id	sid	tid	token	lemma	pos
iajs-3045	1	1	ihjpas	ihjpas	PROPN
iajs-3045	1	2	.	.	PUNCT
iajs-3045	2	1	36(2)2023	36(2)2023	NUM
iajs-3045	2	2	407	407	NUM
iajs-3045	2	3	this	this	DET
iajs-3045	2	4	work	work	NOUN
iajs-3045	2	5	is	be	AUX
iajs-3045	2	6	licensed	license	VERB
iajs-3045	2	7	under	under	ADP
iajs-3045	2	8	a	a	DET
iajs-3045	2	9	creative	creative	ADJ
iajs-3045	2	10	commons	common	NOUN
iajs-3045	2	11	attribution	attribution	NOUN
iajs-3045	2	12	4.0	4.0	NUM
iajs-3045	2	13	international	international	ADJ
iajs-3045	2	14	license	license	NOUN
iajs-3045	2	15	abstract	abstract	NOUN
iajs-3045	2	16	the	the	DET
iajs-3045	2	17	concept	concept	NOUN
iajs-3045	2	18	of	of	ADP
iajs-3045	2	19	the	the	DET
iajs-3045	2	20	extend	extend	NOUN
iajs-3045	2	21	nearly	nearly	ADV
iajs-3045	2	22	pseudo	pseudo	ADJ
iajs-3045	2	23	quasi-2	quasi-2	ADJ
iajs-3045	2	24	-	-	PUNCT
iajs-3045	2	25	absorbing	absorbing	ADJ
iajs-3045	2	26	submodules	submodule	NOUN
iajs-3045	2	27	was	be	AUX
iajs-3045	2	28	recently	recently	ADV
iajs-3045	2	29	introduced	introduce	VERB
iajs-3045	2	30	by	by	ADP
iajs-3045	2	31	omar	omar	PROPN
iajs-3045	2	32	a.	a.	PROPN
iajs-3045	2	33	abdullah	abdullah	PROPN
iajs-3045	2	34	and	and	CCONJ
iajs-3045	2	35	haibat	haibat	PROPN
iajs-3045	2	36	k.	k.	PROPN
iajs-3045	2	37	mohammadali	mohammadali	PROPN
iajs-3045	2	38	in	in	ADP
iajs-3045	2	39	2022	2022	NUM
iajs-3045	2	40	,	,	PUNCT
iajs-3045	2	41	where	where	SCONJ
iajs-3045	2	42	he	he	PRON
iajs-3045	2	43	studies	study	VERB
iajs-3045	2	44	this	this	DET
iajs-3045	2	45	concept	concept	NOUN
iajs-3045	2	46	and	and	CCONJ
iajs-3045	2	47	it	it	PRON
iajs-3045	2	48	is	be	AUX
iajs-3045	2	49	relationship	relationship	NOUN
iajs-3045	2	50	to	to	ADP
iajs-3045	2	51	previous	previous	ADJ
iajs-3045	2	52	generalizationsm	generalizationsm	NOUN
iajs-3045	2	53	especially	especially	ADV
iajs-3045	2	54	2	2	NUM
iajs-3045	2	55	-	-	PUNCT
iajs-3045	2	56	absorbing	absorb	VERB
iajs-3045	2	57	submodule	submodule	NOUN
iajs-3045	2	58	and	and	CCONJ
iajs-3045	2	59	quasi-2absorbing	quasi-2absorbe	VERB
iajs-3045	2	60	submodule	submodule	NOUN
iajs-3045	2	61	,	,	PUNCT
iajs-3045	2	62	in	in	ADP
iajs-3045	2	63	addition	addition	NOUN
iajs-3045	2	64	to	to	ADP
iajs-3045	2	65	studying	study	VERB
iajs-3045	2	66	the	the	DET
iajs-3045	2	67	most	most	ADV
iajs-3045	2	68	important	important	ADJ
iajs-3045	2	69	propositions	proposition	NOUN
iajs-3045	2	70	,	,	PUNCT
iajs-3045	2	71	charactarizations	charactarization	NOUN
iajs-3045	2	72	and	and	CCONJ
iajs-3045	2	73	examples	example	NOUN
iajs-3045	2	74	.	.	PUNCT
iajs-3045	3	1	now	now	ADV
iajs-3045	3	2	in	in	ADP
iajs-3045	3	3	this	this	DET
iajs-3045	3	4	research	research	NOUN
iajs-3045	3	5	,	,	PUNCT
iajs-3045	3	6	which	which	PRON
iajs-3045	3	7	is	be	AUX
iajs-3045	3	8	considered	consider	VERB
iajs-3045	3	9	a	a	DET
iajs-3045	3	10	continuation	continuation	NOUN
iajs-3045	3	11	of	of	ADP
iajs-3045	3	12	the	the	DET
iajs-3045	3	13	definition	definition	NOUN
iajs-3045	3	14	that	that	PRON
iajs-3045	3	15	was	be	AUX
iajs-3045	3	16	presented	present	VERB
iajs-3045	3	17	earlier	early	ADV
iajs-3045	3	18	,	,	PUNCT
iajs-3045	3	19	which	which	PRON
iajs-3045	3	20	is	be	AUX
iajs-3045	3	21	the	the	DET
iajs-3045	3	22	extend	extend	NOUN
iajs-3045	3	23	nearly	nearly	ADV
iajs-3045	3	24	pseudo	pseudo	ADJ
iajs-3045	3	25	quasi-2	quasi-2	ADJ
iajs-3045	3	26	-	-	PUNCT
iajs-3045	3	27	absorbing	absorbing	ADJ
iajs-3045	3	28	submodules	submodule	NOUN
iajs-3045	3	29	,	,	PUNCT
iajs-3045	3	30	we	we	PRON
iajs-3045	3	31	have	have	AUX
iajs-3045	3	32	completed	complete	VERB
iajs-3045	3	33	the	the	DET
iajs-3045	3	34	study	study	NOUN
iajs-3045	3	35	of	of	ADP
iajs-3045	3	36	this	this	DET
iajs-3045	3	37	concept	concept	NOUN
iajs-3045	3	38	in	in	ADP
iajs-3045	3	39	multiplication	multiplication	NOUN
iajs-3045	3	40	modules	module	NOUN
iajs-3045	3	41	.	.	PUNCT
iajs-3045	4	1	and	and	CCONJ
iajs-3045	4	2	the	the	DET
iajs-3045	4	3	relationship	relationship	NOUN
iajs-3045	4	4	between	between	ADP
iajs-3045	4	5	the	the	DET
iajs-3045	4	6	extend	extend	NOUN
iajs-3045	4	7	nearly	nearly	ADV
iajs-3045	4	8	pseudo	pseudo	ADJ
iajs-3045	4	9	quasi-2	quasi-2	ADJ
iajs-3045	4	10	-	-	PUNCT
iajs-3045	4	11	absorbing	absorbing	ADJ
iajs-3045	4	12	submodule	submodule	NOUN
iajs-3045	4	13	and	and	CCONJ
iajs-3045	4	14	extend	extend	VERB
iajs-3045	4	15	nearly	nearly	ADV
iajs-3045	4	16	pseudo	pseudo	NOUN
iajs-3045	4	17	quasi-2absorbing	quasi-2absorbe	VERB
iajs-3045	4	18	ideal	ideal	ADJ
iajs-3045	4	19	.	.	PUNCT
iajs-3045	5	1	we	we	PRON
iajs-3045	5	2	also	also	ADV
iajs-3045	5	3	studied	study	VERB
iajs-3045	5	4	more	more	ADJ
iajs-3045	5	5	result	result	NOUN
iajs-3045	5	6	of	of	ADP
iajs-3045	5	7	extend	extend	NOUN
iajs-3045	5	8	nearly	nearly	ADV
iajs-3045	5	9	pseudo	pseudo	ADJ
iajs-3045	5	10	quasi-2	quasi-2	ADJ
iajs-3045	5	11	-	-	PUNCT
iajs-3045	5	12	absorbing	absorbing	ADJ
iajs-3045	5	13	submodule	submodule	NOUN
iajs-3045	5	14	in	in	ADP
iajs-3045	5	15	multiplication	multiplication	NOUN
iajs-3045	5	16	module	module	NOUN
iajs-3045	5	17	.	.	PUNCT
iajs-3045	6	1	in	in	ADP
iajs-3045	6	2	the	the	DET
iajs-3045	6	3	end	end	NOUN
iajs-3045	6	4	,	,	PUNCT
iajs-3045	6	5	we	we	PRON
iajs-3045	6	6	obtained	obtain	VERB
iajs-3045	6	7	new	new	ADJ
iajs-3045	6	8	propositions	proposition	NOUN
iajs-3045	6	9	and	and	CCONJ
iajs-3045	6	10	distinguished	distinguished	ADJ
iajs-3045	6	11	results	result	NOUN
iajs-3045	6	12	in	in	ADP
iajs-3045	6	13	studying	study	VERB
iajs-3045	6	14	this	this	DET
iajs-3045	6	15	concept	concept	NOUN
iajs-3045	6	16	.	.	PUNCT
iajs-3045	7	1	keywords	keyword	NOUN
iajs-3045	7	2	:	:	PUNCT
iajs-3045	7	3	exnpq-2	exnpq-2	NUM
iajs-3045	7	4	-	-	PUNCT
iajs-3045	7	5	absorbing	absorb	VERB
iajs-3045	7	6	submodule	submodule	NOUN
iajs-3045	7	7	,	,	PUNCT
iajs-3045	7	8	multiplication	multiplication	NOUN
iajs-3045	7	9	modules	module	NOUN
iajs-3045	7	10	,	,	PUNCT
iajs-3045	7	11	non	non	ADJ
iajs-3045	7	12	-	-	ADJ
iajs-3045	7	13	singular	singular	ADJ
iajs-3045	7	14	modules	module	NOUN
iajs-3045	7	15	,	,	PUNCT
iajs-3045	7	16	faithful	faithful	ADJ
iajs-3045	7	17	module	module	NOUN
iajs-3045	7	18	,	,	PUNCT
iajs-3045	7	19	projective	projective	ADJ
iajs-3045	7	20	module	module	NOUN
iajs-3045	7	21	,	,	PUNCT
iajs-3045	7	22	good	good	ADJ
iajs-3045	7	23	rings	ring	NOUN
iajs-3045	7	24	and	and	CCONJ
iajs-3045	7	25	local	local	ADJ
iajs-3045	7	26	rings	ring	NOUN
iajs-3045	7	27	.	.	PUNCT
iajs-3045	8	1	1	1	X
iajs-3045	8	2	.	.	X
iajs-3045	8	3	introduction	introduction	NOUN
iajs-3045	8	4	in	in	ADP
iajs-3045	8	5	recent	recent	ADJ
iajs-3045	8	6	years	year	NOUN
iajs-3045	8	7	,	,	PUNCT
iajs-3045	8	8	many	many	ADJ
iajs-3045	8	9	generalizations	generalization	NOUN
iajs-3045	8	10	have	have	AUX
iajs-3045	8	11	appeared	appear	VERB
iajs-3045	8	12	about	about	ADP
iajs-3045	8	13	the	the	DET
iajs-3045	8	14	concept	concept	NOUN
iajs-3045	8	15	of	of	ADP
iajs-3045	8	16	the	the	DET
iajs-3045	8	17	2	2	NUM
iajs-3045	8	18	-	-	PUNCT
iajs-3045	8	19	absorbing	absorb	VERB
iajs-3045	8	20	submodule	submodule	NOUN
iajs-3045	8	21	such	such	ADJ
iajs-3045	8	22	as	as	ADP
iajs-3045	8	23	(	(	PUNCT
iajs-3045	8	24	pseudo	pseudo	NOUN
iajs-3045	8	25	quasi-2	quasi-2	NOUN
iajs-3045	8	26	-	-	PUNCT
iajs-3045	8	27	absorbing	absorbing	ADJ
iajs-3045	8	28	,	,	PUNCT
iajs-3045	8	29	nearly	nearly	ADV
iajs-3045	8	30	quasi-2	quasi-2	NUM
iajs-3045	8	31	-	-	PUNCT
iajs-3045	8	32	absorbing	absorbing	ADJ
iajs-3045	8	33	and	and	CCONJ
iajs-3045	8	34	soc	soc	NOUN
iajs-3045	8	35	-	-	PUNCT
iajs-3045	8	36	qp2absorbing	qp2absorbing	NOUN
iajs-3045	8	37	)	)	PUNCT
iajs-3045	8	38	submodules	submodule	NOUN
iajs-3045	8	39	see	see	VERB
iajs-3045	9	1	[	[	X
iajs-3045	9	2	1	1	NUM
iajs-3045	9	3	,	,	PUNCT
iajs-3045	9	4	2	2	NUM
iajs-3045	9	5	and	and	CCONJ
iajs-3045	9	6	3	3	NUM
iajs-3045	9	7	]	]	PUNCT
iajs-3045	9	8	.	.	PUNCT
iajs-3045	10	1	the	the	DET
iajs-3045	10	2	concept	concept	NOUN
iajs-3045	10	3	of	of	ADP
iajs-3045	10	4	the	the	DET
iajs-3045	10	5	extend	extend	NOUN
iajs-3045	10	6	nearly	nearly	ADV
iajs-3045	10	7	pseudo	pseudo	NOUN
iajs-3045	10	8	quasi-2absorbing	quasi-2absorbe	VERB
iajs-3045	10	9	submodules	submodule	NOUN
iajs-3045	10	10	is	be	AUX
iajs-3045	10	11	one	one	NUM
iajs-3045	10	12	of	of	ADP
iajs-3045	10	13	the	the	DET
iajs-3045	10	14	recent	recent	ADJ
iajs-3045	10	15	generalizations	generalization	NOUN
iajs-3045	10	16	that	that	PRON
iajs-3045	10	17	were	be	AUX
iajs-3045	10	18	recently	recently	ADV
iajs-3045	10	19	introduced	introduce	VERB
iajs-3045	10	20	by	by	ADP
iajs-3045	10	21	us	we	PRON
iajs-3045	10	22	,	,	PUNCT
iajs-3045	10	23	researchers	researcher	NOUN
iajs-3045	10	24	,	,	PUNCT
iajs-3045	10	25	omar	omar	PROPN
iajs-3045	10	26	and	and	CCONJ
iajs-3045	10	27	haibat	haibat	NOUN
iajs-3045	10	28	see	see	VERB
iajs-3045	10	29	[	[	X
iajs-3045	10	30	4	4	NUM
iajs-3045	10	31	]	]	PUNCT
iajs-3045	10	32	.	.	PUNCT
iajs-3045	11	1	where	where	SCONJ
iajs-3045	11	2	we	we	PRON
iajs-3045	11	3	dealt	deal	VERB
iajs-3045	11	4	with	with	ADP
iajs-3045	11	5	in	in	ADP
iajs-3045	11	6	the	the	DET
iajs-3045	11	7	previous	previous	ADJ
iajs-3045	11	8	research	research	NOUN
iajs-3045	11	9	basic	basic	ADJ
iajs-3045	11	10	properties	property	NOUN
iajs-3045	11	11	with	with	ADP
iajs-3045	11	12	relationships	relationship	NOUN
iajs-3045	11	13	.	.	PUNCT
iajs-3045	12	1	the	the	DET
iajs-3045	12	2	present	present	ADJ
iajs-3045	12	3	work	work	NOUN
iajs-3045	12	4	is	be	AUX
iajs-3045	12	5	divided	divide	VERB
iajs-3045	12	6	into	into	ADP
iajs-3045	12	7	three	three	NUM
iajs-3045	12	8	parts	part	NOUN
iajs-3045	12	9	.	.	PUNCT
iajs-3045	13	1	part	part	NOUN
iajs-3045	13	2	one	one	NUM
iajs-3045	13	3	is	be	AUX
iajs-3045	13	4	preliminaries	preliminary	NOUN
iajs-3045	13	5	part	part	NOUN
iajs-3045	13	6	,	,	PUNCT
iajs-3045	13	7	we	we	PRON
iajs-3045	13	8	present	present	VERB
iajs-3045	13	9	in	in	ADP
iajs-3045	13	10	this	this	DET
iajs-3045	13	11	part	part	NOUN
iajs-3045	13	12	of	of	ADP
iajs-3045	13	13	the	the	DET
iajs-3045	13	14	work	work	NOUN
iajs-3045	13	15	the	the	DET
iajs-3045	13	16	necessary	necessary	ADJ
iajs-3045	13	17	background	background	NOUN
iajs-3045	13	18	needed	need	VERB
iajs-3045	13	19	later	later	ADV
iajs-3045	13	20	consisting	consist	VERB
iajs-3045	13	21	of	of	ADP
iajs-3045	13	22	definitions	definition	NOUN
iajs-3045	13	23	,	,	PUNCT
iajs-3045	13	24	propositions	proposition	NOUN
iajs-3045	13	25	and	and	CCONJ
iajs-3045	13	26	remarks	remark	NOUN
iajs-3045	13	27	(	(	PUNCT
iajs-3045	13	28	without	without	ADP
iajs-3045	13	29	proof	proof	NOUN
iajs-3045	13	30	)	)	PUNCT
iajs-3045	13	31	and	and	CCONJ
iajs-3045	13	32	in	in	ADP
iajs-3045	13	33	the	the	DET
iajs-3045	13	34	second	second	ADJ
iajs-3045	13	35	part	part	NOUN
iajs-3045	13	36	we	we	PRON
iajs-3045	13	37	introduced	introduce	VERB
iajs-3045	13	38	and	and	CCONJ
iajs-3045	13	39	studied	study	VERB
iajs-3045	13	40	the	the	DET
iajs-3045	13	41	concept	concept	NOUN
iajs-3045	13	42	of	of	ADP
iajs-3045	13	43	the	the	DET
iajs-3045	13	44	extend	extend	NOUN
iajs-3045	13	45	nearly	nearly	ADV
iajs-3045	13	46	pseudo	pseudo	ADJ
iajs-3045	13	47	quasi-2	quasi-2	ADJ
iajs-3045	13	48	-	-	PUNCT
iajs-3045	13	49	absorbing	absorbing	ADJ
iajs-3045	13	50	submodule	submodule	NOUN
iajs-3045	13	51	in	in	ADP
iajs-3045	13	52	multiplication	multiplication	NOUN
iajs-3045	13	53	module	module	NOUN
iajs-3045	13	54	.	.	PUNCT
iajs-3045	14	1	also	also	ADV
iajs-3045	14	2	we	we	PRON
iajs-3045	14	3	got	get	VERB
iajs-3045	14	4	a	a	DET
iajs-3045	14	5	lot	lot	NOUN
iajs-3045	14	6	of	of	ADP
iajs-3045	14	7	important	important	ADJ
iajs-3045	14	8	results	result	NOUN
iajs-3045	14	9	like	like	ADP
iajs-3045	14	10	propositions	proposition	NOUN
iajs-3045	14	11	3.2	3.2	NUM
iajs-3045	14	12	,	,	PUNCT
iajs-3045	14	13	3.6	3.6	NUM
iajs-3045	14	14	and	and	CCONJ
iajs-3045	14	15	3.7	3.7	NUM
iajs-3045	14	16	.	.	PUNCT
iajs-3045	15	1	in	in	ADP
iajs-3045	15	2	the	the	DET
iajs-3045	15	3	end	end	NOUN
iajs-3045	15	4	we	we	PRON
iajs-3045	15	5	doi.org/10.30526/36.2.3045	doi.org/10.30526/36.2.3045	ADV
iajs-3045	15	6	article	article	NOUN
iajs-3045	15	7	history	history	NOUN
iajs-3045	15	8	:	:	PUNCT
iajs-3045	15	9	received	receive	VERB
iajs-3045	15	10	16	16	NUM
iajs-3045	15	11	september	september	PROPN
iajs-3045	15	12	2022	2022	NUM
iajs-3045	15	13	,	,	PUNCT
iajs-3045	15	14	accepted	accept	VERB
iajs-3045	15	15	6	6	NUM
iajs-3045	15	16	november	november	PROPN
iajs-3045	15	17	2022	2022	NUM
iajs-3045	15	18	,	,	PUNCT
iajs-3045	15	19	published	publish	VERB
iajs-3045	15	20	in	in	ADP
iajs-3045	15	21	april	april	PROPN
iajs-3045	15	22	2023	2023	NUM
iajs-3045	15	23	.	.	PUNCT
iajs-3045	16	1	ibn	ibn	PROPN
iajs-3045	16	2	al	al	PROPN
iajs-3045	16	3	-	-	PUNCT
iajs-3045	16	4	haitham	haitham	PROPN
iajs-3045	16	5	journal	journal	PROPN
iajs-3045	16	6	for	for	ADP
iajs-3045	16	7	pure	pure	ADJ
iajs-3045	16	8	and	and	CCONJ
iajs-3045	16	9	applied	applied	ADJ
iajs-3045	16	10	sciences	sciences	PROPN
iajs-3045	16	11	journal	journal	PROPN
iajs-3045	16	12	homepage	homepage	NOUN
iajs-3045	16	13	:	:	PUNCT
iajs-3045	16	14	jih.uobaghdad.edu.iq	jih.uobaghdad.edu.iq	NOUN
iajs-3045	16	15	extend	extend	VERB
iajs-3045	16	16	nearly	nearly	ADV
iajs-3045	16	17	pseudo	pseudo	ADJ
iajs-3045	16	18	quasi-2	quasi-2	ADJ
iajs-3045	16	19	-	-	PUNCT
iajs-3045	16	20	absorbing	absorbing	ADJ
iajs-3045	16	21	submodules(ii	submodules(ii	NOUN
iajs-3045	16	22	)	)	PUNCT
iajs-3045	16	23	omar	omar	PROPN
iajs-3045	16	24	a.	a.	PROPN
iajs-3045	16	25	abdullah	abdullah	PROPN
iajs-3045	16	26	department	department	PROPN
iajs-3045	16	27	of	of	ADP
iajs-3045	16	28	mathematics	mathematics	PROPN
iajs-3045	16	29	college	college	PROPN
iajs-3045	16	30	of	of	ADP
iajs-3045	16	31	computer	computer	NOUN
iajs-3045	16	32	science	science	NOUN
iajs-3045	16	33	and	and	CCONJ
iajs-3045	16	34	mathematics	mathematics	PROPN
iajs-3045	16	35	tikrit	tikrit	PROPN
iajs-3045	16	36	university	university	PROPN
iajs-3045	16	37	/	/	SYM
iajs-3045	16	38	iraq	iraq	PROPN
iajs-3045	16	39	.	.	PUNCT
iajs-3045	17	1	omer.a.abdullah35383@st.tu.edu.iq	omer.a.abdullah35383@st.tu.edu.iq	ADJ
iajs-3045	17	2	haibat	haibat	PROPN
iajs-3045	17	3	k.	k.	PROPN
iajs-3045	17	4	mohammadali	mohammadali	PROPN
iajs-3045	17	5	department	department	PROPN
iajs-3045	17	6	of	of	ADP
iajs-3045	17	7	mathematics	mathematics	PROPN
iajs-3045	17	8	college	college	PROPN
iajs-3045	17	9	of	of	ADP
iajs-3045	17	10	computer	computer	NOUN
iajs-3045	17	11	science	science	NOUN
iajs-3045	17	12	and	and	CCONJ
iajs-3045	17	13	mathematics	mathematics	PROPN
iajs-3045	17	14	tikrit	tikrit	PROPN
iajs-3045	17	15	university	university	PROPN
iajs-3045	17	16	/	/	SYM
iajs-3045	17	17	iraq	iraq	PROPN
iajs-3045	17	18	.	.	PUNCT
iajs-3045	18	1	h.mohammadali@tu.edu.iq	h.mohammadali@tu.edu.iq	PROPN
iajs-3045	18	2	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
iajs-3045	18	3	mailto:omer.a.abdullah35383@st.tu.edu.iq	mailto:omer.a.abdullah35383@st.tu.edu.iq	PROPN
iajs-3045	18	4	mailto:h.mohammadali@tu.edu.iq	mailto:h.mohammadali@tu.edu.iq	PROPN
iajs-3045	18	5	mailto:h.mohammadali@tu.edu.iq	mailto:h.mohammadali@tu.edu.iq	PROPN
iajs-3045	18	6	ihjpas	ihjpas	PROPN
iajs-3045	18	7	.	.	PUNCT
iajs-3045	19	1	36(2)2023	36(2)2023	NUM
iajs-3045	19	2	408	408	NUM
iajs-3045	19	3	presented	present	VERB
iajs-3045	19	4	more	more	ADJ
iajs-3045	19	5	result	result	NOUN
iajs-3045	19	6	of	of	ADP
iajs-3045	19	7	extend	extend	NOUN
iajs-3045	19	8	nearly	nearly	ADV
iajs-3045	19	9	pseudo	pseudo	ADJ
iajs-3045	19	10	quasi-2	quasi-2	ADJ
iajs-3045	19	11	-	-	PUNCT
iajs-3045	19	12	absorbing	absorbing	ADJ
iajs-3045	19	13	submodule	submodule	NOUN
iajs-3045	19	14	in	in	ADP
iajs-3045	19	15	multiplication	multiplication	NOUN
iajs-3045	19	16	modules	module	NOUN
iajs-3045	19	17	.	.	PUNCT
iajs-3045	20	1	see	see	VERB
iajs-3045	20	2	propositions	proposition	NOUN
iajs-3045	20	3	4.1	4.1	NUM
iajs-3045	20	4	,	,	PUNCT
iajs-3045	20	5	4.2	4.2	NUM
iajs-3045	20	6	and	and	CCONJ
iajs-3045	20	7	4.10	4.10	NUM
iajs-3045	20	8	.	.	NOUN
iajs-3045	21	1	2	2	NUM
iajs-3045	21	2	.	.	NOUN
iajs-3045	21	3	preliminaries	preliminary	NOUN
iajs-3045	21	4	the	the	DET
iajs-3045	21	5	following	follow	VERB
iajs-3045	21	6	list	list	NOUN
iajs-3045	21	7	some	some	DET
iajs-3045	21	8	fundamental	fundamental	ADJ
iajs-3045	21	9	definitions	definition	NOUN
iajs-3045	21	10	and	and	CCONJ
iajs-3045	21	11	notations	notation	NOUN
iajs-3045	21	12	that	that	PRON
iajs-3045	21	13	will	will	AUX
iajs-3045	21	14	be	be	AUX
iajs-3045	21	15	utilized	utilize	VERB
iajs-3045	21	16	in	in	ADP
iajs-3045	21	17	this	this	DET
iajs-3045	21	18	paper	paper	NOUN
iajs-3045	21	19	.	.	PUNCT
iajs-3045	22	1	definition	definition	NOUN
iajs-3045	22	2	2.1[4	2.1[4	NUM
iajs-3045	22	3	]	]	PUNCT
iajs-3045	22	4	.	.	PUNCT
iajs-3045	23	1	a	a	DET
iajs-3045	23	2	proper	proper	ADJ
iajs-3045	23	3	submodule	submodule	NOUN
iajs-3045	23	4	𝑉	𝑉	PROPN
iajs-3045	23	5	of	of	ADP
iajs-3045	23	6	an	an	DET
iajs-3045	23	7	ʀ	ʀ	NOUN
iajs-3045	23	8	-	-	PUNCT
iajs-3045	23	9	module	module	NOUN
iajs-3045	23	10	ѡ	ѡ	NOUN
iajs-3045	23	11	is	be	AUX
iajs-3045	23	12	said	say	VERB
iajs-3045	23	13	to	to	PART
iajs-3045	23	14	be	be	AUX
iajs-3045	23	15	extend	extend	VERB
iajs-3045	23	16	nearly	nearly	ADV
iajs-3045	23	17	pseudo	pseudo	ADJ
iajs-3045	23	18	quasi-2	quasi-2	NOUN
iajs-3045	23	19	-	-	PUNCT
iajs-3045	23	20	absorbing	absorbing	ADJ
iajs-3045	23	21	(	(	PUNCT
iajs-3045	23	22	for	for	ADP
iajs-3045	23	23	short	short	ADJ
iajs-3045	23	24	exnpq2ab	exnpq2ab	PROPN
iajs-3045	23	25	)	)	PUNCT
iajs-3045	23	26	submodule	submodule	NOUN
iajs-3045	23	27	of	of	ADP
iajs-3045	23	28	ѡ	ѡ	PROPN
iajs-3045	23	29	if	if	SCONJ
iajs-3045	23	30	whenever	whenever	SCONJ
iajs-3045	23	31	ɑɓ𝑐ӽ	ɑɓ𝑐ӽ	NOUN
iajs-3045	23	32	∈	∈	PROPN
iajs-3045	23	33	𝑉	𝑉	PROPN
iajs-3045	23	34	,	,	PUNCT
iajs-3045	23	35	where	where	SCONJ
iajs-3045	23	36	ɑ	ɑ	PROPN
iajs-3045	23	37	,	,	PUNCT
iajs-3045	23	38	ɓ	ɓ	PROPN
iajs-3045	23	39	,	,	PUNCT
iajs-3045	23	40	𝑐	𝑐	PROPN
iajs-3045	23	41	∈	∈	PROPN
iajs-3045	23	42	ʀ	ʀ	NOUN
iajs-3045	23	43	,	,	PUNCT
iajs-3045	23	44	ӽ	ӽ	PRON
iajs-3045	23	45	∈	∈	PROPN
iajs-3045	23	46	ѡ	ѡ	ADP
iajs-3045	23	47	,	,	PUNCT
iajs-3045	23	48	implies	imply	VERB
iajs-3045	23	49	that	that	SCONJ
iajs-3045	23	50	either	either	CCONJ
iajs-3045	23	51	ɑ𝑐ӽ	ɑ𝑐ӽ	PRON
iajs-3045	23	52	∈	∈	PROPN
iajs-3045	23	53	𝑉	𝑉	PROPN
iajs-3045	23	54	+	+	CCONJ
iajs-3045	23	55	𝑠𝑜𝑐(ѡ)+𝐽(ѡ	𝑠𝑜𝑐(ѡ)+𝐽(ѡ	PROPN
iajs-3045	23	56	)	)	PUNCT
iajs-3045	23	57	or	or	CCONJ
iajs-3045	23	58	ɓ𝑐ӽ	ɓ𝑐ӽ	NOUN
iajs-3045	23	59	∈	∈	PROPN
iajs-3045	23	60	𝑉	𝑉	PROPN
iajs-3045	23	61	+	+	CCONJ
iajs-3045	23	62	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	23	63	)	)	PUNCT
iajs-3045	23	64	+	+	PUNCT
iajs-3045	24	1	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	24	2	)	)	PUNCT
iajs-3045	24	3	or	or	CCONJ
iajs-3045	24	4	ɑɓӽ	ɑɓӽ	PROPN
iajs-3045	24	5	∈	∈	PROPN
iajs-3045	24	6	𝑉	𝑉	PROPN
iajs-3045	24	7	+	+	CCONJ
iajs-3045	24	8	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	24	9	)	)	PUNCT
iajs-3045	24	10	+	+	PUNCT
iajs-3045	24	11	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	24	12	)	)	PUNCT
iajs-3045	24	13	.	.	PUNCT
iajs-3045	25	1	and	and	CCONJ
iajs-3045	25	2	an	an	DET
iajs-3045	25	3	ideal	ideal	ADJ
iajs-3045	25	4	ƥ	ƥ	NOUN
iajs-3045	25	5	of	of	ADP
iajs-3045	25	6	a	a	DET
iajs-3045	25	7	ring	ring	NOUN
iajs-3045	25	8	ʀ	ʀ	NOUN
iajs-3045	25	9	is	be	AUX
iajs-3045	25	10	called	call	VERB
iajs-3045	25	11	exnpq2ab	exnpq2ab	PROPN
iajs-3045	25	12	ideal	ideal	NOUN
iajs-3045	25	13	of	of	ADP
iajs-3045	25	14	ʀ	ʀ	NOUN
iajs-3045	25	15	,	,	PUNCT
iajs-3045	25	16	if	if	SCONJ
iajs-3045	25	17	ƥ	ƥ	PRON
iajs-3045	25	18	is	be	AUX
iajs-3045	25	19	an	an	DET
iajs-3045	25	20	exnpq2ab	exnpq2ab	PROPN
iajs-3045	25	21	ʀsubmodule	ʀsubmodule	NOUN
iajs-3045	25	22	of	of	ADP
iajs-3045	25	23	an	an	DET
iajs-3045	25	24	ʀ	ʀ	NOUN
iajs-3045	25	25	-	-	PUNCT
iajs-3045	25	26	module	module	NOUN
iajs-3045	25	27	ʀ	ʀ	NOUN
iajs-3045	25	28	.	.	NOUN
iajs-3045	25	29	definition	definition	NOUN
iajs-3045	25	30	2.2[5	2.2[5	NUM
iajs-3045	25	31	]	]	PUNCT
iajs-3045	25	32	.	.	PUNCT
iajs-3045	26	1	an	an	DET
iajs-3045	26	2	ʀ	ʀ	NOUN
iajs-3045	26	3	-	-	PUNCT
iajs-3045	26	4	module	module	NOUN
iajs-3045	26	5	ѡ	ѡ	NOUN
iajs-3045	26	6	is	be	AUX
iajs-3045	26	7	multiplicatiion	multiplicatiion	NOUN
iajs-3045	26	8	,	,	PUNCT
iajs-3045	26	9	if	if	SCONJ
iajs-3045	26	10	every	every	DET
iajs-3045	26	11	submodule	submodule	NOUN
iajs-3045	26	12	𝑉	𝑉	PROPN
iajs-3045	26	13	of	of	ADP
iajs-3045	26	14	ѡ	ѡ	PROPN
iajs-3045	26	15	is	be	AUX
iajs-3045	26	16	of	of	ADP
iajs-3045	26	17	the	the	DET
iajs-3045	26	18	form	form	NOUN
iajs-3045	26	19	𝑉	𝑉	PROPN
iajs-3045	26	20	=	=	NOUN
iajs-3045	26	21	ƥѡ	ƥѡ	PROPN
iajs-3045	26	22	for	for	ADP
iajs-3045	26	23	some	some	DET
iajs-3045	26	24	ideal	ideal	ADJ
iajs-3045	26	25	ƥ	ƥ	PROPN
iajs-3045	26	26	of	of	ADP
iajs-3045	26	27	ʀ	ʀ	X
iajs-3045	26	28	.	.	X
iajs-3045	26	29	equivalently	equivalently	PROPN
iajs-3045	26	30	ѡ	ѡ	PROPN
iajs-3045	26	31	is	be	AUX
iajs-3045	26	32	a	a	DET
iajs-3045	26	33	multiplicatiion	multiplicatiion	NOUN
iajs-3045	26	34	ʀ	ʀ	NOUN
iajs-3045	26	35	-	-	PUNCT
iajs-3045	26	36	module	module	NOUN
iajs-3045	26	37	if	if	SCONJ
iajs-3045	26	38	every	every	DET
iajs-3045	26	39	submodule	submodule	NOUN
iajs-3045	26	40	𝑉	𝑉	PROPN
iajs-3045	26	41	of	of	ADP
iajs-3045	26	42	ѡ	ѡ	PROPN
iajs-3045	26	43	of	of	ADP
iajs-3045	26	44	the	the	DET
iajs-3045	26	45	form	form	NOUN
iajs-3045	26	46	𝑉	𝑉	NOUN
iajs-3045	26	47	=	=	PUNCT
iajs-3045	27	1	[	[	X
iajs-3045	27	2	𝑉:ʀ	𝑉:ʀ	NOUN
iajs-3045	27	3	ѡ]ѡ.	ѡ]ѡ.	NUM
iajs-3045	27	4	definition	definition	NOUN
iajs-3045	27	5	2.3[6	2.3[6	NUM
iajs-3045	27	6	]	]	PUNCT
iajs-3045	27	7	.	.	PUNCT
iajs-3045	28	1	an	an	DET
iajs-3045	28	2	ʀ	ʀ	NOUN
iajs-3045	28	3	-	-	PUNCT
iajs-3045	28	4	module	module	NOUN
iajs-3045	28	5	ѡ	ѡ	NOUN
iajs-3045	28	6	is	be	AUX
iajs-3045	28	7	faithful	faithful	ADJ
iajs-3045	28	8	if	if	SCONJ
iajs-3045	28	9	𝑎𝑛𝑛ʀ(ѡ	𝑎𝑛𝑛ʀ(ѡ	NOUN
iajs-3045	28	10	)	)	PUNCT
iajs-3045	28	11	=	=	SYM
iajs-3045	28	12	(	(	PUNCT
iajs-3045	28	13	0	0	NUM
iajs-3045	28	14	)	)	PUNCT
iajs-3045	28	15	,	,	PUNCT
iajs-3045	28	16	where	where	SCONJ
iajs-3045	28	17	𝑎𝑛𝑛ʀ(ѡ	𝑎𝑛𝑛ʀ(ѡ	NOUN
iajs-3045	28	18	)	)	PUNCT
iajs-3045	28	19	=	=	SYM
iajs-3045	29	1	{	{	PUNCT
iajs-3045	29	2	𝑟	𝑟	X
iajs-3045	29	3	∈	∈	PROPN
iajs-3045	29	4	ʀ	ʀ	NOUN
iajs-3045	29	5	:	:	PUNCT
iajs-3045	29	6	𝑟𝑤	𝑟𝑤	VERB
iajs-3045	29	7	=	=	SYM
iajs-3045	29	8	(	(	PUNCT
iajs-3045	29	9	0	0	NUM
iajs-3045	29	10	)	)	PUNCT
iajs-3045	29	11	}	}	PUNCT
iajs-3045	29	12	.	.	PUNCT
iajs-3045	30	1	definition	definition	NOUN
iajs-3045	30	2	2.4[6	2.4[6	NUM
iajs-3045	30	3	]	]	PUNCT
iajs-3045	30	4	.	.	PUNCT
iajs-3045	31	1	an	an	DET
iajs-3045	31	2	ʀ	ʀ	NOUN
iajs-3045	31	3	-	-	PUNCT
iajs-3045	31	4	module	module	NOUN
iajs-3045	31	5	ѡ	ѡ	NOUN
iajs-3045	31	6	is	be	AUX
iajs-3045	31	7	finitely	finitely	ADV
iajs-3045	31	8	generated	generate	VERB
iajs-3045	31	9	if	if	SCONJ
iajs-3045	31	10	ѡ	ѡ	PROPN
iajs-3045	31	11	=	=	SYM
iajs-3045	31	12	ʀ𝑥1	ʀ𝑥1	PROPN
iajs-3045	31	13	+	+	CCONJ
iajs-3045	31	14	ʀ𝑥2	ʀ𝑥2	PROPN
iajs-3045	32	1	+	+	CCONJ
iajs-3045	32	2	⋯	⋯	VERB
iajs-3045	33	1	+	+	CCONJ
iajs-3045	33	2	ʀ𝑥𝑛	ʀ𝑥𝑛	VERB
iajs-3045	33	3	for	for	ADP
iajs-3045	33	4	𝑥1	𝑥1	NOUN
iajs-3045	33	5	,	,	PUNCT
iajs-3045	33	6	𝑥2	𝑥2	NOUN
iajs-3045	33	7	,	,	PUNCT
iajs-3045	33	8	…	…	PUNCT
iajs-3045	33	9	..	..	PUNCT
iajs-3045	33	10	,	,	PUNCT
iajs-3045	33	11	𝑥𝑛	𝑥𝑛	PROPN
iajs-3045	33	12	∈	∈	PROPN
iajs-3045	33	13	ѡ.	ѡ.	NOUN
iajs-3045	33	14	definition	definition	NOUN
iajs-3045	33	15	2.5[7	2.5[7	NOUN
iajs-3045	33	16	]	]	X
iajs-3045	33	17	.	.	PUNCT
iajs-3045	34	1	an	an	DET
iajs-3045	34	2	ʀ	ʀ	NOUN
iajs-3045	34	3	-	-	PUNCT
iajs-3045	34	4	module	module	NOUN
iajs-3045	34	5	ѡ	ѡ	NOUN
iajs-3045	34	6	is	be	AUX
iajs-3045	34	7	called	call	VERB
iajs-3045	34	8	concellation	concellation	NOUN
iajs-3045	34	9	module	module	NOUN
iajs-3045	34	10	if	if	SCONJ
iajs-3045	34	11	ƥѡ	ƥѡ	X
iajs-3045	34	12	=	=	SYM
iajs-3045	34	13	ɓѡ	ɓѡ	PROPN
iajs-3045	34	14	for	for	ADP
iajs-3045	34	15	any	any	DET
iajs-3045	34	16	ideals	ideal	NOUN
iajs-3045	34	17	ƥ	ƥ	PROPN
iajs-3045	34	18	and	and	CCONJ
iajs-3045	34	19	ɓ	ɓ	PRON
iajs-3045	34	20	of	of	ADP
iajs-3045	34	21	ʀ	ʀ	PROPN
iajs-3045	34	22	implies	imply	VERB
iajs-3045	35	1	that	that	SCONJ
iajs-3045	35	2	ƥ	ƥ	PROPN
iajs-3045	35	3	=	=	SYM
iajs-3045	35	4	ɓ	ɓ	X
iajs-3045	35	5	.	.	PUNCT
iajs-3045	36	1	lemma	lemma	PROPN
iajs-3045	36	2	2.6	2.6	NUM
iajs-3045	36	3	[	[	SYM
iajs-3045	36	4	5	5	NUM
iajs-3045	36	5	,	,	PUNCT
iajs-3045	36	6	coro	coro	X
iajs-3045	36	7	.	.	PUNCT
iajs-3045	37	1	(	(	PUNCT
iajs-3045	37	2	2.14	2.14	NUM
iajs-3045	37	3	)	)	PUNCT
iajs-3045	37	4	(	(	PUNCT
iajs-3045	37	5	i	i	NOUN
iajs-3045	37	6	)	)	PUNCT
iajs-3045	37	7	]	]	PUNCT
iajs-3045	37	8	.	.	PUNCT
iajs-3045	38	1	let	let	VERB
iajs-3045	38	2	ѡ	ѡ	PRON
iajs-3045	38	3	be	be	AUX
iajs-3045	38	4	faithful	faithful	ADJ
iajs-3045	38	5	multiplication	multiplication	NOUN
iajs-3045	38	6	ʀ	ʀ	NOUN
iajs-3045	38	7	-	-	PUNCT
iajs-3045	38	8	module	module	NOUN
iajs-3045	38	9	,	,	PUNCT
iajs-3045	38	10	then	then	ADV
iajs-3045	38	11	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	PROPN
iajs-3045	38	12	=	=	SYM
iajs-3045	38	13	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	38	14	)	)	PUNCT
iajs-3045	38	15	.	.	PUNCT
iajs-3045	39	1	lemma	lemma	PROPN
iajs-3045	39	2	2.7	2.7	NUM
iajs-3045	39	3	[	[	PUNCT
iajs-3045	39	4	8	8	NUM
iajs-3045	39	5	,	,	PUNCT
iajs-3045	39	6	coro	coro	X
iajs-3045	39	7	.	.	PUNCT
iajs-3045	40	1	(	(	PUNCT
iajs-3045	40	2	2.14	2.14	NUM
iajs-3045	40	3	)	)	PUNCT
iajs-3045	40	4	(	(	PUNCT
iajs-3045	40	5	i	i	NOUN
iajs-3045	40	6	)	)	PUNCT
iajs-3045	40	7	]	]	PUNCT
iajs-3045	40	8	.	.	PUNCT
iajs-3045	41	1	let	let	VERB
iajs-3045	41	2	ѡ	ѡ	PRON
iajs-3045	41	3	be	be	AUX
iajs-3045	41	4	faithful	faithful	ADJ
iajs-3045	41	5	multiplication	multiplication	NOUN
iajs-3045	41	6	ʀ	ʀ	NOUN
iajs-3045	41	7	-	-	PUNCT
iajs-3045	41	8	module	module	NOUN
iajs-3045	41	9	,	,	PUNCT
iajs-3045	41	10	then	then	ADV
iajs-3045	41	11	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	VERB
iajs-3045	41	12	=	=	PRON
iajs-3045	41	13	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	41	14	)	)	PUNCT
iajs-3045	41	15	.	.	PUNCT
iajs-3045	42	1	definition	definition	NOUN
iajs-3045	42	2	2.8[6	2.8[6	NUM
iajs-3045	42	3	]	]	PUNCT
iajs-3045	42	4	.	.	PUNCT
iajs-3045	43	1	an	an	DET
iajs-3045	43	2	ʀ	ʀ	NOUN
iajs-3045	43	3	-	-	PUNCT
iajs-3045	43	4	module	module	NOUN
iajs-3045	43	5	ѡ	ѡ	NOUN
iajs-3045	43	6	is	be	AUX
iajs-3045	43	7	a	a	DET
iajs-3045	43	8	projective	projective	NOUN
iajs-3045	43	9	if	if	SCONJ
iajs-3045	43	10	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3045	43	11	𝑎𝑛𝑦	𝑎𝑛𝑦	VERB
iajs-3045	43	12	ʀ	ʀ	NOUN
iajs-3045	43	13	-	-	PUNCT
iajs-3045	43	14	epimorphism	epimorphism	NOUN
iajs-3045	43	15	𝑓	𝑓	PRON
iajs-3045	43	16	from	from	ADP
iajs-3045	43	17	an	an	DET
iajs-3045	43	18	ʀ	ʀ	NOUN
iajs-3045	43	19	-	-	PUNCT
iajs-3045	43	20	module	module	NOUN
iajs-3045	43	21	ѡ	ѡ	NOUN
iajs-3045	43	22	on	on	ADP
iajs-3045	43	23	to	to	ADP
iajs-3045	43	24	an	an	DET
iajs-3045	43	25	ʀmodule	ʀmodule	NOUN
iajs-3045	43	26	ѡ̅	ѡ̅	PROPN
iajs-3045	43	27	and	and	CCONJ
iajs-3045	43	28	for	for	ADP
iajs-3045	43	29	any	any	DET
iajs-3045	43	30	homomorphism	homomorphism	NOUN
iajs-3045	43	31	𝑔	𝑔	NOUN
iajs-3045	43	32	from	from	ADP
iajs-3045	43	33	an	an	DET
iajs-3045	43	34	ʀ	ʀ	NOUN
iajs-3045	43	35	-	-	PUNCT
iajs-3045	43	36	module	module	NOUN
iajs-3045	43	37	ѡ̿	ѡ̿	NUM
iajs-3045	43	38	to	to	ADP
iajs-3045	43	39	ѡ̅	ѡ̅	PROPN
iajs-3045	43	40	,	,	PUNCT
iajs-3045	43	41	there	there	PRON
iajs-3045	43	42	exists	exist	VERB
iajs-3045	43	43	a	a	DET
iajs-3045	43	44	homomorphism	homomorphism	NOUN
iajs-3045	43	45	ℎ	ℎ	NOUN
iajs-3045	43	46	from	from	ADP
iajs-3045	43	47	ѡ̿	ѡ̿	NUM
iajs-3045	43	48	to	to	ADP
iajs-3045	43	49	ѡ	ѡ	SYM
iajs-3045	43	50	such	such	ADJ
iajs-3045	43	51	that	that	SCONJ
iajs-3045	43	52	𝑓	𝑓	DET
iajs-3045	43	53	∘	∘	NOUN
iajs-3045	43	54	ℎ	ℎ	X
iajs-3045	43	55	=	=	SYM
iajs-3045	43	56	𝑔.	𝑔.	PROPN
iajs-3045	43	57	lemma	lemma	PROPN
iajs-3045	43	58	2.9	2.9	NUM
iajs-3045	43	59	[	[	SYM
iajs-3045	43	60	6	6	NUM
iajs-3045	43	61	,	,	PUNCT
iajs-3045	43	62	theo	theo	PROPN
iajs-3045	43	63	.	.	PUNCT
iajs-3045	44	1	(	(	PUNCT
iajs-3045	44	2	9.2.1	9.2.1	NUM
iajs-3045	44	3	)	)	PUNCT
iajs-3045	44	4	(	(	PUNCT
iajs-3045	44	5	g	g	NOUN
iajs-3045	44	6	)	)	PUNCT
iajs-3045	44	7	]	]	PUNCT
iajs-3045	44	8	.	.	PUNCT
iajs-3045	45	1	for	for	ADP
iajs-3045	45	2	any	any	DET
iajs-3045	45	3	projective	projective	ADJ
iajs-3045	45	4	ʀ	ʀ	NOUN
iajs-3045	45	5	-	-	PUNCT
iajs-3045	45	6	module	module	NOUN
iajs-3045	45	7	ѡ	ѡ	NOUN
iajs-3045	45	8	,	,	PUNCT
iajs-3045	45	9	we	we	PRON
iajs-3045	45	10	have	have	VERB
iajs-3045	45	11	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	NOUN
iajs-3045	45	12	=	=	SYM
iajs-3045	45	13	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	45	14	)	)	PUNCT
iajs-3045	45	15	.	.	PUNCT
iajs-3045	46	1	lemma	lemma	PROPN
iajs-3045	46	2	2.10	2.10	NUM
iajs-3045	46	3	[	[	SYM
iajs-3045	46	4	8	8	NUM
iajs-3045	46	5	,	,	PUNCT
iajs-3045	46	6	prop	prop	NOUN
iajs-3045	46	7	.	.	PUNCT
iajs-3045	47	1	(	(	PUNCT
iajs-3045	47	2	3.24	3.24	NUM
iajs-3045	47	3	)	)	PUNCT
iajs-3045	47	4	]	]	PUNCT
iajs-3045	47	5	.	.	PUNCT
iajs-3045	48	1	for	for	ADP
iajs-3045	48	2	any	any	DET
iajs-3045	48	3	projective	projective	ADJ
iajs-3045	48	4	ʀ	ʀ	NOUN
iajs-3045	48	5	-	-	PUNCT
iajs-3045	48	6	module	module	NOUN
iajs-3045	48	7	ѡ	ѡ	NOUN
iajs-3045	48	8	,	,	PUNCT
iajs-3045	48	9	we	we	PRON
iajs-3045	48	10	have	have	VERB
iajs-3045	48	11	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	48	12	=	=	SYM
iajs-3045	48	13	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	48	14	)	)	PUNCT
iajs-3045	48	15	.	.	PUNCT
iajs-3045	49	1	remark	remark	VERB
iajs-3045	49	2	2.11[6	2.11[6	NOUN
iajs-3045	49	3	]	]	X
iajs-3045	49	4	.	.	PUNCT
iajs-3045	50	1	ʀ	ʀ	PROPN
iajs-3045	50	2	is	be	AUX
iajs-3045	50	3	a	a	DET
iajs-3045	50	4	good	good	ADJ
iajs-3045	50	5	ring	ring	NOUN
iajs-3045	50	6	if	if	SCONJ
iajs-3045	50	7	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	NOUN
iajs-3045	50	8	=	=	ADJ
iajs-3045	50	9	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	50	10	)	)	PUNCT
iajs-3045	50	11	.	.	PUNCT
iajs-3045	51	1	definition	definition	NOUN
iajs-3045	51	2	2.12[9	2.12[9	NUM
iajs-3045	51	3	]	]	X
iajs-3045	51	4	.	.	PUNCT
iajs-3045	52	1	aring	are	VERB
iajs-3045	52	2	ʀ	ʀ	PRON
iajs-3045	52	3	is	be	AUX
iajs-3045	52	4	artinian	artinian	ADJ
iajs-3045	52	5	if	if	SCONJ
iajs-3045	52	6	ʀ	ʀ	PROPN
iajs-3045	52	7	satisfies	satisfie	NOUN
iajs-3045	52	8	(	(	PUNCT
iajs-3045	52	9	dcc	dcc	PROPN
iajs-3045	52	10	)	)	PUNCT
iajs-3045	52	11	is	be	AUX
iajs-3045	52	12	an	an	DET
iajs-3045	52	13	ideals	ideal	NOUN
iajs-3045	52	14	of	of	ADP
iajs-3045	52	15	ʀ	ʀ	NOUN
iajs-3045	52	16	,	,	PUNCT
iajs-3045	52	17	that	that	PRON
iajs-3045	52	18	is	be	AUX
iajs-3045	52	19	if	if	SCONJ
iajs-3045	52	20	{	{	PUNCT
iajs-3045	52	21	ƥ∝}∝∈⋀	ƥ∝}∝∈⋀	PROPN
iajs-3045	52	22	is	be	AUX
iajs-3045	52	23	a	a	DET
iajs-3045	52	24	family	family	NOUN
iajs-3045	52	25	of	of	ADP
iajs-3045	52	26	ideals	ideal	NOUN
iajs-3045	52	27	of	of	ADP
iajs-3045	52	28	ʀ	ʀ	NOUN
iajs-3045	52	29	such	such	ADJ
iajs-3045	52	30	that	that	SCONJ
iajs-3045	52	31	ƥ1	ƥ1	NOUN
iajs-3045	52	32	⊇	⊇	ADJ
iajs-3045	52	33	ƥ2	ƥ2	PROPN
iajs-3045	52	34	⊇	⊇	NOUN
iajs-3045	52	35	⋯	⋯	PROPN
iajs-3045	52	36	,	,	PUNCT
iajs-3045	52	37	then	then	ADV
iajs-3045	52	38	∃ɱ	∃ɱ	PROPN
iajs-3045	52	39	∈	∈	PROPN
iajs-3045	52	40	ȥ+	ȥ+	VERB
iajs-3045	52	41	such	such	ADJ
iajs-3045	52	42	that	that	SCONJ
iajs-3045	52	43	ƥ𝑛	ƥ𝑛	NOUN
iajs-3045	52	44	=	=	SYM
iajs-3045	52	45	ƥɱ	ƥɱ	NOUN
iajs-3045	52	46	for	for	ADP
iajs-3045	52	47	any	any	DET
iajs-3045	52	48	𝑛	𝑛	DET
iajs-3045	52	49	≥	≥	NUM
iajs-3045	52	50	ɱ	ɱ	PROPN
iajs-3045	52	51	.	.	PUNCT
iajs-3045	53	1	definition	definition	NOUN
iajs-3045	53	2	2.13[10	2.13[10	NUM
iajs-3045	53	3	]	]	PUNCT
iajs-3045	53	4	.	.	PUNCT
iajs-3045	54	1	aring	are	VERB
iajs-3045	54	2	ʀ	ʀ	PRON
iajs-3045	54	3	is	be	AUX
iajs-3045	54	4	said	say	VERB
iajs-3045	54	5	to	to	PART
iajs-3045	54	6	be	be	AUX
iajs-3045	54	7	local	local	ADJ
iajs-3045	54	8	ring	ring	NOUN
iajs-3045	54	9	ʀ	ʀ	NOUN
iajs-3045	54	10	if	if	SCONJ
iajs-3045	54	11	ʀ	ʀ	PROPN
iajs-3045	54	12	has	have	VERB
iajs-3045	54	13	a	a	DET
iajs-3045	54	14	unique	unique	ADJ
iajs-3045	54	15	maximal	maximal	ADJ
iajs-3045	54	16	ideal	ideal	NOUN
iajs-3045	54	17	.	.	PUNCT
iajs-3045	55	1	ihjpas	ihjpas	PROPN
iajs-3045	55	2	.	.	PUNCT
iajs-3045	56	1	36(2)2023	36(2)2023	NUM
iajs-3045	56	2	409	409	NUM
iajs-3045	56	3	lemma	lemma	PROPN
iajs-3045	56	4	2.14	2.14	NUM
iajs-3045	56	5	[	[	SYM
iajs-3045	56	6	6	6	NUM
iajs-3045	56	7	,	,	PUNCT
iajs-3045	56	8	coro	coro	X
iajs-3045	56	9	.	.	PUNCT
iajs-3045	57	1	(	(	PUNCT
iajs-3045	57	2	9.7.3	9.7.3	NUM
iajs-3045	57	3	)	)	PUNCT
iajs-3045	57	4	(	(	PUNCT
iajs-3045	57	5	b	b	NOUN
iajs-3045	57	6	)	)	PUNCT
iajs-3045	57	7	]	]	PUNCT
iajs-3045	57	8	.	.	PUNCT
iajs-3045	58	1	if	if	SCONJ
iajs-3045	58	2	ʀ	ʀ	NOUN
iajs-3045	58	3	is	be	AUX
iajs-3045	58	4	an	an	DET
iajs-3045	58	5	artinian	artinian	ADJ
iajs-3045	58	6	ring	ring	NOUN
iajs-3045	58	7	,	,	PUNCT
iajs-3045	58	8	then	then	ADV
iajs-3045	58	9	ʀ	ʀ	PROPN
iajs-3045	58	10	is	be	AUX
iajs-3045	58	11	a	a	DET
iajs-3045	58	12	good	good	ADJ
iajs-3045	58	13	ring	ring	NOUN
iajs-3045	58	14	.	.	PUNCT
iajs-3045	59	1	lemma	lemma	PROPN
iajs-3045	59	2	2.15	2.15	NUM
iajs-3045	59	3	[	[	PUNCT
iajs-3045	59	4	11	11	NUM
iajs-3045	59	5	,	,	PUNCT
iajs-3045	59	6	prop	prop	NOUN
iajs-3045	59	7	.	.	PUNCT
iajs-3045	60	1	(	(	PUNCT
iajs-3045	60	2	1.12	1.12	NUM
iajs-3045	60	3	)	)	PUNCT
iajs-3045	60	4	]	]	PUNCT
iajs-3045	60	5	.	.	PUNCT
iajs-3045	61	1	if	if	SCONJ
iajs-3045	61	2	ѡ	ѡ	PROPN
iajs-3045	61	3	is	be	AUX
iajs-3045	61	4	an	an	DET
iajs-3045	61	5	ʀ	ʀ	NOUN
iajs-3045	61	6	-	-	PUNCT
iajs-3045	61	7	module	module	NOUN
iajs-3045	61	8	over	over	ADP
iajs-3045	61	9	local	local	ADJ
iajs-3045	61	10	ring	ring	NOUN
iajs-3045	61	11	ʀ	ʀ	NOUN
iajs-3045	61	12	,	,	PUNCT
iajs-3045	61	13	then	then	ADV
iajs-3045	61	14	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	VERB
iajs-3045	61	15	=	=	PRON
iajs-3045	61	16	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	61	17	)	)	PUNCT
iajs-3045	61	18	.	.	PUNCT
iajs-3045	62	1	definition	definition	NOUN
