id	sid	tid	token	lemma	pos
ejpam-5210	1	1	european	european	PROPN
ejpam-5210	1	2	journal	journal	PROPN
ejpam-5210	1	3	of	of	ADP
ejpam-5210	1	4	pure	pure	ADJ
ejpam-5210	1	5	and	and	CCONJ
ejpam-5210	1	6	applied	apply	VERB
ejpam-5210	1	7	mathematics	mathematic	NOUN
ejpam-5210	1	8	vol	vol	NOUN
ejpam-5210	1	9	.	.	PROPN
ejpam-5210	2	1	17	17	NUM
ejpam-5210	2	2	,	,	PUNCT
ejpam-5210	2	3	no	no	INTJ
ejpam-5210	2	4	.	.	NOUN
ejpam-5210	2	5	3	3	NUM
ejpam-5210	2	6	,	,	PUNCT
ejpam-5210	2	7	2024	2024	NUM
ejpam-5210	2	8	,	,	PUNCT
ejpam-5210	2	9	2264	2264	NUM
ejpam-5210	2	10	-	-	SYM
ejpam-5210	2	11	2275	2275	NUM
ejpam-5210	2	12	issn	issn	VERB
ejpam-5210	2	13	1307	1307	NUM
ejpam-5210	2	14	-	-	SYM
ejpam-5210	2	15	5543	5543	NUM
ejpam-5210	2	16	–	–	PUNCT
ejpam-5210	2	17	ejpam.com	ejpam.com	X
ejpam-5210	2	18	published	publish	VERB
ejpam-5210	2	19	by	by	ADP
ejpam-5210	2	20	new	new	PROPN
ejpam-5210	2	21	york	york	PROPN
ejpam-5210	2	22	business	business	PROPN
ejpam-5210	2	23	global	global	PROPN
ejpam-5210	2	24	d	d	ADJ
ejpam-5210	2	25	-	-	PUNCT
ejpam-5210	2	26	semiprime	semiprime	NOUN
ejpam-5210	2	27	rings	ring	NOUN
ejpam-5210	2	28	maram	maram	PROPN
ejpam-5210	2	29	alosaimi1,∗	alosaimi1,∗	PROPN
ejpam-5210	2	30	,	,	PUNCT
ejpam-5210	2	31	ahmad	ahmad	PROPN
ejpam-5210	2	32	al	al	PROPN
ejpam-5210	2	33	khalaf1	khalaf1	PROPN
ejpam-5210	2	34	,	,	PUNCT
ejpam-5210	2	35	rohaidah	rohaidah	PROPN
ejpam-5210	2	36	masri2	masri2	PROPN
ejpam-5210	2	37	,	,	PUNCT
ejpam-5210	2	38	iman	iman	NOUN
ejpam-5210	2	39	taha1	taha1	NOUN
ejpam-5210	2	40	1	1	NUM
ejpam-5210	2	41	department	department	NOUN
ejpam-5210	2	42	of	of	ADP
ejpam-5210	2	43	mathematics	mathematic	NOUN
ejpam-5210	2	44	and	and	CCONJ
ejpam-5210	2	45	statistic	statistic	NOUN
ejpam-5210	2	46	,	,	PUNCT
ejpam-5210	2	47	faculty	faculty	NOUN
ejpam-5210	2	48	of	of	ADP
ejpam-5210	2	49	sciences	science	NOUN
ejpam-5210	2	50	,	,	PUNCT
ejpam-5210	2	51	imam	imam	PROPN
ejpam-5210	2	52	mohammad	mohammad	PROPN
ejpam-5210	2	53	ibn	ibn	PROPN
ejpam-5210	2	54	saud	saud	PROPN
ejpam-5210	2	55	islamic	islamic	PROPN
ejpam-5210	2	56	university	university	PROPN
ejpam-5210	2	57	,	,	PUNCT
ejpam-5210	2	58	riyadh	riyadh	PROPN
ejpam-5210	2	59	,	,	PUNCT
ejpam-5210	2	60	riyadh	riyadh	PROPN
ejpam-5210	2	61	,	,	PUNCT
ejpam-5210	2	62	saudi	saudi	PROPN
ejpam-5210	2	63	arabia	arabia	PROPN
ejpam-5210	2	64	2	2	NUM
ejpam-5210	2	65	department	department	NOUN
ejpam-5210	2	66	of	of	ADP
ejpam-5210	2	67	mathematics	mathematic	NOUN
ejpam-5210	2	68	,	,	PUNCT
ejpam-5210	2	69	faculty	faculty	NOUN
ejpam-5210	2	70	of	of	ADP
ejpam-5210	2	71	sciences	science	NOUN
ejpam-5210	2	72	and	and	CCONJ
ejpam-5210	2	73	mathematics	mathematic	NOUN
ejpam-5210	2	74	,	,	PUNCT
ejpam-5210	2	75	sultan	sultan	PROPN
ejpam-5210	2	76	idris	idris	PROPN
ejpam-5210	2	77	universiti	universiti	PROPN
ejpam-5210	2	78	,	,	PUNCT
ejpam-5210	2	79	tanjong	tanjong	PROPN
ejpam-5210	2	80	malim	malim	PROPN
ejpam-5210	2	81	,	,	PUNCT
ejpam-5210	2	82	perak	perak	PROPN
ejpam-5210	2	83	,	,	PUNCT
ejpam-5210	2	84	malaysia	malaysia	PROPN
ejpam-5210	2	85	abstract	abstract	NOUN
ejpam-5210	2	86	.	.	PUNCT
ejpam-5210	3	1	let	let	VERB
ejpam-5210	3	2	r	r	PRON
ejpam-5210	3	3	be	be	AUX
ejpam-5210	3	4	an	an	DET
ejpam-5210	3	5	associative	associative	ADJ
ejpam-5210	3	6	and	and	CCONJ
ejpam-5210	3	7	2	2	NUM
ejpam-5210	3	8	-	-	PUNCT
ejpam-5210	3	9	torsion	torsion	NOUN
ejpam-5210	3	10	-	-	PUNCT
ejpam-5210	3	11	free	free	ADJ
ejpam-5210	3	12	ring	ring	NOUN
ejpam-5210	3	13	with	with	ADP
ejpam-5210	3	14	an	an	DET
ejpam-5210	3	15	identity	identity	NOUN
ejpam-5210	3	16	.	.	PUNCT
ejpam-5210	4	1	in	in	ADP
ejpam-5210	4	2	this	this	DET
ejpam-5210	4	3	work	work	NOUN
ejpam-5210	4	4	,	,	PUNCT
ejpam-5210	4	5	we	we	PRON
ejpam-5210	4	6	will	will	AUX
ejpam-5210	4	7	generaliz	generaliz	VERB
ejpam-5210	4	8	the	the	DET
ejpam-5210	4	9	results	result	NOUN
ejpam-5210	4	10	of	of	ADP
ejpam-5210	4	11	differentially	differentially	ADV
ejpam-5210	4	12	prime	prime	ADJ
ejpam-5210	4	13	rings	ring	NOUN
ejpam-5210	4	14	in	in	ADP
ejpam-5210	4	15	[	[	X
ejpam-5210	4	16	18	18	NUM
ejpam-5210	4	17	]	]	PUNCT
ejpam-5210	4	18	by	by	ADP
ejpam-5210	4	19	applying	apply	VERB
ejpam-5210	4	20	the	the	DET
ejpam-5210	4	21	hypotheses	hypothesis	NOUN
ejpam-5210	4	22	in	in	ADP
ejpam-5210	4	23	a	a	DET
ejpam-5210	4	24	differentially	differentially	ADV
ejpam-5210	4	25	semiprime	semiprime	NOUN
ejpam-5210	4	26	rings	ring	NOUN
ejpam-5210	4	27	.	.	PUNCT
ejpam-5210	5	1	in	in	ADP
ejpam-5210	5	2	particular	particular	ADJ
ejpam-5210	5	3	,	,	PUNCT
ejpam-5210	5	4	we	we	PRON
ejpam-5210	5	5	have	have	AUX
