id	sid	tid	token	lemma	pos
iajs-1999	1	1	microsoft	microsoft	PROPN
iajs-1999	1	2	word	word	NOUN
iajs-1999	1	3	102	102	NUM
iajs-1999	1	4	-	-	SYM
iajs-1999	1	5	108	108	NUM
iajs-1999	1	6	mathematics	mathematic	NOUN
iajs-1999	1	7	|	|	ADV
iajs-1999	1	8	102	102	NUM
iajs-1999	1	9	ibn	ibn	PROPN
iajs-1999	1	10	al	al	PROPN
iajs-1999	1	11	-	-	PUNCT
iajs-1999	1	12	haitham	haitham	PROPN
iajs-1999	1	13	jour	jour	X
iajs-1999	1	14	.	.	PROPN
iajs-1999	2	1	for	for	ADP
iajs-1999	2	2	pure	pure	ADJ
iajs-1999	2	3	&	&	CCONJ
iajs-1999	2	4	appl	appl	PROPN
iajs-1999	2	5	.	.	PUNCT
iajs-1999	3	1	sci	sci	PROPN
iajs-1999	3	2	.	.	PROPN
iajs-1999	3	3	ihjpas	ihjpa	VERB
iajs-1999	3	4	https://doi.org/10.30526/	https://doi.org/10.30526/	PROPN
iajs-1999	3	5	31.3.1999	31.3.1999	NUM
iajs-1999	3	6	vol	vol	NOUN
iajs-1999	3	7	.	.	PUNCT
iajs-1999	4	1	31	31	NUM
iajs-1999	5	1	(	(	PUNCT
iajs-1999	5	2	3	3	NUM
iajs-1999	5	3	)	)	PUNCT
iajs-1999	5	4	2018	2018	NUM
iajs-1999	5	5	for	for	ADP
iajs-1999	5	6	some	some	DET
iajs-1999	5	7	results	result	NOUN
iajs-1999	5	8	of	of	ADP
iajs-1999	5	9	semisecond	semisecond	ADJ
iajs-1999	5	10	submodules	submodule	NOUN
iajs-1999	5	11	rasha	rasha	PROPN
iajs-1999	5	12	i.	i.	PROPN
iajs-1999	5	13	khalaf	khalaf	PROPN
iajs-1999	5	14	department	department	PROPN
iajs-1999	5	15	,	,	PUNCT
iajs-1999	5	16	of	of	ADP
iajs-1999	5	17	mathematics	mathematic	NOUN
iajs-1999	5	18	,	,	PUNCT
iajs-1999	5	19	college	college	NOUN
iajs-1999	5	20	of	of	ADP
iajs-1999	5	21	education	education	NOUN
iajs-1999	5	22	for	for	ADP
iajs-1999	5	23	pure	pure	ADJ
iajs-1999	5	24	science	science	NOUN
iajs-1999	5	25	ibn	ibn	PROPN
iajs-1999	5	26	al	al	PROPN
iajs-1999	5	27	-	-	PUNCT
iajs-1999	5	28	haitham	haitham	PROPN
iajs-1999	5	29	,	,	PUNCT
iajs-1999	5	30	university	university	PROPN
iajs-1999	5	31	of	of	ADP
iajs-1999	5	32	baghdad	baghdad	PROPN
iajs-1999	5	33	,	,	PUNCT
iajs-1999	5	34	baghdad	baghdad	PROPN
iajs-1999	5	35	,	,	PUNCT
iajs-1999	5	36	iraq	iraq	PROPN
iajs-1999	5	37	.	.	PUNCT
iajs-1999	6	1	rasha_sin79@yahoo.com	rasha_sin79@yahoo.com	PROPN
iajs-1999	6	2	article	article	NOUN
iajs-1999	6	3	history	history	NOUN
iajs-1999	6	4	:	:	PUNCT
iajs-1999	6	5	received	receive	VERB
iajs-1999	6	6	15	15	NUM
iajs-1999	6	7	may	may	PROPN
iajs-1999	6	8	2018	2018	NUM
iajs-1999	6	9	,	,	PUNCT
iajs-1999	6	10	accepted	accept	VERB
iajs-1999	6	11	19	19	NUM
iajs-1999	6	12	september	september	PROPN
iajs-1999	6	13	2018	2018	NUM
iajs-1999	6	14	,	,	PUNCT
iajs-1999	6	15	published	publish	VERB
iajs-1999	6	16	december	december	PROPN
iajs-1999	6	17	2018	2018	NUM
iajs-1999	6	18	abstract	abstract	ADV
iajs-1999	6	19	let	let	VERB
iajs-1999	6	20	ℛ	ℛ	PROPN
iajs-1999	6	21	be	be	AUX
iajs-1999	6	22	a	a	DET
iajs-1999	6	23	commutative	commutative	ADJ
iajs-1999	6	24	ring	ring	NOUN
iajs-1999	6	25	with	with	ADP
iajs-1999	6	26	unity	unity	NOUN
iajs-1999	6	27	and	and	CCONJ
iajs-1999	6	28	let	let	VERB
iajs-1999	6	29	ℬ	ℬ	PRON
iajs-1999	6	30	be	be	AUX
iajs-1999	6	31	a	a	DET
iajs-1999	6	32	unitary	unitary	ADJ
iajs-1999	6	33	r	r	NOUN
iajs-1999	6	34	-	-	PUNCT
iajs-1999	6	35	module	module	NOUN
iajs-1999	6	36	.	.	PUNCT
iajs-1999	7	1	let	let	VERB
iajs-1999	7	2	ℵ	ℵ	NOUN
iajs-1999	7	3	be	be	AUX
iajs-1999	7	4	a	a	DET
iajs-1999	7	5	proper	proper	ADJ
iajs-1999	7	6	submodule	submodule	NOUN
iajs-1999	7	7	of	of	ADP
iajs-1999	7	8	ℬ	ℬ	PROPN
iajs-1999	7	9	,	,	PUNCT
iajs-1999	7	10	ℵ	ℵ	PRON
iajs-1999	7	11	is	be	AUX
iajs-1999	7	12	called	call	VERB
iajs-1999	7	13	semisecond	semisecond	ADJ
iajs-1999	7	14	submodule	submodule	NOUN
iajs-1999	7	15	if	if	SCONJ
iajs-1999	7	16	for	for	ADP
iajs-1999	7	17	any	any	DET
iajs-1999	7	18	r∈ℛ	r∈ℛ	NOUN
iajs-1999	7	19	,	,	PUNCT
iajs-1999	7	20	r≠0	r≠0	NOUN
iajs-1999	7	21	,	,	PUNCT
iajs-1999	7	22	n∈z+	n∈z+	ADJ
iajs-1999	7	23	,	,	PUNCT
iajs-1999	7	24	either	either	CCONJ
iajs-1999	7	25	rnℵ=0	rnℵ=0	NOUN
iajs-1999	7	26	or	or	CCONJ
iajs-1999	7	27	rnℵ=rℵ.	rnℵ=rℵ.	PROPN
iajs-1999	7	28	in	in	ADP
iajs-1999	7	29	this	this	DET
iajs-1999	7	30	work	work	NOUN
iajs-1999	7	31	,	,	PUNCT
iajs-1999	7	32	we	we	PRON
iajs-1999	7	33	introduce	introduce	VERB
iajs-1999	7	34	the	the	DET
iajs-1999	7	35	concept	concept	NOUN
iajs-1999	7	36	of	of	ADP
iajs-1999	7	37	semisecond	semisecond	ADJ
iajs-1999	7	38	submodule	submodule	NOUN
iajs-1999	7	39	and	and	CCONJ
iajs-1999	7	40	confer	confer	VERB
iajs-1999	7	41	numerous	numerous	ADJ
iajs-1999	7	42	properties	property	NOUN
iajs-1999	7	43	concerning	concern	VERB
iajs-1999	7	44	with	with	ADP
iajs-1999	7	45	this	this	DET
iajs-1999	7	46	notion	notion	NOUN
iajs-1999	7	47	.	.	PUNCT
iajs-1999	8	1	also	also	ADV
iajs-1999	8	2	we	we	PRON
iajs-1999	8	3	study	study	VERB
iajs-1999	8	4	semisecond	semisecond	ADJ
iajs-1999	8	5	modules	module	NOUN
iajs-1999	8	6	as	as	ADP
iajs-1999	8	7	a	a	DET
iajs-1999	8	8	popularization	popularization	NOUN
iajs-1999	8	9	of	of	ADP
iajs-1999	8	10	second	second	ADJ
iajs-1999	8	11	modules	module	NOUN
iajs-1999	8	12	,	,	PUNCT
iajs-1999	8	13	where	where	SCONJ
iajs-1999	8	14	an	an	DET
iajs-1999	8	15	ℛ-module	ℛ-module	PROPN
iajs-1999	8	16	ℬ	ℬ	NOUN
iajs-1999	8	17	is	be	AUX
iajs-1999	8	18	called	call	VERB
iajs-1999	8	19	semisecond	semisecond	ADJ
iajs-1999	8	20	,	,	PUNCT
iajs-1999	8	21	if	if	SCONJ
iajs-1999	8	22	ℬ	ℬ	NOUN
iajs-1999	8	23	is	be	AUX
iajs-1999	8	24	semisecond	semisecond	ADJ
iajs-1999	8	25	submodul	submodul	NOUN
iajs-1999	8	26	of	of	ADP
iajs-1999	8	27	ℬ.	ℬ.	PROPN
iajs-1999	8	28	keywords	keyword	NOUN
iajs-1999	8	29	:	:	PUNCT
iajs-1999	8	30	semisecond	semisecond	ADJ
iajs-1999	8	31	submodules	submodule	NOUN
iajs-1999	8	32	,	,	PUNCT
iajs-1999	8	33	second	second	ADJ
iajs-1999	8	34	submodules	submodule	NOUN
iajs-1999	8	35	,	,	PUNCT
iajs-1999	8	36	secondary	secondary	ADJ
iajs-1999	8	37	submodules	submodule	NOUN
iajs-1999	8	38	.	.	PUNCT
iajs-1999	9	1	1	1	X
iajs-1999	9	2	.	.	X
iajs-1999	9	3	introduction	introduction	NOUN
iajs-1999	9	4	let	let	VERB
iajs-1999	9	5	ℛ	ℛ	PROPN
iajs-1999	9	6	be	be	AUX
iajs-1999	9	7	a	a	DET
iajs-1999	9	8	commutative	commutative	ADJ
iajs-1999	9	9	ring	ring	NOUN
iajs-1999	9	10	with	with	ADP
iajs-1999	9	11	unity	unity	NOUN
iajs-1999	9	12	and	and	CCONJ
iajs-1999	9	13	let	let	VERB
iajs-1999	9	14	ℬ	ℬ	PRON
iajs-1999	9	15	be	be	AUX
iajs-1999	9	16	a	a	DET
iajs-1999	9	17	unitary	unitary	ADJ
iajs-1999	9	18	ℛ	ℛ	ADJ
iajs-1999	9	19	-module	-module	NOUN
iajs-1999	9	20	.	.	PUNCT
iajs-1999	10	1	s.yass	s.yass	NOUN
iajs-1999	10	2	in	in	ADP
iajs-1999	10	3	[	[	X
iajs-1999	10	4	1	1	NUM
iajs-1999	10	5	]	]	PUNCT
iajs-1999	10	6	introduced	introduce	VERB
iajs-1999	10	7	the	the	DET
iajs-1999	10	8	notation	notation	NOUN
iajs-1999	10	9	of	of	ADP
iajs-1999	10	10	second	second	ADJ
iajs-1999	10	11	submodule	submodule	NOUN
iajs-1999	10	12	and	and	CCONJ
iajs-1999	10	13	second	second	ADJ
iajs-1999	10	14	module	module	NOUN
iajs-1999	10	15	where	where	SCONJ
iajs-1999	10	16	a	a	DET
iajs-1999	10	17	submodule	submodule	NOUN
iajs-1999	10	18	ℵ	ℵ	NOUN
iajs-1999	10	19	of	of	ADP
iajs-1999	10	20	an	an	DET
iajs-1999	10	21	ℛ	ℛ	ADJ
iajs-1999	10	22	-module	-module	NOUN
iajs-1999	10	23	ℬ	ℬ	NOUN
iajs-1999	10	24	is	be	AUX
iajs-1999	10	25	called	call	VERB
iajs-1999	10	26	second	second	ADJ
iajs-1999	10	27	submodule	submodule	NOUN
iajs-1999	10	28	if	if	SCONJ
iajs-1999	10	29	for	for	ADP
iajs-1999	10	30	every	every	DET
iajs-1999	10	31	r∈ℛ	r∈ℛ	NOUN
iajs-1999	10	32	,	,	PUNCT
iajs-1999	10	33	r≠0	r≠0	PROPN
iajs-1999	10	34	,	,	PUNCT
iajs-1999	10	35	either	either	CCONJ
iajs-1999	10	36	rℵ	rℵ	PROPN
iajs-1999	10	37	=	=	SYM
iajs-1999	10	38	ℵ	ℵ	NOUN
iajs-1999	10	39	or	or	CCONJ
iajs-1999	10	40	rℵ	rℵ	PROPN
iajs-1999	10	41	=	=	SYM
iajs-1999	10	42	0	0	PROPN
iajs-1999	10	43	and	and	CCONJ
iajs-1999	10	44	a	a	DET
iajs-1999	10	45	module	module	NOUN
iajs-1999	10	46	ℬ	ℬ	NOUN
iajs-1999	10	47	is	be	AUX
iajs-1999	10	48	called	call	VERB
iajs-1999	10	49	semisecond	semisecond	ADJ
iajs-1999	10	50	if	if	SCONJ
iajs-1999	10	51	ℬ	ℬ	NOUN
iajs-1999	10	52	is	be	AUX
iajs-1999	10	53	semisecond	semisecond	ADJ
iajs-1999	10	54	submodule	submodule	NOUN
iajs-1999	10	55	of	of	ADP
iajs-1999	10	56	ℬ.	ℬ.	PROPN
iajs-1999	10	57	this	this	DET
iajs-1999	10	58	definition	definition	NOUN
iajs-1999	10	59	leads	lead	VERB
iajs-1999	10	60	us	we	PRON
iajs-1999	10	61	to	to	PART
iajs-1999	10	62	introduce	introduce	VERB
iajs-1999	10	63	the	the	DET
iajs-1999	10	64	notion	notion	NOUN
iajs-1999	10	65	of	of	ADP
iajs-1999	10	66	semisecond	semisecond	ADJ
iajs-1999	10	67	submodule	submodule	PROPN
iajs-1999	10	68	and	and	CCONJ
iajs-1999	10	69	semisecond	semisecond	ADJ
iajs-1999	10	70	module	module	NOUN
iajs-1999	10	71	as	as	ADP
iajs-1999	10	72	a	a	DET
iajs-1999	10	73	generalization	generalization	NOUN
iajs-1999	10	74	of	of	ADP
iajs-1999	10	75	second	second	ADJ
iajs-1999	10	76	submodule	submodule	NOUN
iajs-1999	10	77	and	and	CCONJ
iajs-1999	10	78	second	second	ADJ
iajs-1999	10	79	module	module	NOUN
iajs-1999	10	80	,	,	PUNCT
iajs-1999	10	81	where	where	SCONJ
iajs-1999	10	82	a	a	DET
iajs-1999	10	83	submodule	submodule	NOUN
iajs-1999	10	84	ℵ	ℵ	NOUN
iajs-1999	10	85	of	of	ADP
iajs-1999	10	86	an	an	DET
iajs-1999	10	87	ℛ-module	ℛ-module	PROPN
iajs-1999	10	88	ℬ	ℬ	NOUN
iajs-1999	10	89	is	be	AUX
iajs-1999	10	90	called	call	VERB
iajs-1999	10	91	semisecond	semisecond	ADJ
iajs-1999	10	92	if	if	SCONJ
iajs-1999	10	93	for	for	ADP
iajs-1999	10	94	every	every	DET
iajs-1999	10	95	r∈	r∈	PROPN
iajs-1999	10	96	ℛ	ℛ	PROPN
iajs-1999	10	97	,	,	PUNCT
iajs-1999	10	98	r≠0	r≠0	NOUN
iajs-1999	10	99	,	,	PUNCT
iajs-1999	10	100	n∈z+	n∈z+	ADJ
iajs-1999	10	101	,	,	PUNCT
iajs-1999	10	102	either	either	CCONJ
iajs-1999	10	103	rnℵ	rnℵ	NOUN
iajs-1999	10	104	=	=	NOUN
iajs-1999	10	105	0	0	NUM
iajs-1999	10	106	or	or	CCONJ
iajs-1999	10	107	rnℵ	rnℵ	NOUN
iajs-1999	10	108	=	=	X
iajs-1999	10	109	rℵ	rℵ	PROPN
iajs-1999	10	110	and	and	CCONJ
iajs-1999	10	111	a	a	DET
iajs-1999	10	112	module	module	NOUN
iajs-1999	10	113	ℬ	ℬ	NOUN
iajs-1999	10	114	is	be	AUX
iajs-1999	10	115	semisecond	semisecond	ADJ
iajs-1999	10	116	if	if	SCONJ
iajs-1999	10	117	ℬ	ℬ	NOUN
iajs-1999	10	118	is	be	AUX
iajs-1999	10	119	semisecond	semisecond	ADJ
iajs-1999	10	120	submodule	submodule	NOUN
iajs-1999	10	121	of	of	ADP
iajs-1999	10	122	ℬ.	ℬ.	PROPN
iajs-1999	10	123	the	the	DET
iajs-1999	10	124	main	main	ADJ
iajs-1999	10	125	aim	aim	NOUN
iajs-1999	10	126	of	of	ADP
iajs-1999	10	127	this	this	DET
iajs-1999	10	128	work	work	NOUN
iajs-1999	10	129	is	be	AUX
iajs-1999	10	130	to	to	PART
iajs-1999	10	131	give	give	VERB
iajs-1999	10	132	basic	basic	ADJ
iajs-1999	10	133	properties	property	NOUN
iajs-1999	10	134	of	of	ADP
iajs-1999	10	135	semisecond	semisecond	ADJ
iajs-1999	10	136	submodules	submodule	NOUN
iajs-1999	10	137	.	.	PUNCT
iajs-1999	11	1	moreover	moreover	ADV
iajs-1999	11	2	,	,	PUNCT
iajs-1999	11	3	we	we	PRON
iajs-1999	11	4	survey	survey	VERB
iajs-1999	11	5	the	the	DET
iajs-1999	11	6	relationships	relationship	NOUN
iajs-1999	11	7	between	between	ADP
iajs-1999	11	8	semisecond	semisecond	ADJ
iajs-1999	11	9	submodules	submodule	NOUN
iajs-1999	11	10	and	and	CCONJ
iajs-1999	11	11	other	other	ADJ
iajs-1999	11	12	submodules	submodule	NOUN
iajs-1999	11	13	.	.	PUNCT
iajs-1999	12	1	over	over	ADP
iajs-1999	12	2	this	this	DET
iajs-1999	12	3	work	work	NOUN
iajs-1999	12	4	we	we	PRON
iajs-1999	12	5	designate	designate	VERB
iajs-1999	12	6	s.r.m	s.r.m	NOUN
iajs-1999	12	7	.	.	PUNCT
iajs-1999	13	1	for	for	ADP
iajs-1999	13	2	submodule	submodule	NOUN
iajs-1999	13	3	of	of	ADP
iajs-1999	13	4	an	an	DET
iajs-1999	13	5	ℛ-module	ℛ-module	PROPN
iajs-1999	13	6	,	,	PUNCT
iajs-1999	13	7	for	for	ADP
iajs-1999	13	8	integral	integral	ADJ
iajs-1999	13	9	domain	domain	NOUN
iajs-1999	13	10	,	,	PUNCT
iajs-1999	13	11	for	for	ADP
iajs-1999	13	12	finitely	finitely	ADV
iajs-1999	13	13	generated	generate	VERB
iajs-1999	13	14	,	,	PUNCT
iajs-1999	13	15	s.t	s.t	PROPN
iajs-1999	13	16	.	.	PROPN
iajs-1999	13	17	for	for	ADP
iajs-1999	13	18	such	such	ADJ
iajs-1999	13	19	that	that	PRON
iajs-1999	13	20	and	and	CCONJ
iajs-1999	13	21	n.z	n.z	PROPN
iajs-1999	13	22	.	.	PROPN
iajs-1999	14	1	for	for	ADP
iajs-1999	14	2	non	non	ADJ
iajs-1999	14	3	-	-	ADJ
iajs-1999	14	4	zero	zero	NUM
iajs-1999	14	5	.	.	PUNCT
iajs-1999	15	1	2	2	NUM
iajs-1999	15	2	.	.	NOUN
iajs-1999	15	3	semisecond	semisecond	ADJ
iajs-1999	15	4	submodules	submodule	NOUN
iajs-1999	15	5	definition	definition	NOUN
iajs-1999	15	6	(	(	PUNCT
iajs-1999	15	7	1):-let	1):-let	NUM
iajs-1999	15	8	ℵ	ℵ	NOUN
iajs-1999	15	9	be	be	AUX
iajs-1999	15	10	a	a	DET
iajs-1999	15	11	s.r.m	s.r.m	NOUN
iajs-1999	15	12	.	.	PUNCT
iajs-1999	16	1	ℬ	ℬ	NOUN
iajs-1999	16	2	,	,	PUNCT
iajs-1999	16	3	ℵ	ℵ	NOUN
iajs-1999	16	4	is	be	AUX
iajs-1999	16	5	semisecond	semisecond	ADJ
iajs-1999	16	6	submodule	submodule	NOUN
iajs-1999	16	7	if	if	SCONJ
iajs-1999	16	8	for	for	ADP
iajs-1999	16	9	every	every	DET
iajs-1999	16	10	r∈ℛ	r∈ℛ	NOUN
iajs-1999	16	11	,	,	PUNCT
iajs-1999	16	12	n∈z+	n∈z+	ADJ
iajs-1999	16	13	,	,	PUNCT
iajs-1999	16	14	either	either	CCONJ
iajs-1999	16	15	rnℵ=0	rnℵ=0	NOUN
iajs-1999	16	16	or	or	CCONJ
iajs-1999	16	17	rnℵ=rℵ.	rnℵ=rℵ.	PROPN
iajs-1999	16	18	an	an	DET
iajs-1999	16	19	ideal	ideal	NOUN
iajs-1999	16	20	i	i	PRON
iajs-1999	16	21	of	of	ADP
iajs-1999	16	22	a	a	DET
iajs-1999	16	23	ring	ring	NOUN
iajs-1999	16	24	ℛ	ℛ	PROPN
iajs-1999	16	25	is	be	AUX
iajs-1999	16	26	semisecond	semisecond	ADJ
iajs-1999	16	27	ideal	ideal	NOUN
iajs-1999	16	28	if	if	SCONJ
iajs-1999	16	29	it	it	PRON
iajs-1999	16	30	is	be	AUX
iajs-1999	16	31	semisecond	semisecond	ADJ
iajs-1999	16	32	submodule	submodule	NOUN
iajs-1999	16	33	of	of	ADP
iajs-1999	16	34	the	the	DET
iajs-1999	16	35	ℛ	ℛ	ADJ
iajs-1999	16	36	-module	-module	NOUN
iajs-1999	16	37	ℛ.	ℛ.	PROPN
iajs-1999	16	38	the	the	DET
iajs-1999	16	39	later	later	ADJ
iajs-1999	16	40	result	result	NOUN
iajs-1999	16	41	is	be	AUX
iajs-1999	16	42	a	a	DET
iajs-1999	16	43	description	description	NOUN
iajs-1999	16	44	of	of	ADP
iajs-1999	16	45	semisecond	semisecond	ADJ
iajs-1999	16	46	submodule	submodule	NOUN
iajs-1999	16	47	.	.	PUNCT
iajs-1999	17	1	proposition	proposition	NOUN
iajs-1999	17	2	(	(	PUNCT
iajs-1999	17	3	2):ℵ	2):ℵ	NUM
iajs-1999	17	4	is	be	AUX
iajs-1999	17	5	s.r.m	s.r.m	NOUN
iajs-1999	17	6	.	.	PUNCT
iajs-1999	18	1	ℬ	ℬ	NOUN
iajs-1999	18	2	is	be	AUX
iajs-1999	18	3	semisecond	semisecond	ADJ
iajs-1999	18	4	iff	iff	PROPN
iajs-1999	18	5	r2	r2	PROPN
iajs-1999	18	6	ℵ	ℵ	PROPN
iajs-1999	18	7	=	=	NOUN
iajs-1999	18	8	0	0	NUM
iajs-1999	18	9	or	or	CCONJ
iajs-1999	18	10	r2	r2	PROPN
iajs-1999	18	11	ℵ	ℵ	NOUN
iajs-1999	18	12	=	=	NOUN
iajs-1999	18	13	r	r	NOUN
iajs-1999	18	14	ℵ	ℵ	NOUN
iajs-1999	18	15	for	for	ADP
iajs-1999	18	16	any	any	DET
iajs-1999	18	17	r∈	r∈	PROPN
iajs-1999	18	18	ℛ	ℛ	PROPN
iajs-1999	18	19	,	,	PUNCT
iajs-1999	18	20	r≠0	r≠0	NOUN
iajs-1999	18	21	.	.	PUNCT
iajs-1999	19	1	proof:-(⟹	proof:-(⟹	NOUN
iajs-1999	19	2	)	)	PUNCT
iajs-1999	19	3	is	be	AUX
iajs-1999	19	4	obvious	obvious	ADJ
iajs-1999	19	5	.	.	PUNCT
iajs-1999	20	1	(	(	PUNCT
