ
iajs-2001	71	47	submodules	submodule	NOUN
iajs-2001	71	48	of	of	ADP
iajs-2001	71	49	a.	a.	NOUN
iajs-2001	71	50	proof	proof	NOUN
iajs-2001	71	51	since	since	SCONJ
iajs-2001	71	52	a	a	DET
iajs-2001	71	53	⊈	⊈	PROPN
iajs-2001	71	54	b	b	NOUN
iajs-2001	71	55	,	,	PUNCT
iajs-2001	71	56	then	then	ADV
iajs-2001	71	57	a	a	DET
iajs-2001	71	58	∩	∩	ADJ
iajs-2001	71	59	b	b	NOUN
iajs-2001	71	60	is	be	AUX
iajs-2001	71	61	a	a	DET
iajs-2001	71	62	proper	proper	ADJ
iajs-2001	71	63	submodule	submodule	NOUN
iajs-2001	71	64	of	of	ADP
iajs-2001	71	65	a	a	PRON
iajs-2001	71	66	,	,	PUNCT
iajs-2001	71	67	let	let	VERB
iajs-2001	71	68	0	0	NUM
iajs-2001	71	69	≠	≠	PROPN
iajs-2001	71	70	aby	aby	NOUN
iajs-2001	71	71	∈	∈	PROPN
iajs-2001	71	72	a	a	DET
iajs-2001	71	73	∩	∩	ADJ
iajs-2001	71	74	b	b	NOUN
iajs-2001	71	75	,	,	PUNCT
iajs-2001	71	76	with	with	ADP
iajs-2001	71	77	a	a	DET
iajs-2001	71	78	,	,	PUNCT
iajs-2001	71	79	b	b	X
iajs-2001	71	80	∈	∈	PROPN
iajs-2001	71	81	r	r	NOUN
iajs-2001	71	82	,	,	PUNCT
iajs-2001	71	83	y	y	PROPN
iajs-2001	71	84	∈	∈	PROPN
iajs-2001	71	85	y.	y.	NOUN
iajs-2001	71	86	then	then	ADV
iajs-2001	71	87	0	0	NUM
iajs-2001	71	88	≠	≠	PROPN
iajs-2001	71	89	aby	aby	NOUN
iajs-2001	71	90	∈	∈	PROPN
iajs-2001	71	91	a	a	DET
iajs-2001	71	92	and	and	CCONJ
iajs-2001	71	93	0	0	NUM
iajs-2001	71	94	≠	≠	PROPN
iajs-2001	71	95	aby	aby	PROPN
iajs-2001	71	96	∈	∈	PROPN
iajs-2001	71	97	b.	b.	PROPN
iajs-2001	71	98	since	since	SCONJ
iajs-2001	71	99	b	b	PROPN
iajs-2001	71	100	wn-2	wn-2	ADP
iajs-2001	71	101	-	-	ADJ
iajs-2001	71	102	absorbing	absorbing	ADJ
iajs-2001	71	103	submodules	submodule	NOUN
iajs-2001	71	104	of	of	ADP
iajs-2001	71	105	y	y	PROPN
iajs-2001	71	106	,	,	PUNCT
iajs-2001	71	107	then	then	ADV
iajs-2001	71	108	either	either	CCONJ
iajs-2001	71	109	ay	ay	PROPN
iajs-2001	71	110	∊	∊	PROPN
iajs-2001	71	111	′b′	′b′	VERB
iajs-2001	71	112	′ȷ	′ȷ	PROPN
iajs-2001	71	113	y	y	PROPN
iajs-2001	71	114	or	or	CCONJ
iajs-2001	71	115	by	by	ADP
iajs-2001	71	116	∊	∊	PROPN
iajs-2001	71	117	′b	′b	PROPN
iajs-2001	71	118	"	"	PUNCT
iajs-2001	71	119	"	"	PUNCT
iajs-2001	71	120	ȷ	ȷ	PROPN
iajs-2001	71	121	y	y	PROPN
iajs-2001	71	122	or	or	CCONJ
iajs-2001	71	123	ab	ab	PROPN
iajs-2001	71	124	∊	∊	PROPN
iajs-2001	71	125	′b	′b	PROPN
iajs-2001	72	1	′ȷ	′ȷ	NOUN
iajs-2001	72	2	y	y	PROPN
iajs-2001	72	3	:	:	PUNCT
iajs-2001	72	4	y	y	PROPN
iajs-2001	72	5	.	.	PUNCT
iajs-2001	73	1	that	that	PRON
iajs-2001	73	2	is	be	AUX
iajs-2001	73	3	either	either	CCONJ
iajs-2001	73	4	either	either	CCONJ
iajs-2001	73	5	ay	ay	PROPN
iajs-2001	73	6	∊	∊	PROPN
iajs-2001	73	7	b	b	PROPN
iajs-2001	73	8	"	"	PUNCT
iajs-2001	73	9	"	"	PUNCT
iajs-2001	73	10	ȷ	ȷ	PROPN
iajs-2001	73	11	y	y	PROPN
iajs-2001	73	12	∩	∩	NOUN
iajs-2001	73	13	a	a	X
iajs-2001	73	14	or	or	CCONJ
iajs-2001	73	15	by	by	ADP
iajs-2001	73	16	∊	∊	PROPN
iajs-2001	73	17	b	b	PROPN
iajs-2001	73	18	"	"	PUNCT
iajs-2001	73	19	ȷ	ȷ	PROPN
iajs-2001	73	20	y	y	PROPN
iajs-2001	73	21	"	"	PUNCT
iajs-2001	73	22	∩	∩	PROPN
iajs-2001	73	23	a	a	NOUN
iajs-2001	73	24	or	or	CCONJ
iajs-2001	73	25	aby	aby	NOUN
iajs-2001	73	26	⊆	⊆	SYM
iajs-2001	73	27	b	b	PROPN
iajs-2001	73	28	ȷ	ȷ	X
iajs-2001	73	29	y	y	PROPN
iajs-2001	73	30	∩	∩	PROPN
iajs-2001	73	31	a	a	PRON
iajs-2001	73	32	,	,	PUNCT
iajs-2001	73	33	hence	hence	ADV
iajs-2001	73	34	by	by	ADP
iajs-2001	73	35	moduler	moduler	ADJ
iajs-2001	73	36	law	law	NOUN
iajs-2001	73	37	we	we	PRON
iajs-2001	73	38	have	have	AUX
iajs-2001	73	39	either	either	CCONJ
iajs-2001	73	40	ay	ay	PROPN
iajs-2001	73	41	∊	∊	PROPN
iajs-2001	73	42	a	a	DET
iajs-2001	73	43	∩	∩	ADJ
iajs-2001	73	44	b	b	NOUN
iajs-2001	73	45	ȷ	ȷ	NOUN
iajs-2001	73	46	a	a	PRON
iajs-2001	73	47	or	or	CCONJ
iajs-2001	73	48	by	by	ADP
iajs-2001	73	49	∊	∊	PROPN
iajs-2001	73	50	a	a	DET
iajs-2001	73	51	∩	∩	ADJ
iajs-2001	73	52	b	b	NOUN
iajs-2001	73	53	ȷ	ȷ	NOUN
iajs-2001	73	54	a	a	NOUN
iajs-2001	73	55	or	or	CCONJ
iajs-2001	73	56	ab	ab	PROPN
iajs-2001	73	57	∊	∊	PROPN
iajs-2001	73	58	a	a	DET
iajs-2001	73	59	∩	∩	ADJ
iajs-2001	73	60	b	b	NOUN
iajs-2001	73	61	ȷ	ȷ	NOUN
iajs-2001	73	62	a	a	PRON
iajs-2001	73	63	:	:	PUNCT
iajs-2001	73	64	y	y	PROPN
iajs-2001	73	65	⊆	⊆	NUM
iajs-2001	73	66	a	a	DET
iajs-2001	73	67	∩	∩	ADJ
iajs-2001	73	68	b	b	NOUN
iajs-2001	73	69	ȷ	ȷ	NOUN
iajs-2001	73	70	a	a	PRON
iajs-2001	73	71	:	:	PUNCT
iajs-2001	73	72	a	a	DET
iajs-2001	73	73	,	,	PUNCT
iajs-2001	73	74	thus	thus	ADV
iajs-2001	73	75	a	a	DET
iajs-2001	73	76	∩	∩	ADJ
iajs-2001	73	77	b	b	NOUN
iajs-2001	73	78	is	be	AUX
iajs-2001	73	79	wn-2	wn-2	ADP
iajs-2001	73	80	-	-	ADJ
iajs-2001	73	81	absorbing	absorbing	ADJ
iajs-2001	73	82	submodules	submodule	NOUN
iajs-2001	73	83	of	of	ADP
iajs-2001	73	84	a.	a.	NOUN
iajs-2001	73	85	as	as	ADP
iajs-2001	73	86	a	a	DET
iajs-2001	73	87	direct	direct	ADJ
iajs-2001	73	88	consequence	consequence	NOUN
iajs-2001	73	89	of	of	ADP
iajs-2001	73	90	proposition	proposition	NOUN
iajs-2001	73	91	2.8	2.8	NUM
iajs-2001	73	92	,	,	PUNCT
iajs-2001	73	93	we	we	PRON
iajs-2001	73	94	get'''the''following"'corollary	get'''the''following"'corollary	ADJ
iajs-2001	73	95	'	'	PUNCT
iajs-2001	73	96	corollary"9	corollary"9	PROPN
iajs-2001	73	97	'	'	PUNCT
iajs-2001	73	98	let"y	let"y	X
iajs-2001	73	99	be'''an"r'-"module	be'''an"r'-"module	PROPN
iajs-2001	73	100	"	"	PUNCT
iajs-2001	73	101	,	,	PUNCT
iajs-2001	73	102	over	over	ADP
iajs-2001	73	103	a"good	a"good	NOUN
iajs-2001	73	104	ring	ring	NOUN
iajs-2001	73	105	and	and	CCONJ
iajs-2001	73	106	a	a	DET
iajs-2001	73	107	,	,	PUNCT
iajs-2001	73	108	b	b	NOUN
iajs-2001	73	109	are"submodules	are"submodule	NOUN
iajs-2001	73	110	of	of	ADP
iajs-2001	73	111	y	y	PROPN
iajs-2001	73	112	,	,	PUNCT
iajs-2001	73	113	a	a	DET
iajs-2001	73	114	⊈	⊈	PROPN
iajs-2001	73	115	b	b	NOUN
iajs-2001	73	116	and	and	CCONJ
iajs-2001	73	117	a	a	PRON
iajs-2001	73	118	is	be	AUX
iajs-2001	73	119	a"maximal"submodule	a"maximal"submodule	ADP
iajs-2001	73	120	of	of	ADP
iajs-2001	73	121	y	y	PROPN
iajs-2001	73	122	,	,	PUNCT
iajs-2001	73	123	if	if	SCONJ
iajs-2001	73	124	b	b	NOUN
iajs-2001	73	125	is	be	AUX
iajs-2001	73	126	wn-2	wn-2	NOUN
iajs-2001	73	127	-	-	PUNCT
iajs-2001	73	128	absorbing"submodules	absorbing"submodule	NOUN
iajs-2001	73	129	of	of	ADP
iajs-2001	73	130	y	y	PROPN
iajs-2001	73	131	,	,	PUNCT
iajs-2001	73	132	then	then	ADV
iajs-2001	73	133	a	a	DET
iajs-2001	73	134	∩	∩	ADJ
iajs-2001	73	135	b	b	NOUN
iajs-2001	73	136	is	be	AUX
iajs-2001	73	137	wn-2	wn-2	ADP
iajs-2001	73	138	-	-	ADJ
iajs-2001	73	139	absorbing	absorbing	ADJ
iajs-2001	73	140	submodules	submodule	NOUN
iajs-2001	73	141	of	of	ADP
iajs-2001	73	142	a.	a.	NOUN
iajs-2001	73	143	proposition	proposition	NOUN
iajs-2001	73	144	10	10	NUM
iajs-2001	73	145	let	let	VERB
iajs-2001	73	146	y'''be''an"r'-'module'',"and''a	y'''be''an"r'-'module'',"and''a	PROPN
iajs-2001	73	147	proper"submodule'"of	proper"submodule'"of	PROPN
iajs-2001	73	148	y.	y.	PROPN
iajs-2001	73	149	''then"a'''is"wn-2"'absorbing'''submodules'''of"y	''then"a'''is"wn-2"'absorbing'''submodules'''of"y	PART
iajs-2001	73	150	if'''and'''only'''if'''for	if'''and'''only'''if'''for	ADP
iajs-2001	73	151	each	each	DET
iajs-2001	73	152	submodule	submodule	PROPN
iajs-2001	73	153	b	b	PROPN
iajs-2001	73	154	of	of	ADP
iajs-2001	73	155	y	y	PROPN
iajs-2001	73	156	with	with	ADP
iajs-2001	73	157	[	[	PUNCT
iajs-2001	73	158	a	a	X
iajs-2001	73	159	:	:	PUNCT
iajs-2001	73	160	y	y	NOUN
iajs-2001	73	161	]	]	X
iajs-2001	74	1	⊆	⊆	NUM
iajs-2001	75	1	[	[	X
iajs-2001	75	2	a	a	DET
iajs-2001	75	3	:	:	PUNCT
iajs-2001	75	4	b	b	NOUN
iajs-2001	75	5	]	]	PUNCT
iajs-2001	75	6	and	and	CCONJ
iajs-2001	75	7	for	for	ADP
iajs-2001	75	8	each	each	DET
iajs-2001	75	9	a	a	NOUN
iajs-2001	75	10	,	,	PUNCT
iajs-2001	75	11	b	b	PROPN
iajs-2001	75	12	∊	∊	NUM
iajs-2001	75	13	r	r	NOUN
iajs-2001	75	14	with	with	ADP
iajs-2001	75	15	0	0	NUM
iajs-2001	75	16	≠	≠	PROPN
iajs-2001	75	17	abb	abb	PROPN
iajs-2001	75	18	⊆	⊆	SYM
iajs-2001	75	19	a	a	PRON
iajs-2001	75	20	,	,	PUNCT
iajs-2001	75	21	implies	imply	VERB
iajs-2001	75	22	that	that	SCONJ
iajs-2001	75	23	either	either	CCONJ
iajs-2001	75	24	ab⊆a	ab⊆a	PROPN
iajs-2001	75	25	+	+	PROPN
iajs-2001	75	26	𝚥(y	𝚥(y	PROPN
iajs-2001	75	27	)	)	PUNCT
iajs-2001	75	28	or	or	CCONJ
iajs-2001	75	29	bb⊆a	bb⊆a	VERB
iajs-2001	75	30	+	+	CCONJ
iajs-2001	75	31	𝚥(y	𝚥(y	PROPN
iajs-2001	75	32	)	)	PUNCT
iajs-2001	75	33	or	or	CCONJ
iajs-2001	75	34	ab	ab	PROPN
iajs-2001	75	35	∊	∊	PROPN
iajs-2001	76	1	[	[	X
iajs-2001	76	2	a+𝚥(y):y	a+𝚥(y):y	PROPN
iajs-2001	76	3	]	]	PUNCT
iajs-2001	76	4	.	.	PUNCT
iajs-2001	77	1	mathematics	mathematic	NOUN
iajs-2001	77	2	|	|	ADV
iajs-2001	77	3	121	121	NUM
iajs-2001	77	4	ibn	ibn	PROPN
iajs-2001	77	5	al	al	PROPN
iajs-2001	77	6	-	-	PUNCT
iajs-2001	77	7	haitham	haitham	PROPN
iajs-2001	77	8	jour	jour	X
iajs-2001	77	9	.	.	PROPN
iajs-2001	77	10	for	for	ADP
iajs-2001	77	11	pure	pure	ADJ
iajs-2001	77	12	&	&	CCONJ
iajs-2001	77	13	appl	appl	PROPN
iajs-2001	77	14	.	.	PUNCT
iajs-2001	78	1	sci	sci	PROPN
iajs-2001	78	2	.	.	PROPN
iajs-2001	78	3	ihjpas	ihjpa	VERB
iajs-2001	78	4	https://doi.org/10.30526/31.3.2001	https://doi.org/10.30526/31.3.2001	PROPN
iajs-2001	78	5	vol	vol	NOUN
iajs-2001	78	6	.	.	PROPN
iajs-2001	79	1	31	31	NUM
iajs-2001	80	1	(	(	PUNCT
iajs-2001	80	2	3	3	NUM
iajs-2001	80	3	)	)	PUNCT
iajs-2001	80	4	2018	2018	NUM
iajs-2001	80	5	proof	proof	NOUN
iajs-2001	80	6	suppose	suppose	VERB
iajs-2001	80	7	that	that	SCONJ
iajs-2001	80	8	0	0	NUM
iajs-2001	80	9	≠	≠	PROPN
iajs-2001	80	10	abb	abb	NOUN
iajs-2001	80	11	⊆	⊆	NUM
iajs-2001	80	12	a	a	PRON
iajs-2001	80	13	for	for	ADP
iajs-2001	80	14	each	each	DET
iajs-2001	80	15	submodule	submodule	PROPN
iajs-2001	80	16	b	b	PROPN
iajs-2001	80	17	of	of	ADP
iajs-2001	80	18	y	y	PROPN
iajs-2001	80	19	and	and	CCONJ
iajs-2001	80	20	a	a	PRON
iajs-2001	80	21	,	,	PUNCT
iajs-2001	80	22	b	b	PROPN
iajs-2001	80	23	∊	∊	PROPN
iajs-2001	80	24	r.	r.	PROPN
iajs-2001	80	25	then	then	ADV
iajs-2001	80	26	0	0	NUM
iajs-2001	80	27	≠	≠	PROPN
iajs-2001	80	28	aby	aby	NOUN
iajs-2001	80	29	∊	∊	NOUN
iajs-2001	80	30	a	a	PRON
iajs-2001	80	31	for	for	ADP
iajs-2001	80	32	each	each	DET
iajs-2001	80	33	y	y	PROPN
iajs-2001	80	34	∊	∊	PROPN
iajs-2001	80	35	b	b	PROPN
iajs-2001	80	36	⊆	⊆	NUM
iajs-2001	80	37	y.	y.	NOUN
iajs-2001	80	38	but	but	CCONJ
iajs-2001	80	39	a	a	PRON
iajs-2001	80	40	is	be	AUX
iajs-2001	80	41	wn-2	wn-2	NOUN
iajs-2001	80	42	-	-	ADJ
iajs-2001	80	43	absorbing	absorbing	ADJ
iajs-2001	80	44	submodules	submodule	NOUN
iajs-2001	80	45	of	of	ADP
iajs-2001	80	46	y	y	PROPN
iajs-2001	80	47	,	,	PUNCT
iajs-2001	80	48	implies	imply	VERB
iajs-2001	80	49	that	that	SCONJ
iajs-2001	80	50	either	either	CCONJ
iajs-2001	80	51	ay	ay	PROPN
iajs-2001	80	52	∊	∊	PROPN
iajs-2001	80	53	a	a	DET
iajs-2001	80	54	+	+	ADJ
iajs-2001	80	55	𝚥(y	𝚥(y	NOUN
iajs-2001	80	56	)	)	PUNCT
iajs-2001	80	57	or	or	CCONJ
iajs-2001	80	58	by	by	ADP
iajs-2001	80	59	∊	∊	PROPN
iajs-2001	80	60	a	a	PRON
iajs-2001	80	61	+	+	ADJ
iajs-2001	80	62	𝚥(y	𝚥(y	NOUN
iajs-2001	80	63	)	)	PUNCT
iajs-2001	80	64	or	or	CCONJ
iajs-2001	80	65	ab	ab	PROPN
iajs-2001	80	66	∊	∊	PROPN
iajs-2001	80	67	[	[	X
iajs-2001	80	68	a+𝚥(y):y	a+𝚥(y):y	PROPN
iajs-2001	80	69	]	]	PUNCT
iajs-2001	80	70	.	.	PUNCT
iajs-2001	81	1	it	it	PRON
iajs-2001	81	2	follows	follow	VERB
iajs-2001	81	3	that	that	SCONJ
iajs-2001	81	4	either	either	CCONJ