ejpam-5210	5	6	proved	prove	VERB
ejpam-5210	5	7	that	that	SCONJ
ejpam-5210	5	8	if	if	SCONJ
ejpam-5210	5	9	r	r	NOUN
ejpam-5210	5	10	is	be	AUX
ejpam-5210	5	11	a	a	DET
ejpam-5210	5	12	d	d	ADJ
ejpam-5210	5	13	-	-	PUNCT
ejpam-5210	5	14	semiprime	semiprime	ADJ
ejpam-5210	5	15	ring	ring	NOUN
ejpam-5210	5	16	,	,	PUNCT
ejpam-5210	5	17	then	then	ADV
ejpam-5210	5	18	either	either	CCONJ
ejpam-5210	5	19	r	r	NOUN
ejpam-5210	5	20	is	be	AUX
ejpam-5210	5	21	a	a	DET
ejpam-5210	5	22	commutative	commutative	ADJ
ejpam-5210	5	23	ring	ring	NOUN
ejpam-5210	5	24	or	or	CCONJ
ejpam-5210	5	25	d	d	NOUN
ejpam-5210	5	26	is	be	AUX
ejpam-5210	5	27	a	a	DET
ejpam-5210	5	28	semiprime	semiprime	NOUN
ejpam-5210	5	29	ring	ring	NOUN
ejpam-5210	5	30	.	.	PUNCT
ejpam-5210	6	1	2020	2020	NUM
ejpam-5210	6	2	mathematics	mathematic	NOUN
ejpam-5210	6	3	subject	subject	NOUN
ejpam-5210	6	4	classifications	classification	NOUN
ejpam-5210	6	5	:	:	PUNCT
ejpam-5210	6	6	16w25	16w25	NUM
ejpam-5210	6	7	,	,	PUNCT
ejpam-5210	6	8	16n60	16n60	NUM
ejpam-5210	6	9	key	key	ADJ
ejpam-5210	6	10	words	word	NOUN
ejpam-5210	6	11	and	and	CCONJ
ejpam-5210	6	12	phrases	phrase	NOUN
ejpam-5210	6	13	:	:	PUNCT
ejpam-5210	6	14	derivation	derivation	NOUN
ejpam-5210	6	15	,	,	PUNCT
ejpam-5210	6	16	semiprime	semiprime	NOUN
ejpam-5210	6	17	ring	ring	NOUN
ejpam-5210	6	18	,	,	PUNCT
ejpam-5210	6	19	δ	δ	PROPN
ejpam-5210	6	20	-	-	PUNCT
ejpam-5210	6	21	semiprime	semiprime	NOUN
ejpam-5210	6	22	ring	ring	NOUN
ejpam-5210	6	23	,	,	PUNCT
ejpam-5210	6	24	δ	δ	PROPN
ejpam-5210	6	25	-	-	PUNCT
ejpam-5210	6	26	ideal	ideal	ADJ
ejpam-5210	6	27	1	1	NUM
ejpam-5210	6	28	.	.	PUNCT
ejpam-5210	7	1	introduction	introduction	NOUN
ejpam-5210	7	2	let	let	VERB
ejpam-5210	7	3	r	r	PRON
ejpam-5210	7	4	be	be	AUX
ejpam-5210	7	5	an	an	DET
ejpam-5210	7	6	associative	associative	ADJ
ejpam-5210	7	7	ring	ring	NOUN
ejpam-5210	7	8	with	with	ADP
ejpam-5210	7	9	an	an	DET
ejpam-5210	7	10	identity	identity	NOUN
ejpam-5210	7	11	element	element	NOUN
ejpam-5210	7	12	.	.	PUNCT
ejpam-5210	8	1	we	we	PRON
ejpam-5210	8	2	say	say	VERB
ejpam-5210	8	3	that	that	SCONJ
ejpam-5210	8	4	r	r	NOUN
ejpam-5210	8	5	is	be	AUX
ejpam-5210	8	6	2	2	NUM
ejpam-5210	8	7	-	-	PUNCT
ejpam-5210	8	8	torsionfree	torsionfree	NOUN
ejpam-5210	8	9	if	if	SCONJ
ejpam-5210	8	10	for	for	ADP
ejpam-5210	8	11	any	any	DET
ejpam-5210	8	12	r	r	NOUN
ejpam-5210	8	13	∈	∈	NOUN
ejpam-5210	8	14	r	r	NOUN
ejpam-5210	8	15	and	and	CCONJ
ejpam-5210	8	16	an	an	DET
ejpam-5210	8	17	integer	integer	NOUN
ejpam-5210	8	18	n	n	CCONJ
ejpam-5210	8	19	,	,	PUNCT
ejpam-5210	9	1	the	the	DET
ejpam-5210	9	2	condition	condition	NOUN
ejpam-5210	9	3	2r	2r	NUM
ejpam-5210	9	4	=	=	SYM
ejpam-5210	9	5	0	0	NUM
ejpam-5210	9	6	holds	hold	VERB
ejpam-5210	9	7	if	if	SCONJ
ejpam-5210	9	8	and	and	CCONJ
ejpam-5210	9	9	only	only	ADV
ejpam-5210	9	10	if	if	SCONJ
ejpam-5210	9	11	r	r	NOUN
ejpam-5210	9	12	=	=	SYM
ejpam-5210	9	13	0	0	NUM
ejpam-5210	9	14	.	.	PUNCT
ejpam-5210	9	15	z(r	z(r	NOUN
ejpam-5210	9	16	)	)	PUNCT
ejpam-5210	9	17	is	be	AUX
ejpam-5210	9	18	the	the	DET
ejpam-5210	9	19	center	center	NOUN
ejpam-5210	9	20	of	of	ADP
ejpam-5210	9	21	r.	r.	PROPN
ejpam-5210	9	22	d	d	PROPN
ejpam-5210	9	23	is	be	AUX
ejpam-5210	9	24	the	the	DET
ejpam-5210	9	25	set	set	NOUN
ejpam-5210	9	26	of	of	ADP
ejpam-5210	9	27	all	all	DET
ejpam-5210	9	28	derivations	derivation	NOUN
ejpam-5210	9	29	in	in	ADP
ejpam-5210	9	30	r	r	NOUN
ejpam-5210	9	31	and	and	CCONJ
ejpam-5210	9	32	�	�	PROPN
ejpam-5210	9	33	is	be	AUX
ejpam-5210	9	34	a	a	DET
ejpam-5210	9	35	non	non	ADJ
ejpam-5210	9	36	-	-	ADJ
ejpam-5210	9	37	empty	empty	ADJ
ejpam-5210	9	38	subset	subset	NOUN
ejpam-5210	9	39	of	of	ADP
ejpam-5210	9	40	d.	d.	PROPN
ejpam-5210	9	41	an	an	DET
ejpam-5210	9	42	additive	additive	ADJ
ejpam-5210	9	43	subgroup	subgroup	NOUN
ejpam-5210	9	44	a	a	PROPN
ejpam-5210	9	45	is	be	AUX
ejpam-5210	9	46	said	say	VERB
ejpam-5210	9	47	to	to	PART
ejpam-5210	9	48	be	be	AUX
ejpam-5210	9	49	a	a	DET
ejpam-5210	9	50	lie	lie	NOUN
ejpam-5210	9	51	ideal	ideal	NOUN
ejpam-5210	9	52	of	of	ADP
ejpam-5210	9	53	r	r	NOUN
ejpam-5210	9	54	if	if	SCONJ
ejpam-5210	9	55	[	[	X
ejpam-5210	9	56	r	r	X
ejpam-5210	9	57	,	,	PUNCT
ejpam-5210	9	58	a	a	PRON
ejpam-5210	9	59	]	]	X
ejpam-5210	9	60	∈	∈	PROPN
ejpam-5210	9	61	a	a	PRON
ejpam-5210	9	62	,	,	PUNCT
ejpam-5210	9	63	for	for	ADP
ejpam-5210	9	64	all	all	DET
ejpam-5210	9	65	r	r	NOUN
ejpam-5210	9	66	∈	∈	NOUN
ejpam-5210	9	67	r	r	NOUN
ejpam-5210	9	68	and	and	CCONJ
ejpam-5210	9	69	a	a	DET
ejpam-5210	9	70	∈	∈	NOUN
ejpam-5210	9	71	a.	a.	NOUN
ejpam-5210	9	72	a	a	DET
ejpam-5210	9	73	lie	lie	NOUN