iajs-1999	20	2	⟸	⟸	NOUN
iajs-1999	20	3	)	)	PUNCT
iajs-1999	20	4	if	if	SCONJ
iajs-1999	20	5	r=3	r=3	PROPN
iajs-1999	20	6	,	,	PUNCT
iajs-1999	20	7	then	then	ADV
iajs-1999	20	8	r3ℵ	r3ℵ	ADJ
iajs-1999	20	9	=	=	NOUN
iajs-1999	20	10	r(r2ℵ	r(r2ℵ	NOUN
iajs-1999	20	11	)	)	PUNCT
iajs-1999	20	12	.	.	PUNCT
iajs-1999	21	1	since	since	SCONJ
iajs-1999	21	2	either	either	DET
iajs-1999	21	3	r2	r2	PROPN
iajs-1999	21	4	ℵ	ℵ	PROPN
iajs-1999	21	5	=	=	NOUN
iajs-1999	21	6	0	0	NUM
iajs-1999	21	7	or	or	CCONJ
iajs-1999	21	8	r2	r2	PROPN
iajs-1999	21	9	ℵ	ℵ	NOUN
iajs-1999	21	10	=	=	NOUN
iajs-1999	21	11	r	r	NOUN
iajs-1999	21	12	ℵ	ℵ	NOUN
iajs-1999	21	13	,	,	PUNCT
iajs-1999	21	14	that	that	PRON
iajs-1999	21	15	is	be	AUX
iajs-1999	21	16	either	either	CCONJ
iajs-1999	21	17	r3ℵ	r3ℵ	PRON
iajs-1999	21	18	=	=	SYM
iajs-1999	21	19	r(0)=0	r(0)=0	ADJ
iajs-1999	21	20	or	or	CCONJ
iajs-1999	21	21	r3ℵ	r3ℵ	NOUN
iajs-1999	21	22	=	=	SYM
iajs-1999	21	23	r(rℵ	r(rℵ	NOUN
iajs-1999	21	24	)	)	PUNCT
iajs-1999	21	25	=	=	NOUN
iajs-1999	21	26	r2ℵ	r2ℵ	NOUN
iajs-1999	21	27	=	=	NOUN
iajs-1999	21	28	rℵ.	rℵ.	NOUN
iajs-1999	21	29	suppose	suppose	VERB
iajs-1999	21	30	that	that	SCONJ
iajs-1999	21	31	rn	rn	PROPN
iajs-1999	21	32	ℵ	ℵ	PROPN
iajs-1999	21	33	=	=	NOUN
iajs-1999	21	34	0	0	NUM
iajs-1999	21	35	or	or	CCONJ
iajs-1999	21	36	rn	rn	PROPN
iajs-1999	21	37	ℵ	ℵ	PROPN
iajs-1999	21	38	=	=	NOUN
iajs-1999	21	39	r	r	NOUN
iajs-1999	21	40	ℵ	ℵ	NOUN
iajs-1999	21	41	is	be	AUX
iajs-1999	21	42	whole	whole	ADJ
iajs-1999	21	43	for	for	ADP
iajs-1999	21	44	n	n	PRON
iajs-1999	21	45	=	=	PROPN
iajs-1999	21	46	k.	k.	NOUN
iajs-1999	21	47	to	to	PART
iajs-1999	21	48	evidence	evidence	NOUN
iajs-1999	21	49	that	that	SCONJ
iajs-1999	21	50	the	the	DET
iajs-1999	21	51	permit	permit	NOUN
iajs-1999	21	52	is	be	AUX
iajs-1999	21	53	whole	whole	ADJ
iajs-1999	21	54	if	if	SCONJ
iajs-1999	21	55	n	n	CCONJ
iajs-1999	21	56	=	=	SYM
iajs-1999	21	57	k+1	k+1	X
iajs-1999	21	58	.	.	PUNCT
iajs-1999	22	1	(	(	PUNCT
iajs-1999	22	2	r)k+1	r)k+1	NOUN
iajs-1999	22	3	ℵ	ℵ	NOUN
iajs-1999	22	4	=	=	NOUN
iajs-1999	22	5	r(rk	r(rk	NOUN
iajs-1999	22	6	ℵ	ℵ	NOUN
iajs-1999	22	7	)	)	PUNCT
iajs-1999	22	8	.	.	PUNCT
iajs-1999	23	1	but	but	CCONJ
iajs-1999	23	2	rk	rk	PROPN
iajs-1999	23	3	ℵ	ℵ	PROPN
iajs-1999	23	4	=	=	NOUN
iajs-1999	23	5	0	0	NUM
iajs-1999	23	6	or	or	CCONJ
iajs-1999	23	7	rkℵ	rkℵ	NOUN
iajs-1999	23	8	=	=	SYM
iajs-1999	23	9	rℵ	rℵ	NOUN
iajs-1999	23	10	,	,	PUNCT
iajs-1999	23	11	that	that	PRON
iajs-1999	23	12	is	be	AUX
iajs-1999	23	13	rk+1ℵ	rk+1ℵ	PROPN
iajs-1999	23	14	=	=	SYM
iajs-1999	23	15	r(0	r(0	PROPN
iajs-1999	23	16	)	)	PUNCT
iajs-1999	24	1	=	=	NOUN
iajs-1999	24	2	0	0	NUM
iajs-1999	24	3	or	or	CCONJ
iajs-1999	24	4	mathematics	mathematic	NOUN
iajs-1999	24	5	|	|	ADV
iajs-1999	24	6	103	103	NUM
iajs-1999	24	7	ibn	ibn	PROPN
iajs-1999	24	8	al	al	PROPN
iajs-1999	24	9	-	-	PUNCT
iajs-1999	24	10	haitham	haitham	PROPN
iajs-1999	24	11	jour	jour	X
iajs-1999	24	12	.	.	PROPN
iajs-1999	24	13	for	for	ADP
iajs-1999	24	14	pure	pure	ADJ
iajs-1999	24	15	&	&	CCONJ
iajs-1999	24	16	appl	appl	PROPN
iajs-1999	24	17	.	.	PUNCT
iajs-1999	25	1	sci	sci	PROPN
iajs-1999	25	2	.	.	PROPN
iajs-1999	25	3	ihjpas	ihjpa	VERB
iajs-1999	25	4	https://doi.org/10.30526/	https://doi.org/10.30526/	PROPN
iajs-1999	25	5	31.3.1999	31.3.1999	NUM
iajs-1999	25	6	vol	vol	NOUN
iajs-1999	25	7	.	.	PUNCT
iajs-1999	26	1	31	31	NUM
iajs-1999	27	1	(	(	PUNCT
iajs-1999	27	2	3	3	NUM
iajs-1999	27	3	)	)	PUNCT
iajs-1999	27	4	2018	2018	NUM
iajs-1999	27	5	(	(	PUNCT
iajs-1999	27	6	r)k+1	r)k+1	NOUN
iajs-1999	27	7	ℵ	ℵ	NOUN
iajs-1999	27	8	=	=	SYM
iajs-1999	27	9	r(r	r(r	NOUN
iajs-1999	27	10	ℵ)=r2	ℵ)=r2	PUNCT
iajs-1999	27	11	ℵ	ℵ	PROPN
iajs-1999	28	1	=	=	NOUN
iajs-1999	28	2	r	r	NOUN
iajs-1999	28	3	ℵ.	ℵ.	NOUN
iajs-1999	28	4	hence	hence	ADV
iajs-1999	28	5	by	by	ADP
iajs-1999	28	6	the	the	DET
iajs-1999	28	7	principle	principle	NOUN
iajs-1999	28	8	of	of	ADP
iajs-1999	28	9	mathematical	mathematical	ADJ
iajs-1999	28	10	induction	induction	NOUN
iajs-1999	28	11	rn	rn	PROPN
iajs-1999	28	12	ℵ	ℵ	PROPN
iajs-1999	28	13	=	=	NOUN
iajs-1999	28	14	0	0	NUM
iajs-1999	28	15	or	or	CCONJ
iajs-1999	28	16	rn	rn	PROPN
iajs-1999	28	17	ℵ	ℵ	NOUN
iajs-1999	28	18	=	=	NOUN
iajs-1999	28	19	r	r	NOUN
iajs-1999	28	20	ℵ	ℵ	NOUN
iajs-1999	28	21	for	for	ADP
iajs-1999	28	22	any	any	DET
iajs-1999	28	23	r∈ℛ	r∈ℛ	NOUN
iajs-1999	28	24	,	,	PUNCT
iajs-1999	28	25	r≠0	r≠0	NOUN
iajs-1999	28	26	,	,	PUNCT
iajs-1999	28	27	n∈z+	n∈z+	ADJ
iajs-1999	28	28	.	.	PUNCT
iajs-1999	29	1	therefore	therefore	ADV
iajs-1999	29	2	,	,	PUNCT
iajs-1999	29	3	ℵ	ℵ	X
iajs-1999	29	4	is	be	AUX
iajs-1999	29	5	semisecond	semisecond	ADJ
iajs-1999	29	6	submodule	submodule	NOUN
iajs-1999	29	7	.	.	PUNCT
iajs-1999	30	1	remarks	remark	NOUN
iajs-1999	30	2	and	and	CCONJ
iajs-1999	30	3	examples	example	NOUN
iajs-1999	30	4	(	(	PUNCT
iajs-1999	30	5	3	3	NUM
iajs-1999	30	6	):	):	PUNCT
iajs-1999	30	7	(	(	PUNCT
iajs-1999	30	8	1	1	X
iajs-1999	30	9	)	)	PUNCT
iajs-1999	30	10	every	every	DET
iajs-1999	30	11	second	second	ADJ
iajs-1999	30	12	submodule	submodule	NOUN
iajs-1999	30	13	is	be	AUX
iajs-1999	30	14	semisecond	semisecond	ADJ
iajs-1999	30	15	.	.	PUNCT
iajs-1999	31	1	proof	proof	NOUN
iajs-1999	31	2	:	:	PUNCT
iajs-1999	31	3	-let	-let	NUM
iajs-1999	31	4	ℵ	ℵ	X
iajs-1999	31	5	be	be	AUX
iajs-1999	31	6	a	a	DET
iajs-1999	31	7	s.r.m	s.r.m	NOUN
iajs-1999	31	8	.	.	PUNCT
iajs-1999	32	1	ℬ	ℬ	DET
iajs-1999	32	2	such	such	ADJ
iajs-1999	32	3	that	that	DET
iajs-1999	32	4	ℵ	ℵ	NOUN
iajs-1999	32	5	is	be	AUX
iajs-1999	32	6	second	second	ADJ
iajs-1999	32	7	submodule	submodule	NOUN
iajs-1999	32	8	,	,	PUNCT
iajs-1999	32	9	that	that	PRON
iajs-1999	32	10	is	is	ADV
iajs-1999	32	11	r	r	NOUN
iajs-1999	32	12	ℵ	ℵ	NOUN
iajs-1999	32	13	=	=	NOUN
iajs-1999	32	14	0	0	NUM
iajs-1999	32	15	or	or	CCONJ
iajs-1999	32	16	r	r	NOUN
iajs-1999	32	17	ℵ	ℵ	NOUN
iajs-1999	32	18	=	=	NOUN
iajs-1999	32	19	ℵ	ℵ	NOUN
iajs-1999	32	20	for	for	ADP
iajs-1999	32	21	every	every	DET
iajs-1999	32	22	r∈ℛ	r∈ℛ	NOUN
iajs-1999	32	23	,	,	PUNCT
iajs-1999	32	24	r≠0	r≠0	NOUN
iajs-1999	32	25	.	.	PUNCT
iajs-1999	33	1	if	if	SCONJ
iajs-1999	33	2	r	r	NOUN
iajs-1999	33	3	ℵ	ℵ	NOUN
iajs-1999	33	4	=	=	NOUN
iajs-1999	33	5	0	0	NUM
iajs-1999	33	6	,	,	PUNCT
iajs-1999	33	7	then	then	ADV
iajs-1999	33	8	r2	r2	PROPN
iajs-1999	33	9	ℵ	ℵ	PROPN
iajs-1999	33	10	=	=	PROPN
iajs-1999	33	11	r(r	r(r	PROPN
iajs-1999	33	12	ℵ)=r(0)=0	ℵ)=r(0)=0	PROPN
iajs-1999	33	13	.	.	PUNCT
iajs-1999	34	1	if	if	SCONJ
iajs-1999	34	2	r	r	NOUN
iajs-1999	34	3	ℵ	ℵ	NOUN
iajs-1999	34	4	=	=	SYM
iajs-1999	34	5	ℵ	ℵ	NOUN
iajs-1999	34	6	,	,	PUNCT
iajs-1999	34	7	then	then	ADV
iajs-1999	34	8	r2	r2	PROPN
iajs-1999	34	9	ℵ	ℵ	PROPN
iajs-1999	34	10	=	=	ADJ
iajs-1999	34	11	r(r	r(r	NOUN
iajs-1999	34	12	ℵ)=r	ℵ)=r	VERB
iajs-1999	34	13	ℵ	ℵ	NOUN
iajs-1999	34	14	,	,	PUNCT
iajs-1999	34	15	that	that	PRON
iajs-1999	34	16	is	is	ADV
iajs-1999	34	17	r2	r2	PROPN
iajs-1999	34	18	ℵ	ℵ	NOUN
iajs-1999	34	19	=	=	NOUN
iajs-1999	34	20	0	0	NUM
iajs-1999	34	21	or	or	CCONJ
iajs-1999	34	22	r2	r2	PROPN
iajs-1999	34	23	ℵ	ℵ	NOUN
iajs-1999	34	24	=	=	NOUN
iajs-1999	34	25	r	r	NOUN
iajs-1999	34	26	ℵ	ℵ	NOUN
iajs-1999	34	27	so	so	ADJ
iajs-1999	34	28	ℵ	ℵ	NOUN
iajs-1999	34	29	is	be	AUX
iajs-1999	34	30	semisecond	semisecond	ADJ
iajs-1999	34	31	by	by	ADP
iajs-1999	34	32	proposition	proposition	NOUN
iajs-1999	34	33	(	(	PUNCT
iajs-1999	34	34	2.2	2.2	NUM
iajs-1999	34	35	)	)	PUNCT
iajs-1999	34	36	.	.	PUNCT
iajs-1999	35	1	the	the	DET
iajs-1999	35	2	converse	converse	NOUN
iajs-1999	35	3	of	of	ADP
iajs-1999	35	4	this	this	DET
iajs-1999	35	5	remark	remark	NOUN
iajs-1999	35	6	is	be	AUX
iajs-1999	35	7	not	not	PART
iajs-1999	35	8	true	true	ADJ
iajs-1999	35	9	in	in	ADP
iajs-1999	35	10	general	general	ADJ
iajs-1999	35	11	for	for	ADP
iajs-1999	35	12	example	example	NOUN
iajs-1999	35	13	:	:	PUNCT
iajs-1999	35	14	consider	consider	VERB
iajs-1999	35	15	the	the	DET
iajs-1999	35	16	z	z	NOUN
iajs-1999	35	17	-	-	PUNCT
iajs-1999	35	18	module	module	NOUN
iajs-1999	35	19	z8	z8	NOUN
iajs-1999	35	20	,	,	PUNCT
iajs-1999	35	21	let	let	VERB
iajs-1999	35	22	ℵ	ℵ	NOUN
iajs-1999	35	23	=	=	SYM
iajs-1999	35	24	<	<	X
iajs-1999	35	25	2	2	NUM
iajs-1999	35	26	>	>	PUNCT
iajs-1999	35	27	,	,	PUNCT
iajs-1999	35	28	take	take	VERB
iajs-1999	35	29	r=2	r=2	PROPN
iajs-1999	35	30	,	,	PUNCT
iajs-1999	35	31	r	r	NOUN
iajs-1999	35	32	ℵ	ℵ	NOUN
iajs-1999	35	33	=	=	NOUN
iajs-1999	35	34	{	{	PUNCT
iajs-1999	35	35	0,4	0,4	NOUN
iajs-1999	35	36	}	}	PUNCT
iajs-1999	35	37	.	.	PUNCT
iajs-1999	36	1	thus	thus	ADV
iajs-1999	36	2	r	r	NOUN
iajs-1999	36	3	ℵ	ℵ	NOUN
iajs-1999	36	4	≠	≠	ADJ
iajs-1999	36	5	ℵ	ℵ	NOUN
iajs-1999	36	6	and	and	CCONJ
iajs-1999	36	7	r	r	NOUN
iajs-1999	36	8	ℵ	ℵ	NOUN
iajs-1999	36	9	≠(0	≠(0	X
iajs-1999	36	10	)	)	PUNCT
iajs-1999	36	11	,	,	PUNCT
iajs-1999	36	12	that	that	PRON
iajs-1999	36	13	is	be	AUX
iajs-1999	36	14	ℵ	ℵ	ADJ
iajs-1999	36	15	is	be	AUX
iajs-1999	36	16	not	not	PART
iajs-1999	36	17	second	second	ADJ
iajs-1999	36	18	submodule	submodule	NOUN
iajs-1999	36	19	,	,	PUNCT
iajs-1999	36	20	while	while	SCONJ
iajs-1999	36	21	for	for	ADP
iajs-1999	36	22	every	every	DET
iajs-1999	36	23	r∈z	r∈z	NOUN
iajs-1999	36	24	,	,	PUNCT
iajs-1999	36	25	r≠0	r≠0	NOUN
iajs-1999	36	26	,	,	PUNCT
iajs-1999	36	27	such	such	ADJ
iajs-1999	36	28	that	that	SCONJ
iajs-1999	36	29	r	r	NOUN
iajs-1999	36	30	is	be	AUX
iajs-1999	36	31	even	even	ADV
iajs-1999	36	32	,	,	PUNCT
iajs-1999	36	33	then	then	ADV
iajs-1999	36	34	r=2k	r=2k	NOUN
iajs-1999	36	35	for	for	ADP
iajs-1999	36	36	some	some	DET
iajs-1999	36	37	k∈z	k∈z	NOUN
iajs-1999	36	38	,	,	PUNCT
iajs-1999	36	39	so	so	SCONJ
iajs-1999	36	40	r2	r2	PROPN
iajs-1999	36	41	ℵ	ℵ	ADP
iajs-1999	36	42	=(	=(	NOUN
iajs-1999	36	43	2k)2	2k)2	NUM
iajs-1999	36	44	ℵ	ℵ	NOUN
iajs-1999	36	45	=	=	NOUN
iajs-1999	36	46	0.also	0.also	NOUN
iajs-1999	36	47	if	if	SCONJ
iajs-1999	36	48	r	r	NOUN
iajs-1999	36	49	is	be	AUX
iajs-1999	36	50	odd	odd	ADJ
iajs-1999	36	51	,	,	PUNCT
iajs-1999	36	52	then	then	ADV
iajs-1999	36	53	r=(2k+1	r=(2k+1	NOUN
iajs-1999	36	54	)	)	PUNCT
iajs-1999	36	55	,	,	PUNCT
iajs-1999	36	56	so	so	SCONJ
iajs-1999	36	57	r2	r2	PROPN
iajs-1999	36	58	ℵ	ℵ	PROPN
iajs-1999	36	59	=(	=(	NOUN
iajs-1999	36	60	4k2	4k2	PROPN
iajs-1999	36	61	+	+	SYM
iajs-1999	36	62	4k+1	4k+1	NOUN
iajs-1999	36	63	)	)	PUNCT
iajs-1999	36	64	ℵ	ℵ	NOUN
iajs-1999	36	65	=	=	SYM
iajs-1999	36	66	ℵ	ℵ	NOUN
iajs-1999	36	67	and	and	CCONJ
iajs-1999	36	68	r	r	NOUN
iajs-1999	36	69	ℵ	ℵ	ADJ
iajs-1999	36	70	=(	=(	ADJ
iajs-1999	36	71	2k+1	2k+1	NOUN
iajs-1999	36	72	)	)	PUNCT
iajs-1999	36	73	ℵ	ℵ	NOUN
iajs-1999	37	1	=	=	NOUN
iajs-1999	37	2	2k	2k	NUM
iajs-1999	37	3	ℵ	ℵ	ADP
iajs-1999	37	4	+	+	CCONJ
iajs-1999	37	5	ℵ	ℵ	NOUN
iajs-1999	37	6	=	=	X
iajs-1999	37	7	ℵ.	ℵ.	NOUN
iajs-1999	37	8	thus	thus	ADV
iajs-1999	37	9	r2	r2	PROPN
iajs-1999	37	10	ℵ	ℵ	PROPN
iajs-1999	37	11	=	=	NOUN
iajs-1999	37	12	r	r	NOUN
iajs-1999	37	13	ℵ.	ℵ.	NOUN
iajs-1999	37	14	thus	thus	ADV
iajs-1999	37	15	ℵ	ℵ	NOUN
iajs-1999	37	16	is	be	AUX
iajs-1999	37	17	semisecond	semisecond	ADJ
iajs-1999	37	18	submodule	submodule	NOUN
iajs-1999	37	19	.	.	PUNCT
iajs-1999	38	1	(	(	PUNCT
iajs-1999	38	2	2	2	X
iajs-1999	38	3	)	)	PUNCT
iajs-1999	38	4	the	the	DET
iajs-1999	38	5	submodule	submodule	PROPN
iajs-1999	38	6	z	z	PROPN
iajs-1999	38	7	of	of	ADP
iajs-1999	38	8	the	the	DET
iajs-1999	38	9	z	z	NOUN
iajs-1999	38	10	-	-	PUNCT
iajs-1999	38	11	module	module	NOUN
iajs-1999	38	12	q	q	NOUN
iajs-1999	38	13	is	be	AUX
iajs-1999	38	14	not	not	PART
iajs-1999	38	15	semisecond	semisecond	ADJ
iajs-1999	38	16	submodule	submodule	NOUN
iajs-1999	38	17	,	,	PUNCT
iajs-1999	38	18	but	but	CCONJ
iajs-1999	38	19	q	q	NOUN
iajs-1999	38	20	is	be	AUX
iajs-1999	38	21	a	a	DET
iajs-1999	38	22	semisecond	semisecond	ADJ
iajs-1999	38	23	submodule	submodule	NOUN
iajs-1999	38	24	of	of	ADP
iajs-1999	38	25	q.	q.	PROPN
iajs-1999	38	26	(	(	PUNCT
iajs-1999	38	27	3	3	NUM
iajs-1999	38	28	)	)	PUNCT
iajs-1999	38	29	any	any	DET
iajs-1999	38	30	submodule	submodule	NOUN
iajs-1999	38	31	of	of	ADP
iajs-1999	38	32	zp∞	zp∞	PROPN
iajs-1999	38	33	as	as	ADP
iajs-1999	38	34	z	z	NOUN
iajs-1999	38	35	-	-	PUNCT
iajs-1999	38	36	module	module	NOUN
iajs-1999	38	37	is	be	AUX
iajs-1999	38	38	not	not	PART
iajs-1999	38	39	semisecond	semisecond	ADJ
iajs-1999	38	40	submodule	submodule	NOUN
iajs-1999	38	41	.	.	PUNCT
iajs-1999	39	1	(	(	PUNCT
iajs-1999	39	2	4	4	X
iajs-1999	39	3	)	)	PUNCT
iajs-1999	39	4	let	let	VERB
iajs-1999	39	5	ℵ	ℵ	NOUN
iajs-1999	39	6	be	be	AUX
iajs-1999	39	7	a	a	DET
iajs-1999	39	8	non	non	ADJ
iajs-1999	39	9	-	-	ADJ
iajs-1999	39	10	zero	zero	NUM
iajs-1999	39	11	s.r.m	s.r.m	NOUN
iajs-1999	39	12	.	.	PUNCT
iajs-1999	40	1	ℬ	ℬ	PROPN
iajs-1999	40	2	s.t	s.t	PROPN
iajs-1999	40	3	.	.	PUNCT
iajs-1999	40	4	ℛ	ℛ	PROPN
iajs-1999	40	5	is	be	AUX
iajs-1999	40	6	a	a	DET
iajs-1999	40	7	field	field	NOUN
iajs-1999	40	8	,	,	PUNCT
iajs-1999	40	9	then	then	ADV
iajs-1999	40	10	ℵ	ℵ	NOUN
iajs-1999	40	11	is	be	AUX
iajs-1999	40	12	semisecond	semisecond	ADJ
iajs-1999	40	13	.	.	PUNCT
iajs-1999	41	1	proof	proof	NOUN
iajs-1999	41	2	:	:	PUNCT
iajs-1999	41	3	let	let	VERB
iajs-1999	41	4	r∈	r∈	PROPN
iajs-1999	41	5	ℛ	ℛ	PROPN
iajs-1999	41	6	,	,	PUNCT
iajs-1999	41	7	r≠0	r≠0	VERB
iajs-1999	41	8	and	and	CCONJ
iajs-1999	41	9	suppose	suppose	VERB
iajs-1999	41	10	r2ℵ≠0	r2ℵ≠0	PROPN
iajs-1999	41	11	.	.	PUNCT
iajs-1999	42	1	to	to	PART
iajs-1999	42	2	prove	prove	VERB
iajs-1999	42	3	r2ℵ=rℵ	r2ℵ=rℵ	PROPN
iajs-1999	42	4	,	,	PUNCT
iajs-1999	42	5	let	let	VERB
iajs-1999	42	6	rn∈rℵ	rn∈rℵ	ADJ
iajs-1999	42	7	,	,	PUNCT
iajs-1999	42	8	then	then	ADV
iajs-1999	42	9	rn	rn	NOUN
iajs-1999	42	10	=	=	NOUN
iajs-1999	42	11	r2(r-1n)∈r2ℵ	r2(r-1n)∈r2ℵ	ADJ
iajs-1999	42	12	,	,	PUNCT
iajs-1999	42	13	hence	hence	ADV
iajs-1999	42	14	rℵ⊆r2ℵ	rℵ⊆r2ℵ	NOUN
iajs-1999	42	15	,	,	PUNCT
iajs-1999	42	16	which	which	PRON
iajs-1999	42	17	implies	imply	VERB
iajs-1999	42	18	that	that	SCONJ
iajs-1999	42	19	r2ℵ=rℵ.	r2ℵ=rℵ.	PROPN
iajs-1999	42	20	thus	thus	ADV
iajs-1999	42	21	ℵ	ℵ	NOUN
iajs-1999	42	22	is	be	AUX
iajs-1999	42	23	a	a	DET
iajs-1999	42	24	semisecond	semisecond	ADJ
iajs-1999	42	25	submodule	submodule	NOUN
iajs-1999	42	26	.	.	PUNCT
iajs-1999	43	1	(	(	PUNCT
iajs-1999	43	2	5	5	X
iajs-1999	43	3	)	)	PUNCT
iajs-1999	43	4	let	let	VERB
iajs-1999	43	5	f	f	PRON
iajs-1999	43	6	:	:	PUNCT
iajs-1999	43	7	ℬ→ℬ	ℬ→ℬ	PROPN
iajs-1999	43	8	'	'	PUNCT
iajs-1999	43	9	be	be	VERB
iajs-1999	43	10	an	an	DET
iajs-1999	43	11	r	r	NOUN
iajs-1999	43	12	-	-	PUNCT
iajs-1999	43	13	homorphism	homorphism	NOUN
iajs-1999	43	14	and	and	CCONJ
iajs-1999	43	15	ℵ	ℵ	NOUN
iajs-1999	43	16	is	be	AUX
iajs-1999	43	17	a	a	DET
iajs-1999	43	18	semisecond	semisecond	ADJ
iajs-1999	43	19	submodule	submodule	NOUN
iajs-1999	43	20	of	of	ADP
iajs-1999	43	21	ℬ	ℬ	PROPN
iajs-1999	43	22	,	,	PUNCT
iajs-1999	43	23	then	then	ADV
iajs-1999	43	24	f(ℵ	f(ℵ	NOUN
iajs-1999	43	25	)	)	PUNCT
iajs-1999	43	26	is	be	AUX
iajs-1999	43	27	a	a	DET
iajs-1999	43	28	semisecond	semisecond	ADJ
iajs-1999	43	29	submodule	submodule	NOUN