iajs-3045	62	2	2.16[12	2.16[12	PROPN
iajs-3045	62	3	]	]	X
iajs-3045	62	4	.	.	PUNCT
iajs-3045	63	1	an	an	DET
iajs-3045	63	2	ʀ	ʀ	NOUN
iajs-3045	63	3	-	-	PUNCT
iajs-3045	63	4	module	module	NOUN
iajs-3045	63	5	ѡ	ѡ	NOUN
iajs-3045	63	6	is	be	AUX
iajs-3045	63	7	𝑛𝑜𝑛-𝑠𝑖𝑛𝑔𝑢𝑙𝑎𝑟	𝑛𝑜𝑛-𝑠𝑖𝑛𝑔𝑢𝑙𝑎𝑟	NOUN
iajs-3045	63	8	if	if	SCONJ
iajs-3045	63	9	ȥ(ѡ	ȥ(ѡ	NOUN
iajs-3045	63	10	)	)	PUNCT
iajs-3045	63	11	=	=	SYM
iajs-3045	63	12	ѡ	ѡ	NOUN
iajs-3045	63	13	,	,	PUNCT
iajs-3045	63	14	where	where	SCONJ
iajs-3045	63	15	ȥ(ѡ	ȥ(ѡ	NOUN
iajs-3045	63	16	)	)	PUNCT
iajs-3045	63	17	=	=	SYM
iajs-3045	63	18	{	{	PUNCT
iajs-3045	63	19	𝑥	𝑥	PUNCT
iajs-3045	63	20	∈	∈	NOUN
iajs-3045	63	21	ѡ	ѡ	NOUN
iajs-3045	63	22	:	:	PUNCT
iajs-3045	63	23	𝑥ƥ	𝑥ƥ	PROPN
iajs-3045	63	24	=	=	SYM
iajs-3045	63	25	(	(	PUNCT
iajs-3045	63	26	0	0	NUM
iajs-3045	63	27	)	)	PUNCT
iajs-3045	63	28	,	,	PUNCT
iajs-3045	63	29	for	for	ADP
iajs-3045	63	30	some	some	DET
iajs-3045	63	31	essential	essential	ADJ
iajs-3045	63	32	ideal	ideal	ADJ
iajs-3045	63	33	ƥ	ƥ	PROPN
iajs-3045	63	34	of	of	ADP
iajs-3045	63	35	r	r	NOUN
iajs-3045	63	36	}	}	PUNCT
iajs-3045	63	37	.	.	PUNCT
iajs-3045	64	1	lemma	lemma	PROPN
iajs-3045	64	2	2.17	2.17	NUM
iajs-3045	64	3	[	[	PUNCT
iajs-3045	64	4	12	12	NUM
iajs-3045	64	5	,	,	PUNCT
iajs-3045	64	6	coro	coro	NOUN
iajs-3045	64	7	.	.	PUNCT
iajs-3045	65	1	(	(	PUNCT
iajs-3045	65	2	1.26	1.26	NUM
iajs-3045	65	3	)	)	PUNCT
iajs-3045	65	4	]	]	PUNCT
iajs-3045	65	5	.	.	PUNCT
iajs-3045	66	1	let	let	VERB
iajs-3045	66	2	ѡ	ѡ	PRON
iajs-3045	66	3	be	be	AUX
iajs-3045	66	4	is	be	AUX
iajs-3045	66	5	a	a	DET
iajs-3045	66	6	non	non	ADJ
iajs-3045	66	7	-	-	ADJ
iajs-3045	66	8	singular	singular	ADJ
iajs-3045	66	9	ʀ	ʀ	NOUN
iajs-3045	66	10	-	-	PUNCT
iajs-3045	66	11	modules	module	NOUN
iajs-3045	66	12	,	,	PUNCT
iajs-3045	66	13	then	then	ADV
iajs-3045	66	14	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	PROPN
iajs-3045	66	15	=	=	SYM
iajs-3045	66	16	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	66	17	)	)	PUNCT
iajs-3045	66	18	.	.	PUNCT
iajs-3045	67	1	lemma	lemma	PROPN
iajs-3045	67	2	2.18	2.18	NUM
iajs-3045	67	3	[	[	PUNCT
iajs-3045	67	4	13	13	NUM
iajs-3045	67	5	,	,	PUNCT
iajs-3045	67	6	coro	coro	NOUN
iajs-3045	67	7	of	of	ADP
iajs-3045	67	8	theo	theo	PROPN
iajs-3045	67	9	.	.	PUNCT
iajs-3045	68	1	(	(	PUNCT
iajs-3045	68	2	9	9	NUM
iajs-3045	68	3	)	)	PUNCT
iajs-3045	68	4	]	]	PUNCT
iajs-3045	68	5	.	.	PUNCT
iajs-3045	69	1	let	let	VERB
iajs-3045	69	2	ѡ	ѡ	PRON
iajs-3045	69	3	be	be	AUX
iajs-3045	69	4	a	a	DET
iajs-3045	69	5	finitely	finitely	ADV
iajs-3045	69	6	generated	generate	VERB
iajs-3045	69	7	multiplication	multiplication	NOUN
iajs-3045	69	8	ʀ	ʀ	NOUN
iajs-3045	69	9	-	-	PUNCT
iajs-3045	69	10	module	module	NOUN
iajs-3045	69	11	ƥ	ƥ	NOUN
iajs-3045	69	12	and	and	CCONJ
iajs-3045	69	13	ɓ	ɓ	PROPN
iajs-3045	69	14	are	be	AUX
iajs-3045	69	15	ideals	ideal	NOUN
iajs-3045	69	16	of	of	ADP
iajs-3045	69	17	ʀ	ʀ	NOUN
iajs-3045	69	18	.	.	PUNCT
iajs-3045	70	1	then	then	ADV
iajs-3045	70	2	ƥѡ	ƥѡ	VERB
iajs-3045	70	3	⊆	⊆	NUM
iajs-3045	70	4	ɓѡ	ɓѡ	NOUN
iajs-3045	70	5	if	if	SCONJ
iajs-3045	71	1	and	and	CCONJ
iajs-3045	71	2	only	only	ADV
iajs-3045	71	3	if	if	SCONJ
iajs-3045	71	4	ƥ	ƥ	PRON
iajs-3045	71	5	⊆	⊆	NUM
iajs-3045	71	6	ɓ	ɓ	NOUN
iajs-3045	71	7	+	+	X
iajs-3045	71	8	𝑎𝑛𝑛ʀ(ѡ	𝑎𝑛𝑛ʀ(ѡ	NOUN
iajs-3045	71	9	)	)	PUNCT
iajs-3045	71	10	.	.	PUNCT
iajs-3045	72	1	definition	definition	NOUN
iajs-3045	72	2	2.19[14	2.19[14	NUM
iajs-3045	72	3	]	]	X
iajs-3045	72	4	.	.	PUNCT
iajs-3045	73	1	an	an	DET
iajs-3045	73	2	ʀ	ʀ	NOUN
iajs-3045	73	3	-	-	PUNCT
iajs-3045	73	4	module	module	NOUN
iajs-3045	73	5	ѡ	ѡ	NOUN
iajs-3045	73	6	is	be	AUX
iajs-3045	73	7	called	call	VERB
iajs-3045	73	8	a	a	DET
iajs-3045	73	9	𝑍-regular	𝑍-regular	PROPN
iajs-3045	73	10	if	if	SCONJ
iajs-3045	73	11	for	for	SCONJ
iajs-3045	73	12	each	each	DET
iajs-3045	73	13	𝑒	𝑒	PROPN
iajs-3045	73	14	∈	∈	PROPN
iajs-3045	73	15	ѡ	ѡ	AUX
iajs-3045	73	16	𝑡ℎ𝑒𝑟𝑒	𝑡ℎ𝑒𝑟𝑒	VERB
iajs-3045	73	17	𝑒𝑥𝑖𝑠𝑡𝑠	𝑒𝑥𝑖𝑠𝑡𝑠	NOUN
iajs-3045	73	18	𝑓	𝑓	PROPN
iajs-3045	73	19	∈	∈	PROPN
iajs-3045	73	20	ѡ′	ѡ′	PROPN
iajs-3045	73	21	=	=	SYM
iajs-3045	73	22	𝐻𝑜𝑚ʀ(ѡ	𝐻𝑜𝑚ʀ(ѡ	ADJ
iajs-3045	73	23	,	,	PUNCT
iajs-3045	73	24	ʀ	ʀ	NOUN
iajs-3045	73	25	)	)	PUNCT
iajs-3045	73	26	such	such	ADJ
iajs-3045	73	27	that	that	SCONJ
iajs-3045	73	28	𝑒	𝑒	PROPN
iajs-3045	73	29	=	=	SYM
iajs-3045	73	30	𝑓(𝑒)𝑒.	𝑓(𝑒)𝑒.	PROPN
iajs-3045	73	31	definition	definition	NOUN
iajs-3045	73	32	2.20[15	2.20[15	NUM
iajs-3045	73	33	]	]	PUNCT
iajs-3045	73	34	.	.	PUNCT
iajs-3045	74	1	an	an	DET
iajs-3045	74	2	ʀ	ʀ	NOUN
iajs-3045	74	3	-	-	PUNCT
iajs-3045	74	4	module	module	NOUN
iajs-3045	74	5	ѡ	ѡ	NOUN
iajs-3045	74	6	is	be	AUX
iajs-3045	74	7	called	call	VERB
iajs-3045	74	8	weak	weak	ADJ
iajs-3045	74	9	cancellation	cancellation	NOUN
iajs-3045	74	10	if	if	SCONJ
iajs-3045	74	11	ɓѡ	ɓѡ	PROPN
iajs-3045	74	12	=	=	SYM
iajs-3045	74	13	ƥѡ	ƥѡ	PROPN
iajs-3045	74	14	,	,	PUNCT
iajs-3045	74	15	implies	imply	VERB
iajs-3045	74	16	that	that	SCONJ
iajs-3045	74	17	ɓ	ɓ	PRON
iajs-3045	74	18	+	+	X
iajs-3045	74	19	𝑎𝑛𝑛ʀ(ѡ	𝑎𝑛𝑛ʀ(ѡ	NOUN
iajs-3045	74	20	)	)	PUNCT
iajs-3045	75	1	=	=	SYM
iajs-3045	75	2	ƥ	ƥ	PROPN
iajs-3045	76	1	+	+	X
iajs-3045	76	2	𝑎𝑛𝑛ʀ(ѡ	𝑎𝑛𝑛ʀ(ѡ	NOUN
iajs-3045	76	3	)	)	PUNCT
iajs-3045	76	4	for	for	ADP
iajs-3045	76	5	ɓ	ɓ	PRON
iajs-3045	76	6	,	,	PUNCT
iajs-3045	76	7	ƥ	ƥ	PROPN
iajs-3045	76	8	are	be	AUX
iajs-3045	76	9	ideals	ideal	NOUN
iajs-3045	76	10	in	in	ADP
iajs-3045	76	11	ʀ	ʀ	PROPN
iajs-3045	76	12	.	.	PUNCT
iajs-3045	77	1	lemma	lemma	PROPN
iajs-3045	77	2	2.21	2.21	NUM
iajs-3045	77	3	[	[	SYM
iajs-3045	77	4	8	8	NUM
iajs-3045	77	5	,	,	PUNCT
iajs-3045	77	6	prop	prop	NOUN
iajs-3045	77	7	.	.	PUNCT
iajs-3045	78	1	(	(	PUNCT
iajs-3045	78	2	3.25	3.25	NUM
iajs-3045	78	3	)	)	PUNCT
iajs-3045	78	4	]	]	PUNCT
iajs-3045	78	5	.	.	PUNCT
iajs-3045	79	1	let	let	VERB
iajs-3045	79	2	ѡ	ѡ	PRON
iajs-3045	79	3	be	be	AUX
iajs-3045	79	4	a	a	DET
iajs-3045	79	5	𝑍-regular	𝑍-regular	ADJ
iajs-3045	79	6	ʀ	ʀ	NOUN
iajs-3045	79	7	-	-	PUNCT
iajs-3045	79	8	module	module	NOUN
iajs-3045	79	9	,	,	PUNCT
iajs-3045	79	10	then	then	ADV
iajs-3045	79	11	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	79	12	)	)	PUNCT
iajs-3045	80	1	=	=	SYM
iajs-3045	80	2	𝑠𝑜𝑐(ʀ)ѡ.	𝑠𝑜𝑐(ʀ)ѡ.	PROPN
iajs-3045	80	3	lemma	lemma	PROPN
iajs-3045	80	4	2.22	2.22	NUM
iajs-3045	80	5	[	[	PUNCT
iajs-3045	80	6	7	7	NUM
iajs-3045	80	7	,	,	PUNCT
iajs-3045	80	8	prop	prop	NOUN
iajs-3045	80	9	.	.	PUNCT
iajs-3045	81	1	(	(	PUNCT
iajs-3045	81	2	3.9	3.9	NUM
iajs-3045	81	3	)	)	PUNCT
iajs-3045	81	4	]	]	PUNCT
iajs-3045	81	5	.	.	PUNCT
iajs-3045	82	1	if	if	SCONJ
iajs-3045	82	2	ѡ	ѡ	PROPN
iajs-3045	82	3	is	be	AUX
iajs-3045	82	4	a	a	DET
iajs-3045	82	5	multiplication	multiplication	NOUN
iajs-3045	82	6	ʀ	ʀ	NOUN
iajs-3045	82	7	-	-	PUNCT
iajs-3045	82	8	module	module	NOUN
iajs-3045	82	9	,	,	PUNCT
iajs-3045	82	10	then	then	ADV
iajs-3045	82	11	ѡ	ѡ	PROPN
iajs-3045	82	12	is	be	AUX
iajs-3045	82	13	finitely	finitely	ADV
iajs-3045	82	14	generated	generate	VERB
iajs-3045	82	15	if	if	SCONJ
iajs-3045	82	16	and	and	CCONJ
iajs-3045	82	17	only	only	ADV
iajs-3045	82	18	if	if	SCONJ
iajs-3045	82	19	ѡ	ѡ	PROPN
iajs-3045	82	20	is	be	AUX
iajs-3045	82	21	weak	weak	ADJ
iajs-3045	82	22	cancellation	cancellation	NOUN
iajs-3045	82	23	.	.	PUNCT
iajs-3045	83	1	lemma	lemma	PROPN
iajs-3045	83	2	2.23	2.23	NUM
iajs-3045	83	3	[	[	PUNCT
iajs-3045	83	4	7	7	NUM
iajs-3045	83	5	,	,	PUNCT
iajs-3045	83	6	prop	prop	NOUN
iajs-3045	83	7	.	.	PUNCT
iajs-3045	84	1	(	(	PUNCT
iajs-3045	84	2	3.1	3.1	NUM
iajs-3045	84	3	)	)	PUNCT
iajs-3045	84	4	]	]	PUNCT
iajs-3045	84	5	.	.	PUNCT
iajs-3045	85	1	if	if	SCONJ
iajs-3045	85	2	ѡ	ѡ	PROPN
iajs-3045	85	3	is	be	AUX
iajs-3045	85	4	a	a	DET
iajs-3045	85	5	multiplication	multiplication	NOUN
iajs-3045	85	6	ʀ	ʀ	NOUN
iajs-3045	85	7	-	-	PUNCT
iajs-3045	85	8	module	module	NOUN
iajs-3045	85	9	,	,	PUNCT
iajs-3045	85	10	then	then	ADV
iajs-3045	85	11	ѡ	ѡ	PROPN
iajs-3045	85	12	is	be	AUX
iajs-3045	85	13	concellation	concellation	NOUN
iajs-3045	85	14	if	if	SCONJ
iajs-3045	85	15	and	and	CCONJ
iajs-3045	85	16	only	only	ADV
iajs-3045	85	17	if	if	SCONJ
iajs-3045	85	18	ѡ	ѡ	PROPN
iajs-3045	85	19	is	be	AUX
iajs-3045	85	20	faithful	faithful	ADJ
iajs-3045	85	21	finitely	finitely	ADV
iajs-3045	85	22	generated	generate	VERB
iajs-3045	85	23	.	.	PUNCT
iajs-3045	86	1	proposition	proposition	NOUN
iajs-3045	86	2	2.24	2.24	NUM
iajs-3045	86	3	[	[	SYM
iajs-3045	86	4	4	4	NUM
iajs-3045	86	5	,	,	PUNCT
iajs-3045	86	6	prop	prop	NOUN
iajs-3045	86	7	.	.	PUNCT
iajs-3045	87	1	(	(	PUNCT
iajs-3045	87	2	3.4	3.4	NUM
iajs-3045	87	3	)	)	PUNCT
iajs-3045	87	4	]	]	PUNCT
iajs-3045	87	5	.	.	PUNCT
iajs-3045	88	1	a	a	DET
iajs-3045	88	2	proper	proper	ADJ
iajs-3045	88	3	submodule	submodule	NOUN
iajs-3045	88	4	𝑉	𝑉	PROPN
iajs-3045	88	5	of	of	ADP
iajs-3045	88	6	ѡ	ѡ	PROPN
iajs-3045	88	7	is	be	AUX
iajs-3045	88	8	exnpq2ab	exnpq2ab	PROPN
iajs-3045	88	9	submodule	submodule	NOUN
iajs-3045	88	10	of	of	ADP
iajs-3045	88	11	ѡ	ѡ	PROPN
iajs-3045	88	12	if	if	SCONJ
iajs-3045	89	1	and	and	CCONJ
iajs-3045	89	2	only	only	ADV
iajs-3045	89	3	if	if	SCONJ
iajs-3045	89	4	ɑɓ𝑐ℒ	ɑɓ𝑐ℒ	X
iajs-3045	89	5	⊆	⊆	NUM
iajs-3045	89	6	𝑉	𝑉	PROPN
iajs-3045	89	7	,	,	PUNCT
iajs-3045	89	8	for	for	ADP
iajs-3045	89	9	ɑ	ɑ	PROPN
iajs-3045	89	10	,	,	PUNCT
iajs-3045	89	11	ɓ	ɓ	PROPN
iajs-3045	89	12	,	,	PUNCT
iajs-3045	89	13	𝑐	𝑐	PROPN
iajs-3045	89	14	∈	∈	PROPN
iajs-3045	89	15	ʀ	ʀ	NOUN
iajs-3045	89	16	and	and	CCONJ
iajs-3045	89	17	ℒ	ℒ	PROPN
iajs-3045	89	18	is	be	AUX
iajs-3045	89	19	a	a	DET
iajs-3045	89	20	submodule	submodule	NOUN
iajs-3045	89	21	of	of	ADP
iajs-3045	89	22	ѡ	ѡ	PROPN
iajs-3045	89	23	,	,	PUNCT
iajs-3045	89	24	implies	imply	VERB
iajs-3045	89	25	that	that	SCONJ
iajs-3045	89	26	either	either	CCONJ
iajs-3045	89	27	ɑ𝑐ℒ	ɑ𝑐ℒ	VERB
iajs-3045	89	28	⊆	⊆	NUM
iajs-3045	89	29	𝑉	𝑉	PROPN
iajs-3045	89	30	+	+	CCONJ
iajs-3045	89	31	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	89	32	)	)	PUNCT
iajs-3045	89	33	+	+	PUNCT
iajs-3045	90	1	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	90	2	)	)	PUNCT
iajs-3045	90	3	or	or	CCONJ
iajs-3045	90	4	ɓ𝑐ℒ	ɓ𝑐ℒ	PROPN
iajs-3045	90	5	⊆	⊆	NUM
iajs-3045	90	6	𝑉	𝑉	PROPN
iajs-3045	90	7	+	+	CCONJ
iajs-3045	90	8	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	90	9	)	)	PUNCT
iajs-3045	90	10	+	+	PUNCT
iajs-3045	90	11	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	90	12	)	)	PUNCT
iajs-3045	90	13	or	or	CCONJ
iajs-3045	90	14	ɑɓℒ	ɑɓℒ	NOUN
iajs-3045	90	15	⊆	⊆	NUM
iajs-3045	90	16	𝑉	𝑉	PROPN
iajs-3045	90	17	+	+	CCONJ
iajs-3045	90	18	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	90	19	)	)	PUNCT
iajs-3045	90	20	+	+	PUNCT
iajs-3045	90	21	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	90	22	)	)	PUNCT
iajs-3045	90	23	.	.	PUNCT
iajs-3045	91	1	proposition	proposition	NOUN
iajs-3045	91	2	2.25	2.25	NUM
iajs-3045	91	3	[	[	PUNCT
iajs-3045	91	4	4	4	NUM
iajs-3045	91	5	,	,	PUNCT
iajs-3045	91	6	prop	prop	NOUN
iajs-3045	91	7	.	.	PUNCT
iajs-3045	92	1	(	(	PUNCT
iajs-3045	92	2	3.5	3.5	NUM
iajs-3045	92	3	)	)	PUNCT
iajs-3045	92	4	]	]	PUNCT
iajs-3045	92	5	.	.	PUNCT
iajs-3045	93	1	let	let	VERB
iajs-3045	93	2	ѡ	ѡ	PRON
iajs-3045	93	3	be	be	AUX
iajs-3045	93	4	module	module	NOUN
iajs-3045	93	5	and	and	CCONJ
iajs-3045	93	6	𝑉	𝑉	PROPN
iajs-3045	93	7	⊂	⊂	PROPN
iajs-3045	93	8	ѡ.	ѡ.	NOUN
iajs-3045	93	9	then	then	ADV
iajs-3045	93	10	𝑉	𝑉	PROPN
iajs-3045	93	11	is	be	AUX
iajs-3045	93	12	exnpq2ab	exnpq2ab	PROPN
iajs-3045	93	13	submodule	submodule	NOUN
iajs-3045	93	14	of	of	ADP
iajs-3045	93	15	ѡ	ѡ	PROPN
iajs-3045	93	16	if	if	SCONJ
iajs-3045	94	1	and	and	CCONJ
iajs-3045	94	2	only	only	ADV
iajs-3045	94	3	if	if	SCONJ
iajs-3045	94	4	for	for	ADP
iajs-3045	94	5	every	every	DET
iajs-3045	94	6	submodule	submodule	NOUN
iajs-3045	94	7	𝐴	𝐴	PROPN
iajs-3045	94	8	of	of	ADP
iajs-3045	94	9	ѡ	ѡ	PROPN
iajs-3045	94	10	and	and	CCONJ
iajs-3045	94	11	for	for	ADP
iajs-3045	94	12	every	every	DET
iajs-3045	94	13	ideals	ideal	NOUN
iajs-3045	94	14	ƥ1	ƥ1	VERB
iajs-3045	94	15	,	,	PUNCT
iajs-3045	94	16	ƥ2	ƥ2	NOUN
iajs-3045	94	17	,	,	PUNCT
iajs-3045	94	18	ƥ3	ƥ3	NOUN
iajs-3045	94	19	of	of	ADP
iajs-3045	94	20	ʀ	ʀ	NOUN
iajs-3045	94	21	such	such	ADJ
iajs-3045	94	22	that	that	SCONJ
iajs-3045	94	23	ƥ1ƥ2ƥ3𝐴	ƥ1ƥ2ƥ3𝐴	PROPN
iajs-3045	94	24	⊆	⊆	NUM
iajs-3045	94	25	𝑉	𝑉	PROPN
iajs-3045	94	26	,	,	PUNCT
iajs-3045	94	27	implies	imply	VERB
iajs-3045	94	28	that	that	SCONJ
iajs-3045	94	29	either	either	CCONJ
iajs-3045	94	30	ƥ1ƥ2𝐴	ƥ1ƥ2𝐴	PROPN
iajs-3045	94	31	⊆	⊆	NUM
iajs-3045	94	32	𝑉	𝑉	PROPN
iajs-3045	94	33	+	+	CCONJ
iajs-3045	94	34	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	94	35	)	)	PUNCT
iajs-3045	94	36	+	+	PUNCT
iajs-3045	95	1	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	95	2	)	)	PUNCT
iajs-3045	95	3	or	or	CCONJ
iajs-3045	95	4	ƥ1ƥ3𝐴	ƥ1ƥ3𝐴	PROPN
iajs-3045	95	5	⊆	⊆	NUM
iajs-3045	95	6	𝑉	𝑉	PROPN
iajs-3045	95	7	+	+	CCONJ
iajs-3045	95	8	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	95	9	)	)	PUNCT
iajs-3045	95	10	+	+	PUNCT
iajs-3045	95	11	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	95	12	)	)	PUNCT
iajs-3045	95	13	or	or	CCONJ
iajs-3045	95	14	ƥ2ƥ3𝐴	ƥ2ƥ3𝐴	PROPN
iajs-3045	95	15	⊆	⊆	NUM
iajs-3045	95	16	𝑉	𝑉	PROPN
iajs-3045	95	17	+	+	CCONJ
iajs-3045	95	18	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	95	19	)	)	PUNCT
iajs-3045	96	1	+	+	PUNCT
iajs-3045	97	1	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	97	2	)	)	PUNCT
iajs-3045	97	3	.	.	PUNCT
iajs-3045	98	1	proposition	proposition	NOUN
iajs-3045	98	2	2.26	2.26	NUM
iajs-3045	98	3	[	[	SYM
iajs-3045	98	4	4	4	NUM
iajs-3045	98	5	,	,	PUNCT
iajs-3045	98	6	coro	coro	X
iajs-3045	98	7	.	.	PUNCT
iajs-3045	99	1	(	(	PUNCT
iajs-3045	99	2	3.7	3.7	NUM
iajs-3045	99	3	)	)	PUNCT
iajs-3045	99	4	]	]	PUNCT
iajs-3045	99	5	.	.	PUNCT
iajs-3045	100	1	let	let	VERB
iajs-3045	100	2	ѡ	ѡ	PRON
iajs-3045	100	3	be	be	AUX
iajs-3045	100	4	an	an	DET
iajs-3045	100	5	ʀ	ʀ	NOUN
iajs-3045	100	6	-	-	PUNCT
iajs-3045	100	7	module	module	NOUN
iajs-3045	100	8	and	and	CCONJ
iajs-3045	100	9	𝑉	𝑉	PROPN
iajs-3045	100	10	⊂	⊂	PROPN
iajs-3045	100	11	ѡ.	ѡ.	NOUN
iajs-3045	100	12	then	then	ADV
iajs-3045	100	13	𝑉	𝑉	PROPN
iajs-3045	100	14	is	be	AUX
iajs-3045	100	15	exnpq2ab	exnpq2ab	PROPN
iajs-3045	100	16	submodule	submodule	NOUN
iajs-3045	100	17	of	of	ADP
iajs-3045	100	18	ѡ	ѡ	PROPN
iajs-3045	100	19	if	if	SCONJ
iajs-3045	101	1	and	and	CCONJ
iajs-3045	101	2	only	only	ADV
iajs-3045	101	3	if	if	SCONJ
iajs-3045	101	4	for	for	ADP
iajs-3045	101	5	each	each	DET
iajs-3045	101	6	𝑟	𝑟	DET
iajs-3045	101	7	∈	∈	PROPN
iajs-3045	101	8	ʀ	ʀ	NOUN
iajs-3045	101	9	,	,	PUNCT
iajs-3045	101	10	𝑥	𝑥	DET
iajs-3045	101	11	∈	∈	NOUN
iajs-3045	101	12	ѡ	ѡ	X
iajs-3045	101	13	and	and	CCONJ
iajs-3045	101	14	every	every	DET
iajs-3045	101	15	ideals	ideal	NOUN
iajs-3045	101	16	ƥ	ƥ	PROPN
iajs-3045	101	17	,	,	PUNCT
iajs-3045	101	18	𝐽	𝐽	PROPN
iajs-3045	101	19	of	of	ADP
iajs-3045	101	20	ʀ	ʀ	NOUN
iajs-3045	101	21	with	with	ADP
iajs-3045	101	22	𝑟ƥ𝐽𝑥	𝑟ƥ𝐽𝑥	DET
iajs-3045	101	23	⊆	⊆	NUM
iajs-3045	101	24	𝑉	𝑉	PROPN
iajs-3045	101	25	,	,	PUNCT
iajs-3045	101	26	implies	imply	VERB
iajs-3045	101	27	that	that	SCONJ
iajs-3045	101	28	either	either	CCONJ
iajs-3045	101	29	𝑟ƥ𝑥	𝑟ƥ𝑥	NOUN
iajs-3045	101	30	⊆	⊆	NUM
iajs-3045	101	31	𝑉	𝑉	PROPN
iajs-3045	101	32	+	+	CCONJ
iajs-3045	101	33	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	101	34	)	)	PUNCT
iajs-3045	102	1	+	+	PUNCT
iajs-3045	103	1	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	103	2	)	)	PUNCT
iajs-3045	103	3	or	or	CCONJ
iajs-3045	103	4	𝑟𝐽𝑥	𝑟𝐽𝑥	NUM
iajs-3045	103	5	⊆	⊆	NUM
iajs-3045	103	6	𝑉	𝑉	PROPN
iajs-3045	103	7	+	+	CCONJ
iajs-3045	103	8	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	103	9	)	)	PUNCT
iajs-3045	103	10	+	+	PUNCT
iajs-3045	103	11	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	103	12	)	)	PUNCT
iajs-3045	103	13	or	or	CCONJ
iajs-3045	103	14	ƥ𝐽𝑥	ƥ𝐽𝑥	VERB
iajs-3045	103	15	⊆	⊆	NUM
iajs-3045	103	16	𝑉	𝑉	PROPN
iajs-3045	103	17	+	+	CCONJ
iajs-3045	103	18	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	103	19	)	)	PUNCT
iajs-3045	103	20	+	+	PUNCT
iajs-3045	103	21	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	103	22	)	)	PUNCT
iajs-3045	103	23	.	.	PUNCT
iajs-3045	104	1	proposition	proposition	NOUN
iajs-3045	104	2	2.27	2.27	NUM
iajs-3045	104	3	[	[	PUNCT
iajs-3045	104	4	4	4	NUM
iajs-3045	104	5	,	,	PUNCT
iajs-3045	104	6	coro	coro	X
iajs-3045	104	7	.	.	PUNCT
iajs-3045	105	1	(	(	PUNCT
iajs-3045	105	2	3.8	3.8	NUM
iajs-3045	105	3	)	)	PUNCT
iajs-3045	105	4	]	]	PUNCT
iajs-3045	105	5	.	.	PUNCT
iajs-3045	106	1	ihjpas	ihjpas	PROPN
iajs-3045	106	2	.	.	PUNCT
iajs-3045	107	1	36(2)2023	36(2)2023	NUM
iajs-3045	107	2	410	410	NUM
iajs-3045	107	3	let	let	VERB
iajs-3045	107	4	ѡ	ѡ	PRON
iajs-3045	107	5	be	be	AUX
iajs-3045	107	6	an	an	DET
iajs-3045	107	7	ʀ	ʀ	NOUN
iajs-3045	107	8	-	-	PUNCT
iajs-3045	107	9	module	module	NOUN
iajs-3045	107	10	and	and	CCONJ
iajs-3045	107	11	𝑉	𝑉	PROPN
iajs-3045	107	12	⊂	⊂	PROPN
iajs-3045	107	13	ѡ.	ѡ.	NOUN
iajs-3045	107	14	then	then	ADV
iajs-3045	107	15	𝑉	𝑉	PROPN
iajs-3045	107	16	is	be	AUX
iajs-3045	107	17	exnpq2ab	exnpq2ab	PROPN
iajs-3045	107	18	submodule	submodule	NOUN
iajs-3045	107	19	of	of	ADP
iajs-3045	107	20	ѡ	ѡ	PROPN
iajs-3045	107	21	if	if	SCONJ
iajs-3045	107	22	and	and	CCONJ
iajs-3045	107	23	only	only	ADV
iajs-3045	107	24	if	if	SCONJ
iajs-3045	107	25	for	for	ADP
iajs-3045	107	26	every	every	DET
iajs-3045	107	27	ideals	ideal	NOUN
iajs-3045	107	28	ƥ1	ƥ1	VERB
iajs-3045	107	29	,	,	PUNCT
iajs-3045	107	30	ƥ2	ƥ2	NOUN
iajs-3045	107	31	,	,	PUNCT
iajs-3045	107	32	ƥ3	ƥ3	PROPN
iajs-3045	107	33	of	of	ADP
iajs-3045	107	34	ʀ	ʀ	PROPN
iajs-3045	107	35	and	and	CCONJ
iajs-3045	107	36	𝑥	𝑥	DET
iajs-3045	107	37	∈	∈	NOUN
iajs-3045	107	38	ѡ	ѡ	ADP
iajs-3045	107	39	such	such	ADJ
iajs-3045	107	40	that	that	SCONJ
iajs-3045	107	41	ƥ1ƥ2ƥ3𝑥	ƥ1ƥ2ƥ3𝑥	PROPN
iajs-3045	107	42	⊆	⊆	NUM
iajs-3045	107	43	𝑉	𝑉	PROPN
iajs-3045	107	44	implies	imply	VERB
iajs-3045	107	45	that	that	SCONJ
iajs-3045	107	46	either	either	CCONJ
iajs-3045	107	47	ƥ1ƥ2𝑥	ƥ1ƥ2𝑥	NUM
iajs-3045	107	48	⊆	⊆	X
iajs-3045	107	49	𝑉	𝑉	PROPN
iajs-3045	107	50	+	+	CCONJ
iajs-3045	107	51	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	107	52	)	)	PUNCT
iajs-3045	108	1	+	+	PUNCT
iajs-3045	109	1	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	109	2	)	)	PUNCT
iajs-3045	109	3	or	or	CCONJ
iajs-3045	109	4	ƥ1ƥ3𝑥	ƥ1ƥ3𝑥	NUM
iajs-3045	109	5	⊆	⊆	NUM
iajs-3045	109	6	𝑉	𝑉	PROPN
iajs-3045	109	7	+	+	CCONJ
iajs-3045	109	8	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	109	9	)	)	PUNCT
iajs-3045	109	10	+	+	PUNCT
iajs-3045	109	11	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	109	12	)	)	PUNCT
iajs-3045	109	13	or	or	CCONJ
iajs-3045	109	14	ƥ2ƥ3𝑥	ƥ2ƥ3𝑥	NUM
iajs-3045	109	15	⊆	⊆	X
iajs-3045	109	16	𝑉	𝑉	PROPN
iajs-3045	109	17	+	+	CCONJ
iajs-3045	109	18	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	109	19	)	)	PUNCT
iajs-3045	109	20	+	+	PUNCT
iajs-3045	109	21	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	109	22	)	)	PUNCT
iajs-3045	109	23	.	.	PUNCT
iajs-3045	110	1	proposition	proposition	NOUN
iajs-3045	110	2	2.28	2.28	NUM
iajs-3045	110	3	[	[	SYM
iajs-3045	110	4	4	4	NUM
iajs-3045	110	5	,	,	PUNCT
iajs-3045	110	6	coro	coro	X
iajs-3045	110	7	.	.	PUNCT
iajs-3045	111	1	(	(	PUNCT
iajs-3045	111	2	3.9	3.9	NUM
iajs-3045	111	3	)	)	PUNCT
iajs-3045	111	4	]	]	PUNCT
iajs-3045	111	5	.	.	PUNCT
iajs-3045	112	1	let	let	VERB
iajs-3045	112	2	ѡ	ѡ	PRON
iajs-3045	112	3	be	be	AUX
iajs-3045	112	4	an	an	DET
iajs-3045	112	5	ʀ	ʀ	NOUN
iajs-3045	112	6	-	-	PUNCT
iajs-3045	112	7	module	module	NOUN
iajs-3045	112	8	and	and	CCONJ
iajs-3045	112	9	𝑉	𝑉	PROPN
iajs-3045	112	10	⊂	⊂	PROPN
iajs-3045	112	11	ѡ.	ѡ.	NOUN
iajs-3045	112	12	then	then	ADV
iajs-3045	112	13	𝑉	𝑉	PROPN
iajs-3045	112	14	is	be	AUX
iajs-3045	112	15	exnpq2ab	exnpq2ab	PROPN
iajs-3045	112	16	submodule	submodule	NOUN
iajs-3045	112	17	of	of	ADP
iajs-3045	112	18	ѡ	ѡ	PROPN
iajs-3045	112	19	if	if	SCONJ
iajs-3045	113	1	and	and	CCONJ
iajs-3045	113	2	only	only	ADV
iajs-3045	113	3	if	if	SCONJ
iajs-3045	113	4	for	for	ADP
iajs-3045	113	5	any	any	DET
iajs-3045	113	6	𝑟	𝑟	NOUN
iajs-3045	113	7	,	,	PUNCT
iajs-3045	113	8	𝑠	𝑠	PROPN
iajs-3045	113	9	∈	∈	PROPN
iajs-3045	113	10	ʀ	ʀ	NOUN
iajs-3045	113	11	and	and	CCONJ
iajs-3045	113	12	any	any	DET
iajs-3045	113	13	ideal	ideal	ADJ
iajs-3045	113	14	ƥ	ƥ	NOUN
iajs-3045	113	15	of	of	ADP
iajs-3045	113	16	ʀ	ʀ	NOUN
iajs-3045	113	17	and	and	CCONJ
iajs-3045	113	18	every	every	PRON
iajs-3045	113	19	submodule	submodule	NOUN
iajs-3045	113	20	𝐴	𝐴	PROPN
iajs-3045	113	21	of	of	ADP
iajs-3045	113	22	ѡ	ѡ	PROPN
iajs-3045	113	23	with	with	ADP
iajs-3045	113	24	𝑟𝑠ƥ𝐴	𝑟𝑠ƥ𝐴	NOUN
iajs-3045	113	25	⊆	⊆	NUM
iajs-3045	113	26	𝑉	𝑉	PROPN
iajs-3045	113	27	implies	imply	VERB
iajs-3045	113	28	that	that	SCONJ
iajs-3045	113	29	either	either	CCONJ
iajs-3045	113	30	𝑟𝑠𝐴	𝑟𝑠𝐴	PROPN
iajs-3045	113	31	⊆	⊆	PROPN
iajs-3045	113	32	𝑉	𝑉	PROPN
iajs-3045	113	33	+	+	CCONJ
iajs-3045	113	34	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	113	35	)	)	PUNCT
iajs-3045	113	36	+	+	PUNCT
iajs-3045	113	37	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	113	38	)	)	PUNCT
iajs-3045	113	39	or	or	CCONJ
iajs-3045	113	40	𝑟ƥ𝐴	𝑟ƥ𝐴	PROPN
iajs-3045	113	41	⊆	⊆	PROPN
iajs-3045	113	42	𝑉	𝑉	PROPN
iajs-3045	113	43	+	+	CCONJ
iajs-3045	113	44	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	113	45	)	)	PUNCT
iajs-3045	113	46	+	+	PUNCT
iajs-3045	113	47	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	113	48	)	)	PUNCT
iajs-3045	113	49	or	or	CCONJ
iajs-3045	113	50	𝑠ƥ𝐴	𝑠ƥ𝐴	NOUN
iajs-3045	113	51	⊆	⊆	NUM
iajs-3045	113	52	𝑉	𝑉	PROPN
iajs-3045	113	53	+	+	CCONJ
iajs-3045	113	54	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	113	55	)	)	PUNCT
iajs-3045	113	56	+	+	PUNCT
iajs-3045	114	1	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	114	2	)	)	PUNCT
iajs-3045	114	3	.	.	PUNCT
iajs-3045	115	1	proposition	proposition	NOUN
iajs-3045	115	2	2.29	2.29	NUM
iajs-3045	115	3	[	[	SYM
iajs-3045	115	4	4	4	NUM
iajs-3045	115	5	,	,	PUNCT
iajs-3045	115	6	coro	coro	X
iajs-3045	115	7	.	.	PUNCT
iajs-3045	116	1	(	(	PUNCT
iajs-3045	116	2	3.10	3.10	NUM
iajs-3045	116	3	)	)	PUNCT
iajs-3045	116	4	]	]	PUNCT
iajs-3045	116	5	.	.	PUNCT
iajs-3045	117	1	let	let	VERB
iajs-3045	117	2	ѡ	ѡ	PRON
iajs-3045	117	3	be	be	AUX
iajs-3045	117	4	an	an	DET
iajs-3045	117	5	ʀ	ʀ	NOUN
iajs-3045	117	6	-	-	PUNCT
iajs-3045	117	7	module	module	NOUN
iajs-3045	117	8	and	and	CCONJ
iajs-3045	117	9	𝑉	𝑉	PROPN
iajs-3045	117	10	⊂	⊂	PROPN
iajs-3045	117	11	ѡ.	ѡ.	NOUN
iajs-3045	117	12	then	then	ADV
iajs-3045	117	13	𝑉	𝑉	PROPN
iajs-3045	117	14	is	be	AUX
iajs-3045	117	15	exnpq2ab	exnpq2ab	PROPN
iajs-3045	117	16	submodule	submodule	NOUN
iajs-3045	117	17	of	of	ADP
iajs-3045	117	18	ѡ	ѡ	PROPN
iajs-3045	117	19	if	if	SCONJ
iajs-3045	118	1	and	and	CCONJ
iajs-3045	118	2	only	only	ADV
iajs-3045	118	3	if	if	SCONJ
iajs-3045	118	4	for	for	ADP
iajs-3045	118	5	each	each	DET
iajs-3045	118	6	𝑟	𝑟	PRON
iajs-3045	118	7	∈	∈	PROPN
iajs-3045	118	8	ʀ	ʀ	NOUN
iajs-3045	118	9	and	and	CCONJ
iajs-3045	118	10	any	any	DET
iajs-3045	118	11	ideals	ideal	NOUN
iajs-3045	118	12	ƥ	ƥ	PROPN
iajs-3045	118	13	,	,	PUNCT
iajs-3045	118	14	𝐽	𝐽	PROPN
iajs-3045	118	15	of	of	ADP
iajs-3045	118	16	ʀ	ʀ	PROPN
iajs-3045	118	17	and	and	CCONJ
iajs-3045	118	18	every	every	PRON
iajs-3045	118	19	submodule	submodule	NOUN
iajs-3045	118	20	𝐴	𝐴	PROPN
iajs-3045	118	21	of	of	ADP
iajs-3045	118	22	ѡ	ѡ	PROPN
iajs-3045	118	23	with	with	ADP
iajs-3045	118	24	𝑟ƥ𝐽𝐴	𝑟ƥ𝐽𝐴	NOUN
iajs-3045	118	25	⊆	⊆	NUM
iajs-3045	118	26	𝑉	𝑉	PROPN
iajs-3045	118	27	implies	imply	VERB
iajs-3045	118	28	that	that	SCONJ
iajs-3045	118	29	either	either	CCONJ
iajs-3045	118	30	𝑟ƥ𝐴	𝑟ƥ𝐴	PROPN
iajs-3045	118	31	⊆	⊆	NUM
iajs-3045	118	32	𝑉	𝑉	PROPN
iajs-3045	118	33	+	+	CCONJ
iajs-3045	118	34	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	118	35	)	)	PUNCT
iajs-3045	118	36	+	+	PUNCT
iajs-3045	118	37	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	118	38	)	)	PUNCT
iajs-3045	118	39	or	or	CCONJ
iajs-3045	118	40	𝑟𝐽𝐴	𝑟𝐽𝐴	NOUN
iajs-3045	118	41	⊆	⊆	NUM
iajs-3045	118	42	𝑉	𝑉	PROPN
iajs-3045	118	43	+	+	CCONJ
iajs-3045	118	44	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	118	45	)	)	PUNCT
iajs-3045	118	46	+	+	PUNCT
iajs-3045	119	1	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	119	2	)	)	PUNCT
iajs-3045	119	3	or	or	CCONJ
iajs-3045	119	4	ƥ𝐽𝐴	ƥ𝐽𝐴	ADJ
iajs-3045	119	5	⊆	⊆	NUM
iajs-3045	119	6	𝑉	𝑉	PROPN
iajs-3045	119	7	+	+	CCONJ
iajs-3045	119	8	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	119	9	)	)	PUNCT
iajs-3045	119	10	+	+	PUNCT
iajs-3045	119	11	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	119	12	)	)	PUNCT
iajs-3045	119	13	.	.	PUNCT
iajs-3045	120	1	3	3	X
iajs-3045	120	2	.	.	X
iajs-3045	120	3	main	main	ADJ
iajs-3045	120	4	results	result	NOUN
iajs-3045	120	5	in	in	ADP
iajs-3045	120	6	this	this	DET
iajs-3045	120	7	part	part	NOUN
iajs-3045	120	8	we	we	PRON
iajs-3045	120	9	introduced	introduce	VERB
iajs-3045	120	10	some	some	DET
iajs-3045	120	11	characterizations	characterization	NOUN
iajs-3045	120	12	of	of	ADP
iajs-3045	120	13	extend	extend	NOUN
iajs-3045	120	14	nearly	nearly	ADV
iajs-3045	120	15	pseudo	pseudo	ADJ
iajs-3045	120	16	quasi-2	quasi-2	ADJ
iajs-3045	120	17	-	-	PUNCT
iajs-3045	120	18	absorbing	absorbing	ADJ
iajs-3045	120	19	submodules	submodule	NOUN
iajs-3045	120	20	in	in	ADP
iajs-3045	120	21	multiplication	multiplication	NOUN
iajs-3045	120	22	modules	module	NOUN
iajs-3045	120	23	.	.	PUNCT
iajs-3045	121	1	proposition	proposition	NOUN
iajs-3045	121	2	3.1	3.1	NUM
iajs-3045	121	3	let	let	VERB
iajs-3045	121	4	ѡ	ѡ	PRON
iajs-3045	121	5	be	be	AUX
iajs-3045	121	6	a	a	DET
iajs-3045	121	7	multiplication	multiplication	NOUN
iajs-3045	121	8	ʀ	ʀ	NOUN
iajs-3045	121	9	-	-	PUNCT
iajs-3045	121	10	module	module	NOUN
iajs-3045	121	11	and	and	CCONJ
iajs-3045	121	12	𝑉	𝑉	PROPN
iajs-3045	121	13	≠	≠	PROPN
iajs-3045	121	14	ѡ.	ѡ.	NOUN
iajs-3045	121	15	then	then	ADV
iajs-3045	121	16	𝑉	𝑉	PROPN
iajs-3045	121	17	is	be	AUX
iajs-3045	121	18	exnpq2ab	exnpq2ab	PROPN
iajs-3045	121	19	submodule	submodule	NOUN
iajs-3045	121	20	of	of	ADP
iajs-3045	121	21	ѡ	ѡ	PROPN
iajs-3045	121	22	if	if	SCONJ
iajs-3045	122	1	and	and	CCONJ
iajs-3045	122	2	only	only	ADV
iajs-3045	122	3	if	if	SCONJ
iajs-3045	122	4	whenever	whenever	SCONJ
iajs-3045	122	5	ӈ1ӈ2ӈ3𝐴	ӈ1ӈ2ӈ3𝐴	NOUN
iajs-3045	122	6	⊆	⊆	NUM
iajs-3045	122	7	𝑉	𝑉	PROPN
iajs-3045	122	8	for	for	ADP
iajs-3045	122	9	some	some	DET
iajs-3045	122	10	submodules	submodule	NOUN
iajs-3045	122	11	ӈ1,ӈ2,ӈ3	ӈ1,ӈ2,ӈ3	NOUN
iajs-3045	122	12	,	,	PUNCT
iajs-3045	122	13	𝐴	𝐴	PROPN
iajs-3045	122	14	of	of	ADP
iajs-3045	122	15	ѡ	ѡ	PROPN
iajs-3045	122	16	,	,	PUNCT
iajs-3045	122	17	implies	imply	VERB
iajs-3045	122	18	that	that	SCONJ
iajs-3045	122	19	either	either	CCONJ
iajs-3045	122	20	ӈ1ӈ2𝐴	ӈ1ӈ2𝐴	PROPN
iajs-3045	122	21	⊆	⊆	NUM
iajs-3045	122	22	𝑉	𝑉	PROPN
iajs-3045	122	23	+	+	CCONJ
iajs-3045	122	24	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	122	25	)	)	PUNCT
iajs-3045	122	26	+	+	PUNCT
iajs-3045	123	1	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	123	2	)	)	PUNCT
iajs-3045	123	3	or	or	CCONJ
iajs-3045	123	4	ӈ1ӈ3𝐴	ӈ1ӈ3𝐴	PROPN
iajs-3045	123	5	⊆	⊆	NUM
iajs-3045	123	6	𝑉	𝑉	PROPN
iajs-3045	123	7	+	+	CCONJ
iajs-3045	123	8	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	123	9	)	)	PUNCT
iajs-3045	123	10	+	+	PUNCT
iajs-3045	123	11	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	123	12	)	)	PUNCT
iajs-3045	123	13	or	or	CCONJ
iajs-3045	123	14	ӈ2ӈ3𝐴	ӈ2ӈ3𝐴	X
iajs-3045	123	15	⊆	⊆	NUM
iajs-3045	123	16	𝑉	𝑉	PROPN
iajs-3045	123	17	+	+	CCONJ
iajs-3045	123	18	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	123	19	)	)	PUNCT
iajs-3045	123	20	+	+	PUNCT
iajs-3045	123	21	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	123	22	)	)	PUNCT
iajs-3045	123	23	.	.	PUNCT
iajs-3045	124	1	proof	proof	NOUN
iajs-3045	124	2	.	.	PUNCT
iajs-3045	125	1	(	(	PUNCT
iajs-3045	125	2	⟹	⟹	X
iajs-3045	125	3	)	)	PUNCT
iajs-3045	125	4	let	let	VERB
iajs-3045	125	5	ӈ1ӈ2ӈ3𝐴	ӈ1ӈ2ӈ3𝐴	NOUN
iajs-3045	125	6	⊆	⊆	NUM
iajs-3045	125	7	𝑉	𝑉	PROPN
iajs-3045	125	8	for	for	ADP
iajs-3045	125	9	some	some	DET
iajs-3045	125	10	submodules	submodule	NOUN
iajs-3045	125	11	ӈ1	ӈ1	NOUN
iajs-3045	125	12	,	,	PUNCT
iajs-3045	125	13	ӈ2	ӈ2	NOUN
iajs-3045	125	14	,	,	PUNCT
iajs-3045	125	15	ӈ3	ӈ3	PROPN
iajs-3045	125	16	,	,	PUNCT
iajs-3045	125	17	𝐴	𝐴	PROPN
iajs-3045	125	18	of	of	ADP
iajs-3045	125	19	ѡ.	ѡ.	NOUN
iajs-3045	125	20	since	since	SCONJ
iajs-3045	125	21	ѡ	ѡ	PROPN
iajs-3045	125	22	is	be	AUX
iajs-3045	125	23	a	a	DET
iajs-3045	125	24	multiplication	multiplication	NOUN
iajs-3045	125	25	,	,	PUNCT
iajs-3045	125	26	then	then	ADV
iajs-3045	125	27	ӈ1	ӈ1	NOUN
iajs-3045	125	28	=	=	SYM
iajs-3045	125	29	ƥ1ѡ	ƥ1ѡ	PROPN
iajs-3045	125	30	,	,	PUNCT
iajs-3045	125	31	ӈ2	ӈ2	NOUN
iajs-3045	125	32	=	=	SYM
iajs-3045	125	33	ƥ2ѡ	ƥ2ѡ	NOUN
iajs-3045	125	34	,	,	PUNCT
iajs-3045	125	35	ӈ3	ӈ3	NOUN
iajs-3045	125	36	=	=	SYM
iajs-3045	125	37	ƥ3ѡ	ƥ3ѡ	PROPN
iajs-3045	125	38	and	and	CCONJ
iajs-3045	125	39	𝐴	𝐴	PROPN
iajs-3045	125	40	=	=	PUNCT
iajs-3045	125	41	ƥ4ѡ	ƥ4ѡ	ADJ
iajs-3045	125	42	for	for	ADP
iajs-3045	125	43	some	some	DET
iajs-3045	125	44	ideals	ideal	NOUN
iajs-3045	125	45	ƥ1	ƥ1	VERB
iajs-3045	125	46	,	,	PUNCT
iajs-3045	125	47	ƥ2	ƥ2	NOUN
iajs-3045	125	48	,	,	PUNCT
iajs-3045	125	49	ƥ3	ƥ3	ADJ
iajs-3045	125	50	and	and	CCONJ
iajs-3045	125	51	ƥ4of	ƥ4of	NOUN
iajs-3045	126	1	ʀ	ʀ	X
iajs-3045	126	2	.	.	NOUN
iajs-3045	127	1	that	that	PRON
iajs-3045	127	2	is	be	AUX
iajs-3045	127	3	ӈ1ӈ2ӈ3𝐴	ӈ1ӈ2ӈ3𝐴	NOUN
iajs-3045	127	4	=	=	PUNCT
iajs-3045	127	5	ƥ1ƥ2ƥ3(ƥ4ѡ	ƥ1ƥ2ƥ3(ƥ4ѡ	X
iajs-3045	127	6	)	)	PUNCT
iajs-3045	127	7	⊆	⊆	NUM
iajs-3045	127	8	𝑉.	𝑉.	NOUN
iajs-3045	127	9	but	but	CCONJ
iajs-3045	127	10	𝑉	𝑉	PROPN
iajs-3045	127	11	is	be	AUX
iajs-3045	127	12	exnpq2ab	exnpq2ab	PROPN
iajs-3045	127	13	submodule	submodule	NOUN
iajs-3045	127	14	of	of	ADP
iajs-3045	127	15	ѡ	ѡ	PROPN
iajs-3045	127	16	,	,	PUNCT
iajs-3045	127	17	hence	hence	ADV
iajs-3045	127	18	from	from	ADP
iajs-3045	127	19	proposition	proposition	NOUN
iajs-3045	127	20	2.25	2.25	NUM
iajs-3045	127	21	we	we	PRON
iajs-3045	127	22	get	get	VERB
iajs-3045	127	23	either	either	CCONJ
iajs-3045	127	24	ƥ1ƥ2(ƥ4ѡ	ƥ1ƥ2(ƥ4ѡ	NOUN
iajs-3045	127	25	)	)	PUNCT
iajs-3045	127	26	⊆	⊆	NUM
iajs-3045	127	27	𝑉	𝑉	PROPN
iajs-3045	127	28	+	+	CCONJ
iajs-3045	127	29	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	127	30	)	)	PUNCT
iajs-3045	127	31	+	+	PUNCT
iajs-3045	128	1	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	128	2	)	)	PUNCT
iajs-3045	128	3	or	or	CCONJ
iajs-3045	128	4	ƥ1ƥ3(ƥ4ѡ	ƥ1ƥ3(ƥ4ѡ	PRON
iajs-3045	128	5	)	)	PUNCT
iajs-3045	128	6	⊆	⊆	NUM
iajs-3045	128	7	𝑉	𝑉	PROPN
iajs-3045	128	8	+	+	CCONJ
iajs-3045	128	9	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	128	10	)	)	PUNCT
iajs-3045	129	1	+	+	PUNCT
iajs-3045	130	1	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	130	2	)	)	PUNCT
iajs-3045	130	3	or	or	CCONJ
iajs-3045	130	4	ƥ2ƥ3(ƥ4ѡ	ƥ2ƥ3(ƥ4ѡ	VERB
iajs-3045	130	5	)	)	PUNCT
iajs-3045	130	6	⊆	⊆	NUM
iajs-3045	130	7	𝑉	𝑉	PROPN
iajs-3045	130	8	+	+	CCONJ
iajs-3045	130	9	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	130	10	)	)	PUNCT
iajs-3045	130	11	+	+	PUNCT
iajs-3045	130	12	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	130	13	)	)	PUNCT
iajs-3045	130	14	.	.	PUNCT
iajs-3045	131	1	next	next	ADJ
iajs-3045	131	2	,	,	PUNCT
iajs-3045	131	3	following	follow	VERB
iajs-3045	131	4	either	either	CCONJ
iajs-3045	131	5	ӈ1ӈ2𝐴	ӈ1ӈ2𝐴	PROPN
iajs-3045	131	6	⊆	⊆	NUM
iajs-3045	131	7	𝑉	𝑉	PROPN
iajs-3045	131	8	+	+	CCONJ
iajs-3045	131	9	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	131	10	)	)	PUNCT
iajs-3045	131	11	+	+	PUNCT
iajs-3045	131	12	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	131	13	)	)	PUNCT
iajs-3045	131	14	or	or	CCONJ
iajs-3045	131	15	ӈ1ӈ3𝐴	ӈ1ӈ3𝐴	PROPN
iajs-3045	131	16	⊆	⊆	NUM
iajs-3045	131	17	𝑉	𝑉	PROPN
iajs-3045	131	18	+	+	CCONJ
iajs-3045	131	19	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	131	20	)	)	PUNCT
iajs-3045	131	21	+	+	PUNCT
iajs-3045	131	22	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	131	23	)	)	PUNCT
iajs-3045	131	24	or	or	CCONJ
iajs-3045	131	25	ӈ2ӈ3𝐴	ӈ2ӈ3𝐴	X
iajs-3045	131	26	⊆	⊆	NUM
iajs-3045	131	27	𝑉	𝑉	PROPN
iajs-3045	131	28	+	+	CCONJ
iajs-3045	131	29	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	131	30	)	)	PUNCT
iajs-3045	131	31	+	+	PUNCT
iajs-3045	131	32	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	131	33	)	)	PUNCT
iajs-3045	131	34	.	.	PUNCT
iajs-3045	132	1	ƥ	ƥ	X
iajs-3045	132	2	(	(	PUNCT
iajs-3045	132	3	⟸	⟸	X
iajs-3045	132	4	)	)	PUNCT
iajs-3045	132	5	let	let	VERB
iajs-3045	132	6	ƥ1ƥ2ƥ3𝐴	ƥ1ƥ2ƥ3𝐴	NOUN
iajs-3045	132	7	⊆	⊆	NUM
iajs-3045	132	8	𝑉	𝑉	PROPN
iajs-3045	132	9	for	for	ADP
iajs-3045	132	10	ƥ1	ƥ1	NOUN
iajs-3045	132	11	,	,	PUNCT
iajs-3045	132	12	ƥ2	ƥ2	NOUN
iajs-3045	132	13	,	,	PUNCT
iajs-3045	132	14	ƥ3	ƥ3	PROPN
iajs-3045	132	15	are	be	AUX
iajs-3045	132	16	ideals	ideal	NOUN
iajs-3045	132	17	of	of	ADP
iajs-3045	132	18	ʀ	ʀ	NOUN
iajs-3045	132	19	and	and	CCONJ
iajs-3045	132	20	𝐴	𝐴	PROPN
iajs-3045	132	21	is	be	AUX
iajs-3045	132	22	a	a	DET
iajs-3045	132	23	submodule	submodule	NOUN
iajs-3045	132	24	of	of	ADP
iajs-3045	132	25	ѡ.	ѡ.	NOUN
iajs-3045	132	26	put	put	VERB
iajs-3045	132	27	ӈ1	ӈ1	NOUN
iajs-3045	132	28	=	=	SYM
iajs-3045	132	29	ƥ1ѡ	ƥ1ѡ	PROPN
iajs-3045	132	30	,	,	PUNCT
iajs-3045	132	31	ӈ2	ӈ2	NOUN
iajs-3045	132	32	=	=	SYM
iajs-3045	132	33	ƥ2ѡ	ƥ2ѡ	NOUN
iajs-3045	132	34	and	and	CCONJ
iajs-3045	132	35	ӈ3	ӈ3	NOUN
iajs-3045	132	36	=	=	PUNCT
iajs-3045	133	1	ƥ3ѡ.	ƥ3ѡ.	ADV
iajs-3045	133	2	that	that	PRON
iajs-3045	133	3	is	be	AUX
iajs-3045	133	4	ӈ1ӈ2ӈ3𝐴	ӈ1ӈ2ӈ3𝐴	VERB
iajs-3045	133	5	⊆	⊆	NUM
iajs-3045	133	6	𝑉.	𝑉.	NOUN
iajs-3045	133	7	now	now	ADV
iajs-3045	133	8	,	,	PUNCT
iajs-3045	133	9	by	by	ADP
iajs-3045	133	10	hypotheses	hypothesis	NOUN
iajs-3045	133	11	either	either	CCONJ
iajs-3045	133	12	ӈ1ӈ2𝐴	ӈ1ӈ2𝐴	PROPN
iajs-3045	133	13	⊆	⊆	NUM
iajs-3045	133	14	𝑉	𝑉	PROPN
iajs-3045	133	15	+	+	CCONJ
iajs-3045	133	16	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	133	17	)	)	PUNCT
iajs-3045	133	18	+	+	PUNCT
iajs-3045	134	1	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	134	2	)	)	PUNCT
iajs-3045	134	3	or	or	CCONJ
iajs-3045	134	4	ӈ1ӈ3𝐴	ӈ1ӈ3𝐴	PROPN
iajs-3045	134	5	⊆	⊆	NUM
iajs-3045	134	6	𝑉	𝑉	PROPN
iajs-3045	134	7	+	+	CCONJ