ejpam-5210	9	74	ideal	ideal	NOUN
ejpam-5210	9	75	a	a	PRON
ejpam-5210	9	76	of	of	ADP
ejpam-5210	9	77	r	r	NOUN
ejpam-5210	9	78	is	be	AUX
ejpam-5210	9	79	called	call	VERB
ejpam-5210	9	80	�	�	NOUN
ejpam-5210	9	81	ideal	ideal	NOUN
ejpam-5210	9	82	if	if	SCONJ
ejpam-5210	9	83	δ(a	δ(a	PROPN
ejpam-5210	9	84	)	)	PUNCT
ejpam-5210	9	85	∈	∈	PROPN
ejpam-5210	9	86	a	a	PRON
ejpam-5210	9	87	,	,	PUNCT
ejpam-5210	9	88	for	for	ADP
ejpam-5210	9	89	all	all	DET
ejpam-5210	9	90	a	a	DET
ejpam-5210	9	91	∈	∈	PROPN
ejpam-5210	10	1	a	a	DET
ejpam-5210	10	2	and	and	CCONJ
ejpam-5210	10	3	δ	δ	PROPN
ejpam-5210	10	4	∈	∈	PROPN
ejpam-5210	10	5	�	�	PROPN
ejpam-5210	10	6	.	.	PUNCT
ejpam-5210	10	7	annt	annt	PROPN
ejpam-5210	10	8	=	=	PRON
ejpam-5210	10	9	{	{	PUNCT
ejpam-5210	10	10	x	x	SYM
ejpam-5210	10	11	∈	∈	PROPN
ejpam-5210	10	12	r	r	NOUN
ejpam-5210	10	13	|	|	NOUN
ejpam-5210	10	14	xt	xt	PUNCT
ejpam-5210	11	1	=	=	SYM
ejpam-5210	11	2	tx	tx	PROPN
ejpam-5210	11	3	=	=	SYM
ejpam-5210	11	4	0	0	NUM
ejpam-5210	11	5	}	}	PUNCT
ejpam-5210	11	6	is	be	AUX
ejpam-5210	11	7	the	the	DET
ejpam-5210	11	8	annihilator	annihilator	NOUN
ejpam-5210	11	9	of	of	ADP
ejpam-5210	11	10	t	t	PROPN
ejpam-5210	11	11	.	.	PUNCT
ejpam-5210	12	1	if	if	SCONJ
ejpam-5210	12	2	a	a	DET
ejpam-5210	12	3	∈	∈	PROPN
ejpam-5210	12	4	r	r	NOUN
ejpam-5210	12	5	,	,	PUNCT
ejpam-5210	12	6	then	then	ADV
ejpam-5210	12	7	∂a(x	∂a(x	PROPN
ejpam-5210	12	8	)	)	PUNCT
ejpam-5210	12	9	=	=	PUNCT
ejpam-5210	13	1	[	[	X
ejpam-5210	13	2	x	x	X
ejpam-5210	13	3	,	,	PUNCT
ejpam-5210	13	4	a	a	X
ejpam-5210	13	5	]	]	X
ejpam-5210	13	6	=	=	SYM
ejpam-5210	13	7	ax	ax	NOUN
ejpam-5210	13	8	−	−	PROPN
ejpam-5210	13	9	xa	xa	PROPN
ejpam-5210	13	10	is	be	AUX
ejpam-5210	13	11	an	an	DET
ejpam-5210	13	12	inner	inner	ADJ
ejpam-5210	13	13	derivation	derivation	NOUN
ejpam-5210	13	14	of	of	ADP
ejpam-5210	13	15	r	r	NOUN
ejpam-5210	13	16	induced	induce	VERB
ejpam-5210	13	17	by	by	ADP
ejpam-5210	13	18	a	a	DET
ejpam-5210	13	19	∈	∈	PROPN
ejpam-5210	13	20	r	r	NOUN
ejpam-5210	13	21	,	,	PUNCT
ejpam-5210	13	22	where	where	SCONJ
ejpam-5210	13	23	∂a	∂a	PROPN
ejpam-5210	13	24	∈	∈	PROPN
ejpam-5210	13	25	d.	d.	PROPN
ejpam-5210	13	26	i	i	PROPN
ejpam-5210	13	27	d	d	PROPN
ejpam-5210	13	28	=	=	PRON
ejpam-5210	13	29	{	{	PUNCT
ejpam-5210	13	30	∂a	∂a	NOUN
ejpam-5210	13	31	|	|	ADV
ejpam-5210	13	32	a	a	DET
ejpam-5210	13	33	∈	∈	NOUN
ejpam-5210	13	34	r	r	NOUN
ejpam-5210	13	35	}	}	PUNCT
ejpam-5210	13	36	is	be	AUX
ejpam-5210	13	37	an	an	DET
ejpam-5210	13	38	ideal	ideal	NOUN
ejpam-5210	13	39	of	of	ADP
ejpam-5210	13	40	a	a	DET
ejpam-5210	13	41	ring	ring	NOUN
ejpam-5210	13	42	d	d	NOUN
ejpam-5210	13	43	,	,	PUNCT
ejpam-5210	13	44	see	see	VERB
ejpam-5210	13	45	[	[	X
ejpam-5210	13	46	13	13	NUM
ejpam-5210	13	47	]	]	PUNCT
ejpam-5210	13	48	.	.	PUNCT
ejpam-5210	14	1	a	a	DET
ejpam-5210	14	2	ring	ring	NOUN
ejpam-5210	14	3	r	r	NOUN
ejpam-5210	14	4	is	be	AUX
ejpam-5210	14	5	called	call	VERB
ejpam-5210	14	6	a	a	DET
ejpam-5210	14	7	�	�	NOUN
ejpam-5210	14	8	-prime	-prime	NOUN
ejpam-5210	14	9	(	(	PUNCT
ejpam-5210	14	10	differentilly	differentilly	ADV
ejpam-5210	14	11	prime	prime	ADJ
ejpam-5210	14	12	)	)	PUNCT
ejpam-5210	14	13	if	if	SCONJ
ejpam-5210	14	14	for	for	ADP
ejpam-5210	14	15	each	each	DET
ejpam-5210	14	16	�	�	NOUN
ejpam-5210	14	17	-ideals	-ideal	NOUN
ejpam-5210	14	18	a	a	PRON
ejpam-5210	14	19	and	and	CCONJ
ejpam-5210	14	20	b	b	NOUN
ejpam-5210	14	21	of	of	ADP
ejpam-5210	14	22	r	r	NOUN
ejpam-5210	14	23	with	with	ADP
ejpam-5210	14	24	ab	ab	PROPN
ejpam-5210	14	25	=	=	SYM
ejpam-5210	14	26	0	0	PROPN
ejpam-5210	14	27	,	,	PUNCT
ejpam-5210	14	28	implies	imply	VERB
ejpam-5210	15	1	that	that	SCONJ
ejpam-5210	15	2	a	a	DET
ejpam-5210	15	3	=	=	SYM
ejpam-5210	15	4	0	0	NUM
ejpam-5210	15	5	or	or	CCONJ
ejpam-5210	15	6	b	b	NOUN
ejpam-5210	15	7	=	=	SYM
ejpam-5210	15	8	0	0	PROPN
ejpam-5210	15	9	.	.	PUNCT
ejpam-5210	16	1	a	a	DET
ejpam-5210	16	2	ring	ring	NOUN
ejpam-5210	16	3	r	r	NOUN
ejpam-5210	16	4	is	be	AUX
ejpam-5210	16	5	said	say	VERB
ejpam-5210	16	6	to	to	PART
ejpam-5210	16	7	be	be	AUX
ejpam-5210	16	8	�	�	PROPN
ejpam-5210	16	9	-semiprime	-semiprime	PROPN
ejpam-5210	16	10	(	(	PUNCT
ejpam-5210	16	11	differentilly	differentilly	ADV
ejpam-5210	16	12	semiprime	semiprime	NOUN
ejpam-5210	16	13	)	)	PUNCT
ejpam-5210	16	14	if	if	SCONJ
ejpam-5210	16	15	for	for	ADP
ejpam-5210	16	16	every	every	DET
ejpam-5210	16	17	�	�	NOUN
ejpam-5210	16	18	-ideal	-ideal	ADJ
ejpam-5210	16	19	i	i	PRON
ejpam-5210	16	20	of	of	ADP
ejpam-5210	16	21	r	r	NOUN
ejpam-5210	16	22	,	,	PUNCT
ejpam-5210	16	23	the	the	DET
ejpam-5210	16	24	condition	condition	NOUN