iajs-1999	43	30	of	of	ADP
iajs-1999	43	31	ℬ	ℬ	NOUN
iajs-1999	43	32	'	'	PUNCT
iajs-1999	43	33	.	.	PUNCT
iajs-1999	44	1	proof	proof	NOUN
iajs-1999	44	2	:	:	PUNCT
iajs-1999	44	3	since	since	SCONJ
iajs-1999	44	4	ℵ	ℵ	NOUN
iajs-1999	44	5	is	be	AUX
iajs-1999	44	6	semisecond	semisecond	ADJ
iajs-1999	44	7	,	,	PUNCT
iajs-1999	44	8	then	then	ADV
iajs-1999	44	9	r2ℵ=rℵ	r2ℵ=rℵ	PROPN
iajs-1999	44	10	or	or	CCONJ
iajs-1999	44	11	r2ℵ=0	r2ℵ=0	PROPN
iajs-1999	44	12	.	.	PUNCT
iajs-1999	45	1	hence	hence	ADV
iajs-1999	45	2	either	either	CCONJ
iajs-1999	45	3	f(r2ℵ)=f(rℵ	f(r2ℵ)=f(rℵ	NOUN
iajs-1999	45	4	)	)	PUNCT
iajs-1999	45	5	or	or	CCONJ
iajs-1999	45	6	f(r2ℵ)=f(0	f(r2ℵ)=f(0	NOUN
iajs-1999	45	7	)	)	PUNCT
iajs-1999	45	8	.	.	PUNCT
iajs-1999	46	1	thus	thus	ADV
iajs-1999	46	2	r2f(ℵ)=rf(ℵ	r2f(ℵ)=rf(ℵ	NOUN
iajs-1999	46	3	)	)	PUNCT
iajs-1999	46	4	or	or	CCONJ
iajs-1999	46	5	r2f(ℵ)=f(0	r2f(ℵ)=f(0	NOUN
iajs-1999	46	6	)	)	PUNCT
iajs-1999	46	7	.	.	PUNCT
iajs-1999	47	1	therefore	therefore	ADV
iajs-1999	47	2	,	,	PUNCT
iajs-1999	47	3	f(ℵ	f(ℵ	PROPN
iajs-1999	47	4	)	)	PUNCT
iajs-1999	47	5	is	be	AUX
iajs-1999	47	6	a	a	DET
iajs-1999	47	7	semisecond	semisecond	ADJ
iajs-1999	47	8	submodule	submodule	NOUN
iajs-1999	47	9	of	of	ADP
iajs-1999	47	10	ℬ	ℬ	NOUN
iajs-1999	47	11	'	'	PUNCT
iajs-1999	47	12	.	.	PUNCT
iajs-1999	48	1	(	(	PUNCT
iajs-1999	48	2	6	6	X
iajs-1999	48	3	)	)	PUNCT
iajs-1999	48	4	the	the	DET
iajs-1999	48	5	inverse	inverse	ADJ
iajs-1999	48	6	image	image	NOUN
iajs-1999	48	7	of	of	ADP
iajs-1999	48	8	semisecond	semisecond	ADJ
iajs-1999	48	9	submodule	submodule	NOUN
iajs-1999	48	10	need	need	AUX
iajs-1999	48	11	not	not	PART
iajs-1999	48	12	to	to	PART
iajs-1999	48	13	be	be	AUX
iajs-1999	48	14	a	a	DET
iajs-1999	48	15	semisecond	semisecond	NOUN
iajs-1999	48	16	,	,	PUNCT
iajs-1999	48	17	for	for	ADP
iajs-1999	48	18	example	example	NOUN
iajs-1999	48	19	:	:	PUNCT
iajs-1999	48	20	let	let	VERB
iajs-1999	48	21	π	π	PRON
iajs-1999	48	22	:	:	PUNCT
iajs-1999	48	23	z	z	NOUN
iajs-1999	48	24	→z/<6>≅z6	→z/<6>≅z6	PRON
iajs-1999	48	25	,	,	PUNCT
iajs-1999	48	26	<	<	X
iajs-1999	48	27	2	2	NUM
iajs-1999	48	28	>	>	X
iajs-1999	48	29	is	be	AUX
iajs-1999	48	30	semisecond	semisecond	ADJ
iajs-1999	48	31	submodule	submodule	NOUN
iajs-1999	48	32	in	in	ADP
iajs-1999	48	33	z6	z6	PROPN
iajs-1999	48	34	but	but	CCONJ
iajs-1999	48	35	π-1(2)=2z	π-1(2)=2z	NOUN
iajs-1999	48	36	is	be	AUX
iajs-1999	48	37	not	not	PART
iajs-1999	48	38	a	a	DET
iajs-1999	48	39	semisecond	semisecond	NOUN
iajs-1999	48	40	.	.	PUNCT
iajs-1999	49	1	the	the	DET
iajs-1999	49	2	opposite	opposite	NOUN
iajs-1999	49	3	of	of	ADP
iajs-1999	49	4	remark	remark	NOUN
iajs-1999	49	5	and	and	CCONJ
iajs-1999	49	6	example	example	NOUN
iajs-1999	49	7	(	(	PUNCT
iajs-1999	49	8	2.3	2.3	NUM
iajs-1999	49	9	.	.	PUNCT
iajs-1999	50	1	(	(	PUNCT
iajs-1999	50	2	1	1	NUM
iajs-1999	50	3	)	)	PUNCT
iajs-1999	50	4	)	)	PUNCT
iajs-1999	50	5	is	be	AUX
iajs-1999	50	6	true	true	ADJ
iajs-1999	50	7	under	under	ADP
iajs-1999	50	8	the	the	DET
iajs-1999	50	9	class	class	NOUN
iajs-1999	50	10	of	of	ADP
iajs-1999	50	11	torsion	torsion	NOUN
iajs-1999	50	12	free	free	ADJ
iajs-1999	50	13	module	module	NOUN
iajs-1999	50	14	over	over	ADP
iajs-1999	50	15	an	an	DET
iajs-1999	50	16	integral	integral	ADJ
iajs-1999	50	17	domain	domain	NOUN
iajs-1999	50	18	,	,	PUNCT
iajs-1999	50	19	where	where	SCONJ
iajs-1999	50	20	a	a	DET
iajs-1999	50	21	module	module	NOUN
iajs-1999	50	22	ℬ	ℬ	NOUN
iajs-1999	50	23	over	over	ADP
iajs-1999	50	24	an	an	DET
iajs-1999	50	25	i.d	i.d	PROPN
iajs-1999	50	26	.	.	PROPN
iajs-1999	50	27	is	be	AUX
iajs-1999	50	28	called	call	VERB
iajs-1999	50	29	torsion	torsion	NOUN
iajs-1999	50	30	free	free	ADJ
iajs-1999	50	31	if	if	SCONJ
iajs-1999	50	32	τ(ℬ)=0	τ(ℬ)=0	NOUN
iajs-1999	50	33	,	,	PUNCT
iajs-1999	50	34	where	where	SCONJ
iajs-1999	50	35	τ(ℬ)=	τ(ℬ)=	X
iajs-1999	50	36	{	{	PUNCT
iajs-1999	50	37	m∈	m∈	PROPN
iajs-1999	50	38	ℬ	ℬ	PROPN
iajs-1999	50	39	;	;	PUNCT
iajs-1999	50	40	r∈ℛ	r∈ℛ	NOUN
iajs-1999	50	41	,	,	PUNCT
iajs-1999	50	42	r≠0	r≠0	NOUN
iajs-1999	50	43	,	,	PUNCT
iajs-1999	50	44	rm=0	rm=0	NOUN
iajs-1999	50	45	}	}	PUNCT
iajs-1999	50	46	,	,	PUNCT
iajs-1999	50	47	see	see	VERB
iajs-1999	50	48	[	[	X
iajs-1999	50	49	2.p.45	2.p.45	X
iajs-1999	50	50	]	]	X
iajs-1999	50	51	.	.	PUNCT
iajs-1999	51	1	proposition	proposition	NOUN
iajs-1999	51	2	(	(	PUNCT
iajs-1999	51	3	4):-if	4):-if	NOUN
iajs-1999	51	4	ℵ	ℵ	NOUN
iajs-1999	51	5	is	be	AUX
iajs-1999	51	6	a	a	DET
iajs-1999	51	7	semisecond	semisecond	ADJ
iajs-1999	51	8	s.r.m	s.r.m	NOUN
iajs-1999	51	9	.	.	PUNCT
iajs-1999	52	1	ℬ	ℬ	PRON
iajs-1999	52	2	such	such	ADJ
iajs-1999	52	3	that	that	SCONJ
iajs-1999	52	4	ℬ	ℬ	NOUN
iajs-1999	52	5	is	be	AUX
iajs-1999	52	6	torsion	torsion	NOUN
iajs-1999	52	7	free	free	ADJ
iajs-1999	52	8	over	over	ADP
iajs-1999	52	9	an	an	DET
iajs-1999	52	10	i.d	i.d	PROPN
iajs-1999	52	11	.	.	PROPN
iajs-1999	53	1	r	r	NOUN
iajs-1999	53	2	,	,	PUNCT
iajs-1999	53	3	then	then	ADV
iajs-1999	53	4	ℵ	ℵ	NOUN
iajs-1999	53	5	is	be	AUX
iajs-1999	53	6	a	a	DET
iajs-1999	53	7	second	second	ADJ
iajs-1999	53	8	submodule	submodule	NOUN
iajs-1999	53	9	.	.	PUNCT
iajs-1999	54	1	proof	proof	NOUN
iajs-1999	54	2	:	:	PUNCT
iajs-1999	54	3	let	let	VERB
iajs-1999	54	4	r∈	r∈	PROPN
iajs-1999	54	5	ℛ	ℛ	PROPN
iajs-1999	54	6	,	,	PUNCT
iajs-1999	54	7	r≠0	r≠0	NOUN
iajs-1999	54	8	.	.	PUNCT
iajs-1999	55	1	since	since	SCONJ
iajs-1999	55	2	ℵ	ℵ	NOUN
iajs-1999	55	3	is	be	AUX
iajs-1999	55	4	semisecond	semisecond	ADJ
iajs-1999	55	5	submodule	submodule	NOUN
iajs-1999	55	6	,	,	PUNCT
iajs-1999	55	7	then	then	ADV
iajs-1999	55	8	r2ℵ=0	r2ℵ=0	PROPN
iajs-1999	55	9	or	or	CCONJ
iajs-1999	55	10	r2ℵ=rℵ.	r2ℵ=rℵ.	PROPN
iajs-1999	55	11	if	if	SCONJ
iajs-1999	55	12	r2ℵ=0	r2ℵ=0	NOUN
iajs-1999	55	13	,	,	PUNCT
iajs-1999	55	14	then	then	ADV
iajs-1999	55	15	r2=0	r2=0	PROPN
iajs-1999	55	16	(	(	PUNCT
iajs-1999	55	17	since	since	SCONJ
iajs-1999	55	18	m	m	PROPN
iajs-1999	55	19	is	be	AUX
iajs-1999	55	20	torsion	torsion	NOUN
iajs-1999	55	21	free	free	ADJ
iajs-1999	55	22	)	)	PUNCT
iajs-1999	55	23	and	and	CCONJ
iajs-1999	55	24	since	since	SCONJ
iajs-1999	55	25	ℛ	ℛ	PROPN
iajs-1999	55	26	is	be	AUX
iajs-1999	55	27	an	an	DET
iajs-1999	55	28	i.d	i.d	PROPN
iajs-1999	55	29	.	.	PROPN
iajs-1999	55	30	,	,	PUNCT
iajs-1999	55	31	then	then	ADV
iajs-1999	55	32	r=0	r=0	PROPN
iajs-1999	55	33	,	,	PUNCT
iajs-1999	55	34	which	which	PRON
iajs-1999	55	35	is	be	AUX
iajs-1999	55	36	contradiction	contradiction	NOUN
iajs-1999	55	37	.	.	PUNCT
iajs-1999	56	1	thus	thus	ADV
iajs-1999	56	2	r2ℵ=rℵ	r2ℵ=rℵ	PROPN
iajs-1999	56	3	and	and	CCONJ
iajs-1999	56	4	for	for	ADP
iajs-1999	56	5	any	any	DET
iajs-1999	56	6	n∈ℵ	n∈ℵ	ADJ
iajs-1999	56	7	,	,	PUNCT
iajs-1999	56	8	hence	hence	ADV
iajs-1999	56	9			PRON
iajs-1999	56	10	�	�	PROPN
iajs-1999	56	11	́	́	NOUN
iajs-1999	56	12	�	�	PROPN
iajs-1999	56	13	∈ℵ	∈ℵ	ADJ
iajs-1999	56	14	s.t	s.t	PROPN
iajs-1999	56	15	.	.	PUNCT
iajs-1999	56	16	r2	r2	PROPN
iajs-1999	56	17	�	�	PROPN
iajs-1999	56	18	́	́	PROPN
iajs-1999	56	19	�	�	NOUN
iajs-1999	56	20	=rn	=rn	NOUN
iajs-1999	56	21	.	.	PUNCT
iajs-1999	57	1	thus	thus	ADV
iajs-1999	57	2	r(n	r(n	PROPN
iajs-1999	57	3	-	-	PUNCT
iajs-1999	57	4	r	r	NOUN
iajs-1999	57	5	�	�	PROPN
iajs-1999	57	6	́	́	NOUN
iajs-1999	57	7	�	�	NOUN
iajs-1999	57	8	)=0	)=0	NUM
iajs-1999	57	9	.	.	PUNCT
iajs-1999	58	1	since	since	SCONJ
iajs-1999	58	2	r≠0	r≠0	NOUN
iajs-1999	58	3	and	and	CCONJ
iajs-1999	58	4	m	m	NOUN
iajs-1999	58	5	is	be	AUX
iajs-1999	58	6	torsion	torsion	NOUN
iajs-1999	58	7	free	free	ADJ
iajs-1999	58	8	,	,	PUNCT
iajs-1999	58	9	then	then	ADV
iajs-1999	58	10	(	(	PUNCT
iajs-1999	58	11	n	n	CCONJ
iajs-1999	58	12	�	�	PROPN
iajs-1999	58	13	́	́	NOUN
iajs-1999	58	14	�	�	NOUN
iajs-1999	58	15	n)=0	n)=0	NOUN
iajs-1999	58	16	,	,	PUNCT
iajs-1999	58	17	that	that	PRON
iajs-1999	58	18	is	be	AUX
iajs-1999	58	19	n	n	PRON
iajs-1999	58	20	=	=	NOUN
iajs-1999	58	21	r	r	NOUN
iajs-1999	58	22	�	�	NOUN
iajs-1999	58	23	́	́	PROPN
iajs-1999	58	24	�	�	PROPN
iajs-1999	58	25	,	,	PUNCT
iajs-1999	58	26	hence	hence	ADV
iajs-1999	58	27	ℵ⊆rℵ	ℵ⊆rℵ	NUM
iajs-1999	58	28	and	and	CCONJ
iajs-1999	58	29	so	so	ADV
iajs-1999	58	30	,	,	PUNCT
iajs-1999	58	31	rℵ=ℵ.	rℵ=ℵ.	VERB
iajs-1999	58	32	therefore	therefore	ADV
iajs-1999	58	33	,	,	PUNCT
iajs-1999	58	34	ℵ	ℵ	PRON
iajs-1999	58	35	is	be	AUX
iajs-1999	58	36	a	a	DET
iajs-1999	58	37	second	second	ADJ
iajs-1999	58	38	submodule	submodule	NOUN
iajs-1999	58	39	.	.	PUNCT
iajs-1999	59	1	corollary	corollary	ADJ
iajs-1999	59	2	(	(	PUNCT
iajs-1999	59	3	5):if	5):if	NUM
iajs-1999	59	4	ℬ	ℬ	NOUN
iajs-1999	59	5	is	be	AUX
iajs-1999	59	6	a	a	DET
iajs-1999	59	7	torsion	torsion	NOUN
iajs-1999	59	8	free	free	ADJ
iajs-1999	59	9	over	over	ADP
iajs-1999	59	10	an	an	DET
iajs-1999	59	11	integral	integral	ADJ
iajs-1999	59	12	domain	domain	NOUN
iajs-1999	59	13	,	,	PUNCT
iajs-1999	59	14	then	then	ADV
iajs-1999	59	15	ℵ	ℵ	NOUN
iajs-1999	59	16	is	be	AUX
iajs-1999	59	17	second	second	ADJ
iajs-1999	59	18	submodule	submodule	NOUN
iajs-1999	59	19	of	of	ADP
iajs-1999	59	20	ℬ	ℬ	PROPN
iajs-1999	59	21	if	if	SCONJ
iajs-1999	60	1	and	and	CCONJ
iajs-1999	60	2	only	only	ADV
iajs-1999	60	3	if	if	SCONJ
iajs-1999	60	4	ℵ	ℵ	NOUN
iajs-1999	60	5	is	be	AUX
iajs-1999	60	6	semisecond	semisecond	ADJ
iajs-1999	60	7	.	.	PUNCT
iajs-1999	61	1	mathematics	mathematic	NOUN
iajs-1999	61	2	|	|	ADV
iajs-1999	61	3	104	104	NUM
iajs-1999	61	4	ibn	ibn	PROPN
iajs-1999	61	5	al	al	PROPN
iajs-1999	61	6	-	-	PUNCT
iajs-1999	61	7	haitham	haitham	PROPN
iajs-1999	61	8	jour	jour	X
iajs-1999	61	9	.	.	PROPN
iajs-1999	62	1	for	for	ADP
iajs-1999	62	2	pure	pure	ADJ
iajs-1999	62	3	&	&	CCONJ
iajs-1999	62	4	appl	appl	PROPN
iajs-1999	62	5	.	.	PUNCT
iajs-1999	63	1	sci	sci	PROPN
iajs-1999	63	2	.	.	PROPN
iajs-1999	63	3	ihjpas	ihjpa	VERB
iajs-1999	63	4	https://doi.org/10.30526/	https://doi.org/10.30526/	PROPN
iajs-1999	63	5	31.3.1999	31.3.1999	NUM
iajs-1999	63	6	vol	vol	NOUN
iajs-1999	63	7	.	.	PUNCT
iajs-1999	64	1	31	31	NUM
iajs-1999	65	1	(	(	PUNCT
iajs-1999	65	2	3	3	NUM
iajs-1999	65	3	)	)	SYM
iajs-1999	65	4	2018	2018	NUM
iajs-1999	65	5	recall	recall	VERB
iajs-1999	65	6	that	that	SCONJ
iajs-1999	65	7	a	a	DET
iajs-1999	65	8	module	module	NOUN
iajs-1999	65	9	ℬ	ℬ	NOUN
iajs-1999	65	10	is	be	AUX
iajs-1999	65	11	called	call	VERB
iajs-1999	65	12	multiplication	multiplication	NOUN
iajs-1999	65	13	if	if	SCONJ
iajs-1999	65	14	every	every	DET
iajs-1999	65	15	submodule	submodule	NOUN
iajs-1999	65	16	ℵ	ℵ	NOUN
iajs-1999	65	17	of	of	ADP
iajs-1999	65	18	ℬ	ℬ	NOUN
iajs-1999	65	19	,	,	PUNCT
iajs-1999	65	20			ADP
iajs-1999	65	21	an	an	DET
iajs-1999	65	22	ideal	ideal	ADJ
iajs-1999	65	23	i	i	PRON
iajs-1999	65	24	of	of	ADP
iajs-1999	65	25	ℛ	ℛ	PROPN
iajs-1999	65	26	s.t	s.t	PROPN
iajs-1999	65	27	.	.	PUNCT
iajs-1999	66	1	i	i	PRON
iajs-1999	66	2	ℬ	ℬ	ADV
iajs-1999	66	3	=	=	SYM
iajs-1999	66	4	ℵ	ℵ	NOUN
iajs-1999	66	5	,	,	PUNCT
iajs-1999	66	6	amounting	amount	VERB
iajs-1999	66	7	to	to	ADP
iajs-1999	66	8	for	for	ADP
iajs-1999	66	9	every	every	DET
iajs-1999	66	10	submodule	submodule	NOUN
iajs-1999	66	11	ℵ	ℵ	NOUN
iajs-1999	66	12	of	of	ADP
iajs-1999	66	13	ℬ	ℬ	NOUN
iajs-1999	66	14	,	,	PUNCT
iajs-1999	66	15	ℵ=[ℵ	ℵ=[ℵ	NOUN
iajs-1999	66	16	:	:	PUNCT
iajs-1999	66	17	ℛ	ℛ	PROPN
iajs-1999	66	18	ℬ	ℬ	NOUN
iajs-1999	66	19	]	]	PUNCT
iajs-1999	66	20	.	.	PUNCT
iajs-1999	67	1	ℬ	ℬ	NOUN
iajs-1999	67	2	,	,	PUNCT
iajs-1999	67	3	see[3	see[3	ADJ
iajs-1999	67	4	]	]	PUNCT
iajs-1999	67	5	.	.	PUNCT
iajs-1999	68	1	proposition	proposition	NOUN
iajs-1999	68	2	(	(	PUNCT
iajs-1999	68	3	6):if	6):if	NUM
iajs-1999	68	4	ℬ	ℬ	NOUN
iajs-1999	68	5	is	be	AUX
iajs-1999	68	6	a	a	DET
iajs-1999	68	7	faithful	faithful	ADJ
iajs-1999	68	8	f.g	f.g	NOUN
iajs-1999	68	9	.	.	NOUN
iajs-1999	68	10	multiplication	multiplication	NOUN
iajs-1999	68	11	r	r	NOUN
iajs-1999	68	12	-	-	PUNCT
iajs-1999	68	13	module	module	NOUN
iajs-1999	68	14	,	,	PUNCT
iajs-1999	68	15	ℵ	ℵ	ADP
iajs-1999	68	16	<	<	X
iajs-1999	68	17	ℬ	ℬ	NOUN
iajs-1999	68	18	,	,	PUNCT
iajs-1999	68	19	then	then	ADV
iajs-1999	68	20	ℵ	ℵ	NOUN
iajs-1999	68	21	is	be	AUX
iajs-1999	68	22	semisecond	semisecond	ADJ
iajs-1999	68	23	iff	iff	PROPN
iajs-1999	69	1	[	[	X
iajs-1999	69	2	ℵ:ℬ	ℵ:ℬ	X
iajs-1999	69	3	]	]	X
iajs-1999	69	4	is	be	AUX
iajs-1999	69	5	semisecond	semisecond	ADJ
iajs-1999	69	6	ideal	ideal	NOUN
iajs-1999	69	7	of	of	ADP
iajs-1999	69	8	r.	r.	PROPN
iajs-1999	69	9	proof:(⟹	proof:(⟹	PROPN
iajs-1999	69	10	)	)	PUNCT
iajs-1999	69	11	if	if	SCONJ
iajs-1999	69	12	ℵ	ℵ	NOUN
iajs-1999	69	13	is	be	AUX
iajs-1999	69	14	a	a	DET
iajs-1999	69	15	semisecond	semisecond	ADJ
iajs-1999	69	16	submodule	submodule	NOUN
iajs-1999	69	17	,	,	PUNCT
iajs-1999	69	18	then	then	ADV
iajs-1999	69	19	for	for	ADP
iajs-1999	69	20	any	any	DET
iajs-1999	69	21	r∈	r∈	PROPN
iajs-1999	69	22	ℛ	ℛ	PROPN
iajs-1999	69	23	,	,	PUNCT
iajs-1999	69	24	r≠0	r≠0	PROPN
iajs-1999	69	25	,	,	PUNCT
iajs-1999	69	26	r2ℵ=rℵ	r2ℵ=rℵ	PROPN
iajs-1999	69	27	or	or	CCONJ
iajs-1999	69	28	r2ℵ=0	r2ℵ=0	PROPN
iajs-1999	69	29	.	.	PUNCT
iajs-1999	70	1	if	if	SCONJ
iajs-1999	70	2	r2ℵ=rℵ	r2ℵ=rℵ	PROPN
iajs-1999	70	3	,	,	PUNCT
iajs-1999	70	4	then	then	ADV
iajs-1999	70	5	r2[ℵ:ℬ	r2[ℵ:ℬ	PROPN
iajs-1999	70	6	]	]	PUNCT
iajs-1999	70	7	.	.	PUNCT
iajs-1999	71	1	ℬ	ℬ	X
iajs-1999	71	2	=	=	NOUN
iajs-1999	71	3	r[ℵ:ℬ	r[ℵ:ℬ	NOUN
iajs-1999	71	4	]	]	PUNCT
iajs-1999	71	5	.	.	PUNCT
iajs-1999	72	1	ℬ	ℬ	NOUN
iajs-1999	72	2	because	because	SCONJ
iajs-1999	72	3	ℬ	ℬ	NOUN
iajs-1999	72	4	is	be	AUX
iajs-1999	72	5	a	a	DET
iajs-1999	72	6	multiplication	multiplication	NOUN
iajs-1999	72	7	module	module	NOUN
iajs-1999	72	8	.	.	PUNCT
iajs-1999	73	1	since	since	SCONJ
iajs-1999	73	2	ℬ	ℬ	NOUN
iajs-1999	73	3	is	be	AUX
iajs-1999	73	4	a	a	DET
iajs-1999	73	5	f.g	f.g	NOUN
iajs-1999	73	6	.	.	PUNCT
iajs-1999	73	7	faithful	faithful	ADJ
iajs-1999	73	8	multiplication	multiplication	NOUN
iajs-1999	73	9	ℛ	ℛ	PROPN
iajs-1999	73	10	-module	-module	NOUN
iajs-1999	73	11	,	,	PUNCT
iajs-1999	73	12	then	then	ADV
iajs-1999	73	13	by	by	ADP
iajs-1999	73	14	[	[	X
iajs-1999	73	15	1	1	NUM
iajs-1999	73	16	]	]	X
iajs-1999	73	17	r2[ℵ	r2[ℵ	NOUN
iajs-1999	73	18	:	:	PUNCT
iajs-1999	73	19	ℬ]=r[ℵ:ℬ	ℬ]=r[ℵ:ℬ	PROPN
iajs-1999	73	20	]	]	X
iajs-1999	73	21	.	.	PUNCT
iajs-1999	74	1	if	if	SCONJ
iajs-1999	74	2	r2ℵ=0	r2ℵ=0	NOUN
iajs-1999	74	3	,	,	PUNCT
iajs-1999	74	4	then	then	ADV
iajs-1999	74	5	r2[ℵ:ℬ	r2[ℵ:ℬ	PROPN
iajs-1999	74	6	]	]	PUNCT
iajs-1999	74	7	.	.	PUNCT
iajs-1999	75	1	ℬ	ℬ	X
iajs-1999	75	2	=	=	SYM
iajs-1999	75	3	0	0	NUM
iajs-1999	75	4	and	and	CCONJ
iajs-1999	75	5	hence	hence	ADV
iajs-1999	75	6	r2[ℵ:ℬ]⊆	r2[ℵ:ℬ]⊆	VERB
iajs-1999	75	7	ℬ	ℬ	NOUN
iajs-1999	75	8	=	=	NOUN
iajs-1999	75	9	0	0	NUM
iajs-1999	75	10	.	.	PUNCT
iajs-1999	76	1	thus	thus	ADV
iajs-1999	76	2	r2	r2	PROPN
iajs-1999	77	1	[	[	X
iajs-1999	77	2	ℵ:ℬ]=0	ℵ:ℬ]=0	PROPN
iajs-1999	77	3	and	and	CCONJ
iajs-1999	77	4	so	so	ADV
iajs-1999	77	5	[	[	X
iajs-1999	77	6	ℵ:ℬ	ℵ:ℬ	X
iajs-1999	77	7	]	]	X
iajs-1999	77	8	is	be	AUX
iajs-1999	77	9	a	a	DET