iajs-3045	134	8	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	134	9	)	)	PUNCT
iajs-3045	134	10	+	+	PUNCT
iajs-3045	134	11	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	134	12	)	)	PUNCT
iajs-3045	134	13	or	or	CCONJ
iajs-3045	134	14	ӈ2ӈ3𝐴	ӈ2ӈ3𝐴	X
iajs-3045	134	15	⊆	⊆	NUM
iajs-3045	134	16	𝑉	𝑉	PROPN
iajs-3045	134	17	+	+	CCONJ
iajs-3045	134	18	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	134	19	)	)	PUNCT
iajs-3045	134	20	+	+	PUNCT
iajs-3045	135	1	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	135	2	)	)	PUNCT
iajs-3045	135	3	,	,	PUNCT
iajs-3045	135	4	thus	thus	ADV
iajs-3045	135	5	ƥ1ƥ2𝐴	ƥ1ƥ2𝐴	PROPN
iajs-3045	135	6	⊆	⊆	NUM
iajs-3045	135	7	𝑉	𝑉	PROPN
iajs-3045	135	8	+	+	CCONJ
iajs-3045	135	9	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	135	10	)	)	PUNCT
iajs-3045	135	11	+	+	PUNCT
iajs-3045	136	1	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	136	2	)	)	PUNCT
iajs-3045	136	3	or	or	CCONJ
iajs-3045	136	4	ƥ1ƥ3𝐴	ƥ1ƥ3𝐴	PROPN
iajs-3045	136	5	⊆	⊆	NUM
iajs-3045	136	6	𝑉	𝑉	PROPN
iajs-3045	136	7	+	+	CCONJ
iajs-3045	136	8	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	136	9	)	)	PUNCT
iajs-3045	136	10	+	+	PUNCT
iajs-3045	136	11	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	136	12	)	)	PUNCT
iajs-3045	136	13	or	or	CCONJ
iajs-3045	136	14	ƥ2ƥ3𝐴	ƥ2ƥ3𝐴	PROPN
iajs-3045	136	15	⊆	⊆	NUM
iajs-3045	136	16	𝑉	𝑉	PROPN
iajs-3045	136	17	+	+	CCONJ
iajs-3045	136	18	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	136	19	)	)	PUNCT
iajs-3045	137	1	+	+	PUNCT
iajs-3045	138	1	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	138	2	)	)	PUNCT
iajs-3045	138	3	.	.	PUNCT
iajs-3045	139	1	therefore	therefore	ADV
iajs-3045	139	2	by	by	ADP
iajs-3045	139	3	proposition	proposition	NOUN
iajs-3045	139	4	2.25	2.25	NUM
iajs-3045	139	5	𝑉	𝑉	PROPN
iajs-3045	139	6	is	be	AUX
iajs-3045	139	7	exnpq2ab	exnpq2ab	PROPN
iajs-3045	139	8	submodule	submodule	NOUN
iajs-3045	139	9	of	of	ADP
iajs-3045	139	10	ѡ.	ѡ.	NOUN
iajs-3045	139	11	proposition	proposition	NOUN
iajs-3045	139	12	3.2	3.2	NUM
iajs-3045	139	13	let	let	VERB
iajs-3045	139	14	ѡ	ѡ	PRON
iajs-3045	139	15	be	be	AUX
iajs-3045	139	16	a	a	DET
iajs-3045	139	17	multiplication	multiplication	NOUN
iajs-3045	139	18	ʀ	ʀ	NOUN
iajs-3045	139	19	-	-	PUNCT
iajs-3045	139	20	module	module	NOUN
iajs-3045	139	21	and	and	CCONJ
iajs-3045	139	22	𝑉	𝑉	PROPN
iajs-3045	139	23	≠	≠	PROPN
iajs-3045	139	24	ѡ.	ѡ.	NOUN
iajs-3045	139	25	then	then	ADV
iajs-3045	139	26	𝑉	𝑉	PROPN
iajs-3045	139	27	is	be	AUX
iajs-3045	139	28	exnpq2ab	exnpq2ab	PROPN
iajs-3045	139	29	submodule	submodule	NOUN
iajs-3045	139	30	of	of	ADP
iajs-3045	139	31	ѡ	ѡ	PROPN
iajs-3045	139	32	if	if	SCONJ
iajs-3045	140	1	and	and	CCONJ
iajs-3045	140	2	only	only	ADV
iajs-3045	140	3	if	if	SCONJ
iajs-3045	140	4	whenever	whenever	SCONJ
iajs-3045	140	5	ƒ1ƒ2ƒ3𝑥	ƒ1ƒ2ƒ3𝑥	PROPN
iajs-3045	140	6	⊆	⊆	NUM
iajs-3045	140	7	𝑉	𝑉	PROPN
iajs-3045	140	8	for	for	ADP
iajs-3045	140	9	some	some	DET
iajs-3045	140	10	submodules	submodule	NOUN
iajs-3045	140	11	ƒ1	ƒ1	NOUN
iajs-3045	140	12	,	,	PUNCT
iajs-3045	140	13	ƒ2	ƒ2	PROPN
iajs-3045	140	14	,	,	PUNCT
iajs-3045	140	15	ƒ3	ƒ3	NOUN
iajs-3045	140	16	of	of	ADP
iajs-3045	140	17	ѡ,𝑥	ѡ,𝑥	ADJ
iajs-3045	140	18	∈	∈	PROPN
iajs-3045	140	19	ѡ	ѡ	NOUN
iajs-3045	140	20	,	,	PUNCT
iajs-3045	140	21	then	then	ADV
iajs-3045	140	22	either	either	CCONJ
iajs-3045	140	23	ƒ1ƒ2𝑥	ƒ1ƒ2𝑥	NUM
iajs-3045	140	24	⊆	⊆	NUM
iajs-3045	140	25	𝑉	𝑉	PROPN
iajs-3045	140	26	+	+	CCONJ
iajs-3045	140	27	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	140	28	)	)	PUNCT
iajs-3045	140	29	+	+	PUNCT
iajs-3045	140	30	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	140	31	)	)	PUNCT
iajs-3045	140	32	or	or	CCONJ
iajs-3045	140	33	ƒ1ƒ3𝑥	ƒ1ƒ3𝑥	NUM
iajs-3045	140	34	⊆	⊆	NUM
iajs-3045	140	35	𝑉	𝑉	PROPN
iajs-3045	140	36	+	+	CCONJ
iajs-3045	140	37	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	140	38	)	)	PUNCT
iajs-3045	140	39	+	+	PUNCT
iajs-3045	140	40	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	140	41	)	)	PUNCT
iajs-3045	140	42	or	or	CCONJ
iajs-3045	140	43	ƒ2ƒ3𝑥	ƒ2ƒ3𝑥	NUM
iajs-3045	140	44	⊆	⊆	NUM
iajs-3045	140	45	𝑉	𝑉	PROPN
iajs-3045	140	46	+	+	CCONJ
iajs-3045	140	47	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	140	48	)	)	PUNCT
iajs-3045	140	49	+	+	PUNCT
iajs-3045	140	50	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	140	51	)	)	PUNCT
iajs-3045	140	52	.	.	PUNCT
iajs-3045	141	1	proof	proof	NOUN
iajs-3045	141	2	.	.	PUNCT
iajs-3045	142	1	(	(	PUNCT
iajs-3045	142	2	⟹	⟹	X
iajs-3045	142	3	)	)	PUNCT
iajs-3045	142	4	let	let	VERB
iajs-3045	142	5	𝑉	𝑉	PROPN
iajs-3045	142	6	is	be	AUX
iajs-3045	142	7	exnpq2ab	exnpq2ab	PROPN
iajs-3045	142	8	submodule	submodule	NOUN
iajs-3045	142	9	of	of	ADP
iajs-3045	142	10	ѡ	ѡ	PROPN
iajs-3045	142	11	and	and	CCONJ
iajs-3045	142	12	ƒ1ƒ2ƒ3𝑥	ƒ1ƒ2ƒ3𝑥	PROPN
iajs-3045	142	13	⊆	⊆	NUM
iajs-3045	142	14	𝑉	𝑉	PROPN
iajs-3045	142	15	for	for	ADP
iajs-3045	142	16	some	some	DET
iajs-3045	142	17	submodules	submodule	NOUN
iajs-3045	142	18	ƒ1,ƒ2	ƒ1,ƒ2	PROPN
iajs-3045	142	19	,	,	PUNCT
iajs-3045	142	20	ƒ3	ƒ3	NOUN
iajs-3045	142	21	of	of	ADP
iajs-3045	142	22	ѡ	ѡ	PROPN
iajs-3045	142	23	and	and	CCONJ
iajs-3045	142	24	𝑥	𝑥	PRON
iajs-3045	142	25	∈	∈	PROPN
iajs-3045	142	26	ѡ.	ѡ.	NOUN
iajs-3045	142	27	since	since	SCONJ
iajs-3045	142	28	ѡ	ѡ	PROPN
iajs-3045	142	29	is	be	AUX
iajs-3045	142	30	a	a	DET
iajs-3045	142	31	multiplication	multiplication	NOUN
iajs-3045	142	32	,	,	PUNCT
iajs-3045	142	33	then	then	ADV
iajs-3045	142	34	ƒ1	ƒ1	NOUN
iajs-3045	142	35	=	=	SYM
iajs-3045	142	36	ƥ1ѡ	ƥ1ѡ	PROPN
iajs-3045	142	37	,	,	PUNCT
iajs-3045	142	38	ƒ2	ƒ2	PROPN
iajs-3045	142	39	=	=	SYM
iajs-3045	142	40	ƥ2ѡ	ƥ2ѡ	NOUN
iajs-3045	142	41	and	and	CCONJ
iajs-3045	142	42	ƒ3	ƒ3	PROPN
iajs-3045	142	43	=	=	SYM
iajs-3045	142	44	ƥ3ѡ	ƥ3ѡ	PROPN
iajs-3045	142	45	for	for	ADP
iajs-3045	142	46	some	some	DET
iajs-3045	142	47	ihjpas	ihjpa	NOUN
iajs-3045	142	48	.	.	PUNCT
iajs-3045	143	1	36(2)2023	36(2)2023	NUM
iajs-3045	143	2	411	411	NUM
iajs-3045	143	3	ideals	ideal	NOUN
iajs-3045	143	4	ƥ1	ƥ1	VERB
iajs-3045	143	5	,	,	PUNCT
iajs-3045	143	6	ƥ2and	ƥ2and	NOUN
iajs-3045	143	7	ƥ3	ƥ3	PROPN
iajs-3045	143	8	of	of	ADP
iajs-3045	143	9	ʀ	ʀ	PROPN
iajs-3045	143	10	.	.	PUNCT
iajs-3045	144	1	that	that	PRON
iajs-3045	144	2	is	be	AUX
iajs-3045	144	3	ƒ1ƒ2ƒ3𝑥	ƒ1ƒ2ƒ3𝑥	PUNCT
iajs-3045	144	4	=	=	PUNCT
iajs-3045	144	5	ƥ1ƥ2ƥ3𝑥	ƥ1ƥ2ƥ3𝑥	PROPN
iajs-3045	144	6	⊆	⊆	NUM
iajs-3045	144	7	𝑉.	𝑉.	NOUN
iajs-3045	144	8	but	but	CCONJ
iajs-3045	144	9	𝑉	𝑉	PROPN
iajs-3045	144	10	is	be	AUX
iajs-3045	144	11	exnpq2ab	exnpq2ab	PROPN
iajs-3045	144	12	submodule	submodule	NOUN
iajs-3045	144	13	of	of	ADP
iajs-3045	144	14	ѡ	ѡ	PROPN
iajs-3045	144	15	,	,	PUNCT
iajs-3045	144	16	hence	hence	ADV
iajs-3045	144	17	from	from	ADP
iajs-3045	144	18	proposition	proposition	NOUN
iajs-3045	144	19	2.27	2.27	NUM
iajs-3045	144	20	we	we	PRON
iajs-3045	144	21	get	get	VERB
iajs-3045	144	22	either	either	CCONJ
iajs-3045	144	23	ƥ1ƥ3𝑥	ƥ1ƥ3𝑥	NUM
iajs-3045	144	24	⊆	⊆	NUM
iajs-3045	144	25	𝑉	𝑉	PROPN
iajs-3045	144	26	+	+	CCONJ
iajs-3045	144	27	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	144	28	)	)	PUNCT
iajs-3045	144	29	+	+	PUNCT
iajs-3045	144	30	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	144	31	)	)	PUNCT
iajs-3045	144	32	or	or	CCONJ
iajs-3045	144	33	ƥ2ƥ3𝑥	ƥ2ƥ3𝑥	NUM
iajs-3045	144	34	⊆	⊆	X
iajs-3045	144	35	𝑉	𝑉	PROPN
iajs-3045	144	36	+	+	CCONJ
iajs-3045	144	37	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	144	38	)	)	PUNCT
iajs-3045	144	39	+	+	PUNCT
iajs-3045	145	1	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	145	2	)	)	PUNCT
iajs-3045	145	3	or	or	CCONJ
iajs-3045	145	4	ƥ1ƥ2𝑥	ƥ1ƥ2𝑥	NUM
iajs-3045	145	5	⊆	⊆	NUM
iajs-3045	145	6	𝑉	𝑉	PROPN
iajs-3045	145	7	+	+	CCONJ
iajs-3045	145	8	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	145	9	)	)	PUNCT
iajs-3045	145	10	+	+	PUNCT
iajs-3045	146	1	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	146	2	)	)	PUNCT
iajs-3045	146	3	.	.	PUNCT
iajs-3045	147	1	next	next	ADJ
iajs-3045	147	2	,	,	PUNCT
iajs-3045	147	3	following	follow	VERB
iajs-3045	147	4	either	either	CCONJ
iajs-3045	147	5	ƒ1ƒ3𝑥	ƒ1ƒ3𝑥	NUM
iajs-3045	147	6	⊆	⊆	NUM
iajs-3045	147	7	𝑉	𝑉	PROPN
iajs-3045	147	8	+	+	CCONJ
iajs-3045	147	9	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	147	10	)	)	PUNCT
iajs-3045	147	11	+	+	PUNCT
iajs-3045	147	12	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	147	13	)	)	PUNCT
iajs-3045	147	14	or	or	CCONJ
iajs-3045	147	15	ƒ2ƒ3𝑥	ƒ2ƒ3𝑥	NUM
iajs-3045	147	16	⊆	⊆	NUM
iajs-3045	147	17	𝑉	𝑉	PROPN
iajs-3045	147	18	+	+	CCONJ
iajs-3045	147	19	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	147	20	)	)	PUNCT
iajs-3045	147	21	+	+	PUNCT
iajs-3045	147	22	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	147	23	)	)	PUNCT
iajs-3045	147	24	or	or	CCONJ
iajs-3045	147	25	ƒ1ƒ2𝑥	ƒ1ƒ2𝑥	NUM
iajs-3045	147	26	⊆	⊆	NUM
iajs-3045	147	27	𝑉	𝑉	PROPN
iajs-3045	147	28	+	+	CCONJ
iajs-3045	147	29	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	147	30	)	)	PUNCT
iajs-3045	147	31	+	+	PUNCT
iajs-3045	148	1	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	148	2	)	)	PUNCT
iajs-3045	148	3	.	.	PUNCT
iajs-3045	149	1	(	(	PUNCT
iajs-3045	149	2	⟸	⟸	ADV
iajs-3045	149	3	)	)	PUNCT
iajs-3045	149	4	let	let	VERB
iajs-3045	149	5	ƥ1ƥ2ƥ3𝑥	ƥ1ƥ2ƥ3𝑥	NOUN
iajs-3045	149	6	⊆	⊆	NUM
iajs-3045	149	7	𝑉	𝑉	PROPN
iajs-3045	149	8	for	for	ADP
iajs-3045	149	9	ƥ1	ƥ1	NOUN
iajs-3045	149	10	,	,	PUNCT
iajs-3045	149	11	ƥ2	ƥ2	NOUN
iajs-3045	149	12	,	,	PUNCT
iajs-3045	149	13	ƥ3	ƥ3	PROPN
iajs-3045	149	14	are	be	AUX
iajs-3045	149	15	ideals	ideal	NOUN
iajs-3045	149	16	of	of	ADP
iajs-3045	149	17	ʀ	ʀ	NOUN
iajs-3045	149	18	and	and	CCONJ
iajs-3045	149	19	𝑥	𝑥	PRON
iajs-3045	149	20	∈	∈	PROPN
iajs-3045	149	21	ѡ.	ѡ.	NOUN
iajs-3045	149	22	put	put	VERB
iajs-3045	149	23	ƒ1	ƒ1	NOUN
iajs-3045	150	1	=	=	PUNCT
iajs-3045	150	2	ƥ1ѡ	ƥ1ѡ	PROPN
iajs-3045	150	3	,	,	PUNCT
iajs-3045	150	4	ƒ2	ƒ2	PROPN
iajs-3045	150	5	=	=	SYM
iajs-3045	150	6	ƥ2ѡ	ƥ2ѡ	NOUN
iajs-3045	150	7	and	and	CCONJ
iajs-3045	150	8	ƒ3	ƒ3	NOUN
iajs-3045	150	9	=	=	SYM
iajs-3045	150	10	ƥ3ѡ.	ƥ3ѡ.	ADV
iajs-3045	150	11	that	that	PRON
iajs-3045	150	12	is	be	AUX
iajs-3045	150	13	ƒ1ƒ2ƒ3𝑥	ƒ1ƒ2ƒ3𝑥	PUNCT
iajs-3045	150	14	⊆	⊆	NUM
iajs-3045	150	15	𝑉.	𝑉.	NOUN
iajs-3045	150	16	now	now	ADV
iajs-3045	150	17	,	,	PUNCT
iajs-3045	150	18	by	by	ADP
iajs-3045	150	19	hypotheses	hypothesis	NOUN
iajs-3045	150	20	either	either	CCONJ
iajs-3045	150	21	ƒ1ƒ2𝑥	ƒ1ƒ2𝑥	NUM
iajs-3045	150	22	⊆	⊆	NUM
iajs-3045	150	23	𝑉	𝑉	PROPN
iajs-3045	150	24	+	+	CCONJ
iajs-3045	150	25	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	150	26	)	)	PUNCT
iajs-3045	150	27	+	+	PUNCT
iajs-3045	150	28	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	150	29	)	)	PUNCT
iajs-3045	150	30	or	or	CCONJ
iajs-3045	150	31	ƒ1ƒ3𝑥	ƒ1ƒ3𝑥	NUM
iajs-3045	150	32	⊆	⊆	NUM
iajs-3045	150	33	𝑉	𝑉	PROPN
iajs-3045	150	34	+	+	CCONJ
iajs-3045	150	35	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	150	36	)	)	PUNCT
iajs-3045	150	37	+	+	PUNCT
iajs-3045	150	38	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	150	39	)	)	PUNCT
iajs-3045	150	40	or	or	CCONJ
iajs-3045	150	41	ƒ2ƒ3𝑥	ƒ2ƒ3𝑥	NUM
iajs-3045	150	42	⊆	⊆	NUM
iajs-3045	150	43	𝑉	𝑉	PROPN
iajs-3045	150	44	+	+	CCONJ
iajs-3045	150	45	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	150	46	)	)	PUNCT
iajs-3045	150	47	+	+	PUNCT
iajs-3045	150	48	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	150	49	)	)	PUNCT
iajs-3045	150	50	,	,	PUNCT
iajs-3045	150	51	thus	thus	ADV
iajs-3045	150	52	ƥ1ƥ2𝑥	ƥ1ƥ2𝑥	ADP
iajs-3045	150	53	⊆	⊆	NUM
iajs-3045	150	54	𝑉	𝑉	PROPN
iajs-3045	150	55	+	+	CCONJ
iajs-3045	150	56	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	150	57	)	)	PUNCT
iajs-3045	150	58	+	+	PUNCT
iajs-3045	150	59	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	150	60	)	)	PUNCT
iajs-3045	150	61	or	or	CCONJ
iajs-3045	150	62	ƥ1ƥ3𝑥	ƥ1ƥ3𝑥	NUM
iajs-3045	150	63	⊆	⊆	NUM
iajs-3045	150	64	𝑉	𝑉	PROPN
iajs-3045	150	65	+	+	CCONJ
iajs-3045	150	66	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	150	67	)	)	PUNCT
iajs-3045	150	68	+	+	PUNCT
iajs-3045	150	69	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	150	70	)	)	PUNCT
iajs-3045	150	71	or	or	CCONJ
iajs-3045	150	72	ƥ2ƥ3𝑥	ƥ2ƥ3𝑥	NUM
iajs-3045	150	73	⊆	⊆	X
iajs-3045	150	74	𝑉	𝑉	PROPN
iajs-3045	150	75	+	+	CCONJ
iajs-3045	150	76	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	150	77	)	)	PUNCT
iajs-3045	150	78	+	+	PUNCT
iajs-3045	150	79	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	150	80	)	)	PUNCT
iajs-3045	150	81	.	.	PUNCT
iajs-3045	151	1	therefore	therefore	ADV
iajs-3045	151	2	by	by	ADP
iajs-3045	151	3	proposition	proposition	NOUN
iajs-3045	151	4	2.27	2.27	NUM
iajs-3045	151	5	𝑉	𝑉	PROPN
iajs-3045	151	6	is	be	AUX
iajs-3045	151	7	exnpq2ab	exnpq2ab	PROPN
iajs-3045	151	8	submodule	submodule	NOUN
iajs-3045	151	9	of	of	ADP
iajs-3045	151	10	ѡ.	ѡ.	NOUN
iajs-3045	151	11	proposition	proposition	NOUN
iajs-3045	151	12	3.3	3.3	NUM
iajs-3045	151	13	let	let	VERB
iajs-3045	151	14	ѡ	ѡ	PRON
iajs-3045	151	15	be	be	AUX
iajs-3045	151	16	a	a	DET
iajs-3045	151	17	multiplication	multiplication	NOUN
iajs-3045	151	18	ʀ	ʀ	NOUN
iajs-3045	151	19	-	-	PUNCT
iajs-3045	151	20	module	module	NOUN
iajs-3045	151	21	and	and	CCONJ
iajs-3045	151	22	𝑉	𝑉	PROPN
iajs-3045	151	23	≠	≠	PROPN
iajs-3045	151	24	ѡ.	ѡ.	NOUN
iajs-3045	151	25	then	then	ADV
iajs-3045	151	26	𝑉	𝑉	PROPN
iajs-3045	151	27	is	be	AUX
iajs-3045	151	28	exnpq2ab	exnpq2ab	PROPN
iajs-3045	151	29	submodule	submodule	NOUN
iajs-3045	151	30	of	of	ADP
iajs-3045	151	31	ѡ	ѡ	PROPN
iajs-3045	151	32	if	if	SCONJ
iajs-3045	152	1	and	and	CCONJ
iajs-3045	152	2	only	only	ADV
iajs-3045	152	3	if	if	SCONJ
iajs-3045	152	4	whenever	whenever	SCONJ
iajs-3045	152	5	ɱ1ɱ2ɱ3ӈ	ɱ1ɱ2ɱ3ӈ	PROPN
iajs-3045	152	6	⊆	⊆	NUM
iajs-3045	152	7	𝑉	𝑉	PROPN
iajs-3045	152	8	for	for	ADP
iajs-3045	152	9	some	some	DET
iajs-3045	152	10	ɱ1,ɱ2,ɱ3	ɱ1,ɱ2,ɱ3	PROPN
iajs-3045	152	11	∈	∈	PROPN
iajs-3045	152	12	ѡ	ѡ	PROPN
iajs-3045	152	13	,	,	PUNCT
iajs-3045	152	14	ӈ	ӈ	PRON
iajs-3045	152	15	is	be	AUX
iajs-3045	152	16	a	a	DET
iajs-3045	152	17	submodule	submodule	NOUN
iajs-3045	152	18	of	of	ADP
iajs-3045	152	19	ѡ	ѡ	PROPN
iajs-3045	152	20	,	,	PUNCT
iajs-3045	152	21	implies	imply	VERB
iajs-3045	152	22	that	that	SCONJ
iajs-3045	152	23	either	either	CCONJ
iajs-3045	152	24	ɱ1ɱ2ӈ	ɱ1ɱ2ӈ	NUM
iajs-3045	152	25	⊆	⊆	NUM
iajs-3045	152	26	𝑉	𝑉	PROPN
iajs-3045	152	27	+	+	CCONJ
iajs-3045	152	28	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	152	29	)	)	PUNCT
iajs-3045	152	30	+	+	PUNCT
iajs-3045	153	1	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	153	2	)	)	PUNCT
iajs-3045	153	3	or	or	CCONJ
iajs-3045	153	4	ɱ1ɱ3ӈ	ɱ1ɱ3ӈ	NUM
iajs-3045	153	5	⊆	⊆	NUM
iajs-3045	153	6	𝑉	𝑉	PROPN
iajs-3045	153	7	+	+	CCONJ
iajs-3045	153	8	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	153	9	)	)	PUNCT
iajs-3045	153	10	+	+	PUNCT
iajs-3045	153	11	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	153	12	)	)	PUNCT
iajs-3045	153	13	or	or	CCONJ
iajs-3045	153	14	ɱ2ɱ3ӈ	ɱ2ɱ3ӈ	NUM
iajs-3045	153	15	⊆	⊆	NUM
iajs-3045	153	16	𝑉	𝑉	PROPN
iajs-3045	153	17	+	+	CCONJ
iajs-3045	153	18	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	153	19	)	)	PUNCT
iajs-3045	153	20	+	+	PUNCT
iajs-3045	153	21	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	153	22	)	)	PUNCT
iajs-3045	153	23	.	.	PUNCT
iajs-3045	154	1	proof	proof	NOUN
iajs-3045	154	2	.	.	PUNCT
iajs-3045	155	1	(	(	PUNCT
iajs-3045	155	2	⟹	⟹	X
iajs-3045	155	3	)	)	PUNCT
iajs-3045	155	4	let	let	VERB
iajs-3045	155	5	ɱ1ɱ2ɱ3ӈ	ɱ1ɱ2ɱ3ӈ	PROPN
iajs-3045	155	6	⊆	⊆	NUM
iajs-3045	155	7	𝑉	𝑉	PROPN
iajs-3045	155	8	for	for	ADP
iajs-3045	155	9	some	some	DET
iajs-3045	155	10	ɱ1,ɱ2,ɱ3	ɱ1,ɱ2,ɱ3	PROPN
iajs-3045	155	11	∈	∈	PROPN
iajs-3045	155	12	ѡ	ѡ	X
iajs-3045	155	13	and	and	CCONJ
iajs-3045	155	14	ӈ	ӈ	PROPN
iajs-3045	155	15	is	be	AUX
iajs-3045	155	16	a	a	DET
iajs-3045	155	17	submodule	submodule	NOUN
iajs-3045	155	18	of	of	ADP
iajs-3045	155	19	ѡ.	ѡ.	NOUN
iajs-3045	155	20	that	that	PRON
iajs-3045	155	21	is	be	AUX
iajs-3045	155	22	(	(	PUNCT
iajs-3045	155	23	ɱ1)(ɱ2)(ɱ3)ӈ	ɱ1)(ɱ2)(ɱ3)ӈ	NOUN
iajs-3045	155	24	⊆	⊆	NUM
iajs-3045	155	25	𝑉	𝑉	PROPN
iajs-3045	155	26	since	since	SCONJ
iajs-3045	155	27	ѡ	ѡ	PROPN
iajs-3045	155	28	is	be	AUX
iajs-3045	155	29	a	a	DET
iajs-3045	155	30	multiplication	multiplication	NOUN
iajs-3045	155	31	,	,	PUNCT
iajs-3045	155	32	then	then	ADV
iajs-3045	155	33	(	(	PUNCT
iajs-3045	155	34	ɱ1	ɱ1	NOUN
iajs-3045	155	35	)	)	PUNCT
iajs-3045	156	1	=	=	SYM
iajs-3045	156	2	ƥ1ѡ	ƥ1ѡ	PROPN
iajs-3045	156	3	,	,	PUNCT
iajs-3045	156	4	(	(	PUNCT
iajs-3045	156	5	ɱ2	ɱ2	NOUN
iajs-3045	156	6	)	)	PUNCT
iajs-3045	156	7	=	=	SYM
iajs-3045	156	8	ƥ2ѡ	ƥ2ѡ	NOUN
iajs-3045	156	9	,	,	PUNCT
iajs-3045	156	10	(	(	PUNCT
iajs-3045	156	11	ɱ3	ɱ3	PROPN
iajs-3045	156	12	)	)	PUNCT
iajs-3045	156	13	=	=	SYM
iajs-3045	156	14	ƥ3ѡ	ƥ3ѡ	PROPN
iajs-3045	156	15	and	and	CCONJ
iajs-3045	156	16	ӈ	ӈ	PRON
iajs-3045	156	17	=	=	NOUN
iajs-3045	156	18	ƥ4ѡ	ƥ4ѡ	ADJ
iajs-3045	156	19	for	for	ADP
iajs-3045	156	20	some	some	DET
iajs-3045	156	21	ideals	ideal	NOUN
iajs-3045	156	22	ƥ1	ƥ1	VERB
iajs-3045	156	23	,	,	PUNCT
iajs-3045	156	24	ƥ2	ƥ2	NOUN
iajs-3045	156	25	,	,	PUNCT
iajs-3045	156	26	ƥ3	ƥ3	NOUN
iajs-3045	156	27	and	and	CCONJ
iajs-3045	156	28	ƥ4	ƥ4	ADV
iajs-3045	156	29	of	of	ADP
iajs-3045	156	30	ʀ	ʀ	PROPN
iajs-3045	156	31	.	.	PUNCT
iajs-3045	157	1	that	that	PRON
iajs-3045	157	2	is	is	ADV
iajs-3045	157	3	(	(	PUNCT
iajs-3045	157	4	ɱ1)(ɱ2)(ɱ3)ӈ	ɱ1)(ɱ2)(ɱ3)ӈ	NOUN
iajs-3045	157	5	=	=	SYM
iajs-3045	157	6	ƥ1ƥ2ƥ3(ƥ4ѡ	ƥ1ƥ2ƥ3(ƥ4ѡ	X
iajs-3045	157	7	)	)	PUNCT
iajs-3045	157	8	⊆	⊆	NUM
iajs-3045	157	9	𝑉.	𝑉.	NOUN
iajs-3045	157	10	but	but	CCONJ
iajs-3045	157	11	𝑉	𝑉	PROPN
iajs-3045	157	12	is	be	AUX
iajs-3045	157	13	exnpq2ab	exnpq2ab	PROPN
iajs-3045	157	14	submodule	submodule	NOUN
iajs-3045	157	15	of	of	ADP
iajs-3045	157	16	ѡ	ѡ	PROPN
iajs-3045	157	17	,	,	PUNCT
iajs-3045	157	18	hence	hence	ADV
iajs-3045	157	19	from	from	ADP
iajs-3045	157	20	proposition	proposition	NOUN
iajs-3045	157	21	2.25	2.25	NUM
iajs-3045	157	22	we	we	PRON
iajs-3045	157	23	get	get	VERB
iajs-3045	157	24	either	either	CCONJ
iajs-3045	157	25	ƥ1ƥ3(ƥ4ѡ	ƥ1ƥ3(ƥ4ѡ	NOUN
iajs-3045	157	26	)	)	PUNCT
iajs-3045	157	27	⊆	⊆	NUM
iajs-3045	157	28	𝑉	𝑉	PROPN
iajs-3045	157	29	+	+	CCONJ
iajs-3045	157	30	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	157	31	)	)	PUNCT
iajs-3045	157	32	+	+	PUNCT
iajs-3045	157	33	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	157	34	)	)	PUNCT
iajs-3045	157	35	or	or	CCONJ
iajs-3045	157	36	ƥ2ƥ3(ƥ4ѡ	ƥ2ƥ3(ƥ4ѡ	VERB
iajs-3045	157	37	)	)	PUNCT
iajs-3045	157	38	⊆	⊆	NUM
iajs-3045	157	39	𝑉	𝑉	PROPN
iajs-3045	157	40	+	+	CCONJ
iajs-3045	157	41	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	157	42	)	)	PUNCT
iajs-3045	157	43	+	+	PUNCT
iajs-3045	158	1	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	158	2	)	)	PUNCT
iajs-3045	158	3	or	or	CCONJ
iajs-3045	158	4	ƥ1ƥ2(ƥ4ѡ	ƥ1ƥ2(ƥ4ѡ	NOUN
iajs-3045	158	5	)	)	PUNCT
iajs-3045	158	6	⊆	⊆	NUM
iajs-3045	158	7	𝑉	𝑉	PROPN
iajs-3045	158	8	+	+	CCONJ
iajs-3045	158	9	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	158	10	)	)	PUNCT
iajs-3045	158	11	+	+	PUNCT
iajs-3045	158	12	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	158	13	)	)	PUNCT
iajs-3045	158	14	.	.	PUNCT
iajs-3045	159	1	next	next	ADJ
iajs-3045	159	2	,	,	PUNCT
iajs-3045	159	3	following	follow	VERB
iajs-3045	159	4	either	either	CCONJ
iajs-3045	159	5	ɱ1ɱ3ӈ	ɱ1ɱ3ӈ	NUM
iajs-3045	159	6	⊆	⊆	NUM
iajs-3045	159	7	𝑉	𝑉	PROPN
iajs-3045	159	8	+	+	CCONJ
iajs-3045	159	9	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	159	10	)	)	PUNCT
iajs-3045	159	11	+	+	PUNCT
iajs-3045	159	12	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	159	13	)	)	PUNCT
iajs-3045	159	14	or	or	CCONJ
iajs-3045	159	15	ɱ2ɱ3ӈ	ɱ2ɱ3ӈ	NUM
iajs-3045	159	16	⊆	⊆	NUM
iajs-3045	159	17	𝑉	𝑉	PROPN
iajs-3045	159	18	+	+	CCONJ
iajs-3045	159	19	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	159	20	)	)	PUNCT
iajs-3045	159	21	+	+	PUNCT
iajs-3045	159	22	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	159	23	)	)	PUNCT
iajs-3045	159	24	or	or	CCONJ
iajs-3045	159	25	ɱ1ɱ2ӈ	ɱ1ɱ2ӈ	NUM
iajs-3045	159	26	⊆	⊆	NUM
iajs-3045	159	27	𝑉	𝑉	PROPN
iajs-3045	159	28	+	+	CCONJ
iajs-3045	159	29	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	159	30	)	)	PUNCT
iajs-3045	159	31	+	+	PUNCT
iajs-3045	159	32	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	159	33	)	)	PUNCT
iajs-3045	159	34	.	.	PUNCT
iajs-3045	160	1	(	(	PUNCT
iajs-3045	160	2	⟸	⟸	ADV
iajs-3045	160	3	)	)	PUNCT
iajs-3045	160	4	let	let	VERB
iajs-3045	160	5	ƥ1ƥ2ƥ3ӈ	ƥ1ƥ2ƥ3ӈ	PROPN
iajs-3045	160	6	⊆	⊆	NUM
iajs-3045	160	7	𝑉	𝑉	PROPN
iajs-3045	160	8	for	for	ADP
iajs-3045	160	9	ƥ1	ƥ1	NOUN
iajs-3045	160	10	,	,	PUNCT
iajs-3045	160	11	ƥ2	ƥ2	NOUN
iajs-3045	160	12	,	,	PUNCT
iajs-3045	160	13	ƥ3	ƥ3	PROPN
iajs-3045	160	14	are	be	AUX
iajs-3045	160	15	ideals	ideal	NOUN
iajs-3045	160	16	of	of	ADP
iajs-3045	160	17	ʀ	ʀ	NOUN
iajs-3045	160	18	and	and	CCONJ
iajs-3045	160	19	ӈ	ӈ	PRON
iajs-3045	160	20	is	be	AUX
iajs-3045	160	21	a	a	DET
iajs-3045	160	22	submodule	submodule	NOUN
iajs-3045	160	23	of	of	ADP
iajs-3045	160	24	ѡ.	ѡ.	NOUN
iajs-3045	160	25	put	put	PROPN
iajs-3045	160	26	(	(	PUNCT
iajs-3045	160	27	ɱ1	ɱ1	PROPN
iajs-3045	160	28	)	)	PUNCT
iajs-3045	161	1	=	=	SYM
iajs-3045	161	2	ƥ1ѡ	ƥ1ѡ	PROPN
iajs-3045	161	3	,	,	PUNCT
iajs-3045	161	4	(	(	PUNCT
iajs-3045	161	5	ɱ2	ɱ2	NOUN
iajs-3045	161	6	)	)	PUNCT
iajs-3045	161	7	=	=	SYM
iajs-3045	161	8	ƥ2ѡ	ƥ2ѡ	NOUN
iajs-3045	161	9	and	and	CCONJ
iajs-3045	161	10	(	(	PUNCT
iajs-3045	161	11	ɱ3	ɱ3	PROPN
iajs-3045	161	12	)	)	PUNCT
iajs-3045	161	13	=	=	PUNCT
iajs-3045	162	1	ƥ3ѡ.	ƥ3ѡ.	ADV
iajs-3045	162	2	that	that	ADV
iajs-3045	162	3	is	be	AUX
iajs-3045	162	4	(	(	PUNCT
iajs-3045	162	5	ɱ1)(ɱ2)(ɱ3)ӈ	ɱ1)(ɱ2)(ɱ3)ӈ	NUM
iajs-3045	162	6	⊆	⊆	NUM
iajs-3045	162	7	𝑉.	𝑉.	NOUN
iajs-3045	162	8	that	that	PRON
iajs-3045	162	9	is	be	AUX
iajs-3045	162	10	ɱ1ɱ2ɱ3ӈ	ɱ1ɱ2ɱ3ӈ	PROPN
iajs-3045	162	11	⊆	⊆	NUM
iajs-3045	162	12	𝑉.	𝑉.	NOUN
iajs-3045	162	13	now	now	ADV
iajs-3045	162	14	,	,	PUNCT
iajs-3045	162	15	by	by	ADP
iajs-3045	162	16	hypotheses	hypothesis	NOUN
iajs-3045	162	17	either	either	CCONJ
iajs-3045	162	18	ɱ1ɱ2ӈ	ɱ1ɱ2ӈ	NUM
iajs-3045	162	19	⊆	⊆	NUM
iajs-3045	162	20	𝑉	𝑉	PROPN
iajs-3045	162	21	+	+	CCONJ
iajs-3045	162	22	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	162	23	)	)	PUNCT
iajs-3045	162	24	+	+	PUNCT
iajs-3045	162	25	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	162	26	)	)	PUNCT
iajs-3045	162	27	or	or	CCONJ
iajs-3045	162	28	ɱ1ɱ3ӈ	ɱ1ɱ3ӈ	NUM
iajs-3045	162	29	⊆	⊆	NUM
iajs-3045	162	30	𝑉	𝑉	PROPN
iajs-3045	162	31	+	+	CCONJ
iajs-3045	162	32	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	162	33	)	)	PUNCT
iajs-3045	162	34	+	+	PUNCT
iajs-3045	162	35	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	162	36	)	)	PUNCT
iajs-3045	162	37	or	or	CCONJ
iajs-3045	162	38	ɱ2ɱ3ӈ	ɱ2ɱ3ӈ	NUM
iajs-3045	162	39	⊆	⊆	NUM
iajs-3045	162	40	𝑉	𝑉	PROPN
iajs-3045	162	41	+	+	CCONJ
iajs-3045	162	42	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	162	43	)	)	PUNCT
iajs-3045	162	44	+	+	PUNCT
iajs-3045	162	45	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	162	46	)	)	PUNCT
iajs-3045	162	47	,	,	PUNCT
iajs-3045	162	48	thus	thus	ADV
iajs-3045	162	49	(	(	PUNCT
iajs-3045	162	50	ɱ1)(ɱ2)ӈ	ɱ1)(ɱ2)ӈ	PROPN
iajs-3045	162	51	⊆	⊆	NUM
iajs-3045	162	52	𝑉	𝑉	PROPN
iajs-3045	162	53	+	+	CCONJ
iajs-3045	162	54	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	162	55	)	)	PUNCT
iajs-3045	162	56	+	+	PUNCT
iajs-3045	162	57	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	162	58	)	)	PUNCT
iajs-3045	162	59	or	or	CCONJ
iajs-3045	162	60	(	(	PUNCT
iajs-3045	162	61	ɱ1)(ɱ3)ӈ	ɱ1)(ɱ3)ӈ	NOUN
iajs-3045	162	62	⊆	⊆	NUM
iajs-3045	162	63	𝑉	𝑉	PROPN
iajs-3045	162	64	+	+	CCONJ
iajs-3045	162	65	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	162	66	)	)	PUNCT
iajs-3045	162	67	+	+	PUNCT
iajs-3045	162	68	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	162	69	)	)	PUNCT
iajs-3045	162	70	or	or	CCONJ
iajs-3045	162	71	(	(	PUNCT
iajs-3045	162	72	ɱ2)(ɱ3)ӈ	ɱ2)(ɱ3)ӈ	PROPN
iajs-3045	162	73	⊆	⊆	NUM
iajs-3045	162	74	𝑉	𝑉	PROPN
iajs-3045	162	75	+	+	CCONJ
iajs-3045	162	76	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	162	77	)	)	PUNCT
iajs-3045	163	1	+	+	PUNCT
iajs-3045	164	1	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	164	2	)	)	PUNCT
iajs-3045	164	3	.	.	PUNCT
iajs-3045	165	1	then	then	ADV
iajs-3045	165	2	ƥ1ƥ2ӈ	ƥ1ƥ2ӈ	VERB
iajs-3045	166	1	⊆	⊆	NUM
iajs-3045	166	2	𝑉	𝑉	PROPN
iajs-3045	166	3	+	+	CCONJ
iajs-3045	166	4	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	166	5	)	)	PUNCT
iajs-3045	166	6	+	+	PUNCT
iajs-3045	166	7	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	166	8	)	)	PUNCT
iajs-3045	166	9	or	or	CCONJ
iajs-3045	166	10	ƥ1ƥ3ӈ	ƥ1ƥ3ӈ	NUM
iajs-3045	166	11	⊆	⊆	X
iajs-3045	166	12	𝑉	𝑉	PROPN
iajs-3045	166	13	+	+	CCONJ
iajs-3045	166	14	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	166	15	)	)	PUNCT
iajs-3045	166	16	+	+	PUNCT
iajs-3045	166	17	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	166	18	)	)	PUNCT
iajs-3045	166	19	or	or	CCONJ
iajs-3045	166	20	ƥ2ƥ3ӈ	ƥ2ƥ3ӈ	NUM
iajs-3045	166	21	⊆	⊆	NUM
iajs-3045	166	22	𝑉	𝑉	PROPN
iajs-3045	166	23	+	+	CCONJ
iajs-3045	166	24	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	166	25	)	)	PUNCT
iajs-3045	166	26	+	+	PUNCT
iajs-3045	166	27	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	166	28	)	)	PUNCT
iajs-3045	166	29	.	.	PUNCT
iajs-3045	167	1	therefore	therefore	ADV
iajs-3045	167	2	by	by	ADP
iajs-3045	167	3	proposition	proposition	NOUN
iajs-3045	167	4	2.25	2.25	NUM
iajs-3045	167	5	𝑉	𝑉	PROPN
iajs-3045	167	6	is	be	AUX
iajs-3045	167	7	exnpq2ab	exnpq2ab	PROPN
iajs-3045	167	8	submodule	submodule	NOUN
iajs-3045	167	9	of	of	ADP
iajs-3045	167	10	ѡ.	ѡ.	NOUN
iajs-3045	167	11	remark	remark	NOUN
iajs-3045	167	12	3.4	3.4	NUM
iajs-3045	167	13	if	if	SCONJ
iajs-3045	167	14	𝑉	𝑉	PROPN
iajs-3045	167	15	is	be	AUX
iajs-3045	167	16	an	an	DET
iajs-3045	167	17	exnpq2ab	exnpq2ab	PROPN
iajs-3045	167	18	submodule	submodule	NOUN
iajs-3045	167	19	of	of	ADP
iajs-3045	167	20	an	an	DET
iajs-3045	167	21	ʀ	ʀ	NOUN
iajs-3045	167	22	-	-	PUNCT
iajs-3045	167	23	module	module	NOUN
iajs-3045	167	24	ѡ	ѡ	NOUN
iajs-3045	167	25	,	,	PUNCT
iajs-3045	167	26	then	then	ADV
iajs-3045	167	27	[	[	X
iajs-3045	167	28	𝑉:ʀ	𝑉:ʀ	NOUN
iajs-3045	167	29	ѡ	ѡ	NOUN
iajs-3045	167	30	]	]	PUNCT
iajs-3045	167	31	need	need	VERB
iajs-3045	167	32	not	not	PART
iajs-3045	167	33	to	to	PART
iajs-3045	167	34	be	be	AUX
iajs-3045	167	35	exnpq2ab	exnpq2ab	PROPN
iajs-3045	167	36	ideal	ideal	NOUN
iajs-3045	167	37	of	of	ADP
iajs-3045	167	38	ʀ	ʀ	NOUN
iajs-3045	167	39	.	.	PUNCT
iajs-3045	168	1	the	the	DET
iajs-3045	168	2	following	follow	VERB
iajs-3045	168	3	example	example	NOUN
iajs-3045	168	4	shows	show	VERB
iajs-3045	168	5	that	that	SCONJ
iajs-3045	168	6	:	:	PUNCT
iajs-3045	168	7	let	let	VERB
iajs-3045	168	8	ѡ	ѡ	X
iajs-3045	168	9	=	=	SYM
iajs-3045	168	10	𝚉48	𝚉48	PROPN
iajs-3045	168	11	,	,	PUNCT
iajs-3045	168	12	ʀ	ʀ	PROPN
iajs-3045	168	13	=	=	SYM
iajs-3045	168	14	𝚉	𝚉	PROPN
iajs-3045	168	15	and	and	CCONJ
iajs-3045	168	16	the	the	DET
iajs-3045	168	17	submodule	submodule	NOUN
iajs-3045	168	18	𝑉	𝑉	PROPN
iajs-3045	168	19	=	=	PROPN
iajs-3045	168	20	〈	〈	PROPN
iajs-3045	168	21	16̅̅̅̅	16̅̅̅̅	PROPN
iajs-3045	168	22	〉	〉	NOUN
iajs-3045	168	23	is	be	AUX
iajs-3045	168	24	exnpq2ab	exnpq2ab	PROPN
iajs-3045	168	25	submodule	submodule	NOUN
iajs-3045	168	26	of	of	ADP
iajs-3045	168	27	ѡ	ѡ	PROPN
iajs-3045	168	28	,	,	PUNCT
iajs-3045	168	29	since	since	SCONJ
iajs-3045	168	30	𝑠𝑜𝑐(𝚉48	𝑠𝑜𝑐(𝚉48	NOUN
iajs-3045	168	31	)	)	PUNCT
iajs-3045	168	32	=	=	PUNCT
iajs-3045	168	33	〈	〈	PROPN
iajs-3045	168	34	2̅	2̅	NOUN
iajs-3045	168	35	〉	〉	NOUN
iajs-3045	168	36	∩	∩	NOUN
iajs-3045	169	1	〈	〈	NOUN
iajs-3045	169	2	3̅	3̅	ADJ
iajs-3045	169	3	〉	〉	NOUN
iajs-3045	169	4	∩	∩	ADJ
iajs-3045	169	5	〈	〈	NOUN
iajs-3045	169	6	8̅	8̅	NUM
iajs-3045	169	7	〉	〉	NOUN
iajs-3045	169	8	∩	∩	NOUN
iajs-3045	169	9	𝚉48	𝚉48	PROPN
iajs-3045	169	10	=	=	PUNCT
iajs-3045	169	11	〈	〈	NOUN
iajs-3045	169	12	8̅	8̅	NUM
iajs-3045	169	13	〉	〉	NOUN
iajs-3045	169	14	and	and	CCONJ
iajs-3045	169	15	𝐽(𝚉48	𝐽(𝚉48	NOUN
iajs-3045	169	16	)	)	PUNCT
iajs-3045	169	17	=	=	PUNCT
iajs-3045	169	18	〈	〈	PROPN
iajs-3045	169	19	2̅	2̅	NOUN
iajs-3045	169	20	〉	〉	NOUN
iajs-3045	169	21	∩	∩	NOUN
iajs-3045	169	22	〈	〈	NOUN
iajs-3045	169	23	3̅	3̅	ADJ
iajs-3045	169	24	〉	〉	NOUN
iajs-3045	169	25	=	=	SYM
iajs-3045	169	26	〈	〈	PROPN
iajs-3045	169	27	6̅	6̅	NOUN
iajs-3045	169	28	〉	〉	NOUN
iajs-3045	169	29	.	.	PUNCT
iajs-3045	170	1	then	then	ADV
iajs-3045	170	2	〈	〈	PROPN
iajs-3045	170	3	16̅̅̅̅	16̅̅̅̅	PROPN
iajs-3045	170	4	〉	〉	NOUN
iajs-3045	170	5	+	+	CCONJ
iajs-3045	170	6	𝑠𝑜𝑐(𝚉48	𝑠𝑜𝑐(𝚉48	SYM
iajs-3045	170	7	)	)	PUNCT
iajs-3045	170	8	+	+	CCONJ
iajs-3045	170	9	𝐽(𝚉48	𝐽(𝚉48	PROPN
iajs-3045	170	10	)	)	PUNCT
iajs-3045	170	11	=	=	PUNCT
iajs-3045	171	1	〈	〈	PROPN
iajs-3045	171	2	16̅̅̅̅	16̅̅̅̅	NUM
iajs-3045	171	3	〉	〉	NOUN
iajs-3045	171	4	+	+	CCONJ
iajs-3045	171	5	〈	〈	NOUN
iajs-3045	171	6	8̅	8̅	NUM
iajs-3045	171	7	〉	〉	NOUN
iajs-3045	171	8	+	+	X
iajs-3045	171	9	〈	〈	NOUN
iajs-3045	171	10	6̅	6̅	ADJ
iajs-3045	171	11	〉	〉	NOUN
iajs-3045	171	12	=	=	PUNCT
iajs-3045	171	13	〈	〈	PROPN
iajs-3045	171	14	2̅	2̅	NOUN
iajs-3045	171	15	〉	〉	NOUN
iajs-3045	171	16	,	,	PUNCT
iajs-3045	171	17	hence	hence	ADV
iajs-3045	171	18	for	for	ADP
iajs-3045	171	19	all	all	DET
iajs-3045	171	20	ɑ	ɑ	PROPN
iajs-3045	171	21	,	,	PUNCT
iajs-3045	171	22	ɓ	ɓ	PROPN
iajs-3045	171	23	,	,	PUNCT
iajs-3045	171	24	𝑒	𝑒	PROPN
iajs-3045	171	25	∈	∈	PROPN
iajs-3045	171	26	𝚉	𝚉	PROPN
iajs-3045	171	27	and	and	CCONJ
iajs-3045	171	28	ɱ	ɱ	PROPN
iajs-3045	171	29	∈	∈	PROPN
iajs-3045	171	30	𝚉48	𝚉48	PROPN
iajs-3045	171	31	𝑠𝑢𝑐ℎ	𝑠𝑢𝑐ℎ	NOUN
iajs-3045	171	32	𝑡ℎ𝑎𝑡	𝑡ℎ𝑎𝑡	PROPN
iajs-3045	171	33	ɑɓ𝑒𝑚	ɑɓ𝑒𝑚	PROPN
iajs-3045	171	34	∈	∈	PROPN
iajs-3045	171	35	〈	〈	PROPN
iajs-3045	171	36	16̅̅̅̅	16̅̅̅̅	PROPN
iajs-3045	171	37	〉	〉	NOUN
iajs-3045	171	38	,	,	PUNCT
iajs-3045	171	39	implies	imply	VERB
iajs-3045	171	40	that	that	SCONJ
iajs-3045	171	41	either	either	CCONJ
iajs-3045	171	42	ɑɓɱ	ɑɓɱ	VERB
iajs-3045	171	43	∈	∈	PROPN
iajs-3045	171	44	〈	〈	NOUN
iajs-3045	171	45	2̅	2̅	NOUN
iajs-3045	171	46	〉	〉	NOUN
iajs-3045	171	47	or	or	CCONJ
iajs-3045	171	48	ɑ𝑒ɱ	ɑ𝑒ɱ	NOUN
iajs-3045	171	49	∈	∈	PROPN
iajs-3045	171	50	〈	〈	NOUN
iajs-3045	171	51	2̅	2̅	NOUN
iajs-3045	171	52	〉	〉	NOUN
iajs-3045	171	53	or	or	CCONJ
iajs-3045	171	54	ɓ𝑒ɱ	ɓ𝑒ɱ	PROPN
iajs-3045	171	55	∈	∈	PROPN
iajs-3045	171	56	〈	〈	PROPN
iajs-3045	171	57	2̅	2̅	NOUN
iajs-3045	171	58	〉	〉	NOUN
iajs-3045	171	59	.	.	PUNCT
iajs-3045	172	1	but	but	CCONJ
iajs-3045	172	2	[	[	X
iajs-3045	172	3	〈	〈	ADJ
iajs-3045	172	4	16̅̅̅̅	16̅̅̅̅	ADJ
iajs-3045	172	5	〉	〉	NOUN
iajs-3045	172	6	:	:	PUNCT
iajs-3045	172	7	ʀ	ʀ	NOUN
iajs-3045	172	8	𝚉48	𝚉48	PROPN
iajs-3045	172	9	]	]	X
iajs-3045	172	10	=	=	SYM
iajs-3045	172	11	16𝚉	16𝚉	NUM
iajs-3045	172	12	is	be	AUX
iajs-3045	172	13	not	not	PART
iajs-3045	172	14	an	an	DET
iajs-3045	172	15	ihjpas	ihjpa	NOUN
iajs-3045	172	16	.	.	PUNCT
iajs-3045	173	1	36(2)2023	36(2)2023	NUM
iajs-3045	173	2	412	412	NUM
iajs-3045	173	3	exnpq2ab	exnpq2ab	PROPN
iajs-3045	173	4	𝑖𝑑𝑒𝑎𝑙	𝑖𝑑𝑒𝑎𝑙	NUM
iajs-3045	173	5	of	of	ADP
iajs-3045	173	6	𝚉	𝚉	PROPN
iajs-3045	173	7	,	,	PUNCT
iajs-3045	173	8	since	since	SCONJ
iajs-3045	173	9	2.4.2.1	2.4.2.1	NUM
iajs-3045	173	10	∈	∈	NOUN
iajs-3045	173	11	16𝚉	16𝚉	NUM
iajs-3045	173	12	,	,	PUNCT
iajs-3045	173	13	for	for	ADP
iajs-3045	173	14	1,2,4	1,2,4	NUM
iajs-3045	173	15	∈	∈	PROPN
iajs-3045	173	16	𝚉	𝚉	PROPN
iajs-3045	173	17	,	,	PUNCT
iajs-3045	173	18	implies	imply	VERB
iajs-3045	173	19	that	that	SCONJ
iajs-3045	173	20	2.4.1	2.4.1	NUM
iajs-3045	173	21	∉	∉	X
iajs-3045	173	22	16𝚉	16𝚉	NUM
iajs-3045	173	23	and	and	CCONJ
iajs-3045	173	24	2.2.1	2.2.1	NUM
iajs-3045	173	25	∉	∉	X
iajs-3045	173	26	16𝚉	16𝚉	NUM
iajs-3045	173	27	and	and	CCONJ
iajs-3045	173	28	4.2.1	4.2.1	NUM
iajs-3045	173	29	∉	∉	ADJ
iajs-3045	173	30	16𝚉.	16𝚉.	NUM
iajs-3045	173	31	under	under	ADP
iajs-3045	173	32	certain	certain	ADJ
iajs-3045	173	33	conditions	condition	NOUN
iajs-3045	173	34	,	,	PUNCT
iajs-3045	173	35	the	the	DET
iajs-3045	173	36	above	above	ADJ
iajs-3045	173	37	observation	observation	NOUN
iajs-3045	173	38	is	be	AUX
iajs-3045	173	39	fulfilled	fulfil	VERB
iajs-3045	173	40	.	.	PUNCT
iajs-3045	174	1	proposition	proposition	NOUN
iajs-3045	174	2	3.5	3.5	NUM
iajs-3045	174	3	let	let	VERB
iajs-3045	174	4	ƒ	ƒ	PRON
iajs-3045	174	5	≠	≠	NOUN
iajs-3045	174	6	ѡ	ѡ	NOUN
iajs-3045	174	7	and	and	CCONJ
iajs-3045	174	8	ѡ	ѡ	PROPN
iajs-3045	174	9	is	be	AUX
iajs-3045	174	10	faithful	faithful	ADJ
iajs-3045	174	11	multiplication	multiplication	NOUN
iajs-3045	174	12	ʀ	ʀ	NOUN
iajs-3045	174	13	-	-	PUNCT
iajs-3045	174	14	module	module	NOUN
iajs-3045	174	15	.	.	PUNCT
iajs-3045	175	1	then	then	ADV
iajs-3045	175	2	ƒ	ƒ	PRON
iajs-3045	175	3	is	be	AUX
iajs-3045	175	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	175	5	submodule	submodule	NOUN
iajs-3045	175	6	of	of	ADP
iajs-3045	175	7	ѡ	ѡ	PROPN
iajs-3045	175	8	if	if	SCONJ
iajs-3045	176	1	and	and	CCONJ
iajs-3045	176	2	only	only	ADV
iajs-3045	176	3	if	if	SCONJ
iajs-3045	176	4	[	[	X
iajs-3045	176	5	ƒ	ƒ	X
iajs-3045	176	6	:	:	PUNCT
iajs-3045	176	7	ʀ	ʀ	PART
iajs-3045	176	8	ѡ	ѡ	X
iajs-3045	176	9	]	]	PUNCT
iajs-3045	176	10	is	be	AUX
iajs-3045	176	11	exnpq2ab	exnpq2ab	PROPN
iajs-3045	176	12	ideal	ideal	NOUN
iajs-3045	176	13	of	of	ADP
iajs-3045	176	14	ʀ	ʀ	NOUN
iajs-3045	176	15	.	.	NOUN
iajs-3045	176	16	proof	proof	NOUN
iajs-3045	176	17	.	.	PUNCT
iajs-3045	177	1	(	(	PUNCT
iajs-3045	177	2	⟹	⟹	X
iajs-3045	177	3	)	)	PUNCT
iajs-3045	177	4	let	let	VERB
iajs-3045	177	5	ƒ	ƒ	PRON
iajs-3045	177	6	is	be	AUX
iajs-3045	177	7	exnpq2ab	exnpq2ab	PROPN
iajs-3045	177	8	submodule	submodule	NOUN
iajs-3045	177	9	of	of	ADP
iajs-3045	177	10	ѡ	ѡ	PROPN
iajs-3045	177	11	,	,	PUNCT
iajs-3045	177	12	and	and	CCONJ
iajs-3045	177	13	ƥ1ƥ2ƥ3ƥ4	ƥ1ƥ2ƥ3ƥ4	VERB
iajs-3045	177	14	⊆	⊆	NUM
iajs-3045	177	15	[	[	X
iajs-3045	177	16	ƒ	ƒ	NUM
iajs-3045	177	17	:	:	PUNCT
iajs-3045	177	18	ʀ	ʀ	NOUN
iajs-3045	177	19	ѡ	ѡ	NOUN
iajs-3045	177	20	]	]	PUNCT
iajs-3045	177	21	for	for	ADP
iajs-3045	177	22	some	some	DET
iajs-3045	177	23	ideals	ideal	NOUN
iajs-3045	177	24	ƥ1	ƥ1	NOUN
iajs-3045	177	25	,	,	PUNCT
iajs-3045	177	26	ƥ2	ƥ2	NOUN