ejpam-5210	16	25	i2	i2	NOUN
ejpam-5210	16	26	=	=	SYM
ejpam-5210	16	27	0	0	NUM
ejpam-5210	16	28	,	,	PUNCT
ejpam-5210	16	29	implies	imply	VERB
ejpam-5210	16	30	that	that	SCONJ
ejpam-5210	16	31	i	i	PRON
ejpam-5210	16	32	=	=	NOUN
ejpam-5210	16	33	0	0	X
ejpam-5210	16	34	.	.	PUNCT
ejpam-5210	17	1	c(r	c(r	NOUN
ejpam-5210	17	2	)	)	PUNCT
ejpam-5210	17	3	is	be	AUX
ejpam-5210	17	4	the	the	DET
ejpam-5210	17	5	commutator	commutator	NOUN
ejpam-5210	17	6	ideal	ideal	NOUN
ejpam-5210	17	7	of	of	ADP
ejpam-5210	17	8	r	r	NOUN
ejpam-5210	17	9	and	and	CCONJ
ejpam-5210	17	10	charr	charr	NOUN
ejpam-5210	17	11	is	be	AUX
ejpam-5210	17	12	the	the	DET
ejpam-5210	17	13	characteristic	characteristic	NOUN
ejpam-5210	17	14	of	of	ADP
ejpam-5210	17	15	a	a	DET
ejpam-5210	17	16	ring	ring	NOUN
ejpam-5210	17	17	r.	r.	NOUN
ejpam-5210	17	18	by	by	ADP
ejpam-5210	17	19	z0(r	z0(r	PROPN
ejpam-5210	17	20	)	)	PUNCT
ejpam-5210	17	21	we	we	PRON
ejpam-5210	17	22	denote	denote	VERB
ejpam-5210	17	23	the	the	DET
ejpam-5210	17	24	ideal	ideal	NOUN
ejpam-5210	17	25	of	of	ADP
ejpam-5210	17	26	r	r	NOUN
ejpam-5210	17	27	generated	generate	VERB
ejpam-5210	17	28	by	by	ADP
ejpam-5210	17	29	its	its	PRON
ejpam-5210	17	30	central	central	ADJ
ejpam-5210	17	31	ideals	ideal	NOUN
ejpam-5210	17	32	.	.	PUNCT
ejpam-5210	18	1	the	the	DET
ejpam-5210	18	2	properties	property	NOUN
ejpam-5210	18	3	of	of	ADP
ejpam-5210	18	4	differentially	differentially	ADV
ejpam-5210	18	5	prime	prime	ADJ
ejpam-5210	18	6	rings	ring	NOUN
ejpam-5210	18	7	were	be	AUX
ejpam-5210	18	8	studied	study	VERB
ejpam-5210	18	9	by	by	ADP
ejpam-5210	18	10	herstein	herstein	NOUN
ejpam-5210	18	11	[	[	X
ejpam-5210	18	12	7	7	NUM
ejpam-5210	18	13	,	,	PUNCT
ejpam-5210	18	14	8	8	NUM
ejpam-5210	18	15	]	]	PUNCT
ejpam-5210	18	16	and	and	CCONJ
ejpam-5210	18	17	also	also	ADV
ejpam-5210	18	18	in	in	ADP
ejpam-5210	18	19	his	his	PRON
ejpam-5210	18	20	book	book	NOUN
ejpam-5210	18	21	[	[	X
ejpam-5210	18	22	9	9	NUM
ejpam-5210	18	23	]	]	PUNCT
ejpam-5210	18	24	.	.	PUNCT
ejpam-5210	19	1	after	after	ADP
ejpam-5210	19	2	that	that	PRON
ejpam-5210	19	3	,	,	PUNCT
ejpam-5210	19	4	many	many	ADJ
ejpam-5210	19	5	authors	author	NOUN
ejpam-5210	19	6	have	have	AUX
ejpam-5210	19	7	proved	prove	VERB
ejpam-5210	19	8	some	some	DET
ejpam-5210	19	9	results	result	NOUN
ejpam-5210	19	10	about	about	ADP
ejpam-5210	19	11	this	this	DET
ejpam-5210	19	12	concept	concept	NOUN
ejpam-5210	19	13	,	,	PUNCT
ejpam-5210	19	14	such	such	ADJ
ejpam-5210	19	15	∗corresponding	∗corresponde	VERB
ejpam-5210	19	16	author	author	NOUN
ejpam-5210	19	17	.	.	PUNCT
ejpam-5210	20	1	doi	doi	NOUN
ejpam-5210	20	2	:	:	PUNCT
ejpam-5210	20	3	https://doi.org/10.29020/nybg.ejpam.v17i3.5210	https://doi.org/10.29020/nybg.ejpam.v17i3.5210	PROPN
ejpam-5210	20	4	email	email	NOUN
ejpam-5210	20	5	addresses	address	NOUN
ejpam-5210	20	6	:	:	PUNCT
ejpam-5210	20	7	mhalosaimi@imamu.edu.sa	mhalosaimi@imamu.edu.sa	PROPN
ejpam-5210	20	8	(	(	PUNCT
ejpam-5210	20	9	m.	m.	NOUN
ejpam-5210	20	10	alosaimi	alosaimi	PROPN
ejpam-5210	20	11	)	)	PUNCT
ejpam-5210	20	12	,	,	PUNCT
ejpam-5210	20	13	ajalkalaf@imamu.edu.sa	ajalkalaf@imamu.edu.sa	NOUN
ejpam-5210	20	14	(	(	PUNCT
ejpam-5210	20	15	a.	a.	PROPN
ejpam-5210	20	16	al	al	PROPN
ejpam-5210	20	17	khalaf	khalaf	PROPN
ejpam-5210	20	18	)	)	PUNCT
ejpam-5210	20	19	,	,	PUNCT
ejpam-5210	20	20	rohaidah@fsmt.upsi.edu.my	rohaidah@fsmt.upsi.edu.my	PROPN
ejpam-5210	20	21	(	(	PUNCT
ejpam-5210	20	22	r.	r.	PROPN
ejpam-5210	20	23	masri	masri	PROPN
ejpam-5210	20	24	)	)	PUNCT
ejpam-5210	20	25	,	,	PUNCT
ejpam-5210	20	26	tfaith80@gmail.com	tfaith80@gmail.com	X
ejpam-5210	20	27	(	(	PUNCT
ejpam-5210	20	28	i.	i.	PROPN
ejpam-5210	20	29	taha	taha	PROPN
ejpam-5210	20	30	)	)	PUNCT
ejpam-5210	20	31	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5210	20	32	2264	2264	NUM
ejpam-5210	20	33	©	©	ADP
ejpam-5210	20	34	2024	2024	NUM
ejpam-5210	20	35	ejpam	ejpam	NOUN
ejpam-5210	20	36	all	all	DET
ejpam-5210	20	37	rights	right	NOUN
ejpam-5210	20	38	reserved	reserve	VERB
ejpam-5210	20	39	.	.	PUNCT
ejpam-5210	21	1	m.	m.	NOUN
ejpam-5210	21	2	alosaimi	alosaimi	PROPN
ejpam-5210	21	3	et	et	PROPN
ejpam-5210	21	4	al	al	PROPN
ejpam-5210	21	5	.	.	PUNCT
ejpam-5210	21	6	/	/	SYM
ejpam-5210	21	7	eur	eur	PROPN
ejpam-5210	21	8	.	.	PUNCT
ejpam-5210	22	1	j.	j.	PROPN
ejpam-5210	22	2	pure	pure	PROPN
ejpam-5210	22	3	appl	appl	PROPN
ejpam-5210	22	4	.	.	PROPN
ejpam-5210	22	5	math	math	PROPN
ejpam-5210	22	6	,	,	PUNCT
ejpam-5210	22	7	17	17	NUM
ejpam-5210	22	8	(	(	PUNCT
ejpam-5210	22	9	3	3	NUM
ejpam-5210	22	10	)	)	PUNCT
ejpam-5210	22	11	(	(	PUNCT
ejpam-5210	22	12	2024	2024	NUM