iajs-1999	77	10	semisecond	semisecond	ADJ
iajs-1999	77	11	ideal	ideal	NOUN
iajs-1999	77	12	.	.	PUNCT
iajs-1999	78	1	now	now	ADV
iajs-1999	78	2	,	,	PUNCT
iajs-1999	78	3	to	to	PART
iajs-1999	78	4	prove	prove	VERB
iajs-1999	78	5	the	the	DET
iajs-1999	78	6	opposite	opposite	NOUN
iajs-1999	78	7	.	.	PUNCT
iajs-1999	79	1	let	let	VERB
iajs-1999	79	2	[	[	X
iajs-1999	79	3	ℵ	ℵ	X
iajs-1999	79	4	:	:	PUNCT
iajs-1999	79	5	ℛ	ℛ	PROPN
iajs-1999	79	6	ℬ	ℬ	NOUN
iajs-1999	79	7	]	]	PUNCT
iajs-1999	79	8	be	be	VERB
iajs-1999	79	9	a	a	DET
iajs-1999	79	10	semisecond	semisecond	ADJ
iajs-1999	79	11	ideal	ideal	NOUN
iajs-1999	79	12	,	,	PUNCT
iajs-1999	79	13	that	that	ADV
iajs-1999	79	14	is	be	AUX
iajs-1999	79	15	[	[	X
iajs-1999	79	16	ℵ	ℵ	NOUN
iajs-1999	79	17	:	:	PUNCT
iajs-1999	79	18	ℛ	ℛ	PROPN
iajs-1999	79	19	m	m	X
iajs-1999	79	20	]	]	X
iajs-1999	79	21	is	be	AUX
iajs-1999	79	22	a	a	DET
iajs-1999	79	23	semisecond	semisecond	ADJ
iajs-1999	79	24	submodule	submodule	NOUN
iajs-1999	79	25	of	of	ADP
iajs-1999	79	26	the	the	DET
iajs-1999	79	27	ℛ	ℛ	PROPN
iajs-1999	79	28	-module	-module	NOUN
iajs-1999	79	29	ℛ.	ℛ.	PROPN
iajs-1999	79	30	then	then	ADV
iajs-1999	79	31	by	by	ADP
iajs-1999	79	32	proposition	proposition	NOUN
iajs-1999	79	33	(	(	PUNCT
iajs-1999	79	34	2.2	2.2	NUM
iajs-1999	79	35	)	)	PUNCT
iajs-1999	79	36			NOUN
iajs-1999	79	37	r∈r	r∈r	NOUN
iajs-1999	79	38	,	,	PUNCT
iajs-1999	79	39	r≠0	r≠0	PROPN
iajs-1999	79	40	,	,	PUNCT
iajs-1999	79	41	r2[ℵ	r2[ℵ	NOUN
iajs-1999	79	42	:	:	PUNCT
iajs-1999	79	43	ℛ	ℛ	ADJ
iajs-1999	79	44	ℬ]=	ℬ]=	ADJ
iajs-1999	79	45	r[ℵ:ℬ	r[ℵ:ℬ	NOUN
iajs-1999	79	46	]	]	PUNCT
iajs-1999	79	47	or	or	CCONJ
iajs-1999	79	48	r2[ℵ:ℬ]=0	r2[ℵ:ℬ]=0	PROPN
iajs-1999	79	49	,	,	PUNCT
iajs-1999	79	50	that	that	PRON
iajs-1999	79	51	is	be	AUX
iajs-1999	79	52	r2[ℵ	r2[ℵ	NOUN
iajs-1999	79	53	:	:	PUNCT
iajs-1999	79	54	ℛ	ℛ	PROPN
iajs-1999	79	55	ℬ	ℬ	NOUN
iajs-1999	79	56	]	]	X
iajs-1999	79	57	ℬ	ℬ	PROPN
iajs-1999	79	58	=	=	NOUN
iajs-1999	79	59	r[ℵ	r[ℵ	PROPN
iajs-1999	79	60	:	:	PUNCT
iajs-1999	79	61	ℛ	ℛ	PROPN
iajs-1999	79	62	ℬ	ℬ	NOUN
iajs-1999	79	63	]	]	PUNCT
iajs-1999	79	64	.	.	PUNCT
iajs-1999	80	1	ℬ	ℬ	NOUN
iajs-1999	80	2	or	or	CCONJ
iajs-1999	80	3	r2[ℵ	r2[ℵ	NOUN
iajs-1999	80	4	:	:	PUNCT
iajs-1999	80	5	ℛ	ℛ	PROPN
iajs-1999	80	6	ℬ]=0	ℬ]=0	PROPN
iajs-1999	80	7	.	.	PUNCT
iajs-1999	81	1	since	since	SCONJ
iajs-1999	81	2	ℬ	ℬ	NOUN
iajs-1999	81	3	is	be	AUX
iajs-1999	81	4	a	a	DET
iajs-1999	81	5	multiplication	multiplication	NOUN
iajs-1999	81	6	module	module	NOUN
iajs-1999	81	7	,	,	PUNCT
iajs-1999	81	8	we	we	PRON
iajs-1999	81	9	have	have	VERB
iajs-1999	81	10	r2ℵ=rℵ	r2ℵ=rℵ	PROPN
iajs-1999	81	11	or	or	CCONJ
iajs-1999	81	12	r2ℵ=0	r2ℵ=0	NOUN
iajs-1999	81	13	for	for	ADP
iajs-1999	81	14	every	every	DET
iajs-1999	81	15	r∈	r∈	PROPN
iajs-1999	81	16	ℛ	ℛ	PROPN
iajs-1999	81	17	,	,	PUNCT
iajs-1999	81	18	r≠0	r≠0	NOUN
iajs-1999	81	19	.	.	PUNCT
iajs-1999	82	1	therefore	therefore	ADV
iajs-1999	82	2	,	,	PUNCT
iajs-1999	82	3	ℵ	ℵ	PRON
iajs-1999	82	4	is	be	AUX
iajs-1999	82	5	a	a	DET
iajs-1999	82	6	semisecond	semisecond	ADJ
iajs-1999	82	7	submodule	submodule	NOUN
iajs-1999	82	8	.	.	PUNCT
iajs-1999	83	1	we	we	PRON
iajs-1999	83	2	notice	notice	VERB
iajs-1999	83	3	that	that	SCONJ
iajs-1999	83	4	the	the	DET
iajs-1999	83	5	provision	provision	NOUN
iajs-1999	83	6	m	m	VERB
iajs-1999	83	7	is	be	AUX
iajs-1999	83	8	faithful	faithful	ADJ
iajs-1999	83	9	can	can	AUX
iajs-1999	83	10	not	not	PART
iajs-1999	83	11	be	be	AUX
iajs-1999	83	12	dropped	drop	VERB
iajs-1999	83	13	from	from	ADP
iajs-1999	83	14	proposition	proposition	NOUN
iajs-1999	83	15	(	(	PUNCT
iajs-1999	83	16	2.6	2.6	NUM
iajs-1999	83	17	)	)	PUNCT
iajs-1999	83	18	for	for	ADP
iajs-1999	83	19	instance	instance	NOUN
iajs-1999	83	20	:	:	PUNCT
iajs-1999	83	21	consider	consider	VERB
iajs-1999	83	22	the	the	DET
iajs-1999	83	23	z	z	NOUN
iajs-1999	83	24	-	-	PUNCT
iajs-1999	83	25	module	module	NOUN
iajs-1999	83	26	z6	z6	NOUN
iajs-1999	83	27	,	,	PUNCT
iajs-1999	83	28	z6	z6	PROPN
iajs-1999	83	29	is	be	AUX
iajs-1999	83	30	f.g	f.g	NOUN
iajs-1999	83	31	.	.	PROPN
iajs-1999	83	32	multiplication	multiplication	NOUN
iajs-1999	84	1	z	z	NOUN
iajs-1999	84	2	-	-	PUNCT
iajs-1999	84	3	module	module	NOUN
iajs-1999	84	4	but	but	CCONJ
iajs-1999	84	5	not	not	PART
iajs-1999	84	6	faithful	faithful	ADJ
iajs-1999	84	7	.	.	PUNCT
iajs-1999	85	1	however	however	ADV
iajs-1999	85	2	,	,	PUNCT
iajs-1999	85	3	the	the	DET
iajs-1999	85	4	submodule	submodule	PROPN
iajs-1999	85	5	ℵ=<3	ℵ=<3	PROPN
iajs-1999	85	6	>	>	X
iajs-1999	85	7	is	be	AUX
iajs-1999	85	8	a	a	DET
iajs-1999	85	9	semisecond	semisecond	ADJ
iajs-1999	85	10	submodule	submodule	NOUN
iajs-1999	85	11	since	since	SCONJ
iajs-1999	85	12	for	for	ADP
iajs-1999	85	13	any	any	DET
iajs-1999	85	14	r2∉	r2∉	ADJ
iajs-1999	85	15	ℛ	ℛ	PROPN
iajs-1999	85	16	ℵ=2z	ℵ=2z	NOUN
iajs-1999	85	17	,	,	PUNCT
iajs-1999	85	18	r2ℵ=rℵ.	r2ℵ=rℵ.	PROPN
iajs-1999	86	1	but	but	CCONJ
iajs-1999	86	2	[	[	X
iajs-1999	86	3	ℵ	ℵ	X
iajs-1999	86	4	:	:	PUNCT
iajs-1999	86	5	ℛ	ℛ	PROPN
iajs-1999	86	6	ℬ]=[(3	ℬ]=[(3	PROPN
iajs-1999	86	7	)	)	PUNCT
iajs-1999	87	1	:	:	PUNCT
iajs-1999	87	2	z6	z6	X
iajs-1999	87	3	]	]	X
iajs-1999	88	1	=	=	SYM
iajs-1999	88	2	3z	3z	NUM
iajs-1999	88	3	is	be	AUX
iajs-1999	88	4	not	not	PART
iajs-1999	88	5	semisecond	semisecond	ADJ
iajs-1999	88	6	in	in	ADP
iajs-1999	88	7	z	z	PROPN
iajs-1999	88	8	,	,	PUNCT
iajs-1999	88	9	since	since	SCONJ
iajs-1999	88	10	for	for	ADP
iajs-1999	88	11	every	every	DET
iajs-1999	88	12	r2∉	r2∉	ADJ
iajs-1999	88	13	ℛ	ℛ	PROPN
iajs-1999	88	14	(	(	PUNCT
iajs-1999	88	15	3z)=0	3z)=0	NUM
iajs-1999	88	16	and	and	CCONJ
iajs-1999	88	17	for	for	ADP
iajs-1999	88	18	each	each	DET
iajs-1999	88	19	r	r	NOUN
iajs-1999	88	20	≠	≠	NOUN
iajs-1999	88	21	∓1	∓1	NUM
iajs-1999	88	22	we	we	PRON
iajs-1999	88	23	have	have	VERB
iajs-1999	88	24	r2	r2	PROPN
iajs-1999	88	25	(	(	PUNCT
iajs-1999	88	26	3z	3z	NUM
iajs-1999	88	27	)	)	PUNCT
iajs-1999	88	28	≠	≠	PROPN
iajs-1999	88	29	r(3z	r(3z	NUM
iajs-1999	88	30	)	)	PUNCT
iajs-1999	88	31	.	.	PUNCT
iajs-1999	89	1	proposition	proposition	NOUN
iajs-1999	89	2	(	(	PUNCT
iajs-1999	89	3	7):n.z	7):n.z	NUM
iajs-1999	89	4	.	.	PUNCT
iajs-1999	89	5	ℵ	ℵ	ADJ
iajs-1999	89	6	s.r.m	s.r.m	NOUN
iajs-1999	89	7	.	.	PUNCT
iajs-1999	90	1	ℬ	ℬ	NOUN
iajs-1999	90	2	is	be	AUX
iajs-1999	90	3	a	a	DET
iajs-1999	90	4	semisecond	semisecond	ADJ
iajs-1999	90	5	ℛ	ℛ	ADJ
iajs-1999	90	6	-submodule	-submodule	NOUN
iajs-1999	90	7	iff	iff	NOUN
iajs-1999	90	8	ℵ	ℵ	NOUN
iajs-1999	90	9	is	be	AUX
iajs-1999	90	10	a	a	DET
iajs-1999	90	11	semisecond	semisecond	ADJ
iajs-1999	90	12	ℛ	ℛ	ADJ
iajs-1999	90	13	/i	/i	NOUN
iajs-1999	90	14	-	-	PUNCT
iajs-1999	90	15	submodule	submodule	NOUN
iajs-1999	90	16	,	,	PUNCT
iajs-1999	90	17	where	where	SCONJ
iajs-1999	90	18	i⊆	i⊆	NOUN
iajs-1999	90	19	ℛ	ℛ	ADJ
iajs-1999	90	20	ℵ.	ℵ.	NOUN
iajs-1999	90	21	proof	proof	NOUN
iajs-1999	90	22	:	:	PUNCT
iajs-1999	90	23	⟹	⟹	NUM
iajs-1999	90	24	let	let	VERB
iajs-1999	90	25	�	�	PRON
iajs-1999	90	26	̅	̅	VERB
iajs-1999	90	27	�	�	NOUN
iajs-1999	90	28	=	=	SYM
iajs-1999	90	29	r+i	r+i	X
iajs-1999	90	30	∈ℛ=	∈ℛ=	NOUN
iajs-1999	90	31	ℛ	ℛ	NOUN
iajs-1999	90	32	/i	/i	NOUN
iajs-1999	90	33	.	.	PUNCT
iajs-1999	91	1	(	(	PUNCT
iajs-1999	91	2	�	�	NOUN
iajs-1999	91	3	̅	̅	NOUN
iajs-1999	91	4	�	�	NOUN
iajs-1999	91	5	)2ℵ=(r+i)2ℵ=	)2ℵ=(r+i)2ℵ=	PUNCT
iajs-1999	91	6	rℵ	rℵ	PROPN
iajs-1999	91	7	.	.	PUNCT
iajs-1999	92	1	but	but	CCONJ
iajs-1999	92	2	r2ℵ=0	r2ℵ=0	PROPN
iajs-1999	92	3	or	or	CCONJ
iajs-1999	92	4	r2ℵ=rℵ	r2ℵ=rℵ	PROPN
iajs-1999	92	5	,	,	PUNCT
iajs-1999	92	6	since	since	SCONJ
iajs-1999	92	7	ℵ	ℵ	NOUN
iajs-1999	92	8	is	be	AUX
iajs-1999	92	9	semisecond	semisecond	ADJ
iajs-1999	92	10	,	,	PUNCT
iajs-1999	92	11	therefore	therefore	ADV
iajs-1999	92	12	(	(	PUNCT
iajs-1999	92	13	�	�	NOUN
iajs-1999	92	14	̅	̅	NOUN
iajs-1999	92	15	�	�	NOUN
iajs-1999	92	16	2ℵ=0	2ℵ=0	NUM
iajs-1999	92	17	or	or	CCONJ
iajs-1999	92	18	(	(	PUNCT
iajs-1999	92	19	�	�	NOUN
iajs-1999	92	20	̅	̅	NOUN
iajs-1999	92	21	�	�	PROPN
iajs-1999	92	22	)2ℵ=	)2ℵ=	PROPN
iajs-1999	92	23	�	�	PROPN
iajs-1999	92	24	̅	̅	NOUN
iajs-1999	92	25	�	�	NOUN
iajs-1999	92	26	ℵ.	ℵ.	PROPN
iajs-1999	92	27	thus	thus	ADV
iajs-1999	92	28	ℵ	ℵ	NOUN
iajs-1999	92	29	is	be	AUX
iajs-1999	92	30	a	a	DET
iajs-1999	92	31	semisecond	semisecond	ADJ
iajs-1999	92	32	ℛ-submodule	ℛ-submodule	PROPN
iajs-1999	92	33	.	.	PUNCT
iajs-1999	93	1	similarly	similarly	ADV
iajs-1999	93	2	,	,	PUNCT
iajs-1999	93	3	we	we	PRON
iajs-1999	93	4	can	can	AUX
iajs-1999	93	5	proof	proof	VERB
iajs-1999	93	6	the	the	DET
iajs-1999	93	7	opposite	opposite	NOUN
iajs-1999	93	8	.	.	PUNCT
iajs-1999	94	1	hence	hence	ADV
iajs-1999	94	2	,	,	PUNCT
iajs-1999	94	3	we	we	PRON
iajs-1999	94	4	have	have	VERB
iajs-1999	94	5	the	the	DET
iajs-1999	94	6	following	follow	VERB
iajs-1999	94	7	result	result	NOUN
iajs-1999	94	8	.	.	PUNCT
iajs-1999	95	1	corollary	corollary	ADJ
iajs-1999	95	2	(	(	PUNCT
iajs-1999	95	3	8):if	8):if	NUM
iajs-1999	95	4	ℵ	ℵ	NOUN
iajs-1999	95	5	is	be	AUX
iajs-1999	95	6	a	a	DET
iajs-1999	95	7	n.z	n.z	PROPN
iajs-1999	95	8	.	.	PROPN
iajs-1999	95	9	s.r.m	s.r.m	PROPN
iajs-1999	95	10	.	.	PUNCT
iajs-1999	96	1	ℬ	ℬ	NOUN
iajs-1999	96	2	is	be	AUX
iajs-1999	96	3	a	a	DET
iajs-1999	96	4	semisecond	semisecond	ADJ
iajs-1999	96	5	submodule	submodule	NOUN
iajs-1999	96	6	iff	iff	PROPN
iajs-1999	96	7	ℵ	ℵ	PROPN
iajs-1999	96	8	is	be	AUX
iajs-1999	96	9	a	a	DET
iajs-1999	96	10	semisecond	semisecond	ADJ
iajs-1999	96	11	submodules	submodule	NOUN
iajs-1999	96	12	ℛ	ℛ	PROPN
iajs-1999	96	13	/	/	SYM
iajs-1999	96	14	ℛ	ℛ	PROPN
iajs-1999	96	15	ℵ	ℵ	NOUN
iajs-1999	96	16	–	–	PUNCT
iajs-1999	96	17	submodule	submodule	NOUN
iajs-1999	96	18	.	.	PUNCT
iajs-1999	97	1	proposition	proposition	NOUN
iajs-1999	97	2	(	(	PUNCT
iajs-1999	97	3	9):let	9):let	NOUN
iajs-1999	97	4	ℵ	ℵ	NOUN
iajs-1999	97	5	be	be	AUX
iajs-1999	97	6	n.z	n.z	NOUN
iajs-1999	97	7	.	.	PUNCT
iajs-1999	97	8	proper	proper	ADJ
iajs-1999	97	9	submodule	submodule	NOUN
iajs-1999	97	10	of	of	ADP
iajs-1999	97	11	ℬ	ℬ	PROPN
iajs-1999	97	12	s.t	s.t	PROPN
iajs-1999	97	13	.	.	PUNCT
iajs-1999	98	1	ℛ	ℛ	PROPN
iajs-1999	98	2	ℵ	ℵ	NOUN
iajs-1999	98	3	is	be	AUX
iajs-1999	98	4	a	a	DET
iajs-1999	98	5	maximal	maximal	ADJ
iajs-1999	98	6	ideal	ideal	NOUN
iajs-1999	98	7	,	,	PUNCT
iajs-1999	98	8	then	then	ADV
iajs-1999	98	9	ℵ	ℵ	NOUN
iajs-1999	98	10	is	be	AUX
iajs-1999	98	11	a	a	DET
iajs-1999	98	12	semisecond	semisecond	ADJ
iajs-1999	98	13	submodule	submodule	NOUN
iajs-1999	98	14	.	.	PUNCT
iajs-1999	99	1	proof	proof	NOUN
iajs-1999	99	2	:	:	PUNCT
iajs-1999	99	3	since	since	SCONJ
iajs-1999	99	4	ℛ	ℛ	ADJ
iajs-1999	99	5	ℵ	ℵ	NOUN
iajs-1999	99	6	is	be	AUX
iajs-1999	99	7	a	a	DET
iajs-1999	99	8	maximal	maximal	ADJ
iajs-1999	99	9	ideal	ideal	NOUN
iajs-1999	99	10	,	,	PUNCT
iajs-1999	99	11	then	then	ADV
iajs-1999	99	12	ℛ/	ℛ/	NUM
iajs-1999	99	13	ℛ	ℛ	ADJ
iajs-1999	99	14	ℵ	ℵ	NOUN
iajs-1999	99	15	is	be	AUX
iajs-1999	99	16	a	a	DET
iajs-1999	99	17	field	field	NOUN
iajs-1999	99	18	and	and	CCONJ
iajs-1999	99	19	by	by	ADP
iajs-1999	99	20	remark	remark	NOUN
iajs-1999	99	21	and	and	CCONJ
iajs-1999	99	22	example	example	NOUN
iajs-1999	99	23	(	(	PUNCT
iajs-1999	99	24	2.3.(4	2.3.(4	NUM
iajs-1999	99	25	)	)	PUNCT
iajs-1999	99	26	)	)	PUNCT
iajs-1999	99	27	ℵ	ℵ	NOUN
iajs-1999	99	28	is	be	AUX
iajs-1999	99	29	semisecond	semisecond	ADJ
iajs-1999	99	30	submodule	submodule	NOUN
iajs-1999	99	31	ℛ	ℛ	PROPN
iajs-1999	99	32	/	/	SYM
iajs-1999	99	33	ℛ	ℛ	NOUN
iajs-1999	99	34	ℵ-submodule	ℵ-submodule	NOUN
iajs-1999	99	35	.	.	PUNCT
iajs-1999	100	1	thus	thus	ADV
iajs-1999	100	2	by	by	ADP
iajs-1999	100	3	corollary	corollary	ADJ
iajs-1999	100	4	(	(	PUNCT
iajs-1999	100	5	2.8	2.8	NUM
iajs-1999	100	6	)	)	PUNCT
iajs-1999	100	7	,	,	PUNCT
iajs-1999	100	8	ℵ	ℵ	PROPN
iajs-1999	100	9	is	be	AUX
iajs-1999	100	10	a	a	DET
iajs-1999	100	11	semisecond	semisecond	ADJ
iajs-1999	100	12	submodule	submodule	NOUN
iajs-1999	100	13	ℛ	ℛ	PROPN
iajs-1999	100	14	-submodule	-submodule	NOUN
iajs-1999	100	15	.	.	PUNCT
iajs-1999	101	1	mathematics	mathematic	NOUN
iajs-1999	101	2	|	|	ADV
iajs-1999	101	3	105	105	NUM
iajs-1999	101	4	ibn	ibn	PROPN
iajs-1999	101	5	al	al	PROPN
iajs-1999	101	6	-	-	PUNCT
iajs-1999	101	7	haitham	haitham	PROPN
iajs-1999	101	8	jour	jour	X
iajs-1999	101	9	.	.	PROPN
iajs-1999	102	1	for	for	ADP
iajs-1999	102	2	pure	pure	ADJ
iajs-1999	102	3	&	&	CCONJ
iajs-1999	102	4	appl	appl	PROPN
iajs-1999	102	5	.	.	PUNCT
iajs-1999	103	1	sci	sci	PROPN
iajs-1999	103	2	.	.	PROPN
iajs-1999	103	3	ihjpas	ihjpa	VERB
iajs-1999	103	4	https://doi.org/10.30526/	https://doi.org/10.30526/	PROPN
iajs-1999	103	5	31.3.1999	31.3.1999	NUM
iajs-1999	103	6	vol	vol	NOUN
iajs-1999	103	7	.	.	PUNCT
iajs-1999	104	1	31	31	NUM
iajs-1999	105	1	(	(	PUNCT
iajs-1999	105	2	3	3	NUM
iajs-1999	105	3	)	)	SYM
iajs-1999	105	4	2018	2018	NUM
iajs-1999	105	5	remark	remark	NOUN
iajs-1999	105	6	(	(	PUNCT
iajs-1999	105	7	10):if	10):if	NUM
iajs-1999	105	8	ℵ=ℵ1⊕ℵ2	ℵ=ℵ1⊕ℵ2	PROPN
iajs-1999	105	9	is	be	AUX
iajs-1999	105	10	semisecond	semisecond	ADJ
iajs-1999	105	11	submodule	submodule	NOUN
iajs-1999	105	12	in	in	ADP
iajs-1999	105	13	ℬ=ℬ1⊕ℬ2	ℬ=ℬ1⊕ℬ2	PROPN
iajs-1999	105	14	,	,	PUNCT
iajs-1999	105	15	then	then	ADV
iajs-1999	105	16	n1	n1	PROPN
iajs-1999	105	17	and	and	CCONJ
iajs-1999	105	18	n2	n2	NOUN
iajs-1999	105	19	are	be	AUX
iajs-1999	105	20	semiseconds	semisecond	NOUN
iajs-1999	105	21	in	in	ADP
iajs-1999	105	22	ℬ	ℬ	PROPN
iajs-1999	105	23	1	1	NUM
iajs-1999	105	24	,	,	PUNCT
iajs-1999	105	25	ℬ	ℬ	ADV
iajs-1999	105	26	2	2	NUM
iajs-1999	105	27	respectively	respectively	ADV
iajs-1999	105	28	.	.	PUNCT
iajs-1999	106	1	proof	proof	NOUN
iajs-1999	106	2	:	:	PUNCT
iajs-1999	106	3	it	it	PRON
iajs-1999	106	4	follows	follow	VERB
iajs-1999	106	5	directly	directly	ADV
iajs-1999	106	6	by	by	ADP
iajs-1999	106	7	remark	remark	NOUN
iajs-1999	106	8	and	and	CCONJ
iajs-1999	106	9	example	example	NOUN
iajs-1999	106	10	(	(	PUNCT
iajs-1999	106	11	2.3.(5	2.3.(5	NUM
iajs-1999	106	12	)	)	PUNCT
iajs-1999	106	13	)	)	PUNCT
iajs-1999	106	14	.	.	PUNCT
iajs-1999	107	1	remark	remark	NOUN
iajs-1999	107	2	(	(	PUNCT
iajs-1999	107	3	11):let	11):let	NOUN
iajs-1999	107	4	ℬ	ℬ	NOUN
iajs-1999	107	5	=	=	SYM
iajs-1999	107	6	ℬ	ℬ	NOUN
iajs-1999	107	7	1⊕	1⊕	NUM
iajs-1999	107	8	ℬ	ℬ	NOUN
iajs-1999	107	9	2	2	NUM
iajs-1999	107	10	.	.	PUNCT
iajs-1999	108	1	if	if	SCONJ
iajs-1999	108	2	ℵ1	ℵ1	PROPN
iajs-1999	108	3	and	and	CCONJ
iajs-1999	108	4	ℵ2	ℵ2	PROPN
iajs-1999	108	5	are	be	AUX
iajs-1999	108	6	semisecond	semisecond	ADJ
iajs-1999	108	7	submodules	submodule	NOUN
iajs-1999	108	8	in	in	ADP
iajs-1999	108	9	ℬ	ℬ	PROPN
iajs-1999	108	10	1	1	NUM
iajs-1999	108	11	and	and	CCONJ
iajs-1999	108	12	ℬ	ℬ	PROPN
iajs-1999	108	13	2	2	NUM
iajs-1999	108	14	respectively	respectively	ADV
iajs-1999	108	15	,	,	PUNCT
iajs-1999	108	16	then	then	ADV
iajs-1999	108	17	it	it	PRON
iajs-1999	108	18	is	be	AUX
iajs-1999	108	19	not	not	PART
iajs-1999	108	20	necessarily	necessarily	ADV
iajs-1999	108	21	that	that	PRON