iajs-3045	177	27	,	,	PUNCT
iajs-3045	177	28	ƥ3	ƥ3	NOUN
iajs-3045	177	29	and	and	CCONJ
iajs-3045	177	30	ƥ4	ƥ4	ADV
iajs-3045	177	31	of	of	ADP
iajs-3045	177	32	ʀ	ʀ	NOUN
iajs-3045	177	33	,	,	PUNCT
iajs-3045	177	34	then	then	ADV
iajs-3045	177	35	ƥ1ƥ2ƥ3ƥ4ѡ	ƥ1ƥ2ƥ3ƥ4ѡ	PRON
iajs-3045	177	36	⊆	⊆	NUM
iajs-3045	177	37	ƒ	ƒ	NOUN
iajs-3045	177	38	.	.	PUNCT
iajs-3045	178	1	but	but	CCONJ
iajs-3045	178	2	ѡ	ѡ	PROPN
iajs-3045	178	3	is	be	AUX
iajs-3045	178	4	a	a	DET
iajs-3045	178	5	multiplication	multiplication	NOUN
iajs-3045	178	6	,	,	PUNCT
iajs-3045	178	7	then	then	ADV
iajs-3045	178	8	ƥ1ƥ2ƥ3ƥ4ѡ	ƥ1ƥ2ƥ3ƥ4ѡ	PROPN
iajs-3045	178	9	=	=	SYM
iajs-3045	178	10	ƒ1ƒ2ƒ3ƒ4	ƒ1ƒ2ƒ3ƒ4	PROPN
iajs-3045	178	11	⊆	⊆	NUM
iajs-3045	178	12	ƒ	ƒ	NUM
iajs-3045	178	13	,	,	PUNCT
iajs-3045	178	14	by	by	ADP
iajs-3045	178	15	taking	take	VERB
iajs-3045	178	16	ƥ1ѡ	ƥ1ѡ	PRON
iajs-3045	178	17	=	=	SYM
iajs-3045	178	18	ƒ1	ƒ1	PROPN
iajs-3045	178	19	,	,	PUNCT
iajs-3045	178	20	ƥ2ѡ	ƥ2ѡ	PROPN
iajs-3045	178	21	=	=	SYM
iajs-3045	178	22	ƒ2	ƒ2	PROPN
iajs-3045	178	23	,	,	PUNCT
iajs-3045	178	24	ƥ3ѡ	ƥ3ѡ	PROPN
iajs-3045	178	25	=	=	PROPN
iajs-3045	178	26	ƒ3	ƒ3	NOUN
iajs-3045	178	27	and	and	CCONJ
iajs-3045	178	28	ƥ4ѡ	ƥ4ѡ	ADJ
iajs-3045	178	29	=	=	SYM
iajs-3045	178	30	ƒ4	ƒ4	NOUN
iajs-3045	178	31	.	.	PUNCT
iajs-3045	179	1	but	but	CCONJ
iajs-3045	179	2	ƒ	ƒ	PRON
iajs-3045	179	3	is	be	AUX
iajs-3045	179	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	179	5	submodule	submodule	NOUN
iajs-3045	179	6	of	of	ADP
iajs-3045	179	7	ѡ	ѡ	PROPN
iajs-3045	179	8	,	,	PUNCT
iajs-3045	179	9	then	then	ADV
iajs-3045	179	10	by	by	ADP
iajs-3045	179	11	proposition	proposition	NOUN
iajs-3045	179	12	3.1	3.1	NUM
iajs-3045	179	13	either	either	CCONJ
iajs-3045	179	14	ƒ1ƒ3ƒ4	ƒ1ƒ3ƒ4	PROPN
iajs-3045	179	15	⊆	⊆	NUM
iajs-3045	179	16	ƒ	ƒ	PROPN
iajs-3045	179	17	+	+	NUM
iajs-3045	179	18	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	179	19	)	)	PUNCT
iajs-3045	179	20	+	+	PUNCT
iajs-3045	180	1	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	180	2	)	)	PUNCT
iajs-3045	180	3	or	or	CCONJ
iajs-3045	180	4	ƒ2ƒ3ƒ4	ƒ2ƒ3ƒ4	VERB
iajs-3045	180	5	⊆	⊆	NUM
iajs-3045	180	6	ƒ	ƒ	NOUN
iajs-3045	180	7	+	+	NUM
iajs-3045	180	8	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	180	9	)	)	PUNCT
iajs-3045	181	1	+	+	NUM
iajs-3045	181	2	𝐽(ѡ)or	𝐽(ѡ)or	ADJ
iajs-3045	181	3	ƒ1ƒ2ƒ4	ƒ1ƒ2ƒ4	X
iajs-3045	181	4	⊆	⊆	NUM
iajs-3045	181	5	ƒ	ƒ	NOUN
iajs-3045	181	6	+	+	NUM
iajs-3045	181	7	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	181	8	)	)	PUNCT
iajs-3045	181	9	+	+	PUNCT
iajs-3045	181	10	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	181	11	)	)	PUNCT
iajs-3045	181	12	.	.	PUNCT
iajs-3045	182	1	since	since	SCONJ
iajs-3045	182	2	ѡ	ѡ	PROPN
iajs-3045	182	3	is	be	AUX
iajs-3045	182	4	multiplication	multiplication	NOUN
iajs-3045	182	5	,	,	PUNCT
iajs-3045	182	6	then	then	ADV
iajs-3045	182	7	ƒ	ƒ	X
iajs-3045	182	8	=	=	PUNCT
iajs-3045	183	1	[	[	X
iajs-3045	183	2	ƒ	ƒ	X
iajs-3045	183	3	:	:	PUNCT
iajs-3045	183	4	ʀ	ʀ	ADJ
iajs-3045	183	5	ѡ]ѡ	ѡ]ѡ	NOUN
iajs-3045	183	6	,	,	PUNCT
iajs-3045	183	7	and	and	CCONJ
iajs-3045	183	8	since	since	SCONJ
iajs-3045	183	9	ѡ	ѡ	PROPN
iajs-3045	183	10	is	be	AUX
iajs-3045	183	11	faithful	faithful	ADJ
iajs-3045	183	12	multiplication	multiplication	NOUN
iajs-3045	183	13	,	,	PUNCT
iajs-3045	183	14	then	then	ADV
iajs-3045	183	15	by	by	ADP
iajs-3045	183	16	lemma	lemma	PROPN
iajs-3045	183	17	2.6	2.6	NUM
iajs-3045	183	18	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	183	19	)	)	PUNCT
iajs-3045	183	20	=	=	SYM
iajs-3045	184	1	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	184	2	and	and	CCONJ
iajs-3045	184	3	by	by	ADP
iajs-3045	184	4	lemma	lemma	PROPN
iajs-3045	184	5	2.7	2.7	NUM
iajs-3045	184	6	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	184	7	)	)	PUNCT
iajs-3045	184	8	=	=	PUNCT
iajs-3045	184	9	𝐽(ʀ)ѡ.	𝐽(ʀ)ѡ.	PROPN
iajs-3045	184	10	thus	thus	ADV
iajs-3045	184	11	either	either	CCONJ
iajs-3045	184	12	ƥ1ƥ3ƥ4ѡ	ƥ1ƥ3ƥ4ѡ	VERB
iajs-3045	185	1	⊆	⊆	NUM
iajs-3045	185	2	[	[	X
iajs-3045	185	3	ƒ	ƒ	X
iajs-3045	185	4	:	:	PUNCT
iajs-3045	185	5	ʀ	ʀ	ADJ
iajs-3045	185	6	ѡ]ѡ	ѡ]ѡ	NOUN
iajs-3045	185	7	+	+	CCONJ
iajs-3045	185	8	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	185	9	+	+	CCONJ
iajs-3045	185	10	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	ADJ
iajs-3045	185	11	or	or	CCONJ
iajs-3045	185	12	ƥ2ƥ3ƥ4ѡ	ƥ2ƥ3ƥ4ѡ	VERB
iajs-3045	185	13	⊆	⊆	NUM
iajs-3045	185	14	[	[	X
iajs-3045	185	15	ƒ	ƒ	X
iajs-3045	185	16	:	:	PUNCT
iajs-3045	185	17	ʀ	ʀ	ADJ
iajs-3045	185	18	ѡ]ѡ	ѡ]ѡ	NOUN
iajs-3045	185	19	+	+	CCONJ
iajs-3045	185	20	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	185	21	+	+	CCONJ
iajs-3045	185	22	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	ADJ
iajs-3045	185	23	or	or	CCONJ
iajs-3045	185	24	ƥ1ƥ2ƥ4ѡ	ƥ1ƥ2ƥ4ѡ	ADJ
iajs-3045	185	25	⊆	⊆	NUM
iajs-3045	186	1	[	[	X
iajs-3045	186	2	ƒ	ƒ	X
iajs-3045	186	3	:	:	PUNCT
iajs-3045	186	4	ʀ	ʀ	ADJ
iajs-3045	186	5	ѡ]ѡ	ѡ]ѡ	NOUN
iajs-3045	186	6	+	+	CCONJ
iajs-3045	186	7	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	186	8	+	+	CCONJ
iajs-3045	186	9	𝐽(ʀ)ѡ.	𝐽(ʀ)ѡ.	NOUN
iajs-3045	186	10	hence	hence	ADV
iajs-3045	186	11	either	either	CCONJ
iajs-3045	186	12	ƥ1ƥ3ƥ4	ƥ1ƥ3ƥ4	NOUN
iajs-3045	186	13	⊆	⊆	NUM
iajs-3045	186	14	[	[	X
iajs-3045	186	15	ƒ	ƒ	X
iajs-3045	186	16	:	:	PUNCT
iajs-3045	186	17	ʀ	ʀ	PART
iajs-3045	186	18	ѡ	ѡ	NOUN
iajs-3045	186	19	]	]	PUNCT
iajs-3045	186	20	+	+	X
iajs-3045	186	21	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	186	22	)	)	PUNCT
iajs-3045	186	23	+	+	NUM
iajs-3045	186	24	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	186	25	)	)	PUNCT
iajs-3045	186	26	or	or	CCONJ
iajs-3045	186	27	ƥ2ƥ3ƥ4	ƥ2ƥ3ƥ4	VERB
iajs-3045	187	1	⊆	⊆	NUM
iajs-3045	188	1	[	[	X
iajs-3045	188	2	ƒ	ƒ	X
iajs-3045	188	3	:	:	PUNCT
iajs-3045	188	4	ʀ	ʀ	PART
iajs-3045	188	5	ѡ	ѡ	NOUN
iajs-3045	188	6	]	]	PUNCT
iajs-3045	188	7	+	+	X
iajs-3045	188	8	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	188	9	)	)	PUNCT
iajs-3045	188	10	+	+	NUM
iajs-3045	188	11	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	188	12	)	)	PUNCT
iajs-3045	188	13	or	or	CCONJ
iajs-3045	188	14	ƥ1ƥ2ƥ4	ƥ1ƥ2ƥ4	NOUN
iajs-3045	188	15	⊆	⊆	NUM
iajs-3045	189	1	[	[	X
iajs-3045	189	2	ƒ	ƒ	X
iajs-3045	189	3	:	:	PUNCT
iajs-3045	189	4	ʀ	ʀ	PART
iajs-3045	189	5	ѡ	ѡ	NOUN
iajs-3045	189	6	]	]	PUNCT
iajs-3045	189	7	+	+	X
iajs-3045	189	8	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	189	9	)	)	PUNCT
iajs-3045	189	10	+	+	NUM
iajs-3045	189	11	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	189	12	)	)	PUNCT
iajs-3045	189	13	.	.	PUNCT
iajs-3045	190	1	therefore	therefore	ADV
iajs-3045	190	2	[	[	X
iajs-3045	190	3	ƒ	ƒ	X
iajs-3045	190	4	:	:	PUNCT
iajs-3045	190	5	ʀ	ʀ	PART
iajs-3045	190	6	ѡ	ѡ	X
iajs-3045	190	7	]	]	PUNCT
iajs-3045	190	8	is	be	AUX
iajs-3045	190	9	exnpq2ab	exnpq2ab	PROPN
iajs-3045	190	10	ideal	ideal	NOUN
iajs-3045	190	11	of	of	ADP
iajs-3045	190	12	ʀ	ʀ	PRON
iajs-3045	190	13	.	.	PUNCT
iajs-3045	190	14	(	(	PUNCT
iajs-3045	190	15	⟸	⟸	ADJ
iajs-3045	190	16	)	)	PUNCT
iajs-3045	190	17	𝑆uppose	𝑆uppose	NOUN
iajs-3045	190	18	that	that	SCONJ
iajs-3045	190	19	[	[	X
iajs-3045	190	20	ƒ	ƒ	X
iajs-3045	190	21	:	:	PUNCT
iajs-3045	190	22	ʀ	ʀ	PART
iajs-3045	190	23	ѡ	ѡ	X
iajs-3045	190	24	]	]	PUNCT
iajs-3045	190	25	is	be	AUX
iajs-3045	190	26	exnpq2ab	exnpq2ab	PROPN
iajs-3045	190	27	ideal	ideal	NOUN
iajs-3045	190	28	of	of	ADP
iajs-3045	190	29	ʀ	ʀ	NOUN
iajs-3045	190	30	,	,	PUNCT
iajs-3045	190	31	and	and	CCONJ
iajs-3045	190	32	𝑟𝑠𝑡𝐴	𝑟𝑠𝑡𝐴	VERB
iajs-3045	190	33	⊆	⊆	NUM
iajs-3045	190	34	ƒ	ƒ	NOUN
iajs-3045	190	35	for	for	ADP
iajs-3045	190	36	𝑟	𝑟	NOUN
iajs-3045	190	37	,	,	PUNCT
iajs-3045	190	38	𝑠	𝑠	PROPN
iajs-3045	190	39	,	,	PUNCT
iajs-3045	190	40	𝑡	𝑡	PROPN
iajs-3045	190	41	∈	∈	PROPN
iajs-3045	190	42	ʀ	ʀ	NOUN
iajs-3045	190	43	and	and	CCONJ
iajs-3045	190	44	𝐴	𝐴	PROPN
iajs-3045	190	45	is	be	AUX
iajs-3045	190	46	a	a	DET
iajs-3045	190	47	submodule	submodule	NOUN
iajs-3045	190	48	of	of	ADP
iajs-3045	190	49	ѡ	ѡ	PROPN
iajs-3045	190	50	,	,	PUNCT
iajs-3045	190	51	since	since	SCONJ
iajs-3045	190	52	ѡ	ѡ	PROPN
iajs-3045	190	53	is	be	AUX
iajs-3045	190	54	a	a	DET
iajs-3045	190	55	multiplication	multiplication	NOUN
iajs-3045	190	56	,	,	PUNCT
iajs-3045	190	57	then	then	ADV
iajs-3045	190	58	𝐴	𝐴	PROPN
iajs-3045	190	59	=	=	PUNCT
iajs-3045	190	60	ƥѡ	ƥѡ	PROPN
iajs-3045	190	61	for	for	ADP
iajs-3045	190	62	some	some	DET
iajs-3045	190	63	ideal	ideal	ADJ
iajs-3045	190	64	ƥ	ƥ	NOUN
iajs-3045	190	65	of	of	ADP
iajs-3045	190	66	ʀ	ʀ	NOUN
iajs-3045	190	67	,	,	PUNCT
iajs-3045	190	68	that	that	PRON
iajs-3045	190	69	is	be	AUX
iajs-3045	190	70	𝑟𝑠𝑡ƥѡ	𝑟𝑠𝑡ƥѡ	NOUN
iajs-3045	190	71	⊆	⊆	NUM
iajs-3045	190	72	ƒ	ƒ	NUM
iajs-3045	190	73	,	,	PUNCT
iajs-3045	190	74	implies	imply	VERB
iajs-3045	190	75	that	that	SCONJ
iajs-3045	190	76	𝑟𝑠𝑡ƥ	𝑟𝑠𝑡ƥ	ADJ
iajs-3045	190	77	⊆	⊆	NUM
iajs-3045	190	78	[	[	X
iajs-3045	190	79	ƒ	ƒ	X
iajs-3045	190	80	:	:	PUNCT
iajs-3045	190	81	ʀ	ʀ	PART
iajs-3045	190	82	ѡ	ѡ	NOUN
iajs-3045	190	83	]	]	PUNCT
iajs-3045	190	84	,	,	PUNCT
iajs-3045	190	85	but	but	CCONJ
iajs-3045	191	1	[	[	X
iajs-3045	191	2	ƒ	ƒ	X
iajs-3045	191	3	:	:	PUNCT
iajs-3045	191	4	ʀ	ʀ	PART
iajs-3045	191	5	ѡ	ѡ	X
iajs-3045	191	6	]	]	PUNCT
iajs-3045	191	7	is	be	AUX
iajs-3045	191	8	exnpq2ab	exnpq2ab	PROPN
iajs-3045	191	9	ideal	ideal	NOUN
iajs-3045	191	10	of	of	ADP
iajs-3045	191	11	ʀ	ʀ	NOUN
iajs-3045	191	12	,	,	PUNCT
iajs-3045	191	13	then	then	ADV
iajs-3045	191	14	by	by	ADP
iajs-3045	191	15	proposition	proposition	NOUN
iajs-3045	191	16	2.24	2.24	NUM
iajs-3045	191	17	either	either	CCONJ
iajs-3045	191	18	𝑟𝑠ƥ	𝑟𝑠ƥ	ADV
iajs-3045	191	19	⊆	⊆	NUM
iajs-3045	191	20	[	[	X
iajs-3045	191	21	ƒ	ƒ	NUM
iajs-3045	191	22	:	:	PUNCT
iajs-3045	191	23	ʀ	ʀ	PART
iajs-3045	191	24	ѡ	ѡ	NOUN
iajs-3045	191	25	]	]	PUNCT
iajs-3045	191	26	+	+	X
iajs-3045	191	27	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	191	28	)	)	PUNCT
iajs-3045	191	29	+	+	NUM
iajs-3045	191	30	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	191	31	)	)	PUNCT
iajs-3045	191	32	or	or	CCONJ
iajs-3045	191	33	𝑟𝑡ƥ	𝑟𝑡ƥ	VERB
iajs-3045	191	34	⊆	⊆	NUM
iajs-3045	191	35	[	[	X
iajs-3045	191	36	ƒ	ƒ	X
iajs-3045	191	37	:	:	PUNCT
iajs-3045	191	38	ʀ	ʀ	PART
iajs-3045	191	39	ѡ	ѡ	NOUN
iajs-3045	191	40	]	]	PUNCT
iajs-3045	191	41	+	+	X
iajs-3045	191	42	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	191	43	)	)	PUNCT
iajs-3045	191	44	+	+	NUM
iajs-3045	191	45	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	191	46	)	)	PUNCT
iajs-3045	191	47	or	or	CCONJ
iajs-3045	191	48	𝑠𝑡ƥ	𝑠𝑡ƥ	VERB
iajs-3045	191	49	⊆	⊆	NUM
iajs-3045	191	50	[	[	X
iajs-3045	191	51	ƒ	ƒ	X
iajs-3045	191	52	:	:	PUNCT
iajs-3045	191	53	ʀ	ʀ	PART
iajs-3045	191	54	ѡ	ѡ	NOUN
iajs-3045	191	55	]	]	PUNCT
iajs-3045	191	56	+	+	X
iajs-3045	191	57	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	191	58	)	)	PUNCT
iajs-3045	191	59	+	+	NUM
iajs-3045	191	60	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	191	61	)	)	PUNCT
iajs-3045	191	62	.	.	PUNCT
iajs-3045	192	1	thus	thus	ADV
iajs-3045	192	2	either	either	CCONJ
iajs-3045	192	3	𝑟𝑠ƥѡ	𝑟𝑠ƥѡ	ADJ
iajs-3045	192	4	⊆	⊆	NUM
iajs-3045	192	5	[	[	X
iajs-3045	192	6	ƒ	ƒ	X
iajs-3045	192	7	:	:	PUNCT
iajs-3045	192	8	ʀ	ʀ	ADJ
iajs-3045	192	9	ѡ]ѡ	ѡ]ѡ	NOUN
iajs-3045	192	10	+	+	CCONJ
iajs-3045	192	11	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	192	12	+	+	CCONJ
iajs-3045	192	13	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	ADJ
iajs-3045	192	14	or	or	CCONJ
iajs-3045	192	15	𝑟𝑡ƥѡ	𝑟𝑡ƥѡ	VERB
iajs-3045	192	16	⊆	⊆	NUM
iajs-3045	193	1	[	[	X
iajs-3045	193	2	ƒ	ƒ	X
iajs-3045	193	3	:	:	PUNCT
iajs-3045	193	4	ʀ	ʀ	ADJ
iajs-3045	193	5	ѡ]ѡ	ѡ]ѡ	NOUN
iajs-3045	193	6	+	+	CCONJ
iajs-3045	193	7	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	193	8	+	+	CCONJ
iajs-3045	193	9	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	ADJ
iajs-3045	193	10	or	or	CCONJ
iajs-3045	193	11	𝑠𝑡ƥѡ	𝑠𝑡ƥѡ	VERB
iajs-3045	193	12	⊆	⊆	NUM
iajs-3045	194	1	[	[	X
iajs-3045	194	2	ƒ	ƒ	X
iajs-3045	194	3	:	:	PUNCT
iajs-3045	194	4	ʀ	ʀ	ADJ
iajs-3045	194	5	ѡ]ѡ	ѡ]ѡ	NOUN
iajs-3045	194	6	+	+	CCONJ
iajs-3045	194	7	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	ADJ
iajs-3045	194	8	+	+	CCONJ
iajs-3045	194	9	𝐽(ʀ)ѡ.	𝐽(ʀ)ѡ.	NOUN
iajs-3045	194	10	since	since	SCONJ
iajs-3045	194	11	ѡ	ѡ	PROPN
iajs-3045	194	12	is	be	AUX
iajs-3045	194	13	a	a	DET
iajs-3045	194	14	faithful	faithful	ADJ
iajs-3045	194	15	multiplication	multiplication	NOUN
iajs-3045	194	16	,	,	PUNCT
iajs-3045	194	17	then	then	ADV
iajs-3045	194	18	[	[	X
iajs-3045	194	19	ƒ	ƒ	X
iajs-3045	194	20	:	:	PUNCT
iajs-3045	194	21	ʀ	ʀ	NOUN
iajs-3045	194	22	ѡ]ѡ	ѡ]ѡ	NOUN
iajs-3045	194	23	=	=	PUNCT
iajs-3045	194	24	ƒ	ƒ	PROPN
iajs-3045	194	25	and	and	CCONJ
iajs-3045	194	26	by	by	ADP
iajs-3045	194	27	lemma	lemma	PROPN
iajs-3045	194	28	2.6	2.6	NUM
iajs-3045	194	29	and	and	CCONJ
iajs-3045	194	30	lemma	lemma	PROPN
iajs-3045	194	31	2.7	2.7	NUM
iajs-3045	194	32	either	either	CCONJ
iajs-3045	194	33	𝑟𝑠𝐴	𝑟𝑠𝐴	PROPN
iajs-3045	194	34	⊆	⊆	NUM
iajs-3045	194	35	ƒ	ƒ	PROPN
iajs-3045	194	36	+	+	NUM
iajs-3045	194	37	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	194	38	)	)	PUNCT
iajs-3045	195	1	+	+	PUNCT
iajs-3045	196	1	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	196	2	)	)	PUNCT
iajs-3045	196	3	or	or	CCONJ
iajs-3045	196	4	𝑟𝑡𝐴	𝑟𝑡𝐴	NUM
iajs-3045	196	5	⊆	⊆	NUM
iajs-3045	196	6	ƒ	ƒ	NOUN
iajs-3045	196	7	+	+	NUM
iajs-3045	196	8	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	196	9	)	)	PUNCT
iajs-3045	196	10	+	+	PUNCT
iajs-3045	196	11	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	196	12	)	)	PUNCT
iajs-3045	196	13	or	or	CCONJ
iajs-3045	196	14	𝑠𝑡𝐴	𝑠𝑡𝐴	VERB
iajs-3045	196	15	⊆	⊆	NUM
iajs-3045	196	16	ƒ	ƒ	PROPN
iajs-3045	196	17	+	+	NUM
iajs-3045	196	18	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	196	19	)	)	PUNCT
iajs-3045	196	20	+	+	PUNCT
iajs-3045	196	21	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	196	22	)	)	PUNCT
iajs-3045	196	23	.	.	PUNCT
iajs-3045	197	1	thus	thus	ADV
iajs-3045	197	2	by	by	ADP
iajs-3045	197	3	proposition	proposition	NOUN
iajs-3045	197	4	2.24	2.24	NUM
iajs-3045	197	5	ƒ	ƒ	NOUN
iajs-3045	197	6	is	be	AUX
iajs-3045	197	7	exnpq2ab	exnpq2ab	PROPN
iajs-3045	197	8	submodule	submodule	NOUN
iajs-3045	197	9	of	of	ADP
iajs-3045	197	10	ѡ.	ѡ.	NOUN
iajs-3045	197	11	proposition	proposition	NOUN
iajs-3045	197	12	3.6	3.6	NUM
iajs-3045	197	13	let	let	VERB
iajs-3045	197	14	ƒ	ƒ	PRON
iajs-3045	197	15	≠	≠	PROPN
iajs-3045	197	16	ѡ	ѡ	NOUN
iajs-3045	197	17	and	and	CCONJ
iajs-3045	197	18	ѡ	ѡ	PROPN
iajs-3045	197	19	is	be	AUX
iajs-3045	197	20	multiplication	multiplication	NOUN
iajs-3045	197	21	projective	projective	ADJ
iajs-3045	197	22	ʀ	ʀ	NOUN
iajs-3045	197	23	-	-	PUNCT
iajs-3045	197	24	module	module	NOUN
iajs-3045	197	25	.	.	PUNCT
iajs-3045	198	1	then	then	ADV
iajs-3045	198	2	ƒ	ƒ	PRON
iajs-3045	198	3	is	be	AUX
iajs-3045	198	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	198	5	submodule	submodule	NOUN
iajs-3045	198	6	of	of	ADP
iajs-3045	198	7	ѡ	ѡ	PROPN
iajs-3045	198	8	if	if	SCONJ
iajs-3045	199	1	and	and	CCONJ
iajs-3045	199	2	only	only	ADV
iajs-3045	199	3	if	if	SCONJ
iajs-3045	199	4	[	[	X
iajs-3045	199	5	ƒ	ƒ	X
iajs-3045	199	6	:	:	PUNCT
iajs-3045	199	7	ʀ	ʀ	PART
iajs-3045	199	8	ѡ	ѡ	X
iajs-3045	199	9	]	]	PUNCT
iajs-3045	199	10	is	be	AUX
iajs-3045	199	11	exnpq2ab	exnpq2ab	PROPN
iajs-3045	199	12	ideal	ideal	NOUN
iajs-3045	199	13	of	of	ADP
iajs-3045	199	14	ʀ	ʀ	NOUN
iajs-3045	199	15	.	.	NOUN
iajs-3045	199	16	proof	proof	NOUN
iajs-3045	199	17	.	.	PUNCT
iajs-3045	200	1	(	(	PUNCT
iajs-3045	200	2	⟹	⟹	X
iajs-3045	200	3	)	)	PUNCT
iajs-3045	200	4	assume	assume	VERB
iajs-3045	200	5	that	that	SCONJ
iajs-3045	200	6	ƒ	ƒ	PRON
iajs-3045	200	7	is	be	AUX
iajs-3045	200	8	exnpq2ab	exnpq2ab	PROPN
iajs-3045	200	9	submodule	submodule	NOUN
iajs-3045	200	10	of	of	ADP
iajs-3045	200	11	ѡ	ѡ	PROPN
iajs-3045	200	12	,	,	PUNCT
iajs-3045	200	13	and	and	CCONJ
iajs-3045	200	14	ƥ1ƥ2ƥ3𝑏	ƥ1ƥ2ƥ3𝑏	PROPN
iajs-3045	200	15	⊆	⊆	NUM
iajs-3045	201	1	[	[	X
iajs-3045	201	2	ƒ	ƒ	X
iajs-3045	201	3	:	:	PUNCT
iajs-3045	201	4	ʀ	ʀ	NOUN
iajs-3045	201	5	ѡ	ѡ	NOUN
iajs-3045	201	6	]	]	PUNCT
iajs-3045	201	7	for	for	ADP
iajs-3045	201	8	some	some	DET
iajs-3045	201	9	ideals	ideal	NOUN
iajs-3045	201	10	ƥ1	ƥ1	NOUN
iajs-3045	201	11	,	,	PUNCT
iajs-3045	201	12	ƥ2	ƥ2	NOUN
iajs-3045	201	13	,	,	PUNCT
iajs-3045	201	14	ƥ3	ƥ3	NOUN
iajs-3045	201	15	of	of	ADP
iajs-3045	201	16	ʀ	ʀ	PROPN
iajs-3045	201	17	and	and	CCONJ
iajs-3045	201	18	𝑏	𝑏	PRON
iajs-3045	201	19	∈	∈	PROPN
iajs-3045	201	20	ʀ	ʀ	NOUN
iajs-3045	201	21	,	,	PUNCT
iajs-3045	201	22	then	then	ADV
iajs-3045	201	23	ƥ1ƥ2ƥ3(𝑏ѡ	ƥ1ƥ2ƥ3(𝑏ѡ	VERB
iajs-3045	201	24	)	)	PUNCT
iajs-3045	201	25	⊆	⊆	NUM
iajs-3045	201	26	ƒ	ƒ	NUM
iajs-3045	201	27	.	.	PUNCT
iajs-3045	202	1	but	but	CCONJ
iajs-3045	202	2	ƒ	ƒ	PRON
iajs-3045	202	3	is	be	AUX
iajs-3045	202	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	202	5	submodule	submodule	NOUN
iajs-3045	202	6	of	of	ADP
iajs-3045	202	7	ѡ	ѡ	PROPN
iajs-3045	202	8	,	,	PUNCT
iajs-3045	202	9	then	then	ADV
iajs-3045	202	10	by	by	ADP
iajs-3045	202	11	proposition	proposition	NOUN
iajs-3045	202	12	2.25	2.25	NUM
iajs-3045	202	13	either	either	CCONJ
iajs-3045	202	14	ƥ1ƥ3𝑏ѡ	ƥ1ƥ3𝑏ѡ	NOUN
iajs-3045	203	1	⊆	⊆	NUM
iajs-3045	203	2	ƒ	ƒ	NOUN
iajs-3045	203	3	+	+	NUM
iajs-3045	203	4	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	203	5	)	)	PUNCT
iajs-3045	203	6	+	+	PUNCT
iajs-3045	203	7	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	203	8	)	)	PUNCT
iajs-3045	203	9	or	or	CCONJ
iajs-3045	203	10	ƥ2ƥ3𝑏ѡ	ƥ2ƥ3𝑏ѡ	VERB
iajs-3045	203	11	⊆	⊆	NUM
iajs-3045	203	12	ƒ	ƒ	NOUN
iajs-3045	203	13	+	+	NUM
iajs-3045	203	14	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	203	15	)	)	PUNCT
iajs-3045	204	1	+	+	PUNCT
iajs-3045	205	1	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	205	2	)	)	PUNCT
iajs-3045	205	3	or	or	CCONJ
iajs-3045	205	4	ƥ1ƥ2𝑏ѡ	ƥ1ƥ2𝑏ѡ	NOUN
iajs-3045	206	1	⊆	⊆	X
iajs-3045	206	2	ƒ	ƒ	PROPN
iajs-3045	206	3	+	+	NUM
iajs-3045	206	4	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	206	5	)	)	PUNCT
iajs-3045	207	1	+	+	PUNCT
iajs-3045	208	1	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	208	2	)	)	PUNCT
iajs-3045	208	3	.	.	PUNCT
iajs-3045	209	1	since	since	SCONJ
iajs-3045	209	2	ѡ	ѡ	PROPN
iajs-3045	209	3	is	be	AUX
iajs-3045	209	4	multiplication	multiplication	NOUN
iajs-3045	209	5	,	,	PUNCT
iajs-3045	209	6	thenƒ	thenƒ	NOUN
iajs-3045	209	7	=	=	PUNCT
iajs-3045	210	1	[	[	X
iajs-3045	210	2	ƒ	ƒ	X
iajs-3045	210	3	:	:	PUNCT
iajs-3045	210	4	ʀ	ʀ	ADJ
iajs-3045	210	5	ѡ]ѡ	ѡ]ѡ	NOUN
iajs-3045	210	6	,	,	PUNCT
iajs-3045	210	7	and	and	CCONJ
iajs-3045	210	8	since	since	SCONJ
iajs-3045	210	9	ѡ	ѡ	PROPN
iajs-3045	210	10	is	be	AUX
iajs-3045	210	11	projective	projective	ADJ
iajs-3045	210	12	ʀ	ʀ	NOUN
iajs-3045	210	13	-	-	PUNCT
iajs-3045	210	14	module	module	NOUN
iajs-3045	210	15	ѡ	ѡ	NOUN
iajs-3045	210	16	,	,	PUNCT
iajs-3045	210	17	then	then	ADV
iajs-3045	210	18	by	by	ADP
iajs-3045	210	19	lemma	lemma	PROPN
iajs-3045	210	20	2.10	2.10	NUM
iajs-3045	210	21	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	210	22	)	)	PUNCT
iajs-3045	210	23	=	=	SYM
iajs-3045	211	1	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	211	2	and	and	CCONJ
iajs-3045	211	3	by	by	ADP
iajs-3045	211	4	lemma	lemma	PROPN
iajs-3045	211	5	2.9	2.9	NUM
iajs-3045	211	6	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	211	7	)	)	PUNCT
iajs-3045	211	8	=	=	PUNCT
iajs-3045	211	9	𝐽(ʀ)ѡ.	𝐽(ʀ)ѡ.	PROPN
iajs-3045	211	10	thus	thus	ADV
iajs-3045	211	11	either	either	CCONJ
iajs-3045	211	12	ƥ1ƥ3𝑏ѡ	ƥ1ƥ3𝑏ѡ	NOUN
iajs-3045	212	1	⊆	⊆	NUM
iajs-3045	212	2	[	[	X
iajs-3045	212	3	ƒ	ƒ	X
iajs-3045	212	4	:	:	PUNCT
iajs-3045	212	5	ʀ	ʀ	ADJ
iajs-3045	212	6	ѡ]ѡ	ѡ]ѡ	NOUN
iajs-3045	212	7	+	+	CCONJ
iajs-3045	212	8	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	212	9	+	+	CCONJ
iajs-3045	212	10	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	ADJ
iajs-3045	212	11	or	or	CCONJ
iajs-3045	212	12	ƥ2ƥ3𝑏ѡ	ƥ2ƥ3𝑏ѡ	VERB
iajs-3045	212	13	⊆	⊆	NUM
iajs-3045	213	1	[	[	X
iajs-3045	213	2	ƒ	ƒ	X
iajs-3045	213	3	:	:	PUNCT
iajs-3045	213	4	ʀ	ʀ	ADJ
iajs-3045	213	5	ѡ]ѡ	ѡ]ѡ	NOUN
iajs-3045	213	6	+	+	CCONJ
iajs-3045	213	7	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	213	8	+	+	CCONJ
iajs-3045	213	9	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	ADJ
iajs-3045	213	10	or	or	CCONJ
iajs-3045	213	11	ƥ1ƥ2𝑏ѡ	ƥ1ƥ2𝑏ѡ	NOUN
iajs-3045	213	12	⊆	⊆	NUM
iajs-3045	213	13	[	[	X
iajs-3045	213	14	ƒ	ƒ	X
iajs-3045	213	15	:	:	PUNCT
iajs-3045	213	16	ʀ	ʀ	ADJ
iajs-3045	213	17	ѡ]ѡ	ѡ]ѡ	NOUN
iajs-3045	213	18	+	+	CCONJ
iajs-3045	213	19	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	ADJ
iajs-3045	213	20	+	+	CCONJ
iajs-3045	213	21	𝐽(ʀ)ѡ.	𝐽(ʀ)ѡ.	NOUN
iajs-3045	213	22	hence	hence	ADV
iajs-3045	213	23	ƥ1ƥ3𝑏	ƥ1ƥ3𝑏	NUM
iajs-3045	214	1	⊆	⊆	NUM
iajs-3045	214	2	[	[	X
iajs-3045	214	3	ƒ	ƒ	X
iajs-3045	214	4	:	:	PUNCT
iajs-3045	214	5	ʀ	ʀ	PART
iajs-3045	214	6	ѡ	ѡ	NOUN
iajs-3045	214	7	]	]	PUNCT
iajs-3045	214	8	+	+	X
iajs-3045	214	9	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	X
iajs-3045	214	10	)	)	PUNCT
iajs-3045	215	1	+	+	CCONJ
iajs-3045	215	2	𝐽(ʀ)or	𝐽(ʀ)or	NOUN
iajs-3045	215	3	ƥ2ƥ3𝑏	ƥ2ƥ3𝑏	PUNCT
iajs-3045	216	1	⊆	⊆	NUM
iajs-3045	216	2	[	[	X
iajs-3045	216	3	ƒ	ƒ	X
iajs-3045	216	4	:	:	PUNCT
iajs-3045	216	5	ʀ	ʀ	PART
iajs-3045	216	6	ѡ	ѡ	NOUN
iajs-3045	216	7	]	]	PUNCT
iajs-3045	216	8	+	+	X
iajs-3045	216	9	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	216	10	)	)	PUNCT
iajs-3045	216	11	+	+	NUM
iajs-3045	216	12	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	216	13	)	)	PUNCT
iajs-3045	216	14	or	or	CCONJ
iajs-3045	216	15	ƥ1ƥ2𝑏	ƥ1ƥ2𝑏	SYM
iajs-3045	216	16	⊆	⊆	NUM
iajs-3045	216	17	[	[	X
iajs-3045	216	18	ƒ	ƒ	X
iajs-3045	216	19	:	:	PUNCT
iajs-3045	216	20	ʀ	ʀ	PART
iajs-3045	216	21	ѡ	ѡ	NOUN
iajs-3045	216	22	]	]	PUNCT
iajs-3045	216	23	+	+	X
iajs-3045	216	24	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	216	25	)	)	PUNCT
iajs-3045	216	26	+	+	NUM
iajs-3045	216	27	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	216	28	)	)	PUNCT
iajs-3045	216	29	.	.	PUNCT
iajs-3045	217	1	therefore	therefore	ADV
iajs-3045	217	2	by	by	ADP
iajs-3045	217	3	proposition	proposition	NOUN
iajs-3045	217	4	2.27	2.27	NUM
iajs-3045	217	5	[	[	X
iajs-3045	217	6	ƒ	ƒ	X
iajs-3045	217	7	:	:	PUNCT
iajs-3045	217	8	ʀ	ʀ	PART
iajs-3045	217	9	ѡ	ѡ	X
iajs-3045	217	10	]	]	PUNCT
iajs-3045	217	11	is	be	AUX
iajs-3045	217	12	exnpq2ab	exnpq2ab	PROPN
iajs-3045	217	13	ideal	ideal	NOUN
iajs-3045	217	14	of	of	ADP
iajs-3045	217	15	ʀ	ʀ	NOUN
iajs-3045	217	16	.	.	PUNCT
iajs-3045	217	17	ihjpas	ihjpas	PROPN
iajs-3045	217	18	.	.	PUNCT
iajs-3045	218	1	36(2)2023	36(2)2023	NUM
iajs-3045	218	2	413	413	NUM
iajs-3045	218	3	(	(	PUNCT
iajs-3045	218	4	⟸	⟸	ADJ
iajs-3045	218	5	)	)	PUNCT
iajs-3045	218	6	𝑆uppose	𝑆uppose	NOUN
iajs-3045	218	7	that	that	SCONJ
iajs-3045	218	8	[	[	X
iajs-3045	218	9	ƒ	ƒ	X
iajs-3045	218	10	:	:	PUNCT
iajs-3045	218	11	ʀ	ʀ	PART
iajs-3045	218	12	ѡ	ѡ	X
iajs-3045	218	13	]	]	PUNCT
iajs-3045	218	14	is	be	AUX
iajs-3045	218	15	exnpq2ab	exnpq2ab	PROPN
iajs-3045	218	16	ideal	ideal	NOUN
iajs-3045	218	17	of	of	ADP
iajs-3045	218	18	ʀ	ʀ	NOUN
iajs-3045	218	19	,	,	PUNCT
iajs-3045	218	20	and	and	CCONJ
iajs-3045	218	21	𝑟𝑠ƥ𝐴	𝑟𝑠ƥ𝐴	NOUN
iajs-3045	218	22	⊆	⊆	NUM
iajs-3045	218	23	ƒ	ƒ	NUM
iajs-3045	218	24	for	for	ADP
iajs-3045	218	25	𝑟	𝑟	NOUN
iajs-3045	218	26	,	,	PUNCT
iajs-3045	218	27	𝑠	𝑠	PROPN
iajs-3045	218	28	∈	∈	PROPN
iajs-3045	218	29	ʀ	ʀ	NOUN
iajs-3045	218	30	and	and	CCONJ
iajs-3045	218	31	some	some	DET
iajs-3045	218	32	submodule	submodule	NOUN
iajs-3045	218	33	𝐴	𝐴	PROPN
iajs-3045	218	34	of	of	ADP
iajs-3045	218	35	ѡ	ѡ	PROPN
iajs-3045	218	36	and	and	CCONJ
iajs-3045	218	37	for	for	ADP
iajs-3045	218	38	some	some	DET
iajs-3045	218	39	ideal	ideal	ADJ
iajs-3045	218	40	ƥ	ƥ	NOUN
iajs-3045	218	41	of	of	ADP
iajs-3045	218	42	ʀ	ʀ	PRON
iajs-3045	218	43	since	since	SCONJ
iajs-3045	218	44	ѡ	ѡ	PROPN
iajs-3045	218	45	is	be	AUX
iajs-3045	218	46	a	a	DET
iajs-3045	218	47	multiplication	multiplication	NOUN
iajs-3045	218	48	,	,	PUNCT
iajs-3045	218	49	then	then	ADV
iajs-3045	218	50	𝐴	𝐴	PROPN
iajs-3045	218	51	=	=	PUNCT
iajs-3045	219	1	𝐽ѡ	𝐽ѡ	PROPN
iajs-3045	219	2	for	for	ADP
iajs-3045	219	3	some	some	DET
iajs-3045	219	4	ideal	ideal	ADJ
iajs-3045	219	5	𝐽	𝐽	PROPN
iajs-3045	219	6	of	of	ADP
iajs-3045	219	7	ʀ	ʀ	NOUN
iajs-3045	219	8	,	,	PUNCT
iajs-3045	219	9	that	that	PRON
iajs-3045	219	10	is	be	AUX
iajs-3045	219	11	𝑟𝑠ƥ𝐽ѡ	𝑟𝑠ƥ𝐽ѡ	PROPN
iajs-3045	219	12	⊆	⊆	NUM
iajs-3045	219	13	ƒ	ƒ	NUM
iajs-3045	219	14	,	,	PUNCT
iajs-3045	219	15	implies	imply	VERB
iajs-3045	219	16	that	that	SCONJ
iajs-3045	219	17	𝑟𝑠ƥ𝐽	𝑟𝑠ƥ𝐽	NOUN
iajs-3045	219	18	⊆	⊆	NUM
iajs-3045	219	19	[	[	X
iajs-3045	219	20	ƒ	ƒ	X
iajs-3045	219	21	:	:	PUNCT
iajs-3045	219	22	ʀ	ʀ	PART
iajs-3045	219	23	ѡ	ѡ	NOUN
iajs-3045	219	24	]	]	PUNCT
iajs-3045	219	25	,	,	PUNCT
iajs-3045	219	26	but	but	CCONJ
iajs-3045	219	27	[	[	X
iajs-3045	219	28	ƒ	ƒ	X
iajs-3045	219	29	:	:	PUNCT
iajs-3045	219	30	ʀ	ʀ	PART
iajs-3045	219	31	ѡ	ѡ	X
iajs-3045	219	32	]	]	PUNCT
iajs-3045	219	33	is	be	AUX
iajs-3045	219	34	exnpq2ab	exnpq2ab	PROPN
iajs-3045	219	35	𝑖𝑑𝑒𝑎𝑙	𝑖𝑑𝑒𝑎𝑙	ADJ
iajs-3045	219	36	of	of	ADP
iajs-3045	219	37	ʀ	ʀ	NOUN
iajs-3045	219	38	,	,	PUNCT
iajs-3045	219	39	then	then	ADV
iajs-3045	219	40	by	by	ADP
iajs-3045	219	41	proposition	proposition	NOUN
iajs-3045	219	42	2.28	2.28	NUM
iajs-3045	219	43	either	either	CCONJ
iajs-3045	219	44	𝑟𝑠𝐽	𝑟𝑠𝐽	NUM
iajs-3045	219	45	⊆	⊆	NUM
iajs-3045	219	46	[	[	X
iajs-3045	219	47	ƒ	ƒ	X
iajs-3045	219	48	:	:	PUNCT
iajs-3045	219	49	ʀ	ʀ	PART
iajs-3045	219	50	ѡ	ѡ	NOUN
iajs-3045	219	51	]	]	PUNCT
iajs-3045	219	52	+	+	X
iajs-3045	219	53	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	219	54	)	)	PUNCT
iajs-3045	219	55	+	+	NUM
iajs-3045	220	1	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	220	2	)	)	PUNCT
iajs-3045	220	3	or	or	CCONJ
iajs-3045	220	4	𝑟ƥ𝐽	𝑟ƥ𝐽	VERB
iajs-3045	220	5	⊆	⊆	NUM
iajs-3045	221	1	[	[	X
iajs-3045	221	2	ƒ	ƒ	X
iajs-3045	221	3	:	:	PUNCT
iajs-3045	221	4	ʀ	ʀ	PART
iajs-3045	221	5	ѡ	ѡ	NOUN
iajs-3045	221	6	]	]	PUNCT
iajs-3045	221	7	+	+	X
iajs-3045	221	8	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	221	9	)	)	PUNCT
iajs-3045	221	10	+	+	NUM
iajs-3045	221	11	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	221	12	)	)	PUNCT
iajs-3045	221	13	or	or	CCONJ
iajs-3045	221	14	𝑠ƥ𝐽	𝑠ƥ𝐽	PROPN
iajs-3045	221	15	⊆	⊆	NUM
iajs-3045	222	1	[	[	X
iajs-3045	222	2	ƒ	ƒ	X
iajs-3045	222	3	:	:	PUNCT
iajs-3045	222	4	ʀ	ʀ	PART
iajs-3045	222	5	ѡ	ѡ	NOUN
iajs-3045	222	6	]	]	PUNCT
iajs-3045	222	7	+	+	X
iajs-3045	222	8	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	222	9	)	)	PUNCT
iajs-3045	222	10	+	+	NUM
iajs-3045	222	11	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	222	12	)	)	PUNCT
iajs-3045	222	13	.	.	PUNCT
iajs-3045	223	1	thus	thus	ADV
iajs-3045	223	2	either	either	CCONJ
iajs-3045	223	3	𝑟𝑠𝐽ѡ	𝑟𝑠𝐽ѡ	NOUN
iajs-3045	223	4	⊆	⊆	NUM
iajs-3045	223	5	[	[	X
iajs-3045	223	6	ƒ	ƒ	X
iajs-3045	223	7	:	:	PUNCT
iajs-3045	223	8	ʀ	ʀ	ADJ
iajs-3045	223	9	ѡ]ѡ	ѡ]ѡ	NOUN
iajs-3045	223	10	+	+	CCONJ
iajs-3045	223	11	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	223	12	+	+	CCONJ
iajs-3045	223	13	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	ADJ
iajs-3045	223	14	or	or	CCONJ
iajs-3045	223	15	𝑟ƥ𝐽ѡ	𝑟ƥ𝐽ѡ	PROPN
iajs-3045	223	16	⊆	⊆	NUM
iajs-3045	223	17	[	[	X
iajs-3045	223	18	ƒ	ƒ	X
iajs-3045	223	19	:	:	PUNCT
iajs-3045	223	20	ʀ	ʀ	ADJ
iajs-3045	223	21	ѡ]ѡ	ѡ]ѡ	NOUN
iajs-3045	223	22	+	+	CCONJ
iajs-3045	223	23	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	223	24	+	+	CCONJ
iajs-3045	223	25	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	ADJ
iajs-3045	223	26	or	or	CCONJ
iajs-3045	223	27	𝑠ƥ𝐽ѡ	𝑠ƥ𝐽ѡ	PROPN
iajs-3045	223	28	⊆	⊆	NUM
iajs-3045	223	29	[	[	X
iajs-3045	223	30	ƒ	ƒ	X
iajs-3045	223	31	:	:	PUNCT
iajs-3045	223	32	ʀ	ʀ	ADJ
iajs-3045	223	33	ѡ]ѡ	ѡ]ѡ	NOUN
iajs-3045	223	34	+	+	CCONJ
iajs-3045	223	35	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	ADJ
iajs-3045	223	36	+	+	CCONJ
iajs-3045	223	37	𝐽(ʀ)ѡ.	𝐽(ʀ)ѡ.	NOUN
iajs-3045	223	38	hence	hence	ADV
iajs-3045	223	39	by	by	ADP
iajs-3045	223	40	lemma	lemma	PROPN
iajs-3045	223	41	2.10	2.10	NUM
iajs-3045	223	42	and	and	CCONJ
iajs-3045	223	43	lemma	lemma	PROPN
iajs-3045	223	44	2.9	2.9	NUM
iajs-3045	223	45	either	either	CCONJ
iajs-3045	223	46	𝑟𝑠𝐴	𝑟𝑠𝐴	PROPN
iajs-3045	223	47	⊆	⊆	NUM
iajs-3045	223	48	ƒ	ƒ	PROPN
iajs-3045	223	49	+	+	NUM
iajs-3045	223	50	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	223	51	)	)	PUNCT
iajs-3045	223	52	+	+	PUNCT
iajs-3045	223	53	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	223	54	)	)	PUNCT
iajs-3045	223	55	or	or	CCONJ
iajs-3045	223	56	𝑟ƥ𝐴	𝑟ƥ𝐴	PROPN
iajs-3045	223	57	⊆	⊆	NUM
iajs-3045	223	58	ƒ	ƒ	PROPN
iajs-3045	223	59	+	+	NUM
iajs-3045	223	60	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	223	61	)	)	PUNCT
iajs-3045	223	62	+	+	PUNCT
iajs-3045	223	63	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	223	64	)	)	PUNCT
iajs-3045	223	65	or	or	CCONJ
iajs-3045	223	66	𝑠ƥ𝐴	𝑠ƥ𝐴	NOUN
iajs-3045	223	67	⊆	⊆	NUM
iajs-3045	223	68	ƒ	ƒ	NOUN
iajs-3045	223	69	+	+	NUM
iajs-3045	223	70	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	223	71	)	)	PUNCT
iajs-3045	223	72	+	+	PUNCT
iajs-3045	223	73	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	223	74	)	)	PUNCT
iajs-3045	223	75	.	.	PUNCT
iajs-3045	224	1	thus	thus	ADV
iajs-3045	224	2	by	by	ADP
iajs-3045	224	3	proposition	proposition	NOUN
iajs-3045	224	4	2.28	2.28	NUM
iajs-3045	224	5	ƒ	ƒ	NOUN
iajs-3045	224	6	is	be	AUX
iajs-3045	224	7	exnpq2ab	exnpq2ab	PROPN
iajs-3045	224	8	submodule	submodule	NOUN
iajs-3045	224	9	of	of	ADP
iajs-3045	224	10	ѡ.	ѡ.	NOUN
iajs-3045	224	11	proposition	proposition	NOUN
iajs-3045	224	12	3.7	3.7	NUM
iajs-3045	224	13	let	let	VERB
iajs-3045	224	14	ƒ	ƒ	PRON
iajs-3045	224	15	≠	≠	PROPN
iajs-3045	224	16	ѡ	ѡ	NOUN
iajs-3045	224	17	and	and	CCONJ
iajs-3045	224	18	ѡ	ѡ	PROPN
iajs-3045	224	19	is	be	AUX
iajs-3045	224	20	non	non	ADJ
iajs-3045	224	21	-	-	ADJ
iajs-3045	224	22	singular	singular	ADJ
iajs-3045	224	23	multiplication	multiplication	NOUN
iajs-3045	224	24	ʀ	ʀ	NOUN
iajs-3045	224	25	-	-	PUNCT
iajs-3045	224	26	module	module	NOUN
iajs-3045	224	27	ѡ	ѡ	NOUN
iajs-3045	224	28	over	over	ADP
iajs-3045	224	29	an	an	DET
iajs-3045	224	30	a	a	DET
iajs-3045	224	31	good	good	ADJ
iajs-3045	224	32	𝑟𝑖𝑛𝑔	𝑟𝑖𝑛𝑔	NOUN
iajs-3045	224	33	ʀ	ʀ	PROPN
iajs-3045	224	34	.	.	PROPN
iajs-3045	225	1	then	then	ADV
iajs-3045	225	2	ƒ	ƒ	PROPN
iajs-3045	225	3	is	be	AUX
iajs-3045	225	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	225	5	submodule	submodule	NOUN
iajs-3045	225	6	of	of	ADP
iajs-3045	225	7	ѡ	ѡ	PROPN
iajs-3045	225	8	if	if	SCONJ
iajs-3045	226	1	and	and	CCONJ
iajs-3045	226	2	only	only	ADV
iajs-3045	226	3	if	if	SCONJ
iajs-3045	226	4	[	[	X
iajs-3045	226	5	ƒ	ƒ	X
iajs-3045	226	6	:	:	PUNCT
iajs-3045	226	7	ʀ	ʀ	PART
iajs-3045	226	8	ѡ	ѡ	X
iajs-3045	226	9	]	]	PUNCT
iajs-3045	226	10	is	be	AUX
iajs-3045	226	11	exnpq2ab	exnpq2ab	PROPN
iajs-3045	226	12	ideal	ideal	NOUN
iajs-3045	226	13	of	of	ADP
iajs-3045	226	14	ʀ	ʀ	NOUN
iajs-3045	226	15	.	.	NOUN
iajs-3045	226	16	proof	proof	NOUN
iajs-3045	226	17	.	.	PUNCT
iajs-3045	227	1	(	(	PUNCT
iajs-3045	227	2	⟹	⟹	X
iajs-3045	227	3	)	)	PUNCT
iajs-3045	227	4	let	let	VERB
iajs-3045	227	5	𝑎𝑏𝑐𝑡	𝑎𝑏𝑐𝑡	NOUN
iajs-3045	227	6	∈	∈	NOUN
iajs-3045	228	1	[	[	X
iajs-3045	228	2	ƒ	ƒ	X
iajs-3045	228	3	:	:	PUNCT
iajs-3045	228	4	ʀ	ʀ	PART
iajs-3045	228	5	ѡ	ѡ	NOUN
iajs-3045	228	6	]	]	X
iajs-3045	228	7	for	for	ADP
iajs-3045	228	8	𝑎	𝑎	PROPN
iajs-3045	228	9	,	,	PUNCT
iajs-3045	228	10	𝑏	𝑏	NOUN
iajs-3045	228	11	,	,	PUNCT
iajs-3045	228	12	𝑐	𝑐	PROPN
iajs-3045	228	13	,	,	PUNCT
iajs-3045	228	14	𝑡	𝑡	PROPN
iajs-3045	228	15	∈	∈	PROPN
iajs-3045	228	16	ʀ	ʀ	NOUN
iajs-3045	228	17	,	,	PUNCT
iajs-3045	228	18	then	then	ADV
iajs-3045	228	19	𝑎𝑏𝑐(𝑡ѡ	𝑎𝑏𝑐(𝑡ѡ	NOUN
iajs-3045	228	20	)	)	PUNCT
iajs-3045	228	21	⊆	⊆	NUM
iajs-3045	228	22	ƒ	ƒ	NUM
iajs-3045	228	23	.	.	PUNCT
iajs-3045	229	1	but	but	CCONJ
iajs-3045	229	2	ƒ	ƒ	PRON
iajs-3045	229	3	is	be	AUX
iajs-3045	229	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	229	5	submodule	submodule	NOUN
iajs-3045	229	6	of	of	ADP
iajs-3045	229	7	ѡ	ѡ	PROPN
iajs-3045	229	8	,	,	PUNCT
iajs-3045	229	9	then	then	ADV
iajs-3045	229	10	by	by	ADP
iajs-3045	229	11	proposition	proposition	NOUN
iajs-3045	229	12	2.24	2.24	NUM
iajs-3045	229	13	either	either	CCONJ
iajs-3045	229	14	𝑎𝑏(𝑡ѡ	𝑎𝑏(𝑡ѡ	NOUN
iajs-3045	229	15	)	)	PUNCT
iajs-3045	229	16	⊆	⊆	NUM
iajs-3045	229	17	ƒ	ƒ	X
iajs-3045	229	18	+	+	PUNCT
iajs-3045	229	19	(	(	PUNCT
iajs-3045	229	20	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	229	21	)	)	PUNCT
iajs-3045	229	22	+	+	PUNCT
iajs-3045	230	1	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	230	2	)	)	PUNCT
iajs-3045	230	3	)	)	PUNCT
iajs-3045	230	4	or	or	CCONJ
iajs-3045	230	5	𝑎𝑐(𝑡ѡ	𝑎𝑐(𝑡ѡ	NOUN
iajs-3045	230	6	)	)	PUNCT
iajs-3045	231	1	⊆	⊆	NUM
iajs-3045	231	2	ƒ	ƒ	X
iajs-3045	231	3	+	+	PUNCT
iajs-3045	231	4	(	(	PUNCT
iajs-3045	231	5	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	231	6	)	)	PUNCT
iajs-3045	231	7	+	+	PUNCT
iajs-3045	231	8	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	231	9	)	)	PUNCT
iajs-3045	231	10	)	)	PUNCT
iajs-3045	231	11	or	or	CCONJ
iajs-3045	231	12	𝑏𝑐(𝑡ѡ	𝑏𝑐(𝑡ѡ	ADJ
iajs-3045	231	13	)	)	PUNCT
iajs-3045	231	14	⊆	⊆	NUM
iajs-3045	231	15	ƒ	ƒ	X
iajs-3045	231	16	+	+	PUNCT
iajs-3045	231	17	(	(	PUNCT
iajs-3045	231	18	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	231	19	)	)	PUNCT
iajs-3045	231	20	+	+	PUNCT
iajs-3045	231	21	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	231	22	)	)	PUNCT
iajs-3045	231	23	)	)	PUNCT
iajs-3045	231	24	.	.	PUNCT
iajs-3045	232	1	since	since	SCONJ
iajs-3045	232	2	ѡ	ѡ	PROPN
iajs-3045	232	3	is	be	AUX
iajs-3045	232	4	multiplication	multiplication	NOUN
iajs-3045	232	5	,	,	PUNCT
iajs-3045	232	6	then	then	ADV
iajs-3045	232	7	ƒ	ƒ	X
iajs-3045	232	8	=	=	PUNCT
iajs-3045	233	1	[	[	X
iajs-3045	233	2	ƒ	ƒ	X
iajs-3045	233	3	:	:	PUNCT
iajs-3045	233	4	ʀ	ʀ	NOUN
iajs-3045	233	5	ѡ]ѡ	ѡ]ѡ	NOUN
iajs-3045	233	6	and	and	CCONJ
iajs-3045	233	7	since	since	SCONJ
iajs-3045	233	8	ѡ	ѡ	PROPN
iajs-3045	233	9	is	be	AUX
iajs-3045	233	10	non	non	ADJ
iajs-3045	233	11	-	-	ADJ
iajs-3045	233	12	singular	singular	ADJ
iajs-3045	233	13	multiplication	multiplication	NOUN