iajs-2001	81	5	ab⊆a	ab⊆a	PROPN
iajs-2001	81	6	+	+	PROPN
iajs-2001	81	7	𝚥(y	𝚥(y	PROPN
iajs-2001	81	8	)	)	PUNCT
iajs-2001	81	9	or	or	CCONJ
iajs-2001	81	10	bb⊆a	bb⊆a	VERB
iajs-2001	81	11	+	+	CCONJ
iajs-2001	81	12	𝚥(y	𝚥(y	PROPN
iajs-2001	81	13	)	)	PUNCT
iajs-2001	81	14	or	or	CCONJ
iajs-2001	81	15	ab	ab	PROPN
iajs-2001	81	16	∊	∊	PROPN
iajs-2001	82	1	[	[	X
iajs-2001	82	2	a+𝚥(y):y	a+𝚥(y):y	PROPN
iajs-2001	82	3	]	]	PUNCT
iajs-2001	82	4	.	.	PUNCT
iajs-2001	83	1	conversely	conversely	ADV
iajs-2001	83	2	:	:	PUNCT
iajs-2001	83	3	let	let	VERB
iajs-2001	83	4	0	0	NUM
iajs-2001	83	5	≠	≠	PROPN
iajs-2001	83	6	aby	aby	NOUN
iajs-2001	83	7	∊	∊	NOUN
iajs-2001	83	8	a	a	PRON
iajs-2001	83	9	for	for	ADP
iajs-2001	83	10	all	all	DET
iajs-2001	83	11	y	y	PROPN
iajs-2001	83	12	∊	∊	PROPN
iajs-2001	83	13	y	y	PROPN
iajs-2001	83	14	,	,	PUNCT
iajs-2001	83	15	a	a	PRON
iajs-2001	83	16	,	,	PUNCT
iajs-2001	83	17	b	b	PROPN
iajs-2001	83	18	∊	∊	NUM
iajs-2001	83	19	r	r	NOUN
iajs-2001	83	20	.	.	PUNCT
iajs-2001	84	1	that	that	PRON
iajs-2001	84	2	is	be	AUX
iajs-2001	84	3	0	0	NUM
iajs-2001	84	4	≠	≠	PROPN
iajs-2001	84	5	aby	aby	NOUN
iajs-2001	84	6	⊆	⊆	SYM
iajs-2001	84	7	a	a	PRON
iajs-2001	84	8	,	,	PUNCT
iajs-2001	84	9	implies	imply	VERB
iajs-2001	84	10	that	that	SCONJ
iajs-2001	84	11	ab	ab	PROPN
iajs-2001	84	12	∊	∊	PROPN
iajs-2001	84	13	[	[	PUNCT
iajs-2001	84	14	a	a	X
iajs-2001	84	15	:	:	PUNCT
iajs-2001	84	16	y	y	NOUN
iajs-2001	84	17	]	]	X
iajs-2001	84	18	⊆	⊆	NUM
iajs-2001	84	19	[	[	X
iajs-2001	84	20	a	a	X
iajs-2001	84	21	:	:	PUNCT
iajs-2001	84	22	b	b	NOUN
iajs-2001	84	23	]	]	X
iajs-2001	84	24	,	,	PUNCT
iajs-2001	84	25	it	it	PRON
iajs-2001	84	26	follows	follow	VERB
iajs-2001	84	27	that	that	SCONJ
iajs-2001	84	28	0	0	NUM
iajs-2001	84	29	≠	≠	PROPN
iajs-2001	84	30	abb	abb	NOUN
iajs-2001	84	31	⊆	⊆	NUM
iajs-2001	84	32	a	a	DET
iajs-2001	84	33	hence	hence	ADV
iajs-2001	84	34	by	by	ADP
iajs-2001	84	35	hypothesis	hypothesis	NOUN
iajs-2001	84	36	either	either	CCONJ
iajs-2001	84	37	ab⊆	ab⊆	PROPN
iajs-2001	84	38	a	a	DET
iajs-2001	84	39	+	+	NOUN
iajs-2001	84	40	𝚥(y	𝚥(y	NOUN
iajs-2001	84	41	)	)	PUNCT
iajs-2001	84	42	or	or	CCONJ
iajs-2001	84	43	bb⊆	bb⊆	VERB
iajs-2001	84	44	a	a	DET
iajs-2001	84	45	+	+	ADJ
iajs-2001	84	46	𝚥(y	𝚥(y	NOUN
iajs-2001	84	47	)	)	PUNCT
iajs-2001	84	48	or	or	CCONJ
iajs-2001	84	49	ab	ab	PROPN
iajs-2001	84	50	∊	∊	PROPN
iajs-2001	85	1	[	[	X
iajs-2001	85	2	a+𝚥(y):y	a+𝚥(y):y	PROPN
iajs-2001	85	3	]	]	PUNCT
iajs-2001	85	4	.	.	PUNCT
iajs-2001	86	1	that	that	PRON
iajs-2001	86	2	is	be	AUX
iajs-2001	86	3	either	either	PRON
iajs-2001	86	4	ay	ay	PROPN
iajs-2001	86	5	∊	∊	PROPN
iajs-2001	86	6	a	a	DET
iajs-2001	86	7	+	+	ADJ
iajs-2001	86	8	𝚥(y	𝚥(y	NOUN
iajs-2001	86	9	)	)	PUNCT
iajs-2001	86	10	or	or	CCONJ
iajs-2001	86	11	by	by	ADP
iajs-2001	86	12	∊	∊	PROPN
iajs-2001	86	13	a	a	PRON
iajs-2001	86	14	+	+	ADJ
iajs-2001	86	15	𝚥(y	𝚥(y	NOUN
iajs-2001	86	16	)	)	PUNCT
iajs-2001	86	17	or	or	CCONJ
iajs-2001	86	18	ab	ab	PROPN
iajs-2001	86	19	∊	∊	PROPN
iajs-2001	87	1	[	[	X
iajs-2001	87	2	a+𝚥(y):y	a+𝚥(y):y	PROPN
iajs-2001	87	3	]	]	PUNCT
iajs-2001	87	4	.	.	PUNCT
iajs-2001	88	1	thus	thus	ADV
iajs-2001	88	2	a	a	PRON
iajs-2001	88	3	is	be	AUX
iajs-2001	88	4	wn-2	wn-2	NOUN
iajs-2001	88	5	-	-	ADJ
iajs-2001	88	6	absorbing	absorbing	ADJ
iajs-2001	88	7	submodules	submodule	NOUN
iajs-2001	88	8	of	of	ADP
iajs-2001	88	9	y.	y.	PROPN
iajs-2001	88	10	'	'	PUNCT
iajs-2001	88	11	proposition	proposition	NOUN
iajs-2001	88	12	11	11	NUM
iajs-2001	88	13	let	let	VERB
iajs-2001	88	14	y	y	PRON
iajs-2001	88	15	be	be	AUX
iajs-2001	88	16	an"r'-'module'''and"a	an"r'-'module'''and"a	X
iajs-2001	88	17	is'''a'''proper'''submodule''of	is'''a'''proper'''submodule''of	PROPN
iajs-2001	88	18	y.	y.	PROPN
iajs-2001	88	19	'if"a''is"wn-2"-'absorbing	'if"a''is"wn-2"-'absorbing	PROPN
iajs-2001	88	20	submodules"'of"y	submodules"'of"y	NOUN
iajs-2001	88	21	,	,	PUNCT
iajs-2001	88	22	'	'	PUNCT
iajs-2001	88	23	then"s	then"s	VERB
iajs-2001	88	24	a	a	DET
iajs-2001	88	25	is"wn-2"-'absorbing"submodules"'of	is"wn-2"-'absorbing"submodules"'of	NOUN
iajs-2001	88	26	an	an	DET
iajs-2001	88	27	'	'	PUNCT
iajs-2001	88	28	s	s	NOUN
iajs-2001	88	29	r-"module	r-"module	NOUN
iajs-2001	88	30	"	"	PUNCT
iajs-2001	88	31	s	s	PROPN
iajs-2001	88	32	y	y	PROPN
iajs-2001	88	33	,	,	PUNCT
iajs-2001	88	34	where	where	SCONJ
iajs-2001	88	35	s	s	AUX
iajs-2001	88	36	is''a'''multiplicatively'''closed'"subset"'of	is''a'''multiplicatively'''closed'"subset"'of	DET
iajs-2001	88	37	'	'	PUNCT
iajs-2001	88	38	r.	r.	NOUN
iajs-2001	88	39	proof	proof	NOUN
iajs-2001	88	40	let	let	VERB
iajs-2001	88	41	0	0	NUM
iajs-2001	88	42	≠	≠	PROPN
iajs-2001	88	43	∈	∈	PROPN
iajs-2001	88	44	s	s	VERB
iajs-2001	88	45	a	a	PRON
iajs-2001	88	46	,	,	PUNCT
iajs-2001	88	47	where	where	SCONJ
iajs-2001	88	48	,	,	PUNCT
iajs-2001	88	49	∊	∊	PROPN
iajs-2001	88	50	s	s	NOUN
iajs-2001	88	51	r	r	NOUN
iajs-2001	88	52	and	and	CCONJ
iajs-2001	88	53	∈	∈	PROPN
iajs-2001	88	54	s	s	PART
iajs-2001	88	55	y	y	NOUN
iajs-2001	88	56	with	with	ADP
iajs-2001	88	57	r	r	NOUN
iajs-2001	88	58	,	,	PUNCT
iajs-2001	88	59	r	r	NOUN
iajs-2001	88	60	∈	∈	PROPN
iajs-2001	88	61	r	r	NOUN
iajs-2001	88	62	,	,	PUNCT
iajs-2001	88	63	s	s	X
iajs-2001	88	64	,	,	PUNCT
iajs-2001	88	65	s	s	PART
iajs-2001	88	66	,	,	PUNCT
iajs-2001	88	67	s	s	PROPN
iajs-2001	88	68	∈	∈	PROPN
iajs-2001	88	69	s	s	X
iajs-2001	88	70	,	,	PUNCT
iajs-2001	88	71	y	y	PROPN
iajs-2001	88	72	∊	∊	PROPN
iajs-2001	88	73	y.	y.	PROPN
iajs-2001	88	74	then	then	ADV
iajs-2001	88	75	0	0	NUM
iajs-2001	88	76	≠	≠	PROPN
iajs-2001	88	77	∈	∈	PROPN
iajs-2001	88	78	s	s	VERB
iajs-2001	88	79	a	a	PRON
iajs-2001	88	80	,	,	PUNCT
iajs-2001	88	81	where	where	SCONJ
iajs-2001	88	82	t	t	NOUN
iajs-2001	88	83	=	=	PUNCT
iajs-2001	88	84	s	s	X
iajs-2001	88	85	s	s	X
iajs-2001	88	86	s	s	X
iajs-2001	88	87	∈	∈	NOUN
iajs-2001	88	88	s	s	PART
iajs-2001	88	89	,	,	PUNCT
iajs-2001	88	90	then	then	ADV
iajs-2001	88	91	there	there	PRON
iajs-2001	88	92	exists	exist	VERB
iajs-2001	88	93	t	t	PROPN
iajs-2001	88	94	∈	∈	PROPN
iajs-2001	88	95	s	s	VERB
iajs-2001	88	96	such	such	ADJ
iajs-2001	88	97	that	that	DET
iajs-2001	88	98	0	0	NUM
iajs-2001	88	99	≠	≠	PROPN
iajs-2001	88	100	t	t	NOUN
iajs-2001	88	101	r	r	NOUN
iajs-2001	88	102	r	r	NOUN
iajs-2001	88	103	y	y	PROPN
iajs-2001	88	104	∈	∈	PROPN
iajs-2001	88	105	a.	a.	NOUN
iajs-2001	88	106	but	but	CCONJ
iajs-2001	88	107	a	a	PRON
iajs-2001	88	108	is	be	AUX
iajs-2001	88	109	wn-2	wn-2	NOUN
iajs-2001	88	110	-	-	ADJ
iajs-2001	88	111	absorbing	absorbing	ADJ
iajs-2001	88	112	submodules	submodule	NOUN
iajs-2001	88	113	of	of	ADP
iajs-2001	88	114	y	y	PROPN
iajs-2001	88	115	,	,	PUNCT
iajs-2001	88	116	then	then	ADV
iajs-2001	88	117	either	either	CCONJ
iajs-2001	88	118	t	t	NOUN
iajs-2001	88	119	r	r	NOUN
iajs-2001	88	120	y	y	PROPN
iajs-2001	88	121	∈	∈	PROPN
iajs-2001	88	122	a	a	DET
iajs-2001	88	123	+	+	ADJ
iajs-2001	88	124	𝚥(y	𝚥(y	NOUN
iajs-2001	88	125	)	)	PUNCT
iajs-2001	88	126	or	or	CCONJ
iajs-2001	88	127	t	t	NOUN
iajs-2001	88	128	r	r	NOUN
iajs-2001	88	129	y	y	PROPN
iajs-2001	88	130	∈	∈	PROPN
iajs-2001	88	131	a	a	DET
iajs-2001	88	132	+	+	ADJ
iajs-2001	88	133	𝚥(y	𝚥(y	NOUN
iajs-2001	88	134	)	)	PUNCT
iajs-2001	88	135	or	or	CCONJ
iajs-2001	88	136	t	t	NOUN
iajs-2001	88	137	r	r	NOUN
iajs-2001	88	138	r	r	NOUN
iajs-2001	88	139	∈	∈	PROPN
iajs-2001	88	140	a	a	DET
iajs-2001	88	141	+	+	NOUN
iajs-2001	88	142	𝚥(y	𝚥(y	NOUN
iajs-2001	88	143	)	)	PUNCT
iajs-2001	88	144	:	:	PUNCT
iajs-2001	89	1	y	y	X
iajs-2001	89	2	]	]	PUNCT
iajs-2001	89	3	.	.	PUNCT
iajs-2001	89	4	implies	imply	VERB
iajs-2001	89	5	that	that	SCONJ
iajs-2001	89	6	∈	∈	PROPN
iajs-2001	89	7	s	s	VERB
iajs-2001	89	8	a	a	PRON
iajs-2001	89	9	ȷ	ȷ	NOUN
iajs-2001	89	10	y	y	PROPN
iajs-2001	89	11	⊆	⊆	NUM
iajs-2001	89	12	s	s	VERB
iajs-2001	89	13	a	a	DET
iajs-2001	89	14	ȷ	ȷ	NOUN
iajs-2001	89	15	s	s	PART
iajs-2001	89	16	y	y	NOUN
iajs-2001	89	17	or	or	CCONJ
iajs-2001	89	18	∈	∈	PROPN
iajs-2001	89	19	s	s	VERB
iajs-2001	89	20	a	a	PRON
iajs-2001	90	1	ȷ	ȷ	NOUN
iajs-2001	90	2	y	y	PROPN
iajs-2001	90	3	⊆	⊆	NUM
iajs-2001	90	4	s	s	VERB
iajs-2001	90	5	a	a	DET
iajs-2001	90	6	ȷ	ȷ	NOUN
iajs-2001	90	7	s	s	PART
iajs-2001	90	8	y	y	NOUN
iajs-2001	90	9	or	or	CCONJ
iajs-2001	90	10	∈	∈	PROPN
iajs-2001	90	11	s	s	VERB
iajs-2001	90	12	a	a	DET
iajs-2001	90	13	+	+	ADJ
iajs-2001	90	14	𝚥(y	𝚥(y	NOUN
iajs-2001	90	15	)	)	PUNCT
iajs-2001	90	16	:	:	PUNCT
iajs-2001	91	1	y	y	X
iajs-2001	91	2	]	]	PUNCT
iajs-2001	91	3	⊆	⊆	NUM
iajs-2001	92	1	[	[	X
iajs-2001	92	2	s	s	X
iajs-2001	92	3	a	a	DET
iajs-2001	92	4	ȷ	ȷ	NOUN
iajs-2001	92	5	s	s	NOUN
iajs-2001	92	6	y	y	NOUN
iajs-2001	92	7	:	:	PUNCT
iajs-2001	92	8	s	s	VERB
iajs-2001	92	9	y	y	PROPN
iajs-2001	92	10	.	.	PUNCT
iajs-2001	93	1	thus	thus	ADV
iajs-2001	93	2	either	either	CCONJ
iajs-2001	93	3	∈	∈	PROPN
iajs-2001	93	4	s	s	VERB
iajs-2001	93	5	a	a	DET
iajs-2001	93	6	ȷ	ȷ	NOUN
iajs-2001	93	7	s	s	PART
iajs-2001	93	8	y	y	NOUN
iajs-2001	93	9	or	or	CCONJ
iajs-2001	93	10	∈	∈	PROPN
iajs-2001	93	11	s	s	VERB
iajs-2001	93	12	a	a	DET
iajs-2001	93	13	ȷ	ȷ	NOUN
iajs-2001	93	14	s	s	PART
iajs-2001	93	15	y	y	NOUN
iajs-2001	93	16	or	or	CCONJ
iajs-2001	93	17	∈	∈	PROPN
iajs-2001	94	1	[	[	X
iajs-2001	94	2	s	s	X
iajs-2001	94	3	a	a	DET
iajs-2001	94	4	ȷ	ȷ	NOUN
iajs-2001	94	5	s	s	NOUN
iajs-2001	94	6	y	y	NOUN
iajs-2001	94	7	:	:	PUNCT
iajs-2001	94	8	s	s	VERB
iajs-2001	94	9	y	y	NOUN
iajs-2001	94	10	.	.	PUNCT
iajs-2001	95	1	hence	hence	ADV
iajs-2001	95	2	s	s	VERB
iajs-2001	95	3	a	a	PRON
iajs-2001	95	4	is	be	AUX
iajs-2001	95	5	wn-2	wn-2	NOUN
iajs-2001	95	6	-	-	ADJ
iajs-2001	95	7	absorbing	absorbing	ADJ
iajs-2001	95	8	submodules	submodule	NOUN
iajs-2001	95	9	of	of	ADP
iajs-2001	95	10	an	an	DET
iajs-2001	95	11	s	s	NOUN
iajs-2001	95	12	rmodule	rmodule	NOUN
iajs-2001	95	13	s	s	PART
iajs-2001	95	14	y.	y.	NOUN
iajs-2001	95	15	proposition	proposition	NOUN
iajs-2001	95	16	12	12	NUM
iajs-2001	95	17	let	let	VERB
iajs-2001	95	18	h	h	NOUN
iajs-2001	95	19	:	:	PUNCT
iajs-2001	95	20	y	y	PROPN
iajs-2001	95	21	→	→	SYM
iajs-2001	95	22	y	y	AUX
iajs-2001	95	23	`	`	PUNCT
iajs-2001	95	24	be	be	AUX
iajs-2001	95	25	a	a	DET
iajs-2001	95	26	small	small	ADJ
iajs-2001	95	27	r	r	NOUN
iajs-2001	95	28	-	-	PUNCT
iajs-2001	95	29	epimorphism	epimorphism	NOUN
iajs-2001	95	30	.	.	PUNCT
iajs-2001	96	1	and	and	CCONJ
iajs-2001	96	2	a	a	PRON
iajs-2001	96	3	is	be	AUX
iajs-2001	96	4	wn-2	wn-2	NOUN
iajs-2001	96	5	-	-	ADJ
iajs-2001	96	6	absorbing	absorbing	ADJ
iajs-2001	96	7	submodules	submodule	NOUN
iajs-2001	96	8	of	of	ADP
iajs-2001	96	9	y	y	PROPN
iajs-2001	96	10	containing	contain	VERB
iajs-2001	96	11	kerh	kerh	PROPN
iajs-2001	96	12	.	.	PUNCT
iajs-2001	97	1	,	,	PUNCT
iajs-2001	97	2	then	then	ADV
iajs-2001	97	3	h	h	PROPN
iajs-2001	97	4	a	a	PRON
iajs-2001	97	5	is	be	AUX
iajs-2001	97	6	wn-2	wn-2	NOUN
iajs-2001	97	7	-	-	ADJ
iajs-2001	97	8	absorbing	absorbing	ADJ
iajs-2001	97	9	submodules	submodule	NOUN
iajs-2001	97	10	of	of	ADP
iajs-2001	97	11	of	of	ADP
iajs-2001	97	12	y	y	PROPN
iajs-2001	97	13	`	`	PUNCT
iajs-2001	97	14	.	.	PUNCT
iajs-2001	98	1	'	'	PUNCT
iajs-2001	98	2	proof	proof	NOUN
iajs-2001	98	3	"	"	PUNCT
iajs-2001	98	4	'	'	PUNCT
iajs-2001	98	5	it''is''clear''that	it''is''clear''that	PRON
iajs-2001	98	6	ℎ(a	ℎ(a	PROPN
iajs-2001	98	7	)	)	PUNCT
iajs-2001	98	8	is'''a'''proper'''submodule''of	is'''a'''proper'''submodule''of	NOUN