ejpam-5210	22	13	)	)	PUNCT
ejpam-5210	22	14	,	,	PUNCT
ejpam-5210	22	15	2264	2264	NUM
ejpam-5210	22	16	-	-	SYM
ejpam-5210	22	17	2275	2275	NUM
ejpam-5210	22	18	2265	2265	NUM
ejpam-5210	22	19	as	as	ADP
ejpam-5210	22	20	bergen	bergen	NOUN
ejpam-5210	22	21	and	and	CCONJ
ejpam-5210	22	22	herstein	herstein	NOUN
ejpam-5210	22	23	[	[	X
ejpam-5210	22	24	3	3	NUM
ejpam-5210	22	25	]	]	PUNCT
ejpam-5210	22	26	,	,	PUNCT
ejpam-5210	22	27	hirano	hirano	PROPN
ejpam-5210	23	1	[	[	X
ejpam-5210	23	2	11	11	NUM
ejpam-5210	23	3	]	]	PUNCT
ejpam-5210	23	4	,	,	PUNCT
ejpam-5210	23	5	hongan	hongan	VERB
ejpam-5210	23	6	and	and	CCONJ
ejpam-5210	23	7	trzepizur	trzepizur	NOUN
ejpam-5210	24	1	[	[	X
ejpam-5210	24	2	12	12	NUM
ejpam-5210	24	3	]	]	PUNCT
ejpam-5210	24	4	,	,	PUNCT
ejpam-5210	24	5	beidar	beidar	NOUN
ejpam-5210	24	6	and	and	CCONJ
ejpam-5210	24	7	mikhalev	mikhalev	NOUN
ejpam-5210	25	1	[	[	X
ejpam-5210	25	2	2	2	NUM
ejpam-5210	25	3	]	]	PUNCT
ejpam-5210	25	4	,	,	PUNCT
ejpam-5210	25	5	chebotar	chebotar	VERB
ejpam-5210	25	6	and	and	CCONJ
ejpam-5210	25	7	lee	lee	PROPN
ejpam-5210	26	1	[	[	X
ejpam-5210	26	2	6	6	NUM
ejpam-5210	26	3	]	]	PUNCT
ejpam-5210	26	4	and	and	CCONJ
ejpam-5210	26	5	could	could	AUX
ejpam-5210	26	6	be	be	AUX
ejpam-5210	26	7	seen	see	VERB
ejpam-5210	26	8	in	in	ADP
ejpam-5210	26	9	lee	lee	PROPN
ejpam-5210	26	10	and	and	CCONJ
ejpam-5210	26	11	liu	liu	PROPN
ejpam-5210	27	1	[	[	X
ejpam-5210	27	2	23	23	NUM
ejpam-5210	27	3	]	]	PUNCT
ejpam-5210	27	4	.	.	PUNCT
ejpam-5210	28	1	al	al	PROPN
ejpam-5210	28	2	khalaf	khalaf	PROPN
ejpam-5210	28	3	and	and	CCONJ
ejpam-5210	28	4	others	other	NOUN
ejpam-5210	28	5	,	,	PUNCT
ejpam-5210	28	6	see	see	VERB
ejpam-5210	28	7	[	[	X
ejpam-5210	28	8	18	18	NUM
ejpam-5210	28	9	,	,	PUNCT
ejpam-5210	28	10	19	19	NUM
ejpam-5210	28	11	,	,	PUNCT
ejpam-5210	28	12	27	27	NUM
ejpam-5210	28	13	,	,	PUNCT
ejpam-5210	28	14	28	28	NUM
ejpam-5210	28	15	]	]	PUNCT
ejpam-5210	28	16	,	,	PUNCT
ejpam-5210	28	17	have	have	AUX
ejpam-5210	28	18	demonstrated	demonstrate	VERB
ejpam-5210	28	19	the	the	DET
ejpam-5210	28	20	differentially	differentially	ADV
ejpam-5210	28	21	prime	prime	ADJ
ejpam-5210	28	22	rings	ring	NOUN
ejpam-5210	28	23	,	,	PUNCT
ejpam-5210	28	24	simple	simple	ADJ
ejpam-5210	28	25	rings	ring	NOUN
ejpam-5210	28	26	,	,	PUNCT
ejpam-5210	28	27	differentially	differentially	ADV
ejpam-5210	28	28	δprime	δprime	NOUN
ejpam-5210	28	29	rings	ring	NOUN
ejpam-5210	28	30	and	and	CCONJ
ejpam-5210	28	31	reverse	reverse	ADJ
ejpam-5210	28	32	derivation	derivation	NOUN
ejpam-5210	28	33	on	on	ADP
ejpam-5210	28	34	δprime	δprime	NOUN
ejpam-5210	28	35	rings	ring	NOUN
ejpam-5210	28	36	.	.	PUNCT
ejpam-5210	29	1	furthermore	furthermore	ADV
ejpam-5210	29	2	,	,	PUNCT
ejpam-5210	29	3	they	they	PRON
ejpam-5210	29	4	discussed	discuss	VERB
ejpam-5210	29	5	the	the	DET
ejpam-5210	29	6	differentially	differentially	ADV
ejpam-5210	29	7	semiprime	semiprime	NOUN
ejpam-5210	29	8	and	and	CCONJ
ejpam-5210	29	9	semiprime	semiprime	NOUN
ejpam-5210	29	10	gamma	gamma	PROPN
ejpam-5210	29	11	rings	rings	PROPN
ejpam-5210	29	12	,	,	PUNCT
ejpam-5210	29	13	see	see	VERB
ejpam-5210	29	14	that	that	SCONJ
ejpam-5210	29	15	in	in	ADP
ejpam-5210	29	16	[	[	X
ejpam-5210	29	17	20	20	NUM
ejpam-5210	29	18	,	,	PUNCT
ejpam-5210	29	19	21	21	NUM
ejpam-5210	29	20	]	]	PUNCT
ejpam-5210	29	21	.	.	PUNCT
ejpam-5210	30	1	many	many	ADJ
ejpam-5210	30	2	authors	author	NOUN
ejpam-5210	30	3	have	have	AUX
ejpam-5210	30	4	investigated	investigate	VERB
ejpam-5210	30	5	lie	lie	NOUN
ejpam-5210	30	6	rings	ring	NOUN
ejpam-5210	30	7	of	of	ADP
ejpam-5210	30	8	differentially	differentially	ADV
ejpam-5210	30	9	semiprime	semiprime	NOUN
ejpam-5210	30	10	rings	ring	NOUN
ejpam-5210	30	11	as	as	ADP
ejpam-5210	30	12	[	[	X
ejpam-5210	30	13	14	14	NUM
ejpam-5210	30	14	]	]	PUNCT
ejpam-5210	30	15	,	,	PUNCT
ejpam-5210	30	16	[	[	X
ejpam-5210	30	17	24	24	NUM
ejpam-5210	30	18	]	]	PUNCT
ejpam-5210	30	19	and	and	CCONJ
ejpam-5210	30	20	jordan	jordan	PROPN
ejpam-5210	30	21	in	in	ADP
ejpam-5210	30	22	[	[	X
ejpam-5210	30	23	15–17	15–17	NUM
ejpam-5210	30	24	]	]	PUNCT
ejpam-5210	30	25	and	and	CCONJ
ejpam-5210	30	26	nowicki	nowicki	PROPN
ejpam-5210	30	27	[	[	X
ejpam-5210	30	28	25	25	NUM
ejpam-5210	30	29	]	]	PUNCT
ejpam-5210	30	30	.	.	PUNCT
ejpam-5210	31	1	the	the	DET
ejpam-5210	31	2	commutative	commutative	ADJ
ejpam-5210	31	3	rings	ring	NOUN
ejpam-5210	31	4	with	with	ADP
ejpam-5210	31	5	semiprime	semiprime	NOUN
ejpam-5210	31	6	lie	lie	NOUN
ejpam-5210	31	7	rings	ring	NOUN
ejpam-5210	31	8	were	be	AUX
ejpam-5210	31	9	studied	study	VERB