iajs-3045	233	14	,	,	PUNCT
iajs-3045	233	15	then	then	ADV
iajs-3045	233	16	by	by	ADP
iajs-3045	233	17	lemma	lemma	PROPN
iajs-3045	233	18	2.17	2.17	NUM
iajs-3045	233	19	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	233	20	)	)	PUNCT
iajs-3045	233	21	=	=	SYM
iajs-3045	234	1	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	234	2	and	and	CCONJ
iajs-3045	234	3	ʀ	ʀ	NOUN
iajs-3045	234	4	is	be	AUX
iajs-3045	234	5	a	a	DET
iajs-3045	234	6	good	good	ADJ
iajs-3045	234	7	ring	ring	NOUN
iajs-3045	234	8	then	then	ADV
iajs-3045	234	9	by	by	ADP
iajs-3045	234	10	remark	remark	NOUN
iajs-3045	234	11	2.11	2.11	NUM
iajs-3045	234	12	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	234	13	)	)	PUNCT
iajs-3045	234	14	=	=	PUNCT
iajs-3045	234	15	𝐽(ʀ)ѡ.	𝐽(ʀ)ѡ.	PROPN
iajs-3045	234	16	thus	thus	ADV
iajs-3045	234	17	either	either	CCONJ
iajs-3045	234	18	𝑎𝑏(𝑡ѡ	𝑎𝑏(𝑡ѡ	NOUN
iajs-3045	234	19	)	)	PUNCT
iajs-3045	235	1	⊆	⊆	NUM
iajs-3045	236	1	[	[	X
iajs-3045	236	2	ƒ	ƒ	X
iajs-3045	236	3	:	:	PUNCT
iajs-3045	236	4	ʀ	ʀ	ADJ
iajs-3045	236	5	ѡ]ѡ	ѡ]ѡ	X
iajs-3045	236	6	+	+	CCONJ
iajs-3045	236	7	(	(	PUNCT
iajs-3045	236	8	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	236	9	+	+	CCONJ
iajs-3045	236	10	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	NOUN
iajs-3045	236	11	)	)	PUNCT
iajs-3045	236	12	or	or	CCONJ
iajs-3045	236	13	(	(	PUNCT
iajs-3045	236	14	𝑡ѡ	𝑡ѡ	NOUN
iajs-3045	236	15	)	)	PUNCT
iajs-3045	236	16	⊆	⊆	NUM
iajs-3045	237	1	[	[	X
iajs-3045	237	2	ƒ	ƒ	X
iajs-3045	237	3	:	:	PUNCT
iajs-3045	237	4	ʀ	ʀ	ADJ
iajs-3045	237	5	ѡ]ѡ	ѡ]ѡ	X
iajs-3045	237	6	+	+	CCONJ
iajs-3045	237	7	(	(	PUNCT
iajs-3045	237	8	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	237	9	+	+	CCONJ
iajs-3045	237	10	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	ADJ
iajs-3045	237	11	)	)	PUNCT
iajs-3045	237	12	or	or	CCONJ
iajs-3045	237	13	𝑎𝑐(𝑡ѡ	𝑎𝑐(𝑡ѡ	NOUN
iajs-3045	237	14	)	)	PUNCT
iajs-3045	238	1	⊆	⊆	NUM
iajs-3045	239	1	[	[	X
iajs-3045	239	2	ƒ	ƒ	X
iajs-3045	239	3	:	:	PUNCT
iajs-3045	239	4	ʀ	ʀ	ADJ
iajs-3045	239	5	ѡ]ѡ	ѡ]ѡ	X
iajs-3045	239	6	+	+	CCONJ
iajs-3045	239	7	(	(	PUNCT
iajs-3045	239	8	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	239	9	+	+	CCONJ
iajs-3045	239	10	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	NOUN
iajs-3045	239	11	)	)	PUNCT
iajs-3045	239	12	,	,	PUNCT
iajs-3045	239	13	then	then	ADV
iajs-3045	239	14	either	either	CCONJ
iajs-3045	239	15	𝑎𝑏𝑡	𝑎𝑏𝑡	ADV
iajs-3045	239	16	∈	∈	PROPN
iajs-3045	240	1	[	[	X
iajs-3045	240	2	ƒ	ƒ	X
iajs-3045	240	3	:	:	PUNCT
iajs-3045	240	4	ʀ	ʀ	NOUN
iajs-3045	240	5	ѡ	ѡ	NOUN
iajs-3045	240	6	]	]	X
iajs-3045	240	7	+	+	CCONJ
iajs-3045	240	8	(	(	PUNCT
iajs-3045	240	9	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	PROPN
iajs-3045	240	10	)	)	PUNCT
iajs-3045	240	11	+	+	NUM
iajs-3045	240	12	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	240	13	)	)	PUNCT
iajs-3045	240	14	)	)	PUNCT
iajs-3045	240	15	or	or	CCONJ
iajs-3045	240	16	𝑏𝑐𝑡	𝑏𝑐𝑡	VERB
iajs-3045	240	17	∈	∈	NOUN
iajs-3045	241	1	[	[	X
iajs-3045	241	2	ƒ	ƒ	X
iajs-3045	241	3	:	:	PUNCT
iajs-3045	241	4	ʀ	ʀ	NOUN
iajs-3045	241	5	ѡ	ѡ	NOUN
iajs-3045	241	6	]	]	X
iajs-3045	241	7	+	+	CCONJ
iajs-3045	241	8	(	(	PUNCT
iajs-3045	241	9	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	PROPN
iajs-3045	241	10	)	)	PUNCT
iajs-3045	241	11	+	+	NUM
iajs-3045	241	12	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	241	13	)	)	PUNCT
iajs-3045	241	14	)	)	PUNCT
iajs-3045	241	15	or	or	CCONJ
iajs-3045	241	16	𝑎𝑐𝑡	𝑎𝑐𝑡	NOUN
iajs-3045	241	17	∈	∈	NOUN
iajs-3045	242	1	[	[	X
iajs-3045	242	2	ƒ	ƒ	X
iajs-3045	242	3	:	:	PUNCT
iajs-3045	242	4	ʀ	ʀ	NOUN
iajs-3045	242	5	ѡ	ѡ	NOUN
iajs-3045	242	6	]	]	X
iajs-3045	242	7	+	+	CCONJ
iajs-3045	242	8	(	(	PUNCT
iajs-3045	242	9	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	PROPN
iajs-3045	242	10	)	)	PUNCT
iajs-3045	242	11	+	+	NUM
iajs-3045	242	12	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	242	13	)	)	PUNCT
iajs-3045	242	14	)	)	PUNCT
iajs-3045	242	15	.	.	PUNCT
iajs-3045	243	1	hence	hence	ADV
iajs-3045	243	2	by	by	ADP
iajs-3045	243	3	proposition	proposition	NOUN
iajs-3045	243	4	2.24	2.24	NUM
iajs-3045	243	5	[	[	X
iajs-3045	243	6	ƒ	ƒ	X
iajs-3045	243	7	:	:	PUNCT
iajs-3045	243	8	ʀ	ʀ	PART
iajs-3045	243	9	ѡ	ѡ	X
iajs-3045	243	10	]	]	PUNCT
iajs-3045	243	11	is	be	AUX
iajs-3045	243	12	exnpq2ab	exnpq2ab	PROPN
iajs-3045	243	13	ideal	ideal	NOUN
iajs-3045	243	14	of	of	ADP
iajs-3045	243	15	ʀ	ʀ	PRON
iajs-3045	243	16	.	.	PUNCT
iajs-3045	243	17	(	(	PUNCT
iajs-3045	243	18	⟸	⟸	ADJ
iajs-3045	243	19	)	)	PUNCT
iajs-3045	243	20	𝑆uppose	𝑆uppose	NOUN
iajs-3045	243	21	that	that	SCONJ
iajs-3045	243	22	[	[	X
iajs-3045	243	23	ƒ	ƒ	X
iajs-3045	243	24	:	:	PUNCT
iajs-3045	243	25	ʀ	ʀ	PART
iajs-3045	243	26	ѡ	ѡ	X
iajs-3045	243	27	]	]	PUNCT
iajs-3045	243	28	is	be	AUX
iajs-3045	243	29	exnpq2ab	exnpq2ab	PROPN
iajs-3045	243	30	ideal	ideal	NOUN
iajs-3045	243	31	of	of	ADP
iajs-3045	243	32	ʀ	ʀ	NOUN
iajs-3045	243	33	,	,	PUNCT
iajs-3045	243	34	and	and	CCONJ
iajs-3045	243	35	𝑎𝑏𝑐𝑥	𝑎𝑏𝑐𝑥	NOUN
iajs-3045	243	36	∈	∈	PROPN
iajs-3045	243	37	ƒ	ƒ	X
iajs-3045	243	38	for	for	ADP
iajs-3045	243	39	𝑎	𝑎	NOUN
iajs-3045	243	40	,	,	PUNCT
iajs-3045	243	41	𝑏	𝑏	NOUN
iajs-3045	243	42	,	,	PUNCT
iajs-3045	243	43	𝑐	𝑐	PROPN
iajs-3045	243	44	∈	∈	PROPN
iajs-3045	243	45	ʀ	ʀ	NOUN
iajs-3045	243	46	,	,	PUNCT
iajs-3045	243	47	𝑥	𝑥	DET
iajs-3045	243	48	∈	∈	PROPN
iajs-3045	243	49	ѡ	ѡ	NOUN
iajs-3045	243	50	,	,	PUNCT
iajs-3045	243	51	hence	hence	ADV
iajs-3045	243	52	𝑎𝑏𝑐(𝑥	𝑎𝑏𝑐(𝑥	PROPN
iajs-3045	243	53	)	)	PUNCT
iajs-3045	243	54	⊆	⊆	NUM
iajs-3045	243	55	ƒ	ƒ	NOUN
iajs-3045	243	56	.	.	PUNCT
iajs-3045	244	1	since	since	SCONJ
iajs-3045	244	2	ѡ	ѡ	PROPN
iajs-3045	244	3	is	be	AUX
iajs-3045	244	4	a	a	DET
iajs-3045	244	5	multiplication	multiplication	NOUN
iajs-3045	244	6	,	,	PUNCT
iajs-3045	244	7	then	then	ADV
iajs-3045	244	8	(	(	PUNCT
iajs-3045	244	9	𝑥	𝑥	NOUN
iajs-3045	244	10	)	)	PUNCT
iajs-3045	244	11	=	=	PUNCT
iajs-3045	245	1	𝐽ѡ	𝐽ѡ	PROPN
iajs-3045	245	2	for	for	ADP
iajs-3045	245	3	some	some	DET
iajs-3045	245	4	ideal	ideal	ADJ
iajs-3045	245	5	𝐽	𝐽	PROPN
iajs-3045	245	6	of	of	ADP
iajs-3045	245	7	ʀ	ʀ	NOUN
iajs-3045	245	8	,	,	PUNCT
iajs-3045	245	9	that	that	PRON
iajs-3045	245	10	is	be	AUX
iajs-3045	245	11	𝑎𝑏𝑐𝐽ѡ	𝑎𝑏𝑐𝐽ѡ	NOUN
iajs-3045	245	12	⊆	⊆	NUM
iajs-3045	245	13	ƒ	ƒ	NUM
iajs-3045	245	14	,	,	PUNCT
iajs-3045	245	15	implies	imply	VERB
iajs-3045	245	16	that	that	SCONJ
iajs-3045	245	17	𝑎𝑏𝑐𝐽	𝑎𝑏𝑐𝐽	VERB
iajs-3045	245	18	⊆	⊆	NUM
iajs-3045	245	19	[	[	X
iajs-3045	245	20	ƒ	ƒ	NUM
iajs-3045	245	21	:	:	PUNCT
iajs-3045	245	22	ʀ	ʀ	PART
iajs-3045	245	23	ѡ	ѡ	NOUN
iajs-3045	245	24	]	]	PUNCT
iajs-3045	245	25	,	,	PUNCT
iajs-3045	245	26	but	but	CCONJ
iajs-3045	246	1	[	[	X
iajs-3045	246	2	ƒ	ƒ	X
iajs-3045	246	3	:	:	PUNCT
iajs-3045	246	4	ʀ	ʀ	PART
iajs-3045	246	5	ѡ	ѡ	X
iajs-3045	246	6	]	]	PUNCT
iajs-3045	246	7	is	be	AUX
iajs-3045	246	8	exnpq2ab	exnpq2ab	PROPN
iajs-3045	246	9	ideal	ideal	NOUN
iajs-3045	246	10	of	of	ADP
iajs-3045	246	11	ʀ	ʀ	NOUN
iajs-3045	246	12	,	,	PUNCT
iajs-3045	246	13	then	then	ADV
iajs-3045	246	14	by	by	ADP
iajs-3045	246	15	definition	definition	NOUN
iajs-3045	246	16	either	either	CCONJ
iajs-3045	246	17	𝑎𝑏𝐽	𝑎𝑏𝐽	NUM
iajs-3045	246	18	⊆	⊆	NUM
iajs-3045	246	19	[	[	X
iajs-3045	246	20	ƒ	ƒ	X
iajs-3045	246	21	:	:	PUNCT
iajs-3045	246	22	ʀ	ʀ	PART
iajs-3045	246	23	ѡ	ѡ	NOUN
iajs-3045	246	24	]	]	PUNCT
iajs-3045	246	25	+	+	X
iajs-3045	246	26	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	246	27	)	)	PUNCT
iajs-3045	246	28	+	+	NUM
iajs-3045	246	29	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	246	30	)	)	PUNCT
iajs-3045	246	31	or	or	CCONJ
iajs-3045	246	32	𝑎𝑐𝐽	𝑎𝑐𝐽	PRON
iajs-3045	246	33	⊆	⊆	NUM
iajs-3045	246	34	[	[	X
iajs-3045	246	35	ƒ	ƒ	X
iajs-3045	246	36	:	:	PUNCT
iajs-3045	246	37	ʀ	ʀ	PART
iajs-3045	246	38	ѡ	ѡ	NOUN
iajs-3045	246	39	]	]	PUNCT
iajs-3045	246	40	+	+	X
iajs-3045	246	41	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	246	42	)	)	PUNCT
iajs-3045	246	43	+	+	NUM
iajs-3045	246	44	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	246	45	)	)	PUNCT
iajs-3045	246	46	or	or	CCONJ
iajs-3045	246	47	𝑏𝑐𝐽	𝑏𝑐𝐽	VERB
iajs-3045	246	48	⊆	⊆	NUM
iajs-3045	246	49	[	[	X
iajs-3045	246	50	ƒ	ƒ	X
iajs-3045	246	51	:	:	PUNCT
iajs-3045	246	52	ʀ	ʀ	PART
iajs-3045	246	53	ѡ	ѡ	NOUN
iajs-3045	246	54	]	]	PUNCT
iajs-3045	246	55	+	+	X
iajs-3045	246	56	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	246	57	)	)	PUNCT
iajs-3045	246	58	+	+	NUM
iajs-3045	246	59	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	246	60	)	)	PUNCT
iajs-3045	246	61	.	.	PUNCT
iajs-3045	247	1	thus	thus	ADV
iajs-3045	247	2	either	either	CCONJ
iajs-3045	247	3	𝑎𝑏𝐽ѡ	𝑎𝑏𝐽ѡ	PROPN
iajs-3045	247	4	⊆	⊆	NUM
iajs-3045	247	5	[	[	X
iajs-3045	247	6	ƒ	ƒ	X
iajs-3045	247	7	:	:	PUNCT
iajs-3045	247	8	ʀ	ʀ	ADJ
iajs-3045	247	9	ѡ]ѡ	ѡ]ѡ	NOUN
iajs-3045	247	10	+	+	CCONJ
iajs-3045	247	11	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	247	12	)	)	PUNCT
iajs-3045	247	13	ѡ	ѡ	NOUN
iajs-3045	247	14	+	+	CCONJ
iajs-3045	247	15	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	247	16	)	)	PUNCT
iajs-3045	247	17	ѡ	ѡ	PRON
iajs-3045	247	18	or	or	CCONJ
iajs-3045	247	19	𝑎𝑐𝐽ѡ	𝑎𝑐𝐽ѡ	VERB
iajs-3045	247	20	⊆	⊆	NUM
iajs-3045	247	21	[	[	X
iajs-3045	247	22	ƒ	ƒ	X
iajs-3045	247	23	:	:	PUNCT
iajs-3045	247	24	ʀ	ʀ	PART
iajs-3045	247	25	ѡ	ѡ	ADP
iajs-3045	247	26	]	]	PUNCT
iajs-3045	247	27	ѡ	ѡ	PROPN
iajs-3045	247	28	+	+	SYM
iajs-3045	247	29	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	PROPN
iajs-3045	247	30	+	+	CCONJ
iajs-3045	247	31	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	ADJ
iajs-3045	247	32	or	or	CCONJ
iajs-3045	247	33	𝑏𝑐𝐽ѡ	𝑏𝑐𝐽ѡ	VERB
iajs-3045	247	34	⊆	⊆	NUM
iajs-3045	247	35	[	[	X
iajs-3045	247	36	ƒ	ƒ	X
iajs-3045	247	37	:	:	PUNCT
iajs-3045	247	38	ʀ	ʀ	ADJ
iajs-3045	247	39	ѡ]ѡ	ѡ]ѡ	NOUN
iajs-3045	247	40	+	+	CCONJ
iajs-3045	247	41	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	ADJ
iajs-3045	247	42	+	+	CCONJ
iajs-3045	247	43	𝐽(ʀ)ѡ.	𝐽(ʀ)ѡ.	NOUN
iajs-3045	247	44	hence	hence	ADV
iajs-3045	247	45	by	by	ADP
iajs-3045	247	46	lemma	lemma	PROPN
iajs-3045	247	47	2.17	2.17	NUM
iajs-3045	247	48	and	and	CCONJ
iajs-3045	247	49	remark	remark	VERB
iajs-3045	247	50	2.11	2.11	NUM
iajs-3045	247	51	either	either	CCONJ
iajs-3045	247	52	𝑎𝑏(𝑥	𝑎𝑏(𝑥	NOUN
iajs-3045	247	53	)	)	PUNCT
iajs-3045	247	54	⊆	⊆	NUM
iajs-3045	247	55	ƒ	ƒ	NOUN
iajs-3045	247	56	+	+	NUM
iajs-3045	247	57	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	247	58	)	)	PUNCT
iajs-3045	247	59	+	+	PUNCT
iajs-3045	247	60	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	247	61	)	)	PUNCT
iajs-3045	247	62	or	or	CCONJ
iajs-3045	247	63	𝑎𝑐(𝑥	𝑎𝑐(𝑥	NUM
iajs-3045	247	64	)	)	PUNCT
iajs-3045	247	65	⊆	⊆	NUM
iajs-3045	247	66	ƒ	ƒ	NOUN
iajs-3045	247	67	+	+	NUM
iajs-3045	247	68	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	247	69	)	)	PUNCT
iajs-3045	247	70	+	+	PUNCT
iajs-3045	247	71	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	247	72	)	)	PUNCT
iajs-3045	247	73	or	or	CCONJ
iajs-3045	247	74	𝑏𝑐(𝑥	𝑏𝑐(𝑥	ADJ
iajs-3045	247	75	)	)	PUNCT
iajs-3045	247	76	⊆	⊆	NUM
iajs-3045	247	77	ƒ	ƒ	NOUN
iajs-3045	247	78	+	+	NUM
iajs-3045	247	79	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	247	80	)	)	PUNCT
iajs-3045	247	81	+	+	PUNCT
iajs-3045	247	82	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	247	83	)	)	PUNCT
iajs-3045	247	84	.	.	PUNCT
iajs-3045	248	1	next	next	ADV
iajs-3045	248	2	,	,	PUNCT
iajs-3045	248	3	follows	follow	VERB
iajs-3045	248	4	either	either	DET
iajs-3045	248	5	𝑎𝑏𝑥	𝑎𝑏𝑥	NOUN
iajs-3045	248	6	∈	∈	NOUN
iajs-3045	248	7	ƒ	ƒ	X
iajs-3045	248	8	+	+	NUM
iajs-3045	248	9	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	248	10	)	)	PUNCT
iajs-3045	249	1	+	+	PUNCT
iajs-3045	250	1	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	250	2	)	)	PUNCT
iajs-3045	250	3	or	or	CCONJ
iajs-3045	250	4	𝑎𝑐𝑥	𝑎𝑐𝑥	VERB
iajs-3045	250	5	∈	∈	PROPN
iajs-3045	251	1	ƒ	ƒ	X
iajs-3045	251	2	+	+	NUM
iajs-3045	251	3	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	251	4	)	)	PUNCT
iajs-3045	252	1	+	+	PUNCT
iajs-3045	253	1	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	253	2	)	)	PUNCT
iajs-3045	253	3	or	or	CCONJ
iajs-3045	253	4	𝑏𝑐𝑥	𝑏𝑐𝑥	VERB
iajs-3045	253	5	∈	∈	PROPN
iajs-3045	253	6	ƒ	ƒ	X
iajs-3045	253	7	+	+	NUM
iajs-3045	253	8	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	253	9	)	)	PUNCT
iajs-3045	253	10	+	+	CCONJ
iajs-3045	253	11	𝐽(ѡ).therefore	𝐽(ѡ).therefore	VERB
iajs-3045	253	12	ƒ	ƒ	NOUN
iajs-3045	253	13	is	be	AUX
iajs-3045	253	14	exnpq2ab	exnpq2ab	PROPN
iajs-3045	253	15	submodule	submodule	NOUN
iajs-3045	253	16	of	of	ADP
iajs-3045	253	17	ѡ.	ѡ.	NOUN
iajs-3045	253	18	as	as	ADP
iajs-3045	253	19	a	a	DET
iajs-3045	253	20	direct	direct	ADJ
iajs-3045	253	21	application	application	NOUN
iajs-3045	253	22	of	of	ADP
iajs-3045	253	23	proposition	proposition	NOUN
iajs-3045	253	24	3.7	3.7	NUM
iajs-3045	253	25	,	,	PUNCT
iajs-3045	253	26	we	we	PRON
iajs-3045	253	27	get	get	VERB
iajs-3045	253	28	the	the	DET
iajs-3045	253	29	following	follow	VERB
iajs-3045	253	30	corollary	corollary	ADJ
iajs-3045	253	31	:	:	PUNCT
iajs-3045	253	32	corollary	corollary	ADJ
iajs-3045	253	33	3.8	3.8	NUM
iajs-3045	253	34	let	let	VERB
iajs-3045	253	35	ƒ	ƒ	PRON
iajs-3045	253	36	≠	≠	PROPN
iajs-3045	253	37	ѡ	ѡ	NOUN
iajs-3045	253	38	and	and	CCONJ
iajs-3045	253	39	ѡ	ѡ	PROPN
iajs-3045	253	40	is	be	AUX
iajs-3045	253	41	non	non	ADJ
iajs-3045	253	42	-	-	ADJ
iajs-3045	253	43	singular	singular	ADJ
iajs-3045	253	44	multiplication	multiplication	NOUN
iajs-3045	253	45	ʀ	ʀ	NOUN
iajs-3045	253	46	-	-	PUNCT
iajs-3045	253	47	module	module	NOUN
iajs-3045	253	48	ѡ	ѡ	NOUN
iajs-3045	253	49	over	over	ADP
iajs-3045	253	50	artinian	artinian	ADJ
iajs-3045	253	51	𝑟𝑖𝑛𝑔	𝑟𝑖𝑛𝑔	PROPN
iajs-3045	253	52	ʀ	ʀ	PROPN
iajs-3045	253	53	.	.	PROPN
iajs-3045	254	1	then	then	ADV
iajs-3045	254	2	ƒ	ƒ	PROPN
iajs-3045	254	3	is	be	AUX
iajs-3045	254	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	254	5	submodule	submodule	NOUN
iajs-3045	254	6	of	of	ADP
iajs-3045	254	7	ѡ	ѡ	PROPN
iajs-3045	254	8	if	if	SCONJ
iajs-3045	255	1	and	and	CCONJ
iajs-3045	255	2	only	only	ADV
iajs-3045	255	3	if	if	SCONJ
iajs-3045	255	4	[	[	X
iajs-3045	255	5	ƒ	ƒ	X
iajs-3045	255	6	:	:	PUNCT
iajs-3045	255	7	ʀ	ʀ	PART
iajs-3045	255	8	ѡ	ѡ	X
iajs-3045	255	9	]	]	PUNCT
iajs-3045	255	10	is	be	AUX
iajs-3045	255	11	exnpq2ab	exnpq2ab	PROPN
iajs-3045	255	12	ideal	ideal	NOUN
iajs-3045	255	13	of	of	ADP
iajs-3045	255	14	ʀ	ʀ	NOUN
iajs-3045	255	15	.	.	PUNCT
iajs-3045	255	16	by	by	ADP
iajs-3045	255	17	proof	proof	NOUN
iajs-3045	255	18	of	of	ADP
iajs-3045	255	19	proposition	proposition	NOUN
iajs-3045	255	20	3.7	3.7	NUM
iajs-3045	255	21	and	and	CCONJ
iajs-3045	255	22	using	use	VERB
iajs-3045	255	23	lemma	lemma	PROPN
iajs-3045	255	24	2.15	2.15	NUM
iajs-3045	255	25	we	we	PRON
iajs-3045	255	26	get	get	VERB
iajs-3045	255	27	:	:	PUNCT
iajs-3045	255	28	proposition	proposition	NOUN
iajs-3045	255	29	3.9	3.9	NUM
iajs-3045	255	30	let	let	VERB
iajs-3045	255	31	ƒ	ƒ	PRON
iajs-3045	255	32	≠	≠	PROPN
iajs-3045	255	33	ѡ	ѡ	NOUN
iajs-3045	255	34	and	and	CCONJ
iajs-3045	255	35	ѡ	ѡ	PROPN
iajs-3045	255	36	is	be	AUX
iajs-3045	255	37	non	non	ADJ
iajs-3045	255	38	-	-	ADJ
iajs-3045	255	39	singular	singular	ADJ
iajs-3045	255	40	multiplication	multiplication	NOUN
iajs-3045	255	41	ʀ	ʀ	NOUN
iajs-3045	255	42	-	-	PUNCT
iajs-3045	255	43	module	module	NOUN
iajs-3045	255	44	ѡ	ѡ	NOUN
iajs-3045	255	45	over	over	ADP
iajs-3045	255	46	local	local	ADJ
iajs-3045	255	47	ring	ring	NOUN
iajs-3045	255	48	ʀ	ʀ	NOUN
iajs-3045	255	49	.	.	PUNCT
iajs-3045	256	1	then	then	ADV
iajs-3045	256	2	ƒ	ƒ	PRON
iajs-3045	256	3	is	be	AUX
iajs-3045	256	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	256	5	submodule	submodule	NOUN
iajs-3045	256	6	of	of	ADP
iajs-3045	256	7	ѡ	ѡ	PROPN
iajs-3045	256	8	if	if	SCONJ
iajs-3045	257	1	and	and	CCONJ
iajs-3045	257	2	only	only	ADV
iajs-3045	257	3	if	if	SCONJ
iajs-3045	257	4	[	[	X
iajs-3045	257	5	ƒ	ƒ	X
iajs-3045	257	6	:	:	PUNCT
iajs-3045	257	7	ʀ	ʀ	PART
iajs-3045	257	8	ѡ	ѡ	X
iajs-3045	257	9	]	]	PUNCT
iajs-3045	257	10	is	be	AUX
iajs-3045	257	11	exnpq2ab	exnpq2ab	PROPN
iajs-3045	257	12	ideal	ideal	NOUN
iajs-3045	257	13	of	of	ADP
iajs-3045	257	14	ʀ	ʀ	NOUN
iajs-3045	257	15	.	.	PUNCT
iajs-3045	257	16	ihjpas	ihjpas	PROPN
iajs-3045	257	17	.	.	PUNCT
iajs-3045	258	1	36(2)2023	36(2)2023	NUM
iajs-3045	258	2	414	414	NUM
iajs-3045	258	3	proposition	proposition	NOUN
iajs-3045	258	4	3.10	3.10	NUM
iajs-3045	258	5	let	let	VERB
iajs-3045	258	6	ƒ	ƒ	PRON
iajs-3045	258	7	≠	≠	PROPN
iajs-3045	258	8	ѡ	ѡ	NOUN
iajs-3045	258	9	and	and	CCONJ
iajs-3045	258	10	ѡ	ѡ	PROPN
iajs-3045	258	11	is	be	AUX
iajs-3045	258	12	𝑍-regular	𝑍-regular	ADJ
iajs-3045	258	13	multiplication	multiplication	NOUN
iajs-3045	258	14	ʀ	ʀ	NOUN
iajs-3045	258	15	-	-	PUNCT
iajs-3045	258	16	module	module	NOUN
iajs-3045	258	17	ѡ	ѡ	NOUN
iajs-3045	258	18	over	over	ADP
iajs-3045	258	19	an	an	DET
iajs-3045	258	20	a	a	DET
iajs-3045	258	21	good	good	ADJ
iajs-3045	258	22	ring	ring	NOUN
iajs-3045	258	23	ʀ	ʀ	NOUN
iajs-3045	258	24	.	.	PUNCT
iajs-3045	259	1	then	then	ADV
iajs-3045	259	2	ƒ	ƒ	PRON
iajs-3045	259	3	is	be	AUX
iajs-3045	259	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	259	5	submodule	submodule	NOUN
iajs-3045	259	6	of	of	ADP
iajs-3045	259	7	ѡ	ѡ	PROPN
iajs-3045	259	8	if	if	SCONJ
iajs-3045	260	1	and	and	CCONJ
iajs-3045	260	2	only	only	ADV
iajs-3045	260	3	if	if	SCONJ
iajs-3045	260	4	[	[	X
iajs-3045	260	5	ƒ	ƒ	X
iajs-3045	260	6	:	:	PUNCT
iajs-3045	260	7	ʀ	ʀ	PART
iajs-3045	260	8	ѡ	ѡ	X
iajs-3045	260	9	]	]	PUNCT
iajs-3045	260	10	is	be	AUX
iajs-3045	260	11	exnpq2ab	exnpq2ab	PROPN
iajs-3045	260	12	ideal	ideal	NOUN
iajs-3045	260	13	of	of	ADP
iajs-3045	260	14	ʀ	ʀ	NOUN
iajs-3045	260	15	.	.	PUNCT
iajs-3045	261	1	ƥ	ƥ	DET
iajs-3045	261	2	proof	proof	NOUN
iajs-3045	261	3	.	.	PUNCT
iajs-3045	262	1	(	(	PUNCT
iajs-3045	262	2	⟹	⟹	X
iajs-3045	262	3	)	)	PUNCT
iajs-3045	262	4	let	let	VERB
iajs-3045	262	5	𝑟𝑠𝑡ƥ	𝑟𝑠𝑡ƥ	ADJ
iajs-3045	262	6	⊆	⊆	NUM
iajs-3045	262	7	[	[	X
iajs-3045	262	8	ƒ	ƒ	NUM
iajs-3045	262	9	:	:	PUNCT
iajs-3045	262	10	ʀ	ʀ	PART
iajs-3045	262	11	ѡ	ѡ	NOUN
iajs-3045	262	12	]	]	PUNCT
iajs-3045	262	13	for	for	ADP
iajs-3045	262	14	𝑟	𝑟	NOUN
iajs-3045	262	15	,	,	PUNCT
iajs-3045	262	16	𝑠	𝑠	PROPN
iajs-3045	262	17	,	,	PUNCT
iajs-3045	262	18	𝑡	𝑡	PROPN
iajs-3045	262	19	∈	∈	PROPN
iajs-3045	262	20	ʀ	ʀ	NOUN
iajs-3045	262	21	and	and	CCONJ
iajs-3045	262	22	ƥ	ƥ	PRON
iajs-3045	262	23	is	be	AUX
iajs-3045	262	24	an	an	DET
iajs-3045	262	25	ideal	ideal	NOUN
iajs-3045	262	26	of	of	ADP
iajs-3045	262	27	ʀ	ʀ	NOUN
iajs-3045	262	28	,	,	PUNCT
iajs-3045	262	29	then	then	ADV
iajs-3045	262	30	𝑟𝑠𝑡ƥѡ	𝑟𝑠𝑡ƥѡ	NOUN
iajs-3045	262	31	⊆	⊆	NUM
iajs-3045	262	32	ƒ	ƒ	NOUN
iajs-3045	262	33	.	.	PUNCT
iajs-3045	263	1	but	but	CCONJ
iajs-3045	263	2	ƒ	ƒ	PRON
iajs-3045	263	3	is	be	AUX
iajs-3045	263	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	263	5	submodule	submodule	NOUN
iajs-3045	263	6	of	of	ADP
iajs-3045	263	7	ѡ	ѡ	PROPN
iajs-3045	263	8	,	,	PUNCT
iajs-3045	263	9	then	then	ADV
iajs-3045	263	10	either	either	CCONJ
iajs-3045	263	11	𝑟𝑠ƥѡ	𝑟𝑠ƥѡ	NUM
iajs-3045	263	12	⊆	⊆	NUM
iajs-3045	263	13	ƒ	ƒ	PRON
iajs-3045	263	14	+	+	PUNCT
iajs-3045	263	15	(	(	PUNCT
iajs-3045	263	16	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	263	17	)	)	PUNCT
iajs-3045	263	18	+	+	PUNCT
iajs-3045	264	1	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	264	2	)	)	PUNCT
iajs-3045	264	3	)	)	PUNCT
iajs-3045	265	1	or	or	CCONJ
iajs-3045	265	2	𝑟𝑡ƥѡ	𝑟𝑡ƥѡ	VERB
iajs-3045	265	3	⊆	⊆	NUM
iajs-3045	265	4	ƒ	ƒ	X
iajs-3045	265	5	+	+	X
iajs-3045	265	6	(	(	PUNCT
iajs-3045	265	7	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	265	8	)	)	PUNCT
iajs-3045	265	9	+	+	PUNCT
iajs-3045	266	1	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	266	2	)	)	PUNCT
iajs-3045	266	3	)	)	PUNCT
iajs-3045	267	1	or	or	CCONJ
iajs-3045	267	2	𝑠𝑡ƥѡ	𝑠𝑡ƥѡ	VERB
iajs-3045	267	3	⊆	⊆	NUM
iajs-3045	267	4	ƒ	ƒ	X
iajs-3045	267	5	+	+	X
iajs-3045	267	6	(	(	PUNCT
iajs-3045	267	7	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	267	8	)	)	PUNCT
iajs-3045	267	9	+	+	PUNCT
iajs-3045	267	10	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	267	11	)	)	PUNCT
iajs-3045	267	12	)	)	PUNCT
iajs-3045	267	13	.	.	PUNCT
iajs-3045	268	1	since	since	SCONJ
iajs-3045	268	2	ѡ	ѡ	PROPN
iajs-3045	268	3	is	be	AUX
iajs-3045	268	4	multiplication	multiplication	NOUN
iajs-3045	268	5	,	,	PUNCT
iajs-3045	268	6	then	then	ADV
iajs-3045	268	7	ƒ	ƒ	X
iajs-3045	268	8	=	=	PUNCT
iajs-3045	269	1	[	[	X
iajs-3045	269	2	ƒ	ƒ	X
iajs-3045	269	3	:	:	PUNCT
iajs-3045	269	4	ʀ	ʀ	NOUN
iajs-3045	269	5	ѡ]ѡ	ѡ]ѡ	NOUN
iajs-3045	269	6	and	and	CCONJ
iajs-3045	269	7	since	since	SCONJ
iajs-3045	269	8	ѡ	ѡ	PROPN
iajs-3045	269	9	is	be	AUX
iajs-3045	269	10	a	a	DET
iajs-3045	269	11	𝑍-regular	𝑍-regular	PROPN
iajs-3045	269	12	,	,	PUNCT
iajs-3045	269	13	then	then	ADV
iajs-3045	269	14	by	by	ADP
iajs-3045	269	15	lemma	lemma	PROPN
iajs-3045	269	16	2.21	2.21	NUM
iajs-3045	269	17	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	269	18	)	)	PUNCT
iajs-3045	269	19	=	=	SYM
iajs-3045	270	1	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	270	2	and	and	CCONJ
iajs-3045	270	3	ʀ	ʀ	NOUN
iajs-3045	270	4	is	be	AUX
iajs-3045	270	5	a	a	DET
iajs-3045	270	6	good	good	ADJ
iajs-3045	270	7	ring	ring	NOUN
iajs-3045	270	8	then	then	ADV
iajs-3045	270	9	by	by	ADP
iajs-3045	270	10	remark	remark	NOUN
iajs-3045	270	11	1.11	1.11	NUM
iajs-3045	270	12	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	270	13	)	)	PUNCT
iajs-3045	270	14	=	=	PUNCT
iajs-3045	270	15	𝐽(ʀ)ѡ.	𝐽(ʀ)ѡ.	PROPN
iajs-3045	270	16	thus	thus	ADV
iajs-3045	271	1	either	either	CCONJ
iajs-3045	271	2	𝑟𝑠ƥѡ	𝑟𝑠ƥѡ	ADJ
iajs-3045	271	3	⊆	⊆	NUM
iajs-3045	271	4	[	[	X
iajs-3045	271	5	ƒ	ƒ	X
iajs-3045	271	6	:	:	PUNCT
iajs-3045	271	7	ʀ	ʀ	ADJ
iajs-3045	271	8	ѡ]ѡ	ѡ]ѡ	X
iajs-3045	271	9	+	+	CCONJ
iajs-3045	271	10	(	(	PUNCT
iajs-3045	271	11	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	271	12	+	+	CCONJ
iajs-3045	271	13	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	NOUN
iajs-3045	271	14	)	)	PUNCT
iajs-3045	271	15	or	or	CCONJ
iajs-3045	271	16	𝑟𝑡ƥѡ	𝑟𝑡ƥѡ	VERB
iajs-3045	271	17	⊆	⊆	NUM
iajs-3045	271	18	[	[	X
iajs-3045	271	19	ƒ	ƒ	X
iajs-3045	271	20	:	:	PUNCT
iajs-3045	271	21	ʀ	ʀ	ADJ
iajs-3045	271	22	ѡ]ѡ	ѡ]ѡ	X
iajs-3045	271	23	+	+	CCONJ
iajs-3045	271	24	(	(	PUNCT
iajs-3045	271	25	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	271	26	+	+	CCONJ
iajs-3045	271	27	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	NOUN
iajs-3045	271	28	)	)	PUNCT
iajs-3045	271	29	or	or	CCONJ
iajs-3045	271	30	𝑠𝑡ƥѡ	𝑠𝑡ƥѡ	VERB
iajs-3045	271	31	⊆	⊆	NUM
iajs-3045	271	32	[	[	X
iajs-3045	271	33	ƒ	ƒ	X
iajs-3045	271	34	:	:	PUNCT
iajs-3045	271	35	ʀ	ʀ	ADJ
iajs-3045	271	36	ѡ]ѡ	ѡ]ѡ	X
iajs-3045	271	37	+	+	CCONJ
iajs-3045	271	38	(	(	PUNCT
iajs-3045	271	39	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	271	40	+	+	CCONJ
iajs-3045	271	41	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	NOUN
iajs-3045	271	42	)	)	PUNCT
iajs-3045	271	43	,	,	PUNCT
iajs-3045	271	44	it	it	PRON
iajs-3045	271	45	follows	follow	VERB
iajs-3045	271	46	that	that	SCONJ
iajs-3045	271	47	either	either	CCONJ
iajs-3045	271	48	𝑟𝑠ƥ	𝑟𝑠ƥ	ADV
iajs-3045	271	49	⊆	⊆	NUM
iajs-3045	271	50	[	[	X
iajs-3045	271	51	ƒ	ƒ	NUM
iajs-3045	271	52	:	:	PUNCT
iajs-3045	271	53	ʀ	ʀ	NOUN
iajs-3045	271	54	ѡ	ѡ	NOUN
iajs-3045	271	55	]	]	X
iajs-3045	271	56	+	+	CCONJ
iajs-3045	271	57	(	(	PUNCT
iajs-3045	271	58	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	PROPN
iajs-3045	271	59	)	)	PUNCT
iajs-3045	271	60	+	+	NUM
iajs-3045	272	1	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	272	2	)	)	PUNCT
iajs-3045	272	3	)	)	PUNCT
iajs-3045	273	1	or	or	CCONJ
iajs-3045	273	2	𝑟𝑡ƥ	𝑟𝑡ƥ	VERB
iajs-3045	273	3	⊆	⊆	NUM
iajs-3045	273	4	[	[	X
iajs-3045	273	5	ƒ	ƒ	X
iajs-3045	273	6	:	:	PUNCT
iajs-3045	273	7	ʀ	ʀ	NOUN
iajs-3045	273	8	ѡ	ѡ	NOUN
iajs-3045	273	9	]	]	X
iajs-3045	273	10	+	+	CCONJ
iajs-3045	273	11	(	(	PUNCT
iajs-3045	273	12	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	PROPN
iajs-3045	273	13	)	)	PUNCT
iajs-3045	273	14	+	+	NUM
iajs-3045	273	15	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	273	16	)	)	PUNCT
iajs-3045	273	17	)	)	PUNCT
iajs-3045	273	18	or	or	CCONJ
iajs-3045	273	19	𝑠𝑡ƥ	𝑠𝑡ƥ	VERB
iajs-3045	273	20	⊆	⊆	NUM
iajs-3045	273	21	[	[	X
iajs-3045	273	22	ƒ	ƒ	X
iajs-3045	273	23	:	:	PUNCT
iajs-3045	273	24	ʀ	ʀ	NOUN
iajs-3045	273	25	ѡ	ѡ	NOUN
iajs-3045	273	26	]	]	X
iajs-3045	273	27	+	+	CCONJ
iajs-3045	273	28	(	(	PUNCT
iajs-3045	273	29	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	PROPN
iajs-3045	273	30	)	)	PUNCT
iajs-3045	273	31	+	+	NUM
iajs-3045	273	32	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	273	33	)	)	PUNCT
iajs-3045	273	34	)	)	PUNCT
iajs-3045	273	35	.	.	PUNCT
iajs-3045	274	1	hence	hence	ADV
iajs-3045	274	2	by	by	ADP
iajs-3045	274	3	proposition	proposition	NOUN
iajs-3045	274	4	2.24	2.24	NUM
iajs-3045	274	5	[	[	X
iajs-3045	274	6	ƒ	ƒ	X
iajs-3045	274	7	:	:	PUNCT
iajs-3045	274	8	ʀ	ʀ	PART
iajs-3045	274	9	ѡ	ѡ	X
iajs-3045	274	10	]	]	PUNCT
iajs-3045	274	11	is	be	AUX
iajs-3045	274	12	exnpq2ab	exnpq2ab	PROPN
iajs-3045	274	13	ideal	ideal	NOUN
iajs-3045	274	14	of	of	ADP
iajs-3045	274	15	ʀ	ʀ	PRON
iajs-3045	274	16	.	.	PUNCT
iajs-3045	274	17	(	(	PUNCT
iajs-3045	274	18	⟸	⟸	ADJ
iajs-3045	274	19	)	)	PUNCT
iajs-3045	274	20	𝑆uppose	𝑆uppose	NOUN
iajs-3045	274	21	that	that	SCONJ
iajs-3045	274	22	[	[	X
iajs-3045	274	23	ƒ	ƒ	X
iajs-3045	274	24	:	:	PUNCT
iajs-3045	274	25	ʀ	ʀ	PART
iajs-3045	274	26	ѡ	ѡ	X
iajs-3045	274	27	]	]	PUNCT
iajs-3045	274	28	is	be	AUX
iajs-3045	274	29	exnpq2ab	exnpq2ab	PROPN
iajs-3045	274	30	ideal	ideal	NOUN
iajs-3045	274	31	of	of	ADP
iajs-3045	274	32	ʀ	ʀ	NOUN
iajs-3045	274	33	,	,	PUNCT
iajs-3045	274	34	and	and	CCONJ
iajs-3045	274	35	𝑟𝑠𝑡𝐴	𝑟𝑠𝑡𝐴	VERB
iajs-3045	274	36	⊆	⊆	NUM
iajs-3045	274	37	ƒ	ƒ	NOUN
iajs-3045	274	38	for	for	ADP
iajs-3045	274	39	𝑟	𝑟	NOUN
iajs-3045	274	40	,	,	PUNCT
iajs-3045	274	41	𝑠	𝑠	PROPN
iajs-3045	274	42	,	,	PUNCT
iajs-3045	274	43	𝑡	𝑡	PROPN
iajs-3045	274	44	∈	∈	PROPN
iajs-3045	274	45	ѡ	ѡ	NOUN
iajs-3045	274	46	and	and	CCONJ
iajs-3045	274	47	𝐴	𝐴	PROPN
iajs-3045	274	48	is	be	AUX
iajs-3045	274	49	a	a	DET
iajs-3045	274	50	submodule	submodule	NOUN
iajs-3045	274	51	of	of	ADP
iajs-3045	274	52	ѡ.	ѡ.	NOUN
iajs-3045	274	53	since	since	SCONJ
iajs-3045	274	54	ѡ	ѡ	PROPN
iajs-3045	274	55	is	be	AUX
iajs-3045	274	56	a	a	DET
iajs-3045	274	57	multiplication	multiplication	NOUN
iajs-3045	274	58	,	,	PUNCT
iajs-3045	274	59	then	then	ADV
iajs-3045	274	60	𝐴	𝐴	PROPN
iajs-3045	274	61	=	=	SYM
iajs-3045	274	62	ƥѡ	ƥѡ	PROPN
iajs-3045	274	63	,	,	PUNCT
iajs-3045	274	64	that	that	PRON
iajs-3045	274	65	is	be	AUX
iajs-3045	274	66	𝑟𝑠𝑡𝐴	𝑟𝑠𝑡𝐴	ADJ
iajs-3045	274	67	=	=	SYM
iajs-3045	274	68	𝑟𝑠𝑡ƥѡ	𝑟𝑠𝑡ƥѡ	NOUN
iajs-3045	274	69	⊆	⊆	NUM
iajs-3045	274	70	ƒ	ƒ	NUM
iajs-3045	274	71	,	,	PUNCT
iajs-3045	274	72	implies	imply	VERB
iajs-3045	274	73	that	that	SCONJ
iajs-3045	274	74	𝑟𝑠𝑡ƥ	𝑟𝑠𝑡ƥ	ADJ
iajs-3045	274	75	⊆	⊆	NUM
iajs-3045	274	76	[	[	X
iajs-3045	274	77	ƒ	ƒ	X
iajs-3045	274	78	:	:	PUNCT
iajs-3045	274	79	ʀ	ʀ	PART
iajs-3045	274	80	ѡ	ѡ	NOUN
iajs-3045	274	81	]	]	PUNCT
iajs-3045	274	82	,	,	PUNCT
iajs-3045	274	83	but	but	CCONJ
iajs-3045	275	1	[	[	X
iajs-3045	275	2	ƒ	ƒ	X
iajs-3045	275	3	:	:	PUNCT
iajs-3045	275	4	ʀ	ʀ	PART
iajs-3045	275	5	ѡ	ѡ	X
iajs-3045	275	6	]	]	PUNCT
iajs-3045	275	7	is	be	AUX
iajs-3045	275	8	exnpq2ab	exnpq2ab	PROPN
iajs-3045	275	9	𝑖𝑑𝑒𝑎𝑙	𝑖𝑑𝑒𝑎𝑙	ADJ
iajs-3045	275	10	of	of	ADP
iajs-3045	275	11	ʀ	ʀ	NOUN
iajs-3045	275	12	,	,	PUNCT
iajs-3045	275	13	then	then	ADV
iajs-3045	275	14	by	by	ADP
iajs-3045	275	15	proposition	proposition	NOUN
iajs-3045	275	16	2.24	2.24	NUM
iajs-3045	275	17	either	either	CCONJ
iajs-3045	275	18	𝑟𝑠ƥ	𝑟𝑠ƥ	ADV
iajs-3045	275	19	⊆	⊆	NUM
iajs-3045	275	20	[	[	X
iajs-3045	275	21	ƒ	ƒ	NUM
iajs-3045	275	22	:	:	PUNCT
iajs-3045	275	23	ʀ	ʀ	PART
iajs-3045	275	24	ѡ	ѡ	NOUN
iajs-3045	275	25	]	]	PUNCT
iajs-3045	275	26	+	+	X
iajs-3045	275	27	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	275	28	)	)	PUNCT
iajs-3045	275	29	+	+	NUM
iajs-3045	275	30	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	275	31	)	)	PUNCT
iajs-3045	275	32	or	or	CCONJ
iajs-3045	275	33	𝑟𝑡ƥ	𝑟𝑡ƥ	VERB
iajs-3045	275	34	⊆	⊆	NUM
iajs-3045	275	35	[	[	X
iajs-3045	275	36	ƒ	ƒ	X
iajs-3045	275	37	:	:	PUNCT
iajs-3045	275	38	ʀ	ʀ	PART
iajs-3045	275	39	ѡ	ѡ	NOUN
iajs-3045	275	40	]	]	PUNCT
iajs-3045	275	41	+	+	X
iajs-3045	275	42	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	275	43	)	)	PUNCT
iajs-3045	275	44	+	+	NUM
iajs-3045	275	45	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	275	46	)	)	PUNCT
iajs-3045	275	47	or	or	CCONJ
iajs-3045	275	48	𝑠𝑡ƥ	𝑠𝑡ƥ	VERB
iajs-3045	275	49	⊆	⊆	NUM
iajs-3045	275	50	[	[	X
iajs-3045	275	51	ƒ	ƒ	X
iajs-3045	275	52	:	:	PUNCT
iajs-3045	275	53	ʀ	ʀ	PART
iajs-3045	275	54	ѡ	ѡ	NOUN
iajs-3045	275	55	]	]	PUNCT
iajs-3045	275	56	+	+	X
iajs-3045	275	57	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	275	58	)	)	PUNCT
iajs-3045	275	59	+	+	CCONJ
iajs-3045	275	60	𝐽(ʀ).thus	𝐽(ʀ).thus	PRON
iajs-3045	275	61	either	either	CCONJ
iajs-3045	275	62	𝑟𝑠ƥѡ	𝑟𝑠ƥѡ	NUM
iajs-3045	275	63	⊆	⊆	NUM
iajs-3045	275	64	[	[	X
iajs-3045	275	65	ƒ	ƒ	X
iajs-3045	275	66	:	:	PUNCT
iajs-3045	275	67	ʀ	ʀ	ADJ
iajs-3045	275	68	ѡ]ѡ	ѡ]ѡ	NOUN
iajs-3045	275	69	+	+	CCONJ
iajs-3045	275	70	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	275	71	)	)	PUNCT
iajs-3045	275	72	ѡ	ѡ	NOUN
iajs-3045	275	73	+	+	CCONJ
iajs-3045	275	74	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	275	75	)	)	PUNCT
iajs-3045	275	76	ѡ	ѡ	PROPN
iajs-3045	275	77	or	or	CCONJ
iajs-3045	275	78	𝑟𝑡ƥѡ	𝑟𝑡ƥѡ	VERB
iajs-3045	275	79	⊆	⊆	NUM
iajs-3045	275	80	[	[	X
iajs-3045	275	81	ƒ	ƒ	X
iajs-3045	275	82	:	:	PUNCT
iajs-3045	275	83	ʀ	ʀ	PART
iajs-3045	275	84	ѡ	ѡ	ADP
iajs-3045	275	85	]	]	PUNCT
iajs-3045	275	86	ѡ	ѡ	PROPN
iajs-3045	275	87	+	+	SYM
iajs-3045	275	88	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	PROPN
iajs-3045	275	89	+	+	CCONJ
iajs-3045	275	90	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	ADJ
iajs-3045	275	91	or	or	CCONJ
iajs-3045	275	92	𝑠𝑡ƥѡ	𝑠𝑡ƥѡ	VERB
iajs-3045	275	93	⊆	⊆	NUM
iajs-3045	275	94	[	[	X
iajs-3045	275	95	ƒ	ƒ	X
iajs-3045	275	96	:	:	PUNCT
iajs-3045	275	97	ʀ	ʀ	ADJ
iajs-3045	275	98	ѡ]ѡ	ѡ]ѡ	NOUN
iajs-3045	275	99	+	+	CCONJ
iajs-3045	275	100	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	ADJ
iajs-3045	275	101	+	+	CCONJ
iajs-3045	275	102	𝐽(ʀ)ѡ.	𝐽(ʀ)ѡ.	NOUN
iajs-3045	275	103	hence	hence	ADV
iajs-3045	275	104	by	by	ADP
iajs-3045	275	105	lemma	lemma	PROPN
iajs-3045	275	106	2.21	2.21	NUM
iajs-3045	275	107	and	and	CCONJ
iajs-3045	275	108	remark	remark	VERB
iajs-3045	275	109	2.11	2.11	NUM
iajs-3045	275	110	either	either	CCONJ
iajs-3045	275	111	𝑟𝑠𝐴	𝑟𝑠𝐴	NOUN
iajs-3045	275	112	⊆	⊆	NUM
iajs-3045	275	113	ƒ	ƒ	PROPN
iajs-3045	275	114	+	+	NUM
iajs-3045	275	115	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	275	116	)	)	PUNCT
iajs-3045	275	117	+	+	PUNCT
iajs-3045	275	118	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	275	119	)	)	PUNCT
iajs-3045	275	120	or	or	CCONJ
iajs-3045	275	121	𝑟𝑡𝐴	𝑟𝑡𝐴	NUM
iajs-3045	275	122	⊆	⊆	NUM
iajs-3045	275	123	ƒ	ƒ	NOUN
iajs-3045	275	124	+	+	NUM
iajs-3045	275	125	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	275	126	)	)	PUNCT
iajs-3045	275	127	+	+	PUNCT
iajs-3045	275	128	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	275	129	)	)	PUNCT
iajs-3045	275	130	or	or	CCONJ
iajs-3045	275	131	𝑠𝑡𝐴	𝑠𝑡𝐴	VERB
iajs-3045	275	132	⊆	⊆	NUM
iajs-3045	275	133	ƒ	ƒ	PROPN
iajs-3045	275	134	+	+	NUM
iajs-3045	275	135	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	275	136	)	)	PUNCT
iajs-3045	275	137	+	+	PUNCT
iajs-3045	275	138	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	275	139	)	)	PUNCT
iajs-3045	275	140	.	.	PUNCT
iajs-3045	276	1	therefore	therefore	ADV
iajs-3045	276	2	ƒ	ƒ	PROPN
iajs-3045	276	3	is	be	AUX
iajs-3045	276	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	276	5	submodule	submodule	NOUN
iajs-3045	276	6	of	of	ADP
iajs-3045	276	7	ѡ.	ѡ.	NOUN
iajs-3045	276	8	as	as	ADP
iajs-3045	276	9	a	a	DET
iajs-3045	276	10	direct	direct	ADJ
iajs-3045	276	11	application	application	NOUN
iajs-3045	276	12	of	of	ADP
iajs-3045	276	13	proposition	proposition	NOUN
iajs-3045	276	14	3.10	3.10	NUM
iajs-3045	276	15	,	,	PUNCT
iajs-3045	276	16	we	we	PRON
iajs-3045	276	17	get	get	VERB
iajs-3045	276	18	the	the	DET
iajs-3045	276	19	following	follow	VERB
iajs-3045	276	20	corollary	corollary	ADJ
iajs-3045	276	21	:	:	PUNCT
iajs-3045	276	22	corollary	corollary	ADJ
iajs-3045	276	23	3.11	3.11	NUM
iajs-3045	276	24	let	let	VERB
iajs-3045	276	25	ƒ	ƒ	PRON
iajs-3045	276	26	≠	≠	PROPN
iajs-3045	276	27	ѡ	ѡ	NOUN
iajs-3045	276	28	and	and	CCONJ
iajs-3045	276	29	ѡ	ѡ	PROPN
iajs-3045	276	30	is	be	AUX
iajs-3045	276	31	𝑍-regular	𝑍-regular	ADJ
iajs-3045	276	32	multiplication	multiplication	NOUN
iajs-3045	276	33	ʀ	ʀ	NOUN
iajs-3045	276	34	-	-	PUNCT
iajs-3045	276	35	module	module	NOUN
iajs-3045	276	36	ѡ	ѡ	NOUN
iajs-3045	276	37	over	over	ADP
iajs-3045	276	38	artinian	artinian	ADJ
iajs-3045	276	39	ring	ring	NOUN
iajs-3045	276	40	ʀ	ʀ	NOUN
iajs-3045	276	41	.	.	PUNCT
iajs-3045	277	1	then	then	ADV
iajs-3045	277	2	ƒ	ƒ	PRON
iajs-3045	277	3	is	be	AUX
iajs-3045	277	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	277	5	submodule	submodule	NOUN
iajs-3045	277	6	of	of	ADP
iajs-3045	277	7	ѡ	ѡ	PROPN
iajs-3045	277	8	if	if	SCONJ
iajs-3045	278	1	and	and	CCONJ
iajs-3045	278	2	only	only	ADV
iajs-3045	278	3	if	if	SCONJ
iajs-3045	278	4	[	[	X
iajs-3045	278	5	ƒ	ƒ	X
iajs-3045	278	6	:	:	PUNCT
iajs-3045	278	7	ʀ	ʀ	PART
iajs-3045	278	8	ѡ	ѡ	X
iajs-3045	278	9	]	]	PUNCT
iajs-3045	278	10	is	be	AUX
iajs-3045	278	11	exnpq2ab	exnpq2ab	PROPN
iajs-3045	278	12	ideal	ideal	NOUN
iajs-3045	278	13	of	of	ADP
iajs-3045	278	14	ʀ	ʀ	NOUN
iajs-3045	278	15	.	.	PUNCT
iajs-3045	278	16	by	by	ADP
iajs-3045	278	17	proof	proof	NOUN
iajs-3045	278	18	of	of	ADP
iajs-3045	278	19	proposition	proposition	NOUN
iajs-3045	278	20	3.10	3.10	NUM
iajs-3045	278	21	and	and	CCONJ
iajs-3045	278	22	using	use	VERB
iajs-3045	278	23	lemma	lemma	PROPN
iajs-3045	278	24	2.15	2.15	NUM
iajs-3045	278	25	we	we	PRON
iajs-3045	278	26	get	get	VERB
iajs-3045	278	27	:	:	PUNCT
iajs-3045	278	28	proposition	proposition	NOUN
iajs-3045	278	29	3.12	3.12	NUM
iajs-3045	278	30	let	let	VERB