iajs-2001	98	9	y	y	PROPN
iajs-2001	98	10	`	`	PUNCT
iajs-2001	98	11	,	,	PUNCT
iajs-2001	98	12	let	let	VERB
iajs-2001	98	13	aby`∈	aby`∈	PROPN
iajs-2001	98	14	h	h	NOUN
iajs-2001	98	15	(	(	PUNCT
iajs-2001	98	16	a	a	NOUN
iajs-2001	98	17	)	)	PUNCT
iajs-2001	98	18	,	,	PUNCT
iajs-2001	98	19	where	where	SCONJ
iajs-2001	98	20	a	a	DET
iajs-2001	98	21	,	,	PUNCT
iajs-2001	98	22	b	b	X
iajs-2001	98	23	∈	∈	PROPN
iajs-2001	98	24	r	r	NOUN
iajs-2001	98	25	,	,	PUNCT
iajs-2001	98	26	y	y	PROPN
iajs-2001	98	27	`	`	PUNCT
iajs-2001	98	28	∈	∈	PROPN
iajs-2001	98	29	y	y	PROPN
iajs-2001	98	30	`	`	PUNCT
iajs-2001	98	31	,	,	PUNCT
iajs-2001	98	32	then	then	ADV
iajs-2001	98	33	h	h	PROPN
iajs-2001	98	34	(	(	PUNCT
iajs-2001	98	35	y	y	NOUN
iajs-2001	98	36	)	)	PUNCT
iajs-2001	98	37	=	=	SYM
iajs-2001	99	1	y	y	PROPN
iajs-2001	99	2	`	`	PUNCT
iajs-2001	99	3	.	.	PUNCT
iajs-2001	100	1	for	for	ADP
iajs-2001	100	2	some	some	DET
iajs-2001	100	3	y	y	PROPN
iajs-2001	100	4	∊	∊	PROPN
iajs-2001	100	5	y.	y.	PROPN
iajs-2001	100	6	thus	thus	ADV
iajs-2001	100	7	0	0	NUM
iajs-2001	100	8	≠	≠	PROPN
iajs-2001	100	9	abh(y	abh(y	PROPN
iajs-2001	100	10	)	)	PUNCT
iajs-2001	100	11	∊	∊	PROPN
iajs-2001	100	12	h(a	h(a	PROPN
iajs-2001	100	13	)	)	PUNCT
iajs-2001	100	14	,	,	PUNCT
iajs-2001	100	15	then	then	ADV
iajs-2001	100	16	h(aby	h(aby	X
iajs-2001	100	17	)	)	PUNCT
iajs-2001	100	18	=	=	SYM
iajs-2001	101	1	h(n	h(n	PROPN
iajs-2001	101	2	)	)	PUNCT
iajs-2001	101	3	for	for	ADP
iajs-2001	101	4	some	some	DET
iajs-2001	101	5	nonzero	nonzero	NOUN
iajs-2001	101	6	n	n	PROPN
iajs-2001	101	7	∊	∊	NOUN
iajs-2001	101	8	a.	a.	NOUN
iajs-2001	101	9	since	since	SCONJ
iajs-2001	101	10	kerh	kerh	NOUN
iajs-2001	101	11	⊆	⊆	PROPN
iajs-2001	101	12	a	a	DET
iajs-2001	101	13	it	it	PRON
iajs-2001	101	14	follows	follow	VERB
iajs-2001	101	15	that	that	SCONJ
iajs-2001	101	16	0	0	NUM
iajs-2001	101	17	≠	≠	PROPN
iajs-2001	101	18	aby	aby	NOUN
iajs-2001	101	19	∊	∊	NOUN
iajs-2001	101	20	a	a	NOUN
iajs-2001	101	21	,	,	PUNCT
iajs-2001	101	22	but	but	CCONJ
iajs-2001	101	23	a	a	PRON
iajs-2001	101	24	is	be	AUX
iajs-2001	101	25	wn-2	wn-2	NOUN
iajs-2001	101	26	-	-	ADJ
iajs-2001	101	27	absorbing	absorbing	ADJ
iajs-2001	101	28	submodules	submodule	NOUN
iajs-2001	101	29	of	of	ADP
iajs-2001	101	30	y	y	PROPN
iajs-2001	101	31	,	,	PUNCT
iajs-2001	101	32	then	then	ADV
iajs-2001	101	33	either	either	CCONJ
iajs-2001	101	34	ay	ay	PROPN
iajs-2001	101	35	∊	∊	PROPN
iajs-2001	101	36	a	a	DET
iajs-2001	101	37	+	+	ADJ
iajs-2001	101	38	𝚥(y	𝚥(y	NOUN
iajs-2001	101	39	)	)	PUNCT
iajs-2001	101	40	or	or	CCONJ
iajs-2001	101	41	by	by	ADP
iajs-2001	101	42	∊	∊	PROPN
iajs-2001	101	43	a	a	PRON
iajs-2001	101	44	+	+	ADJ
iajs-2001	101	45	𝚥(y	𝚥(y	NOUN
iajs-2001	101	46	)	)	PUNCT
iajs-2001	101	47	or	or	CCONJ
iajs-2001	101	48	ab	ab	PROPN
iajs-2001	101	49	∊	∊	PROPN
iajs-2001	101	50	[	[	PUNCT
iajs-2001	101	51	a	a	DET
iajs-2001	101	52	+	+	X
iajs-2001	101	53	𝚥(y	𝚥(y	NOUN
iajs-2001	101	54	)	)	PUNCT
iajs-2001	101	55	:	:	PUNCT
iajs-2001	101	56	y	y	X
iajs-2001	101	57	]	]	PUNCT
iajs-2001	101	58	.	.	PUNCT
iajs-2001	102	1	thus	thus	ADV
iajs-2001	102	2	either	either	CCONJ
iajs-2001	102	3	ah(y	ah(y	NUM
iajs-2001	102	4	)	)	PUNCT
iajs-2001	102	5	∊	∊	PROPN
iajs-2001	102	6	h(a	h(a	PROPN
iajs-2001	102	7	)	)	PUNCT
iajs-2001	103	1	+	+	NOUN
iajs-2001	103	2	h	h	NOUN
iajs-2001	103	3	(	(	PUNCT
iajs-2001	103	4	𝚥(y	𝚥(y	PROPN
iajs-2001	103	5	)	)	PUNCT
iajs-2001	103	6	)	)	PUNCT
iajs-2001	103	7	or	or	CCONJ
iajs-2001	103	8	bh(y	bh(y	NUM
iajs-2001	103	9	)	)	PUNCT
iajs-2001	103	10	∊	∊	PROPN
iajs-2001	103	11	h(a	h(a	PROPN
iajs-2001	103	12	)	)	PUNCT
iajs-2001	104	1	+	+	CCONJ
iajs-2001	104	2	h(𝚥(y	h(𝚥(y	PROPN
iajs-2001	104	3	)	)	PUNCT
iajs-2001	104	4	)	)	PUNCT
iajs-2001	104	5	or	or	CCONJ
iajs-2001	104	6	abh(y	abh(y	X
iajs-2001	104	7	)	)	PUNCT
iajs-2001	104	8	⊆	⊆	NUM
iajs-2001	104	9	h(a	h(a	PROPN
iajs-2001	104	10	)	)	PUNCT
iajs-2001	105	1	+	+	NOUN
iajs-2001	105	2	h	h	NOUN
iajs-2001	105	3	(	(	PUNCT
iajs-2001	105	4	𝚥(y	𝚥(y	PROPN
iajs-2001	105	5	)	)	PUNCT
iajs-2001	105	6	)	)	PUNCT
iajs-2001	105	7	.	.	PUNCT
iajs-2001	106	1	but	but	CCONJ
iajs-2001	106	2	h	h	NOUN
iajs-2001	106	3	is	be	AUX
iajs-2001	106	4	small	small	ADJ
iajs-2001	106	5	epimorphism	epimorphism	NOUN
iajs-2001	106	6	then	then	ADV
iajs-2001	106	7	either	either	CCONJ
iajs-2001	106	8	ay`∈	ay`∈	ADP
iajs-2001	106	9	h	h	PROPN
iajs-2001	106	10	(	(	PUNCT
iajs-2001	106	11	a	a	NOUN
iajs-2001	106	12	)	)	PUNCT
iajs-2001	106	13	+	+	CCONJ
iajs-2001	106	14	(	(	PUNCT
iajs-2001	106	15	y	y	NOUN
iajs-2001	106	16	`	`	PUNCT
iajs-2001	106	17	)	)	PUNCT
iajs-2001	106	18	or	or	CCONJ
iajs-2001	106	19	by`∈	by`∈	PROPN
iajs-2001	106	20	h	h	PROPN
iajs-2001	106	21	(	(	PUNCT
iajs-2001	106	22	a	a	NOUN
iajs-2001	106	23	)	)	PUNCT
iajs-2001	106	24	+	+	CCONJ
iajs-2001	106	25	(	(	PUNCT
iajs-2001	106	26	y	y	NOUN
iajs-2001	106	27	`	`	PUNCT
iajs-2001	106	28	)	)	PUNCT
iajs-2001	106	29	or	or	CCONJ
iajs-2001	106	30	aby	aby	ADJ
iajs-2001	106	31	`	`	PROPN
iajs-2001	106	32	⊆	⊆	NUM
iajs-2001	106	33	h	h	NOUN
iajs-2001	106	34	(	(	PUNCT
iajs-2001	106	35	a	a	NOUN
iajs-2001	106	36	)	)	PUNCT
iajs-2001	106	37	+	+	CCONJ
iajs-2001	106	38	𝚥(y	𝚥(y	PROPN
iajs-2001	106	39	`	`	PUNCT
iajs-2001	106	40	)	)	PUNCT
iajs-2001	106	41	.	.	PUNCT
iajs-2001	107	1	hence	hence	ADV
iajs-2001	107	2	h	h	NOUN
iajs-2001	107	3	a	a	PRON
iajs-2001	107	4	is	be	AUX
iajs-2001	107	5	wn-2	wn-2	NOUN
iajs-2001	107	6	-	-	ADJ
iajs-2001	107	7	absorbing	absorbing	ADJ
iajs-2001	107	8	submodules	submodule	NOUN
iajs-2001	107	9	of	of	ADP
iajs-2001	107	10	of	of	ADP
iajs-2001	107	11	y	y	PROPN
iajs-2001	107	12	`	`	PUNCT
iajs-2001	107	13	.	.	PUNCT
iajs-2001	108	1	proposition	proposition	NOUN
iajs-2001	108	2	13	13	NUM
iajs-2001	108	3	let	let	VERB
iajs-2001	108	4	h	h	NOUN
iajs-2001	108	5	:	:	PUNCT
iajs-2001	108	6	y	y	PROPN
iajs-2001	108	7	→	→	SYM
iajs-2001	108	8	y	y	AUX
iajs-2001	108	9	`	`	PUNCT
iajs-2001	108	10	be	be	AUX
iajs-2001	108	11	a	a	DET
iajs-2001	108	12	small	small	ADJ
iajs-2001	108	13	r	r	NOUN
iajs-2001	108	14	-	-	PUNCT
iajs-2001	108	15	epimorphism	epimorphism	NOUN
iajs-2001	108	16	.	.	PUNCT
iajs-2001	109	1	and	and	CCONJ
iajs-2001	109	2	a	a	PRON
iajs-2001	109	3	is	be	AUX
iajs-2001	109	4	wn-2	wn-2	NOUN
iajs-2001	109	5	-	-	ADJ
iajs-2001	109	6	absorbing	absorbing	ADJ
iajs-2001	109	7	submodules	submodule	NOUN
iajs-2001	109	8	of	of	ADP
iajs-2001	109	9	y	y	PROPN
iajs-2001	109	10	`	`	PUNCT
iajs-2001	109	11	then	then	ADV
iajs-2001	109	12	ℎ	ℎ	PROPN
iajs-2001	109	13	𝐴	𝐴	PROPN
iajs-2001	109	14	is	be	AUX
iajs-2001	109	15	wn-2	wn-2	ADP
iajs-2001	109	16	-	-	ADJ
iajs-2001	109	17	absorbing	absorbing	ADJ
iajs-2001	109	18	submodules	submodule	NOUN
iajs-2001	109	19	of	of	ADP
iajs-2001	109	20	y.	y.	PROPN
iajs-2001	109	21	proof	proof	NOUN
iajs-2001	109	22	let	let	VERB
iajs-2001	109	23	0	0	NUM
iajs-2001	109	24	≠	≠	PROPN
iajs-2001	109	25	aby	aby	NOUN
iajs-2001	109	26	∊	∊	NOUN
iajs-2001	109	27	ℎ	ℎ	PROPN
iajs-2001	109	28	𝐴	𝐴	PROPN
iajs-2001	109	29	,	,	PUNCT
iajs-2001	109	30	where	where	SCONJ
iajs-2001	109	31	a	a	DET
iajs-2001	109	32	,	,	PUNCT
iajs-2001	109	33	b	b	X
iajs-2001	109	34	∈	∈	PROPN
iajs-2001	109	35	r	r	NOUN
iajs-2001	109	36	,	,	PUNCT
iajs-2001	109	37	y	y	PROPN
iajs-2001	109	38	∊	∊	PROPN
iajs-2001	109	39	y	y	PROPN
iajs-2001	109	40	,	,	PUNCT
iajs-2001	109	41	with	with	ADP
iajs-2001	109	42	ay	ay	PROPN
iajs-2001	109	43	∉	∉	PROPN
iajs-2001	109	44	ℎ	ℎ	PROPN
iajs-2001	109	45	𝐴	𝐴	PROPN
iajs-2001	109	46	+	+	CCONJ
iajs-2001	109	47	𝚥(y)and	𝚥(y)and	PROPN
iajs-2001	109	48	by	by	ADP
iajs-2001	109	49	∉	∉	PROPN
iajs-2001	109	50	ℎ	ℎ	PROPN
iajs-2001	109	51	𝐴	𝐴	PROPN
iajs-2001	109	52	+	+	CCONJ
iajs-2001	109	53	𝚥(y	𝚥(y	PROPN
iajs-2001	109	54	)	)	PUNCT
iajs-2001	109	55	.	.	PUNCT
iajs-2001	110	1	it	it	PRON
iajs-2001	110	2	follows	follow	VERB
iajs-2001	110	3	that	that	SCONJ
iajs-2001	110	4	ah(y	ah(y	NOUN
iajs-2001	110	5	)	)	PUNCT
iajs-2001	111	1	∉	∉	PROPN
iajs-2001	111	2	h	h	PROPN
iajs-2001	111	3	(	(	PUNCT
iajs-2001	111	4	ℎ	ℎ	PROPN
iajs-2001	111	5	𝐴	𝐴	PROPN
iajs-2001	111	6	+	+	CCONJ
iajs-2001	111	7	𝚥(y	𝚥(y	PROPN
iajs-2001	111	8	)	)	PUNCT
iajs-2001	111	9	)	)	PUNCT
iajs-2001	111	10	=	=	PUNCT
iajs-2001	112	1	a	a	DET
iajs-2001	112	2	+	+	ADJ
iajs-2001	112	3	𝚥(y	𝚥(y	PROPN
iajs-2001	112	4	`	`	PUNCT
iajs-2001	112	5	)	)	PUNCT
iajs-2001	112	6	and	and	CCONJ
iajs-2001	112	7	bh(y	bh(y	NUM
iajs-2001	112	8	)	)	PUNCT
iajs-2001	112	9	∉	∉	PROPN
iajs-2001	112	10	h	h	PROPN
iajs-2001	112	11	(	(	PUNCT
iajs-2001	112	12	ℎ	ℎ	PROPN
iajs-2001	112	13	𝐴	𝐴	PROPN
iajs-2001	112	14	+	+	CCONJ
iajs-2001	112	15	𝚥(y	𝚥(y	PROPN
iajs-2001	112	16	)	)	PUNCT
iajs-2001	112	17	)	)	PUNCT
iajs-2001	113	1	=	=	PUNCT
iajs-2001	113	2	a	a	DET
iajs-2001	113	3	mathematics	mathematic	NOUN
iajs-2001	113	4	|	|	ADV
iajs-2001	113	5	122	122	NUM
iajs-2001	113	6	ibn	ibn	PROPN
iajs-2001	113	7	al	al	PROPN
iajs-2001	113	8	-	-	PUNCT
iajs-2001	113	9	haitham	haitham	PROPN
iajs-2001	113	10	jour	jour	X
iajs-2001	113	11	.	.	PROPN
iajs-2001	114	1	for	for	ADP
iajs-2001	114	2	pure	pure	ADJ
iajs-2001	114	3	&	&	CCONJ
iajs-2001	114	4	appl	appl	PROPN
iajs-2001	114	5	.	.	PUNCT
iajs-2001	115	1	sci	sci	PROPN
iajs-2001	115	2	.	.	PROPN
iajs-2001	115	3	ihjpas	ihjpa	VERB
iajs-2001	115	4	https://doi.org/10.30526/31.3.2001	https://doi.org/10.30526/31.3.2001	PROPN
iajs-2001	115	5	vol	vol	NOUN
iajs-2001	115	6	.	.	PROPN
iajs-2001	116	1	31	31	NUM
iajs-2001	117	1	(	(	PUNCT
iajs-2001	117	2	3	3	NUM
iajs-2001	117	3	)	)	PUNCT
iajs-2001	117	4	2018	2018	NUM
iajs-2001	117	5	+	+	CCONJ
iajs-2001	118	1	𝚥(y	𝚥(y	PROPN
iajs-2001	118	2	`	`	PUNCT
iajs-2001	118	3	)	)	PUNCT
iajs-2001	118	4	because	because	SCONJ
iajs-2001	118	5	h	h	NOUN
iajs-2001	118	6	is	be	AUX
iajs-2001	118	7	a	a	DET
iajs-2001	118	8	small	small	ADJ
iajs-2001	118	9	epimorphism	epimorphism	NOUN
iajs-2001	118	10	.	.	PUNCT
iajs-2001	119	1	we	we	PRON
iajs-2001	119	2	have	have	VERB
iajs-2001	119	3	0	0	NUM
iajs-2001	119	4	≠	≠	PROPN
iajs-2001	119	5	aby	aby	NOUN
iajs-2001	119	6	∊	∊	NOUN
iajs-2001	119	7	ℎ	ℎ	PROPN
iajs-2001	119	8	𝐴	𝐴	PROPN
iajs-2001	119	9	,	,	PUNCT
iajs-2001	119	10	implies	imply	VERB
iajs-2001	119	11	that	that	SCONJ
iajs-2001	119	12	0	0	NUM
iajs-2001	119	13	≠	≠	PROPN
iajs-2001	119	14	abh(y	abh(y	PROPN
iajs-2001	119	15	)	)	PUNCT
iajs-2001	119	16	∊	∊	PROPN
iajs-2001	119	17	𝐴	𝐴	PROPN
iajs-2001	119	18	,	,	PUNCT
iajs-2001	119	19	but	but	CCONJ
iajs-2001	119	20	a	a	PRON
iajs-2001	119	21	is	be	AUX
iajs-2001	119	22	wn-2	wn-2	NOUN
iajs-2001	119	23	-	-	ADJ
iajs-2001	119	24	absorbing	absorbing	ADJ
iajs-2001	119	25	submodules	submodule	NOUN
iajs-2001	119	26	of	of	ADP
iajs-2001	119	27	y	y	PROPN
iajs-2001	119	28	`	`	PUNCT
iajs-2001	119	29	,	,	PUNCT
iajs-2001	119	30	then	then	ADV
iajs-2001	119	31	ab	ab	PROPN
iajs-2001	119	32	∊	∊	PROPN
iajs-2001	120	1	[	[	X
iajs-2001	120	2	a	a	X
iajs-2001	120	3	+	+	ADJ
iajs-2001	120	4	𝚥(y	𝚥(y	PROPN
iajs-2001	120	5	`	`	PUNCT
iajs-2001	120	6	):	):	PUNCT
iajs-2001	120	7	y	y	X
iajs-2001	120	8	`	`	X
iajs-2001	120	9	]	]	PUNCT
iajs-2001	120	10	that	that	PRON
iajs-2001	120	11	is	be	AUX
iajs-2001	120	12	aby	aby	X
iajs-2001	120	13	`	`	PUNCT
iajs-2001	120	14	⊆	⊆	SYM
iajs-2001	120	15	a	a	DET
iajs-2001	120	16	+	+	ADJ
iajs-2001	120	17	𝚥(y	𝚥(y	PROPN
iajs-2001	120	18	`	`	PUNCT
iajs-2001	120	19	)	)	PUNCT