ejpam-5210	31	10	by	by	ADP
ejpam-5210	31	11	passman	passman	PROPN
ejpam-5210	31	12	[	[	X
ejpam-5210	31	13	26	26	NUM
ejpam-5210	31	14	]	]	PUNCT
ejpam-5210	31	15	and	and	CCONJ
ejpam-5210	31	16	bresar	bresar	VERB
ejpam-5210	31	17	[	[	PUNCT
ejpam-5210	31	18	4	4	NUM
ejpam-5210	31	19	]	]	PUNCT
ejpam-5210	31	20	.	.	PUNCT
ejpam-5210	32	1	finally	finally	ADV
ejpam-5210	32	2	,	,	PUNCT
ejpam-5210	32	3	all	all	DET
ejpam-5210	32	4	other	other	ADJ
ejpam-5210	32	5	definitions	definition	NOUN
ejpam-5210	32	6	and	and	CCONJ
ejpam-5210	32	7	facts	fact	NOUN
ejpam-5210	32	8	are	be	AUX
ejpam-5210	32	9	standard	standard	ADJ
ejpam-5210	32	10	,	,	PUNCT
ejpam-5210	32	11	which	which	PRON
ejpam-5210	32	12	were	be	AUX
ejpam-5210	32	13	be	be	AUX
ejpam-5210	32	14	found	find	VERB
ejpam-5210	32	15	in	in	ADP
ejpam-5210	32	16	[	[	X
ejpam-5210	32	17	1	1	NUM
ejpam-5210	32	18	,	,	PUNCT
ejpam-5210	32	19	13	13	NUM
ejpam-5210	32	20	]	]	PUNCT
ejpam-5210	32	21	and	and	CCONJ
ejpam-5210	32	22	also	also	ADV
ejpam-5210	32	23	in	in	ADP
ejpam-5210	32	24	[	[	X
ejpam-5210	32	25	10	10	NUM
ejpam-5210	32	26	]	]	PUNCT
ejpam-5210	32	27	.	.	PUNCT
ejpam-5210	33	1	2	2	X
ejpam-5210	33	2	.	.	X
ejpam-5210	33	3	preliminaries	preliminary	NOUN
ejpam-5210	33	4	for	for	ADP
ejpam-5210	33	5	any	any	DET
ejpam-5210	33	6	associative	associative	ADJ
ejpam-5210	33	7	lie	lie	NOUN
ejpam-5210	33	8	ring	ring	NOUN
ejpam-5210	33	9	r	r	NOUN
ejpam-5210	33	10	,	,	PUNCT
ejpam-5210	33	11	the	the	DET
ejpam-5210	33	12	commutator	commutator	NOUN
ejpam-5210	33	13	[	[	X
ejpam-5210	33	14	r	r	X
ejpam-5210	33	15	,	,	PUNCT
ejpam-5210	33	16	r	r	NOUN
ejpam-5210	33	17	]	]	X
ejpam-5210	33	18	is	be	AUX
ejpam-5210	33	19	a	a	DET
ejpam-5210	33	20	subgroup	subgroup	NOUN
ejpam-5210	33	21	of	of	ADP
ejpam-5210	33	22	r	r	NOUN
ejpam-5210	33	23	,	,	PUNCT
ejpam-5210	33	24	which	which	PRON
ejpam-5210	33	25	is	be	AUX
ejpam-5210	33	26	an	an	DET
ejpam-5210	33	27	additive	additive	ADJ
ejpam-5210	33	28	subgroup	subgroup	NOUN
ejpam-5210	33	29	generated	generate	VERB
ejpam-5210	33	30	by	by	ADP
ejpam-5210	33	31	all	all	PRON
ejpam-5210	33	32	[	[	X
ejpam-5210	33	33	s	s	X
ejpam-5210	33	34	,	,	PUNCT
ejpam-5210	33	35	t	t	PROPN
ejpam-5210	33	36	]	]	PUNCT
ejpam-5210	33	37	with	with	ADP
ejpam-5210	33	38	s	s	PROPN
ejpam-5210	33	39	,	,	PUNCT
ejpam-5210	33	40	t	t	PROPN
ejpam-5210	33	41	∈	∈	PROPN
ejpam-5210	33	42	r.	r.	PROPN
ejpam-5210	33	43	for	for	ADP
ejpam-5210	33	44	what	what	PRON
ejpam-5210	33	45	we	we	PRON
ejpam-5210	33	46	will	will	AUX
ejpam-5210	33	47	prove	prove	VERB
ejpam-5210	33	48	,	,	PUNCT
ejpam-5210	33	49	we	we	PRON
ejpam-5210	33	50	need	need	VERB
ejpam-5210	33	51	some	some	DET
ejpam-5210	33	52	lemmas	lemma	NOUN
ejpam-5210	33	53	.	.	PUNCT
ejpam-5210	34	1	lemma	lemma	PROPN
ejpam-5210	34	2	1	1	NUM
ejpam-5210	34	3	.	.	PUNCT
ejpam-5210	35	1	the	the	DET
ejpam-5210	35	2	following	follow	VERB
ejpam-5210	35	3	conditions	condition	NOUN
ejpam-5210	35	4	are	be	AUX
ejpam-5210	35	5	equivalent	equivalent	ADJ
ejpam-5210	35	6	:	:	PUNCT
ejpam-5210	35	7	(	(	PUNCT
ejpam-5210	35	8	1	1	X
ejpam-5210	35	9	)	)	PUNCT
ejpam-5210	35	10	r	r	NOUN
ejpam-5210	35	11	is	be	AUX
ejpam-5210	35	12	�	�	PROPN
ejpam-5210	35	13	semiprime	semiprime	NOUN
ejpam-5210	35	14	ring	ring	NOUN
ejpam-5210	35	15	,	,	PUNCT
ejpam-5210	35	16	(	(	PUNCT
ejpam-5210	35	17	2	2	NUM
ejpam-5210	35	18	)	)	PUNCT
ejpam-5210	35	19	for	for	ADP
ejpam-5210	35	20	any	any	DET
ejpam-5210	35	21	�	�	NOUN
ejpam-5210	35	22	-ideals	-ideal	NOUN
ejpam-5210	35	23	a	a	PRON
ejpam-5210	35	24	and	and	CCONJ
ejpam-5210	35	25	b	b	NOUN
ejpam-5210	35	26	of	of	ADP
ejpam-5210	35	27	r	r	NOUN
ejpam-5210	35	28	,	,	PUNCT
ejpam-5210	35	29	the	the	DET
ejpam-5210	35	30	implication	implication	NOUN
ejpam-5210	35	31	ab	ab	PROPN
ejpam-5210	35	32	=	=	SYM
ejpam-5210	35	33	0	0	PROPN
ejpam-5210	35	34	⇒	⇒	VERB
ejpam-5210	35	35	a	a	DET
ejpam-5210	35	36	∩b	∩b	NOUN
ejpam-5210	35	37	=	=	SYM
ejpam-5210	35	38	0	0	NUM
ejpam-5210	35	39	is	be	AUX
ejpam-5210	35	40	true	true	ADJ
ejpam-5210	35	41	.	.	PUNCT
ejpam-5210	36	1	(	(	PUNCT
ejpam-5210	36	2	3	3	X
ejpam-5210	36	3	)	)	PUNCT
ejpam-5210	36	4	if	if	SCONJ
ejpam-5210	36	5	a	a	DET
ejpam-5210	36	6	∈	∈	PROPN
ejpam-5210	36	7	r	r	NOUN
ejpam-5210	36	8	,	,	PUNCT
ejpam-5210	36	9	such	such	ADJ
ejpam-5210	36	10	that	that	DET
ejpam-5210	36	11	arδm1	arδm1	NOUN
ejpam-5210	36	12	1	1	NUM
ejpam-5210	36	13	...	...	PUNCT
ejpam-5210	36	14	δmn	δmn	NOUN
ejpam-5210	36	15	n	n	INTJ
ejpam-5210	36	16	(	(	PUNCT
ejpam-5210	36	17	a	a	X
ejpam-5210	36	18	)	)	PUNCT
ejpam-5210	36	19	=	=	SYM
ejpam-5210	36	20	0	0	NUM
ejpam-5210	36	21	,	,	PUNCT
ejpam-5210	36	22	for	for	ADP