iajs-3045	278	31	ƒ	ƒ	PRON
iajs-3045	278	32	≠	≠	NOUN
iajs-3045	278	33	ѡ	ѡ	NOUN
iajs-3045	278	34	and	and	CCONJ
iajs-3045	278	35	ѡ	ѡ	ADP
iajs-3045	278	36	𝑍-regular	𝑍-regular	ADJ
iajs-3045	278	37	multiplication	multiplication	NOUN
iajs-3045	278	38	ʀ	ʀ	NOUN
iajs-3045	278	39	-	-	PUNCT
iajs-3045	278	40	module	module	NOUN
iajs-3045	278	41	ѡ	ѡ	NOUN
iajs-3045	278	42	over	over	ADP
iajs-3045	278	43	local	local	ADJ
iajs-3045	278	44	ring	ring	NOUN
iajs-3045	278	45	ʀ	ʀ	NOUN
iajs-3045	278	46	.	.	PUNCT
iajs-3045	279	1	then	then	ADV
iajs-3045	279	2	ƒ	ƒ	PRON
iajs-3045	279	3	is	be	AUX
iajs-3045	279	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	279	5	submodule	submodule	NOUN
iajs-3045	279	6	of	of	ADP
iajs-3045	279	7	ѡ	ѡ	PROPN
iajs-3045	279	8	if	if	SCONJ
iajs-3045	280	1	and	and	CCONJ
iajs-3045	280	2	only	only	ADV
iajs-3045	280	3	if	if	SCONJ
iajs-3045	280	4	[	[	X
iajs-3045	280	5	ƒ	ƒ	X
iajs-3045	280	6	:	:	PUNCT
iajs-3045	280	7	ʀ	ʀ	PART
iajs-3045	280	8	ѡ	ѡ	X
iajs-3045	280	9	]	]	PUNCT
iajs-3045	280	10	is	be	AUX
iajs-3045	280	11	exnpq2ab	exnpq2ab	PROPN
iajs-3045	280	12	ideal	ideal	NOUN
iajs-3045	280	13	of	of	ADP
iajs-3045	280	14	ʀ	ʀ	NOUN
iajs-3045	280	15	.	.	NOUN
iajs-3045	280	16	4	4	NUM
iajs-3045	280	17	.	.	X
iajs-3045	281	1	more	more	ADJ
iajs-3045	281	2	result	result	NOUN
iajs-3045	281	3	of	of	ADP
iajs-3045	281	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	281	5	submodules	submodule	NOUN
iajs-3045	281	6	in	in	ADP
iajs-3045	281	7	multiplication	multiplication	NOUN
iajs-3045	281	8	modules	module	NOUN
iajs-3045	281	9	.	.	PUNCT
iajs-3045	282	1	in	in	ADP
iajs-3045	282	2	this	this	DET
iajs-3045	282	3	part	part	NOUN
iajs-3045	282	4	we	we	PRON
iajs-3045	282	5	studied	study	VERB
iajs-3045	282	6	more	more	ADJ
iajs-3045	282	7	result	result	NOUN
iajs-3045	282	8	of	of	ADP
iajs-3045	282	9	exnpq2ab	exnpq2ab	PROPN
iajs-3045	282	10	submodules	submodule	NOUN
iajs-3045	282	11	in	in	ADP
iajs-3045	282	12	multiplication	multiplication	NOUN
iajs-3045	282	13	modules	module	NOUN
iajs-3045	282	14	.	.	PUNCT
iajs-3045	283	1	and	and	CCONJ
iajs-3045	283	2	we	we	PRON
iajs-3045	283	3	got	get	VERB
iajs-3045	283	4	the	the	DET
iajs-3045	283	5	most	most	ADV
iajs-3045	283	6	important	important	ADJ
iajs-3045	283	7	results	result	NOUN
iajs-3045	283	8	.	.	PUNCT
iajs-3045	284	1	proposition	proposition	NOUN
iajs-3045	284	2	4.1	4.1	NUM
iajs-3045	284	3	let	let	VERB
iajs-3045	284	4	ѡ	ѡ	PRON
iajs-3045	284	5	be	be	AUX
iajs-3045	284	6	a	a	DET
iajs-3045	284	7	finitely	finitely	ADV
iajs-3045	284	8	generated	generate	VERB
iajs-3045	284	9	multiplication	multiplication	NOUN
iajs-3045	284	10	projective	projective	ADJ
iajs-3045	284	11	ʀ	ʀ	NOUN
iajs-3045	284	12	-	-	PUNCT
iajs-3045	284	13	module	module	NOUN
iajs-3045	284	14	,	,	PUNCT
iajs-3045	284	15	and	and	CCONJ
iajs-3045	284	16	ɓ	ɓ	PRON
iajs-3045	284	17	is	be	AUX
iajs-3045	284	18	an	an	DET
iajs-3045	284	19	ideal	ideal	NOUN
iajs-3045	284	20	of	of	ADP
iajs-3045	284	21	ʀ	ʀ	NOUN
iajs-3045	284	22	with	with	ADP
iajs-3045	284	23	𝑎𝑛𝑛ʀ(ѡ	𝑎𝑛𝑛ʀ(ѡ	NOUN
iajs-3045	284	24	)	)	PUNCT
iajs-3045	284	25	⊆	⊆	NUM
iajs-3045	284	26	ɓ	ɓ	NOUN
iajs-3045	284	27	.	.	PUNCT
iajs-3045	285	1	then	then	ADV
iajs-3045	285	2	ɓ	ɓ	PROPN
iajs-3045	285	3	is	be	AUX
iajs-3045	285	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	285	5	ideal	ideal	NOUN
iajs-3045	285	6	of	of	ADP
iajs-3045	285	7	ʀ	ʀ	PRON
iajs-3045	285	8	if	if	NOUN
iajs-3045	285	9	and	and	CCONJ
iajs-3045	285	10	only	only	ADV
iajs-3045	285	11	if	if	SCONJ
iajs-3045	285	12	ɓѡ	ɓѡ	PROPN
iajs-3045	285	13	is	be	AUX
iajs-3045	285	14	exnpq2ab	exnpq2ab	PROPN
iajs-3045	285	15	submodule	submodule	NOUN
iajs-3045	285	16	of	of	ADP
iajs-3045	285	17	ѡ.	ѡ.	NOUN
iajs-3045	285	18	proof	proof	NOUN
iajs-3045	285	19	.	.	PUNCT
iajs-3045	286	1	ihjpas	ihjpas	PROPN
iajs-3045	286	2	.	.	PUNCT
iajs-3045	287	1	36(2)2023	36(2)2023	NUM
iajs-3045	287	2	415	415	NUM
iajs-3045	287	3	(	(	PUNCT
iajs-3045	287	4	⟹	⟹	X
iajs-3045	287	5	)	)	PUNCT
iajs-3045	287	6	let	let	VERB
iajs-3045	287	7	𝐻1𝐻2𝐻3𝐴	𝐻1𝐻2𝐻3𝐴	PROPN
iajs-3045	287	8	⊆	⊆	NUM
iajs-3045	287	9	ɓѡ	ɓѡ	NOUN
iajs-3045	287	10	for	for	ADP
iajs-3045	287	11	some	some	DET
iajs-3045	287	12	submodules	submodule	NOUN
iajs-3045	287	13	𝐻1,𝐻2,𝐻3	𝐻1,𝐻2,𝐻3	NOUN
iajs-3045	287	14	,	,	PUNCT
iajs-3045	287	15	𝐴	𝐴	PROPN
iajs-3045	287	16	of	of	ADP
iajs-3045	287	17	ѡ.	ѡ.	NOUN
iajs-3045	287	18	since	since	SCONJ
iajs-3045	287	19	ѡ	ѡ	PROPN
iajs-3045	287	20	is	be	AUX
iajs-3045	287	21	a	a	DET
iajs-3045	287	22	multiplication	multiplication	NOUN
iajs-3045	287	23	,	,	PUNCT
iajs-3045	287	24	then	then	ADV
iajs-3045	287	25	𝐻1	𝐻1	PROPN
iajs-3045	287	26	=	=	SYM
iajs-3045	287	27	𝚥1ѡ	𝚥1ѡ	PROPN
iajs-3045	287	28	,	,	PUNCT
iajs-3045	287	29	𝐻2	𝐻2	NOUN
iajs-3045	287	30	=	=	SYM
iajs-3045	287	31	𝚥2ѡ	𝚥2ѡ	NOUN
iajs-3045	287	32	,	,	PUNCT
iajs-3045	287	33	𝐻3	𝐻3	PROPN
iajs-3045	287	34	=	=	SYM
iajs-3045	287	35	𝚥3ѡ	𝚥3ѡ	X
iajs-3045	287	36	and	and	CCONJ
iajs-3045	287	37	𝐴	𝐴	PROPN
iajs-3045	287	38	=	=	PUNCT
iajs-3045	287	39	𝚥4ѡ	𝚥4ѡ	X
iajs-3045	287	40	for	for	ADP
iajs-3045	287	41	some	some	DET
iajs-3045	287	42	ideals	ideal	NOUN
iajs-3045	287	43	𝚥1	𝚥1	NOUN
iajs-3045	287	44	,	,	PUNCT
iajs-3045	287	45	𝚥2	𝚥2	NOUN
iajs-3045	287	46	,	,	PUNCT
iajs-3045	287	47	𝚥3	𝚥3	ADJ
iajs-3045	287	48	and	and	CCONJ
iajs-3045	287	49	𝚥4of	𝚥4of	PUNCT
iajs-3045	287	50	ʀ	ʀ	VERB
iajs-3045	287	51	.	.	NOUN
iajs-3045	287	52	that	that	PRON
iajs-3045	287	53	is	be	AUX
iajs-3045	287	54	𝐻1𝐻2𝐻3𝐴	𝐻1𝐻2𝐻3𝐴	PROPN
iajs-3045	287	55	=	=	SYM
iajs-3045	287	56	𝚥1𝚥2𝚥3𝚥4ѡ	𝚥1𝚥2𝚥3𝚥4ѡ	VERB
iajs-3045	287	57	⊆	⊆	NUM
iajs-3045	287	58	ɓѡ.	ɓѡ.	NOUN
iajs-3045	287	59	but	but	CCONJ
iajs-3045	287	60	ѡ	ѡ	PROPN
iajs-3045	287	61	is	be	AUX
iajs-3045	287	62	a	a	DET
iajs-3045	287	63	finitely	finitely	ADV
iajs-3045	287	64	generated	generate	VERB
iajs-3045	287	65	multiplication	multiplication	NOUN
iajs-3045	287	66	ʀ	ʀ	NOUN
iajs-3045	287	67	-	-	PUNCT
iajs-3045	287	68	module	module	NOUN
iajs-3045	287	69	then	then	ADV
iajs-3045	287	70	by	by	ADP
iajs-3045	287	71	lemma	lemma	PROPN
iajs-3045	287	72	2.18	2.18	NUM
iajs-3045	287	73	𝚥1𝚥2𝚥3𝚥4	𝚥1𝚥2𝚥3𝚥4	PROPN
iajs-3045	287	74	⊆	⊆	NUM
iajs-3045	287	75	ɓ	ɓ	NOUN
iajs-3045	287	76	+	+	X
iajs-3045	287	77	𝑎𝑛𝑛ʀ(ѡ	𝑎𝑛𝑛ʀ(ѡ	NOUN
iajs-3045	287	78	)	)	PUNCT
iajs-3045	287	79	,	,	PUNCT
iajs-3045	287	80	but	but	CCONJ
iajs-3045	287	81	𝑎𝑛𝑛ʀ(ѡ	𝑎𝑛𝑛ʀ(ѡ	NOUN
iajs-3045	287	82	)	)	PUNCT
iajs-3045	287	83	⊆	⊆	NUM
iajs-3045	287	84	ɓ	ɓ	NUM
iajs-3045	287	85	,	,	PUNCT
iajs-3045	287	86	implies	imply	VERB
iajs-3045	287	87	that	that	SCONJ
iajs-3045	287	88	ɓ	ɓ	PRON
iajs-3045	287	89	+	+	X
iajs-3045	287	90	𝑎𝑛𝑛ʀ(ѡ	𝑎𝑛𝑛ʀ(ѡ	NOUN
iajs-3045	287	91	)	)	PUNCT
iajs-3045	287	92	=	=	SYM
iajs-3045	288	1	ɓ	ɓ	NOUN
iajs-3045	288	2	,	,	PUNCT
iajs-3045	288	3	thus	thus	ADV
iajs-3045	288	4	𝚥1𝚥2𝚥3𝚥4	𝚥1𝚥2𝚥3𝚥4	PROPN
iajs-3045	288	5	⊆	⊆	NUM
iajs-3045	288	6	ɓ	ɓ	NOUN
iajs-3045	288	7	.	.	PUNCT
iajs-3045	289	1	now	now	ADV
iajs-3045	289	2	,	,	PUNCT
iajs-3045	289	3	by	by	ADP
iajs-3045	289	4	assumption	assumption	NOUN
iajs-3045	289	5	ɓ	ɓ	PRON
iajs-3045	289	6	is	be	AUX
iajs-3045	289	7	exnpq2ab	exnpq2ab	PROPN
iajs-3045	289	8	ideal	ideal	NOUN
iajs-3045	289	9	of	of	ADP
iajs-3045	289	10	ʀ	ʀ	PROPN
iajs-3045	289	11	then	then	ADV
iajs-3045	289	12	by	by	ADP
iajs-3045	289	13	proposition	proposition	NOUN
iajs-3045	289	14	3.2	3.2	NUM
iajs-3045	289	15	either	either	CCONJ
iajs-3045	289	16	𝚥1𝚥3𝚥4	𝚥1𝚥3𝚥4	PROPN
iajs-3045	289	17	⊆	⊆	NUM
iajs-3045	289	18	ɓ	ɓ	NOUN
iajs-3045	289	19	+	+	CCONJ
iajs-3045	289	20	(	(	PUNCT
iajs-3045	289	21	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	PROPN
iajs-3045	289	22	)	)	PUNCT
iajs-3045	289	23	+	+	NUM
iajs-3045	289	24	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	289	25	)	)	PUNCT
iajs-3045	289	26	)	)	PUNCT
iajs-3045	289	27	or	or	CCONJ
iajs-3045	289	28	𝚥2𝚥3𝚥4	𝚥2𝚥3𝚥4	ADP
iajs-3045	289	29	⊆	⊆	NUM
iajs-3045	289	30	ɓ	ɓ	NOUN
iajs-3045	289	31	+	+	CCONJ
iajs-3045	289	32	(	(	PUNCT
iajs-3045	289	33	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	PROPN
iajs-3045	289	34	)	)	PUNCT
iajs-3045	289	35	+	+	NUM
iajs-3045	290	1	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	290	2	)	)	PUNCT
iajs-3045	290	3	)	)	PUNCT
iajs-3045	290	4	or	or	CCONJ
iajs-3045	290	5	𝚥1𝚥2𝚥4	𝚥1𝚥2𝚥4	ADJ
iajs-3045	290	6	⊆	⊆	NUM
iajs-3045	290	7	ɓ	ɓ	NOUN
iajs-3045	290	8	+	+	CCONJ
iajs-3045	290	9	(	(	PUNCT
iajs-3045	290	10	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	PROPN
iajs-3045	290	11	)	)	PUNCT
iajs-3045	290	12	+	+	NUM
iajs-3045	290	13	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	290	14	)	)	PUNCT
iajs-3045	290	15	)	)	PUNCT
iajs-3045	291	1	,	,	PUNCT
iajs-3045	291	2	hence	hence	ADV
iajs-3045	291	3	either	either	CCONJ
iajs-3045	291	4	𝚥1𝚥3𝚥4ѡ	𝚥1𝚥3𝚥4ѡ	PROPN
iajs-3045	291	5	⊆	⊆	NUM
iajs-3045	291	6	ɓѡ	ɓѡ	X
iajs-3045	291	7	+	+	CCONJ
iajs-3045	291	8	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	291	9	+	+	CCONJ
iajs-3045	291	10	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	ADJ
iajs-3045	291	11	or	or	CCONJ
iajs-3045	291	12	𝚥2𝚥3𝚥4ѡ	𝚥2𝚥3𝚥4ѡ	X
iajs-3045	291	13	⊆	⊆	NUM
iajs-3045	291	14	ɓѡ	ɓѡ	NOUN
iajs-3045	291	15	+	+	CCONJ
iajs-3045	291	16	(	(	PUNCT
iajs-3045	291	17	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	291	18	+	+	CCONJ
iajs-3045	291	19	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	NOUN
iajs-3045	291	20	)	)	PUNCT
iajs-3045	291	21	or	or	CCONJ
iajs-3045	291	22	𝚥1𝚥2𝚥4ѡ	𝚥1𝚥2𝚥4ѡ	PROPN
iajs-3045	291	23	⊆	⊆	NUM
iajs-3045	291	24	ɓѡ	ɓѡ	NOUN
iajs-3045	291	25	+	+	CCONJ
iajs-3045	291	26	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	291	27	+	+	CCONJ
iajs-3045	291	28	𝐽(ʀ)ѡ.	𝐽(ʀ)ѡ.	NOUN
iajs-3045	291	29	since	since	SCONJ
iajs-3045	291	30	ѡ	ѡ	PROPN
iajs-3045	291	31	is	be	AUX
iajs-3045	291	32	a	a	DET
iajs-3045	291	33	projective	projective	NOUN
iajs-3045	291	34	then	then	ADV
iajs-3045	291	35	by	by	ADP
iajs-3045	291	36	lemma	lemma	PROPN
iajs-3045	291	37	2.10	2.10	NUM
iajs-3045	291	38	and	and	CCONJ
iajs-3045	291	39	lemma	lemma	PROPN
iajs-3045	291	40	2.9	2.9	NUM
iajs-3045	291	41	(	(	PUNCT
iajs-3045	291	42	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	291	43	)	)	PUNCT
iajs-3045	291	44	+	+	PUNCT
iajs-3045	291	45	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	291	46	)	)	PUNCT
iajs-3045	291	47	)	)	PUNCT
iajs-3045	291	48	=	=	PRON
iajs-3045	292	1	(	(	PUNCT
iajs-3045	292	2	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	292	3	+	+	CCONJ
iajs-3045	292	4	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	NOUN
iajs-3045	292	5	)	)	PUNCT
iajs-3045	292	6	,	,	PUNCT
iajs-3045	292	7	thus	thus	ADV
iajs-3045	292	8	either	either	PRON
iajs-3045	292	9	𝐻1𝐻3𝐴	𝐻1𝐻3𝐴	NUM
iajs-3045	293	1	⊆	⊆	NUM
iajs-3045	293	2	ɓѡ	ɓѡ	NOUN
iajs-3045	293	3	+	+	CCONJ
iajs-3045	293	4	(	(	PUNCT
iajs-3045	293	5	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	293	6	)	)	PUNCT
iajs-3045	293	7	+	+	PUNCT
iajs-3045	293	8	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	293	9	)	)	PUNCT
iajs-3045	293	10	)	)	PUNCT
iajs-3045	293	11	or	or	CCONJ
iajs-3045	293	12	𝐻2𝐻3𝐴	𝐻2𝐻3𝐴	VERB
iajs-3045	293	13	⊆	⊆	NUM
iajs-3045	293	14	ɓѡ	ɓѡ	NOUN
iajs-3045	293	15	+	+	CCONJ
iajs-3045	293	16	(	(	PUNCT
iajs-3045	293	17	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	293	18	)	)	PUNCT
iajs-3045	293	19	+	+	PUNCT
iajs-3045	293	20	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	293	21	)	)	PUNCT
iajs-3045	293	22	)	)	PUNCT
iajs-3045	293	23	or	or	CCONJ
iajs-3045	293	24	𝐻1𝐻2𝐴	𝐻1𝐻2𝐴	X
iajs-3045	293	25	⊆	⊆	NUM
iajs-3045	293	26	ɓѡ	ɓѡ	NOUN
iajs-3045	293	27	+	+	CCONJ
iajs-3045	293	28	(	(	PUNCT
iajs-3045	293	29	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	293	30	)	)	PUNCT
iajs-3045	293	31	+	+	PUNCT
iajs-3045	293	32	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	293	33	)	)	PUNCT
iajs-3045	293	34	)	)	PUNCT
iajs-3045	293	35	.	.	PUNCT
iajs-3045	294	1	hence	hence	ADV
iajs-3045	294	2	by	by	ADP
iajs-3045	294	3	proposition	proposition	NOUN
iajs-3045	294	4	3.2	3.2	NUM
iajs-3045	294	5	ɓѡ	ɓѡ	NOUN
iajs-3045	294	6	is	be	AUX
iajs-3045	294	7	exnpq2ab	exnpq2ab	PROPN
iajs-3045	294	8	submodule	submodule	NOUN
iajs-3045	294	9	of	of	ADP
iajs-3045	294	10	ѡ.	ѡ.	NOUN
iajs-3045	294	11	(	(	PUNCT
iajs-3045	294	12	⟸	⟸	ADJ
iajs-3045	294	13	)	)	PUNCT
iajs-3045	294	14	let	let	VERB
iajs-3045	294	15	ƥ1ƥ2ƥ3ƥ4	ƥ1ƥ2ƥ3ƥ4	VERB
iajs-3045	294	16	⊆	⊆	NUM
iajs-3045	294	17	ɓ	ɓ	NOUN
iajs-3045	294	18	,	,	PUNCT
iajs-3045	294	19	for	for	ADP
iajs-3045	294	20	ƥ1	ƥ1	NOUN
iajs-3045	294	21	,	,	PUNCT
iajs-3045	294	22	ƥ2	ƥ2	NOUN
iajs-3045	294	23	,	,	PUNCT
iajs-3045	294	24	ƥ3	ƥ3	PROPN
iajs-3045	294	25	and	and	CCONJ
iajs-3045	294	26	ƥ4	ƥ4	ADV
iajs-3045	294	27	are	be	AUX
iajs-3045	294	28	ideals	ideal	NOUN
iajs-3045	294	29	in	in	ADP
iajs-3045	294	30	ʀ	ʀ	NOUN
iajs-3045	294	31	,	,	PUNCT
iajs-3045	294	32	implies	imply	VERB
iajs-3045	294	33	that	that	SCONJ
iajs-3045	294	34	ƥ1ƥ2ƥ3(ƥ4ѡ	ƥ1ƥ2ƥ3(ƥ4ѡ	X
iajs-3045	294	35	)	)	PUNCT
iajs-3045	294	36	⊆	⊆	NUM
iajs-3045	294	37	ɓѡ.	ɓѡ.	NOUN
iajs-3045	294	38	but	but	CCONJ
iajs-3045	294	39	ɓѡ	ɓѡ	PROPN
iajs-3045	294	40	is	be	AUX
iajs-3045	294	41	exnpq2ab	exnpq2ab	PROPN
iajs-3045	294	42	submodule	submodule	NOUN
iajs-3045	294	43	of	of	ADP
iajs-3045	294	44	ѡ	ѡ	PROPN
iajs-3045	294	45	,	,	PUNCT
iajs-3045	294	46	then	then	ADV
iajs-3045	294	47	by	by	ADP
iajs-3045	294	48	proposition	proposition	NOUN
iajs-3045	294	49	2.25	2.25	NUM
iajs-3045	294	50	either	either	CCONJ
iajs-3045	294	51	ƥ1ƥ2(ƥ4ѡ	ƥ1ƥ2(ƥ4ѡ	X
iajs-3045	294	52	)	)	PUNCT
iajs-3045	294	53	⊆	⊆	NUM
iajs-3045	294	54	ɓѡ	ɓѡ	NOUN
iajs-3045	294	55	+	+	CCONJ
iajs-3045	294	56	(	(	PUNCT
iajs-3045	294	57	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	294	58	)	)	PUNCT
iajs-3045	295	1	+	+	CCONJ
iajs-3045	295	2	𝐽(ѡ))or	𝐽(ѡ))or	NOUN
iajs-3045	295	3	ƥ1ƥ3(ƥ4ѡ	ƥ1ƥ3(ƥ4ѡ	PRON
iajs-3045	295	4	)	)	PUNCT
iajs-3045	295	5	⊆	⊆	NUM
iajs-3045	295	6	ɓѡ	ɓѡ	NOUN
iajs-3045	295	7	+	+	CCONJ
iajs-3045	295	8	(	(	PUNCT
iajs-3045	295	9	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	295	10	)	)	PUNCT
iajs-3045	295	11	+	+	PUNCT
iajs-3045	295	12	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	295	13	)	)	PUNCT
iajs-3045	295	14	)	)	PUNCT
iajs-3045	295	15	or	or	CCONJ
iajs-3045	295	16	ƥ2ƥ3(ƥ4ѡ	ƥ2ƥ3(ƥ4ѡ	VERB
iajs-3045	295	17	)	)	PUNCT
iajs-3045	295	18	⊆	⊆	NUM
iajs-3045	295	19	ɓѡ	ɓѡ	NOUN
iajs-3045	295	20	+	+	CCONJ
iajs-3045	295	21	(	(	PUNCT
iajs-3045	295	22	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	295	23	)	)	PUNCT
iajs-3045	295	24	+	+	PUNCT
iajs-3045	295	25	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	295	26	)	)	PUNCT
iajs-3045	295	27	)	)	PUNCT
iajs-3045	295	28	.	.	PUNCT
iajs-3045	296	1	but	but	CCONJ
iajs-3045	296	2	ѡ	ѡ	PROPN
iajs-3045	296	3	is	be	AUX
iajs-3045	296	4	a	a	DET
iajs-3045	296	5	projective	projective	NOUN
iajs-3045	296	6	then	then	ADV
iajs-3045	296	7	(	(	PUNCT
iajs-3045	296	8	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	296	9	)	)	PUNCT
iajs-3045	296	10	+	+	PUNCT
iajs-3045	296	11	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	296	12	)	)	PUNCT
iajs-3045	296	13	)	)	PUNCT
iajs-3045	297	1	=	=	PRON
iajs-3045	297	2	(	(	PUNCT
iajs-3045	297	3	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	297	4	+	+	CCONJ
iajs-3045	297	5	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	NOUN
iajs-3045	297	6	)	)	PUNCT
iajs-3045	297	7	.	.	PUNCT
iajs-3045	298	1	thus	thus	ADV
iajs-3045	298	2	either	either	CCONJ
iajs-3045	298	3	ƥ1ƥ2ƥ4ѡ	ƥ1ƥ2ƥ4ѡ	PROPN
iajs-3045	298	4	⊆	⊆	NUM
iajs-3045	298	5	ɓѡ	ɓѡ	NOUN
iajs-3045	298	6	+	+	CCONJ
iajs-3045	298	7	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	298	8	+	+	CCONJ
iajs-3045	298	9	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	ADJ
iajs-3045	298	10	or	or	CCONJ
iajs-3045	298	11	ƥ1ƥ3ƥ4ѡ	ƥ1ƥ3ƥ4ѡ	VERB
iajs-3045	298	12	⊆	⊆	NUM
iajs-3045	298	13	ɓѡ	ɓѡ	X
iajs-3045	298	14	+	+	CCONJ
iajs-3045	298	15	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	298	16	+	+	CCONJ
iajs-3045	298	17	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	ADJ
iajs-3045	298	18	or	or	CCONJ
iajs-3045	298	19	ƥ2ƥ3ƥ4ѡ	ƥ2ƥ3ƥ4ѡ	VERB
iajs-3045	298	20	⊆	⊆	NUM
iajs-3045	298	21	ɓѡ	ɓѡ	NOUN
iajs-3045	298	22	+	+	CCONJ
iajs-3045	298	23	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	298	24	+	+	CCONJ
iajs-3045	298	25	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	ADJ
iajs-3045	298	26	,	,	PUNCT
iajs-3045	298	27	hence	hence	ADV
iajs-3045	298	28	either	either	CCONJ
iajs-3045	298	29	ƥ1ƥ2ƥ4	ƥ1ƥ2ƥ4	PROPN
iajs-3045	298	30	⊆	⊆	NUM
iajs-3045	298	31	ɓ	ɓ	NOUN
iajs-3045	298	32	+	+	X
iajs-3045	298	33	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	298	34	)	)	PUNCT
iajs-3045	298	35	+	+	NUM
iajs-3045	298	36	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	298	37	)	)	PUNCT
iajs-3045	298	38	or	or	CCONJ
iajs-3045	298	39	ƥ1ƥ3ƥ4	ƥ1ƥ3ƥ4	NOUN
iajs-3045	298	40	⊆	⊆	NUM
iajs-3045	298	41	ɓ	ɓ	NOUN
iajs-3045	298	42	+	+	X
iajs-3045	298	43	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	298	44	)	)	PUNCT
iajs-3045	298	45	+	+	NUM
iajs-3045	298	46	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	298	47	)	)	PUNCT
iajs-3045	298	48	or	or	CCONJ
iajs-3045	298	49	ƥ2ƥ3ƥ4	ƥ2ƥ3ƥ4	VERB
iajs-3045	298	50	⊆	⊆	NUM
iajs-3045	298	51	ɓ	ɓ	NOUN
iajs-3045	298	52	+	+	X
iajs-3045	298	53	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	298	54	)	)	PUNCT
iajs-3045	298	55	+	+	NUM
iajs-3045	298	56	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	298	57	)	)	PUNCT
iajs-3045	298	58	.	.	PUNCT
iajs-3045	299	1	then	then	ADV
iajs-3045	299	2	by	by	ADP
iajs-3045	299	3	proposition	proposition	NOUN
iajs-3045	299	4	2.25	2.25	NUM
iajs-3045	299	5	ɓ	ɓ	PRON
iajs-3045	299	6	is	be	AUX
iajs-3045	299	7	exnpq2ab	exnpq2ab	PROPN
iajs-3045	299	8	ideal	ideal	NOUN
iajs-3045	299	9	of	of	ADP
iajs-3045	299	10	ʀ	ʀ	NOUN
iajs-3045	299	11	.	.	PUNCT
iajs-3045	299	12	proposition	proposition	NOUN
iajs-3045	299	13	4.2	4.2	NUM
iajs-3045	299	14	let	let	VERB
iajs-3045	299	15	ѡ	ѡ	PRON
iajs-3045	299	16	be	be	AUX
iajs-3045	299	17	a	a	DET
iajs-3045	299	18	faithful	faithful	ADJ
iajs-3045	299	19	finitely	finitely	ADV
iajs-3045	299	20	generated	generate	VERB
iajs-3045	299	21	multiplication	multiplication	NOUN
iajs-3045	299	22	ʀ	ʀ	NOUN
iajs-3045	299	23	-	-	PUNCT
iajs-3045	299	24	module	module	NOUN
iajs-3045	299	25	,	,	PUNCT
iajs-3045	299	26	and	and	CCONJ
iajs-3045	299	27	ɓ	ɓ	PRON
iajs-3045	299	28	is	be	AUX
iajs-3045	299	29	an	an	DET
iajs-3045	299	30	ideal	ideal	NOUN
iajs-3045	299	31	of	of	ADP
iajs-3045	299	32	ʀ	ʀ	X
iajs-3045	299	33	.	.	PUNCT
iajs-3045	300	1	then	then	ADV
iajs-3045	300	2	ɓ	ɓ	PROPN
iajs-3045	300	3	is	be	AUX
iajs-3045	300	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	300	5	ideal	ideal	NOUN
iajs-3045	300	6	of	of	ADP
iajs-3045	300	7	ʀ	ʀ	PRON
iajs-3045	300	8	if	if	NOUN
iajs-3045	300	9	and	and	CCONJ
iajs-3045	300	10	only	only	ADV
iajs-3045	300	11	if	if	SCONJ
iajs-3045	300	12	ɓѡ	ɓѡ	PROPN
iajs-3045	300	13	is	be	AUX
iajs-3045	300	14	exnpq2ab	exnpq2ab	PROPN
iajs-3045	300	15	submodule	submodule	NOUN
iajs-3045	300	16	of	of	ADP
iajs-3045	300	17	ѡ.	ѡ.	NOUN
iajs-3045	300	18	proof	proof	NOUN
iajs-3045	300	19	.	.	PUNCT
iajs-3045	301	1	(	(	PUNCT
iajs-3045	301	2	⟹	⟹	X
iajs-3045	301	3	)	)	PUNCT
iajs-3045	301	4	let	let	VERB
iajs-3045	301	5	𝑟ƥ𝐽𝑥	𝑟ƥ𝐽𝑥	ADV
iajs-3045	301	6	⊆	⊆	NUM
iajs-3045	301	7	ɓѡ	ɓѡ	NOUN
iajs-3045	301	8	for	for	ADP
iajs-3045	301	9	any	any	DET
iajs-3045	301	10	𝑟	𝑟	PRON
iajs-3045	301	11	∈	∈	PROPN
iajs-3045	301	12	ʀ	ʀ	NOUN
iajs-3045	301	13	,	,	PUNCT
iajs-3045	301	14	𝑥	𝑥	DET
iajs-3045	301	15	∈	∈	NOUN
iajs-3045	301	16	ѡ	ѡ	X
iajs-3045	301	17	and	and	CCONJ
iajs-3045	301	18	ƥ	ƥ	PROPN
iajs-3045	301	19	,	,	PUNCT
iajs-3045	301	20	𝐽	𝐽	PROPN
iajs-3045	301	21	are	be	AUX
iajs-3045	301	22	ideals	ideal	NOUN
iajs-3045	301	23	of	of	ADP
iajs-3045	301	24	ʀ	ʀ	NOUN
iajs-3045	301	25	.	.	PROPN
iajs-3045	301	26	next	next	ADV
iajs-3045	301	27	,	,	PUNCT
iajs-3045	301	28	follows	follow	VERB
iajs-3045	301	29	𝑟ƥ𝐽(𝑥	𝑟ƥ𝐽(𝑥	PROPN
iajs-3045	301	30	)	)	PUNCT
iajs-3045	301	31	⊆	⊆	NUM
iajs-3045	301	32	ɓѡ.	ɓѡ.	NOUN
iajs-3045	301	33	since	since	SCONJ
iajs-3045	301	34	ѡ	ѡ	PROPN
iajs-3045	301	35	is	be	AUX
iajs-3045	301	36	a	a	DET
iajs-3045	301	37	multiplication	multiplication	NOUN
iajs-3045	301	38	,	,	PUNCT
iajs-3045	301	39	then	then	ADV
iajs-3045	301	40	(	(	PUNCT
iajs-3045	301	41	𝑥	𝑥	NOUN
iajs-3045	301	42	)	)	PUNCT
iajs-3045	301	43	=	=	PUNCT
iajs-3045	302	1	ƥ1ѡ	ƥ1ѡ	NOUN
iajs-3045	302	2	for	for	ADP
iajs-3045	302	3	some	some	DET
iajs-3045	302	4	ideal	ideal	ADJ
iajs-3045	302	5	ƥ1	ƥ1	NOUN
iajs-3045	302	6	of	of	ADP
iajs-3045	302	7	ʀ	ʀ	NOUN
iajs-3045	302	8	,	,	PUNCT
iajs-3045	302	9	that	that	PRON
iajs-3045	302	10	is	be	AUX
iajs-3045	302	11	𝑟ƥ𝐽ƥ1ѡ	𝑟ƥ𝐽ƥ1ѡ	NUM
iajs-3045	302	12	⊆	⊆	NUM
iajs-3045	302	13	ɓѡ.	ɓѡ.	NOUN
iajs-3045	302	14	thus	thus	ADV
iajs-3045	302	15	by	by	ADP
iajs-3045	302	16	lemma	lemma	PROPN
iajs-3045	302	17	2.18	2.18	NUM
iajs-3045	302	18	we	we	PRON
iajs-3045	302	19	get	get	VERB
iajs-3045	302	20	𝑟ƥ𝐽ƥ1	𝑟ƥ𝐽ƥ1	PUNCT
iajs-3045	302	21	⊆	⊆	NUM
iajs-3045	302	22	ɓ	ɓ	NOUN
iajs-3045	302	23	+	+	CCONJ
iajs-3045	302	24	𝑎𝑛𝑛(ѡ	𝑎𝑛𝑛(ѡ	PROPN
iajs-3045	302	25	)	)	PUNCT
iajs-3045	302	26	,	,	PUNCT
iajs-3045	302	27	but	but	CCONJ
iajs-3045	302	28	ѡ	ѡ	PROPN
iajs-3045	302	29	is	be	AUX
iajs-3045	302	30	faithful	faithful	ADJ
iajs-3045	302	31	,	,	PUNCT
iajs-3045	302	32	then	then	ADV
iajs-3045	302	33	𝑎𝑛𝑛(ѡ	𝑎𝑛𝑛(ѡ	PROPN
iajs-3045	302	34	)	)	PUNCT
iajs-3045	302	35	=	=	PRON
iajs-3045	302	36	{	{	PUNCT
iajs-3045	302	37	0	0	NUM
iajs-3045	302	38	}	}	PUNCT
iajs-3045	302	39	,	,	PUNCT
iajs-3045	302	40	that	that	ADV
iajs-3045	302	41	is	is	ADV
iajs-3045	302	42	𝑟ƥ𝐽ƥ1	𝑟ƥ𝐽ƥ1	PUNCT
iajs-3045	302	43	⊆	⊆	NUM
iajs-3045	302	44	ɓ	ɓ	NOUN
iajs-3045	302	45	.	.	PUNCT
iajs-3045	303	1	since	since	SCONJ
iajs-3045	303	2	ɓ	ɓ	PRON
iajs-3045	303	3	is	be	AUX
iajs-3045	303	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	303	5	ideal	ideal	NOUN
iajs-3045	303	6	of	of	ADP
iajs-3045	303	7	ʀ	ʀ	NOUN
iajs-3045	303	8	,	,	PUNCT
iajs-3045	303	9	then	then	ADV
iajs-3045	303	10	by	by	ADP
iajs-3045	303	11	proposition	proposition	NOUN
iajs-3045	303	12	2.27	2.27	NUM
iajs-3045	303	13	either	either	CCONJ
iajs-3045	303	14	𝑟ƥƥ1	𝑟ƥƥ1	VERB
iajs-3045	303	15	⊆	⊆	NUM
iajs-3045	303	16	ɓ	ɓ	NOUN
iajs-3045	303	17	+	+	X
iajs-3045	303	18	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	303	19	)	)	PUNCT
iajs-3045	303	20	+	+	NUM
iajs-3045	303	21	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	303	22	)	)	PUNCT
iajs-3045	303	23	or	or	CCONJ
iajs-3045	303	24	𝑟𝐽ƥ1	𝑟𝐽ƥ1	X
iajs-3045	303	25	⊆	⊆	NUM
iajs-3045	303	26	ɓ	ɓ	NOUN
iajs-3045	303	27	+	+	X
iajs-3045	303	28	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	303	29	)	)	PUNCT
iajs-3045	303	30	+	+	NUM
iajs-3045	303	31	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	303	32	)	)	PUNCT
iajs-3045	303	33	or	or	CCONJ
iajs-3045	303	34	ƥ𝐽ƥ1	ƥ𝐽ƥ1	PRON
iajs-3045	303	35	⊆	⊆	NUM
iajs-3045	303	36	ɓ	ɓ	NOUN
iajs-3045	303	37	+	+	X
iajs-3045	303	38	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	303	39	)	)	PUNCT
iajs-3045	303	40	+	+	NUM
iajs-3045	303	41	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	303	42	)	)	PUNCT
iajs-3045	303	43	,	,	PUNCT
iajs-3045	303	44	hence	hence	ADV
iajs-3045	303	45	either	either	CCONJ
iajs-3045	303	46	𝑟ƥƥ1ѡ	𝑟ƥƥ1ѡ	NUM
iajs-3045	303	47	⊆	⊆	NUM
iajs-3045	303	48	ɓѡ	ɓѡ	NOUN
iajs-3045	303	49	+	+	CCONJ
iajs-3045	303	50	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	303	51	+	+	CCONJ
iajs-3045	303	52	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	ADJ
iajs-3045	303	53	or	or	CCONJ
iajs-3045	303	54	𝑟𝐽ƥ1ѡ	𝑟𝐽ƥ1ѡ	X
iajs-3045	303	55	⊆	⊆	NUM
iajs-3045	303	56	ɓѡ	ɓѡ	NOUN
iajs-3045	303	57	+	+	CCONJ
iajs-3045	303	58	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	303	59	+	+	CCONJ
iajs-3045	303	60	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	ADJ
iajs-3045	303	61	or	or	CCONJ
iajs-3045	303	62	ƥ𝐽ƥ1ѡ	ƥ𝐽ƥ1ѡ	NOUN
iajs-3045	303	63	⊆	⊆	NUM
iajs-3045	303	64	ɓѡ	ɓѡ	NOUN
iajs-3045	303	65	+	+	CCONJ
iajs-3045	303	66	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	303	67	+	+	CCONJ
iajs-3045	303	68	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	ADJ
iajs-3045	303	69	,	,	PUNCT
iajs-3045	303	70	hence	hence	ADV
iajs-3045	303	71	by	by	ADP
iajs-3045	303	72	lemma	lemma	PROPN
iajs-3045	303	73	2.6	2.6	NUM
iajs-3045	303	74	and	and	CCONJ
iajs-3045	303	75	lemma	lemma	PROPN
iajs-3045	303	76	2.7	2.7	NUM
iajs-3045	303	77	either	either	CCONJ
iajs-3045	303	78	𝑟ƥ(𝑥	𝑟ƥ(𝑥	NOUN
iajs-3045	303	79	)	)	PUNCT
iajs-3045	303	80	⊆	⊆	NUM
iajs-3045	303	81	ɓѡ	ɓѡ	NOUN
iajs-3045	303	82	+	+	CCONJ
iajs-3045	303	83	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	303	84	)	)	PUNCT
iajs-3045	303	85	+	+	PUNCT
iajs-3045	303	86	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	303	87	)	)	PUNCT
iajs-3045	303	88	or	or	CCONJ
iajs-3045	303	89	𝑟𝐽(𝑥	𝑟𝐽(𝑥	NUM
iajs-3045	303	90	)	)	PUNCT
iajs-3045	303	91	⊆	⊆	NUM
iajs-3045	303	92	ɓѡ	ɓѡ	NOUN
iajs-3045	303	93	+	+	CCONJ
iajs-3045	303	94	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	303	95	)	)	PUNCT
iajs-3045	303	96	+	+	PUNCT
iajs-3045	303	97	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	303	98	)	)	PUNCT
iajs-3045	303	99	or	or	CCONJ
iajs-3045	303	100	ƥ𝐽(𝑥	ƥ𝐽(𝑥	NUM
iajs-3045	303	101	)	)	PUNCT
iajs-3045	303	102	⊆	⊆	NUM
iajs-3045	303	103	ɓѡ	ɓѡ	NOUN
iajs-3045	303	104	+	+	CCONJ
iajs-3045	303	105	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	303	106	)	)	PUNCT
iajs-3045	303	107	+	+	PUNCT
iajs-3045	303	108	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	303	109	)	)	PUNCT
iajs-3045	303	110	.	.	PUNCT
iajs-3045	304	1	that	that	PRON
iajs-3045	304	2	is	be	AUX
iajs-3045	304	3	either	either	CCONJ
iajs-3045	304	4	𝑟ƥ𝑥	𝑟ƥ𝑥	NOUN
iajs-3045	304	5	⊆	⊆	NUM
iajs-3045	304	6	ɓѡ	ɓѡ	NOUN
iajs-3045	304	7	+	+	CCONJ
iajs-3045	304	8	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	304	9	)	)	PUNCT
iajs-3045	305	1	+	+	PUNCT
iajs-3045	306	1	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	306	2	)	)	PUNCT
iajs-3045	306	3	or	or	CCONJ
iajs-3045	306	4	𝑟𝐽𝑥	𝑟𝐽𝑥	NUM
iajs-3045	306	5	⊆	⊆	NUM
iajs-3045	306	6	ɓѡ	ɓѡ	NOUN
iajs-3045	306	7	+	+	CCONJ
iajs-3045	306	8	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	306	9	)	)	PUNCT
iajs-3045	306	10	+	+	PUNCT
iajs-3045	306	11	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	306	12	)	)	PUNCT
iajs-3045	306	13	or	or	CCONJ
iajs-3045	306	14	ƥ𝐽𝑥	ƥ𝐽𝑥	VERB
iajs-3045	306	15	⊆	⊆	NUM
iajs-3045	306	16	ɓѡ	ɓѡ	X
iajs-3045	306	17	+	+	CCONJ
iajs-3045	306	18	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	306	19	)	)	PUNCT
iajs-3045	306	20	+	+	PUNCT
iajs-3045	306	21	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	306	22	)	)	PUNCT
iajs-3045	306	23	.	.	PUNCT
iajs-3045	307	1	hence	hence	ADV
iajs-3045	307	2	by	by	ADP
iajs-3045	307	3	proposition	proposition	NOUN
iajs-3045	307	4	2.26	2.26	NUM
iajs-3045	307	5	ɓѡ	ɓѡ	NOUN
iajs-3045	307	6	is	be	AUX
iajs-3045	307	7	an	an	DET
iajs-3045	307	8	exnpq2ab	exnpq2ab	PROPN
iajs-3045	307	9	submodule	submodule	NOUN
iajs-3045	307	10	of	of	ADP
iajs-3045	307	11	ѡ.	ѡ.	NOUN
iajs-3045	307	12	(	(	PUNCT
iajs-3045	307	13	⟸	⟸	ADJ
iajs-3045	307	14	)	)	PUNCT
iajs-3045	307	15	let	let	VERB
iajs-3045	307	16	𝑟𝑠𝑡ƥ	𝑟𝑠𝑡ƥ	ADJ
iajs-3045	307	17	⊆	⊆	NUM
iajs-3045	307	18	ɓ	ɓ	NOUN
iajs-3045	307	19	for	for	ADP
iajs-3045	307	20	𝑟	𝑟	NOUN
iajs-3045	307	21	,	,	PUNCT
iajs-3045	307	22	𝑠	𝑠	PROPN
iajs-3045	307	23	,	,	PUNCT
iajs-3045	307	24	𝑡	𝑡	PROPN
iajs-3045	307	25	∈	∈	PROPN
iajs-3045	307	26	ʀ	ʀ	NOUN
iajs-3045	307	27	and	and	CCONJ
iajs-3045	307	28	ƥ	ƥ	DET
iajs-3045	307	29	ideal	ideal	NOUN
iajs-3045	307	30	of	of	ADP
iajs-3045	307	31	ʀ	ʀ	NOUN
iajs-3045	307	32	,	,	PUNCT
iajs-3045	307	33	hence	hence	ADV
iajs-3045	307	34	𝑟𝑠𝑡(ƥѡ	𝑟𝑠𝑡(ƥѡ	NOUN
iajs-3045	307	35	)	)	PUNCT
iajs-3045	307	36	⊆	⊆	NUM
iajs-3045	307	37	ɓѡ	ɓѡ	NOUN
iajs-3045	307	38	,	,	PUNCT
iajs-3045	307	39	but	but	CCONJ
iajs-3045	307	40	ɓѡ	ɓѡ	X
iajs-3045	307	41	is	be	AUX
iajs-3045	307	42	an	an	DET
iajs-3045	307	43	exnpq2ab	exnpq2ab	PROPN
iajs-3045	307	44	submodule	submodule	NOUN
iajs-3045	307	45	of	of	ADP
iajs-3045	307	46	ѡ	ѡ	PROPN
iajs-3045	307	47	,	,	PUNCT
iajs-3045	307	48	then	then	ADV
iajs-3045	307	49	either	either	CCONJ
iajs-3045	307	50	𝑟𝑠(ƥѡ	𝑟𝑠(ƥѡ	NOUN
iajs-3045	307	51	)	)	PUNCT
iajs-3045	307	52	⊆	⊆	NUM
iajs-3045	307	53	ɓѡ	ɓѡ	NOUN
iajs-3045	307	54	+	+	CCONJ
iajs-3045	307	55	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	307	56	)	)	PUNCT
iajs-3045	308	1	+	+	PUNCT
iajs-3045	308	2	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	308	3	)	)	PUNCT
iajs-3045	308	4	or	or	CCONJ
iajs-3045	308	5	𝑟𝑡(ƥѡ	𝑟𝑡(ƥѡ	NOUN
iajs-3045	308	6	)	)	PUNCT
iajs-3045	308	7	⊆	⊆	NUM
iajs-3045	308	8	ɓѡ	ɓѡ	NOUN
iajs-3045	308	9	+	+	CCONJ
iajs-3045	308	10	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	308	11	)	)	PUNCT
iajs-3045	309	1	+	+	PUNCT
iajs-3045	309	2	𝐽(ѡ	𝐽(ѡ	X
iajs-3045	309	3	)	)	PUNCT
iajs-3045	309	4	or	or	CCONJ
iajs-3045	309	5	𝑠𝑡(ƥѡ	𝑠𝑡(ƥѡ	ADJ
iajs-3045	309	6	)	)	PUNCT
iajs-3045	309	7	⊆	⊆	NUM
iajs-3045	309	8	ɓѡ	ɓѡ	NOUN
iajs-3045	309	9	+	+	CCONJ
iajs-3045	309	10	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	309	11	)	)	PUNCT
iajs-3045	309	12	+	+	PUNCT
iajs-3045	309	13	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	309	14	)	)	PUNCT
iajs-3045	309	15	.	.	PUNCT
iajs-3045	310	1	thus	thus	ADV
iajs-3045	310	2	by	by	ADP
iajs-3045	310	3	lemma	lemma	PROPN
iajs-3045	310	4	2.6	2.6	NUM
iajs-3045	310	5	and	and	CCONJ
iajs-3045	310	6	lemma	lemma	PROPN
iajs-3045	310	7	2.7	2.7	NUM
iajs-3045	310	8	either	either	CCONJ
iajs-3045	310	9	𝑟𝑠ƥѡ	𝑟𝑠ƥѡ	NUM
iajs-3045	310	10	⊆	⊆	NUM
iajs-3045	310	11	ɓѡ	ɓѡ	NOUN
iajs-3045	310	12	+	+	CCONJ
iajs-3045	310	13	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	310	14	+	+	CCONJ
iajs-3045	310	15	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	ADJ
iajs-3045	310	16	or	or	CCONJ
iajs-3045	310	17	𝑟𝑡ƥѡ	𝑟𝑡ƥѡ	VERB
iajs-3045	310	18	⊆	⊆	NUM
iajs-3045	310	19	ɓѡ	ɓѡ	NOUN
iajs-3045	310	20	+	+	CCONJ
iajs-3045	310	21	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	310	22	+	+	CCONJ
iajs-3045	310	23	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	ADJ
iajs-3045	310	24	or	or	CCONJ
iajs-3045	310	25	𝑠𝑡ƥѡ	𝑠𝑡ƥѡ	VERB
iajs-3045	310	26	⊆	⊆	NUM
iajs-3045	310	27	ɓѡ	ɓѡ	NOUN
iajs-3045	310	28	+	+	CCONJ
iajs-3045	310	29	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	310	30	+	+	CCONJ
iajs-3045	310	31	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	ADJ
iajs-3045	310	32	,	,	PUNCT
iajs-3045	310	33	hence	hence	ADV
iajs-3045	310	34	either	either	CCONJ
iajs-3045	310	35	𝑟𝑠ƥ	𝑟𝑠ƥ	ADV
iajs-3045	310	36	⊆	⊆	NUM
iajs-3045	310	37	ɓ	ɓ	NOUN
iajs-3045	310	38	+	+	X
iajs-3045	310	39	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	310	40	)	)	PUNCT
iajs-3045	310	41	+	+	NUM
iajs-3045	310	42	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	310	43	)	)	PUNCT
iajs-3045	310	44	or	or	CCONJ
iajs-3045	310	45	𝑟𝑡ƥ	𝑟𝑡ƥ	VERB
iajs-3045	310	46	⊆	⊆	NUM
iajs-3045	310	47	ɓ	ɓ	NOUN
iajs-3045	310	48	+	+	X
iajs-3045	310	49	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	310	50	)	)	PUNCT
iajs-3045	310	51	+	+	NUM
iajs-3045	310	52	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	310	53	)	)	PUNCT
iajs-3045	310	54	or	or	CCONJ
iajs-3045	310	55	𝑠𝑡ƥ	𝑠𝑡ƥ	VERB
iajs-3045	310	56	⊆	⊆	NUM
iajs-3045	310	57	ɓ	ɓ	NOUN
iajs-3045	310	58	+	+	X
iajs-3045	310	59	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	310	60	)	)	PUNCT
iajs-3045	310	61	+	+	NUM
iajs-3045	310	62	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	310	63	)	)	PUNCT
iajs-3045	310	64	.	.	PUNCT
iajs-3045	311	1	therefore	therefore	ADV
iajs-3045	311	2	ɓ	ɓ	PRON
iajs-3045	311	3	is	be	AUX
iajs-3045	311	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	311	5	ideal	ideal	NOUN
iajs-3045	311	6	of	of	ADP
iajs-3045	311	7	ʀ	ʀ	NOUN
iajs-3045	311	8	.	.	PUNCT
iajs-3045	311	9	ihjpas	ihjpas	PROPN
iajs-3045	311	10	.	.	PUNCT
iajs-3045	312	1	36(2)2023	36(2)2023	NUM
iajs-3045	312	2	416	416	NUM
iajs-3045	312	3	proposition	proposition	NOUN
iajs-3045	312	4	4.3	4.3	NUM
iajs-3045	312	5	let	let	VERB
iajs-3045	312	6	ѡ	ѡ	PRON
iajs-3045	312	7	be	be	AUX
iajs-3045	312	8	a	a	DET
iajs-3045	312	9	finitely	finitely	ADV
iajs-3045	312	10	generated	generate	VERB
iajs-3045	312	11	non	non	ADJ
iajs-3045	312	12	-	-	ADJ
iajs-3045	312	13	singular	singular	ADJ
iajs-3045	312	14	multiplication	multiplication	NOUN
iajs-3045	312	15	module	module	NOUN
iajs-3045	312	16	over	over	ADP
iajs-3045	312	17	good	good	ADJ
iajs-3045	312	18	ring	ring	NOUN
iajs-3045	312	19	ʀ	ʀ	NOUN
iajs-3045	312	20	and	and	CCONJ
iajs-3045	312	21	ɓ	ɓ	PRON
iajs-3045	312	22	is	be	AUX
iajs-3045	312	23	an	an	DET
iajs-3045	312	24	ideal	ideal	NOUN
iajs-3045	312	25	of	of	ADP
iajs-3045	312	26	ʀ	ʀ	NOUN
iajs-3045	312	27	with	with	ADP
iajs-3045	312	28	𝑎𝑛𝑛ʀ(ѡ	𝑎𝑛𝑛ʀ(ѡ	NOUN
iajs-3045	312	29	)	)	PUNCT
iajs-3045	312	30	⊆	⊆	NUM
iajs-3045	312	31	ɓ	ɓ	NOUN
iajs-3045	312	32	.	.	PUNCT
iajs-3045	313	1	then	then	ADV
iajs-3045	313	2	ɓ	ɓ	PROPN
iajs-3045	313	3	is	be	AUX
iajs-3045	313	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	313	5	ideal	ideal	NOUN
iajs-3045	313	6	of	of	ADP
iajs-3045	313	7	ʀ	ʀ	PRON
iajs-3045	313	8	if	if	NOUN
iajs-3045	313	9	and	and	CCONJ
iajs-3045	313	10	only	only	ADV
iajs-3045	313	11	if	if	SCONJ
iajs-3045	313	12	ɓѡ	ɓѡ	PROPN
iajs-3045	313	13	is	be	AUX
iajs-3045	313	14	exnpq2ab	exnpq2ab	PROPN
iajs-3045	313	15	submodule	submodule	NOUN
iajs-3045	313	16	of	of	ADP
iajs-3045	313	17	ѡ.	ѡ.	NOUN
iajs-3045	313	18	proof	proof	NOUN
iajs-3045	313	19	.	.	PUNCT
iajs-3045	314	1	(	(	PUNCT
iajs-3045	314	2	⟹	⟹	X
iajs-3045	314	3	)	)	PUNCT
iajs-3045	314	4	let	let	VERB
iajs-3045	314	5	𝑟𝑠ƥ𝐴	𝑟𝑠ƥ𝐴	NOUN
iajs-3045	314	6	⊆	⊆	NUM
iajs-3045	314	7	ɓѡ	ɓѡ	NOUN
iajs-3045	314	8	,	,	PUNCT
iajs-3045	314	9	for	for	ADP
iajs-3045	314	10	𝑟	𝑟	NOUN
iajs-3045	314	11	,	,	PUNCT
iajs-3045	314	12	𝑠	𝑠	PROPN
iajs-3045	314	13	∈	∈	PROPN
iajs-3045	314	14	ʀ	ʀ	PROPN
iajs-3045	314	15	,	,	PUNCT
iajs-3045	314	16	ƥ	ƥ	PRON
iajs-3045	314	17	is	be	AUX
iajs-3045	314	18	an	an	DET
iajs-3045	314	19	ideal	ideal	NOUN
iajs-3045	314	20	of	of	ADP
iajs-3045	314	21	ʀ	ʀ	NOUN
iajs-3045	314	22	and	and	CCONJ
iajs-3045	314	23	𝐴	𝐴	PROPN
iajs-3045	314	24	is	be	AUX
iajs-3045	314	25	a	a	DET
iajs-3045	314	26	submodule	submodule	NOUN
iajs-3045	314	27	of	of	ADP
iajs-3045	314	28	ѡ.	ѡ.	NOUN
iajs-3045	314	29	since	since	SCONJ
iajs-3045	314	30	ѡ	ѡ	PROPN
iajs-3045	314	31	is	be	AUX