iajs-2001	120	20	,	,	PUNCT
iajs-2001	120	21	implies	imply	VERB
iajs-2001	120	22	that	that	SCONJ
iajs-2001	120	23	abh(y	abh(y	X
iajs-2001	120	24	)	)	PUNCT
iajs-2001	120	25	⊆	⊆	NUM
iajs-2001	120	26	a	a	DET
iajs-2001	120	27	+	+	ADJ
iajs-2001	120	28	𝚥(y	𝚥(y	PROPN
iajs-2001	120	29	`	`	PUNCT
iajs-2001	120	30	)	)	PUNCT
iajs-2001	120	31	,	,	PUNCT
iajs-2001	120	32	hence	hence	ADV
iajs-2001	120	33	aby	aby	VERB
iajs-2001	120	34	⊆	⊆	X
iajs-2001	120	35	ℎ	ℎ	PROPN
iajs-2001	120	36	𝐴	𝐴	PROPN
iajs-2001	120	37	𝚥	𝚥	PROPN
iajs-2001	120	38	𝑌	𝑌	PROPN
iajs-2001	120	39	`	`	NUM
iajs-2001	120	40	⊆	⊆	NUM
iajs-2001	120	41	ℎ	ℎ	PROPN
iajs-2001	120	42	𝐴	𝐴	VERB
iajs-2001	120	43	𝚥	𝚥	PROPN
iajs-2001	120	44	𝑌	𝑌	PROPN
iajs-2001	120	45	.	.	PUNCT
iajs-2001	121	1	thus	thus	ADV
iajs-2001	121	2	ab	ab	PROPN
iajs-2001	121	3	∊	∊	PROPN
iajs-2001	122	1	[	[	X
iajs-2001	122	2	ℎ	ℎ	X
iajs-2001	122	3	𝐴	𝐴	VERB
iajs-2001	122	4	𝚥	𝚥	PROPN
iajs-2001	122	5	𝑌	𝑌	PROPN
iajs-2001	122	6	:	:	PUNCT
iajs-2001	122	7	y	y	X
iajs-2001	122	8	]	]	X
iajs-2001	122	9	.	.	PUNCT
iajs-2001	123	1	3	3	X
iajs-2001	123	2	.	.	NUM
iajs-2001	123	3	wns-2	wns-2	AUX
iajs-2001	123	4	-	-	PUNCT
iajs-2001	123	5	absorbing	absorb	VERB
iajs-2001	123	6	submodules	submodule	NOUN
iajs-2001	123	7	and	and	CCONJ
iajs-2001	123	8	related	relate	VERB
iajs-2001	123	9	concept	concept	NOUN
iajs-2001	123	10	this	this	DET
iajs-2001	123	11	section	section	NOUN
iajs-2001	123	12	devoted	devote	VERB
iajs-2001	123	13	to'''introduce'''and'''study''the'''concept'''of	to'''introduce'''and'''study''the'''concept'''of	PROPN
iajs-2001	123	14	wns"-2"absorbing'''submodules''''as"a'''generalization'''of'''a''weakly"semi"2-'absorbing'''submodule	wns"-2"absorbing'''submodules''''as"a'''generalization'''of'''a''weakly"semi"2-'absorbing'''submodule	NOUN
iajs-2001	123	15	.	.	PUNCT
iajs-2001	124	1	definition	definition	NOUN
iajs-2001	124	2	14	14	NUM
iajs-2001	124	3	a''proper'''submodule'''b'''of"an"r'-'module"y"is'"said''to'be"a"wns'-'2''absorbing'''submodule'''of''y	a''proper'''submodule'''b'''of"an"r'-'module"y"is'"said''to'be"a"wns'-'2''absorbing'''submodule'''of''y	NOUN
iajs-2001	124	4	,	,	PUNCT
iajs-2001	124	5	'	'	PUNCT
iajs-2001	124	6	if"whenever	if"whenever	NOUN
iajs-2001	124	7	'	'	PUNCT
iajs-2001	124	8	0	0	NUM
iajs-2001	124	9	≠	≠	PROPN
iajs-2001	125	1	𝑎	𝑎	PUNCT
iajs-2001	125	2	𝑦	𝑦	NUM
iajs-2001	125	3	∈	∈	NOUN
iajs-2001	125	4	"	"	PUNCT
iajs-2001	125	5	𝐵	𝐵	NOUN
iajs-2001	125	6	,	,	PUNCT
iajs-2001	125	7	where	where	SCONJ
iajs-2001	125	8	a	a	DET
iajs-2001	125	9	∊"r	∊"r	PROPN
iajs-2001	125	10	,	,	PUNCT
iajs-2001	125	11	y	y	PROPN
iajs-2001	125	12	∊y	∊y	NOUN
iajs-2001	125	13	,	,	PUNCT
iajs-2001	125	14	implies	imply	VERB
iajs-2001	125	15	that	that	SCONJ
iajs-2001	125	16	either	either	CCONJ
iajs-2001	125	17	ay	ay	PROPN
iajs-2001	125	18	∊	∊	PROPN
iajs-2001	125	19	b	b	PROPN
iajs-2001	125	20	+	+	CCONJ
iajs-2001	125	21	𝚥(y	𝚥(y	NOUN
iajs-2001	125	22	)	)	PUNCT
iajs-2001	125	23	or	or	CCONJ
iajs-2001	125	24	𝑎	𝑎	PROPN
iajs-2001	125	25	∈	∈	PROPN
iajs-2001	125	26	[	[	PUNCT
iajs-2001	125	27	b	b	NOUN
iajs-2001	125	28	+	+	CCONJ
iajs-2001	125	29	𝚥(y	𝚥(y	PROPN
iajs-2001	125	30	)	)	PUNCT
iajs-2001	125	31	:	:	PUNCT
iajs-2001	125	32	y	y	X
iajs-2001	125	33	]	]	PUNCT
iajs-2001	125	34	.	.	PUNCT
iajs-2001	126	1	an"'ideal	an"'ideal	NOUN
iajs-2001	126	2	i'"of"a"'ring	i'"of"a"'re	VERB
iajs-2001	126	3	r"'is'''called'''a'''wns;-"2'absorbing'''ideal	r"'is'''called'''a'''wns;-"2'absorbing'''ideal	ADJ
iajs-2001	126	4	if"i	if"i	NOUN
iajs-2001	126	5	is"a	is"a	NOUN
iajs-2001	126	6	wns-'2-'absorbing'r-'submodule'''of	wns-'2-'absorbing'r-'submodule'''of	PROPN
iajs-2001	126	7	an''r'-'module"r	an''r'-'module"r	NOUN
iajs-2001	126	8	.	.	PUNCT
iajs-2001	126	9	remarks''and'examples	remarks''and'example	VERB
iajs-2001	126	10	15	15	NUM
iajs-2001	126	11	1	1	NUM
iajs-2001	126	12	.	.	PUNCT
iajs-2001	127	1	it	it	PRON
iajs-2001	127	2	is	be	AUX
iajs-2001	127	3	clear	clear	ADJ
iajs-2001	127	4	that'''every'''weakly'''semi'''2-'absorbing"submodule"'of	that'''every'''weakly'''semi'''2-'absorbing"submodule"'of	ADJ
iajs-2001	127	5	an"r-'module"y	an"r-'module"y	NOUN
iajs-2001	127	6	is'a	is'a	PRON
iajs-2001	127	7	wns-"2-'absorbing"submodule	wns-"2-'absorbing"submodule	NOUN
iajs-2001	127	8	of	of	ADP
iajs-2001	127	9	y	y	PROPN
iajs-2001	127	10	while'''the'''converse	while'''the'''converse	PROPN
iajs-2001	127	11	'	'	PUNCT
iajs-2001	127	12	is'''not'''true	is'''not'''true	NUM
iajs-2001	127	13	2	2	NUM
iajs-2001	127	14	.	.	PUNCT
iajs-2001	128	1	in	in	ADP
iajs-2001	128	2	the	the	DET
iajs-2001	128	3	z-'module	z-'module	NOUN
iajs-2001	128	4	"	"	PUNCT
iajs-2001	128	5	𝑍	𝑍	NOUN
iajs-2001	128	6	,	,	PUNCT
iajs-2001	128	7	the	the	DET
iajs-2001	128	8	submodule	submodule	PROPN
iajs-2001	128	9	b	b	PROPN
iajs-2001	128	10	=	=	PUNCT
iajs-2001	128	11	〈	〈	PROPN
iajs-2001	128	12	8	8	NUM
iajs-2001	128	13	〉	〉	NOUN
iajs-2001	128	14	is'a	is'a	AUX
iajs-2001	128	15	wns-2'-"absorbing	wns-2'-"absorbe	VERB
iajs-2001	128	16	submodule'"'of	submodule'"'of	ADJ
iajs-2001	128	17	y	y	NOUN
iajs-2001	128	18	,	,	PUNCT
iajs-2001	128	19	'	'	PUNCT
iajs-2001	128	20	but"'not''weakly"'semi	but"'not''weakly"'semi	PROPN
iajs-2001	128	21	2	2	NUM
iajs-2001	128	22	-	-	PUNCT
iajs-2001	128	23	absorbing	absorbing	NOUN
iajs-2001	128	24	of	of	ADP
iajs-2001	128	25	y	y	PROPN
iajs-2001	128	26	since	since	SCONJ
iajs-2001	128	27	0	0	NUM
iajs-2001	128	28	≠	≠	PROPN
iajs-2001	128	29	2	2	NUM
iajs-2001	128	30	2	2	NUM
iajs-2001	128	31	∈	∈	PROPN
iajs-2001	128	32	𝐵	𝐵	NOUN
iajs-2001	128	33	,	,	PUNCT
iajs-2001	128	34	but	but	CCONJ
iajs-2001	128	35	2	2	NUM
iajs-2001	128	36	∉	∉	PROPN
iajs-2001	128	37	𝐵	𝐵	NOUN
iajs-2001	128	38	and	and	CCONJ
iajs-2001	128	39	2	2	NUM
iajs-2001	128	40	∉	∉	PROPN
iajs-2001	129	1	[	[	X
iajs-2001	129	2	b	b	X
iajs-2001	129	3	:	:	PUNCT
iajs-2001	129	4	y	y	NOUN
iajs-2001	129	5	]	]	X
iajs-2001	129	6	.	.	PUNCT
iajs-2001	130	1	3	3	X
iajs-2001	130	2	.	.	X
iajs-2001	131	1	if	if	SCONJ
iajs-2001	131	2	y	y	PRON
iajs-2001	131	3	be	be	VERB
iajs-2001	131	4	an	an	DET
iajs-2001	131	5	r	r	NOUN
iajs-2001	131	6	-	-	PUNCT
iajs-2001	131	7	module	module	NOUN
iajs-2001	131	8	,	,	PUNCT
iajs-2001	131	9	with	with	ADP
iajs-2001	131	10	𝚥(y	𝚥(y	NOUN
iajs-2001	131	11	)	)	PUNCT
iajs-2001	131	12	=	=	SYM
iajs-2001	131	13	0	0	NUM
iajs-2001	131	14	,	,	PUNCT
iajs-2001	131	15	then	then	ADV
iajs-2001	131	16	a	a	DET
iajs-2001	131	17	wns-2	wns-2	ADV
iajs-2001	131	18	-	-	PUNCT
iajs-2001	131	19	absorbing	absorb	VERB
iajs-2001	131	20	submodule	submodule	NOUN
iajs-2001	131	21	of	of	ADP
iajs-2001	131	22	y	y	PROPN
iajs-2001	131	23	,	,	PUNCT
iajs-2001	131	24	equivalent	equivalent	ADJ
iajs-2001	131	25	with	with	ADP
iajs-2001	131	26	a	a	DET
iajs-2001	131	27	weakly	weakly	ADJ
iajs-2001	131	28	semi	semi	ADJ
iajs-2001	131	29	2	2	NUM
iajs-2001	131	30	-	-	PUNCT
iajs-2001	131	31	absorbing	absorb	VERB
iajs-2001	131	32	submodule	submodule	NOUN
iajs-2001	131	33	of	of	ADP
iajs-2001	131	34	y.	y.	PROPN
iajs-2001	131	35	4	4	NUM
iajs-2001	131	36	.	.	PUNCT
iajs-2001	132	1	if	if	SCONJ
iajs-2001	132	2	y	y	PROPN
iajs-2001	132	3	is	be	AUX
iajs-2001	132	4	semi	semi	ADV
iajs-2001	132	5	simple	simple	ADJ
iajs-2001	132	6	(	(	PUNCT
iajs-2001	132	7	regular	regular	ADJ
iajs-2001	132	8	)	)	PUNCT
iajs-2001	132	9	r	r	NOUN
iajs-2001	132	10	-	-	PUNCT
iajs-2001	132	11	module	module	NOUN
iajs-2001	132	12	,	,	PUNCT
iajs-2001	132	13	then	then	ADV
iajs-2001	132	14	a	a	DET
iajs-2001	132	15	wns-2	wns-2	ADV
iajs-2001	132	16	-	-	PUNCT
iajs-2001	132	17	absorbing	absorb	VERB
iajs-2001	132	18	submodule	submodule	NOUN
iajs-2001	132	19	of	of	ADP
iajs-2001	132	20	y	y	PROPN
iajs-2001	132	21	and	and	CCONJ
iajs-2001	132	22	weakly	weakly	ADJ
iajs-2001	132	23	semi	semi	ADJ
iajs-2001	132	24	2	2	NUM
iajs-2001	132	25	-	-	PUNCT
iajs-2001	132	26	absorbing	absorb	VERB
iajs-2001	132	27	submodule	submodule	NOUN
iajs-2001	132	28	of	of	ADP
iajs-2001	132	29	y	y	PROPN
iajs-2001	132	30	are	be	AUX
iajs-2001	132	31	equivalent	equivalent	ADJ
iajs-2001	132	32	.	.	PUNCT
iajs-2001	133	1	5	5	X
iajs-2001	133	2	.	.	X
iajs-2001	134	1	if	if	SCONJ
iajs-2001	134	2	y	y	PROPN
iajs-2001	134	3	is	be	AUX
iajs-2001	134	4	a	a	DET
iajs-2001	134	5	r-"module	r-"module	NOUN
iajs-2001	134	6	"	"	PUNCT
iajs-2001	134	7	,	,	PUNCT
iajs-2001	134	8	and	and	CCONJ
iajs-2001	134	9	b"a	b"a	PROPN
iajs-2001	134	10	proper"submodule"of	proper"submodule"of	NOUN
iajs-2001	134	11	y	y	PROPN
iajs-2001	134	12	,	,	PUNCT
iajs-2001	134	13	with	with	ADP
iajs-2001	134	14	𝚥(y	𝚥(y	PROPN
iajs-2001	134	15	)	)	PUNCT
iajs-2001	134	16	⊆	⊆	NUM
iajs-2001	134	17	b.	b.	NOUN
iajs-2001	134	18	then	then	ADV
iajs-2001	134	19	b	b	X
iajs-2001	134	20	is"a	is"a	NOUN
iajs-2001	134	21	wns-2"-absorbing"submodule"of	wns-2"-absorbing"submodule"of	PROPN
iajs-2001	134	22	y	y	PROPN
iajs-2001	134	23	if"and"only"if	if"and"only"if	PROPN
iajs-2001	134	24	b	b	PROPN
iajs-2001	134	25	is	be	AUX
iajs-2001	134	26	a"weakly	a"weakly	ADV
iajs-2001	134	27	semi	semi	ADV
iajs-2001	134	28	2"absorbing"submodule"of	2"absorbing"submodule"of	NUM
iajs-2001	134	29	y.	y.	NOUN
iajs-2001	134	30	6	6	NUM
iajs-2001	134	31	.	.	PUNCT
iajs-2001	135	1	if	if	SCONJ
iajs-2001	135	2	b	b	PROPN
iajs-2001	135	3	is	be	AUX
iajs-2001	135	4	a	a	DET
iajs-2001	135	5	proper	proper	ADJ
iajs-2001	135	6	submodule	submodule	NOUN
iajs-2001	135	7	of	of	ADP
iajs-2001	135	8	y	y	PROPN
iajs-2001	135	9	,	,	PUNCT
iajs-2001	135	10	with	with	ADP
iajs-2001	135	11	b	b	PROPN
iajs-2001	135	12	+	+	CCONJ
iajs-2001	135	13	𝚥(y	𝚥(y	NOUN
iajs-2001	135	14	)	)	PUNCT
iajs-2001	135	15	is	be	AUX
iajs-2001	135	16	a	a	DET
iajs-2001	135	17	wns-2"-'absorbing	wns-2"-'absorbe	VERB
iajs-2001	135	18	'	'	PUNCT
iajs-2001	135	19	submodule'"of''y	submodule'"of''y	NOUN
iajs-2001	135	20	,	,	PUNCT
iajs-2001	135	21	then''b	then''b	NOUN
iajs-2001	135	22	is'"a	is'"a	VERB
iajs-2001	135	23	wns-"2"-'absorbing"'submodule"of	wns-"2"-'absorbing"'submodule"of	PRON
iajs-2001	135	24	'	'	PUNCT
iajs-2001	135	25	y.	y.	NOUN
iajs-2001	135	26	'	'	PUNCT
iajs-2001	135	27	proposition	proposition	NOUN
iajs-2001	135	28	16	16	NUM
iajs-2001	135	29	let	let	VERB
iajs-2001	135	30	y	y	PROPN
iajs-2001	135	31	be"an"r'-'module'''and	be"an"r'-'module'''and	PROPN
iajs-2001	135	32	b	b	PROPN
iajs-2001	135	33	be'a''proper''submodule'"of'y	be'a''proper''submodule'"of'y	PROPN
iajs-2001	135	34	then	then	ADV
iajs-2001	135	35	b	b	PROPN
iajs-2001	135	36	+	+	CCONJ
iajs-2001	135	37	𝚥(y)'is	𝚥(y)'is	NOUN
iajs-2001	135	38	a'"wns-2'"absorbing"'submodule"of	a'"wns-2'"absorbing"'submodule"of	NOUN
iajs-2001	135	39	'	'	PART
iajs-2001	135	40	y	y	NOUN
iajs-2001	135	41	'	'	PUNCT
iajs-2001	135	42	if"and"only"if"for	if"and"only"if"for	VERB
iajs-2001	135	43	each	each	DET
iajs-2001	135	44	non	non	NOUN
iajs-2001	135	45	-	-	NOUN
iajs-2001	135	46	zero	zero	NUM
iajs-2001	136	1	a	a	PRON
iajs-2001	136	2	∊	∊	NOUN
iajs-2001	136	3	r	r	NOUN
iajs-2001	137	1	[	[	X
iajs-2001	137	2	b	b	X
iajs-2001	137	3	+	+	CCONJ
iajs-2001	137	4	𝚥(y	𝚥(y	NOUN
iajs-2001	137	5	)	)	PUNCT
iajs-2001	137	6	:	:	PUNCT
iajs-2001	138	1	𝑎	𝑎	X
iajs-2001	138	2	𝑦	𝑦	NOUN
iajs-2001	138	3	]	]	X
iajs-2001	138	4	=	=	PUNCT
iajs-2001	139	1	[	[	X
iajs-2001	139	2	b	b	X
iajs-2001	139	3	+	+	CCONJ
iajs-2001	139	4	𝚥(y	𝚥(y	NOUN
iajs-2001	139	5	)	)	PUNCT
iajs-2001	139	6	:	:	PUNCT