iajs-1999	108	22	ℵ	ℵ	ADP
iajs-1999	108	23	1⊕	1⊕	NUM
iajs-1999	108	24	ℵ	ℵ	DET
iajs-1999	108	25	2	2	NUM
iajs-1999	108	26	is	be	AUX
iajs-1999	108	27	semisecond	semisecond	ADJ
iajs-1999	108	28	submodule	submodule	NOUN
iajs-1999	108	29	in	in	ADP
iajs-1999	108	30	m	m	PROPN
iajs-1999	108	31	for	for	ADP
iajs-1999	108	32	example	example	NOUN
iajs-1999	108	33	:	:	PUNCT
iajs-1999	108	34	let	let	VERB
iajs-1999	108	35	ℬ	ℬ	PROPN
iajs-1999	108	36	=	=	PROPN
iajs-1999	108	37	z6⊕z16	z6⊕z16	PROPN
iajs-1999	108	38	,	,	PUNCT
iajs-1999	108	39	let	let	VERB
iajs-1999	108	40	ℵ=<3>⊕<2	ℵ=<3>⊕<2	PROPN
iajs-1999	108	41	>	>	X
iajs-1999	108	42	,	,	PUNCT
iajs-1999	108	43	<3	<3	X
iajs-1999	108	44	>	>	X
iajs-1999	108	45	is	be	AUX
iajs-1999	108	46	semisecond	semisecond	ADJ
iajs-1999	108	47	submodule	submodule	NOUN
iajs-1999	108	48	in	in	ADP
iajs-1999	108	49	z6	z6	PROPN
iajs-1999	108	50	,	,	PUNCT
iajs-1999	108	51	<	<	X
iajs-1999	108	52	2	2	NUM
iajs-1999	108	53	>	>	X
iajs-1999	108	54	is	be	AUX
iajs-1999	108	55	semisecond	semisecond	ADJ
iajs-1999	108	56	submodule	submodule	NOUN
iajs-1999	108	57	in	in	ADP
iajs-1999	108	58	z16	z16	PROPN
iajs-1999	108	59	.	.	PUNCT
iajs-1999	109	1	however	however	ADV
iajs-1999	109	2	2	2	NUM
iajs-1999	109	3	ℵ	ℵ	NOUN
iajs-1999	109	4	=	=	NOUN
iajs-1999	109	5	<	<	X
iajs-1999	109	6	0	0	NUM
iajs-1999	109	7	>	>	PUNCT
iajs-1999	109	8	+	+	NOUN
iajs-1999	109	9	<	<	X
iajs-1999	109	10	4	4	NUM
iajs-1999	109	11	>	>	PUNCT
iajs-1999	109	12	,	,	PUNCT
iajs-1999	109	13	(	(	PUNCT
iajs-1999	109	14	22	22	NUM
iajs-1999	109	15	)	)	PUNCT
iajs-1999	109	16	ℵ	ℵ	NOUN
iajs-1999	109	17	=	=	SYM
iajs-1999	109	18	4	4	NUM
iajs-1999	109	19	ℵ	ℵ	NOUN
iajs-1999	109	20	=	=	NOUN
iajs-1999	109	21	<	<	X
iajs-1999	109	22	0	0	NUM
iajs-1999	109	23	>	>	PUNCT
iajs-1999	109	24	⊕	⊕	PROPN
iajs-1999	109	25	<	<	X
iajs-1999	109	26	8	8	NUM
iajs-1999	109	27	>	>	PUNCT
iajs-1999	109	28	,	,	PUNCT
iajs-1999	109	29	then	then	ADV
iajs-1999	109	30	22	22	NUM
iajs-1999	109	31	ℵ	ℵ	NOUN
iajs-1999	109	32	≠2	≠2	NOUN
iajs-1999	109	33	ℵ	ℵ	NOUN
iajs-1999	109	34	and	and	CCONJ
iajs-1999	109	35	22ℵ≠<0>⊕<0	22ℵ≠<0>⊕<0	NUM
iajs-1999	109	36	>	>	PUNCT
iajs-1999	109	37	.	.	PUNCT
iajs-1999	110	1	the	the	DET
iajs-1999	110	2	following	follow	VERB
iajs-1999	110	3	result	result	NOUN
iajs-1999	110	4	shows	show	VERB
iajs-1999	110	5	the	the	DET
iajs-1999	110	6	direct	direct	ADJ
iajs-1999	110	7	sum	sum	NOUN
iajs-1999	110	8	of	of	ADP
iajs-1999	110	9	two	two	NUM
iajs-1999	110	10	semisecond	semisecond	ADJ
iajs-1999	110	11	submodules	submodule	NOUN
iajs-1999	110	12	under	under	ADP
iajs-1999	110	13	certain	certain	ADJ
iajs-1999	110	14	condition	condition	NOUN
iajs-1999	110	15	.	.	PUNCT
iajs-1999	111	1	proposition	proposition	NOUN
iajs-1999	111	2	(	(	PUNCT
iajs-1999	111	3	12):let	12):let	NUM
iajs-1999	111	4	ℵ1	ℵ1	PROPN
iajs-1999	111	5	and	and	CCONJ
iajs-1999	111	6	ℵ2	ℵ2	PROPN
iajs-1999	111	7	be	be	AUX
iajs-1999	111	8	semisecond	semisecond	ADJ
iajs-1999	111	9	submodules	submodule	NOUN
iajs-1999	111	10	in	in	ADP
iajs-1999	111	11	ℬ1	ℬ1	NOUN
iajs-1999	111	12	and	and	CCONJ
iajs-1999	111	13	ℬ2	ℬ2	NOUN
iajs-1999	111	14	respectively	respectively	ADV
iajs-1999	111	15	such	such	ADJ
iajs-1999	111	16	that	that	SCONJ
iajs-1999	111	17	ℛ	ℛ	PROPN
iajs-1999	111	18	ℵ1	ℵ1	NOUN
iajs-1999	111	19	=	=	SYM
iajs-1999	111	20	ℛ	ℛ	ADJ
iajs-1999	111	21	ℵ2	ℵ2	NOUN
iajs-1999	111	22	.	.	PUNCT
iajs-1999	112	1	then	then	ADV
iajs-1999	112	2	ℵ1⊕ℵ2	ℵ1⊕ℵ2	PROPN
iajs-1999	112	3	is	be	AUX
iajs-1999	112	4	semisecond	semisecond	ADJ
iajs-1999	112	5	submodule	submodule	NOUN
iajs-1999	112	6	in	in	ADP
iajs-1999	112	7	ℬ=ℬ1⊕ℬ2	ℬ=ℬ1⊕ℬ2	PROPN
iajs-1999	112	8	.	.	PUNCT
iajs-1999	113	1	proof	proof	NOUN
iajs-1999	113	2	:	:	PUNCT
iajs-1999	113	3	let	let	VERB
iajs-1999	113	4	r∈	r∈	PROPN
iajs-1999	113	5	ℛ	ℛ	PROPN
iajs-1999	113	6	,	,	PUNCT
iajs-1999	113	7	r≠0	r≠0	PROPN
iajs-1999	113	8	,	,	PUNCT
iajs-1999	113	9	then	then	ADV
iajs-1999	113	10	(	(	PUNCT
iajs-1999	113	11	r2ℵ1	r2ℵ1	NOUN
iajs-1999	113	12	=	=	NOUN
iajs-1999	113	13	rℵ1	rℵ1	NOUN
iajs-1999	113	14	or	or	CCONJ
iajs-1999	113	15	r2ℵ1=0	r2ℵ1=0	ADJ
iajs-1999	113	16	)	)	PUNCT
iajs-1999	113	17	and	and	CCONJ
iajs-1999	113	18	(	(	PUNCT
iajs-1999	113	19	r2ℵ2	r2ℵ2	X
iajs-1999	113	20	=	=	NOUN
iajs-1999	113	21	rℵ2	rℵ2	NOUN
iajs-1999	113	22	or	or	CCONJ
iajs-1999	113	23	r2ℵ2=0	r2ℵ2=0	NUM
iajs-1999	113	24	)	)	PUNCT
iajs-1999	113	25	.	.	PUNCT
iajs-1999	114	1	suppose	suppose	VERB
iajs-1999	115	1	r2ℵ1=0	r2ℵ1=0	ADJ
iajs-1999	115	2	,	,	PUNCT
iajs-1999	115	3	then	then	ADV
iajs-1999	115	4	r2ℵ2=0	r2ℵ2=0	PUNCT
iajs-1999	115	5	since	since	SCONJ
iajs-1999	115	6	ℛ	ℛ	PROPN
iajs-1999	115	7	ℵ1	ℵ1	NOUN
iajs-1999	115	8	=	=	SYM
iajs-1999	115	9	ℛ	ℛ	PROPN
iajs-1999	115	10	ℵ2	ℵ2	ADJ
iajs-1999	115	11	and	and	CCONJ
iajs-1999	115	12	so	so	ADV
iajs-1999	115	13	r2(ℵ1⊕ℵ2	r2(ℵ1⊕ℵ2	NOUN
iajs-1999	115	14	)	)	PUNCT
iajs-1999	115	15	=	=	NOUN
iajs-1999	115	16	0	0	NUM
iajs-1999	115	17	.	.	PUNCT
iajs-1999	116	1	if	if	SCONJ
iajs-1999	116	2	r2ℵ1	r2ℵ1	NOUN
iajs-1999	116	3	=	=	NOUN
iajs-1999	116	4	rℵ1	rℵ1	NOUN
iajs-1999	116	5	and	and	CCONJ
iajs-1999	116	6	r2ℵ1≠0	r2ℵ1≠0	NOUN
iajs-1999	116	7	,	,	PUNCT
iajs-1999	116	8	hence	hence	ADV
iajs-1999	116	9	r2ℵ2≠0	r2ℵ2≠0	VERB
iajs-1999	116	10	so	so	SCONJ
iajs-1999	116	11	r2ℵ2	r2ℵ2	X
iajs-1999	116	12	=	=	NOUN
iajs-1999	116	13	rℵ2	rℵ2	NOUN
iajs-1999	116	14	.	.	PUNCT
iajs-1999	117	1	it	it	PRON
iajs-1999	117	2	follows	follow	VERB
iajs-1999	117	3	that	that	SCONJ
iajs-1999	117	4	r2(ℵ1⊕ℵ2	r2(ℵ1⊕ℵ2	VERB
iajs-1999	117	5	)	)	PUNCT
iajs-1999	117	6	=	=	SYM
iajs-1999	117	7	r2ℵ1⊕r2ℵ2	r2ℵ1⊕r2ℵ2	NOUN
iajs-1999	117	8	=	=	NOUN
iajs-1999	117	9	rℵ1⊕rℵ2	rℵ1⊕rℵ2	PROPN
iajs-1999	117	10	.	.	PUNCT
iajs-1999	118	1	now	now	ADV
iajs-1999	118	2	,	,	PUNCT
iajs-1999	118	3	we	we	PRON
iajs-1999	118	4	survey	survey	VERB
iajs-1999	118	5	the	the	DET
iajs-1999	118	6	relationships	relationship	NOUN
iajs-1999	118	7	between	between	ADP
iajs-1999	118	8	semisecond	semisecond	ADJ
iajs-1999	118	9	submodules	submodule	NOUN
iajs-1999	118	10	and	and	CCONJ
iajs-1999	118	11	some	some	DET
iajs-1999	118	12	kind	kind	NOUN
iajs-1999	118	13	of	of	ADP
iajs-1999	118	14	submodules	submodule	NOUN
iajs-1999	118	15	.	.	PUNCT
iajs-1999	119	1	a	a	DET
iajs-1999	119	2	submodule	submodule	NOUN
iajs-1999	119	3	ℵ	ℵ	NOUN
iajs-1999	119	4	of	of	ADP
iajs-1999	119	5	a	a	DET
iajs-1999	119	6	module	module	NOUN
iajs-1999	119	7	ℬ	ℬ	NOUN
iajs-1999	119	8	is	be	AUX
iajs-1999	119	9	rendering	render	VERB
iajs-1999	119	10	semiprime	semiprime	NOUN
iajs-1999	119	11	if	if	SCONJ
iajs-1999	119	12	ℵ≠ℬ	ℵ≠ℬ	NOUN
iajs-1999	119	13	and	and	CCONJ
iajs-1999	119	14	r∈ℛ	r∈ℛ	NOUN
iajs-1999	119	15	,	,	PUNCT
iajs-1999	119	16	m∈ℬ	m∈ℬ	PROPN
iajs-1999	119	17	,	,	PUNCT
iajs-1999	119	18	k∈z+	k∈z+	VERB
iajs-1999	119	19	with	with	ADP
iajs-1999	119	20	rkm∈ℵ	rkm∈ℵ	NOUN
iajs-1999	119	21	,	,	PUNCT
iajs-1999	119	22	then	then	ADV
iajs-1999	119	23	rm∈ℵ,see[4].equivalently	rm∈ℵ,see[4].equivalently	ADV
iajs-1999	119	24	ℵ	ℵ	ADJ
iajs-1999	119	25	is	be	AUX
iajs-1999	119	26	semiprime	semiprime	NOUN
iajs-1999	119	27	if	if	SCONJ
iajs-1999	119	28	whenever	whenever	SCONJ
iajs-1999	119	29	r∈ℛ	r∈ℛ	X
iajs-1999	119	30	,	,	PUNCT
iajs-1999	119	31	m∈ℬ	m∈ℬ	PROPN
iajs-1999	119	32	,	,	PUNCT
iajs-1999	119	33	r2m∈ℵ	r2m∈ℵ	NOUN
iajs-1999	119	34	,	,	PUNCT
iajs-1999	119	35	then	then	ADV
iajs-1999	119	36	rm∈ℵ,see[3,prop.(1.2	rm∈ℵ,see[3,prop.(1.2	NOUN
iajs-1999	119	37	)	)	PUNCT
iajs-1999	119	38	]	]	PUNCT
iajs-1999	119	39	.	.	PUNCT
iajs-1999	120	1	an	an	DET
iajs-1999	120	2	r	r	NOUN
iajs-1999	120	3	-	-	PUNCT
iajs-1999	120	4	module	module	NOUN
iajs-1999	120	5	ℬ	ℬ	NOUN
iajs-1999	120	6	is	be	AUX
iajs-1999	120	7	rendering	render	VERB
iajs-1999	120	8	semiprime	semiprime	NOUN
iajs-1999	120	9	if	if	SCONJ
iajs-1999	120	10	(	(	PUNCT
iajs-1999	120	11	0	0	X
iajs-1999	120	12	)	)	PUNCT
iajs-1999	120	13	is	be	AUX
iajs-1999	120	14	a	a	DET
iajs-1999	120	15	semiprime	semiprime	NOUN
iajs-1999	120	16	submodule	submodule	NOUN
iajs-1999	120	17	of	of	ADP
iajs-1999	120	18	ℬ.	ℬ.	PROPN
iajs-1999	120	19	proposition	proposition	NOUN
iajs-1999	120	20	(	(	PUNCT
iajs-1999	120	21	13):let	13):let	NUM
iajs-1999	120	22	ℬ	ℬ	PROPN
iajs-1999	120	23	be	be	AUX
iajs-1999	120	24	a	a	DET
iajs-1999	120	25	semiprime	semiprime	NOUN
iajs-1999	120	26	ℛ-module	ℛ-module	PROPN
iajs-1999	120	27	,	,	PUNCT
iajs-1999	120	28	ℵ	ℵ	PRON
iajs-1999	120	29	submodule	submodule	NOUN
iajs-1999	120	30	of	of	ADP
iajs-1999	120	31	ℬ	ℬ	PROPN
iajs-1999	120	32	if	if	SCONJ
iajs-1999	120	33	ℵ	ℵ	NOUN
iajs-1999	120	34	is	be	AUX
iajs-1999	120	35	semisecond	semisecond	ADJ
iajs-1999	120	36	,	,	PUNCT
iajs-1999	120	37	then	then	ADV
iajs-1999	120	38	ℵ	ℵ	NOUN
iajs-1999	120	39	is	be	AUX
iajs-1999	120	40	semiprime	semiprime	NOUN
iajs-1999	120	41	submodule	submodule	NOUN
iajs-1999	120	42	of	of	ADP
iajs-1999	120	43	ℬ.	ℬ.	PROPN
iajs-1999	120	44	proof	proof	NOUN
iajs-1999	120	45	:	:	PUNCT
iajs-1999	120	46	let	let	VERB
iajs-1999	120	47	a2x∈ℵ	a2x∈ℵ	VERB
iajs-1999	120	48	,	,	PUNCT
iajs-1999	120	49	where	where	SCONJ
iajs-1999	120	50	a∈ℛ	a∈ℛ	PROPN
iajs-1999	120	51	,	,	PUNCT
iajs-1999	120	52	x∈ℬ.	x∈ℬ.	PROPN
iajs-1999	120	53	to	to	PART
iajs-1999	120	54	prove	prove	VERB
iajs-1999	120	55	ax∈ℵ.	ax∈ℵ.	NOUN
iajs-1999	120	56	since	since	SCONJ
iajs-1999	120	57	ℵ	ℵ	NOUN
iajs-1999	120	58	is	be	AUX
iajs-1999	120	59	semisecond	semisecond	ADJ
iajs-1999	120	60	,	,	PUNCT
iajs-1999	120	61	then	then	ADV
iajs-1999	120	62	either	either	CCONJ
iajs-1999	120	63	a2ℵ=0	a2ℵ=0	NOUN
iajs-1999	120	64	or	or	CCONJ
iajs-1999	120	65	a2ℵ=aℵ.	a2ℵ=aℵ.	PROPN
iajs-1999	120	66	assume	assume	VERB
iajs-1999	120	67	a2ℵ=(0	a2ℵ=(0	NUM
iajs-1999	120	68	)	)	PUNCT
iajs-1999	120	69	.	.	PUNCT
iajs-1999	121	1	put	put	VERB
iajs-1999	121	2	a2x	a2x	PROPN
iajs-1999	121	3	=	=	NOUN
iajs-1999	121	4	n	n	X
iajs-1999	121	5	for	for	ADP
iajs-1999	121	6	some	some	DET
iajs-1999	121	7	n∈ℵ.	n∈ℵ.	NOUN
iajs-1999	121	8	then	then	ADV
iajs-1999	121	9	a4x	a4x	VERB
iajs-1999	121	10	=	=	SYM
iajs-1999	121	11	a2n∈a2ℵ=0	a2n∈a2ℵ=0	NOUN
iajs-1999	121	12	,	,	PUNCT
iajs-1999	121	13	hence	hence	ADV
iajs-1999	121	14	ax=0∈ℵ	ax=0∈ℵ	PROPN
iajs-1999	121	15	(	(	PUNCT
iajs-1999	121	16	since	since	SCONJ
iajs-1999	121	17	ℬ	ℬ	PROPN
iajs-1999	121	18	is	be	AUX
iajs-1999	121	19	semiprime	semiprime	NOUN
iajs-1999	121	20	)	)	PUNCT
iajs-1999	121	21	.	.	PUNCT
iajs-1999	122	1	assume	assume	VERB
iajs-1999	122	2	a2ℵ=aℵ.	a2ℵ=aℵ.	PROPN
iajs-1999	122	3	since	since	SCONJ
iajs-1999	122	4	a2x	a2x	PROPN
iajs-1999	122	5	=	=	SYM
iajs-1999	122	6	n∈ℵ	n∈ℵ	ADJ
iajs-1999	122	7	,	,	PUNCT
iajs-1999	122	8	then	then	ADV
iajs-1999	122	9	a3x	a3x	NOUN
iajs-1999	122	10	=	=	SYM
iajs-1999	122	11	an∈aℵ=a2ℵ	an∈aℵ=a2ℵ	NOUN
iajs-1999	122	12	,	,	PUNCT
iajs-1999	122	13	so	so	SCONJ
iajs-1999	122	14	that	that	SCONJ
iajs-1999	122	15	a3x	a3x	NOUN
iajs-1999	122	16	=	=	SYM
iajs-1999	122	17	a2n1for	a2n1for	ADP
iajs-1999	122	18	some	some	DET
iajs-1999	122	19	n1∈ℵ.	n1∈ℵ.	NOUN
iajs-1999	122	20	hence	hence	ADV
iajs-1999	122	21	a2(ax	a2(ax	PROPN
iajs-1999	122	22	-	-	SYM
iajs-1999	122	23	n1)=0.as	n1)=0.as	ADJ
iajs-1999	122	24	ℬ	ℬ	PROPN
iajs-1999	122	25	is	be	AUX
iajs-1999	122	26	semiprime	semiprime	NOUN
iajs-1999	122	27	a(ax1	a(ax1	PROPN
iajs-1999	122	28	-	-	ADJ
iajs-1999	122	29	n1)=0	n1)=0	ADJ
iajs-1999	122	30	and	and	CCONJ
iajs-1999	122	31	so	so	SCONJ
iajs-1999	122	32	that	that	SCONJ
iajs-1999	122	33	a2x	a2x	PROPN
iajs-1999	122	34	=	=	PRON
iajs-1999	122	35	an	an	DET
iajs-1999	122	36	∈aℵ=a2ℵ.	∈aℵ=a2ℵ.	PROPN
iajs-1999	122	37	thus	thus	ADV
iajs-1999	122	38	a2ℵ=a2n2	a2ℵ=a2n2	VERB
iajs-1999	122	39	for	for	ADP
iajs-1999	122	40	some	some	DET
iajs-1999	122	41	n2∈ℵ.	n2∈ℵ.	NOUN
iajs-1999	122	42	this	this	PRON
iajs-1999	122	43	implies	imply	VERB
iajs-1999	122	44	a2(x	a2(x	PROPN
iajs-1999	122	45	-	-	ADJ
iajs-1999	122	46	n2	n2	ADJ
iajs-1999	122	47	)	)	PUNCT
iajs-1999	123	1	=	=	NOUN
iajs-1999	123	2	0	0	NUM
iajs-1999	123	3	.	.	PUNCT
iajs-1999	124	1	but	but	CCONJ
iajs-1999	124	2	ℬ	ℬ	PROPN
iajs-1999	124	3	is	be	AUX
iajs-1999	124	4	semiprime	semiprime	NOUN
iajs-1999	124	5	,	,	PUNCT
iajs-1999	124	6	so	so	SCONJ
iajs-1999	124	7	that	that	SCONJ
iajs-1999	124	8	a(x	a(x	NOUN
iajs-1999	124	9	-	-	PUNCT
iajs-1999	124	10	n2	n2	NOUN
iajs-1999	124	11	)	)	PUNCT
iajs-1999	124	12	=	=	NOUN
iajs-1999	124	13	0	0	NUM
iajs-1999	124	14	.	.	PUNCT
iajs-1999	125	1	it	it	PRON
iajs-1999	125	2	follows	follow	VERB
iajs-1999	125	3	that	that	SCONJ
iajs-1999	125	4	ax	ax	NOUN
iajs-1999	125	5	=	=	NOUN
iajs-1999	125	6	an2∈ℵ.	an2∈ℵ.	NOUN
iajs-1999	125	7	therefore	therefore	ADV
iajs-1999	125	8	,	,	PUNCT
iajs-1999	125	9	ℵ	ℵ	X
iajs-1999	125	10	is	be	AUX
iajs-1999	125	11	a	a	DET
iajs-1999	125	12	semiprime	semiprime	NOUN
iajs-1999	125	13	submodule	submodule	NOUN
iajs-1999	125	14	.	.	PUNCT
iajs-1999	126	1	note	note	VERB
iajs-1999	126	2	that	that	SCONJ
iajs-1999	126	3	the	the	DET
iajs-1999	126	4	opposite	opposite	NOUN
iajs-1999	126	5	of	of	ADP
iajs-1999	126	6	previous	previous	ADJ
iajs-1999	126	7	proposition	proposition	NOUN
iajs-1999	126	8	is	be	AUX
iajs-1999	126	9	not	not	PART
iajs-1999	126	10	hold	hold	VERB
iajs-1999	126	11	in	in	ADP
iajs-1999	126	12	public	public	NOUN
iajs-1999	126	13	for	for	ADP
iajs-1999	126	14	instance	instance	NOUN
iajs-1999	126	15	:	:	PUNCT
iajs-1999	126	16	take	take	VERB
iajs-1999	126	17	ℬ=z	ℬ=z	NOUN
iajs-1999	126	18	as	as	ADP
iajs-1999	126	19	z	z	NOUN
iajs-1999	126	20	-	-	PUNCT
iajs-1999	126	21	module	module	NOUN
iajs-1999	126	22	.	.	PUNCT
iajs-1999	127	1	ℬ	ℬ	NOUN
iajs-1999	127	2	is	be	AUX
iajs-1999	127	3	prime	prime	ADJ
iajs-1999	127	4	so	so	SCONJ
iajs-1999	127	5	it	it	PRON
iajs-1999	127	6	is	be	AUX
iajs-1999	127	7	semiprime	semiprime	NOUN
iajs-1999	127	8	.	.	PUNCT
iajs-1999	128	1	let	let	VERB
iajs-1999	128	2	ℵ=<6	ℵ=<6	NOUN
iajs-1999	128	3	>	>	X
iajs-1999	128	4	is	be	AUX
iajs-1999	128	5	semiprime	semiprime	NOUN
iajs-1999	128	6	,	,	PUNCT
iajs-1999	128	7	but	but	CCONJ
iajs-1999	128	8	n	n	PRON
iajs-1999	128	9	is	be	AUX
iajs-1999	128	10	not	not	PART
iajs-1999	128	11	semisecond	semisecond	ADJ
iajs-1999	128	12	since	since	SCONJ
iajs-1999	128	13	for	for	ADP
iajs-1999	128	14	every	every	DET
iajs-1999	128	15	r∈z	r∈z	NOUN
iajs-1999	128	16	,	,	PUNCT
iajs-1999	128	17	r≠0	r≠0	NOUN
iajs-1999	128	18	,	,	PUNCT
iajs-1999	128	19	r2ℵ≠(0	r2ℵ≠(0	NOUN
iajs-1999	128	20	)	)	PUNCT
iajs-1999	128	21	and	and	CCONJ
iajs-1999	128	22	r2ℵ≠rℵ.	r2ℵ≠rℵ.	PROPN
iajs-1999	128	23	reminiscence	reminiscence	NOUN
iajs-1999	128	24	that	that	SCONJ
iajs-1999	128	25	a	a	DET
iajs-1999	128	26	module	module	NOUN
iajs-1999	128	27	ℬ	ℬ	NOUN
iajs-1999	128	28	is	be	AUX
iajs-1999	128	29	rendering	render	VERB
iajs-1999	128	30	coprime	coprime	ADV
iajs-1999	128	31	if	if	SCONJ
iajs-1999	128	32	ℛ	ℛ	PROPN
iajs-1999	128	33	ℬ=	ℬ=	NOUN
iajs-1999	128	34	ℛ	ℛ	NOUN
iajs-1999	128	35	ℬ	ℬ	NOUN
iajs-1999	128	36	ℵ	ℵ	NOUN
iajs-1999	128	37	for	for	ADP
iajs-1999	128	38	every	every	DET
iajs-1999	128	39	proper	proper	ADJ
iajs-1999	128	40	submodule	submodule	NOUN