ejpam-5210	36	23	any	any	DET
ejpam-5210	36	24	integers	integer	NOUN
ejpam-5210	36	25	n	n	PRON
ejpam-5210	36	26	≥	≥	NOUN
ejpam-5210	36	27	1,mi	1,mi	NUM
ejpam-5210	36	28	≥	≥	NOUN
ejpam-5210	36	29	0	0	NUM
ejpam-5210	36	30	and	and	CCONJ
ejpam-5210	36	31	any	any	DET
ejpam-5210	36	32	derivation	derivation	NOUN
ejpam-5210	36	33	δi	δi	ADP
ejpam-5210	36	34	∈	∈	PROPN
ejpam-5210	36	35	�	�	PROPN
ejpam-5210	36	36	,	,	PUNCT
ejpam-5210	36	37	where	where	SCONJ
ejpam-5210	36	38	i	i	PRON
ejpam-5210	36	39	=	=	NOUN
ejpam-5210	36	40	1	1	NUM
ejpam-5210	36	41	,	,	PUNCT
ejpam-5210	36	42	...	...	PUNCT
ejpam-5210	36	43	,	,	PUNCT
ejpam-5210	36	44	n	n	CCONJ
ejpam-5210	36	45	,	,	PUNCT
ejpam-5210	36	46	then	then	ADV
ejpam-5210	36	47	a	a	DET
ejpam-5210	36	48	=	=	NOUN
ejpam-5210	36	49	0	0	NUM
ejpam-5210	36	50	.	.	PUNCT
ejpam-5210	37	1	proof	proof	NOUN
ejpam-5210	37	2	.	.	PUNCT
ejpam-5210	38	1	a	a	DET
ejpam-5210	38	2	simple	simple	ADJ
ejpam-5210	38	3	modification	modification	NOUN
ejpam-5210	38	4	of	of	ADP
ejpam-5210	38	5	proposition	proposition	NOUN
ejpam-5210	38	6	2	2	NUM
ejpam-5210	38	7	from	from	ADP
ejpam-5210	38	8	[	[	X
ejpam-5210	38	9	22	22	NUM
ejpam-5210	38	10	]	]	PUNCT
ejpam-5210	38	11	.	.	PUNCT
ejpam-5210	39	1	lemma	lemma	PROPN
ejpam-5210	39	2	2	2	NUM
ejpam-5210	39	3	.	.	PUNCT
ejpam-5210	40	1	[	[	X
ejpam-5210	40	2	1	1	X
ejpam-5210	40	3	]	]	PUNCT
ejpam-5210	40	4	let	let	VERB
ejpam-5210	40	5	a	a	PRON
ejpam-5210	40	6	be	be	AUX
ejpam-5210	40	7	a	a	DET
ejpam-5210	40	8	lie	lie	NOUN
ejpam-5210	41	1	�	�	NOUN
ejpam-5210	41	2	-ideal	-ideal	NOUN
ejpam-5210	41	3	of	of	ADP
ejpam-5210	41	4	a	a	DET
ejpam-5210	41	5	�	�	PROPN
ejpam-5210	41	6	-semiprime	-semiprime	NOUN
ejpam-5210	41	7	ring	ring	NOUN
ejpam-5210	41	8	r	r	NOUN
ejpam-5210	41	9	of	of	ADP
ejpam-5210	41	10	charr	charr	NOUN
ejpam-5210	41	11	̸=	̸=	PROPN
ejpam-5210	41	12	2	2	NUM
ejpam-5210	41	13	.	.	PUNCT
ejpam-5210	42	1	if	if	SCONJ
ejpam-5210	42	2	[	[	X
ejpam-5210	42	3	a	a	X
ejpam-5210	42	4	,	,	PUNCT
ejpam-5210	42	5	a	a	PRON
ejpam-5210	42	6	]	]	X
ejpam-5210	42	7	⊆	⊆	NUM
ejpam-5210	42	8	z(r	z(r	NOUN
ejpam-5210	42	9	)	)	PUNCT
ejpam-5210	42	10	,	,	PUNCT
ejpam-5210	42	11	then	then	ADV
ejpam-5210	42	12	a	a	DET
ejpam-5210	42	13	⊆	⊆	NUM
ejpam-5210	42	14	z(r	z(r	NUM
ejpam-5210	42	15	)	)	PUNCT
ejpam-5210	42	16	.	.	PUNCT
ejpam-5210	43	1	lemma	lemma	PROPN
ejpam-5210	43	2	3	3	X
ejpam-5210	43	3	.	.	PUNCT
ejpam-5210	44	1	[	[	X
ejpam-5210	44	2	1	1	X
ejpam-5210	44	3	]	]	PUNCT
ejpam-5210	44	4	let	let	VERB
ejpam-5210	44	5	r	r	PRON
ejpam-5210	44	6	be	be	AUX
ejpam-5210	44	7	a	a	DET
ejpam-5210	44	8	2	2	NUM
ejpam-5210	44	9	-	-	PUNCT
ejpam-5210	44	10	torsionfree	torsionfree	NUM
ejpam-5210	44	11	�	�	PROPN
ejpam-5210	44	12	-semiprime	-semiprime	NOUN
ejpam-5210	44	13	ring	ring	NOUN
ejpam-5210	44	14	and	and	CCONJ
ejpam-5210	44	15	a	a	DET
ejpam-5210	44	16	a	a	DET
ejpam-5210	44	17	nonzero	nonzero	NOUN
ejpam-5210	44	18	lie	lie	NOUN
ejpam-5210	44	19	�	�	NOUN
ejpam-5210	44	20	-ideal	-ideal	NOUN
ejpam-5210	44	21	of	of	ADP
ejpam-5210	44	22	r	r	NOUN
ejpam-5210	44	23	and	and	CCONJ
ejpam-5210	44	24	an	an	DET
ejpam-5210	44	25	associative	associative	ADJ
ejpam-5210	44	26	subring	subring	NOUN
ejpam-5210	44	27	.	.	PUNCT
ejpam-5210	45	1	then	then	ADV
ejpam-5210	45	2	a	a	DET
ejpam-5210	45	3	⊆	⊆	NUM
ejpam-5210	45	4	z(r	z(r	NOUN
ejpam-5210	45	5	)	)	PUNCT
ejpam-5210	45	6	or	or	CCONJ
ejpam-5210	45	7	a	a	PRON
ejpam-5210	45	8	contains	contain	VERB
ejpam-5210	45	9	a	a	DET
ejpam-5210	45	10	non	non	ADJ
ejpam-5210	45	11	-	-	ADJ
ejpam-5210	45	12	central	central	ADJ
ejpam-5210	45	13	associative	associative	ADJ
ejpam-5210	45	14	�	�	NOUN
ejpam-5210	45	15	-ideal	-ideal	NOUN
ejpam-5210	45	16	of	of	ADP
ejpam-5210	45	17	r.	r.	PROPN
ejpam-5210	45	18	lemma	lemma	PROPN
ejpam-5210	46	1	4	4	X
ejpam-5210	46	2	.	.	PUNCT
ejpam-5210	47	1	[	[	X
ejpam-5210	47	2	20	20	NUM
ejpam-5210	47	3	]	]	X
ejpam-5210	47	4	if	if	SCONJ
ejpam-5210	47	5	r	r	NOUN
ejpam-5210	47	6	is	be	AUX
ejpam-5210	47	7	a	a	DET
ejpam-5210	47	8	d	d	ADJ
ejpam-5210	47	9	-	-	PUNCT
ejpam-5210	47	10	semiprime	semiprime	ADJ
ejpam-5210	47	11	ring	ring	NOUN
ejpam-5210	47	12	,	,	PUNCT
ejpam-5210	47	13	φ	φ	X
ejpam-5210	47	14	an	an	DET
ejpam-5210	47	15	ideal	ideal	NOUN
ejpam-5210	47	16	of	of	ADP
ejpam-5210	47	17	d.	d.	PROPN
ejpam-5210	48	1	then	then	ADV
ejpam-5210	48	2	[	[	X
ejpam-5210	48	3	φ	φ	NUM
ejpam-5210	48	4	,	,	PUNCT
ejpam-5210	48	5	i	i	PROPN
ejpam-5210	48	6	d	d	NOUN
ejpam-5210	48	7	]	]	X
ejpam-5210	48	8	=	=	SYM
ejpam-5210	48	9	0	0	NUM
ejpam-5210	48	10	⇔	⇔	PROPN
ejpam-5210	48	11	φ	φ	PROPN
ejpam-5210	48	12	∩	∩	PROPN