iajs-3045	314	32	a	a	DET
iajs-3045	314	33	multiplication	multiplication	NOUN
iajs-3045	314	34	,	,	PUNCT
iajs-3045	314	35	then	then	ADV
iajs-3045	314	36	𝐴	𝐴	PROPN
iajs-3045	314	37	=	=	SYM
iajs-3045	314	38	ƥ1ѡ	ƥ1ѡ	PROPN
iajs-3045	314	39	,	,	PUNCT
iajs-3045	314	40	for	for	ADP
iajs-3045	314	41	some	some	DET
iajs-3045	314	42	ideal	ideal	ADJ
iajs-3045	314	43	ƥ1	ƥ1	NOUN
iajs-3045	314	44	of	of	ADP
iajs-3045	314	45	ʀ	ʀ	NOUN
iajs-3045	314	46	,	,	PUNCT
iajs-3045	314	47	then	then	ADV
iajs-3045	314	48	𝑟𝑠ƥƥ1ѡ	𝑟𝑠ƥƥ1ѡ	PROPN
iajs-3045	314	49	⊆	⊆	NUM
iajs-3045	314	50	ɓѡ.	ɓѡ.	PROPN
iajs-3045	314	51	but	but	CCONJ
iajs-3045	314	52	ѡ	ѡ	PROPN
iajs-3045	314	53	is	be	AUX
iajs-3045	314	54	a	a	DET
iajs-3045	314	55	finitely	finitely	ADV
iajs-3045	314	56	generated	generate	VERB
iajs-3045	314	57	multiplication	multiplication	NOUN
iajs-3045	314	58	ʀ	ʀ	NOUN
iajs-3045	314	59	-	-	PUNCT
iajs-3045	314	60	module	module	NOUN
iajs-3045	314	61	then	then	ADV
iajs-3045	314	62	by	by	ADP
iajs-3045	314	63	lemma	lemma	PROPN
iajs-3045	314	64	2.18	2.18	NUM
iajs-3045	314	65	𝑟𝑠ƥƥ1	𝑟𝑠ƥƥ1	NOUN
iajs-3045	314	66	⊆	⊆	NUM
iajs-3045	314	67	ɓ	ɓ	NOUN
iajs-3045	314	68	+	+	X
iajs-3045	314	69	𝑎𝑛𝑛ʀ(ѡ	𝑎𝑛𝑛ʀ(ѡ	NOUN
iajs-3045	314	70	)	)	PUNCT
iajs-3045	314	71	,	,	PUNCT
iajs-3045	314	72	since	since	SCONJ
iajs-3045	314	73	𝑎𝑛𝑛ʀ(ѡ	𝑎𝑛𝑛ʀ(ѡ	NOUN
iajs-3045	314	74	)	)	PUNCT
iajs-3045	314	75	⊆	⊆	NUM
iajs-3045	314	76	ɓ	ɓ	NUM
iajs-3045	314	77	,	,	PUNCT
iajs-3045	314	78	implies	imply	VERB
iajs-3045	314	79	that	that	SCONJ
iajs-3045	314	80	ɓ	ɓ	PRON
iajs-3045	314	81	+	+	X
iajs-3045	314	82	𝑎𝑛𝑛ʀ(ѡ	𝑎𝑛𝑛ʀ(ѡ	NOUN
iajs-3045	314	83	)	)	PUNCT
iajs-3045	314	84	=	=	SYM
iajs-3045	314	85	ɓ	ɓ	NOUN
iajs-3045	314	86	,	,	PUNCT
iajs-3045	314	87	hence	hence	ADV
iajs-3045	314	88	𝑟𝑠ƥƥ1	𝑟𝑠ƥƥ1	NOUN
iajs-3045	314	89	⊆	⊆	NUM
iajs-3045	314	90	ɓ	ɓ	NOUN
iajs-3045	314	91	.	.	PUNCT
iajs-3045	315	1	but	but	CCONJ
iajs-3045	315	2	ɓ	ɓ	PRON
iajs-3045	315	3	is	be	AUX
iajs-3045	315	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	315	5	ideal	ideal	NOUN
iajs-3045	315	6	of	of	ADP
iajs-3045	315	7	ʀ	ʀ	PROPN
iajs-3045	315	8	then	then	ADV
iajs-3045	315	9	by	by	ADP
iajs-3045	315	10	proposition	proposition	NOUN
iajs-3045	315	11	2.28	2.28	NUM
iajs-3045	315	12	either	either	CCONJ
iajs-3045	315	13	𝑟𝑠ƥ1	𝑟𝑠ƥ1	PROPN
iajs-3045	315	14	⊆	⊆	NUM
iajs-3045	315	15	ɓ	ɓ	NOUN
iajs-3045	315	16	+	+	CCONJ
iajs-3045	315	17	(	(	PUNCT
iajs-3045	315	18	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	PROPN
iajs-3045	315	19	)	)	PUNCT
iajs-3045	315	20	+	+	NUM
iajs-3045	315	21	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	315	22	)	)	PUNCT
iajs-3045	315	23	)	)	PUNCT
iajs-3045	315	24	or	or	CCONJ
iajs-3045	315	25	𝑟ƥƥ1	𝑟ƥƥ1	VERB
iajs-3045	315	26	⊆	⊆	NUM
iajs-3045	315	27	ɓ	ɓ	NOUN
iajs-3045	315	28	+	+	CCONJ
iajs-3045	315	29	(	(	PUNCT
iajs-3045	315	30	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	PROPN
iajs-3045	315	31	)	)	PUNCT
iajs-3045	315	32	+	+	NUM
iajs-3045	316	1	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	316	2	)	)	PUNCT
iajs-3045	316	3	)	)	PUNCT
iajs-3045	316	4	or	or	CCONJ
iajs-3045	316	5	𝑠ƥƥ1	𝑠ƥƥ1	PROPN
iajs-3045	316	6	⊆	⊆	NUM
iajs-3045	316	7	ɓ	ɓ	NOUN
iajs-3045	316	8	+	+	CCONJ
iajs-3045	316	9	(	(	PUNCT
iajs-3045	316	10	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	PROPN
iajs-3045	316	11	)	)	PUNCT
iajs-3045	316	12	+	+	NUM
iajs-3045	316	13	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	316	14	)	)	PUNCT
iajs-3045	316	15	)	)	PUNCT
iajs-3045	316	16	.	.	PUNCT
iajs-3045	317	1	thus	thus	ADV
iajs-3045	317	2	either	either	CCONJ
iajs-3045	317	3	𝑟𝑠ƥ1ѡ	𝑟𝑠ƥ1ѡ	PROPN
iajs-3045	317	4	⊆	⊆	NUM
iajs-3045	317	5	ɓѡ	ɓѡ	NOUN
iajs-3045	317	6	+	+	CCONJ
iajs-3045	317	7	(	(	PUNCT
iajs-3045	317	8	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	317	9	+	+	CCONJ
iajs-3045	317	10	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	NOUN
iajs-3045	317	11	)	)	PUNCT
iajs-3045	317	12	or	or	CCONJ
iajs-3045	317	13	𝑟ƥƥ1ѡ	𝑟ƥƥ1ѡ	NUM
iajs-3045	317	14	⊆	⊆	NUM
iajs-3045	317	15	ɓѡ	ɓѡ	NOUN
iajs-3045	317	16	+	+	CCONJ
iajs-3045	317	17	(	(	PUNCT
iajs-3045	317	18	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	317	19	+	+	CCONJ
iajs-3045	317	20	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	NOUN
iajs-3045	317	21	)	)	PUNCT
iajs-3045	317	22	or	or	CCONJ
iajs-3045	317	23	𝑠ƥƥ1ѡ	𝑠ƥƥ1ѡ	NUM
iajs-3045	317	24	⊆	⊆	NUM
iajs-3045	317	25	ɓѡ	ɓѡ	NOUN
iajs-3045	317	26	+	+	CCONJ
iajs-3045	317	27	(	(	PUNCT
iajs-3045	317	28	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	317	29	+	+	CCONJ
iajs-3045	317	30	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	NOUN
iajs-3045	317	31	)	)	PUNCT
iajs-3045	317	32	.	.	PUNCT
iajs-3045	318	1	since	since	SCONJ
iajs-3045	318	2	ѡ	ѡ	PROPN
iajs-3045	318	3	is	be	AUX
iajs-3045	318	4	non	non	ADJ
iajs-3045	318	5	-	-	ADJ
iajs-3045	318	6	singular	singular	ADJ
iajs-3045	318	7	,	,	PUNCT
iajs-3045	318	8	then	then	ADV
iajs-3045	318	9	by	by	ADP
iajs-3045	318	10	lemma	lemma	PROPN
iajs-3045	318	11	2.17	2.17	NUM
iajs-3045	318	12	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	318	13	=	=	SYM
iajs-3045	318	14	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	318	15	)	)	PUNCT
iajs-3045	318	16	and	and	CCONJ
iajs-3045	318	17	ʀ	ʀ	NOUN
iajs-3045	318	18	is	be	AUX
iajs-3045	318	19	good	good	ADJ
iajs-3045	318	20	ring	ring	NOUN
iajs-3045	318	21	then	then	ADV
iajs-3045	318	22	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	VERB
iajs-3045	318	23	=	=	PUNCT
iajs-3045	318	24	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	318	25	)	)	PUNCT
iajs-3045	318	26	.	.	PUNCT
iajs-3045	319	1	hence	hence	ADV
iajs-3045	319	2	either	either	CCONJ
iajs-3045	319	3	𝑟𝑠ƥ1ѡ	𝑟𝑠ƥ1ѡ	PROPN
iajs-3045	319	4	⊆	⊆	NUM
iajs-3045	319	5	ɓѡ	ɓѡ	NOUN
iajs-3045	319	6	+	+	CCONJ
iajs-3045	319	7	(	(	PUNCT
iajs-3045	319	8	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	319	9	)	)	PUNCT
iajs-3045	319	10	+	+	PUNCT
iajs-3045	319	11	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	319	12	)	)	PUNCT
iajs-3045	319	13	)	)	PUNCT
iajs-3045	319	14	or	or	CCONJ
iajs-3045	319	15	𝑟ƥƥ1ѡ	𝑟ƥƥ1ѡ	NUM
iajs-3045	319	16	⊆	⊆	NUM
iajs-3045	319	17	ɓѡ	ɓѡ	NOUN
iajs-3045	319	18	+	+	CCONJ
iajs-3045	319	19	(	(	PUNCT
iajs-3045	319	20	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	319	21	)	)	PUNCT
iajs-3045	319	22	+	+	PUNCT
iajs-3045	319	23	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	319	24	)	)	PUNCT
iajs-3045	319	25	)	)	PUNCT
iajs-3045	319	26	or	or	CCONJ
iajs-3045	319	27	𝑠ƥƥ1ѡ	𝑠ƥƥ1ѡ	NUM
iajs-3045	319	28	⊆	⊆	NUM
iajs-3045	319	29	ɓѡ	ɓѡ	NOUN
iajs-3045	319	30	+	+	CCONJ
iajs-3045	319	31	(	(	PUNCT
iajs-3045	319	32	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	319	33	)	)	PUNCT
iajs-3045	319	34	+	+	PUNCT
iajs-3045	319	35	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	319	36	)	)	PUNCT
iajs-3045	319	37	)	)	PUNCT
iajs-3045	319	38	.	.	PUNCT
iajs-3045	320	1	that	that	PRON
iajs-3045	320	2	is	be	AUX
iajs-3045	320	3	either	either	CCONJ
iajs-3045	320	4	𝑟𝑠𝐴	𝑟𝑠𝐴	PROPN
iajs-3045	320	5	⊆	⊆	NUM
iajs-3045	320	6	ɓѡ	ɓѡ	NOUN
iajs-3045	321	1	+	+	CCONJ
iajs-3045	321	2	(	(	PUNCT
iajs-3045	321	3	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	321	4	)	)	PUNCT
iajs-3045	321	5	+	+	PUNCT
iajs-3045	321	6	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	321	7	)	)	PUNCT
iajs-3045	321	8	)	)	PUNCT
iajs-3045	321	9	or	or	CCONJ
iajs-3045	321	10	𝑟ƥ𝐴	𝑟ƥ𝐴	PROPN
iajs-3045	321	11	⊆	⊆	NUM
iajs-3045	321	12	ɓѡ	ɓѡ	NOUN
iajs-3045	321	13	+	+	CCONJ
iajs-3045	321	14	(	(	PUNCT
iajs-3045	321	15	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	321	16	)	)	PUNCT
iajs-3045	321	17	+	+	PUNCT
iajs-3045	321	18	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	321	19	)	)	PUNCT
iajs-3045	321	20	)	)	PUNCT
iajs-3045	321	21	or	or	CCONJ
iajs-3045	321	22	𝑠ƥ𝐴	𝑠ƥ𝐴	NOUN
iajs-3045	321	23	⊆	⊆	NUM
iajs-3045	321	24	ɓѡ	ɓѡ	NOUN
iajs-3045	321	25	+	+	CCONJ
iajs-3045	321	26	(	(	PUNCT
iajs-3045	321	27	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	321	28	)	)	PUNCT
iajs-3045	321	29	+	+	PUNCT
iajs-3045	321	30	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	321	31	)	)	PUNCT
iajs-3045	321	32	)	)	PUNCT
iajs-3045	321	33	.	.	PUNCT
iajs-3045	322	1	therefore	therefore	ADV
iajs-3045	322	2	by	by	ADP
iajs-3045	322	3	proposition	proposition	NOUN
iajs-3045	322	4	2.28	2.28	NUM
iajs-3045	322	5	ɓѡ	ɓѡ	NOUN
iajs-3045	322	6	is	be	AUX
iajs-3045	322	7	exnpq2ab	exnpq2ab	PROPN
iajs-3045	322	8	submodule	submodule	NOUN
iajs-3045	322	9	of	of	ADP
iajs-3045	322	10	ѡ.	ѡ.	NOUN
iajs-3045	322	11	(	(	PUNCT
iajs-3045	322	12	⟸	⟸	ADV
iajs-3045	322	13	)	)	PUNCT
iajs-3045	322	14	let	let	VERB
iajs-3045	322	15	𝑟ƥ1ƥ2ƥ3	𝑟ƥ1ƥ2ƥ3	PROPN
iajs-3045	322	16	⊆	⊆	NUM
iajs-3045	322	17	ɓ	ɓ	NOUN
iajs-3045	322	18	,	,	PUNCT
iajs-3045	322	19	for	for	ADP
iajs-3045	322	20	𝑟	𝑟	DET
iajs-3045	322	21	∈	∈	PROPN
iajs-3045	322	22	ʀ	ʀ	NOUN
iajs-3045	322	23	,	,	PUNCT
iajs-3045	322	24	and	and	CCONJ
iajs-3045	322	25	ƥ1	ƥ1	NOUN
iajs-3045	322	26	,	,	PUNCT
iajs-3045	322	27	ƥ2	ƥ2	NOUN
iajs-3045	322	28	,	,	PUNCT
iajs-3045	322	29	ƥ3	ƥ3	PROPN
iajs-3045	322	30	are	be	AUX
iajs-3045	322	31	ideals	ideal	NOUN
iajs-3045	322	32	of	of	ADP
iajs-3045	322	33	ʀ	ʀ	NOUN
iajs-3045	322	34	,	,	PUNCT
iajs-3045	322	35	implies	imply	VERB
iajs-3045	322	36	that	that	SCONJ
iajs-3045	322	37	𝑟ƥ1ƥ2(ƥ3ѡ	𝑟ƥ1ƥ2(ƥ3ѡ	NOUN
iajs-3045	322	38	)	)	PUNCT
iajs-3045	322	39	⊆	⊆	NUM
iajs-3045	322	40	ɓѡ.	ɓѡ.	NOUN
iajs-3045	322	41	since	since	SCONJ
iajs-3045	322	42	ɓѡ	ɓѡ	PROPN
iajs-3045	322	43	is	be	AUX
iajs-3045	322	44	exnpq2ab	exnpq2ab	PROPN
iajs-3045	322	45	submodule	submodule	NOUN
iajs-3045	322	46	of	of	ADP
iajs-3045	322	47	ѡ	ѡ	PROPN
iajs-3045	322	48	,	,	PUNCT
iajs-3045	322	49	then	then	ADV
iajs-3045	322	50	by	by	ADP
iajs-3045	322	51	proposition	proposition	NOUN
iajs-3045	322	52	2.29	2.29	NUM
iajs-3045	322	53	either	either	CCONJ
iajs-3045	322	54	𝑟ƥ1(ƥ3ѡ	𝑟ƥ1(ƥ3ѡ	PROPN
iajs-3045	322	55	)	)	PUNCT
iajs-3045	322	56	⊆	⊆	NUM
iajs-3045	322	57	ɓѡ	ɓѡ	NOUN
iajs-3045	322	58	+	+	CCONJ
iajs-3045	322	59	(	(	PUNCT
iajs-3045	322	60	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	322	61	)	)	PUNCT
iajs-3045	322	62	+	+	PUNCT
iajs-3045	322	63	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	322	64	)	)	PUNCT
iajs-3045	322	65	)	)	PUNCT
iajs-3045	322	66	or	or	CCONJ
iajs-3045	322	67	𝑟ƥ2(ƥ3ѡ	𝑟ƥ2(ƥ3ѡ	NOUN
iajs-3045	322	68	)	)	PUNCT
iajs-3045	322	69	⊆	⊆	NUM
iajs-3045	322	70	ɓѡ	ɓѡ	NOUN
iajs-3045	322	71	+	+	CCONJ
iajs-3045	322	72	(	(	PUNCT
iajs-3045	322	73	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	322	74	)	)	PUNCT
iajs-3045	322	75	+	+	PUNCT
iajs-3045	323	1	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	323	2	)	)	PUNCT
iajs-3045	323	3	)	)	PUNCT
iajs-3045	323	4	or	or	CCONJ
iajs-3045	323	5	ƥ1ƥ2(ƥ3ѡ	ƥ1ƥ2(ƥ3ѡ	NOUN
iajs-3045	323	6	)	)	PUNCT
iajs-3045	323	7	⊆	⊆	NUM
iajs-3045	323	8	ɓѡ	ɓѡ	NOUN
iajs-3045	323	9	+	+	CCONJ
iajs-3045	323	10	(	(	PUNCT
iajs-3045	323	11	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	323	12	)	)	PUNCT
iajs-3045	323	13	+	+	PUNCT
iajs-3045	324	1	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	324	2	)	)	PUNCT
iajs-3045	324	3	)	)	PUNCT
iajs-3045	324	4	.	.	PUNCT
iajs-3045	325	1	but	but	CCONJ
iajs-3045	325	2	ѡ	ѡ	PROPN
iajs-3045	325	3	is	be	AUX
iajs-3045	325	4	non	non	ADJ
iajs-3045	325	5	-	-	ADJ
iajs-3045	325	6	singular	singular	ADJ
iajs-3045	325	7	and	and	CCONJ
iajs-3045	325	8	ʀ	ʀ	NOUN
iajs-3045	325	9	is	be	AUX
iajs-3045	325	10	good	good	ADJ
iajs-3045	325	11	ring	ring	NOUN
iajs-3045	325	12	then	then	ADV
iajs-3045	325	13	(	(	PUNCT
iajs-3045	325	14	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	325	15	)	)	PUNCT
iajs-3045	326	1	+	+	PUNCT
iajs-3045	327	1	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	327	2	)	)	PUNCT
iajs-3045	327	3	)	)	PUNCT
iajs-3045	328	1	=	=	PRON
iajs-3045	328	2	(	(	PUNCT
iajs-3045	328	3	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	328	4	+	+	CCONJ
iajs-3045	328	5	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	NOUN
iajs-3045	328	6	)	)	PUNCT
iajs-3045	328	7	.	.	PUNCT
iajs-3045	329	1	thus	thus	ADV
iajs-3045	329	2	either	either	CCONJ
iajs-3045	329	3	𝑟ƥ1ƥ3ѡ	𝑟ƥ1ƥ3ѡ	PROPN
iajs-3045	329	4	⊆	⊆	NUM
iajs-3045	329	5	ɓѡ	ɓѡ	NOUN
iajs-3045	329	6	+	+	CCONJ
iajs-3045	329	7	(	(	PUNCT
iajs-3045	329	8	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	329	9	+	+	CCONJ
iajs-3045	329	10	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	NOUN
iajs-3045	329	11	)	)	PUNCT
iajs-3045	329	12	or	or	CCONJ
iajs-3045	329	13	𝑟ƥ2ƥ3ѡ	𝑟ƥ2ƥ3ѡ	PROPN
iajs-3045	329	14	⊆	⊆	NUM
iajs-3045	329	15	ɓѡ	ɓѡ	NOUN
iajs-3045	329	16	+	+	CCONJ
iajs-3045	329	17	(	(	PUNCT
iajs-3045	329	18	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	329	19	+	+	CCONJ
iajs-3045	329	20	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	NOUN
iajs-3045	329	21	)	)	PUNCT
iajs-3045	329	22	or	or	CCONJ
iajs-3045	329	23	ƥ1ƥ2ƥ3ѡ	ƥ1ƥ2ƥ3ѡ	PROPN
iajs-3045	329	24	⊆	⊆	NUM
iajs-3045	329	25	ɓѡ	ɓѡ	NOUN
iajs-3045	329	26	+	+	CCONJ
iajs-3045	329	27	(	(	PUNCT
iajs-3045	329	28	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	329	29	+	+	CCONJ
iajs-3045	329	30	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	NOUN
iajs-3045	329	31	)	)	PUNCT
iajs-3045	329	32	,	,	PUNCT
iajs-3045	329	33	then	then	ADV
iajs-3045	329	34	either	either	CCONJ
iajs-3045	329	35	𝑟ƥ1ƥ3	𝑟ƥ1ƥ3	PROPN
iajs-3045	329	36	⊆	⊆	NUM
iajs-3045	329	37	ɓ	ɓ	PRON
iajs-3045	329	38	+	+	X
iajs-3045	329	39	(	(	PUNCT
iajs-3045	329	40	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	PROPN
iajs-3045	329	41	)	)	PUNCT
iajs-3045	329	42	+	+	NUM
iajs-3045	329	43	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	329	44	)	)	PUNCT
iajs-3045	329	45	)	)	PUNCT
iajs-3045	329	46	or	or	CCONJ
iajs-3045	329	47	𝑟ƥ2ƥ3	𝑟ƥ2ƥ3	PROPN
iajs-3045	329	48	⊆	⊆	NUM
iajs-3045	329	49	ɓ	ɓ	DET
iajs-3045	329	50	+	+	CCONJ
iajs-3045	329	51	(	(	PUNCT
iajs-3045	329	52	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	PROPN
iajs-3045	329	53	)	)	PUNCT
iajs-3045	329	54	+	+	NUM
iajs-3045	329	55	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	329	56	)	)	PUNCT
iajs-3045	329	57	)	)	PUNCT
iajs-3045	329	58	or	or	CCONJ
iajs-3045	329	59	ƥ1ƥ2ƥ3	ƥ1ƥ2ƥ3	ADV
iajs-3045	329	60	⊆	⊆	NUM
iajs-3045	329	61	ɓ	ɓ	NOUN
iajs-3045	329	62	+	+	X
iajs-3045	329	63	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	329	64	)	)	PUNCT
iajs-3045	329	65	+	+	NUM
iajs-3045	329	66	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	329	67	)	)	PUNCT
iajs-3045	329	68	.	.	PUNCT
iajs-3045	330	1	hence	hence	ADV
iajs-3045	330	2	by	by	ADP
iajs-3045	330	3	proposition	proposition	NOUN
iajs-3045	330	4	2.28	2.28	NUM
iajs-3045	330	5	ɓ	ɓ	PRON
iajs-3045	330	6	is	be	AUX
iajs-3045	330	7	exnpq2ab	exnpq2ab	PROPN
iajs-3045	330	8	ideal	ideal	NOUN
iajs-3045	330	9	of	of	ADP
iajs-3045	330	10	ʀ	ʀ	NOUN
iajs-3045	330	11	.	.	NOUN
iajs-3045	330	12	corollary	corollary	NOUN
iajs-3045	330	13	4.4	4.4	NUM
iajs-3045	330	14	let	let	VERB
iajs-3045	330	15	ѡ	ѡ	PRON
iajs-3045	330	16	be	be	AUX
iajs-3045	330	17	a	a	DET
iajs-3045	330	18	finitely	finitely	ADV
iajs-3045	330	19	generated	generate	VERB
iajs-3045	330	20	non	non	ADJ
iajs-3045	330	21	-	-	ADJ
iajs-3045	330	22	singular	singular	ADJ
iajs-3045	330	23	multiplication	multiplication	NOUN
iajs-3045	330	24	module	module	NOUN
iajs-3045	330	25	over	over	ADP
iajs-3045	330	26	artinian	artinian	ADJ
iajs-3045	330	27	ring	ring	NOUN
iajs-3045	330	28	ʀ	ʀ	NOUN
iajs-3045	330	29	and	and	CCONJ
iajs-3045	330	30	ɓ	ɓ	PRON
iajs-3045	330	31	is	be	AUX
iajs-3045	330	32	an	an	DET
iajs-3045	330	33	ideal	ideal	NOUN
iajs-3045	330	34	of	of	ADP
iajs-3045	330	35	ʀ	ʀ	NOUN
iajs-3045	330	36	with	with	ADP
iajs-3045	330	37	𝑎𝑛𝑛ʀ(ѡ	𝑎𝑛𝑛ʀ(ѡ	NOUN
iajs-3045	330	38	)	)	PUNCT
iajs-3045	330	39	⊆	⊆	NUM
iajs-3045	330	40	ɓ	ɓ	NOUN
iajs-3045	330	41	.	.	PUNCT
iajs-3045	331	1	then	then	ADV
iajs-3045	331	2	ɓ	ɓ	PROPN
iajs-3045	331	3	is	be	AUX
iajs-3045	331	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	331	5	ideal	ideal	NOUN
iajs-3045	331	6	of	of	ADP
iajs-3045	331	7	ʀ	ʀ	PRON
iajs-3045	331	8	if	if	NOUN
iajs-3045	331	9	and	and	CCONJ
iajs-3045	331	10	only	only	ADV
iajs-3045	331	11	if	if	SCONJ
iajs-3045	331	12	ɓѡ	ɓѡ	PROPN
iajs-3045	331	13	is	be	AUX
iajs-3045	331	14	exnpq2ab	exnpq2ab	PROPN
iajs-3045	331	15	submodule	submodule	NOUN
iajs-3045	331	16	of	of	ADP
iajs-3045	331	17	ѡ.	ѡ.	NOUN
iajs-3045	331	18	proposition	proposition	NOUN
iajs-3045	331	19	4.5	4.5	NUM
iajs-3045	331	20	let	let	VERB
iajs-3045	331	21	ѡ	ѡ	PRON
iajs-3045	331	22	be	be	AUX
iajs-3045	331	23	a	a	DET
iajs-3045	331	24	finitely	finitely	ADV
iajs-3045	331	25	generated	generate	VERB
iajs-3045	331	26	non	non	ADJ
iajs-3045	331	27	-	-	ADJ
iajs-3045	331	28	singular	singular	ADJ
iajs-3045	331	29	multiplication	multiplication	NOUN
iajs-3045	331	30	module	module	NOUN
iajs-3045	331	31	over	over	ADP
iajs-3045	331	32	local	local	ADJ
iajs-3045	331	33	ring	ring	NOUN
iajs-3045	331	34	ʀ	ʀ	NOUN
iajs-3045	331	35	and	and	CCONJ
iajs-3045	331	36	ɓ	ɓ	PRON
iajs-3045	331	37	is	be	AUX
iajs-3045	331	38	an	an	DET
iajs-3045	331	39	ideal	ideal	NOUN
iajs-3045	331	40	of	of	ADP
iajs-3045	331	41	ʀ	ʀ	NOUN
iajs-3045	331	42	with	with	ADP
iajs-3045	331	43	𝑎𝑛𝑛ʀ(ѡ	𝑎𝑛𝑛ʀ(ѡ	NOUN
iajs-3045	331	44	)	)	PUNCT
iajs-3045	331	45	⊆	⊆	NUM
iajs-3045	331	46	ɓ	ɓ	NOUN
iajs-3045	331	47	.	.	PUNCT
iajs-3045	332	1	then	then	ADV
iajs-3045	332	2	ɓ	ɓ	PROPN
iajs-3045	332	3	is	be	AUX
iajs-3045	332	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	332	5	ideal	ideal	NOUN
iajs-3045	332	6	of	of	ADP
iajs-3045	332	7	ʀ	ʀ	PRON
iajs-3045	332	8	if	if	NOUN
iajs-3045	332	9	and	and	CCONJ
iajs-3045	332	10	only	only	ADV
iajs-3045	332	11	if	if	SCONJ
iajs-3045	332	12	ɓѡ	ɓѡ	PROPN
iajs-3045	332	13	is	be	AUX
iajs-3045	332	14	exnpq2ab	exnpq2ab	PROPN
iajs-3045	332	15	submodule	submodule	NOUN
iajs-3045	332	16	of	of	ADP
iajs-3045	332	17	ѡ.	ѡ.	NOUN
iajs-3045	332	18	proof	proof	NOUN
iajs-3045	332	19	.	.	PUNCT
iajs-3045	333	1	similarly	similarly	ADV
iajs-3045	333	2	to	to	ADP
iajs-3045	333	3	the	the	DET
iajs-3045	333	4	proof	proof	NOUN
iajs-3045	333	5	of	of	ADP
iajs-3045	333	6	proposition	proposition	NOUN
iajs-3045	333	7	4.3	4.3	NUM
iajs-3045	333	8	by	by	ADP
iajs-3045	333	9	using	use	VERB
iajs-3045	333	10	lemma	lemma	PROPN
iajs-3045	333	11	2.15	2.15	NUM
iajs-3045	333	12	.	.	PUNCT
iajs-3045	334	1	proposition	proposition	NOUN
iajs-3045	334	2	4.6	4.6	NUM
iajs-3045	334	3	let	let	VERB
iajs-3045	334	4	ѡ	ѡ	PRON
iajs-3045	334	5	be	be	AUX
iajs-3045	334	6	a	a	DET
iajs-3045	334	7	finitely	finitely	ADV
iajs-3045	334	8	generated	generate	VERB
iajs-3045	334	9	multiplication	multiplication	NOUN
iajs-3045	334	10	𝑍-regular	𝑍-regular	ADJ
iajs-3045	334	11	module	module	NOUN
iajs-3045	334	12	over	over	ADP
iajs-3045	334	13	good	good	ADJ
iajs-3045	334	14	ring	ring	NOUN
iajs-3045	334	15	ʀ	ʀ	NOUN
iajs-3045	334	16	and	and	CCONJ
iajs-3045	334	17	ɓ	ɓ	PRON
iajs-3045	334	18	is	be	AUX
iajs-3045	334	19	an	an	DET
iajs-3045	334	20	ideal	ideal	NOUN
iajs-3045	334	21	of	of	ADP
iajs-3045	334	22	ʀ	ʀ	NOUN
iajs-3045	334	23	with	with	ADP
iajs-3045	334	24	𝑎𝑛𝑛ʀ(ѡ	𝑎𝑛𝑛ʀ(ѡ	NOUN
iajs-3045	334	25	)	)	PUNCT
iajs-3045	334	26	⊆	⊆	NUM
iajs-3045	334	27	ɓ	ɓ	NOUN
iajs-3045	334	28	.	.	PUNCT
iajs-3045	335	1	then	then	ADV
iajs-3045	335	2	ɓ	ɓ	PROPN
iajs-3045	335	3	is	be	AUX
iajs-3045	335	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	335	5	ideal	ideal	NOUN
iajs-3045	335	6	of	of	ADP
iajs-3045	335	7	ʀ	ʀ	PRON
iajs-3045	335	8	if	if	NOUN
iajs-3045	335	9	and	and	CCONJ
iajs-3045	335	10	only	only	ADV
iajs-3045	335	11	if	if	SCONJ
iajs-3045	335	12	ɓѡ	ɓѡ	PROPN
iajs-3045	335	13	is	be	AUX
iajs-3045	335	14	exnpq2ab	exnpq2ab	PROPN
iajs-3045	335	15	submodule	submodule	NOUN
iajs-3045	335	16	of	of	ADP
iajs-3045	335	17	ѡ.	ѡ.	NOUN
iajs-3045	335	18	ihjpas	ihjpas	PROPN
iajs-3045	335	19	.	.	PUNCT
iajs-3045	336	1	36(2)2023	36(2)2023	NUM
iajs-3045	336	2	417	417	NUM
iajs-3045	336	3	proof	proof	NOUN
iajs-3045	336	4	.	.	PUNCT
iajs-3045	337	1	(	(	PUNCT
iajs-3045	337	2	⟹	⟹	X
iajs-3045	337	3	)	)	PUNCT
iajs-3045	337	4	let	let	VERB
iajs-3045	337	5	𝑟𝑠𝑡𝑥	𝑟𝑠𝑡𝑥	NOUN
iajs-3045	337	6	∈	∈	NOUN
iajs-3045	337	7	ɓѡ	ɓѡ	NOUN
iajs-3045	337	8	for	for	ADP
iajs-3045	337	9	𝑟	𝑟	NOUN
iajs-3045	337	10	,	,	PUNCT
iajs-3045	337	11	𝑠	𝑠	PROPN
iajs-3045	337	12	,	,	PUNCT
iajs-3045	337	13	𝑡	𝑡	PROPN
iajs-3045	337	14	∈	∈	PROPN
iajs-3045	337	15	ʀ	ʀ	NOUN
iajs-3045	337	16	and	and	CCONJ
iajs-3045	337	17	𝑥	𝑥	ADP
iajs-3045	337	18	∈	∈	ADJ
iajs-3045	337	19	ѡ	ѡ	AUX
iajs-3045	337	20	,	,	PUNCT
iajs-3045	337	21	that	that	PRON
iajs-3045	337	22	is	be	AUX
iajs-3045	337	23	𝑟𝑠𝑡〈𝑥	𝑟𝑠𝑡〈𝑥	NUM
iajs-3045	337	24	〉	〉	NOUN
iajs-3045	337	25	⊆	⊆	NUM
iajs-3045	337	26	ɓѡ.	ɓѡ.	NOUN
iajs-3045	337	27	since	since	SCONJ
iajs-3045	337	28	ѡ	ѡ	PROPN
iajs-3045	337	29	is	be	AUX
iajs-3045	337	30	a	a	DET
iajs-3045	337	31	multiplication	multiplication	NOUN
iajs-3045	337	32	,	,	PUNCT
iajs-3045	337	33	then	then	ADV
iajs-3045	337	34	〈	〈	PROPN
iajs-3045	337	35	𝑥	𝑥	NOUN
iajs-3045	337	36	〉	〉	NOUN
iajs-3045	337	37	=	=	SYM
iajs-3045	337	38	ƥѡ	ƥѡ	NOUN
iajs-3045	337	39	for	for	ADP
iajs-3045	337	40	some	some	DET
iajs-3045	337	41	ideal	ideal	ADJ
iajs-3045	337	42	ƥ	ƥ	NOUN
iajs-3045	337	43	of	of	ADP
iajs-3045	337	44	ʀ	ʀ	NOUN
iajs-3045	337	45	,	,	PUNCT
iajs-3045	337	46	that	that	PRON
iajs-3045	337	47	is	be	AUX
iajs-3045	337	48	𝑟𝑠𝑡ƥѡ	𝑟𝑠𝑡ƥѡ	ADJ
iajs-3045	337	49	⊆	⊆	NUM
iajs-3045	337	50	ɓѡ.	ɓѡ.	PROPN
iajs-3045	338	1	but	but	CCONJ
iajs-3045	338	2	ѡ	ѡ	PROPN
iajs-3045	338	3	is	be	AUX
iajs-3045	338	4	a	a	DET
iajs-3045	338	5	finitely	finitely	ADV
iajs-3045	338	6	generated	generate	VERB
iajs-3045	338	7	multiplication	multiplication	NOUN
iajs-3045	338	8	ʀ	ʀ	NOUN
iajs-3045	338	9	-	-	PUNCT
iajs-3045	338	10	module	module	NOUN
iajs-3045	338	11	then	then	ADV
iajs-3045	338	12	by	by	ADP
iajs-3045	338	13	lemma	lemma	PROPN
iajs-3045	338	14	2.18	2.18	NUM
iajs-3045	338	15	𝑟𝑠𝑡ƥ	𝑟𝑠𝑡ƥ	NOUN
iajs-3045	338	16	⊆	⊆	NUM
iajs-3045	338	17	ɓ	ɓ	NOUN
iajs-3045	338	18	.	.	PUNCT
iajs-3045	339	1	but	but	CCONJ
iajs-3045	339	2	ɓ	ɓ	PRON
iajs-3045	339	3	is	be	AUX
iajs-3045	339	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	339	5	ideal	ideal	NOUN
iajs-3045	339	6	of	of	ADP
iajs-3045	339	7	ʀ	ʀ	PRON
iajs-3045	339	8	then	then	ADV
iajs-3045	339	9	by	by	ADP
iajs-3045	339	10	proposition	proposition	NOUN
iajs-3045	339	11	2.24	2.24	NUM
iajs-3045	339	12	either	either	CCONJ
iajs-3045	339	13	𝑟𝑠ƥ	𝑟𝑠ƥ	ADV
iajs-3045	339	14	⊆	⊆	NUM
iajs-3045	339	15	ɓ	ɓ	NOUN
iajs-3045	339	16	+	+	CCONJ
iajs-3045	339	17	(	(	PUNCT
iajs-3045	339	18	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	PROPN
iajs-3045	339	19	)	)	PUNCT
iajs-3045	339	20	+	+	NUM
iajs-3045	339	21	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	339	22	)	)	PUNCT
iajs-3045	339	23	)	)	PUNCT
iajs-3045	339	24	or	or	CCONJ
iajs-3045	339	25	𝑟𝑡ƥ	𝑟𝑡ƥ	VERB
iajs-3045	339	26	⊆	⊆	NUM
iajs-3045	339	27	ɓ	ɓ	NOUN
iajs-3045	339	28	+	+	CCONJ
iajs-3045	339	29	(	(	PUNCT
iajs-3045	339	30	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	PROPN
iajs-3045	339	31	)	)	PUNCT
iajs-3045	339	32	+	+	NUM
iajs-3045	339	33	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	339	34	)	)	PUNCT
iajs-3045	339	35	)	)	PUNCT
iajs-3045	339	36	or	or	CCONJ
iajs-3045	339	37	𝑠𝑡ƥ	𝑠𝑡ƥ	VERB
iajs-3045	339	38	⊆	⊆	NUM
iajs-3045	339	39	ɓ	ɓ	NOUN
iajs-3045	339	40	+	+	CCONJ
iajs-3045	339	41	(	(	PUNCT
iajs-3045	339	42	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	PROPN
iajs-3045	339	43	)	)	PUNCT
iajs-3045	339	44	+	+	NUM
iajs-3045	339	45	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	339	46	)	)	PUNCT
iajs-3045	339	47	)	)	PUNCT
iajs-3045	339	48	.	.	PUNCT
iajs-3045	340	1	thus	thus	ADV
iajs-3045	340	2	either	either	CCONJ
iajs-3045	340	3	𝑟𝑠ƥѡ	𝑟𝑠ƥѡ	NUM
iajs-3045	340	4	⊆	⊆	NUM
iajs-3045	340	5	ɓѡ	ɓѡ	NOUN
iajs-3045	340	6	+	+	CCONJ
iajs-3045	340	7	(	(	PUNCT
iajs-3045	340	8	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	340	9	+	+	CCONJ
iajs-3045	340	10	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	NOUN
iajs-3045	340	11	)	)	PUNCT
iajs-3045	340	12	or	or	CCONJ
iajs-3045	340	13	𝑟𝑡ƥѡ	𝑟𝑡ƥѡ	VERB
iajs-3045	340	14	⊆	⊆	NUM
iajs-3045	340	15	ɓѡ	ɓѡ	NOUN
iajs-3045	340	16	+	+	CCONJ
iajs-3045	340	17	(	(	PUNCT
iajs-3045	340	18	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	340	19	+	+	CCONJ
iajs-3045	340	20	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	NOUN
iajs-3045	340	21	)	)	PUNCT
iajs-3045	340	22	or	or	CCONJ
iajs-3045	340	23	𝑠𝑡ƥѡ	𝑠𝑡ƥѡ	VERB
iajs-3045	340	24	⊆	⊆	NUM
iajs-3045	340	25	ɓѡ	ɓѡ	NOUN
iajs-3045	340	26	+	+	CCONJ
iajs-3045	340	27	(	(	PUNCT
iajs-3045	340	28	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	340	29	+	+	CCONJ
iajs-3045	340	30	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	NOUN
iajs-3045	340	31	)	)	PUNCT
iajs-3045	340	32	.	.	PUNCT
iajs-3045	341	1	since	since	SCONJ
iajs-3045	341	2	ѡ	ѡ	PROPN
iajs-3045	341	3	is	be	AUX
iajs-3045	341	4	𝑍-𝑟𝑒𝑔𝑢𝑙𝑎𝑟	𝑍-𝑟𝑒𝑔𝑢𝑙𝑎𝑟	PROPN
iajs-3045	341	5	then	then	ADV
iajs-3045	341	6	by	by	ADP
iajs-3045	341	7	lemma	lemma	PROPN
iajs-3045	341	8	2.21	2.21	NUM
iajs-3045	341	9	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	341	10	=	=	SYM
iajs-3045	341	11	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	341	12	)	)	PUNCT
iajs-3045	341	13	and	and	CCONJ
iajs-3045	341	14	ʀ	ʀ	NOUN
iajs-3045	341	15	is	be	AUX
iajs-3045	341	16	good	good	ADJ
iajs-3045	341	17	ring	ring	NOUN
iajs-3045	341	18	then	then	ADV
iajs-3045	341	19	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	VERB
iajs-3045	341	20	=	=	PUNCT
iajs-3045	341	21	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	341	22	)	)	PUNCT
iajs-3045	341	23	.	.	PUNCT
iajs-3045	342	1	hence	hence	ADV
iajs-3045	342	2	either	either	CCONJ
iajs-3045	342	3	𝑟𝑠ƥѡ	𝑟𝑠ƥѡ	NUM
iajs-3045	342	4	⊆	⊆	NUM
iajs-3045	342	5	ɓѡ	ɓѡ	NOUN
iajs-3045	342	6	+	+	CCONJ
iajs-3045	342	7	(	(	PUNCT
iajs-3045	342	8	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	342	9	)	)	PUNCT
iajs-3045	342	10	+	+	PUNCT
iajs-3045	342	11	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	342	12	)	)	PUNCT
iajs-3045	342	13	)	)	PUNCT
iajs-3045	342	14	or	or	CCONJ
iajs-3045	342	15	𝑟𝑡ƥѡ	𝑟𝑡ƥѡ	VERB
iajs-3045	342	16	⊆	⊆	NUM
iajs-3045	342	17	ɓѡ	ɓѡ	NOUN
iajs-3045	342	18	+	+	CCONJ
iajs-3045	342	19	(	(	PUNCT
iajs-3045	342	20	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	342	21	)	)	PUNCT
iajs-3045	342	22	+	+	PUNCT
iajs-3045	342	23	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	342	24	)	)	PUNCT
iajs-3045	342	25	)	)	PUNCT
iajs-3045	342	26	or	or	CCONJ
iajs-3045	342	27	𝑠𝑡ƥѡ	𝑠𝑡ƥѡ	VERB
iajs-3045	342	28	⊆	⊆	NUM
iajs-3045	342	29	ɓѡ	ɓѡ	NOUN
iajs-3045	342	30	+	+	CCONJ
iajs-3045	342	31	(	(	PUNCT
iajs-3045	342	32	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	342	33	)	)	PUNCT
iajs-3045	342	34	+	+	PUNCT
iajs-3045	342	35	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	342	36	)	)	PUNCT
iajs-3045	342	37	)	)	PUNCT
iajs-3045	342	38	.	.	PUNCT
iajs-3045	343	1	that	that	PRON
iajs-3045	343	2	is	be	AUX
iajs-3045	343	3	either	either	CCONJ
iajs-3045	343	4	𝑟𝑠〈𝑥	𝑟𝑠〈𝑥	NOUN
iajs-3045	343	5	〉	〉	NOUN
iajs-3045	343	6	⊆	⊆	NUM
iajs-3045	343	7	ɓѡ	ɓѡ	NOUN
iajs-3045	343	8	+	+	CCONJ
iajs-3045	343	9	(	(	PUNCT
iajs-3045	343	10	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	343	11	)	)	PUNCT
iajs-3045	343	12	+	+	PUNCT
iajs-3045	343	13	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	343	14	)	)	PUNCT
iajs-3045	343	15	)	)	PUNCT
iajs-3045	343	16	or	or	CCONJ
iajs-3045	343	17	𝑟𝑡〈𝑥	𝑟𝑡〈𝑥	NUM
iajs-3045	343	18	〉	〉	NOUN
iajs-3045	343	19	⊆	⊆	NUM
iajs-3045	343	20	ɓѡ	ɓѡ	NOUN
iajs-3045	343	21	+	+	CCONJ
iajs-3045	343	22	(	(	PUNCT
iajs-3045	343	23	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	343	24	)	)	PUNCT
iajs-3045	343	25	+	+	PUNCT
iajs-3045	343	26	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	343	27	)	)	PUNCT
iajs-3045	343	28	)	)	PUNCT
iajs-3045	343	29	or	or	CCONJ
iajs-3045	343	30	𝑠𝑡〈𝑥	𝑠𝑡〈𝑥	NOUN
iajs-3045	343	31	〉	〉	NOUN
iajs-3045	343	32	⊆	⊆	NUM
iajs-3045	343	33	ɓѡ	ɓѡ	NOUN
iajs-3045	343	34	+	+	CCONJ
iajs-3045	343	35	(	(	PUNCT
iajs-3045	343	36	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	343	37	)	)	PUNCT
iajs-3045	343	38	+	+	PUNCT
iajs-3045	343	39	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	343	40	)	)	PUNCT
iajs-3045	343	41	)	)	PUNCT
iajs-3045	343	42	,	,	PUNCT
iajs-3045	343	43	thus	thus	ADV
iajs-3045	343	44	either	either	CCONJ
iajs-3045	343	45	𝑟𝑠𝑥	𝑟𝑠𝑥	NUM
iajs-3045	343	46	∈	∈	PROPN
iajs-3045	343	47	ɓѡ	ɓѡ	X
iajs-3045	344	1	+	+	CCONJ
iajs-3045	344	2	(	(	PUNCT
iajs-3045	344	3	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	344	4	)	)	PUNCT
iajs-3045	344	5	+	+	PUNCT
iajs-3045	344	6	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	344	7	)	)	PUNCT
iajs-3045	344	8	)	)	PUNCT
iajs-3045	344	9	or	or	CCONJ
iajs-3045	344	10	𝑟𝑡𝑥	𝑟𝑡𝑥	PRON
iajs-3045	344	11	∈	∈	NOUN
iajs-3045	344	12	ɓѡ	ɓѡ	X
iajs-3045	345	1	+	+	CCONJ
iajs-3045	345	2	(	(	PUNCT
iajs-3045	345	3	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	345	4	)	)	PUNCT
iajs-3045	345	5	+	+	PUNCT
iajs-3045	345	6	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	345	7	)	)	PUNCT
iajs-3045	345	8	)	)	PUNCT
iajs-3045	345	9	or	or	CCONJ
iajs-3045	345	10	𝑠𝑡𝑥	𝑠𝑡𝑥	NOUN
iajs-3045	345	11	∈	∈	PROPN
iajs-3045	345	12	ɓѡ	ɓѡ	PROPN
iajs-3045	346	1	+	+	CCONJ
iajs-3045	346	2	(	(	PUNCT
iajs-3045	346	3	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	346	4	)	)	PUNCT
iajs-3045	346	5	+	+	PUNCT
iajs-3045	346	6	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	346	7	)	)	PUNCT
iajs-3045	346	8	)	)	PUNCT
iajs-3045	346	9	.	.	PUNCT
iajs-3045	347	1	therefore	therefore	ADV
iajs-3045	347	2	ɓѡ	ɓѡ	PROPN
iajs-3045	347	3	is	be	AUX
iajs-3045	347	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	347	5	submodule	submodule	NOUN
iajs-3045	347	6	of	of	ADP
iajs-3045	347	7	ѡ.	ѡ.	NOUN
iajs-3045	347	8	(	(	PUNCT
iajs-3045	347	9	⟸	⟸	ADV
iajs-3045	347	10	)	)	PUNCT
iajs-3045	347	11	let	let	VERB
iajs-3045	347	12	𝑎𝑏𝑐𝑑	𝑎𝑏𝑐𝑑	NOUN
iajs-3045	347	13	∈	∈	PROPN
iajs-3045	347	14	ɓ	ɓ	PROPN
iajs-3045	347	15	,	,	PUNCT
iajs-3045	347	16	for	for	ADP
iajs-3045	347	17	𝑎	𝑎	NOUN
iajs-3045	347	18	,	,	PUNCT
iajs-3045	347	19	𝑏	𝑏	NOUN
iajs-3045	347	20	,	,	PUNCT
iajs-3045	347	21	𝑐	𝑐	PROPN
iajs-3045	347	22	,	,	PUNCT
iajs-3045	347	23	𝑑	𝑑	PROPN
iajs-3045	347	24	∈	∈	PROPN
iajs-3045	347	25	ʀ	ʀ	NOUN
iajs-3045	347	26	,	,	PUNCT
iajs-3045	347	27	implies	imply	VERB
iajs-3045	347	28	that	that	SCONJ
iajs-3045	347	29	𝑎𝑏𝑐(𝑑ѡ	𝑎𝑏𝑐(𝑑ѡ	X
iajs-3045	347	30	)	)	PUNCT
iajs-3045	347	31	⊆	⊆	NUM
iajs-3045	347	32	ɓѡ.	ɓѡ.	NOUN
iajs-3045	347	33	since	since	SCONJ
iajs-3045	347	34	ɓѡ	ɓѡ	PROPN
iajs-3045	347	35	is	be	AUX
iajs-3045	347	36	exnpq2ab	exnpq2ab	PROPN
iajs-3045	347	37	submodule	submodule	NOUN
iajs-3045	347	38	of	of	ADP
iajs-3045	347	39	ѡ	ѡ	PROPN
iajs-3045	347	40	,	,	PUNCT
iajs-3045	347	41	then	then	ADV
iajs-3045	347	42	by	by	ADP
iajs-3045	347	43	proposition	proposition	NOUN
iajs-3045	347	44	2.24	2.24	NUM
iajs-3045	347	45	either	either	CCONJ
iajs-3045	347	46	𝑎𝑏(𝑑ѡ	𝑎𝑏(𝑑ѡ	PROPN
iajs-3045	347	47	)	)	PUNCT
iajs-3045	347	48	⊆	⊆	NUM
iajs-3045	347	49	ɓѡ	ɓѡ	NOUN
iajs-3045	347	50	+	+	CCONJ
iajs-3045	347	51	(	(	PUNCT
iajs-3045	347	52	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	347	53	)	)	PUNCT
iajs-3045	347	54	+	+	PUNCT
iajs-3045	347	55	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	347	56	)	)	PUNCT
iajs-3045	347	57	)	)	PUNCT
iajs-3045	347	58	or	or	CCONJ
iajs-3045	347	59	𝑎𝑐(𝑑ѡ	𝑎𝑐(𝑑ѡ	NUM
iajs-3045	347	60	)	)	PUNCT
iajs-3045	347	61	⊆	⊆	NUM
iajs-3045	347	62	ɓѡ	ɓѡ	NOUN
iajs-3045	347	63	+	+	CCONJ
iajs-3045	347	64	(	(	PUNCT
iajs-3045	347	65	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	347	66	)	)	PUNCT
iajs-3045	347	67	+	+	PUNCT
iajs-3045	347	68	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	347	69	)	)	PUNCT
iajs-3045	347	70	)	)	PUNCT
iajs-3045	347	71	or	or	CCONJ
iajs-3045	347	72	𝑏𝑐(𝑑ѡ	𝑏𝑐(𝑑ѡ	NOUN
iajs-3045	347	73	)	)	PUNCT
iajs-3045	347	74	⊆	⊆	NUM
iajs-3045	347	75	ɓѡ	ɓѡ	NOUN
iajs-3045	347	76	+	+	CCONJ
iajs-3045	347	77	(	(	PUNCT
iajs-3045	347	78	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	347	79	)	)	PUNCT
iajs-3045	347	80	+	+	PUNCT
iajs-3045	347	81	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	347	82	)	)	PUNCT
iajs-3045	347	83	)	)	PUNCT
iajs-3045	347	84	.	.	PUNCT
iajs-3045	348	1	but	but	CCONJ
iajs-3045	348	2	ѡ	ѡ	PROPN
iajs-3045	348	3	is	be	AUX
iajs-3045	348	4	𝑍-regular	𝑍-regular	ADJ
iajs-3045	348	5	and	and	CCONJ
iajs-3045	348	6	ʀ	ʀ	NOUN
iajs-3045	348	7	is	be	AUX
iajs-3045	348	8	good	good	ADJ
iajs-3045	348	9	ring	ring	NOUN
iajs-3045	348	10	,	,	PUNCT
iajs-3045	348	11	then	then	ADV
iajs-3045	348	12	(	(	PUNCT
iajs-3045	348	13	𝑠𝑜𝑐(ѡ	𝑠𝑜𝑐(ѡ	PROPN
iajs-3045	348	14	)	)	PUNCT
iajs-3045	348	15	+	+	PUNCT
iajs-3045	348	16	𝐽(ѡ	𝐽(ѡ	NOUN
iajs-3045	348	17	)	)	PUNCT
iajs-3045	348	18	)	)	PUNCT
iajs-3045	349	1	=	=	PRON
iajs-3045	349	2	(	(	PUNCT
iajs-3045	349	3	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	349	4	+	+	CCONJ
iajs-3045	349	5	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	NOUN
iajs-3045	349	6	)	)	PUNCT
iajs-3045	349	7	.	.	PUNCT
iajs-3045	350	1	thus	thus	ADV
iajs-3045	350	2	either	either	CCONJ
iajs-3045	350	3	𝑎𝑏𝑑ѡ	𝑎𝑏𝑑ѡ	NOUN
iajs-3045	350	4	⊆	⊆	NUM
iajs-3045	350	5	ɓѡ	ɓѡ	NOUN
iajs-3045	350	6	+	+	CCONJ
iajs-3045	350	7	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	350	8	+	+	CCONJ
iajs-3045	350	9	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	ADJ
iajs-3045	350	10	or	or	CCONJ
iajs-3045	350	11	𝑎𝑐𝑑ѡ	𝑎𝑐𝑑ѡ	VERB
iajs-3045	350	12	⊆	⊆	NUM
iajs-3045	350	13	ɓѡ	ɓѡ	NOUN
iajs-3045	350	14	+	+	CCONJ
iajs-3045	350	15	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	350	16	+	+	CCONJ
iajs-3045	350	17	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	ADJ
iajs-3045	350	18	or	or	CCONJ
iajs-3045	350	19	𝑏𝑐𝑑ѡ	𝑏𝑐𝑑ѡ	VERB
iajs-3045	350	20	⊆	⊆	NUM
iajs-3045	350	21	ɓѡ	ɓѡ	NOUN
iajs-3045	350	22	+	+	CCONJ
iajs-3045	350	23	𝑠𝑜𝑐(ʀ)ѡ	𝑠𝑜𝑐(ʀ)ѡ	NOUN
iajs-3045	350	24	+	+	CCONJ
iajs-3045	350	25	𝐽(ʀ)ѡ	𝐽(ʀ)ѡ	PROPN
iajs-3045	350	26	,	,	PUNCT
iajs-3045	350	27	then	then	ADV
iajs-3045	350	28	either	either	CCONJ
iajs-3045	350	29	𝑎𝑏𝑑	𝑎𝑏𝑑	PROPN
iajs-3045	350	30	∈	∈	PROPN
iajs-3045	350	31	ɓ	ɓ	X
iajs-3045	350	32	+	+	NOUN
iajs-3045	350	33	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	350	34	)	)	PUNCT
iajs-3045	350	35	+	+	NUM
iajs-3045	350	36	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	350	37	)	)	PUNCT
iajs-3045	350	38	or	or	CCONJ