iajs-1999	128	41	ℵ	ℵ	NOUN
iajs-1999	128	42	of	of	ADP
iajs-1999	128	43	ℬ	ℬ	NOUN
iajs-1999	128	44	,	,	PUNCT
iajs-1999	128	45	see	see	VERB
iajs-1999	128	46	[	[	X
iajs-1999	128	47	5	5	NUM
iajs-1999	128	48	]	]	PUNCT
iajs-1999	128	49	.	.	PUNCT
iajs-1999	129	1	equivalently	equivalently	ADV
iajs-1999	129	2	ℬ	ℬ	PROPN
iajs-1999	129	3	is	be	AUX
iajs-1999	129	4	coprime	coprime	NOUN
iajs-1999	129	5	module	module	NOUN
iajs-1999	129	6	if	if	SCONJ
iajs-1999	129	7	and	and	CCONJ
iajs-1999	129	8	only	only	ADV
iajs-1999	129	9	if	if	SCONJ
iajs-1999	129	10	ℬ	ℬ	NOUN
iajs-1999	129	11	is	be	AUX
iajs-1999	129	12	second	second	ADJ
iajs-1999	129	13	module	module	NOUN
iajs-1999	129	14	,	,	PUNCT
iajs-1999	129	15	see	see	VERB
iajs-1999	129	16	[	[	X
iajs-1999	129	17	6	6	NUM
iajs-1999	129	18	,	,	PUNCT
iajs-1999	129	19	th.(2.1.6	th.(2.1.6	NOUN
iajs-1999	129	20	)	)	PUNCT
iajs-1999	129	21	]	]	PUNCT
iajs-1999	129	22	.	.	PUNCT
iajs-1999	130	1	mathematics	mathematic	NOUN
iajs-1999	130	2	|	|	ADV
iajs-1999	130	3	106	106	NUM
iajs-1999	130	4	ibn	ibn	PROPN
iajs-1999	130	5	al	al	PROPN
iajs-1999	130	6	-	-	PUNCT
iajs-1999	130	7	haitham	haitham	PROPN
iajs-1999	130	8	jour	jour	X
iajs-1999	130	9	.	.	PROPN
iajs-1999	130	10	for	for	ADP
iajs-1999	130	11	pure	pure	ADJ
iajs-1999	130	12	&	&	CCONJ
iajs-1999	130	13	appl	appl	PROPN
iajs-1999	130	14	.	.	PUNCT
iajs-1999	131	1	sci	sci	PROPN
iajs-1999	131	2	.	.	PROPN
iajs-1999	131	3	ihjpas	ihjpa	VERB
iajs-1999	131	4	https://doi.org/10.30526/	https://doi.org/10.30526/	PROPN
iajs-1999	131	5	31.3.1999	31.3.1999	NUM
iajs-1999	131	6	vol	vol	NOUN
iajs-1999	131	7	.	.	PUNCT
iajs-1999	132	1	31	31	NUM
iajs-1999	133	1	(	(	PUNCT
iajs-1999	133	2	3	3	NUM
iajs-1999	133	3	)	)	PUNCT
iajs-1999	133	4	2018	2018	NUM
iajs-1999	133	5	a	a	DET
iajs-1999	133	6	submodule	submodule	NOUN
iajs-1999	133	7	ℵ	ℵ	NOUN
iajs-1999	133	8	of	of	ADP
iajs-1999	133	9	an	an	DET
iajs-1999	133	10	ℛ-module	ℛ-module	PROPN
iajs-1999	133	11	ℬ	ℬ	NOUN
iajs-1999	133	12	is	be	AUX
iajs-1999	133	13	rendering	render	VERB
iajs-1999	133	14	irreducible	irreducible	ADJ
iajs-1999	133	15	if	if	SCONJ
iajs-1999	133	16	ℵ	ℵ	NOUN
iajs-1999	133	17	can	can	AUX
iajs-1999	133	18	not	not	PART
iajs-1999	133	19	be	be	AUX
iajs-1999	133	20	expressed	express	VERB
iajs-1999	133	21	as	as	ADP
iajs-1999	133	22	a	a	DET
iajs-1999	133	23	finite	finite	ADJ
iajs-1999	133	24	intersection	intersection	NOUN
iajs-1999	133	25	of	of	ADP
iajs-1999	133	26	proper	proper	ADJ
iajs-1999	133	27	divisors	divisor	NOUN
iajs-1999	133	28	of	of	ADP
iajs-1999	133	29	ℵ	ℵ	NOUN
iajs-1999	133	30	,	,	PUNCT
iajs-1999	133	31	see	see	VERB
iajs-1999	133	32	[	[	X
iajs-1999	133	33	4	4	NUM
iajs-1999	133	34	]	]	PUNCT
iajs-1999	133	35	.	.	PUNCT
iajs-1999	134	1	proposition	proposition	NOUN
iajs-1999	134	2	(	(	PUNCT
iajs-1999	134	3	14):let	14):let	NUM
iajs-1999	134	4	ℬ	ℬ	PROPN
iajs-1999	134	5	be	be	VERB
iajs-1999	134	6	a	a	DET
iajs-1999	134	7	coprime	coprime	NOUN
iajs-1999	134	8	module	module	NOUN
iajs-1999	134	9	,	,	PUNCT
iajs-1999	134	10	let	let	VERB
iajs-1999	134	11	n	n	PRON
iajs-1999	134	12	be	be	AUX
iajs-1999	134	13	a	a	DET
iajs-1999	134	14	submodule	submodule	NOUN
iajs-1999	134	15	of	of	ADP
iajs-1999	134	16	ℬ	ℬ	NOUN
iajs-1999	134	17	such	such	ADJ
iajs-1999	134	18	that	that	DET
iajs-1999	134	19	ℵ	ℵ	NOUN
iajs-1999	134	20	is	be	AUX
iajs-1999	134	21	irreducible	irreducible	ADJ
iajs-1999	134	22	.	.	PUNCT
iajs-1999	135	1	if	if	SCONJ
iajs-1999	135	2	ℵ	ℵ	NOUN
iajs-1999	135	3	is	be	AUX
iajs-1999	135	4	semiprime	semiprime	NOUN
iajs-1999	135	5	,	,	PUNCT
iajs-1999	135	6	then	then	ADV
iajs-1999	135	7	ℵ	ℵ	NOUN
iajs-1999	135	8	is	be	AUX
iajs-1999	135	9	second	second	ADJ
iajs-1999	135	10	and	and	CCONJ
iajs-1999	135	11	hence	hence	ADV
iajs-1999	135	12	semisecond	semisecond	ADJ
iajs-1999	135	13	.	.	PUNCT
iajs-1999	136	1	proof	proof	NOUN
iajs-1999	136	2	:	:	PUNCT
iajs-1999	136	3	let	let	VERB
iajs-1999	136	4	ℵ	ℵ	PART
iajs-1999	136	5	be	be	AUX
iajs-1999	136	6	a	a	DET
iajs-1999	136	7	semiprime	semiprime	NOUN
iajs-1999	136	8	ℛ-submodule	ℛ-submodule	PROPN
iajs-1999	136	9	,	,	PUNCT
iajs-1999	136	10	since	since	SCONJ
iajs-1999	136	11	ℵ	ℵ	NOUN
iajs-1999	136	12	is	be	AUX
iajs-1999	136	13	irreducible	irreducible	ADJ
iajs-1999	136	14	,	,	PUNCT
iajs-1999	136	15	then	then	ADV
iajs-1999	136	16	by	by	ADP
iajs-1999	136	17	[	[	PUNCT
iajs-1999	136	18	3,prop.(1	3,prop.(1	NUM
iajs-1999	136	19	-	-	SYM
iajs-1999	136	20	10	10	NUM
iajs-1999	136	21	)	)	PUNCT
iajs-1999	136	22	]	]	PUNCT
iajs-1999	136	23	ℵ	ℵ	NOUN
iajs-1999	136	24	is	be	AUX
iajs-1999	136	25	prime	prime	ADJ
iajs-1999	136	26	,	,	PUNCT
iajs-1999	136	27	but	but	CCONJ
iajs-1999	136	28	ℬ	ℬ	NOUN
iajs-1999	136	29	is	be	AUX
iajs-1999	136	30	coprime	coprime	NOUN
iajs-1999	136	31	module	module	NOUN
iajs-1999	136	32	,	,	PUNCT
iajs-1999	136	33	then	then	ADV
iajs-1999	136	34	by	by	ADP
iajs-1999	136	35	[	[	X
iajs-1999	136	36	6,prop(2.4.7	6,prop(2.4.7	NUM
iajs-1999	136	37	)	)	PUNCT
iajs-1999	136	38	]	]	PUNCT
iajs-1999	136	39	ℵ	ℵ	PROPN
iajs-1999	136	40	is	be	AUX
iajs-1999	136	41	second	second	ADJ
iajs-1999	136	42	,	,	PUNCT
iajs-1999	136	43	hence	hence	ADV
iajs-1999	136	44	ℵ	ℵ	NOUN
iajs-1999	136	45	is	be	AUX
iajs-1999	136	46	semisecond	semisecond	ADJ
iajs-1999	136	47	.	.	PUNCT
iajs-1999	137	1	corollary	corollary	ADJ
iajs-1999	137	2	(	(	PUNCT
iajs-1999	137	3	15):let	15):let	NOUN
iajs-1999	137	4	ℬ	ℬ	NOUN
iajs-1999	137	5	be	be	VERB
iajs-1999	137	6	a	a	DET
iajs-1999	137	7	prime	prime	ADJ
iajs-1999	137	8	module	module	NOUN
iajs-1999	137	9	over	over	ADP
iajs-1999	137	10	regular	regular	ADJ
iajs-1999	137	11	ring	ring	NOUN
iajs-1999	137	12	ℛ	ℛ	PROPN
iajs-1999	137	13	(	(	PUNCT
iajs-1999	137	14	in	in	ADP
iajs-1999	137	15	sense	sense	NOUN
iajs-1999	137	16	of	of	ADP
iajs-1999	137	17	von	von	PROPN
iajs-1999	137	18	neuman	neuman	PROPN
iajs-1999	137	19	)	)	PUNCT
iajs-1999	137	20	,	,	PUNCT
iajs-1999	137	21	let	let	VERB
iajs-1999	137	22	ℵ	ℵ	NOUN
iajs-1999	137	23	be	be	AUX
iajs-1999	137	24	a	a	DET
iajs-1999	137	25	submodule	submodule	NOUN
iajs-1999	137	26	of	of	ADP
iajs-1999	137	27	ℬ	ℬ	NOUN
iajs-1999	137	28	such	such	ADJ
iajs-1999	137	29	that	that	DET
iajs-1999	137	30	ℵ	ℵ	NOUN
iajs-1999	137	31	is	be	AUX
iajs-1999	137	32	irreducible.then	irreducible.then	PRON
iajs-1999	137	33	n	n	ADJ
iajs-1999	137	34	is	be	AUX
iajs-1999	137	35	semisecond	semisecond	ADJ
iajs-1999	137	36	if	if	SCONJ
iajs-1999	137	37	and	and	CCONJ
iajs-1999	137	38	only	only	ADV
iajs-1999	137	39	if	if	SCONJ
iajs-1999	137	40	ℵ	ℵ	NOUN
iajs-1999	137	41	is	be	AUX
iajs-1999	137	42	semiprime	semiprime	NOUN
iajs-1999	137	43	.	.	PUNCT
iajs-1999	138	1	proof:(⟹	proof:(⟹	NOUN
iajs-1999	138	2	)	)	PUNCT
iajs-1999	138	3	since	since	SCONJ
iajs-1999	138	4	ℬ	ℬ	NOUN
iajs-1999	138	5	is	be	AUX
iajs-1999	138	6	prime	prime	ADJ
iajs-1999	138	7	,	,	PUNCT
iajs-1999	138	8	so	so	CCONJ
iajs-1999	138	9	it	it	PRON
iajs-1999	138	10	is	be	AUX
iajs-1999	138	11	semiprime	semiprime	NOUN
iajs-1999	138	12	.	.	PUNCT
iajs-1999	139	1	thus	thus	ADV
iajs-1999	139	2	we	we	PRON
iajs-1999	139	3	have	have	VERB
iajs-1999	139	4	the	the	DET
iajs-1999	139	5	result	result	NOUN
iajs-1999	139	6	by	by	ADP
iajs-1999	139	7	proposition	proposition	NOUN
iajs-1999	139	8	(	(	PUNCT
iajs-1999	139	9	2.13	2.13	NUM
iajs-1999	139	10	)	)	PUNCT
iajs-1999	139	11	.	.	PUNCT
iajs-1999	140	1	(	(	PUNCT
iajs-1999	140	2	⟸	⟸	ADJ
iajs-1999	140	3	)	)	PUNCT
iajs-1999	140	4	since	since	SCONJ
iajs-1999	140	5	ℬ	ℬ	NOUN
iajs-1999	140	6	is	be	AUX
iajs-1999	140	7	prime	prime	ADJ
iajs-1999	140	8	module	module	NOUN
iajs-1999	140	9	over	over	ADP
iajs-1999	140	10	regular	regular	ADJ
iajs-1999	140	11	ring	ring	NOUN
iajs-1999	140	12	,	,	PUNCT
iajs-1999	140	13	then	then	ADV
iajs-1999	140	14	by	by	ADP
iajs-1999	140	15	[	[	X
iajs-1999	140	16	6	6	NUM
iajs-1999	140	17	,	,	PUNCT
iajs-1999	140	18	corollary	corollary	ADJ
iajs-1999	140	19	(	(	PUNCT
iajs-1999	140	20	2.4.3	2.4.3	NUM
iajs-1999	140	21	)	)	PUNCT
iajs-1999	140	22	]	]	PUNCT
iajs-1999	140	23	ℬ	ℬ	NOUN
iajs-1999	140	24	is	be	AUX
iajs-1999	140	25	coprime	coprime	ADJ
iajs-1999	140	26	,	,	PUNCT
iajs-1999	140	27	hence	hence	ADV
iajs-1999	140	28	we	we	PRON
iajs-1999	140	29	have	have	VERB
iajs-1999	140	30	the	the	DET
iajs-1999	140	31	result	result	NOUN
iajs-1999	140	32	by	by	ADP
iajs-1999	140	33	proposition	proposition	NOUN
iajs-1999	140	34	(	(	PUNCT
iajs-1999	140	35	2.14	2.14	NUM
iajs-1999	140	36	)	)	PUNCT
iajs-1999	140	37	.	.	PUNCT
iajs-1999	141	1	reminiscence	reminiscence	VERB
iajs-1999	141	2	that	that	SCONJ
iajs-1999	141	3	a	a	DET
iajs-1999	141	4	submodule	submodule	NOUN
iajs-1999	141	5	ℵ	ℵ	NOUN
iajs-1999	141	6	of	of	ADP
iajs-1999	141	7	a	a	DET
iajs-1999	141	8	module	module	NOUN
iajs-1999	141	9	ℬ	ℬ	NOUN
iajs-1999	141	10	is	be	AUX
iajs-1999	141	11	rendering	render	VERB
iajs-1999	141	12	secondary	secondary	ADJ
iajs-1999	141	13	(	(	PUNCT
iajs-1999	141	14	dual	dual	ADJ
iajs-1999	141	15	notion	notion	NOUN
iajs-1999	141	16	of	of	ADP
iajs-1999	141	17	primary	primary	ADJ
iajs-1999	141	18	module	module	NOUN
iajs-1999	141	19	)	)	PUNCT
iajs-1999	141	20	if	if	SCONJ
iajs-1999	141	21	for	for	ADP
iajs-1999	141	22	each	each	DET
iajs-1999	141	23	r∈r	r∈r	NOUN
iajs-1999	141	24	,	,	PUNCT
iajs-1999	141	25	the	the	DET
iajs-1999	141	26	homothety	homothety	NOUN
iajs-1999	141	27	r	r	NOUN
iajs-1999	141	28	*	*	NOUN
iajs-1999	141	29	on	on	ADP
iajs-1999	141	30	ℵ	ℵ	PRON
iajs-1999	141	31	is	be	AUX
iajs-1999	141	32	either	either	CCONJ
iajs-1999	141	33	surjective	surjective	ADJ
iajs-1999	141	34	or	or	CCONJ
iajs-1999	141	35	nilpotent	nilpotent	ADJ
iajs-1999	141	36	,	,	PUNCT
iajs-1999	141	37	where	where	SCONJ
iajs-1999	141	38	r	r	NOUN
iajs-1999	141	39	*	*	NOUN
iajs-1999	141	40	is	be	AUX
iajs-1999	141	41	nilpotent	nilpotent	ADJ
iajs-1999	141	42	if	if	SCONJ
iajs-1999	141	43	there	there	PRON
iajs-1999	141	44	exist	exist	VERB
iajs-1999	141	45	k∈z+	k∈z+	NOUN
iajs-1999	141	46	,	,	PUNCT
iajs-1999	141	47	such	such	ADJ
iajs-1999	141	48	that	that	SCONJ
iajs-1999	141	49	(	(	PUNCT
iajs-1999	141	50	r*)k=0	r*)k=0	PROPN
iajs-1999	141	51	,	,	PUNCT
iajs-1999	141	52	see[7].it	see[7].it	PROPN
iajs-1999	141	53	is	be	AUX
iajs-1999	141	54	obvious	obvious	ADJ
iajs-1999	141	55	that	that	SCONJ
iajs-1999	141	56	every	every	DET
iajs-1999	141	57	second	second	ADJ
iajs-1999	141	58	submodule	submodule	NOUN
iajs-1999	141	59	is	be	AUX
iajs-1999	141	60	secondary	secondary	ADJ
iajs-1999	141	61	,	,	PUNCT
iajs-1999	141	62	but	but	CCONJ
iajs-1999	141	63	the	the	DET
iajs-1999	141	64	opposite	opposite	NOUN
iajs-1999	141	65	is	be	AUX
iajs-1999	141	66	not	not	PART
iajs-1999	141	67	whole	whole	ADJ
iajs-1999	141	68	in	in	ADP
iajs-1999	141	69	public	public	ADJ
iajs-1999	141	70	.	.	PUNCT
iajs-1999	142	1	the	the	DET
iajs-1999	142	2	next	next	ADJ
iajs-1999	142	3	lemma	lemma	PROPN
iajs-1999	142	4	explains	explain	VERB
iajs-1999	142	5	that	that	SCONJ
iajs-1999	142	6	the	the	DET
iajs-1999	142	7	opposite	opposite	NOUN
iajs-1999	142	8	is	be	AUX
iajs-1999	142	9	whole	whole	ADJ
iajs-1999	142	10	under	under	ADP
iajs-1999	142	11	certain	certain	ADJ
iajs-1999	142	12	condition	condition	NOUN
iajs-1999	142	13	.	.	PUNCT
iajs-1999	143	1	lemma	lemma	PROPN
iajs-1999	143	2	(	(	PUNCT
iajs-1999	143	3	16):let	16):let	NUM
iajs-1999	143	4	ℵ	ℵ	NOUN
iajs-1999	143	5	be	be	AUX
iajs-1999	143	6	an	an	DET
iajs-1999	143	7	ℛ-submodule	ℛ-submodule	PROPN
iajs-1999	143	8	such	such	ADJ
iajs-1999	143	9	that	that	SCONJ
iajs-1999	143	10	ℛ	ℛ	ADJ
iajs-1999	143	11	ℵ	ℵ	NOUN
iajs-1999	143	12	is	be	AUX
iajs-1999	143	13	semiprime	semiprime	NOUN
iajs-1999	143	14	ideal	ideal	ADJ
iajs-1999	143	15	.	.	PUNCT
iajs-1999	144	1	if	if	SCONJ
iajs-1999	144	2	ℵ	ℵ	NOUN
iajs-1999	144	3	is	be	AUX
iajs-1999	144	4	secondary	secondary	ADJ
iajs-1999	144	5	,	,	PUNCT
iajs-1999	144	6	then	then	ADV
iajs-1999	144	7	ℵ	ℵ	NOUN
iajs-1999	144	8	is	be	AUX
iajs-1999	144	9	second	second	ADJ
iajs-1999	144	10	submodule	submodule	NOUN
iajs-1999	144	11	and	and	CCONJ
iajs-1999	144	12	hence	hence	ADV
iajs-1999	144	13	semisecond	semisecond	NOUN
iajs-1999	144	14	.	.	PUNCT
iajs-1999	145	1	proof	proof	NOUN
iajs-1999	145	2	:	:	PUNCT
iajs-1999	145	3	since	since	SCONJ
iajs-1999	145	4	ℵ	ℵ	NOUN
iajs-1999	145	5	is	be	AUX
iajs-1999	145	6	secondary	secondary	ADJ
iajs-1999	145	7	,	,	PUNCT
iajs-1999	145	8	then	then	ADV
iajs-1999	145	9	for	for	ADP
iajs-1999	145	10	any	any	DET
iajs-1999	145	11	r∈ℛ	r∈ℛ	NOUN
iajs-1999	145	12	,	,	PUNCT
iajs-1999	145	13	r≠0	r≠0	PROPN
iajs-1999	145	14	,	,	PUNCT
iajs-1999	145	15	rℵ=ℵ	rℵ=ℵ	NOUN
iajs-1999	145	16	or	or	CCONJ
iajs-1999	145	17	rnℵ=0	rnℵ=0	NOUN
iajs-1999	145	18	;	;	PUNCT
iajs-1999	145	19	n∈z+	n∈z+	ADJ
iajs-1999	145	20	.	.	PUNCT
iajs-1999	146	1	if	if	SCONJ
iajs-1999	146	2	rℵ=ℵ	rℵ=ℵ	PROPN
iajs-1999	146	3	,	,	PUNCT
iajs-1999	146	4	then	then	ADV
iajs-1999	146	5	there	there	PRON
iajs-1999	146	6	is	be	VERB
iajs-1999	146	7	nothing	nothing	PRON
iajs-1999	146	8	to	to	PART
iajs-1999	146	9	prove	prove	VERB
iajs-1999	146	10	.	.	PUNCT
iajs-1999	147	1	if	if	SCONJ
iajs-1999	147	2	rnℵ=0	rnℵ=0	NOUN
iajs-1999	147	3	,	,	PUNCT
iajs-1999	147	4	then	then	ADV
iajs-1999	147	5	rn∈	rn∈	PROPN
iajs-1999	147	6	ℛ	ℛ	PROPN
iajs-1999	147	7	ℵ.	ℵ.	NOUN
iajs-1999	147	8	but	but	CCONJ
iajs-1999	147	9	ℛ	ℛ	PROPN
iajs-1999	147	10	ℵ	ℵ	NOUN
iajs-1999	147	11	is	be	AUX
iajs-1999	147	12	semiprime	semiprime	NOUN
iajs-1999	147	13	,	,	PUNCT
iajs-1999	147	14	so	so	ADV
iajs-1999	147	15	r∈	r∈	PROPN
iajs-1999	147	16	ℛ	ℛ	PROPN
iajs-1999	147	17	ℵ.	ℵ.	PROPN
iajs-1999	147	18	thus	thus	ADV
iajs-1999	147	19	rℵ=0	rℵ=0	PROPN
iajs-1999	147	20	and	and	CCONJ
iajs-1999	147	21	hence	hence	ADV
iajs-1999	147	22	ℵ	ℵ	NOUN
iajs-1999	147	23	is	be	AUX
iajs-1999	147	24	second	second	ADJ
iajs-1999	147	25	.	.	PUNCT
iajs-1999	148	1	corollary	corollary	ADJ
iajs-1999	148	2	(	(	PUNCT
iajs-1999	148	3	17	17	NUM
iajs-1999	148	4	):	):	PUNCT
iajs-1999	148	5	let	let	VERB
iajs-1999	148	6	ℵ	ℵ	PART
iajs-1999	148	7	be	be	AUX
iajs-1999	148	8	a	a	DET
iajs-1999	148	9	s.r.m	s.r.m	NOUN
iajs-1999	148	10	.	.	PUNCT
iajs-1999	149	1	ℬ	ℬ	PRON
iajs-1999	149	2	such	such	ADJ
iajs-1999	149	3	that	that	SCONJ
iajs-1999	149	4	ℛ	ℛ	ADJ
iajs-1999	149	5	ℵ	ℵ	NOUN
iajs-1999	149	6	is	be	AUX
iajs-1999	149	7	semiprime	semiprime	NOUN
iajs-1999	149	8	,	,	PUNCT
iajs-1999	149	9	then	then	ADV
iajs-1999	149	10	ℵ	ℵ	NOUN
iajs-1999	149	11	is	be	AUX
iajs-1999	149	12	secondary	secondary	ADJ
iajs-1999	149	13	if	if	SCONJ
iajs-1999	149	14	and	and	CCONJ
iajs-1999	149	15	only	only	ADV
iajs-1999	149	16	if	if	SCONJ
iajs-1999	149	17	ℵ	ℵ	NOUN
iajs-1999	149	18	is	be	AUX
iajs-1999	149	19	second	second	ADJ
iajs-1999	149	20	.	.	PUNCT
iajs-1999	150	1	the	the	DET
iajs-1999	150	2	opposite	opposite	NOUN
iajs-1999	150	3	of	of	ADP
iajs-1999	150	4	corollary	corollary	ADJ
iajs-1999	150	5	(	(	PUNCT
iajs-1999	150	6	17	17	NUM
iajs-1999	150	7	)	)	PUNCT
iajs-1999	150	8	need	need	VERB
iajs-1999	150	9	not	not	PART
iajs-1999	150	10	to	to	PART
iajs-1999	150	11	be	be	AUX