iajs-2001	140	1	ay	ay	X
iajs-2001	140	2	]	]	X
iajs-2001	140	3	or	or	CCONJ
iajs-2001	140	4	𝑎	𝑎	DET
iajs-2001	140	5	∊	∊	NOUN
iajs-2001	140	6	[	[	X
iajs-2001	140	7	b	b	X
iajs-2001	140	8	+	+	CCONJ
iajs-2001	140	9	𝚥(y	𝚥(y	NOUN
iajs-2001	140	10	)	)	PUNCT
iajs-2001	140	11	:	:	PUNCT
iajs-2001	141	1	y	y	X
iajs-2001	141	2	]	]	PUNCT
iajs-2001	141	3	.	.	PUNCT
iajs-2001	142	1	proof	proof	NOUN
iajs-2001	142	2	⟹	⟹	PROPN
iajs-2001	142	3	suppose	suppose	VERB
iajs-2001	142	4	that	that	SCONJ
iajs-2001	142	5	𝑎	𝑎	PROPN
iajs-2001	142	6	∉	∉	PROPN
iajs-2001	142	7	[	[	X
iajs-2001	142	8	b	b	X
iajs-2001	142	9	+	+	CCONJ
iajs-2001	142	10	𝚥(y	𝚥(y	NOUN
iajs-2001	142	11	)	)	PUNCT
iajs-2001	142	12	:	:	PUNCT
iajs-2001	142	13	y	y	X
iajs-2001	142	14	]	]	X
iajs-2001	142	15	,	,	PUNCT
iajs-2001	142	16	and	and	CCONJ
iajs-2001	142	17	let	let	VERB
iajs-2001	142	18	c	c	NOUN
iajs-2001	142	19	∊	∊	VERB
iajs-2001	143	1	[	[	X
iajs-2001	143	2	b	b	X
iajs-2001	143	3	+	+	CCONJ
iajs-2001	143	4	𝚥(y	𝚥(y	NOUN
iajs-2001	143	5	)	)	PUNCT
iajs-2001	143	6	:	:	PUNCT
iajs-2001	144	1	𝑎	𝑎	X
iajs-2001	144	2	𝑦	𝑦	NOUN
iajs-2001	144	3	]	]	PUNCT
iajs-2001	144	4	,	,	PUNCT
iajs-2001	144	5	implies	imply	VERB
iajs-2001	144	6	that	that	SCONJ
iajs-2001	144	7	0	0	NUM
iajs-2001	144	8	≠	≠	NOUN
iajs-2001	144	9	𝑎	𝑎	PRON
iajs-2001	144	10	𝑐𝑦	𝑐𝑦	NOUN
iajs-2001	144	11	∊	∊	PROPN
iajs-2001	144	12	b	b	PROPN
iajs-2001	144	13	+	+	CCONJ
iajs-2001	144	14	𝚥(y	𝚥(y	PROPN
iajs-2001	144	15	)	)	PUNCT
iajs-2001	144	16	,	,	PUNCT
iajs-2001	144	17	but	but	CCONJ
iajs-2001	144	18	b	b	X
iajs-2001	144	19	+	+	CCONJ
iajs-2001	144	20	𝚥(y	𝚥(y	NOUN
iajs-2001	144	21	)	)	PUNCT
iajs-2001	144	22	is	be	AUX
iajs-2001	144	23	a	a	DET
iajs-2001	144	24	wns-2	wns-2	ADV
iajs-2001	144	25	-	-	PUNCT
iajs-2001	144	26	absorbing	absorb	VERB
iajs-2001	144	27	submodule	submodule	NOUN
iajs-2001	144	28	of	of	ADP
iajs-2001	144	29	y	y	PROPN
iajs-2001	144	30	and	and	CCONJ
iajs-2001	144	31	𝑎	𝑎	DET
iajs-2001	144	32	∉	∉	PROPN
iajs-2001	145	1	[	[	X
iajs-2001	145	2	b	b	X
iajs-2001	145	3	+	+	CCONJ
iajs-2001	145	4	𝚥(y	𝚥(y	NOUN
iajs-2001	145	5	)	)	PUNCT
iajs-2001	145	6	:	:	PUNCT
iajs-2001	146	1	y	y	X
iajs-2001	146	2	]	]	PUNCT
iajs-2001	146	3	,	,	PUNCT
iajs-2001	146	4	then	then	ADV
iajs-2001	146	5	acy	acy	VERB
iajs-2001	146	6	∊	∊	PROPN
iajs-2001	146	7	b	b	PROPN
iajs-2001	146	8	+	+	CCONJ
iajs-2001	146	9	𝚥(y	𝚥(y	PROPN
iajs-2001	146	10	)	)	PUNCT
iajs-2001	146	11	,	,	PUNCT
iajs-2001	146	12	implies	imply	VERB
iajs-2001	146	13	that	that	SCONJ
iajs-2001	147	1	c	c	PROPN
iajs-2001	147	2	∊	∊	PROPN
iajs-2001	148	1	[	[	X
iajs-2001	148	2	b	b	X
iajs-2001	148	3	+	+	CCONJ
iajs-2001	148	4	𝚥(y	𝚥(y	NOUN
iajs-2001	148	5	)	)	PUNCT
iajs-2001	148	6	:	:	PUNCT
iajs-2001	149	1	ay	ay	X
iajs-2001	149	2	]	]	X
iajs-2001	149	3	.	.	PUNCT
iajs-2001	150	1	thus	thus	ADV
iajs-2001	150	2	[	[	X
iajs-2001	150	3	b	b	X
iajs-2001	150	4	+	+	CCONJ
iajs-2001	150	5	𝚥(y	𝚥(y	NOUN
iajs-2001	150	6	)	)	PUNCT
iajs-2001	150	7	:	:	PUNCT
iajs-2001	151	1	𝑎	𝑎	X
iajs-2001	151	2	𝑦	𝑦	X
iajs-2001	151	3	]	]	PUNCT
iajs-2001	151	4	⊆	⊆	NUM
iajs-2001	151	5	[	[	X
iajs-2001	151	6	b	b	X
iajs-2001	151	7	+	+	CCONJ
iajs-2001	151	8	𝚥(y	𝚥(y	NOUN
iajs-2001	151	9	)	)	PUNCT
iajs-2001	151	10	:	:	PUNCT
iajs-2001	151	11	ay	ay	X
iajs-2001	151	12	]	]	X
iajs-2001	151	13	.	.	PUNCT
iajs-2001	152	1	clearly	clearly	ADV
iajs-2001	152	2	[	[	X
iajs-2001	152	3	b	b	X
iajs-2001	152	4	+	+	CCONJ
iajs-2001	152	5	𝚥(y	𝚥(y	NOUN
iajs-2001	152	6	)	)	PUNCT
iajs-2001	152	7	:	:	PUNCT
iajs-2001	153	1	ay	ay	X
iajs-2001	153	2	]	]	X
iajs-2001	153	3	⊆	⊆	NUM
iajs-2001	153	4	[	[	X
iajs-2001	153	5	b	b	X
iajs-2001	153	6	+	+	CCONJ
iajs-2001	153	7	𝚥(y	𝚥(y	NOUN
iajs-2001	153	8	)	)	PUNCT
iajs-2001	153	9	:	:	PUNCT
iajs-2001	154	1	𝑎	𝑎	X
iajs-2001	154	2	𝑦	𝑦	NOUN
iajs-2001	154	3	]	]	X
iajs-2001	154	4	.	.	PUNCT
iajs-2001	155	1	hence	hence	ADV
iajs-2001	155	2	[	[	X
iajs-2001	155	3	b	b	X
iajs-2001	155	4	+	+	CCONJ
iajs-2001	155	5	𝚥(y	𝚥(y	NOUN
iajs-2001	155	6	)	)	PUNCT
iajs-2001	155	7	:	:	PUNCT
iajs-2001	156	1	𝑎	𝑎	X
iajs-2001	156	2	𝑦	𝑦	NOUN
iajs-2001	156	3	]	]	X
iajs-2001	156	4	=	=	PUNCT
iajs-2001	157	1	[	[	X
iajs-2001	157	2	b	b	X
iajs-2001	157	3	+	+	CCONJ
iajs-2001	157	4	𝚥(y	𝚥(y	NOUN
iajs-2001	157	5	)	)	PUNCT
iajs-2001	157	6	:	:	PUNCT
iajs-2001	158	1	ay	ay	X
iajs-2001	158	2	]	]	X
iajs-2001	158	3	.	.	PUNCT
iajs-2001	159	1	⟸	⟸	X
iajs-2001	159	2	)	)	PUNCT
iajs-2001	159	3	let	let	VERB
iajs-2001	159	4	0	0	NUM
iajs-2001	159	5	≠	≠	PROPN
iajs-2001	159	6	𝑎	𝑎	PRON
iajs-2001	159	7	𝑦	𝑦	NUM
iajs-2001	159	8	∊	∊	NUM
iajs-2001	159	9	b	b	NOUN
iajs-2001	159	10	+	+	CCONJ
iajs-2001	159	11	𝚥(y	𝚥(y	PROPN
iajs-2001	159	12	)	)	PUNCT
iajs-2001	159	13	,	,	PUNCT
iajs-2001	159	14	where	where	SCONJ
iajs-2001	159	15	a	a	DET
iajs-2001	159	16	∈	∈	PROPN
iajs-2001	159	17	r	r	NOUN
iajs-2001	159	18	,	,	PUNCT
iajs-2001	159	19	y	y	PROPN
iajs-2001	159	20	∈	∈	PROPN
iajs-2001	159	21	y.	y.	NOUN
iajs-2001	159	22	by	by	ADP
iajs-2001	159	23	hypothesis	hypothesis	NOUN
iajs-2001	159	24	,	,	PUNCT
iajs-2001	159	25	if	if	SCONJ
iajs-2001	159	26	[	[	X
iajs-2001	159	27	b	b	X
iajs-2001	159	28	+	+	CCONJ
iajs-2001	159	29	𝚥(y	𝚥(y	NOUN
iajs-2001	159	30	)	)	PUNCT
iajs-2001	159	31	:	:	PUNCT
iajs-2001	160	1	𝑎	𝑎	X
iajs-2001	160	2	𝑦	𝑦	NOUN
iajs-2001	160	3	]	]	X
iajs-2001	160	4	=	=	PUNCT
iajs-2001	161	1	[	[	X
iajs-2001	161	2	b	b	X
iajs-2001	161	3	+	+	CCONJ
iajs-2001	161	4	𝚥(y	𝚥(y	NOUN
iajs-2001	161	5	)	)	PUNCT
iajs-2001	161	6	:	:	PUNCT
iajs-2001	162	1	ay	ay	X
iajs-2001	162	2	]	]	X
iajs-2001	162	3	and	and	CCONJ
iajs-2001	162	4	0	0	NUM
iajs-2001	162	5	≠	≠	PROPN
iajs-2001	162	6	𝑎	𝑎	X
iajs-2001	162	7	𝑦	𝑦	NUM
iajs-2001	162	8	∊	∊	NUM
iajs-2001	162	9	b	b	NOUN
iajs-2001	162	10	+	+	CCONJ
iajs-2001	162	11	𝚥(y	𝚥(y	PROPN
iajs-2001	162	12	)	)	PUNCT
iajs-2001	162	13	,	,	PUNCT
iajs-2001	162	14	implies	imply	VERB
iajs-2001	162	15	that	that	SCONJ
iajs-2001	162	16	[	[	X
iajs-2001	162	17	b	b	X
iajs-2001	162	18	+	+	CCONJ
iajs-2001	162	19	𝚥(y	𝚥(y	NOUN
iajs-2001	162	20	)	)	PUNCT
iajs-2001	162	21	:	:	PUNCT
iajs-2001	162	22	𝑎	𝑎	X
iajs-2001	162	23	𝑦	𝑦	NOUN
iajs-2001	162	24	]	]	X
iajs-2001	162	25	=	=	SYM
iajs-2001	162	26	r	r	NOUN
iajs-2001	162	27	implies	imply	VERB
iajs-2001	162	28	that	that	SCONJ
iajs-2001	162	29	[	[	X
iajs-2001	162	30	b	b	X
iajs-2001	162	31	+	+	CCONJ
iajs-2001	162	32	𝚥(y	𝚥(y	NOUN
iajs-2001	162	33	)	)	PUNCT
iajs-2001	162	34	:	:	PUNCT
iajs-2001	162	35	ay	ay	X
iajs-2001	162	36	]	]	X
iajs-2001	162	37	=	=	SYM
iajs-2001	162	38	r	r	NOUN
iajs-2001	162	39	,	,	PUNCT
iajs-2001	162	40	hence	hence	ADV
iajs-2001	162	41	ay	ay	PROPN
iajs-2001	162	42	∈	∈	PROPN
iajs-2001	162	43	b	b	PROPN
iajs-2001	162	44	+	+	NOUN
iajs-2001	162	45	'	'	NOUN
iajs-2001	162	46	𝚥(y	𝚥(y	NOUN
iajs-2001	162	47	)	)	PUNCT
iajs-2001	162	48	'	'	PUNCT
iajs-2001	162	49	.	.	PUNCT
iajs-2001	163	1	mathematics	mathematic	NOUN
iajs-2001	163	2	|	|	ADV
iajs-2001	163	3	123	123	NUM
iajs-2001	163	4	ibn	ibn	PROPN
iajs-2001	163	5	al	al	PROPN
iajs-2001	163	6	-	-	PUNCT
iajs-2001	163	7	haitham	haitham	PROPN
iajs-2001	163	8	jour	jour	X
iajs-2001	163	9	.	.	PROPN
iajs-2001	164	1	for	for	ADP
iajs-2001	164	2	pure	pure	ADJ
iajs-2001	164	3	&	&	CCONJ
iajs-2001	164	4	appl	appl	PROPN
iajs-2001	164	5	.	.	PUNCT
iajs-2001	165	1	sci	sci	PROPN
iajs-2001	165	2	.	.	PROPN
iajs-2001	165	3	ihjpas	ihjpa	VERB
iajs-2001	165	4	https://doi.org/10.30526/31.3.2001	https://doi.org/10.30526/31.3.2001	PROPN
iajs-2001	165	5	vol	vol	NOUN
iajs-2001	165	6	.	.	PROPN
iajs-2001	166	1	31	31	NUM
iajs-2001	167	1	(	(	PUNCT
iajs-2001	167	2	3	3	NUM
iajs-2001	167	3	)	)	SYM
iajs-2001	167	4	2018	2018	NUM
iajs-2001	167	5	proposition	proposition	NOUN
iajs-2001	167	6	17	17	NUM
iajs-2001	167	7	'	'	NOUN
iajs-2001	167	8	let"y	let"y	NOUN
iajs-2001	167	9	'	'	PUNCT
iajs-2001	167	10	be"'an"r'-'module"'and"a',"b	be"'an"r'-'module"'and"a',"b	NOUN
iajs-2001	167	11	are	be	AUX
iajs-2001	167	12	submodules	submodule	NOUN
iajs-2001	167	13	of	of	ADP
iajs-2001	167	14	y	y	PROPN
iajs-2001	167	15	,	,	PUNCT
iajs-2001	167	16	with	with	ADP
iajs-2001	167	17	a	a	PRON
iajs-2001	167	18	is	be	AUX
iajs-2001	167	19	a	a	DET
iajs-2001	167	20	subset	subset	NOUN
iajs-2001	167	21	of	of	ADP
iajs-2001	167	22	b.	b.	PROPN
iajs-2001	167	23	if	if	SCONJ
iajs-2001	167	24	a	a	PRON
iajs-2001	167	25	is	be	AUX
iajs-2001	167	26	a	a	DET
iajs-2001	167	27	wns-2-"absorbing"submodule"of	wns-2-"absorbing"submodule"of	ADJ
iajs-2001	167	28	y	y	NOUN
iajs-2001	167	29	and	and	CCONJ
iajs-2001	167	30	𝚥(y	𝚥(y	PROPN
iajs-2001	167	31	)	)	PUNCT
iajs-2001	167	32	⊆	⊆	NUM
iajs-2001	167	33	(	(	PUNCT
iajs-2001	167	34	b	b	NOUN
iajs-2001	167	35	)	)	PUNCT
iajs-2001	167	36	,	,	PUNCT
iajs-2001	167	37	then	then	ADV
iajs-2001	167	38	a	a	PRON
iajs-2001	167	39	is	be	AUX
iajs-2001	167	40	a	a	DET
iajs-2001	167	41	wns-2	wns-2	ADV
iajs-2001	167	42	-	-	PUNCT
iajs-2001	167	43	absorbing	absorb	VERB
iajs-2001	167	44	submodule	submodule	NOUN
iajs-2001	167	45	of	of	ADP
iajs-2001	167	46	b.	b.	PROPN
iajs-2001	167	47	proof	proof	NOUN
iajs-2001	167	48	similarly	similarly	ADV
iajs-2001	167	49	as	as	ADP
iajs-2001	167	50	in	in	ADP
iajs-2001	167	51	proposition	proposition	NOUN
iajs-2001	167	52	2.4	2.4	NUM
iajs-2001	167	53	proposition	proposition	NOUN
iajs-2001	167	54	18	18	NUM
iajs-2001	167	55	let	let	VERB
iajs-2001	167	56	y	y	PRON
iajs-2001	167	57	be	be	AUX
iajs-2001	167	58	an"r'-'module"over"a	an"r'-'module"over"a	PROPN
iajs-2001	167	59	good"ring''r	good"ring''r	NOUN
iajs-2001	167	60	"	"	PUNCT
iajs-2001	167	61	and"a	and"a	PROPN
iajs-2001	167	62	'	'	PUNCT
iajs-2001	167	63	,	,	PUNCT
iajs-2001	167	64	b	b	PROPN
iajs-2001	167	65	are"proper'"submodules	are"proper'"submodule	NOUN
iajs-2001	167	66	of"y	of"y	ADV
iajs-2001	167	67	.	.	PUNCT
iajs-2001	168	1	if	if	SCONJ
iajs-2001	168	2	a	a	PRON
iajs-2001	168	3	is	be	AUX
iajs-2001	168	4	a	a	DET
iajs-2001	168	5	wns-"2-'absorbing'"submodule'"'of"y"then	wns-"2-'absorbing'"submodule'"'of"y"then	NOUN
iajs-2001	168	6	'	'	PUNCT
iajs-2001	168	7	a	a	PRON
iajs-2001	168	8	is	be	AUX
iajs-2001	168	9	'	'	PUNCT
iajs-2001	168	10	a	a	NOUN
iajs-2001	168	11	;	;	PUNCT
iajs-2001	168	12	wns-'2-'absorbing''submodule''of	wns-'2-'absorbing''submodule''of	PROPN
iajs-2001	168	13	b.	b.	PROPN
iajs-2001	169	1	'	'	PUNCT
iajs-2001	169	2	proof	proof	NOUN
iajs-2001	169	3	"	"	PUNCT
iajs-2001	169	4	"	"	PUNCT
iajs-2001	169	5	'	'	PUNCT
iajs-2001	169	6	let"0	let"0	NOUN
iajs-2001	169	7	≠	≠	PROPN
iajs-2001	169	8	b	b	X
iajs-2001	169	9	y	y	PROPN
iajs-2001	169	10	∊	∊	PROPN
iajs-2001	169	11	b	b	PROPN
iajs-2001	169	12	,	,	PUNCT
iajs-2001	169	13	for	for	ADP
iajs-2001	169	14	b	b	PROPN
iajs-2001	169	15	∈	∈	PROPN
iajs-2001	169	16	r	r	NOUN
iajs-2001	169	17	,	,	PUNCT
iajs-2001	169	18	y	y	PROPN
iajs-2001	169	19	∈	∈	PROPN
iajs-2001	169	20	b	b	PROPN
iajs-2001	169	21	⊆	⊆	NUM
iajs-2001	169	22	y	y	PROPN
iajs-2001	169	23	,	,	PUNCT
iajs-2001	169	24	it	it	PRON
iajs-2001	169	25	follows	follow	VERB
iajs-2001	169	26	that	that	SCONJ
iajs-2001	169	27	either	either	CCONJ
iajs-2001	169	28	by	by	ADP
iajs-2001	169	29	∈	∈	PROPN
iajs-2001	169	30	a	a	DET
iajs-2001	169	31	+	+	ADJ
iajs-2001	169	32	𝚥(y	𝚥(y	NOUN
iajs-2001	169	33	)	)	PUNCT
iajs-2001	169	34	or	or	CCONJ
iajs-2001	169	35	b	b	X
iajs-2001	169	36	∊	∊	NOUN
iajs-2001	170	1	[	[	X
iajs-2001	170	2	a	a	X
iajs-2001	170	3	+	+	NUM
iajs-2001	170	4	𝚥(y):y	𝚥(y):y	NOUN
iajs-2001	170	5	]	]	PUNCT