iajs-3045	350	39	𝑎𝑐𝑑	𝑎𝑐𝑑	NOUN
iajs-3045	350	40	∈	∈	PROPN
iajs-3045	350	41	ɓ	ɓ	X
iajs-3045	350	42	+	+	X
iajs-3045	350	43	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	350	44	)	)	PUNCT
iajs-3045	350	45	+	+	NUM
iajs-3045	350	46	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	350	47	)	)	PUNCT
iajs-3045	350	48	or	or	CCONJ
iajs-3045	350	49	𝑏𝑐𝑑	𝑏𝑐𝑑	NUM
iajs-3045	350	50	∈	∈	PROPN
iajs-3045	350	51	ɓ	ɓ	X
iajs-3045	350	52	+	+	NOUN
iajs-3045	350	53	𝑠𝑜𝑐(ʀ	𝑠𝑜𝑐(ʀ	NOUN
iajs-3045	350	54	)	)	PUNCT
iajs-3045	350	55	+	+	NUM
iajs-3045	350	56	𝐽(ʀ	𝐽(ʀ	NUM
iajs-3045	350	57	)	)	PUNCT
iajs-3045	350	58	.	.	PUNCT
iajs-3045	351	1	hence	hence	ADV
iajs-3045	351	2	ɓ	ɓ	PROPN
iajs-3045	351	3	is	be	AUX
iajs-3045	351	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	351	5	ideal	ideal	NOUN
iajs-3045	351	6	of	of	ADP
iajs-3045	351	7	ʀ	ʀ	NOUN
iajs-3045	351	8	.	.	PUNCT
iajs-3045	351	9	corollary	corollary	NOUN
iajs-3045	351	10	4.7	4.7	NUM
iajs-3045	351	11	let	let	VERB
iajs-3045	351	12	ѡ	ѡ	PRON
iajs-3045	351	13	be	be	AUX
iajs-3045	351	14	a	a	DET
iajs-3045	351	15	finitely	finitely	ADV
iajs-3045	351	16	generated	generate	VERB
iajs-3045	351	17	multiplication	multiplication	NOUN
iajs-3045	351	18	𝑍-regular	𝑍-regular	ADJ
iajs-3045	351	19	module	module	NOUN
iajs-3045	351	20	over	over	ADP
iajs-3045	351	21	artinian	artinian	ADJ
iajs-3045	351	22	ring	ring	NOUN
iajs-3045	351	23	ʀ	ʀ	NOUN
iajs-3045	351	24	and	and	CCONJ
iajs-3045	351	25	ɓ	ɓ	PRON
iajs-3045	351	26	is	be	AUX
iajs-3045	351	27	an	an	DET
iajs-3045	351	28	ideal	ideal	NOUN
iajs-3045	351	29	of	of	ADP
iajs-3045	351	30	ʀ	ʀ	NOUN
iajs-3045	351	31	with	with	ADP
iajs-3045	351	32	𝑎𝑛𝑛ʀ(ѡ	𝑎𝑛𝑛ʀ(ѡ	NOUN
iajs-3045	351	33	)	)	PUNCT
iajs-3045	351	34	⊆	⊆	NUM
iajs-3045	351	35	ɓ	ɓ	NOUN
iajs-3045	351	36	.	.	PUNCT
iajs-3045	352	1	then	then	ADV
iajs-3045	352	2	ɓ	ɓ	PROPN
iajs-3045	352	3	is	be	AUX
iajs-3045	352	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	352	5	ideal	ideal	NOUN
iajs-3045	352	6	of	of	ADP
iajs-3045	352	7	ʀ	ʀ	PRON
iajs-3045	352	8	if	if	NOUN
iajs-3045	352	9	and	and	CCONJ
iajs-3045	352	10	only	only	ADV
iajs-3045	352	11	if	if	SCONJ
iajs-3045	352	12	ɓѡ	ɓѡ	PROPN
iajs-3045	352	13	is	be	AUX
iajs-3045	352	14	exnpq2ab	exnpq2ab	PROPN
iajs-3045	352	15	submodule	submodule	NOUN
iajs-3045	352	16	of	of	ADP
iajs-3045	352	17	ѡ.	ѡ.	NOUN
iajs-3045	352	18	proposition	proposition	NOUN
iajs-3045	352	19	4.8	4.8	NUM
iajs-3045	352	20	let	let	VERB
iajs-3045	352	21	ѡ	ѡ	PART
iajs-3045	352	22	be	be	AUX
iajs-3045	352	23	a	a	DET
iajs-3045	352	24	finitely	finitely	ADV
iajs-3045	352	25	generated	generate	VERB
iajs-3045	352	26	multiplication	multiplication	NOUN
iajs-3045	352	27	𝑍-regular	𝑍-regular	ADJ
iajs-3045	352	28	module	module	NOUN
iajs-3045	352	29	over	over	ADP
iajs-3045	352	30	local	local	ADJ
iajs-3045	352	31	ring	ring	NOUN
iajs-3045	352	32	ʀ	ʀ	NOUN
iajs-3045	352	33	and	and	CCONJ
iajs-3045	352	34	ɓ	ɓ	PRON
iajs-3045	352	35	is	be	AUX
iajs-3045	352	36	an	an	DET
iajs-3045	352	37	ideal	ideal	NOUN
iajs-3045	352	38	of	of	ADP
iajs-3045	352	39	ʀ	ʀ	NOUN
iajs-3045	352	40	with	with	ADP
iajs-3045	352	41	𝑎𝑛𝑛ʀ(ѡ	𝑎𝑛𝑛ʀ(ѡ	NOUN
iajs-3045	352	42	)	)	PUNCT
iajs-3045	352	43	⊆	⊆	NUM
iajs-3045	352	44	ɓ	ɓ	NOUN
iajs-3045	352	45	.	.	PUNCT
iajs-3045	353	1	then	then	ADV
iajs-3045	353	2	ɓ	ɓ	PROPN
iajs-3045	353	3	is	be	AUX
iajs-3045	353	4	exnpq2ab	exnpq2ab	PROPN
iajs-3045	353	5	ideal	ideal	NOUN
iajs-3045	353	6	of	of	ADP
iajs-3045	353	7	ʀ	ʀ	PRON
iajs-3045	353	8	if	if	NOUN
iajs-3045	353	9	and	and	CCONJ
iajs-3045	353	10	only	only	ADV
iajs-3045	353	11	if	if	SCONJ
iajs-3045	353	12	ɓѡ	ɓѡ	PROPN
iajs-3045	353	13	is	be	AUX
iajs-3045	353	14	exnpq2ab	exnpq2ab	PROPN
iajs-3045	353	15	submodule	submodule	NOUN
iajs-3045	353	16	of	of	ADP
iajs-3045	353	17	ѡ.	ѡ.	NOUN
iajs-3045	353	18	proof	proof	NOUN
iajs-3045	353	19	.	.	PUNCT
iajs-3045	354	1	similar	similar	ADJ
iajs-3045	354	2	to	to	ADP
iajs-3045	354	3	the	the	DET
iajs-3045	354	4	proof	proof	NOUN
iajs-3045	354	5	of	of	ADP
iajs-3045	354	6	proposition	proposition	NOUN
iajs-3045	354	7	4.6	4.6	NUM
iajs-3045	354	8	by	by	ADP
iajs-3045	354	9	using	use	VERB
iajs-3045	354	10	lemma	lemma	PROPN
iajs-3045	354	11	2.15	2.15	NUM
iajs-3045	354	12	.	.	PUNCT
iajs-3045	355	1	proposition	proposition	NOUN
iajs-3045	355	2	4.9	4.9	NUM
iajs-3045	355	3	let	let	VERB
iajs-3045	355	4	ѡ	ѡ	PRON
iajs-3045	355	5	be	be	AUX
iajs-3045	355	6	a	a	DET
iajs-3045	355	7	faithful	faithful	ADJ
iajs-3045	355	8	finitely	finitely	ADV
iajs-3045	355	9	generated	generate	VERB
iajs-3045	355	10	multiplication	multiplication	NOUN
iajs-3045	355	11	ʀ	ʀ	NOUN
iajs-3045	355	12	-	-	PUNCT
iajs-3045	355	13	module	module	NOUN
iajs-3045	355	14	and	and	CCONJ
iajs-3045	355	15	𝑉	𝑉	PROPN
iajs-3045	355	16	≠	≠	PROPN
iajs-3045	355	17	ѡ	ѡ	NOUN
iajs-3045	355	18	,	,	PUNCT
iajs-3045	355	19	in	in	ADP
iajs-3045	355	20	which	which	DET
iajs-3045	355	21	case	case	NOUN
iajs-3045	355	22	the	the	DET
iajs-3045	355	23	following	follow	VERB
iajs-3045	355	24	claims	claim	NOUN
iajs-3045	355	25	are	be	AUX
iajs-3045	355	26	equivalent	equivalent	ADJ
iajs-3045	355	27	:	:	PUNCT
iajs-3045	355	28	1	1	X
iajs-3045	355	29	.	.	X
iajs-3045	355	30	𝑉	𝑉	PROPN
iajs-3045	355	31	is	be	AUX
iajs-3045	355	32	exnpq2ab	exnpq2ab	PROPN
iajs-3045	355	33	submodule	submodule	NOUN
iajs-3045	355	34	of	of	ADP
iajs-3045	355	35	ѡ.	ѡ.	NOUN
iajs-3045	355	36	2	2	NUM
iajs-3045	355	37	.	.	PUNCT
iajs-3045	356	1	[	[	X
iajs-3045	356	2	𝑉:ʀ	𝑉:ʀ	NOUN
iajs-3045	356	3	ѡ	ѡ	X
iajs-3045	356	4	]	]	PUNCT
iajs-3045	356	5	is	be	AUX
iajs-3045	356	6	exnpq2ab	exnpq2ab	PROPN
iajs-3045	356	7	ideal	ideal	NOUN
iajs-3045	356	8	of	of	ADP
iajs-3045	356	9	ʀ	ʀ	NOUN
iajs-3045	356	10	.	.	NOUN
iajs-3045	357	1	3	3	X
iajs-3045	357	2	.	.	X
iajs-3045	357	3	𝑉	𝑉	PROPN
iajs-3045	357	4	=	=	PUNCT
iajs-3045	357	5	ɓѡ	ɓѡ	PROPN
iajs-3045	357	6	for	for	ADP
iajs-3045	357	7	some	some	DET
iajs-3045	357	8	exnpq2ab	exnpq2ab	PROPN
iajs-3045	357	9	ideal	ideal	NOUN
iajs-3045	357	10	ɓ	ɓ	PROPN
iajs-3045	357	11	of	of	ADP
iajs-3045	357	12	ʀ	ʀ	NOUN
iajs-3045	357	13	.	.	NOUN
iajs-3045	357	14	proof	proof	NOUN
iajs-3045	357	15	.	.	PUNCT
iajs-3045	358	1	(	(	PUNCT
iajs-3045	358	2	𝟏	𝟏	PROPN
iajs-3045	358	3	⇔	⇔	X
iajs-3045	358	4	𝟐	𝟐	NUM
iajs-3045	358	5	)	)	PUNCT
iajs-3045	358	6	by	by	ADP
iajs-3045	358	7	proposition	proposition	NOUN
iajs-3045	358	8	3.6	3.6	NUM
iajs-3045	358	9	.	.	PUNCT
iajs-3045	359	1	ihjpas	ihjpas	PROPN
iajs-3045	359	2	.	.	PUNCT
iajs-3045	360	1	36(2)2023	36(2)2023	NUM
iajs-3045	360	2	418	418	NUM
iajs-3045	360	3	(	(	PUNCT
iajs-3045	360	4	𝟐	𝟐	NUM
iajs-3045	360	5	⇒	⇒	NOUN
iajs-3045	360	6	𝟑	𝟑	NUM
iajs-3045	360	7	)	)	PUNCT
iajs-3045	360	8	since	since	SCONJ
iajs-3045	360	9	[	[	X
iajs-3045	360	10	𝑉:ʀ	𝑉:ʀ	NOUN
iajs-3045	360	11	ѡ	ѡ	NOUN
iajs-3045	360	12	]	]	PUNCT
iajs-3045	360	13	is	be	AUX
iajs-3045	360	14	exnpq2ab	exnpq2ab	PROPN
iajs-3045	360	15	ideal	ideal	NOUN
iajs-3045	360	16	of	of	ADP
iajs-3045	360	17	ʀ	ʀ	PROPN
iajs-3045	360	18	and	and	CCONJ
iajs-3045	360	19	ѡ	ѡ	PROPN
iajs-3045	360	20	is	be	AUX
iajs-3045	360	21	a	a	DET
iajs-3045	360	22	faithful	faithful	ADJ
iajs-3045	360	23	,	,	PUNCT
iajs-3045	360	24	that	that	ADV
iajs-3045	360	25	is	is	ADV
iajs-3045	360	26	(	(	PUNCT
iajs-3045	360	27	0	0	NUM
iajs-3045	360	28	)	)	PUNCT
iajs-3045	361	1	=	=	SYM
iajs-3045	361	2	ɑ𝑛𝑛ʀ(ѡ	ɑ𝑛𝑛ʀ(ѡ	ADJ
iajs-3045	361	3	)	)	PUNCT
iajs-3045	361	4	=	=	PUNCT
iajs-3045	362	1	[	[	X
iajs-3045	362	2	0	0	NUM
iajs-3045	362	3	:	:	PUNCT
iajs-3045	362	4	ʀ	ʀ	PART
iajs-3045	362	5	ѡ	ѡ	ADP
iajs-3045	362	6	]	]	PUNCT
iajs-3045	362	7	⊆	⊆	NUM
iajs-3045	362	8	[	[	X
iajs-3045	362	9	𝑉:ʀ	𝑉:ʀ	NOUN
iajs-3045	362	10	ѡ	ѡ	NOUN
iajs-3045	362	11	]	]	PUNCT
iajs-3045	362	12	and	and	CCONJ
iajs-3045	362	13	ѡ	ѡ	PROPN
iajs-3045	362	14	is	be	AUX
iajs-3045	362	15	a	a	DET
iajs-3045	362	16	multiplication	multiplication	NOUN
iajs-3045	362	17	,	,	PUNCT
iajs-3045	362	18	so	so	ADV
iajs-3045	362	19	v	v	ADP
iajs-3045	362	20	=	=	SYM
iajs-3045	363	1	[	[	X
iajs-3045	363	2	𝑉:ʀ	𝑉:ʀ	NOUN
iajs-3045	363	3	ѡ]ѡ	ѡ]ѡ	NOUN
iajs-3045	363	4	,	,	PUNCT
iajs-3045	363	5	implies	imply	VERB
iajs-3045	363	6	that	that	SCONJ
iajs-3045	363	7	v	v	NOUN
iajs-3045	363	8	=	=	SYM
iajs-3045	363	9	jѡ	jѡ	ADJ
iajs-3045	363	10	for	for	ADP
iajs-3045	363	11	some	some	DET
iajs-3045	363	12	exnpq2ab	exnpq2ab	PROPN
iajs-3045	363	13	ideal	ideal	NOUN
iajs-3045	363	14	j	j	PROPN
iajs-3045	364	1	=	=	PUNCT
iajs-3045	365	1	[	[	X
iajs-3045	365	2	𝑉:ʀ	𝑉:ʀ	NOUN
iajs-3045	365	3	ѡ	ѡ	NOUN
iajs-3045	365	4	]	]	PUNCT
iajs-3045	365	5	of	of	ADP
iajs-3045	365	6	ʀ	ʀ	PRON
iajs-3045	365	7	.	.	PUNCT
iajs-3045	365	8	(	(	PUNCT
iajs-3045	365	9	𝟑	𝟑	X
iajs-3045	365	10	⇒	⇒	NOUN
iajs-3045	365	11	𝟐	𝟐	NUM
iajs-3045	365	12	)	)	PUNCT
iajs-3045	365	13	suppose	suppose	VERB
iajs-3045	365	14	that	that	SCONJ
iajs-3045	365	15	𝑉	𝑉	PROPN
iajs-3045	365	16	=	=	PUNCT
iajs-3045	365	17	𝐽ѡ	𝐽ѡ	PROPN
iajs-3045	365	18	for	for	ADP
iajs-3045	365	19	some	some	DET
iajs-3045	365	20	exnpq2ab	exnpq2ab	PROPN
iajs-3045	365	21	ideal	ideal	NOUN
iajs-3045	365	22	𝐽	𝐽	PROPN
iajs-3045	365	23	of	of	ADP
iajs-3045	365	24	ʀ	ʀ	PROPN
iajs-3045	365	25	.	.	PUNCT
iajs-3045	365	26	since	since	SCONJ
iajs-3045	365	27	ѡ	ѡ	PROPN
iajs-3045	365	28	is	be	AUX
iajs-3045	365	29	multiplication	multiplication	NOUN
iajs-3045	365	30	,	,	PUNCT
iajs-3045	365	31	then	then	ADV
iajs-3045	365	32	v	v	NOUN
iajs-3045	365	33	=	=	PUNCT
iajs-3045	366	1	[	[	X
iajs-3045	366	2	v	v	ADP
iajs-3045	366	3	:	:	PUNCT
iajs-3045	366	4	ʀ	ʀ	PROPN
iajs-3045	366	5	ѡ]ѡ.	ѡ]ѡ.	PROPN
iajs-3045	366	6	that	that	PRON
iajs-3045	366	7	is	be	AUX
iajs-3045	366	8	𝐽ѡ	𝐽ѡ	PROPN
iajs-3045	366	9	=	=	PUNCT
iajs-3045	367	1	[	[	X
iajs-3045	367	2	v	v	ADP
iajs-3045	367	3	:	:	PUNCT
iajs-3045	367	4	ʀ	ʀ	ADJ
iajs-3045	367	5	ѡ]ѡ	ѡ]ѡ	NOUN
iajs-3045	367	6	,	,	PUNCT
iajs-3045	367	7	but	but	CCONJ
iajs-3045	367	8	ѡ	ѡ	PROPN
iajs-3045	367	9	is	be	AUX
iajs-3045	367	10	faithful	faithful	ADJ
iajs-3045	367	11	finitely	finitely	ADV
iajs-3045	367	12	generated	generate	VERB
iajs-3045	367	13	multiplication	multiplication	NOUN
iajs-3045	367	14	then	then	ADV
iajs-3045	367	15	by	by	ADP
iajs-3045	367	16	lemma	lemma	PROPN
iajs-3045	367	17	2.23	2.23	NUM
iajs-3045	367	18	we	we	PRON
iajs-3045	367	19	get	get	VERB
iajs-3045	367	20	[	[	X
iajs-3045	367	21	v	v	NOUN
iajs-3045	367	22	:	:	PUNCT
iajs-3045	367	23	ʀ	ʀ	PART
iajs-3045	367	24	ѡ	ѡ	NOUN
iajs-3045	367	25	]	]	PUNCT
iajs-3045	367	26	=	=	PUNCT
iajs-3045	367	27	j.	j.	X
iajs-3045	368	1	thus	thus	ADV
iajs-3045	368	2	[	[	X
iajs-3045	368	3	v	v	NOUN
iajs-3045	368	4	:	:	PUNCT
iajs-3045	368	5	ʀ	ʀ	PART
iajs-3045	368	6	ѡ	ѡ	X
iajs-3045	368	7	]	]	PUNCT
iajs-3045	368	8	is	be	AUX
iajs-3045	368	9	exnpq2ab	exnpq2ab	PROPN
iajs-3045	368	10	ideal	ideal	NOUN
iajs-3045	368	11	of	of	ADP
iajs-3045	368	12	ʀ	ʀ	NOUN
iajs-3045	368	13	.	.	PUNCT
iajs-3045	368	14	proposition	proposition	NOUN
iajs-3045	368	15	4.10	4.10	NUM
iajs-3045	368	16	let	let	VERB
iajs-3045	368	17	ѡ	ѡ	PRON
iajs-3045	368	18	be	be	AUX
iajs-3045	368	19	a	a	DET
iajs-3045	368	20	finitely	finitely	ADV
iajs-3045	368	21	generated	generate	VERB
iajs-3045	368	22	multiplication	multiplication	NOUN
iajs-3045	368	23	projective	projective	ADJ
iajs-3045	368	24	ʀ	ʀ	NOUN
iajs-3045	368	25	-	-	PUNCT
iajs-3045	368	26	module	module	NOUN
iajs-3045	368	27	and	and	CCONJ
iajs-3045	368	28	𝑉	𝑉	PROPN
iajs-3045	368	29	≠	≠	PROPN
iajs-3045	368	30	ѡ	ѡ	NOUN
iajs-3045	368	31	,	,	PUNCT
iajs-3045	368	32	in	in	ADP
iajs-3045	368	33	which	which	DET
iajs-3045	368	34	case	case	NOUN
iajs-3045	368	35	the	the	DET
iajs-3045	368	36	following	follow	VERB
iajs-3045	368	37	claims	claim	NOUN
iajs-3045	368	38	are	be	AUX
iajs-3045	368	39	equivalent	equivalent	ADJ
iajs-3045	368	40	:	:	PUNCT
iajs-3045	368	41	1	1	X
iajs-3045	368	42	.	.	X
iajs-3045	368	43	𝑉	𝑉	PROPN
iajs-3045	368	44	is	be	AUX
iajs-3045	368	45	exnpq2ab	exnpq2ab	PROPN
iajs-3045	368	46	submodule	submodule	NOUN
iajs-3045	368	47	of	of	ADP
iajs-3045	368	48	ѡ.	ѡ.	NOUN
iajs-3045	368	49	2	2	NUM
iajs-3045	368	50	.	.	PUNCT
iajs-3045	369	1	[	[	X
iajs-3045	369	2	𝑉:ʀ	𝑉:ʀ	NOUN
iajs-3045	369	3	ѡ	ѡ	X
iajs-3045	369	4	]	]	PUNCT
iajs-3045	369	5	is	be	AUX
iajs-3045	369	6	exnpq2ab	exnpq2ab	PROPN
iajs-3045	369	7	ideal	ideal	NOUN
iajs-3045	369	8	of	of	ADP
iajs-3045	369	9	ʀ	ʀ	NOUN
iajs-3045	369	10	.	.	NOUN
iajs-3045	370	1	3	3	X
iajs-3045	370	2	.	.	X
iajs-3045	370	3	𝑉	𝑉	PROPN
iajs-3045	370	4	=	=	PUNCT
iajs-3045	370	5	ɓѡ	ɓѡ	PROPN
iajs-3045	370	6	for	for	ADP
iajs-3045	370	7	some	some	DET
iajs-3045	370	8	exnpq2ab	exnpq2ab	PROPN
iajs-3045	370	9	ideal	ideal	NOUN
iajs-3045	370	10	ɓ	ɓ	PROPN
iajs-3045	370	11	of	of	ADP
iajs-3045	370	12	ʀ	ʀ	NOUN
iajs-3045	370	13	.	.	NOUN
iajs-3045	370	14	proof	proof	NOUN
iajs-3045	370	15	.	.	PUNCT
iajs-3045	371	1	(	(	PUNCT
iajs-3045	371	2	𝟏	𝟏	PROPN
iajs-3045	371	3	⇔	⇔	X
iajs-3045	371	4	𝟐	𝟐	NUM
iajs-3045	371	5	)	)	PUNCT
iajs-3045	371	6	by	by	ADP
iajs-3045	371	7	proposition	proposition	NOUN
iajs-3045	371	8	3.7	3.7	NUM
iajs-3045	371	9	.	.	PUNCT
iajs-3045	372	1	(	(	PUNCT
iajs-3045	372	2	𝟐	𝟐	NUM
iajs-3045	372	3	⇒	⇒	NOUN
iajs-3045	372	4	𝟑	𝟑	NUM
iajs-3045	372	5	)	)	PUNCT
iajs-3045	372	6	clear	clear	ADJ
iajs-3045	372	7	.	.	PUNCT
iajs-3045	373	1	(	(	PUNCT
iajs-3045	373	2	𝟑	𝟑	X
iajs-3045	373	3	⇒	⇒	NOUN
iajs-3045	373	4	𝟐	𝟐	NUM
iajs-3045	373	5	)	)	PUNCT
iajs-3045	373	6	assume	assume	VERB
iajs-3045	373	7	that	that	SCONJ
iajs-3045	373	8	𝑉	𝑉	PROPN
iajs-3045	373	9	=	=	SYM
iajs-3045	373	10	ɓѡ	ɓѡ	X
iajs-3045	373	11	…	…	PUNCT
iajs-3045	373	12	..	..	PUNCT
iajs-3045	373	13	(	(	PUNCT
iajs-3045	373	14	1	1	X
iajs-3045	373	15	)	)	PUNCT
iajs-3045	373	16	for	for	ADP
iajs-3045	373	17	some	some	DET
iajs-3045	373	18	exnpq2ab	exnpq2ab	PROPN
iajs-3045	373	19	ideal	ideal	NOUN
iajs-3045	373	20	ɓ	ɓ	PROPN
iajs-3045	373	21	of	of	ADP
iajs-3045	373	22	ʀ	ʀ	NOUN
iajs-3045	373	23	with	with	ADP
iajs-3045	373	24	𝑎𝑛𝑛ʀ(ѡ	𝑎𝑛𝑛ʀ(ѡ	NOUN
iajs-3045	373	25	)	)	PUNCT
iajs-3045	373	26	⊆	⊆	NUM
iajs-3045	373	27	ɓ	ɓ	NOUN
iajs-3045	373	28	.	.	PUNCT
iajs-3045	374	1	while	while	SCONJ
iajs-3045	374	2	ѡ	ѡ	PROPN
iajs-3045	374	3	is	be	AUX
iajs-3045	374	4	a	a	DET
iajs-3045	374	5	multiplication	multiplication	NOUN
iajs-3045	374	6	,	,	PUNCT
iajs-3045	374	7	then	then	ADV
iajs-3045	374	8	𝑉	𝑉	PROPN
iajs-3045	374	9	=	=	PUNCT
iajs-3045	375	1	[	[	X
iajs-3045	375	2	𝑉:ʀ	𝑉:ʀ	NOUN
iajs-3045	375	3	ѡ]ѡ	ѡ]ѡ	NOUN
iajs-3045	375	4	…	…	PUNCT
iajs-3045	375	5	..	..	PUNCT
iajs-3045	375	6	(2	(2	NUM
iajs-3045	375	7	)	)	PUNCT
iajs-3045	375	8	,	,	PUNCT
iajs-3045	375	9	from	from	ADP
iajs-3045	375	10	(	(	PUNCT
iajs-3045	375	11	1	1	NUM
iajs-3045	375	12	)	)	PUNCT
iajs-3045	375	13	and	and	CCONJ
iajs-3045	375	14	(	(	PUNCT
iajs-3045	375	15	2	2	X
iajs-3045	375	16	)	)	PUNCT
iajs-3045	375	17	we	we	PRON
iajs-3045	375	18	have	have	VERB
iajs-3045	375	19	[	[	PUNCT
iajs-3045	375	20	𝑉:ʀ	𝑉:ʀ	NOUN
iajs-3045	375	21	ѡ]ѡ	ѡ]ѡ	NOUN
iajs-3045	375	22	=	=	PUNCT
iajs-3045	376	1	ɓѡ.	ɓѡ.	INTJ
iajs-3045	376	2	since	since	SCONJ
iajs-3045	376	3	ѡ	ѡ	PROPN
iajs-3045	376	4	is	be	AUX
iajs-3045	376	5	a	a	DET
iajs-3045	376	6	finitely	finitely	ADV
iajs-3045	376	7	generated	generate	VERB
iajs-3045	376	8	,	,	PUNCT
iajs-3045	376	9	then	then	ADV
iajs-3045	376	10	by	by	ADP
iajs-3045	376	11	lemma	lemma	PROPN
iajs-3045	376	12	2.22	2.22	NUM
iajs-3045	376	13	ѡ	ѡ	NOUN
iajs-3045	376	14	is	be	AUX
iajs-3045	376	15	weak	weak	ADJ
iajs-3045	376	16	cancellation	cancellation	NOUN
iajs-3045	376	17	,	,	PUNCT
iajs-3045	376	18	then	then	ADV
iajs-3045	376	19	[	[	X
iajs-3045	376	20	𝑉:ʀ	𝑉:ʀ	NOUN
iajs-3045	376	21	ѡ	ѡ	X
iajs-3045	376	22	]	]	X
iajs-3045	376	23	+	+	CCONJ
iajs-3045	376	24	𝑎𝑛𝑛ʀ(ѡ	𝑎𝑛𝑛ʀ(ѡ	NOUN
iajs-3045	376	25	)	)	PUNCT
iajs-3045	376	26	=	=	SYM
iajs-3045	376	27	ɓ	ɓ	PROPN
iajs-3045	376	28	+	+	NUM
iajs-3045	376	29	𝑎𝑛𝑛ʀ(ѡ	𝑎𝑛𝑛ʀ(ѡ	NOUN
iajs-3045	376	30	)	)	PUNCT
iajs-3045	376	31	,	,	PUNCT
iajs-3045	376	32	but	but	CCONJ
iajs-3045	376	33	𝑎𝑛𝑛ʀ(ѡ	𝑎𝑛𝑛ʀ(ѡ	NOUN
iajs-3045	376	34	)	)	PUNCT
iajs-3045	376	35	⊆	⊆	NUM
iajs-3045	376	36	ɓ	ɓ	NOUN
iajs-3045	376	37	,	,	PUNCT
iajs-3045	376	38	and	and	CCONJ
iajs-3045	376	39	𝑎𝑛𝑛ʀ(ѡ	𝑎𝑛𝑛ʀ(ѡ	NOUN
iajs-3045	376	40	)	)	PUNCT
iajs-3045	376	41	⊆	⊆	NUM
iajs-3045	377	1	[	[	X
iajs-3045	377	2	𝑉:ʀ	𝑉:ʀ	NOUN
iajs-3045	377	3	ѡ	ѡ	NOUN
iajs-3045	377	4	]	]	PUNCT
iajs-3045	377	5	,	,	PUNCT
iajs-3045	377	6	implies	imply	VERB
iajs-3045	377	7	that	that	SCONJ
iajs-3045	377	8	𝑎𝑛𝑛ʀ(ѡ	𝑎𝑛𝑛ʀ(ѡ	NOUN
iajs-3045	377	9	)	)	PUNCT
iajs-3045	378	1	+	+	NUM
iajs-3045	378	2	ɓ	ɓ	X
iajs-3045	378	3	=	=	SYM
iajs-3045	378	4	ɓ	ɓ	NOUN
iajs-3045	378	5	and	and	CCONJ
iajs-3045	378	6	[	[	X
iajs-3045	378	7	𝑉:ʀ	𝑉:ʀ	NOUN
iajs-3045	378	8	ѡ	ѡ	NOUN
iajs-3045	378	9	]	]	X
iajs-3045	378	10	+	+	CCONJ
iajs-3045	378	11	𝑎𝑛𝑛ʀ(ѡ	𝑎𝑛𝑛ʀ(ѡ	NOUN
iajs-3045	378	12	)	)	PUNCT
iajs-3045	378	13	=	=	NOUN
iajs-3045	379	1	[	[	X
iajs-3045	379	2	𝑉:ʀ	𝑉:ʀ	NOUN
iajs-3045	379	3	ѡ	ѡ	NOUN
iajs-3045	379	4	]	]	PUNCT
iajs-3045	379	5	.	.	PUNCT
iajs-3045	380	1	thus	thus	ADV
iajs-3045	380	2	ɓ	ɓ	X
iajs-3045	380	3	=	=	SYM
iajs-3045	381	1	[	[	X
iajs-3045	381	2	𝑉:ʀ	𝑉:ʀ	NOUN
iajs-3045	381	3	ѡ	ѡ	NOUN
iajs-3045	381	4	]	]	PUNCT
iajs-3045	381	5	,	,	PUNCT
iajs-3045	381	6	but	but	CCONJ
iajs-3045	381	7	ɓ	ɓ	PRON
iajs-3045	381	8	is	be	AUX
iajs-3045	381	9	exnpq2ab	exnpq2ab	PROPN
iajs-3045	381	10	ideal	ideal	NOUN
iajs-3045	381	11	of	of	ADP
iajs-3045	381	12	ʀ	ʀ	NOUN
iajs-3045	381	13	,	,	PUNCT
iajs-3045	381	14	hence	hence	ADV
iajs-3045	381	15	[	[	X
iajs-3045	381	16	𝑉:ʀ	𝑉:ʀ	NOUN
iajs-3045	381	17	ѡ	ѡ	AUX
iajs-3045	381	18	]	]	PUNCT
iajs-3045	381	19	is	be	AUX
iajs-3045	381	20	exnpq2ab	exnpq2ab	PROPN
iajs-3045	381	21	ideal	ideal	NOUN
iajs-3045	381	22	of	of	ADP
iajs-3045	381	23	ʀ	ʀ	NOUN
iajs-3045	381	24	.	.	NOUN
iajs-3045	381	25	5	5	NUM
iajs-3045	381	26	.	.	X
iajs-3045	381	27	conclusion	conclusion	NOUN
iajs-3045	381	28	.	.	PUNCT
iajs-3045	382	1	in	in	ADP
iajs-3045	382	2	this	this	DET
iajs-3045	382	3	paper	paper	NOUN
iajs-3045	382	4	,	,	PUNCT
iajs-3045	382	5	we	we	PRON
iajs-3045	382	6	introduced	introduce	VERB
iajs-3045	382	7	the	the	DET
iajs-3045	382	8	some	some	DET
iajs-3045	382	9	characterizations	characterization	NOUN
iajs-3045	382	10	in	in	ADP
iajs-3045	382	11	class	class	NOUN
iajs-3045	382	12	of	of	ADP
iajs-3045	382	13	multiplication	multiplication	NOUN
iajs-3045	382	14	modules	module	NOUN
iajs-3045	382	15	.	.	PUNCT
iajs-3045	383	1	and	and	CCONJ
iajs-3045	383	2	,	,	PUNCT
iajs-3045	383	3	we	we	PRON
iajs-3045	383	4	show	show	VERB
iajs-3045	383	5	by	by	ADP
iajs-3045	383	6	example	example	NOUN
iajs-3045	383	7	the	the	DET
iajs-3045	383	8	residual	residual	ADJ
iajs-3045	383	9	of	of	ADP
iajs-3045	383	10	extend	extend	NOUN
iajs-3045	383	11	nearly	nearly	ADV
iajs-3045	383	12	pseudo	pseudo	ADJ
iajs-3045	383	13	quasi-2	quasi-2	ADJ
iajs-3045	383	14	-	-	PUNCT
iajs-3045	383	15	absorbing	absorb	VERB
iajs-3045	383	16	submodule	submodule	NOUN
iajs-3045	383	17	need	need	AUX
iajs-3045	383	18	not	not	PART
iajs-3045	383	19	to	to	PART
iajs-3045	383	20	be	be	AUX
iajs-3045	383	21	extend	extend	VERB
iajs-3045	383	22	nearly	nearly	ADV
iajs-3045	383	23	pseudo	pseudo	ADJ
iajs-3045	383	24	quasi-2	quasi-2	ADJ
iajs-3045	383	25	-	-	PUNCT
iajs-3045	383	26	absorbing	absorbing	ADJ
iajs-3045	383	27	ideal	ideal	NOUN
iajs-3045	383	28	;	;	PUNCT
iajs-3045	383	29	we	we	PRON
iajs-3045	383	30	gave	give	VERB
iajs-3045	383	31	an	an	DET
iajs-3045	383	32	example	example	NOUN
iajs-3045	383	33	of	of	ADP
iajs-3045	383	34	that	that	PRON
iajs-3045	383	35	.	.	PUNCT
iajs-3045	384	1	under	under	ADP
iajs-3045	384	2	a	a	DET
iajs-3045	384	3	certain	certain	ADJ
iajs-3045	384	4	condition	condition	NOUN
iajs-3045	384	5	it	it	PRON
iajs-3045	384	6	is	be	AUX
iajs-3045	384	7	equivalent	equivalent	ADJ
iajs-3045	384	8	.	.	PUNCT
iajs-3045	385	1	also	also	ADV
iajs-3045	385	2	,	,	PUNCT
iajs-3045	385	3	we	we	PRON
iajs-3045	385	4	studied	study	VERB
iajs-3045	385	5	the	the	DET
iajs-3045	385	6	characterized	characterized	ADJ
iajs-3045	385	7	extend	extend	VERB
iajs-3045	385	8	nearly	nearly	ADV
iajs-3045	385	9	pseudo	pseudo	ADJ
iajs-3045	385	10	quasi2	quasi2	NOUN
iajs-3045	385	11	-	-	PUNCT
iajs-3045	385	12	absorbing	absorbing	ADJ
iajs-3045	385	13	ideals	ideal	NOUN
iajs-3045	385	14	by	by	ADP
iajs-3045	385	15	extend	extend	NOUN
iajs-3045	385	16	nearly	nearly	ADV
iajs-3045	385	17	pseudo	pseudo	ADJ
iajs-3045	385	18	quasi-2	quasi-2	ADJ
iajs-3045	385	19	-	-	PUNCT
iajs-3045	385	20	absorbing	absorbing	ADJ
iajs-3045	385	21	submodules	submodule	NOUN
iajs-3045	385	22	.	.	PUNCT
iajs-3045	386	1	in	in	ADP
iajs-3045	386	2	the	the	DET
iajs-3045	386	3	end	end	NOUN
iajs-3045	386	4	,	,	PUNCT
iajs-3045	386	5	we	we	PRON
iajs-3045	386	6	got	get	VERB
iajs-3045	386	7	a	a	DET
iajs-3045	386	8	lot	lot	NOUN
iajs-3045	386	9	of	of	ADP
iajs-3045	386	10	important	important	ADJ
iajs-3045	386	11	results	result	NOUN
iajs-3045	386	12	.	.	PUNCT
iajs-3045	387	1	references	reference	NOUN
iajs-3045	387	2	1	1	NUM
iajs-3045	387	3	.	.	PUNCT
iajs-3045	387	4	haibat	haibat	PROPN
iajs-3045	387	5	,	,	PUNCT
iajs-3045	387	6	k.	k.	PROPN
iajs-3045	387	7	;	;	PUNCT
iajs-3045	387	8	mohammedali	mohammedali	PROPN
iajs-3045	387	9	.	.	PUNCT
iajs-3045	387	10	;	;	PUNCT
iajs-3045	388	1	omar	omar	PROPN
iajs-3045	388	2	,	,	PUNCT
iajs-3045	388	3	a.	a.	NOUN
iajs-3045	388	4	abdalla	abdalla	PROPN
iajs-3045	388	5	.	.	PUNCT
iajs-3045	388	6	,	,	PUNCT
iajs-3045	388	7	pseudo	pseudo	NOUN
iajs-3045	388	8	quasi-2	quasi-2	NOUN
iajs-3045	388	9	-	-	PUNCT
iajs-3045	388	10	absorbing	absorbing	ADJ
iajs-3045	388	11	submodules	submodule	NOUN
iajs-3045	388	12	and	and	CCONJ
iajs-3045	388	13	some	some	DET
iajs-3045	388	14	related	related	ADJ
iajs-3045	388	15	concepts	concept	NOUN
iajs-3045	388	16	,	,	PUNCT
iajs-3045	388	17	ibn	ibn	PROPN
iajs-3045	388	18	al	al	PROPN
iajs-3045	388	19	-	-	PUNCT
iajs-3045	388	20	haitham	haitham	PROPN
iajs-3045	388	21	journal	journal	PROPN
iajs-3045	388	22	for	for	ADP
iajs-3045	388	23	pure	pure	ADJ
iajs-3045	388	24	and	and	CCONJ
iajs-3045	388	25	applied	applied	ADJ
iajs-3045	388	26	science	science	NOUN
iajs-3045	388	27	,	,	PUNCT
iajs-3045	388	28	2019	2019	NUM
iajs-3045	388	29	,	,	PUNCT
iajs-3045	388	30	32(2	32(2	NUM
iajs-3045	388	31	)	)	PUNCT
iajs-3045	388	32	,	,	PUNCT
iajs-3045	388	33	114122	114122	NUM
iajs-3045	388	34	.	.	PUNCT
iajs-3045	389	1	2	2	X
iajs-3045	389	2	.	.	X
iajs-3045	389	3	haibat	haibat	PROPN
iajs-3045	389	4	,	,	PUNCT
iajs-3045	389	5	k.	k.	PROPN
iajs-3045	389	6	;	;	PUNCT
iajs-3045	389	7	mohammedali	mohammedali	PROPN
iajs-3045	389	8	;	;	PUNCT
iajs-3045	389	9	khalaf	khalaf	PROPN
iajs-3045	389	10	,	,	PUNCT
iajs-3045	389	11	h.	h.	PROPN
iajs-3045	389	12	alhabeeb	alhabeeb	PROPN
iajs-3045	389	13	.	.	PROPN
iajs-3045	389	14	,	,	PUNCT
iajs-3045	389	15	nearly	nearly	ADV
iajs-3045	389	16	quasi2	quasi2	NOUN
iajs-3045	389	17	-	-	PUNCT
iajs-3045	389	18	absorbing	absorbing	ADJ
iajs-3045	389	19	submodules	submodule	NOUN
iajs-3045	389	20	,	,	PUNCT
iajs-3045	389	21	tikrit	tikrit	NOUN
iajs-3045	389	22	journal	journal	NOUN
iajs-3045	389	23	for	for	ADP
iajs-3045	389	24	pure	pure	ADJ
iajs-3045	389	25	.	.	PUNCT
iajs-3045	390	1	sci,2018	sci,2018	NOUN
iajs-3045	390	2	,	,	PUNCT
iajs-3045	390	3	23(9	23(9	NUM
iajs-3045	390	4	)	)	PUNCT
iajs-3045	390	5	,	,	PUNCT
iajs-3045	390	6	99	99	NUM
iajs-3045	390	7	-	-	SYM
iajs-3045	390	8	102	102	NUM
iajs-3045	390	9	.	.	PUNCT
iajs-3045	391	1	3	3	X
iajs-3045	391	2	.	.	X
iajs-3045	391	3	reem	reem	PROPN
iajs-3045	391	4	,	,	PUNCT
iajs-3045	391	5	t.	t.	PROPN
iajs-3045	391	6	abdulqader	abdulqader	NOUN
iajs-3045	391	7	.	.	PUNCT
iajs-3045	391	8	;	;	PUNCT
iajs-3045	391	9	zinah	zinah	PROPN
iajs-3045	391	10	,	,	PUNCT
iajs-3045	391	11	t.	t.	PROPN
iajs-3045	391	12	abdulqader	abdulqader	NOUN
iajs-3045	391	13	.	.	PUNCT
iajs-3045	391	14	;	;	PUNCT
iajs-3045	392	1	haibat	haibat	PROPN
iajs-3045	392	2	,	,	PUNCT
iajs-3045	392	3	k.	k.	PROPN
iajs-3045	392	4	mohammedali	mohammedali	PROPN
iajs-3045	392	5	.	.	PUNCT
iajs-3045	392	6	;	;	PUNCT
iajs-3045	392	7	akram	akram	PROPN
iajs-3045	392	8	,	,	PUNCT
iajs-3045	392	9	s.	s.	PROPN
iajs-3045	392	10	mohammed	mohammed	PROPN
iajs-3045	392	11	.	.	PROPN
iajs-3045	392	12	,	,	PUNCT
iajs-3045	392	13	soc	soc	PROPN
iajs-3045	392	14	-	-	PUNCT
iajs-3045	392	15	qp2	qp2	NOUN
iajs-3045	392	16	-	-	PUNCT
iajs-3045	392	17	absorbing	absorb	VERB
iajs-3045	392	18	submodules	submodule	NOUN
iajs-3045	392	19	,	,	PUNCT
iajs-3045	392	20	turkish	turkish	ADJ
iajs-3045	392	21	journal	journal	NOUN
iajs-3045	392	22	of	of	ADP
iajs-3045	392	23	computer	computer	NOUN
iajs-3045	392	24	and	and	CCONJ
iajs-3045	392	25	mathematics	mathematic	NOUN
iajs-3045	392	26	education	education	NOUN
iajs-3045	392	27	,	,	PUNCT
iajs-3045	392	28	2022	2022	NUM
iajs-3045	392	29	,	,	PUNCT
iajs-3045	392	30	13(3	13(3	NUM
iajs-3045	392	31	)	)	PUNCT
iajs-3045	392	32	,	,	PUNCT
iajs-3045	392	33	761	761	NUM
iajs-3045	392	34	-	-	SYM
iajs-3045	392	35	770	770	NUM
iajs-3045	392	36	.	.	NOUN
iajs-3045	392	37	4	4	NUM
iajs-3045	392	38	.	.	X
iajs-3045	392	39	omar	omar	PROPN
iajs-3045	392	40	,	,	PUNCT
iajs-3045	392	41	a.abdalla	a.abdalla	ADV
iajs-3045	392	42	.	.	PUNCT
iajs-3045	392	43	;	;	PUNCT
iajs-3045	392	44	haibat	haibat	PROPN
iajs-3045	392	45	,	,	PUNCT
iajs-3045	392	46	k.	k.	PROPN
iajs-3045	392	47	mohammedali	mohammedali	PROPN
iajs-3045	392	48	.	.	PUNCT
iajs-3045	393	1	,	,	PUNCT
iajs-3045	393	2	extend	extend	VERB
iajs-3045	393	3	nearly	nearly	ADV
iajs-3045	393	4	pseudo	pseudo	ADJ
iajs-3045	393	5	quasi-2	quasi-2	ADJ
iajs-3045	393	6	-	-	PUNCT
iajs-3045	393	7	absorbing	absorbing	ADJ
iajs-3045	393	8	submodules(i	submodules(i	NOUN
iajs-3045	393	9	)	)	PUNCT
iajs-3045	393	10	,	,	PUNCT
iajs-3045	393	11	ibn	ibn	PROPN
iajs-3045	393	12	al	al	PROPN
iajs-3045	393	13	-	-	PUNCT
iajs-3045	393	14	haitham	haitham	PROPN
iajs-3045	393	15	journal	journal	PROPN
iajs-3045	393	16	for	for	ADP
iajs-3045	393	17	pure	pure	ADJ
iajs-3045	393	18	and	and	CCONJ
iajs-3045	393	19	applied	applied	ADJ
iajs-3045	393	20	scince	scince	NOUN
iajs-3045	393	21	,	,	PUNCT
iajs-3045	393	22	2022	2022	NUM
iajs-3045	393	23	,	,	PUNCT
iajs-3045	393	24	too	too	ADV
iajs-3045	393	25	aper	aper	ADV
iajs-3045	393	26	.	.	PUNCT
iajs-3045	394	1	ihjpas	ihjpas	PROPN
iajs-3045	394	2	.	.	PUNCT
iajs-3045	395	1	36(2)2023	36(2)2023	NUM
iajs-3045	395	2	419	419	NUM
iajs-3045	395	3	5	5	NUM
iajs-3045	395	4	.	.	PUNCT
iajs-3045	396	1	el	el	NOUN
iajs-3045	396	2	-	-	PUNCT
iajs-3045	396	3	bast	bast	NOUN
iajs-3045	396	4	,	,	PUNCT
iajs-3045	396	5	z.	z.	PROPN
iajs-3045	397	1	a	a	PRON
iajs-3045	397	2	.;smith	.;smith	PROPN
iajs-3045	397	3	,	,	PUNCT
iajs-3045	397	4	p.	p.	PROPN
iajs-3045	397	5	f.	f.	PROPN
iajs-3045	397	6	,	,	PUNCT
iajs-3045	397	7	multiplication	multiplication	NOUN
iajs-3045	397	8	modules	module	NOUN
iajs-3045	397	9	,	,	PUNCT
iajs-3045	397	10	comm	comm	NOUN
iajs-3045	397	11	.	.	PUNCT
iajs-3045	398	1	in	in	ADP
iajs-3045	398	2	algebra	algebra	NOUN
iajs-3045	398	3	,	,	PUNCT
iajs-3045	398	4	1988	1988	NUM
iajs-3045	398	5	,	,	PUNCT
iajs-3045	398	6	16(4	16(4	NUM
iajs-3045	398	7	)	)	PUNCT
iajs-3045	398	8	,	,	PUNCT
iajs-3045	398	9	755	755	NUM
iajs-3045	398	10	-	-	SYM
iajs-3045	398	11	779	779	NUM
iajs-3045	398	12	.	.	PROPN
iajs-3045	398	13	6	6	NUM
iajs-3045	398	14	.	.	X
iajs-3045	398	15	kash	kash	PROPN
iajs-3045	398	16	,	,	PUNCT
iajs-3045	398	17	f.	f.	PROPN
iajs-3045	398	18	modules	modules	PROPN
iajs-3045	398	19	.	.	PUNCT
iajs-3045	398	20	,	,	PUNCT
iajs-3045	398	21	rings	rings	PROPN
iajs-3045	398	22	.	.	PUNCT
iajs-3045	399	1	london	london	PROPN
iajs-3045	399	2	math	math	PROPN
iajs-3045	399	3	.	.	PUNCT
iajs-3045	400	1	soc	soc	PROPN
iajs-3045	400	2	.	.	PUNCT
iajs-3045	401	1	monographs	monographs	PROPN
iajs-3045	401	2	new	new	PROPN
iajs-3045	401	3	york	york	PROPN
iajs-3045	401	4	,	,	PUNCT
iajs-3045	401	5	academic	academic	ADJ
iajs-3045	401	6	press	press	NOUN
iajs-3045	401	7	,	,	PUNCT
iajs-3045	401	8	1982	1982	NUM
iajs-3045	401	9	,	,	PUNCT
iajs-3045	401	10	370	370	NUM
iajs-3045	401	11	.	.	X
iajs-3045	402	1	7	7	X
iajs-3045	402	2	.	.	X
iajs-3045	402	3	ali	ali	PROPN
iajs-3045	402	4	,	,	PUNCT
iajs-3045	402	5	s.	s.	PROPN
iajs-3045	402	6	m.	m.	PROPN
iajs-3045	402	7	,	,	PUNCT
iajs-3045	402	8	on	on	ADP
iajs-3045	402	9	cancellation	cancellation	NOUN
iajs-3045	402	10	modules	module	NOUN
iajs-3045	402	11	,	,	PUNCT
iajs-3045	402	12	m.sc	m.sc	PROPN
iajs-3045	402	13	.	.	PUNCT
iajs-3045	403	1	thesis	thesis	NOUN
iajs-3045	403	2	,	,	PUNCT
iajs-3045	403	3	university	university	NOUN
iajs-3045	403	4	of	of	ADP
iajs-3045	403	5	baghdad	baghdad	PROPN
iajs-3045	403	6	.	.	PUNCT
iajs-3045	404	1	1992	1992	NUM
iajs-3045	404	2	.	.	PUNCT
iajs-3045	405	1	8	8	X
iajs-3045	405	2	.	.	X
iajs-3045	405	3	nuha	nuha	PROPN
iajs-3045	405	4	,	,	PUNCT
iajs-3045	405	5	h.h	h.h	PROPN
iajs-3045	405	6	.	.	PROPN
iajs-3045	405	7	,	,	PUNCT
iajs-3045	405	8	the	the	DET
iajs-3045	405	9	radicals	radical	NOUN
iajs-3045	405	10	of	of	ADP
iajs-3045	405	11	modules	module	NOUN
iajs-3045	405	12	,	,	PUNCT
iajs-3045	405	13	m.sc	m.sc	PROPN
iajs-3045	405	14	.	.	PUNCT
iajs-3045	406	1	thesis	thesis	NOUN
iajs-3045	406	2	,	,	PUNCT
iajs-3045	406	3	university	university	NOUN
iajs-3045	406	4	of	of	ADP
iajs-3045	406	5	baghdad	baghdad	PROPN
iajs-3045	406	6	.	.	PUNCT
iajs-3045	407	1	1996	1996	NUM
iajs-3045	407	2	.	.	PUNCT
iajs-3045	408	1	9	9	X
iajs-3045	408	2	.	.	X
iajs-3045	408	3	barnard	barnard	PROPN
iajs-3045	408	4	,	,	PUNCT
iajs-3045	408	5	a.	a.	NOUN
iajs-3045	408	6	multiplication	multiplication	NOUN
iajs-3045	408	7	modules	module	NOUN
iajs-3045	408	8	,	,	PUNCT
iajs-3045	408	9	journal	journal	NOUN
iajs-3045	408	10	of	of	ADP
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iajs-3045	408	12	,	,	PUNCT
iajs-3045	408	13	(	(	PUNCT
iajs-3045	408	14	7	7	NUM
iajs-3045	408	15	)	)	PUNCT
iajs-3045	408	16	,	,	PUNCT
iajs-3045	408	17	174	174	NUM
iajs-3045	408	18	178	178	NUM
iajs-3045	408	19	.	.	PUNCT
iajs-3045	408	20	1981	1981	NUM
iajs-3045	408	21	.	.	PUNCT
iajs-3045	409	1	7	7	NUM
iajs-3045	409	2	.	.	SYM
iajs-3045	409	3	10	10	NUM
iajs-3045	409	4	.	.	PUNCT
iajs-3045	410	1	behboodi	behboodi	NOUN
iajs-3045	410	2	,	,	PUNCT
iajs-3045	410	3	m.	m.	NOUN
iajs-3045	410	4	;	;	PUNCT
iajs-3045	410	5	koohi	koohi	PROPN
iajs-3045	410	6	,	,	PUNCT
iajs-3045	410	7	h.	h.	NOUN
iajs-3045	410	8	,	,	PUNCT
iajs-3045	410	9	weakly	weakly	ADJ
iajs-3045	410	10	prime	prime	ADJ
iajs-3045	410	11	modules	module	NOUN
iajs-3045	410	12	,	,	PUNCT
iajs-3045	410	13	vietnam	vietnam	PROPN
iajs-3045	410	14	j.	j.	PROPN
iajs-3045	410	15	of	of	ADP
iajs-3045	410	16	math	math	PROPN
iajs-3045	410	17	.	.	PUNCT
iajs-3045	410	18	,	,	PUNCT
iajs-3045	410	19	2004,32(2	2004,32(2	ADJ
iajs-3045	410	20	)	)	PUNCT
iajs-3045	410	21	,	,	PUNCT
iajs-3045	410	22	185	185	NUM
iajs-3045	410	23	-	-	SYM
iajs-3045	410	24	195	195	NUM
iajs-3045	410	25	.	.	NOUN
iajs-3045	410	26	11	11	NUM
iajs-3045	410	27	.	.	PUNCT
iajs-3045	411	1	payman	payman	NOUN
iajs-3045	411	2	,	,	PUNCT
iajs-3045	411	3	m.	m.	PROPN
iajs-3045	411	4	h.	h.	PROPN
iajs-3045	411	5	,	,	PUNCT
iajs-3045	411	6	hollow	hollow	ADJ
iajs-3045	411	7	modules	module	NOUN
iajs-3045	411	8	and	and	CCONJ
iajs-3045	411	9	semi	semi	ADJ
iajs-3045	411	10	hollow	hollow	ADJ
iajs-3045	411	11	modules	module	NOUN
iajs-3045	411	12	,	,	PUNCT
iajs-3045	411	13	m.sc	m.sc	PROPN
iajs-3045	411	14	.	.	PUNCT
iajs-3045	412	1	thesis	thesis	NOUN
iajs-3045	412	2	,	,	PUNCT
iajs-3045	412	3	university	university	NOUN
iajs-3045	412	4	of	of	ADP
iajs-3045	412	5	baghdad	baghdad	PROPN
iajs-3045	412	6	.	.	PUNCT
iajs-3045	413	1	2005	2005	NUM
iajs-3045	413	2	.	.	PUNCT
iajs-3045	414	1	12	12	NUM
iajs-3045	414	2	.	.	PUNCT
iajs-3045	415	1	goodearl	goodearl	PROPN
iajs-3045	415	2	,	,	PUNCT
iajs-3045	415	3	k.	k.	PROPN
iajs-3045	415	4	r.	r.	PROPN
iajs-3045	415	5	,	,	PUNCT
iajs-3045	415	6	ring	ring	NOUN
iajs-3045	415	7	theory	theory	NOUN
iajs-3045	415	8	,	,	PUNCT
iajs-3045	415	9	marcel	marcel	PROPN
iajs-3045	415	10	dekker	dekker	PROPN
iajs-3045	415	11	,	,	PUNCT
iajs-3045	415	12	inc	inc	PROPN
iajs-3045	415	13	.	.	PROPN
iajs-3045	415	14	new	new	PROPN
iajs-3045	415	15	york	york	PROPN
iajs-3045	415	16	and	and	CCONJ
iajs-3045	415	17	basel	basel	PROPN
iajs-3045	415	18	,	,	PUNCT
iajs-3045	415	19	206	206	NUM
iajs-3045	415	20	.	.	PUNCT
iajs-3045	415	21	1976	1976	NUM
iajs-3045	415	22	.	.	PUNCT
iajs-3045	416	1	13	13	NUM
iajs-3045	416	2	.	.	X
iajs-3045	416	3	smith	smith	PROPN
iajs-3045	416	4	,	,	PUNCT
iajs-3045	416	5	p.f	p.f	PROPN
iajs-3045	416	6	.	.	PUNCT
iajs-3045	417	1	some	some	DET
iajs-3045	417	2	remarks	remark	NOUN
iajs-3045	417	3	of	of	ADP
iajs-3045	417	4	multiplication	multiplication	NOUN
iajs-3045	417	5	modules	module	NOUN
iajs-3045	417	6	,	,	PUNCT
iajs-3045	417	7	arch	arch	NOUN
iajs-3045	417	8	.	.	PUNCT
iajs-3045	418	1	math	math	NOUN
iajs-3045	418	2	,	,	PUNCT
iajs-3045	418	3	1986	1986	NUM
iajs-3045	418	4	,	,	PUNCT
iajs-3045	418	5	(	(	PUNCT
iajs-3045	418	6	50	50	NUM
iajs-3045	418	7	)	)	PUNCT
iajs-3045	418	8	,	,	PUNCT
iajs-3045	418	9	223	223	NUM
iajs-3045	418	10	-	-	SYM
iajs-3045	418	11	225	225	NUM
iajs-3045	418	12	.	.	PUNCT
iajs-3045	419	1	14	14	NUM
iajs-3045	419	2	.	.	PUNCT
iajs-3045	420	1	zelmanowitz	zelmanowitz	PROPN
iajs-3045	420	2	,	,	PUNCT
iajs-3045	420	3	j.	j.	PROPN
iajs-3045	420	4	,	,	PUNCT
iajs-3045	420	5	regular	regular	ADJ
iajs-3045	420	6	modules	module	NOUN
iajs-3045	420	7	,	,	PUNCT
iajs-3045	420	8	trans	tran	NOUN
iajs-3045	420	9	.	.	PROPN
iajs-3045	421	1	amerecan	amerecan	PROPN
iajs-3045	421	2	,	,	PUNCT
iajs-3045	421	3	math	math	NOUN
iajs-3045	421	4	.	.	PUNCT
iajs-3045	422	1	soc	soc	PROPN
iajs-3045	422	2	.	.	PUNCT
iajs-3045	423	1	,	,	PUNCT
iajs-3045	423	2	1973	1973	NUM
iajs-3045	423	3	(	(	PUNCT
iajs-3045	423	4	163	163	NUM
iajs-3045	423	5	)	)	PUNCT
iajs-3045	423	6	,	,	PUNCT
iajs-3045	423	7	341	341	NUM
iajs-3045	423	8	-	-	SYM
iajs-3045	423	9	355	355	NUM
iajs-3045	423	10	.	.	PUNCT
iajs-3045	424	1	15	15	NUM
iajs-3045	424	2	.	.	X
iajs-3045	425	1	mijbass	mijbass	PROPN
iajs-3045	425	2	,	,	PUNCT
iajs-3045	425	3	a.	a.	NOUN
iajs-3045	425	4	s.	s.	PROPN
iajs-3045	425	5	on	on	ADP
iajs-3045	425	6	cancellation	cancellation	NOUN
iajs-3045	425	7	modules	module	NOUN
iajs-3045	425	8	,	,	PUNCT
iajs-3045	425	9	m.sc	m.sc	PROPN
iajs-3045	425	10	.	.	PUNCT
iajs-3045	426	1	theses	theses	PROPN
iajs-3045	426	2	,	,	PUNCT
iajs-3045	426	3	university	university	NOUN
iajs-3045	426	4	of	of	ADP
iajs-3045	426	5	baghdad	baghdad	PROPN
iajs-3045	426	6	.	.	PUNCT
iajs-3045	427	1	1993	1993	NUM
iajs-3045	427	2	.	.	PUNCT