iajs-1999	150	12	whole	whole	ADJ
iajs-1999	150	13	in	in	ADP
iajs-1999	150	14	public	public	NOUN
iajs-1999	150	15	for	for	ADP
iajs-1999	150	16	example	example	NOUN
iajs-1999	150	17	:	:	PUNCT
iajs-1999	150	18	in	in	ADP
iajs-1999	150	19	z8	z8	NOUN
iajs-1999	150	20	as	as	ADP
iajs-1999	150	21	z	z	NOUN
iajs-1999	150	22	-	-	PUNCT
iajs-1999	150	23	module	module	NOUN
iajs-1999	150	24	,	,	PUNCT
iajs-1999	150	25	<	<	X
iajs-1999	150	26	2	2	NUM
iajs-1999	150	27	>	>	X
iajs-1999	150	28	is	be	AUX
iajs-1999	150	29	semisecond	semisecond	ADJ
iajs-1999	150	30	and	and	CCONJ
iajs-1999	150	31	not	not	PART
iajs-1999	150	32	secondary	secondary	ADJ
iajs-1999	150	33	.	.	PUNCT
iajs-1999	151	1	the	the	DET
iajs-1999	151	2	opposite	opposite	NOUN
iajs-1999	151	3	is	be	AUX
iajs-1999	151	4	whole	whole	ADJ
iajs-1999	151	5	under	under	ADP
iajs-1999	151	6	the	the	DET
iajs-1999	151	7	class	class	NOUN
iajs-1999	151	8	of	of	ADP
iajs-1999	151	9	torsion	torsion	NOUN
iajs-1999	151	10	free	free	ADJ
iajs-1999	151	11	module	module	NOUN
iajs-1999	151	12	over	over	ADP
iajs-1999	151	13	regular	regular	ADJ
iajs-1999	151	14	ring	ring	NOUN
iajs-1999	151	15	.	.	PUNCT
iajs-1999	152	1	remark	remark	NOUN
iajs-1999	152	2	(	(	PUNCT
iajs-1999	152	3	18	18	NUM
iajs-1999	152	4	):	):	PUNCT
iajs-1999	152	5	if	if	SCONJ
iajs-1999	152	6	ℵ	ℵ	NOUN
iajs-1999	152	7	is	be	AUX
iajs-1999	152	8	semisecond	semisecond	ADJ
iajs-1999	152	9	submodules	submodule	NOUN
iajs-1999	152	10	of	of	ADP
iajs-1999	152	11	torsion	torsion	NOUN
iajs-1999	152	12	free	free	ADJ
iajs-1999	152	13	module	module	NOUN
iajs-1999	152	14	ℬ	ℬ	NOUN
iajs-1999	152	15	over	over	ADP
iajs-1999	152	16	regular	regular	ADJ
iajs-1999	152	17	ring	ring	NOUN
iajs-1999	152	18	,	,	PUNCT
iajs-1999	152	19	then	then	ADV
iajs-1999	152	20	ℵ	ℵ	NOUN
iajs-1999	152	21	is	be	AUX
iajs-1999	152	22	secondary	secondary	ADJ
iajs-1999	152	23	.	.	PUNCT
iajs-1999	153	1	proof	proof	NOUN
iajs-1999	153	2	:	:	PUNCT
iajs-1999	153	3	the	the	DET
iajs-1999	153	4	proof	proof	NOUN
iajs-1999	153	5	directly	directly	ADV
iajs-1999	153	6	by	by	ADP
iajs-1999	153	7	proposition	proposition	NOUN
iajs-1999	153	8	(	(	PUNCT
iajs-1999	153	9	4	4	NUM
iajs-1999	153	10	)	)	PUNCT
iajs-1999	153	11	.	.	PUNCT
iajs-1999	154	1	corollary	corollary	ADJ
iajs-1999	154	2	(	(	PUNCT
iajs-1999	154	3	19	19	NUM
iajs-1999	154	4	)	)	PUNCT
iajs-1999	154	5	:	:	PUNCT
iajs-1999	154	6	let	let	VERB
iajs-1999	154	7	ℬ	ℬ	PRON
iajs-1999	154	8	be	be	AUX
iajs-1999	154	9	torsion	torsion	NOUN
iajs-1999	154	10	free	free	ADJ
iajs-1999	154	11	over	over	ADP
iajs-1999	154	12	regular	regular	ADJ
iajs-1999	154	13	ring	ring	NOUN
iajs-1999	154	14	,	,	PUNCT
iajs-1999	154	15	let	let	VERB
iajs-1999	154	16	ℵ	ℵ	NOUN
iajs-1999	154	17	be	be	AUX
iajs-1999	154	18	submodule	submodule	NOUN
iajs-1999	154	19	of	of	ADP
iajs-1999	154	20	ℬ	ℬ	NOUN
iajs-1999	154	21	such	such	ADJ
iajs-1999	154	22	that	that	SCONJ
iajs-1999	154	23	ℛ	ℛ	ADJ
iajs-1999	154	24	ℵ	ℵ	NOUN
iajs-1999	154	25	is	be	AUX
iajs-1999	154	26	semiprime	semiprime	NOUN
iajs-1999	154	27	,	,	PUNCT
iajs-1999	154	28	then	then	ADV
iajs-1999	154	29	ℵ	ℵ	NOUN
iajs-1999	154	30	is	be	AUX
iajs-1999	154	31	secondary	secondary	ADJ
iajs-1999	154	32	if	if	SCONJ
iajs-1999	154	33	and	and	CCONJ
iajs-1999	154	34	only	only	ADV
iajs-1999	154	35	if	if	SCONJ
iajs-1999	154	36	ℵ	ℵ	NOUN
iajs-1999	154	37	is	be	AUX
iajs-1999	154	38	semisecond	semisecond	ADJ
iajs-1999	154	39	.	.	PUNCT
iajs-1999	155	1	now	now	ADV
iajs-1999	155	2	,	,	PUNCT
iajs-1999	155	3	we	we	PRON
iajs-1999	155	4	turn	turn	VERB
iajs-1999	155	5	our	our	PRON
iajs-1999	155	6	attention	attention	NOUN
iajs-1999	155	7	to	to	ADP
iajs-1999	155	8	the	the	DET
iajs-1999	155	9	localization	localization	NOUN
iajs-1999	155	10	of	of	ADP
iajs-1999	155	11	semisecond	semisecond	NOUN
iajs-1999	155	12	.	.	PUNCT
iajs-1999	156	1	proposition	proposition	NOUN
iajs-1999	156	2	(	(	PUNCT
iajs-1999	156	3	20):let	20):let	NUM
iajs-1999	156	4	ℵ	ℵ	NOUN
iajs-1999	156	5	be	be	AUX
iajs-1999	156	6	a	a	DET
iajs-1999	156	7	semisecond	semisecond	ADJ
iajs-1999	156	8	submodule	submodule	NOUN
iajs-1999	156	9	of	of	ADP
iajs-1999	156	10	an	an	DET
iajs-1999	156	11	ℛ-module	ℛ-module	PROPN
iajs-1999	156	12	ℬ	ℬ	PROPN
iajs-1999	156	13	,	,	PUNCT
iajs-1999	156	14	then	then	ADV
iajs-1999	156	15	ℵs	ℵs	NOUN
iajs-1999	156	16	is	be	AUX
iajs-1999	156	17	semisecond	semisecond	ADJ
iajs-1999	156	18	ℛs	ℛs	PROPN
iajs-1999	156	19	-	-	PUNCT
iajs-1999	156	20	submodule	submodule	NOUN
iajs-1999	156	21	of	of	ADP
iajs-1999	156	22	ℬs	ℬs	PROPN
iajs-1999	156	23	,	,	PUNCT
iajs-1999	156	24	s.t	s.t	PROPN
iajs-1999	156	25	.	.	PROPN
iajs-1999	156	26	s	s	PROPN
iajs-1999	156	27	is	be	AUX
iajs-1999	156	28	a	a	DET
iajs-1999	156	29	multiplicatively	multiplicatively	ADV
iajs-1999	156	30	closed	close	VERB
iajs-1999	156	31	subset	subset	NOUN
iajs-1999	156	32	of	of	ADP
iajs-1999	156	33	r.	r.	PROPN
iajs-1999	156	34	proof	proof	NOUN
iajs-1999	156	35	:	:	PUNCT
iajs-1999	156	36	let	let	VERB
iajs-1999	156	37	�	�	PRON
iajs-1999	156	38	̅	̅	VERB
iajs-1999	156	39	�	�	NOUN
iajs-1999	156	40	∈ℛs	∈ℛs	PROPN
iajs-1999	156	41	,	,	PUNCT
iajs-1999	156	42	�	�	NOUN
iajs-1999	156	43	̅	̅	NOUN
iajs-1999	156	44	�	�	NOUN
iajs-1999	156	45	=	=	SYM
iajs-1999	156	46	,	,	PUNCT
iajs-1999	156	47	where	where	SCONJ
iajs-1999	156	48	r∈ℛ	r∈ℛ	NOUN
iajs-1999	156	49	,	,	PUNCT
iajs-1999	156	50	s∈s	s∈s	NOUN
iajs-1999	156	51	.	.	PUNCT
iajs-1999	156	52	assume	assume	VERB
iajs-1999	156	53	that	that	SCONJ
iajs-1999	156	54	(	(	PUNCT
iajs-1999	156	55	�	�	NOUN
iajs-1999	156	56	̅	̅	NOUN
iajs-1999	156	57	�	�	NOUN
iajs-1999	156	58	)2∉	)2∉	ADJ
iajs-1999	156	59	ℵs	ℵs	PROPN
iajs-1999	156	60	.to	.to	PUNCT
iajs-1999	156	61	prove	prove	PROPN
iajs-1999	156	62	(	(	PUNCT
iajs-1999	156	63	�	�	NOUN
iajs-1999	156	64	̅	̅	NOUN
iajs-1999	156	65	�	�	PROPN
iajs-1999	156	66	2ℵs=	2ℵs=	NUM
iajs-1999	156	67	�	�	NOUN
iajs-1999	156	68	̅	̅	NOUN
iajs-1999	156	69	�	�	NOUN
iajs-1999	156	70	ℵs	ℵs	NOUN
iajs-1999	156	71	.	.	PUNCT
iajs-1999	157	1	since	since	SCONJ
iajs-1999	157	2	(	(	PUNCT
iajs-1999	157	3	�	�	NOUN
iajs-1999	157	4	̅	̅	NOUN
iajs-1999	157	5	�	�	NOUN
iajs-1999	157	6	)2∉	)2∉	ADJ
iajs-1999	157	7	ℵs	ℵs	PROPN
iajs-1999	157	8	,	,	PUNCT
iajs-1999	157	9	then	then	ADV
iajs-1999	157	10	(	(	PUNCT
iajs-1999	157	11	2	2	X
iajs-1999	157	12	.	.	PUNCT
iajs-1999	157	13	(	(	PUNCT
iajs-1999	157	14	)	)	PUNCT
iajs-1999	157	15	≠	≠	PROPN
iajs-1999	157	16	for	for	ADP
iajs-1999	157	17	some	some	DET
iajs-1999	157	18	n∈ℵ	n∈ℵ	ADJ
iajs-1999	157	19	,	,	PUNCT
iajs-1999	157	20	a∈s	a∈s	PROPN
iajs-1999	157	21	.	.	PUNCT
iajs-1999	158	1	(	(	PUNCT
iajs-1999	158	2	)	)	PUNCT
iajs-1999	158	3	≠	≠	PROPN
iajs-1999	158	4	,	,	PUNCT
iajs-1999	158	5	that	that	PRON
iajs-1999	158	6	is	be	AUX
iajs-1999	158	7	for	for	ADP
iajs-1999	158	8	any	any	DET
iajs-1999	158	9	t∈s	t∈s	NOUN
iajs-1999	158	10	,	,	PUNCT
iajs-1999	158	11	r2tn≠0	r2tn≠0	PROPN
iajs-1999	158	12	.	.	PUNCT
iajs-1999	159	1	thus	thus	ADV
iajs-1999	159	2	mathematics	mathematic	NOUN
iajs-1999	159	3	|	|	ADV
iajs-1999	159	4	107	107	NUM
iajs-1999	159	5	ibn	ibn	PROPN
iajs-1999	159	6	al	al	PROPN
iajs-1999	159	7	-	-	PUNCT
iajs-1999	159	8	haitham	haitham	PROPN
iajs-1999	159	9	jour	jour	X
iajs-1999	159	10	.	.	PROPN
iajs-1999	160	1	for	for	ADP
iajs-1999	160	2	pure	pure	ADJ
iajs-1999	160	3	&	&	CCONJ
iajs-1999	160	4	appl	appl	PROPN
iajs-1999	160	5	.	.	PUNCT
iajs-1999	161	1	sci	sci	PROPN
iajs-1999	161	2	.	.	PROPN
iajs-1999	161	3	ihjpas	ihjpa	VERB
iajs-1999	161	4	https://doi.org/10.30526/	https://doi.org/10.30526/	PROPN
iajs-1999	161	5	31.3.1999	31.3.1999	NUM
iajs-1999	161	6	vol	vol	NOUN
iajs-1999	161	7	.	.	PUNCT
iajs-1999	162	1	31	31	NUM
iajs-1999	163	1	(	(	PUNCT
iajs-1999	163	2	3	3	NUM
iajs-1999	163	3	)	)	PUNCT
iajs-1999	163	4	2018	2018	NUM
iajs-1999	163	5	r2t∉	r2t∉	NOUN
iajs-1999	163	6	ℛ	ℛ	PROPN
iajs-1999	163	7	ℵ	ℵ	NOUN
iajs-1999	163	8	which	which	PRON
iajs-1999	163	9	implies	imply	VERB
iajs-1999	163	10	that	that	SCONJ
iajs-1999	163	11	r2∉	r2∉	NOUN
iajs-1999	163	12	ℛ	ℛ	ADJ
iajs-1999	163	13	ℵ	ℵ	NOUN
iajs-1999	163	14	,	,	PUNCT
iajs-1999	163	15	so	so	SCONJ
iajs-1999	163	16	r2ℵ≠0.but	r2ℵ≠0.but	NOUN
iajs-1999	163	17	ℵ	ℵ	NOUN
iajs-1999	163	18	is	be	AUX
iajs-1999	163	19	semisecond	semisecond	ADJ
iajs-1999	163	20	,	,	PUNCT
iajs-1999	163	21	hence	hence	ADV
iajs-1999	163	22	r2ℵ=rℵ.	r2ℵ=rℵ.	VERB
iajs-1999	163	23	therefore	therefore	ADV
iajs-1999	163	24	(	(	PUNCT
iajs-1999	163	25	r2ℵ)s	r2ℵ)s	NOUN
iajs-1999	163	26	=(	=(	NOUN
iajs-1999	163	27	rℵ)s	rℵ)s	NOUN
iajs-1999	163	28	.	.	PUNCT
iajs-1999	164	1	thus	thus	ADV
iajs-1999	164	2	(	(	PUNCT
iajs-1999	164	3	r2)s	r2)s	VERB
iajs-1999	164	4	ℵs	ℵs	NOUN
iajs-1999	164	5	=	=	PUNCT
iajs-1999	164	6	(	(	PUNCT
iajs-1999	164	7	r)sℵs	r)sℵs	NOUN
iajs-1999	164	8	and	and	CCONJ
iajs-1999	164	9	so	so	ADV
iajs-1999	164	10	(	(	PUNCT
iajs-1999	164	11	�	�	NOUN
iajs-1999	164	12	̅	̅	NOUN
iajs-1999	164	13	�	�	PROPN
iajs-1999	164	14	)2ℵs=	)2ℵs=	NOUN
iajs-1999	164	15	�	�	PROPN
iajs-1999	164	16	̅	̅	NOUN
iajs-1999	164	17	�	�	PROPN
iajs-1999	164	18	ℵs	ℵs	NOUN
iajs-1999	164	19	.	.	NOUN
iajs-1999	164	20	corollary	corollary	NOUN
iajs-1999	164	21	(	(	PUNCT
iajs-1999	164	22	21):let	21):let	PROPN
iajs-1999	164	23	ℵ	ℵ	NOUN
iajs-1999	164	24	be	be	AUX
iajs-1999	164	25	a	a	DET
iajs-1999	164	26	semisecond	semisecond	ADJ
iajs-1999	164	27	submodule	submodule	NOUN
iajs-1999	164	28	of	of	ADP
iajs-1999	164	29	an	an	DET
iajs-1999	164	30	r	r	NOUN
iajs-1999	164	31	-	-	PUNCT
iajs-1999	164	32	module	module	NOUN
iajs-1999	164	33	ℬ	ℬ	NOUN
iajs-1999	164	34	,	,	PUNCT
iajs-1999	164	35	then	then	ADV
iajs-1999	164	36	ℵp	ℵp	PROPN
iajs-1999	164	37	is	be	AUX
iajs-1999	164	38	semisecond	semisecond	ADJ
iajs-1999	164	39	ℛp	ℛp	PROPN
iajs-1999	164	40	-	-	NOUN
iajs-1999	164	41	submodule	submodule	NOUN
iajs-1999	164	42	of	of	ADP
iajs-1999	164	43	ℬp	ℬp	PROPN
iajs-1999	164	44	for	for	ADP
iajs-1999	164	45	any	any	DET
iajs-1999	164	46	prome	prome	NOUN
iajs-1999	164	47	ideal	ideal	NOUN
iajs-1999	164	48	p	p	NOUN
iajs-1999	164	49	of	of	ADP
iajs-1999	164	50	ℛ.	ℛ.	PROPN
iajs-1999	164	51	3	3	NUM
iajs-1999	164	52	.	.	NOUN
iajs-1999	164	53	semisecond	semisecond	PROPN
iajs-1999	164	54	modules	modules	PROPN
iajs-1999	164	55	yass	yass	PROPN
iajs-1999	164	56	in	in	ADP
iajs-1999	164	57	[	[	X
iajs-1999	164	58	1	1	NUM
iajs-1999	164	59	]	]	PUNCT
iajs-1999	164	60	introduced	introduce	VERB
iajs-1999	164	61	the	the	DET
iajs-1999	164	62	notion	notion	NOUN
iajs-1999	164	63	of	of	ADP
iajs-1999	164	64	second	second	ADJ
iajs-1999	164	65	module	module	NOUN
iajs-1999	164	66	(	(	PUNCT
iajs-1999	164	67	where	where	SCONJ
iajs-1999	164	68	ℬ	ℬ	NOUN
iajs-1999	164	69	is	be	AUX
iajs-1999	164	70	second	second	ADJ
iajs-1999	164	71	if	if	SCONJ
iajs-1999	164	72	for	for	ADP
iajs-1999	164	73	every	every	DET
iajs-1999	164	74	r∈ℛ	r∈ℛ	NOUN
iajs-1999	164	75	,	,	PUNCT
iajs-1999	164	76	r≠0	r≠0	NOUN
iajs-1999	164	77	,	,	PUNCT
iajs-1999	164	78	rℬ=0	rℬ=0	NOUN
iajs-1999	164	79	or	or	CCONJ
iajs-1999	164	80	rℬ=ℬ	rℬ=ℬ	NUM
iajs-1999	164	81	)	)	PUNCT
iajs-1999	164	82	.	.	PUNCT
iajs-1999	165	1	equivalently	equivalently	ADV
iajs-1999	165	2	ℬ	ℬ	PROPN
iajs-1999	165	3	is	be	AUX
iajs-1999	165	4	second	second	ADJ
iajs-1999	165	5	module	module	NOUN
iajs-1999	165	6	if	if	SCONJ
iajs-1999	165	7	ℬ	ℬ	NOUN
iajs-1999	165	8	is	be	AUX
iajs-1999	165	9	second	second	ADJ
iajs-1999	165	10	submodule	submodule	NOUN
iajs-1999	165	11	of	of	ADP
iajs-1999	165	12	ℬ.	ℬ.	PROPN
iajs-1999	165	13	in	in	ADP
iajs-1999	165	14	this	this	DET
iajs-1999	165	15	section	section	NOUN
iajs-1999	165	16	we	we	PRON
iajs-1999	165	17	introduce	introduce	VERB
iajs-1999	165	18	the	the	DET
iajs-1999	165	19	notion	notion	NOUN
iajs-1999	165	20	of	of	ADP
iajs-1999	165	21	semisecond	semisecond	ADJ
iajs-1999	165	22	module	module	NOUN
iajs-1999	165	23	as	as	ADP
iajs-1999	165	24	a	a	DET
iajs-1999	165	25	generalization	generalization	NOUN
iajs-1999	165	26	of	of	ADP
iajs-1999	165	27	second	second	ADJ
iajs-1999	165	28	module	module	NOUN
iajs-1999	165	29	.	.	PUNCT
iajs-1999	166	1	we	we	PRON
iajs-1999	166	2	give	give	VERB
iajs-1999	166	3	some	some	DET
iajs-1999	166	4	properties	property	NOUN
iajs-1999	166	5	of	of	ADP
iajs-1999	166	6	semisecond	semisecond	ADJ
iajs-1999	166	7	module	module	NOUN
iajs-1999	166	8	.	.	PUNCT
iajs-1999	167	1	definition	definition	NOUN
iajs-1999	167	2	(	(	PUNCT
iajs-1999	167	3	22):let	22):let	NOUN
iajs-1999	167	4	ℬ	ℬ	NOUN
iajs-1999	167	5	be	be	VERB
iajs-1999	167	6	an	an	DET
iajs-1999	167	7	r	r	NOUN
iajs-1999	167	8	-	-	PUNCT
iajs-1999	167	9	module	module	NOUN
iajs-1999	167	10	,	,	PUNCT
iajs-1999	167	11	ℬ	ℬ	PRON
iajs-1999	167	12	is	be	AUX
iajs-1999	167	13	rendering	render	VERB
iajs-1999	167	14	semisecond	semisecond	NOUN
iajs-1999	167	15	if	if	SCONJ
iajs-1999	167	16	ℬ	ℬ	NOUN
iajs-1999	167	17	is	be	AUX
iajs-1999	167	18	semisecond	semisecond	ADJ
iajs-1999	167	19	submodule	submodule	NOUN
iajs-1999	167	20	,	,	PUNCT
iajs-1999	167	21	that	that	PRON
iajs-1999	167	22	is	be	AUX
iajs-1999	167	23	for	for	ADP
iajs-1999	167	24	any	any	DET
iajs-1999	167	25	r∈ℛ	r∈ℛ	NOUN
iajs-1999	167	26	,	,	PUNCT
iajs-1999	167	27	r≠0	r≠0	PROPN
iajs-1999	167	28	,	,	PUNCT
iajs-1999	167	29	r2ℬ=rℬ	r2ℬ=rℬ	NOUN
iajs-1999	167	30	or	or	CCONJ
iajs-1999	167	31	r2m=0	r2m=0	NUM
iajs-1999	167	32	.	.	PUNCT
iajs-1999	168	1	remarks	remark	NOUN
iajs-1999	168	2	and	and	CCONJ
iajs-1999	168	3	examples	example	NOUN
iajs-1999	168	4	(	(	PUNCT
iajs-1999	168	5	23	23	NUM
iajs-1999	168	6	)	)	PUNCT
iajs-1999	168	7	(	(	PUNCT
iajs-1999	168	8	1	1	X
iajs-1999	168	9	)	)	PUNCT
iajs-1999	168	10	it	it	PRON
iajs-1999	168	11	is	be	AUX
iajs-1999	168	12	obvious	obvious	ADJ
iajs-1999	168	13	that	that	SCONJ
iajs-1999	168	14	every	every	DET
iajs-1999	168	15	second	second	ADJ
iajs-1999	168	16	module	module	NOUN
iajs-1999	168	17	is	be	AUX
iajs-1999	168	18	semisecond	semisecond	ADJ
iajs-1999	168	19	,	,	PUNCT
iajs-1999	168	20	by	by	ADP
iajs-1999	168	21	remark	remark	NOUN
iajs-1999	168	22	and	and	CCONJ
iajs-1999	168	23	example	example	NOUN
iajs-1999	168	24	(	(	PUNCT
iajs-1999	168	25	2.3.(1)).the	2.3.(1)).the	DET
iajs-1999	168	26	opposite	opposite	NOUN
iajs-1999	168	27	is	be	AUX
iajs-1999	168	28	not	not	PART
iajs-1999	168	29	whole	whole	ADJ
iajs-1999	168	30	in	in	ADP
iajs-1999	168	31	public	public	NOUN
iajs-1999	168	32	for	for	ADP
iajs-1999	168	33	instance	instance	NOUN
iajs-1999	168	34	:	:	PUNCT
iajs-1999	168	35	z4	z4	PROPN
iajs-1999	168	36	as	as	SCONJ
iajs-1999	168	37	z	z	NOUN
iajs-1999	168	38	-	-	PUNCT
iajs-1999	168	39	module	module	NOUN
iajs-1999	168	40	is	be	AUX
iajs-1999	168	41	not	not	PART
iajs-1999	168	42	second	second	ADJ
iajs-1999	168	43	since	since	SCONJ
iajs-1999	168	44	2z4≠z4	2z4≠z4	NUM
iajs-1999	168	45	and	and	CCONJ
iajs-1999	168	46	2z4≠(0	2z4≠(0	NUM
iajs-1999	168	47	)	)	PUNCT
iajs-1999	168	48	but	but	CCONJ
iajs-1999	168	49	z4	z4	PROPN
iajs-1999	168	50	is	be	AUX
iajs-1999	168	51	semisecond	semisecond	ADJ
iajs-1999	168	52	module	module	NOUN