iajs-2001	170	6	,	,	PUNCT
iajs-2001	170	7	implies	imply	VERB
iajs-2001	170	8	that	that	SCONJ
iajs-2001	170	9	by	by	ADP
iajs-2001	170	10	∈	∈	PROPN
iajs-2001	170	11	(	(	PUNCT
iajs-2001	170	12	a	a	DET
iajs-2001	170	13	+	+	NUM
iajs-2001	170	14	𝚥(y))∩	𝚥(y))∩	PROPN
iajs-2001	170	15	b	b	PROPN
iajs-2001	170	16	or	or	CCONJ
iajs-2001	170	17	b	b	PROPN
iajs-2001	170	18	y	y	PROPN
iajs-2001	170	19	∊	∊	PROPN
iajs-2001	170	20	(	(	PUNCT
iajs-2001	170	21	a	a	DET
iajs-2001	170	22	+	+	X
iajs-2001	170	23	𝚥(y	𝚥(y	NOUN
iajs-2001	170	24	)	)	PUNCT
iajs-2001	170	25	)	)	PUNCT
iajs-2001	170	26	∩	∩	PROPN
iajs-2001	170	27	b	b	X
iajs-2001	170	28	,	,	PUNCT
iajs-2001	170	29	for	for	ADP
iajs-2001	170	30	each	each	DET
iajs-2001	170	31	y	y	PROPN
iajs-2001	170	32	∈	∈	PROPN
iajs-2001	170	33	b.	b.	PROPN
iajs-2001	170	34	thus	thus	ADV
iajs-2001	170	35	by	by	ADP
iajs-2001	170	36	modular	modular	ADJ
iajs-2001	170	37	law	law	NOUN
iajs-2001	170	38	,	,	PUNCT
iajs-2001	170	39	by	by	ADP
iajs-2001	170	40	∈	∈	PROPN
iajs-2001	170	41	(	(	PUNCT
iajs-2001	170	42	a∩b	a∩b	PROPN
iajs-2001	170	43	)	)	PUNCT
iajs-2001	171	1	+	+	CCONJ
iajs-2001	171	2	(	(	PUNCT
iajs-2001	171	3	𝚥(y)∩	𝚥(y)∩	PROPN
iajs-2001	171	4	b	b	NOUN
iajs-2001	171	5	)	)	PUNCT
iajs-2001	171	6	.	.	PUNCT
iajs-2001	172	1	but	but	CCONJ
iajs-2001	172	2	r	r	NOUN
iajs-2001	172	3	is	be	AUX
iajs-2001	172	4	a	a	DET
iajs-2001	172	5	good	good	ADJ
iajs-2001	172	6	ring	ring	NOUN
iajs-2001	172	7	,	,	PUNCT
iajs-2001	172	8	then	then	ADV
iajs-2001	172	9	𝚥(y)∩	𝚥(y)∩	PROPN
iajs-2001	172	10	b	b	X
iajs-2001	172	11	=	=	PUNCT
iajs-2001	172	12	𝚥(b	𝚥(b	NOUN
iajs-2001	172	13	)	)	PUNCT
iajs-2001	172	14	and''a	and''a	NOUN
iajs-2001	172	15	∩'b'is'''a'''proper'''subset'of"a	∩'b'is'''a'''proper'''subset'of"a	PROPN
iajs-2001	172	16	,	,	PUNCT
iajs-2001	172	17	hence	hence	ADV
iajs-2001	172	18	either	either	CCONJ
iajs-2001	172	19	by	by	ADP
iajs-2001	172	20	∈	∈	PROPN
iajs-2001	172	21	'	'	PUNCT
iajs-2001	172	22	a"+"𝚥('b	a"+"𝚥('b	NOUN
iajs-2001	172	23	"	"	PUNCT
iajs-2001	172	24	)	)	PUNCT
iajs-2001	172	25	or	or	CCONJ
iajs-2001	172	26	b	b	X
iajs-2001	172	27	y	y	PROPN
iajs-2001	172	28	∊	∊	PROPN
iajs-2001	172	29	a	a	DET
iajs-2001	172	30	+	+	ADJ
iajs-2001	172	31	(	(	PUNCT
iajs-2001	172	32	b	b	NOUN
iajs-2001	172	33	)	)	PUNCT
iajs-2001	172	34	for	for	ADP
iajs-2001	172	35	each	each	DET
iajs-2001	172	36	y	y	PROPN
iajs-2001	172	37	∈	∈	PROPN
iajs-2001	172	38	b.	b.	PROPN
iajs-2001	172	39	thus	thus	ADV
iajs-2001	172	40	either	either	CCONJ
iajs-2001	172	41	by	by	ADP
iajs-2001	172	42	∈	∈	PROPN
iajs-2001	172	43	a	a	DET
iajs-2001	172	44	+	+	X
iajs-2001	172	45	(	(	PUNCT
iajs-2001	172	46	b	b	NOUN
iajs-2001	172	47	)	)	PUNCT
iajs-2001	172	48	or	or	CCONJ
iajs-2001	172	49	b	b	NOUN
iajs-2001	172	50	y	y	PROPN
iajs-2001	172	51	∊	∊	PROPN
iajs-2001	173	1	[	[	X
iajs-2001	173	2	a	a	X
iajs-2001	173	3	+	+	X
iajs-2001	173	4	(	(	PUNCT
iajs-2001	173	5	b):b	b):b	X
iajs-2001	173	6	]	]	PUNCT
iajs-2001	173	7	.	.	PUNCT
iajs-2001	174	1	hence	hence	ADV
iajs-2001	174	2	a	a	PRON
iajs-2001	174	3	is	be	AUX
iajs-2001	174	4	a	a	DET
iajs-2001	174	5	wns-2	wns-2	ADV
iajs-2001	174	6	-	-	PUNCT
iajs-2001	174	7	absorbing	absorb	VERB
iajs-2001	174	8	submodule	submodule	NOUN
iajs-2001	174	9	of	of	ADP
iajs-2001	174	10	b.	b.	PROPN
iajs-2001	174	11	'	'	PUNCT
iajs-2001	174	12	remark	remark	NOUN
iajs-2001	174	13	19	19	NUM
iajs-2001	174	14	the	the	DET
iajs-2001	174	15	intersection	intersection	NOUN
iajs-2001	174	16	of	of	ADP
iajs-2001	174	17	two	two	NUM
iajs-2001	174	18	wns-2"-"absorbing	wns-2"-"absorbing	NOUN
iajs-2001	174	19	'	'	PUNCT
iajs-2001	174	20	submodules	submodule	NOUN
iajs-2001	174	21	'	'	PUNCT
iajs-2001	174	22	of	of	ADP
iajs-2001	174	23	'	'	PUNCT
iajs-2001	174	24	an"r-"'module	an"r-"'module	PROPN
iajs-2001	174	25	y"is	y"is	NOUN
iajs-2001	174	26	'	'	PUNCT
iajs-2001	174	27	not	not	PART
iajs-2001	174	28	necessary	necessary	ADJ
iajs-2001	174	29	wns-"2-"absorbing'"submodules"of	wns-"2-"absorbing'"submodules"of	PROPN
iajs-2001	174	30	y.	y.	NOUN
iajs-2001	175	1	the	the	DET
iajs-2001	175	2	following	follow	VERB
iajs-2001	175	3	example	example	NOUN
iajs-2001	175	4	explain	explain	VERB
iajs-2001	175	5	that	that	SCONJ
iajs-2001	175	6	:	:	PUNCT
iajs-2001	175	7	let	let	VERB
iajs-2001	175	8	y	y	PROPN
iajs-2001	175	9	=	=	PROPN
iajs-2001	175	10	z	z	PROPN
iajs-2001	175	11	,	,	PUNCT
iajs-2001	175	12	r	r	NOUN
iajs-2001	175	13	=	=	SYM
iajs-2001	175	14	z	z	NOUN
iajs-2001	175	15	and	and	CCONJ
iajs-2001	175	16	a=2z	a=2z	NOUN
iajs-2001	175	17	,	,	PUNCT
iajs-2001	175	18	b=25z	b=25z	PROPN
iajs-2001	175	19	,	,	PUNCT
iajs-2001	175	20	clearly	clearly	ADV
iajs-2001	175	21	a	a	DET
iajs-2001	175	22	,	,	PUNCT
iajs-2001	175	23	b	b	NOUN
iajs-2001	175	24	are	be	AUX
iajs-2001	175	25	wns-2	wns-2	ADV
iajs-2001	175	26	-	-	PUNCT
iajs-2001	175	27	absorbing	absorb	VERB
iajs-2001	175	28	submodules	submodule	NOUN
iajs-2001	175	29	of	of	ADP
iajs-2001	175	30	y	y	PROPN
iajs-2001	175	31	,	,	PUNCT
iajs-2001	175	32	but	but	CCONJ
iajs-2001	175	33	a	a	DET
iajs-2001	175	34	∩	∩	ADJ
iajs-2001	175	35	b	b	NOUN
iajs-2001	175	36	=	=	PUNCT
iajs-2001	175	37	50z	50z	NOUN
iajs-2001	175	38	,	,	PUNCT
iajs-2001	175	39	is	be	AUX
iajs-2001	175	40	not	not	PART
iajs-2001	175	41	wns-2	wns-2	ADV
iajs-2001	175	42	-	-	PUNCT
iajs-2001	175	43	absorbing	absorb	VERB
iajs-2001	175	44	submodule	submodule	NOUN
iajs-2001	175	45	of	of	ADP
iajs-2001	175	46	y.	y.	PROPN
iajs-2001	175	47	'	'	PUNCT
iajs-2001	175	48	proposition	proposition	NOUN
iajs-2001	175	49	20	20	NUM
iajs-2001	175	50	let	let	VERB
iajs-2001	175	51	y	y	PRON
iajs-2001	175	52	be"'an"r'-"module'","and'"a	be"'an"r'-"module'","and'"a	VERB
iajs-2001	175	53	'	'	PROPN
iajs-2001	175	54	,	,	PUNCT
iajs-2001	175	55	b	b	X
iajs-2001	175	56	are"proper	are"proper	NOUN
iajs-2001	175	57	'	'	PUNCT
iajs-2001	175	58	submodules"of	submodules"of	PROPN
iajs-2001	175	59	y	y	PROPN
iajs-2001	175	60	with	with	ADP
iajs-2001	175	61	𝚥(y	𝚥(y	PROPN
iajs-2001	175	62	)	)	PUNCT
iajs-2001	175	63	⊆	⊆	NUM
iajs-2001	175	64	a	a	PRON
iajs-2001	175	65	,	,	PUNCT
iajs-2001	175	66	or	or	CCONJ
iajs-2001	175	67	𝚥(y	𝚥(y	PROPN
iajs-2001	175	68	)	)	PUNCT
iajs-2001	175	69	⊆	⊆	NUM
iajs-2001	175	70	b	b	NOUN
iajs-2001	175	71	,	,	PUNCT
iajs-2001	175	72	if	if	SCONJ
iajs-2001	175	73	a	a	PRON
iajs-2001	175	74	and	and	CCONJ
iajs-2001	175	75	b	b	NOUN
iajs-2001	175	76	are	be	AUX
iajs-2001	175	77	wns"-2-'absorbing'"submodules"of'"y',"then	wns"-2-'absorbing'"submodules"of'"y',"then	ADV
iajs-2001	175	78	'	'	PUNCT
iajs-2001	175	79	a	a	DET
iajs-2001	175	80	∩	∩	NOUN
iajs-2001	175	81	b]is	b]i	VERB
iajs-2001	175	82	a''wns"-2"absorbing'"submodule'"of;;y	a''wns"-2"absorbing'"submodule'"of;;y	NOUN
iajs-2001	175	83	.	.	PUNCT
iajs-2001	175	84	proof	proof	NOUN
iajs-2001	175	85	"	"	PUNCT
iajs-2001	175	86	"	"	PUNCT
iajs-2001	175	87	let	let	VERB
iajs-2001	175	88	'	'	PRON
iajs-2001	175	89	0	0	NUM
iajs-2001	175	90	"	"	PUNCT
iajs-2001	175	91	≠	≠	PROPN
iajs-2001	175	92	r	r	NOUN
iajs-2001	175	93	y	y	PROPN
iajs-2001	175	94	∈	∈	PROPN
iajs-2001	175	95	a	a	DET
iajs-2001	175	96	∩	∩	ADJ
iajs-2001	175	97	b	b	NOUN
iajs-2001	175	98	,	,	PUNCT
iajs-2001	175	99	where	where	SCONJ
iajs-2001	175	100	r	r	NOUN
iajs-2001	175	101	∈	∈	PROPN
iajs-2001	175	102	r	r	NOUN
iajs-2001	175	103	,	,	PUNCT
iajs-2001	175	104	y	y	PROPN
iajs-2001	175	105	∈	∈	PROPN
iajs-2001	175	106	y	y	PROPN
iajs-2001	175	107	,	,	PUNCT
iajs-2001	175	108	then	then	ADV
iajs-2001	175	109	0	0	NUM
iajs-2001	175	110	≠	≠	PROPN
iajs-2001	175	111	r	r	NOUN
iajs-2001	175	112	y	y	PROPN
iajs-2001	175	113	∈	∈	PROPN
iajs-2001	175	114	a	a	PRON
iajs-2001	175	115	and	and	CCONJ
iajs-2001	175	116	0	0	NUM
iajs-2001	175	117	≠	≠	PROPN
iajs-2001	175	118	r	r	NOUN
iajs-2001	175	119	y	y	PROPN
iajs-2001	175	120	∈	∈	PROPN
iajs-2001	175	121	b	b	PROPN
iajs-2001	175	122	,	,	PUNCT
iajs-2001	175	123	but	but	CCONJ
iajs-2001	175	124	both	both	DET
iajs-2001	175	125	a	a	PRON
iajs-2001	175	126	and	and	CCONJ
iajs-2001	175	127	b	b	NOUN
iajs-2001	175	128	are	be	AUX
iajs-2001	175	129	wns-2	wns-2	ADV
iajs-2001	175	130	-	-	PUNCT
iajs-2001	175	131	absorbing	absorb	VERB
iajs-2001	175	132	submodules	submodule	NOUN
iajs-2001	175	133	of	of	ADP
iajs-2001	175	134	y	y	PROPN
iajs-2001	175	135	then	then	ADV
iajs-2001	175	136	either	either	CCONJ
iajs-2001	175	137	ry	ry	PROPN
iajs-2001	175	138	∈	∈	PROPN
iajs-2001	175	139	a	a	DET
iajs-2001	175	140	+	+	ADJ
iajs-2001	175	141	𝚥(y	𝚥(y	NOUN
iajs-2001	175	142	)	)	PUNCT
iajs-2001	175	143	or	or	CCONJ
iajs-2001	175	144	r	r	NOUN
iajs-2001	175	145	∈	∈	PROPN
iajs-2001	175	146	a	a	DET
iajs-2001	175	147	ȷ	ȷ	NOUN
iajs-2001	175	148	y	y	NOUN
iajs-2001	175	149	:	:	PUNCT
iajs-2001	175	150	y	y	PROPN
iajs-2001	175	151	and	and	CCONJ
iajs-2001	175	152	either	either	CCONJ
iajs-2001	175	153	ry	ry	PROPN
iajs-2001	175	154	∈	∈	PROPN
iajs-2001	175	155	b	b	PROPN
iajs-2001	175	156	+	+	CCONJ
iajs-2001	175	157	𝚥(y	𝚥(y	NOUN
iajs-2001	175	158	)	)	PUNCT
iajs-2001	175	159	or	or	CCONJ
iajs-2001	175	160	r	r	NOUN
iajs-2001	175	161	∈	∈	PROPN
iajs-2001	175	162	b	b	NOUN
iajs-2001	175	163	ȷ	ȷ	X
iajs-2001	175	164	y	y	NOUN
iajs-2001	175	165	:	:	PUNCT
iajs-2001	175	166	y	y	PROPN
iajs-2001	175	167	.	.	PUNCT
iajs-2001	175	168	implies	imply	VERB
iajs-2001	175	169	that	that	SCONJ
iajs-2001	175	170	ry	ry	PROPN
iajs-2001	175	171	∈	∈	PROPN
iajs-2001	175	172	a	a	PRON
iajs-2001	176	1	+	+	ADJ
iajs-2001	176	2	𝚥(y	𝚥(y	NOUN
iajs-2001	176	3	)	)	PUNCT
iajs-2001	176	4	∩	∩	PROPN
iajs-2001	176	5	b	b	PROPN
iajs-2001	176	6	+	+	CCONJ
iajs-2001	176	7	𝚥(y	𝚥(y	NOUN
iajs-2001	176	8	)	)	PUNCT
iajs-2001	176	9	or	or	CCONJ
iajs-2001	176	10	r	r	NOUN
iajs-2001	176	11	y	y	PROPN
iajs-2001	176	12	⊆	⊆	NUM
iajs-2001	176	13	a	a	DET
iajs-2001	176	14	ȷ	ȷ	X
iajs-2001	176	15	y	y	PROPN
iajs-2001	176	16	∩	∩	PROPN
iajs-2001	176	17	b	b	X
iajs-2001	176	18	ȷ	ȷ	X
iajs-2001	176	19	y	y	PROPN
iajs-2001	176	20	if	if	SCONJ
iajs-2001	176	21	𝚥(y	𝚥(y	NOUN
iajs-2001	176	22	)	)	PUNCT
iajs-2001	176	23	⊆	⊆	NUM
iajs-2001	176	24	b	b	NOUN
iajs-2001	176	25	,	,	PUNCT
iajs-2001	176	26	then	then	ADV
iajs-2001	176	27	b	b	PROPN
iajs-2001	176	28	+	+	CCONJ
iajs-2001	176	29	𝚥(y	𝚥(y	PROPN
iajs-2001	176	30	)	)	PUNCT
iajs-2001	176	31	=	=	SYM
iajs-2001	176	32	b	b	NOUN
iajs-2001	176	33	,	,	PUNCT
iajs-2001	176	34	thus	thus	ADV
iajs-2001	176	35	either	either	CCONJ
iajs-2001	176	36	ry	ry	PROPN
iajs-2001	176	37	∈	∈	PROPN
iajs-2001	176	38	a	a	DET
iajs-2001	176	39	+	+	ADJ
iajs-2001	176	40	𝚥(y	𝚥(y	NOUN
iajs-2001	176	41	)	)	PUNCT
iajs-2001	176	42	∩	∩	PROPN
iajs-2001	176	43	b	b	NOUN
iajs-2001	176	44	or	or	CCONJ
iajs-2001	176	45	r	r	NOUN
iajs-2001	176	46	y	y	PROPN
iajs-2001	176	47	⊆	⊆	NUM
iajs-2001	176	48	a	a	DET
iajs-2001	176	49	ȷ	ȷ	X
iajs-2001	176	50	y	y	PROPN
iajs-2001	176	51	∩	∩	ADJ
iajs-2001	176	52	b	b	X
iajs-2001	176	53	,	,	PUNCT
iajs-2001	176	54	it	it	PRON
iajs-2001	176	55	follows	follow	VERB
iajs-2001	176	56	that	that	SCONJ
iajs-2001	176	57	either	either	CCONJ
iajs-2001	176	58	ry	ry	PROPN
iajs-2001	176	59	∈	∈	PROPN
iajs-2001	176	60	a∩	a∩	PROPN
iajs-2001	176	61	b	b	PROPN
iajs-2001	176	62	+	+	CCONJ
iajs-2001	176	63	𝚥(y	𝚥(y	PROPN
iajs-2001	176	64	)	)	PUNCT
iajs-2001	176	65	or	or	CCONJ
iajs-2001	176	66	r	r	NOUN
iajs-2001	176	67	y	y	PROPN
iajs-2001	176	68	⊆	⊆	NUM
iajs-2001	176	69	a	a	DET
iajs-2001	176	70	∩	∩	ADJ