iajs-1999	168	53	.	.	PUNCT
iajs-1999	169	1	(	(	PUNCT
iajs-1999	169	2	2	2	X
iajs-1999	169	3	)	)	PUNCT
iajs-1999	169	4	z	z	NOUN
iajs-1999	169	5	as	as	SCONJ
iajs-1999	169	6	z	z	NOUN
iajs-1999	169	7	-	-	PUNCT
iajs-1999	169	8	module	module	NOUN
iajs-1999	169	9	is	be	AUX
iajs-1999	169	10	not	not	PART
iajs-1999	169	11	semisecond	semisecond	ADJ
iajs-1999	169	12	,	,	PUNCT
iajs-1999	169	13	since	since	SCONJ
iajs-1999	169	14	for	for	ADP
iajs-1999	169	15	any	any	DET
iajs-1999	169	16	r∈ℛ	r∈ℛ	NOUN
iajs-1999	169	17	,	,	PUNCT
iajs-1999	169	18	r≠0	r≠0	NOUN
iajs-1999	169	19	,	,	PUNCT
iajs-1999	169	20	r2z≠(0	r2z≠(0	NOUN
iajs-1999	169	21	)	)	PUNCT
iajs-1999	169	22	and	and	CCONJ
iajs-1999	169	23	r2z≠rz	r2z≠rz	PROPN
iajs-1999	169	24	.	.	PUNCT
iajs-1999	170	1	(	(	PUNCT
iajs-1999	170	2	3	3	X
iajs-1999	170	3	)	)	PUNCT
iajs-1999	170	4	consider	consider	VERB
iajs-1999	170	5	the	the	DET
iajs-1999	170	6	z	z	NOUN
iajs-1999	170	7	-	-	PUNCT
iajs-1999	170	8	module	module	NOUN
iajs-1999	170	9	zp∞	zp∞	PROPN
iajs-1999	170	10	,	,	PUNCT
iajs-1999	170	11	zp∞	zp∞	PROPN
iajs-1999	170	12	=	=	SYM
iajs-1999	170	13	0	0	PROPN
iajs-1999	170	14	,	,	PUNCT
iajs-1999	170	15	that	that	PRON
iajs-1999	170	16	is	be	AUX
iajs-1999	170	17	for	for	ADP
iajs-1999	170	18	all	all	DET
iajs-1999	170	19	r∈z	r∈z	NOUN
iajs-1999	170	20	,	,	PUNCT
iajs-1999	170	21	r≠0	r≠0	NOUN
iajs-1999	170	22	,	,	PUNCT
iajs-1999	170	23	r2zp∞≠(0	r2zp∞≠(0	PROPN
iajs-1999	170	24	)	)	PUNCT
iajs-1999	170	25	.	.	PUNCT
iajs-1999	171	1	but	but	CCONJ
iajs-1999	171	2	zp∞	zp∞	PROPN
iajs-1999	171	3	is	be	AUX
iajs-1999	171	4	divisible	divisible	ADJ
iajs-1999	171	5	z	z	NOUN
iajs-1999	171	6	-	-	PUNCT
iajs-1999	171	7	module	module	NOUN
iajs-1999	171	8	,	,	PUNCT
iajs-1999	171	9	so	so	SCONJ
iajs-1999	171	10	r2zp∞=rzp∞	r2zp∞=rzp∞	NOUN
iajs-1999	171	11	;	;	PUNCT
iajs-1999	171	12	for	for	ADP
iajs-1999	171	13	all	all	DET
iajs-1999	171	14	r∈z	r∈z	NOUN
iajs-1999	171	15	,	,	PUNCT
iajs-1999	171	16	r≠0	r≠0	VERB
iajs-1999	171	17	,	,	PUNCT
iajs-1999	171	18	then	then	ADV
iajs-1999	171	19	zp∞	zp∞	PROPN
iajs-1999	171	20	is	be	AUX
iajs-1999	171	21	semisecond	semisecond	ADJ
iajs-1999	171	22	.	.	PUNCT
iajs-1999	172	1	(	(	PUNCT
iajs-1999	172	2	4	4	X
iajs-1999	172	3	)	)	PUNCT
iajs-1999	172	4	q	q	NOUN
iajs-1999	172	5	as	as	SCONJ
iajs-1999	172	6	z	z	NOUN
iajs-1999	172	7	-	-	PUNCT
iajs-1999	172	8	module	module	NOUN
iajs-1999	172	9	is	be	AUX
iajs-1999	172	10	semisecond	semisecond	ADJ
iajs-1999	172	11	module	module	NOUN
iajs-1999	172	12	.	.	PUNCT
iajs-1999	173	1	(	(	PUNCT
iajs-1999	173	2	5	5	NUM
iajs-1999	173	3	)	)	PUNCT
iajs-1999	173	4	if	if	SCONJ
iajs-1999	173	5	n	n	PRON
iajs-1999	173	6	is	be	AUX
iajs-1999	173	7	a	a	DET
iajs-1999	173	8	prime	prime	ADJ
iajs-1999	173	9	number	number	NOUN
iajs-1999	173	10	,	,	PUNCT
iajs-1999	173	11	then	then	ADV
iajs-1999	173	12	zn	zn	PROPN
iajs-1999	173	13	is	be	AUX
iajs-1999	173	14	semisecond	semisecond	ADJ
iajs-1999	173	15	z	z	NOUN
iajs-1999	173	16	-	-	PUNCT
iajs-1999	173	17	module	module	NOUN
iajs-1999	173	18	,	,	PUNCT
iajs-1999	173	19	but	but	CCONJ
iajs-1999	173	20	the	the	DET
iajs-1999	173	21	opposite	opposite	NOUN
iajs-1999	173	22	is	be	AUX
iajs-1999	173	23	not	not	PART
iajs-1999	173	24	whole	whole	ADJ
iajs-1999	173	25	in	in	ADP
iajs-1999	173	26	public	public	NOUN
iajs-1999	173	27	for	for	ADP
iajs-1999	173	28	example	example	NOUN
iajs-1999	173	29	z6	z6	PROPN
iajs-1999	173	30	is	be	AUX
iajs-1999	173	31	semisecond	semisecond	ADJ
iajs-1999	173	32	but	but	CCONJ
iajs-1999	173	33	6	6	NUM
iajs-1999	173	34	is	be	AUX
iajs-1999	173	35	not	not	PART
iajs-1999	173	36	prime	prime	ADJ
iajs-1999	173	37	.	.	PUNCT
iajs-1999	174	1	(	(	PUNCT
iajs-1999	174	2	6	6	NUM
iajs-1999	174	3	)	)	PUNCT
iajs-1999	174	4	a	a	DET
iajs-1999	174	5	module	module	NOUN
iajs-1999	174	6	ℬ	ℬ	NOUN
iajs-1999	174	7	is	be	AUX
iajs-1999	174	8	semisecond	semisecond	ADJ
iajs-1999	174	9	ℛ-module	ℛ-module	PROPN
iajs-1999	174	10	iff	iff	NOUN
iajs-1999	174	11	ℬ	ℬ	PROPN
iajs-1999	174	12	is	be	AUX
iajs-1999	174	13	semisecond	semisecond	ADJ
iajs-1999	174	14	ℛ/i	ℛ/i	PROPN
iajs-1999	174	15	-	-	PUNCT
iajs-1999	174	16	module	module	NOUN
iajs-1999	174	17	,	,	PUNCT
iajs-1999	174	18	where	where	SCONJ
iajs-1999	174	19	i⊆	i⊆	PROPN
iajs-1999	174	20	ℛ	ℛ	ADJ
iajs-1999	174	21	ℬ.	ℬ.	NOUN
iajs-1999	174	22	proof	proof	NOUN
iajs-1999	174	23	:	:	PUNCT
iajs-1999	174	24	it	it	PRON
iajs-1999	174	25	follows	follow	VERB
iajs-1999	174	26	by	by	ADP
iajs-1999	174	27	proposition	proposition	NOUN
iajs-1999	174	28	(	(	PUNCT
iajs-1999	174	29	7	7	NUM
iajs-1999	174	30	)	)	PUNCT
iajs-1999	174	31	.	.	PUNCT
iajs-1999	175	1	(	(	PUNCT
iajs-1999	175	2	7	7	X
iajs-1999	175	3	)	)	PUNCT
iajs-1999	175	4	a	a	DET
iajs-1999	175	5	module	module	NOUN
iajs-1999	175	6	ℬ	ℬ	NOUN
iajs-1999	175	7	is	be	AUX
iajs-1999	175	8	semisecond	semisecond	ADJ
iajs-1999	175	9	ℛ-module	ℛ-module	PROPN
iajs-1999	175	10	iff	iff	NOUN
iajs-1999	175	11	ℬ	ℬ	PROPN
iajs-1999	175	12	is	be	AUX
iajs-1999	175	13	semisecond	semisecond	ADJ
iajs-1999	175	14	ℛ/	ℛ/	X
iajs-1999	175	15	ℛ	ℛ	PROPN
iajs-1999	175	16	ℬ-module	ℬ-module	PROPN
iajs-1999	175	17	.	.	PUNCT
iajs-1999	176	1	proof	proof	NOUN
iajs-1999	176	2	:	:	PUNCT
iajs-1999	176	3	it	it	PRON
iajs-1999	176	4	follows	follow	VERB
iajs-1999	176	5	by	by	ADP
iajs-1999	176	6	corollary	corollary	ADJ
iajs-1999	176	7	(	(	PUNCT
iajs-1999	176	8	8)	8)	NUM
iajs-1999	176	9	.	.	PUNCT
iajs-1999	177	1	(	(	PUNCT
iajs-1999	177	2	8)let	8)let	NUM
iajs-1999	177	3	f:ℬ⟶	f:ℬ⟶	PROPN
iajs-1999	177	4	ℬ	ℬ	NOUN
iajs-1999	177	5	'	'	PUNCT
iajs-1999	177	6	be	be	AUX
iajs-1999	177	7	an	an	DET
iajs-1999	177	8	r	r	NOUN
iajs-1999	177	9	-	-	PUNCT
iajs-1999	177	10	homomorphism	homomorphism	NOUN
iajs-1999	177	11	,	,	PUNCT
iajs-1999	177	12	if	if	SCONJ
iajs-1999	177	13	ℬ	ℬ	NOUN
iajs-1999	177	14	is	be	AUX
iajs-1999	177	15	semisecond	semisecond	ADJ
iajs-1999	177	16	module	module	NOUN
iajs-1999	177	17	,	,	PUNCT
iajs-1999	177	18	then	then	ADV
iajs-1999	177	19	f(ℬ	f(ℬ	NOUN
iajs-1999	177	20	)	)	PUNCT
iajs-1999	177	21	is	be	AUX
iajs-1999	177	22	semisecond	semisecond	ADJ
iajs-1999	177	23	ℬ'-module	ℬ'-module	NOUN
iajs-1999	177	24	.	.	PUNCT
iajs-1999	178	1	(	(	PUNCT
iajs-1999	178	2	9	9	X
iajs-1999	178	3	)	)	PUNCT
iajs-1999	178	4	let	let	VERB
iajs-1999	178	5	ℬ	ℬ	PRON
iajs-1999	178	6	be	be	AUX
iajs-1999	178	7	a	a	DET
iajs-1999	178	8	semisecond	semisecond	ADJ
iajs-1999	178	9	ℛ-module	ℛ-module	PROPN
iajs-1999	178	10	,	,	PUNCT
iajs-1999	178	11	then	then	ADV
iajs-1999	178	12	ℬs	ℬs	PROPN
iajs-1999	178	13	is	be	AUX
iajs-1999	178	14	semisecond	semisecond	ADJ
iajs-1999	178	15	ℛs	ℛs	NOUN
iajs-1999	178	16	-	-	PUNCT
iajs-1999	178	17	module	module	NOUN
iajs-1999	178	18	,	,	PUNCT
iajs-1999	178	19	s.t	s.t	PROPN
iajs-1999	178	20	.	.	PROPN
iajs-1999	178	21	s	s	PROPN
iajs-1999	178	22	is	be	AUX
iajs-1999	178	23	a	a	DET
iajs-1999	178	24	multiplicatively	multiplicatively	ADV
iajs-1999	178	25	closed	close	VERB
iajs-1999	178	26	subset	subset	NOUN
iajs-1999	178	27	of	of	ADP
iajs-1999	178	28	ℛ.	ℛ.	PROPN
iajs-1999	178	29	proof	proof	NOUN
iajs-1999	178	30	:	:	PUNCT
iajs-1999	178	31	it	it	PRON
iajs-1999	178	32	holds	hold	VERB
iajs-1999	178	33	by	by	ADP
iajs-1999	178	34	proposition	proposition	NOUN
iajs-1999	178	35	(	(	PUNCT
iajs-1999	178	36	20	20	NUM
iajs-1999	178	37	)	)	PUNCT
iajs-1999	178	38	.	.	PUNCT
iajs-1999	179	1	(	(	PUNCT
iajs-1999	179	2	10	10	NUM
iajs-1999	179	3	)	)	PUNCT
iajs-1999	179	4	let	let	VERB
iajs-1999	179	5	ℬ	ℬ	PRON
iajs-1999	179	6	be	be	AUX
iajs-1999	179	7	a	a	DET
iajs-1999	179	8	semisecond	semisecond	ADJ
iajs-1999	179	9	ℛ-module	ℛ-module	PROPN
iajs-1999	179	10	,	,	PUNCT
iajs-1999	179	11	then	then	ADV
iajs-1999	179	12	ℬp	ℬp	PROPN
iajs-1999	179	13	is	be	AUX
iajs-1999	179	14	a	a	DET
iajs-1999	179	15	semisecond	semisecond	ADJ
iajs-1999	179	16	ℛp	ℛp	NOUN
iajs-1999	179	17	-	-	NOUN
iajs-1999	179	18	module	module	NOUN
iajs-1999	179	19	for	for	ADP
iajs-1999	179	20	any	any	DET
iajs-1999	179	21	prime	prime	ADJ
iajs-1999	179	22	ideal	ideal	NOUN
iajs-1999	179	23	p	p	NOUN
iajs-1999	179	24	of	of	ADP
iajs-1999	179	25	ℛ.	ℛ.	PROPN
iajs-1999	179	26	proof	proof	NOUN
iajs-1999	179	27	:	:	PUNCT
iajs-1999	179	28	it	it	PRON
iajs-1999	179	29	follows	follow	VERB
iajs-1999	179	30	by	by	ADP
iajs-1999	179	31	corollary	corollary	ADJ
iajs-1999	179	32	(	(	PUNCT
iajs-1999	179	33	21	21	NUM
iajs-1999	179	34	)	)	PUNCT
iajs-1999	179	35	.	.	PUNCT
iajs-1999	180	1	mathematics	mathematic	NOUN
iajs-1999	180	2	|	|	ADV
iajs-1999	180	3	108	108	NUM
iajs-1999	180	4	ibn	ibn	PROPN
iajs-1999	180	5	al	al	PROPN
iajs-1999	180	6	-	-	PUNCT
iajs-1999	180	7	haitham	haitham	PROPN
iajs-1999	180	8	jour	jour	X
iajs-1999	180	9	.	.	PROPN
iajs-1999	180	10	for	for	ADP
iajs-1999	180	11	pure	pure	ADJ
iajs-1999	180	12	&	&	CCONJ
iajs-1999	180	13	appl	appl	PROPN
iajs-1999	180	14	.	.	PUNCT
iajs-1999	181	1	sci	sci	PROPN
iajs-1999	181	2	.	.	PROPN
iajs-1999	181	3	ihjpas	ihjpa	VERB
iajs-1999	181	4	https://doi.org/10.30526/	https://doi.org/10.30526/	PROPN
iajs-1999	181	5	31.3.1999	31.3.1999	NUM
iajs-1999	181	6	vol	vol	NOUN
iajs-1999	181	7	.	.	PUNCT
iajs-1999	182	1	31	31	NUM
iajs-1999	183	1	(	(	PUNCT
iajs-1999	183	2	3	3	NUM
iajs-1999	183	3	)	)	SYM
iajs-1999	183	4	2018	2018	NUM
iajs-1999	183	5	references	reference	NOUN
iajs-1999	183	6	1	1	NUM
iajs-1999	183	7	.	.	PUNCT
iajs-1999	183	8	yassemi	yassemi	NOUN
iajs-1999	183	9	,	,	PUNCT
iajs-1999	183	10	s.	s.	PROPN
iajs-1999	183	11	the	the	DET
iajs-1999	183	12	dual	dual	ADJ
iajs-1999	183	13	notion	notion	NOUN
iajs-1999	183	14	of	of	ADP
iajs-1999	183	15	prime	prime	ADJ
iajs-1999	183	16	submodules	submodule	NOUN
iajs-1999	183	17	.	.	PUNCT
iajs-1999	184	1	arch	arch	NOUN
iajs-1999	184	2	.	.	PUNCT
iajs-1999	185	1	math	math	NOUN
iajs-1999	185	2	.	.	PUNCT
iajs-1999	186	1	(	(	PUNCT
iajs-1999	186	2	born	bear	VERB
iajs-1999	186	3	)	)	PUNCT
iajs-1999	186	4	.	.	PUNCT
iajs-1999	187	1	2001	2001	NUM
iajs-1999	187	2	,	,	PUNCT
iajs-1999	187	3	73	73	NUM
iajs-1999	187	4	,	,	PUNCT
iajs-1999	187	5	273	273	NUM
iajs-1999	187	6	-	-	SYM
iajs-1999	187	7	278	278	NUM
iajs-1999	187	8	.	.	PUNCT
iajs-1999	188	1	2	2	X
iajs-1999	188	2	.	.	X
iajs-1999	188	3	abdul	abdul	NOUN
iajs-1999	188	4	-	-	PUNCT
iajs-1999	188	5	baste	baste	NOUN
iajs-1999	188	6	,	,	PUNCT
iajs-1999	188	7	z.	z.	PROPN
iajs-1999	188	8	;	;	PUNCT
iajs-1999	188	9	smith	smith	PROPN
iajs-1999	188	10	,	,	PUNCT
iajs-1999	188	11	p.f	p.f	PROPN
iajs-1999	188	12	.	.	PROPN
iajs-1999	188	13	multiplication	multiplication	NOUN
iajs-1999	188	14	modules	module	NOUN
iajs-1999	188	15	.	.	PUNCT
iajs-1999	189	1	comm	comm	NOUN
iajs-1999	189	2	.	.	PUNCT
iajs-1999	190	1	in	in	ADP
iajs-1999	190	2	algebra	algebra	PROPN
iajs-1999	190	3	.	.	PUNCT
iajs-1999	191	1	1988	1988	NUM
iajs-1999	191	2	,	,	PUNCT
iajs-1999	191	3	16	16	NUM
iajs-1999	191	4	,	,	PUNCT
iajs-1999	191	5	755	755	NUM
iajs-1999	191	6	-	-	SYM
iajs-1999	191	7	779	779	NUM
iajs-1999	191	8	.	.	NOUN
iajs-1999	192	1	3	3	NUM
iajs-1999	192	2	.	.	NOUN
iajs-1999	192	3	athab	athab	PROPN
iajs-1999	192	4	,	,	PUNCT
iajs-1999	192	5	i.a	i.a	PROPN
iajs-1999	192	6	.	.	PROPN
iajs-1999	192	7	prime	prime	ADJ
iajs-1999	192	8	submodules	submodule	NOUN
iajs-1999	192	9	and	and	CCONJ
iajs-1999	192	10	semiprime	semiprime	NOUN
iajs-1999	192	11	submodules	submodule	NOUN
iajs-1999	192	12	.	.	PUNCT
iajs-1999	193	1	m.sc	m.sc	PROPN
iajs-1999	193	2	.	.	PUNCT
iajs-1999	194	1	thesis	thesis	NOUN
iajs-1999	194	2	,	,	PUNCT
iajs-1999	194	3	university	university	NOUN
iajs-1999	194	4	of	of	ADP
iajs-1999	194	5	baghdad	baghdad	PROPN
iajs-1999	194	6	.	.	PUNCT
iajs-1999	195	1	1996	1996	NUM
iajs-1999	195	2	.	.	PUNCT
iajs-1999	196	1	4	4	X
iajs-1999	196	2	.	.	X
iajs-1999	196	3	dauns	daun	NOUN
iajs-1999	196	4	,	,	PUNCT
iajs-1999	196	5	j.	j.	PROPN
iajs-1999	196	6	prime	prime	PROPN
iajs-1999	196	7	modules	module	NOUN
iajs-1999	196	8	and	and	CCONJ
iajs-1999	196	9	one	one	NUM
iajs-1999	196	10	sided	sided	ADJ
iajs-1999	196	11	ideals	ideal	NOUN
iajs-1999	196	12	in	in	ADP
iajs-1999	196	13	ring	ring	NOUN
iajs-1999	196	14	theory	theory	NOUN
iajs-1999	196	15	and	and	CCONJ
iajs-1999	196	16	algebra	algebra	PROPN
iajs-1999	196	17	iii	iii	PROPN
iajs-1999	196	18	.	.	PUNCT
iajs-1999	197	1	proceedings	proceeding	NOUN
iajs-1999	197	2	of	of	ADP
iajs-1999	197	3	the	the	DET
iajs-1999	197	4	third	third	ADJ
iajs-1999	197	5	oklahomo	oklahomo	NOUN
iajs-1999	197	6	conference	conference	NOUN
iajs-1999	197	7	.	.	PUNCT
iajs-1999	198	1	1980	1980	NUM
iajs-1999	198	2	,	,	PUNCT
iajs-1999	198	3	301	301	NUM
iajs-1999	198	4	-	-	SYM
iajs-1999	198	5	344	344	NUM
iajs-1999	198	6	.	.	NOUN
iajs-1999	199	1	5	5	NUM
iajs-1999	199	2	.	.	PUNCT
iajs-1999	199	3	annin	annin	PROPN
iajs-1999	199	4	,	,	PUNCT
iajs-1999	199	5	s.	s.	PROPN
iajs-1999	199	6	associated	associate	VERB
iajs-1999	199	7	and	and	CCONJ
iajs-1999	199	8	attached	attach	VERB
iajs-1999	199	9	primes	prime	NOUN
iajs-1999	199	10	over	over	ADP
iajs-1999	199	11	non	non	ADJ
iajs-1999	199	12	commutative	commutative	ADJ
iajs-1999	199	13	rings	ring	NOUN
iajs-1999	199	14	.	.	PUNCT
iajs-1999	200	1	ph	ph	PROPN
iajs-1999	200	2	.	.	PUNCT
iajs-1999	201	1	d	d	X
iajs-1999	201	2	thesis	thesis	NOUN
iajs-1999	201	3	,	,	PUNCT
iajs-1999	201	4	university	university	NOUN
iajs-1999	201	5	of	of	ADP
iajs-1999	201	6	berkeley	berkeley	PROPN
iajs-1999	201	7	.	.	PUNCT
iajs-1999	202	1	2002	2002	NUM
iajs-1999	202	2	.	.	PUNCT
iajs-1999	203	1	6	6	X
iajs-1999	203	2	.	.	X
iajs-1999	203	3	rasha	rasha	PROPN
iajs-1999	203	4	,	,	PUNCT
iajs-1999	203	5	i.k	i.k	PROPN
iajs-1999	203	6	.	.	PROPN
iajs-1999	203	7	dual	dual	ADJ
iajs-1999	203	8	notions	notion	NOUN
iajs-1999	203	9	of	of	ADP
iajs-1999	203	10	prime	prime	ADJ
iajs-1999	203	11	submodules	submodule	NOUN
iajs-1999	203	12	and	and	CCONJ
iajs-1999	203	13	prime	prime	ADJ
iajs-1999	203	14	modules	module	NOUN
iajs-1999	203	15	.	.	PUNCT
iajs-1999	204	1	m.sc	m.sc	NOUN
iajs-1999	204	2	.	.	PUNCT
iajs-1999	205	1	thesis	thesis	NOUN
iajs-1999	205	2	.	.	PUNCT
iajs-1999	206	1	university	university	NOUN
iajs-1999	206	2	of	of	ADP
iajs-1999	206	3	baghdad	baghdad	PROPN
iajs-1999	206	4	.	.	PUNCT
iajs-1999	207	1	2009	2009	NUM
iajs-1999	207	2	.	.	PUNCT
iajs-1999	208	1	7	7	X
iajs-1999	208	2	.	.	X
iajs-1999	208	3	macdonald	macdonald	PROPN
iajs-1999	208	4	,	,	PUNCT
iajs-1999	208	5	l.g	l.g	PROPN
iajs-1999	208	6	.	.	PROPN
iajs-1999	208	7	secondary	secondary	ADJ
iajs-1999	208	8	representation	representation	NOUN
iajs-1999	208	9	of	of	ADP
iajs-1999	208	10	modules	module	NOUN
iajs-1999	208	11	over	over	ADP
iajs-1999	208	12	commutattive	commutattive	ADJ
iajs-1999	208	13	ring	ring	NOUN
iajs-1999	208	14	.	.	PUNCT
iajs-1999	209	1	sympos	sympos	PROPN
iajs-1999	209	2	.	.	PUNCT
iajs-1999	210	1	math	math	NOUN
iajs-1999	210	2	.	.	PUNCT
iajs-1999	211	1	xi	xi	PROPN
iajs-1999	211	2	,	,	PUNCT
iajs-1999	211	3	1973	1973	NUM
iajs-1999	211	4	,	,	PUNCT
iajs-1999	211	5	33	33	NUM
iajs-1999	211	6	-	-	SYM
iajs-1999	211	7	43	43	NUM
iajs-1999	211	8	.	.	PUNCT