iajs-2001	176	71	b	b	NOUN
iajs-2001	176	72	ȷ	ȷ	X
iajs-2001	176	73	y	y	NOUN
iajs-2001	176	74	.	.	PUNCT
iajs-2001	177	1	that	that	PRON
iajs-2001	177	2	is	be	AUX
iajs-2001	177	3	ry	ry	PROPN
iajs-2001	177	4	∈	∈	PROPN
iajs-2001	177	5	a∩	a∩	PROPN
iajs-2001	177	6	b	b	PROPN
iajs-2001	177	7	+	+	CCONJ
iajs-2001	177	8	𝚥(y	𝚥(y	PROPN
iajs-2001	177	9	)	)	PUNCT
iajs-2001	177	10	or	or	CCONJ
iajs-2001	177	11	r	r	NOUN
iajs-2001	177	12	∈	∈	PROPN
iajs-2001	177	13	a	a	DET
iajs-2001	177	14	∩	∩	ADJ
iajs-2001	177	15	b	b	NOUN
iajs-2001	177	16	ȷ	ȷ	X
iajs-2001	177	17	y	y	NOUN
iajs-2001	177	18	:	:	PUNCT
iajs-2001	178	1	y	y	PROPN
iajs-2001	178	2	.	.	PUNCT
iajs-2001	179	1	hence	hence	ADV
iajs-2001	179	2	a	a	DET
iajs-2001	179	3	∩	∩	ADJ
iajs-2001	179	4	b	b	NOUN
iajs-2001	179	5	is	be	AUX
iajs-2001	179	6	a	a	DET
iajs-2001	179	7	wns-2	wns-2	ADV
iajs-2001	179	8	-	-	PUNCT
iajs-2001	179	9	absorbing	absorb	VERB
iajs-2001	179	10	submodule	submodule	NOUN
iajs-2001	179	11	of	of	ADP
iajs-2001	179	12	y.	y.	PROPN
iajs-2001	179	13	similarly	similarly	ADV
iajs-2001	179	14	if	if	SCONJ
iajs-2001	179	15	𝚥(y	𝚥(y	NOUN
iajs-2001	179	16	)	)	PUNCT
iajs-2001	179	17	⊆	⊆	NUM
iajs-2001	179	18	a	a	PRON
iajs-2001	179	19	,	,	PUNCT
iajs-2001	179	20	we	we	PRON
iajs-2001	179	21	get	get	VERB
iajs-2001	179	22	a	a	DET
iajs-2001	179	23	∩	∩	ADJ
iajs-2001	179	24	b	b	NOUN
iajs-2001	179	25	is	be	AUX
iajs-2001	179	26	a	a	DET
iajs-2001	179	27	wns-2absorbing	wns-2absorbe	VERB
iajs-2001	179	28	submodule	submodule	NOUN
iajs-2001	179	29	of	of	ADP
iajs-2001	179	30	y.	y.	PROPN
iajs-2001	179	31	proposition"21	proposition"21	NOUN
iajs-2001	179	32	"	"	PUNCT
iajs-2001	179	33	let"y	let"y	NOUN
iajs-2001	179	34	be	be	AUX
iajs-2001	179	35	'	'	PUNCT
iajs-2001	179	36	an"r-"module	an"r-"module	NOUN
iajs-2001	179	37	'	'	PUNCT
iajs-2001	179	38	,	,	PUNCT
iajs-2001	179	39	"	"	PUNCT
iajs-2001	179	40	and'"a	and'"a	NOUN
iajs-2001	179	41	'	'	PUNCT
iajs-2001	179	42	is	be	AUX
iajs-2001	179	43	a"proper'"submodule'"of	a"proper'"submodule'"of	PRON
iajs-2001	179	44	'	'	PUNCT
iajs-2001	179	45	y	y	NOUN
iajs-2001	179	46	with	with	ADP
iajs-2001	179	47	𝚥(y	𝚥(y	PROPN
iajs-2001	179	48	)	)	PUNCT
iajs-2001	179	49	⊆	⊆	NUM
iajs-2001	179	50	a	a	DET
iajs-2001	179	51	,	,	PUNCT
iajs-2001	179	52	"	"	PUNCT
iajs-2001	179	53	'	'	PUNCT
iajs-2001	179	54	if	if	SCONJ
iajs-2001	179	55	a'"is	a'"is	PROPN
iajs-2001	179	56	'	'	PART
iajs-2001	179	57	a'"wns-2-"absorbing	a'"wns-2-"absorbing	NOUN
iajs-2001	179	58	'	'	PUNCT
iajs-2001	179	59	submodules'"of	submodules'"of	NOUN
iajs-2001	179	60	y	y	PROPN
iajs-2001	179	61	,	,	PUNCT
iajs-2001	179	62	then'"['a	then'"['a	NUM
iajs-2001	179	63	:	:	PUNCT
iajs-2001	179	64	y	y	X
iajs-2001	179	65	]	]	X
iajs-2001	179	66	is"a	is"a	NOUN
iajs-2001	179	67	wns-2-"absorbing''"ideal	wns-2-"absorbing''"ideal	NOUN
iajs-2001	179	68	'	'	PUNCT
iajs-2001	179	69	of'''r	of'''r	NOUN
iajs-2001	179	70	.	.	PUNCT
iajs-2001	180	1	mathematics	mathematic	NOUN
iajs-2001	180	2	|	|	ADV
iajs-2001	180	3	124	124	NUM
iajs-2001	180	4	ibn	ibn	PROPN
iajs-2001	180	5	al	al	PROPN
iajs-2001	180	6	-	-	PUNCT
iajs-2001	180	7	haitham	haitham	PROPN
iajs-2001	180	8	jour	jour	X
iajs-2001	180	9	.	.	PROPN
iajs-2001	181	1	for	for	ADP
iajs-2001	181	2	pure	pure	ADJ
iajs-2001	181	3	&	&	CCONJ
iajs-2001	181	4	appl	appl	PROPN
iajs-2001	181	5	.	.	PUNCT
iajs-2001	182	1	sci	sci	PROPN
iajs-2001	182	2	.	.	PROPN
iajs-2001	182	3	ihjpas	ihjpa	VERB
iajs-2001	182	4	https://doi.org/10.30526/31.3.2001	https://doi.org/10.30526/31.3.2001	PROPN
iajs-2001	182	5	vol	vol	NOUN
iajs-2001	182	6	.	.	PROPN
iajs-2001	183	1	31	31	NUM
iajs-2001	184	1	(	(	PUNCT
iajs-2001	184	2	3	3	NUM
iajs-2001	184	3	)	)	PUNCT
iajs-2001	184	4	2018	2018	NUM
iajs-2001	184	5	proof	proof	NOUN
iajs-2001	184	6	since	since	SCONJ
iajs-2001	184	7	a	a	PRON
iajs-2001	184	8	is	be	AUX
iajs-2001	184	9	a	a	DET
iajs-2001	184	10	wns-2	wns-2	ADV
iajs-2001	184	11	-	-	PUNCT
iajs-2001	184	12	absorbing	absorb	VERB
iajs-2001	184	13	submodules	submodule	NOUN
iajs-2001	184	14	of	of	ADP
iajs-2001	184	15	y	y	PROPN
iajs-2001	184	16	,	,	PUNCT
iajs-2001	184	17	and	and	CCONJ
iajs-2001	184	18	𝚥(y	𝚥(y	NOUN
iajs-2001	184	19	)	)	PUNCT
iajs-2001	184	20	⊆	⊆	NUM
iajs-2001	184	21	a	a	PRON
iajs-2001	184	22	,	,	PUNCT
iajs-2001	184	23	then	then	ADV
iajs-2001	184	24	by	by	ADP
iajs-2001	184	25	remarks	remark	NOUN
iajs-2001	184	26	and	and	CCONJ
iajs-2001	184	27	examples	example	NOUN
iajs-2001	184	28	15	15	NUM
iajs-2001	184	29	(	(	PUNCT
iajs-2001	184	30	5	5	NUM
iajs-2001	184	31	)	)	PUNCT
iajs-2001	184	32	,	,	PUNCT
iajs-2001	184	33	a	a	DET
iajs-2001	184	34	a	a	DET
iajs-2001	184	35	weakly	weakly	ADJ
iajs-2001	184	36	semi	semi	ADJ
iajs-2001	184	37	2	2	NUM
iajs-2001	184	38	-	-	PUNCT
iajs-2001	184	39	absorbing	absorb	VERB
iajs-2001	184	40	submodule	submodule	NOUN
iajs-2001	184	41	of	of	ADP
iajs-2001	184	42	y	y	PROPN
iajs-2001	184	43	,	,	PUNCT
iajs-2001	184	44	then	then	ADV
iajs-2001	184	45	by	by	ADP
iajs-2001	184	46	[	[	X
iajs-2001	184	47	2,prop	2,prop	NUM
iajs-2001	184	48	.	.	NOUN
iajs-2001	184	49	4	4	NUM
iajs-2001	184	50	]	]	PUNCT
iajs-2001	184	51	,	,	PUNCT
iajs-2001	184	52	[	[	PUNCT
iajs-2001	184	53	a	a	X
iajs-2001	184	54	:	:	PUNCT
iajs-2001	184	55	y	y	NOUN
iajs-2001	184	56	]	]	PUNCT
iajs-2001	184	57	is	be	AUX
iajs-2001	184	58	a	a	DET
iajs-2001	184	59	weakly	weakly	ADJ
iajs-2001	184	60	semi	semi	ADJ
iajs-2001	184	61	2-"absorbing'"ideal	2-"absorbing'"ideal	NUM
iajs-2001	184	62	'	'	PUNCT
iajs-2001	184	63	of'"r	of'"r	ADJ
iajs-2001	184	64	,	,	PUNCT
iajs-2001	184	65	so	so	ADV
iajs-2001	184	66	by	by	ADP
iajs-2001	184	67	(	(	PUNCT
iajs-2001	184	68	1	1	X
iajs-2001	184	69	)	)	PUNCT
iajs-2001	185	1	[	[	X
iajs-2001	185	2	'	'	X
iajs-2001	185	3	a':y	a':y	X
iajs-2001	185	4	]	]	PUNCT
iajs-2001	185	5	is"a	is"a	NOUN
iajs-2001	185	6	wns-'2-"absorbing'"ideal	wns-'2-"absorbing'"ideal	NOUN
iajs-2001	185	7	'	'	PART
iajs-2001	185	8	of'r	of'r	NOUN
iajs-2001	185	9	'	'	PUNCT
iajs-2001	185	10	.	.	PUNCT
iajs-2001	186	1	proposition	proposition	NOUN
iajs-2001	186	2	22	22	NUM
iajs-2001	186	3	let	let	VERB
iajs-2001	186	4	y	y	PRON
iajs-2001	186	5	'	'	PUNCT
iajs-2001	186	6	be"'a	be"'a	ADV
iajs-2001	186	7	cyclic	cyclic	ADJ
iajs-2001	186	8	r'-"module	r'-"module	NOUN
iajs-2001	186	9	,	,	PUNCT
iajs-2001	186	10	'	'	PUNCT
iajs-2001	186	11	and"a	and"a	PROPN
iajs-2001	186	12	is"a'"proper'"submodule'"of	is"a'"proper'"submodule'"of	PROPN
iajs-2001	186	13	'	'	PUNCT
iajs-2001	186	14	y	y	NOUN
iajs-2001	186	15	if'"[a	if'"[a	NOUN
iajs-2001	186	16	:	:	PUNCT
iajs-2001	186	17	y]"'is	y]"'is	NUM
iajs-2001	186	18	a'"wns"-2-"absorbing'"ideal	a'"wns"-2-"absorbing'"ideal	NOUN
iajs-2001	186	19	'	'	PUNCT
iajs-2001	186	20	'	'	PUNCT
iajs-2001	186	21	of"r	of"r	PROPN
iajs-2001	186	22	'	'	PUNCT
iajs-2001	186	23	with	with	ADP
iajs-2001	186	24	'	'	PUNCT
iajs-2001	186	25	𝚥(r	𝚥(r	NOUN
iajs-2001	186	26	)	)	PUNCT
iajs-2001	186	27	⊆	⊆	NUM
iajs-2001	187	1	[	[	X
iajs-2001	187	2	'	'	PUNCT
iajs-2001	187	3	a	a	X
iajs-2001	187	4	:	:	PUNCT
iajs-2001	187	5	y	y	NOUN
iajs-2001	187	6	]	]	X
iajs-2001	187	7	,	,	PUNCT
iajs-2001	187	8	then"a"is	then"a"is	PROPN
iajs-2001	187	9	a	a	DET
iajs-2001	187	10	wns-2"absorbing'"submodules'"of	wns-2"absorbing'"submodules'"of	NOUN
iajs-2001	187	11	''	''	PUNCT
iajs-2001	187	12	y.	y.	NOUN
iajs-2001	187	13	proof	proof	NOUN
iajs-2001	187	14	follows	follow	VERB
iajs-2001	187	15	by	by	ADP
iajs-2001	187	16	remarks	remark	NOUN
iajs-2001	187	17	and	and	CCONJ
iajs-2001	187	18	examples	example	NOUN
iajs-2001	187	19	15(5)(1	15(5)(1	NUM
iajs-2001	187	20	)	)	PUNCT
iajs-2001	187	21	and	and	CCONJ
iajs-2001	187	22	corollary	corollary	ADJ
iajs-2001	187	23	[	[	X
iajs-2001	187	24	2	2	NUM
iajs-2001	187	25	,	,	PUNCT
iajs-2001	187	26	coro	coro	NOUN
iajs-2001	187	27	.	.	PUNCT
iajs-2001	188	1	2.5	2.5	NUM
iajs-2001	188	2	]	]	PUNCT
iajs-2001	188	3	.	.	PUNCT
iajs-2001	189	1	proposition	proposition	NOUN
iajs-2001	189	2	23	23	NUM
iajs-2001	189	3	let	let	VERB
iajs-2001	189	4	g	g	NOUN
iajs-2001	189	5	:	:	PUNCT
iajs-2001	189	6	y	y	PROPN
iajs-2001	189	7	→	→	SYM
iajs-2001	189	8	y	y	AUX
iajs-2001	189	9	`	`	PUNCT
iajs-2001	189	10	be	be	AUX
iajs-2001	189	11	small	small	ADJ
iajs-2001	189	12	r	r	NOUN
iajs-2001	189	13	-	-	PUNCT
iajs-2001	189	14	epimorphism	epimorphism	NOUN
iajs-2001	189	15	and	and	CCONJ
iajs-2001	189	16	a	a	DET
iajs-2001	189	17	proper	proper	ADJ
iajs-2001	189	18	submodules	submodule	NOUN
iajs-2001	189	19	of	of	ADP
iajs-2001	189	20	y	y	PROPN
iajs-2001	189	21	,	,	PUNCT
iajs-2001	189	22	with	with	ADP
iajs-2001	189	23	kerg	kerg	PROPN
iajs-2001	189	24	⊆	⊆	NUM
iajs-2001	189	25	a.	a.	NOUN
iajs-2001	189	26	if	if	SCONJ
iajs-2001	189	27	a	a	PRON
iajs-2001	189	28	is	be	AUX
iajs-2001	189	29	a	a	DET
iajs-2001	189	30	wns-'2-"absorbing'''submodules"of'"y',"then'"g(a	wns-'2-"absorbing'''submodules"of'"y',"then'"g(a	NOUN
iajs-2001	189	31	)	)	PUNCT
iajs-2001	189	32	is'"a	is'"a	NOUN
iajs-2001	189	33	'	'	PART
iajs-2001	189	34	wns-"2-"absorbing	wns-"2-"absorbing	ADJ
iajs-2001	189	35	''	''	PUNCT
iajs-2001	189	36	submodules	submodule	NOUN
iajs-2001	189	37	'	'	PUNCT
iajs-2001	189	38	of	of	ADP
iajs-2001	189	39	"	"	PUNCT
iajs-2001	189	40	y	y	PROPN
iajs-2001	189	41	`	`	PUNCT
iajs-2001	189	42	.	.	PUNCT
iajs-2001	190	1	proof	proof	NOUN
iajs-2001	190	2	"	"	PUNCT
iajs-2001	190	3	"	"	PUNCT
iajs-2001	190	4	similarly,"'as"'in"'proposition'"2.12	similarly,"'as"'in"'proposition'"2.12	PROPN
iajs-2001	190	5	.	.	PROPN
iajs-2001	190	6	proposition	proposition	NOUN
iajs-2001	190	7	24	24	NUM
iajs-2001	190	8	'	'	NUM
iajs-2001	190	9	let′	let′	PROPN
iajs-2001	190	10	𝑔	𝑔	PROPN
iajs-2001	190	11	∶	∶	NOUN
iajs-2001	190	12	"	"	PUNCT
iajs-2001	190	13	𝑌′	𝑌′	PROPN
iajs-2001	190	14	→	→	SYM
iajs-2001	190	15	′𝑌	′𝑌	PROPN
iajs-2001	190	16	`	`	PUNCT
iajs-2001	190	17	"	"	PUNCT
iajs-2001	190	18	"	"	PUNCT
iajs-2001	190	19	be	be	AUX
iajs-2001	190	20	small"r'-epimorphism	small"r'-epimorphism	NOUN
iajs-2001	190	21	and	and	CCONJ
iajs-2001	190	22	𝐴	𝐴	PROPN
iajs-2001	190	23	`	`	PUNCT
iajs-2001	190	24	proper	proper	ADJ
iajs-2001	190	25	submodules	submodule	NOUN
iajs-2001	190	26	of	of	ADP
iajs-2001	190	27	𝑌	𝑌	PROPN
iajs-2001	190	28	`	`	PUNCT
iajs-2001	190	29	.	.	PUNCT
iajs-2001	191	1	if	if	SCONJ
iajs-2001	191	2	𝐴	𝐴	PROPN
iajs-2001	191	3	`	`	PUNCT
iajs-2001	191	4	is	be	AUX
iajs-2001	191	5	a	a	DET
iajs-2001	191	6	wns-2"-"absorbing	wns-2"-"absorbing	NOUN
iajs-2001	191	7	''	''	PUNCT
iajs-2001	191	8	submodules	submodule	NOUN
iajs-2001	191	9	'	'	PUNCT
iajs-2001	191	10	of	of	ADP
iajs-2001	191	11	"	"	PUNCT
iajs-2001	191	12	'	'	PUNCT
iajs-2001	191	13	𝑌′	𝑌′	NOUN
iajs-2001	191	14	`	`	PUNCT
iajs-2001	191	15	,	,	PUNCT
iajs-2001	191	16	"	"	PUNCT
iajs-2001	191	17	then	then	ADV
iajs-2001	191	18	'	'	PUNCT
iajs-2001	191	19	𝑔	𝑔	PROPN
iajs-2001	191	20	𝐴	𝐴	PROPN
iajs-2001	191	21	`	`	PUNCT
iajs-2001	191	22	'	'	PUNCT
iajs-2001	191	23	is"a''wns-"2-"absorbing	is"a''wns-"2-"absorbing	NOUN
iajs-2001	191	24	''	''	PUNCT
iajs-2001	191	25	submodules"of'"y	submodules"of'"y	PROPN
iajs-2001	191	26	.	.	PUNCT
iajs-2001	192	1	proof	proof	NOUN
iajs-2001	192	2	"	"	PUNCT
iajs-2001	192	3	"	"	PUNCT
iajs-2001	192	4	"	"	PUNCT
iajs-2001	192	5	similarly,"'as"'in"'proposition'"2'.13	similarly,"'as"'in"'proposition'"2'.13	PROPN
iajs-2001	192	6	.	.	PUNCT
iajs-2001	193	1	proposition"25	proposition"25	NOUN
iajs-2001	193	2	'	'	PUNCT
iajs-2001	193	3	let"y"be"an"r'-"module	let"y"be"an"r'-"module	ADJ
iajs-2001	193	4	,	,	PUNCT
iajs-2001	193	5	"	"	PUNCT
iajs-2001	193	6	and	and	CCONJ
iajs-2001	193	7	'	'	PUNCT
iajs-2001	193	8	a	a	DET
iajs-2001	193	9	is'"a'"proper'"submodule'"of	is'"a'"proper'"submodule'"of	NOUN
iajs-2001	193	10	''	''	PUNCT
iajs-2001	193	11	y"if	y"if	NOUN
iajs-2001	193	12	''	''	PUNCT
iajs-2001	193	13	a	a	DET
iajs-2001	193	14	is"a	is"a	NOUN
iajs-2001	193	15	wns-2"absorbing'"submodules"'of	wns-2"absorbing'"submodules"'of	PUNCT
iajs-2001	194	1	y,"then	y,"then	NOUN
iajs-2001	194	2	'	'	PUNCT
iajs-2001	194	3	s	s	VERB
iajs-2001	194	4	a	a	DET
iajs-2001	194	5	"	"	PUNCT
iajs-2001	194	6	is	be	AUX
iajs-2001	194	7	'	'	PUNCT
iajs-2001	194	8	is	be	AUX
iajs-2001	194	9	a	a	DET
iajs-2001	194	10	wns'-2	wns'-2	NOUN
iajs-2001	194	11	-	-	PUNCT
iajs-2001	194	12	absorbing	absorbing	ADJ
iajs-2001	194	13	'	'	PUNCT
iajs-2001	194	14	submodules	submodule	NOUN
iajs-2001	194	15	of	of	ADP
iajs-2001	194	16	s	s	NOUN
iajs-2001	194	17	r	r	NOUN
iajs-2001	194	18	module	module	NOUN
iajs-2001	194	19	s	s	PART
iajs-2001	194	20	y.	y.	NOUN
iajs-2001	194	21	proof	proof	NOUN
iajs-2001	194	22	similarly	similarly	ADV
iajs-2001	194	23	as	as	ADP
iajs-2001	194	24	in	in	ADP
iajs-2001	194	25	proposition	proposition	NOUN
iajs-2001	194	26	2'.11	2'.11	NUM
iajs-2001	194	27	proposition	proposition	NOUN
iajs-2001	194	28	26	26	NUM
iajs-2001	194	29	'	'	SYM
iajs-2001	194	30	let"y"be"an"r'-"module"and'"a"is"a'"wn-2-"absorbing'"submodules"'of"y	let"y"be"an"r'-"module"and'"a"is"a'"wn-2-"absorbing'"submodules"'of"y	NOUN
iajs-2001	194	31	,	,	PUNCT
iajs-2001	194	32	then'"a	then'"a	NUM
iajs-2001	194	33	is'"a	is'"a	ADJ
iajs-2001	194	34	'	'	PART
iajs-2001	194	35	wns-"2-"absorbing	wns-"2-"absorbing	ADJ
iajs-2001	194	36	''	''	PUNCT
iajs-2001	194	37	submodules'"of'"y	submodules'"of'"y	PROPN
iajs-2001	194	38	.	.	PUNCT
iajs-2001	195	1	proof	proof	NOUN
iajs-2001	195	2	"	"	PUNCT
iajs-2001	195	3	let"0	let"0	PROPN
iajs-2001	195	4	'	'	PUNCT
iajs-2001	195	5	≠	≠	PROPN
iajs-2001	195	6	a	a	DET
iajs-2001	195	7	y	y	PROPN
iajs-2001	195	8	∈	∈	PROPN
iajs-2001	195	9	a	a	DET
iajs-2001	195	10	,	,	PUNCT
iajs-2001	195	11	where	where	SCONJ
iajs-2001	195	12	a	a	DET
iajs-2001	195	13	∈	∈	PROPN
iajs-2001	195	14	r	r	NOUN
iajs-2001	195	15	,	,	PUNCT
iajs-2001	195	16	y	y	PROPN
iajs-2001	195	17	∈	∈	PROPN
iajs-2001	195	18	y	y	PROPN
iajs-2001	195	19	that	that	PRON
iajs-2001	195	20	is	be	AUX
iajs-2001	195	21	0	0	NUM
iajs-2001	195	22	≠	≠	PROPN
iajs-2001	195	23	a.	a.	NOUN
iajs-2001	195	24	a	a	PRON
iajs-2001	195	25	y	y	PROPN
iajs-2001	195	26	∈	∈	PROPN
iajs-2001	195	27	a	a	PRON
iajs-2001	195	28	.	.	PUNCT
iajs-2001	196	1	since	since	SCONJ
iajs-2001	196	2	a	a	PRON
iajs-2001	196	3	is	be	AUX
iajs-2001	196	4	a	a	DET
iajs-2001	196	5	wn-2absorbing	wn-2absorbing	NOUN
iajs-2001	196	6	.	.	PUNCT
iajs-2001	197	1	then	then	ADV
iajs-2001	197	2	either	either	CCONJ
iajs-2001	197	3	ay	ay	PROPN
iajs-2001	197	4	∈	∈	PROPN
iajs-2001	197	5	a	a	DET
iajs-2001	197	6	+	+	ADJ
iajs-2001	197	7	𝚥(y	𝚥(y	NOUN
iajs-2001	197	8	)	)	PUNCT
iajs-2001	197	9	or	or	CCONJ
iajs-2001	197	10	a	a	DET
iajs-2001	197	11	∈	∈	PROPN
iajs-2001	197	12	a	a	DET
iajs-2001	197	13	ȷ	ȷ	NOUN
iajs-2001	197	14	y	y	NOUN
iajs-2001	197	15	:	:	PUNCT
iajs-2001	197	16	y	y	PROPN
iajs-2001	197	17	.	.	PUNCT
iajs-2001	198	1	thus	thus	ADV
iajs-2001	198	2	a	a	DET
iajs-2001	198	3	is	be	AUX
iajs-2001	198	4	wns-2	wns-2	NOUN
iajs-2001	198	5	-	-	PUNCT
iajs-2001	198	6	absorbing	absorb	VERB
iajs-2001	198	7	submodules	submodule	NOUN
iajs-2001	198	8	of	of	ADP
iajs-2001	198	9	y.	y.	PROPN
iajs-2001	198	10	'	'	PART
iajs-2001	198	11	the"'converse"'of"'proposition	the"'converse"'of"'proposition	NOUN
iajs-2001	198	12	'	'	NUM
iajs-2001	198	13	3.13	3.13	NUM
iajs-2001	198	14	is"'not"'true	is"'not"'true	X
iajs-2001	198	15	.	.	PUNCT
iajs-2001	199	1	'	'	PUNCT
iajs-2001	199	2	in"'general	in"'general	ADJ
iajs-2001	199	3	'	'	PUNCT
iajs-2001	199	4	as"'the"'following"examples	as"'the"'following"example	NOUN
iajs-2001	199	5	'	'	PART
iajs-2001	199	6	shows	show	NOUN
iajs-2001	199	7	'	'	PUNCT
iajs-2001	199	8	that	that	SCONJ
iajs-2001	199	9	:	:	PUNCT
iajs-2001	199	10	mathematics	mathematic	NOUN
iajs-2001	199	11	|	|	ADV
iajs-2001	199	12	125	125	NUM
iajs-2001	199	13	ibn	ibn	PROPN
iajs-2001	199	14	al	al	PROPN
iajs-2001	199	15	-	-	PUNCT
iajs-2001	199	16	haitham	haitham	PROPN
iajs-2001	199	17	jour	jour	X
iajs-2001	199	18	.	.	PROPN
iajs-2001	200	1	for	for	ADP
iajs-2001	200	2	pure	pure	ADJ
iajs-2001	200	3	&	&	CCONJ
iajs-2001	200	4	appl	appl	PROPN
iajs-2001	200	5	.	.	PUNCT
iajs-2001	201	1	sci	sci	PROPN
iajs-2001	201	2	.	.	PROPN
iajs-2001	201	3	ihjpas	ihjpa	VERB
iajs-2001	201	4	https://doi.org/10.30526/31.3.2001	https://doi.org/10.30526/31.3.2001	PROPN
iajs-2001	201	5	vol	vol	NOUN
iajs-2001	201	6	.	.	PROPN
iajs-2001	202	1	31	31	NUM
iajs-2001	203	1	(	(	PUNCT
iajs-2001	203	2	3	3	NUM
iajs-2001	203	3	)	)	SYM
iajs-2001	203	4	2018	2018	NUM
iajs-2001	203	5	example"27	example"27	NOUN
iajs-2001	203	6	let	let	VERB
iajs-2001	203	7	y	y	PROPN
iajs-2001	203	8	=	=	PUNCT
iajs-2001	203	9	z	z	PROPN
iajs-2001	203	10	⊕	⊕	PROPN
iajs-2001	203	11	z	z	NOUN
iajs-2001	203	12	,	,	PUNCT
iajs-2001	203	13	r	r	NOUN
iajs-2001	203	14	=	=	SYM
iajs-2001	203	15	r	r	NOUN
iajs-2001	203	16	,	,	PUNCT
iajs-2001	203	17	a	a	DET
iajs-2001	203	18	=	=	SYM
iajs-2001	203	19	15z	15z	PROPN
iajs-2001	203	20	⊕	⊕	PROPN
iajs-2001	203	21	(	(	PUNCT
iajs-2001	203	22	0	0	NUM
iajs-2001	203	23	)	)	PUNCT
iajs-2001	203	24	,	,	PUNCT
iajs-2001	203	25	a	a	PRON
iajs-2001	203	26	is	be	AUX
iajs-2001	203	27	wns-2	wns-2	NOUN
iajs-2001	203	28	-	-	PUNCT
iajs-2001	203	29	absorbing	absorb	VERB
iajs-2001	203	30	submodules	submodule	NOUN
iajs-2001	203	31	of	of	ADP
iajs-2001	203	32	y	y	PROPN
iajs-2001	203	33	,	,	PUNCT
iajs-2001	203	34	but	but	CCONJ
iajs-2001	203	35	not	not	PART
iajs-2001	203	36	is	be	AUX
iajs-2001	203	37	wn-2	wn-2	NOUN
iajs-2001	203	38	-	-	ADJ
iajs-2001	203	39	absorbing	absorbing	ADJ
iajs-2001	203	40	.	.	PUNCT
iajs-2001	204	1	since	since	SCONJ
iajs-2001	204	2	0	0	NUM
iajs-2001	204	3	≠	≠	PROPN
iajs-2001	204	4	3.5(1,0	3.5(1,0	NUM
iajs-2001	204	5	)	)	PUNCT
iajs-2001	204	6	∊	∊	PROPN
iajs-2001	204	7	a	a	NOUN
iajs-2001	204	8	,	,	PUNCT
iajs-2001	204	9	but	but	CCONJ
iajs-2001	204	10	3(1,0	3(1,0	NUM
iajs-2001	204	11	)	)	PUNCT
iajs-2001	204	12	∉	∉	PROPN
iajs-2001	204	13	a	a	DET
iajs-2001	204	14	+	+	X
iajs-2001	204	15	𝚥(y	𝚥(y	NOUN
iajs-2001	204	16	)	)	PUNCT
iajs-2001	204	17	and	and	CCONJ
iajs-2001	204	18	5(1,0	5(1,0	NUM
iajs-2001	204	19	)	)	PUNCT
iajs-2001	204	20	∉	∉	PROPN
iajs-2001	204	21	a	a	DET
iajs-2001	204	22	+	+	X
iajs-2001	204	23	𝚥(y	𝚥(y	NOUN
iajs-2001	204	24	)	)	PUNCT
iajs-2001	204	25	and	and	CCONJ
iajs-2001	204	26	3.5	3.5	NUM
iajs-2001	204	27	∉	∉	X
iajs-2001	204	28	[	[	X
iajs-2001	204	29	a	a	X
iajs-2001	204	30	+	+	ADJ
iajs-2001	204	31	𝚥(y	𝚥(y	NOUN
iajs-2001	204	32	)	)	PUNCT
iajs-2001	204	33	:	:	PUNCT
iajs-2001	204	34	y	y	X
iajs-2001	204	35	]	]	X
iajs-2001	204	36	=	=	PUNCT
iajs-2001	204	37	(	(	PUNCT
iajs-2001	204	38	0	0	NUM
iajs-2001	204	39	)	)	PUNCT
iajs-2001	204	40	.	.	PUNCT
iajs-2001	205	1	references	reference	NOUN
iajs-2001	205	2	1	1	NUM
iajs-2001	205	3	.	.	PUNCT
iajs-2001	205	4	darani	darani	PROPN
iajs-2001	205	5	'	'	PUNCT
iajs-2001	205	6	.	.	PUNCT
iajs-2001	206	1	a.	a.	NOUN
iajs-2001	206	2	"y	"y	PROPN
iajs-2001	206	3	.	.	PUNCT
iajs-2001	206	4	'	'	PUNCT
iajs-2001	206	5	;	;	PUNCT
iajs-2001	207	1	soheilnia	soheilnia	PROPN
iajs-2001	207	2	,	,	PUNCT
iajs-2001	207	3	f.	f.	PROPN
iajs-2001	207	4	2'-"absorbing"and"weakly'"2-"absorbing	2'-"absorbing"and"weakly'"2-"absorbe	VERB
iajs-2001	207	5	'	'	PUNCT
iajs-2001	207	6	"	"	PUNCT
iajs-2001	207	7	submodules'.taحhi	submodules'.taحhi	VERB
iajs-2001	207	8	journal"math	journal"math	PROPN
iajs-2001	207	9	.	.	PUNCT
iajs-2001	208	1	"2011	"2011	NOUN
iajs-2001	208	2	,	,	PUNCT
iajs-2001	208	3	9',"577	9',"577	NUM
iajs-2001	208	4	'	'	PUNCT
iajs-2001	208	5	–	–	PUNCT
iajs-2001	208	6	584	584	NUM
iajs-2001	208	7	.	.	NOUN
iajs-2001	208	8	2	2	NUM
iajs-2001	208	9	.	.	X
iajs-2001	208	10	haibat	haibat	NOUN
iajs-2001	208	11	.	.	PUNCT
iajs-2001	209	1	m.	m.	PROPN
iajs-2001	209	2	k.	k.	PROPN
iajs-2001	209	3	;	;	PUNCT
iajs-2001	209	4	khalaf	khalaf	PROPN
iajs-2001	209	5	,	,	PUNCT
iajs-2001	209	6	a.h	a.h	PROPN
iajs-2001	209	7	.	.	PROPN
iajs-2001	209	8	weakly	weakly	ADJ
iajs-2001	209	9	semi	semi	ADJ
iajs-2001	209	10	2	2	NUM
iajs-2001	209	11	-	-	PUNCT
iajs-2001	209	12	absorbing	absorbing	ADJ
iajs-2001	209	13	submodules	submodule	NOUN
iajs-2001	209	14	.	.	PUNCT
iajs-2001	210	1	journal	journal	NOUN
iajs-2001	210	2	of	of	ADP
iajs-2001	210	3	alanbar	alanbar	PROPN
iajs-2001	210	4	university	university	NOUN
iajs-2001	210	5	for	for	ADP
iajs-2001	210	6	pure	pure	ADJ
iajs-2001	210	7	science	science	NOUN
iajs-2001	210	8	.	.	PUNCT
iajs-2001	211	1	in	in	ADP
iajs-2001	211	2	press	press	NOUN
iajs-2001	211	3	2018	2018	NUM
iajs-2001	211	4	.	.	PUNCT
iajs-2001	212	1	3	3	X
iajs-2001	212	2	.	.	X
iajs-2001	212	3	kasch	kasch	PROPN
iajs-2001	212	4	,	,	PUNCT
iajs-2001	212	5	f.	f.	PROPN
iajs-2001	212	6	modules	module	NOUN
iajs-2001	212	7	and	and	CCONJ
iajs-2001	212	8	rings	ring	NOUN
iajs-2001	212	9	.	.	PUNCT
iajs-2001	213	1	london	london	PROPN
iajs-2001	213	2	math	math	PROPN
iajs-2001	213	3	.	.	PUNCT
iajs-2001	214	1	soc	soc	PROPN
iajs-2001	214	2	.	.	PUNCT
iajs-2001	215	1	monographs	monograph	NOUN
iajs-2001	215	2	.	.	PUNCT
iajs-2001	216	1	17	17	NUM
iajs-2001	216	2	,	,	PUNCT
iajs-2001	216	3	new	new	PROPN
iajs-2001	216	4	york	york	PROPN
iajs-2001	216	5	,	,	PUNCT
iajs-2001	216	6	academic	academic	ADJ
iajs-2001	216	7	press	press	NOUN
iajs-2001	216	8	.	.	PUNCT
iajs-2001	217	1	1982	1982	NUM
iajs-2001	217	2	.	.	PUNCT
iajs-2001	218	1	4	4	X
iajs-2001	218	2	.	.	X
iajs-2001	218	3	yaseen	yaseen	PROPN
iajs-2001	218	4	,	,	PUNCT
iajs-2001	218	5	m.	m.	NOUN
iajs-2001	218	6	"s	"s	PROPN
iajs-2001	218	7	.	.	PUNCT
iajs-2001	219	1	f-'regular''modules'	f-'regular''modules'	PROPN
iajs-2001	219	2	.	.	PUNCT
iajs-2001	219	3	"m	"m	PROPN
iajs-2001	219	4	.	.	PUNCT
iajs-2001	220	1	sc"	sc"	NOUN
iajs-2001	220	2	.	.	PUNCT
iajs-2001	221	1	'thesis''university"of	'thesis''university"of	PROPN
iajs-2001	221	2	'	'	PART
iajs-2001	221	3	baghdad	baghdad	PROPN
iajs-2001	221	4	'	'	PUNCT
iajs-2001	221	5	.	.	PUNCT
iajs-2001	222	1	1993	1993	NUM
iajs-2001	222	2	.	.	PUNCT
iajs-2001	223	1	5	5	X
iajs-2001	223	2	.	.	X
iajs-2001	223	3	larsen	larsen	PROPN
iajs-2001	223	4	,	,	PUNCT
iajs-2001	223	5	d.	d.	PROPN
iajs-2001	223	6	m	m	PROPN
iajs-2001	223	7	;	;	PUNCT
iajs-2001	223	8	mc	mc	PROPN
iajs-2001	223	9	carthy	carthy	PROPN
iajs-2001	223	10	,	,	PUNCT
iajs-2001	223	11	g.	g.	PROPN
iajs-2001	223	12	p.	p.	NOUN
iajs-2001	223	13	multiplication	multiplication	NOUN
iajs-2001	223	14	theory	theory	NOUN
iajs-2001	223	15	of	of	ADP
iajs-2001	223	16	ideals	ideal	NOUN
iajs-2001	223	17	.	.	PUNCT
iajs-2001	224	1	academic	academic	ADJ
iajs-2001	224	2	press	press	NOUN
iajs-2001	224	3	new	new	PROPN
iajs-2001	224	4	-	-	PUNCT
iajs-2001	224	5	york	york	PROPN
iajs-2001	224	6	and	and	CCONJ
iajs-2001	224	7	london	london	PROPN
iajs-2001	224	8	.	.	PUNCT
iajs-2001	225	1	1971	1971	NUM
iajs-2001	225	2	.	.	PUNCT