ejpam-5210	48	13	i	i	PROPN
ejpam-5210	48	14	d	d	NOUN
ejpam-5210	48	15	=	=	SYM
ejpam-5210	48	16	0	0	X
ejpam-5210	48	17	.	.	PUNCT
ejpam-5210	49	1	m.	m.	NOUN
ejpam-5210	49	2	alosaimi	alosaimi	PROPN
ejpam-5210	49	3	et	et	PROPN
ejpam-5210	49	4	al	al	PROPN
ejpam-5210	49	5	.	.	PUNCT
ejpam-5210	49	6	/	/	SYM
ejpam-5210	49	7	eur	eur	PROPN
ejpam-5210	49	8	.	.	PUNCT
ejpam-5210	50	1	j.	j.	PROPN
ejpam-5210	50	2	pure	pure	PROPN
ejpam-5210	50	3	appl	appl	PROPN
ejpam-5210	50	4	.	.	PROPN
ejpam-5210	50	5	math	math	PROPN
ejpam-5210	50	6	,	,	PUNCT
ejpam-5210	50	7	17	17	NUM
ejpam-5210	50	8	(	(	PUNCT
ejpam-5210	50	9	3	3	NUM
ejpam-5210	50	10	)	)	PUNCT
ejpam-5210	50	11	(	(	PUNCT
ejpam-5210	50	12	2024	2024	NUM
ejpam-5210	50	13	)	)	PUNCT
ejpam-5210	50	14	,	,	PUNCT
ejpam-5210	50	15	2264	2264	NUM
ejpam-5210	50	16	-	-	SYM
ejpam-5210	50	17	2275	2275	NUM
ejpam-5210	50	18	2266	2266	NUM
ejpam-5210	50	19	3	3	NUM
ejpam-5210	50	20	.	.	X
ejpam-5210	51	1	lie	lie	NOUN
ejpam-5210	51	2	ideals	ideal	NOUN
ejpam-5210	51	3	in	in	ADP
ejpam-5210	51	4	d	d	ADJ
ejpam-5210	51	5	-	-	PUNCT
ejpam-5210	51	6	semiprime	semiprime	NOUN
ejpam-5210	51	7	rings	ring	NOUN
ejpam-5210	51	8	lemma	lemma	PROPN
ejpam-5210	51	9	5	5	X
ejpam-5210	51	10	.	.	PUNCT
ejpam-5210	52	1	let	let	VERB
ejpam-5210	52	2	r	r	PRON
ejpam-5210	52	3	be	be	AUX
ejpam-5210	52	4	a	a	DET
ejpam-5210	52	5	2	2	NUM
ejpam-5210	52	6	-	-	PUNCT
ejpam-5210	52	7	torsion	torsion	NOUN
ejpam-5210	52	8	-	-	PUNCT
ejpam-5210	52	9	free	free	ADJ
ejpam-5210	52	10	�	�	PROPN
ejpam-5210	52	11	-semiprime	-semiprime	PROPN
ejpam-5210	52	12	ring	ring	NOUN
ejpam-5210	52	13	,	,	PUNCT
ejpam-5210	52	14	u	u	NOUN
ejpam-5210	52	15	its	its	PRON
ejpam-5210	52	16	nonzero	nonzero	NOUN
ejpam-5210	52	17	lie	lie	NOUN
ejpam-5210	52	18	�	�	NOUN
ejpam-5210	52	19	-ideal	-ideal	NOUN
ejpam-5210	52	20	and	and	CCONJ
ejpam-5210	52	21	an	an	DET
ejpam-5210	52	22	associative	associative	ADJ
ejpam-5210	52	23	subring	subring	NOUN
ejpam-5210	52	24	,	,	PUNCT
ejpam-5210	52	25	where	where	SCONJ
ejpam-5210	52	26	a	a	DET
ejpam-5210	52	27	∈	∈	PROPN
ejpam-5210	52	28	r.	r.	NOUN
ejpam-5210	53	1	if	if	SCONJ
ejpam-5210	53	2	[	[	X
ejpam-5210	53	3	δm1	δm1	NOUN
ejpam-5210	53	4	1	1	NUM
ejpam-5210	53	5	·	·	PUNCT
ejpam-5210	53	6	·	·	PUNCT
ejpam-5210	53	7	·	·	PUNCT
ejpam-5210	54	1	δmk	δmk	PRON
ejpam-5210	55	1	k	k	X
ejpam-5210	55	2	(	(	PUNCT
ejpam-5210	55	3	a	a	NOUN
ejpam-5210	55	4	)	)	PUNCT
ejpam-5210	55	5	,	,	PUNCT
ejpam-5210	55	6	[	[	X
ejpam-5210	55	7	a	a	X
ejpam-5210	55	8	,	,	PUNCT
ejpam-5210	55	9	u	u	NOUN
ejpam-5210	55	10	]	]	X
ejpam-5210	55	11	]	]	X
ejpam-5210	55	12	=	=	SYM
ejpam-5210	55	13	0	0	NUM
ejpam-5210	55	14	,	,	PUNCT
ejpam-5210	55	15	for	for	ADP
ejpam-5210	55	16	any	any	DET
ejpam-5210	55	17	integers	integer	NOUN
ejpam-5210	55	18	mi	mi	PROPN
ejpam-5210	55	19	≥	≥	PROPN
ejpam-5210	55	20	0	0	NUM
ejpam-5210	55	21	,	,	PUNCT
ejpam-5210	55	22	k	k	PROPN
ejpam-5210	55	23	≥	≥	NUM
ejpam-5210	55	24	1	1	NUM
ejpam-5210	55	25	and	and	CCONJ
ejpam-5210	55	26	derivations	derivation	NOUN
ejpam-5210	55	27	δi	δi	PROPN
ejpam-5210	55	28	∈	∈	PROPN
ejpam-5210	55	29	�	�	PROPN
ejpam-5210	55	30	,	,	PUNCT
ejpam-5210	55	31	where	where	SCONJ
ejpam-5210	55	32	i	i	PRON
ejpam-5210	55	33	=	=	NOUN
ejpam-5210	55	34	1	1	NUM
ejpam-5210	55	35	,	,	PUNCT
ejpam-5210	55	36	.	.	PUNCT
ejpam-5210	55	37	.	.	PUNCT
ejpam-5210	55	38	.	.	PUNCT
ejpam-5210	56	1	,	,	PUNCT
ejpam-5210	56	2	k	k	NOUN
ejpam-5210	56	3	,	,	PUNCT
ejpam-5210	56	4	then	then	ADV
ejpam-5210	56	5	a	a	DET
ejpam-5210	56	6	∈	∈	PROPN
ejpam-5210	56	7	z(r	z(r	PROPN
ejpam-5210	56	8	)	)	PUNCT
ejpam-5210	56	9	.	.	PUNCT
ejpam-5210	57	1	proof	proof	NOUN
ejpam-5210	57	2	.	.	PUNCT
ejpam-5210	58	1	let	let	VERB
ejpam-5210	58	2	xa	xa	PROPN
ejpam-5210	59	1	=	=	PRON
ejpam-5210	60	1	{	{	PUNCT
ejpam-5210	61	1	[	[	X
ejpam-5210	61	2	δm1	δm1	NOUN
ejpam-5210	61	3	1	1	NUM
ejpam-5210	61	4	·	·	PUNCT
ejpam-5210	61	5	·	·	PUNCT
ejpam-5210	61	6	·	·	PUNCT
ejpam-5210	62	1	δmk	δmk	PRON
ejpam-5210	63	1	k	k	X
ejpam-5210	63	2	(	(	PUNCT
ejpam-5210	63	3	a	a	NOUN
ejpam-5210	63	4	)	)	PUNCT
ejpam-5210	63	5	,	,	PUNCT
ejpam-5210	63	6	x	x	X
ejpam-5210	63	7	]	]	X
ejpam-5210	63	8	}	}	PUNCT
ejpam-5210	63	9	,	,	PUNCT
ejpam-5210	63	10	x	x	X
ejpam-5210	63	11	,	,	PUNCT
ejpam-5210	63	12	a	a	DET
ejpam-5210	63	13	∈	∈	PROPN
ejpam-5210	63	14	r	r	NOUN
ejpam-5210	63	15	,	,	PUNCT
ejpam-5210	63	16	δi	δi	PROPN
ejpam-5210	63	17	∈	∈	PROPN
ejpam-5210	63	18	�	�	PROPN
ejpam-5210	63	19	,	,	PUNCT
ejpam-5210	63	20	mi	mi	PROPN
ejpam-5210	63	21	≥	≥	PROPN
ejpam-5210	63	22	0	0	NUM
ejpam-5210	63	23	and	and	CCONJ
ejpam-5210	63	24	x	x	NOUN
ejpam-5210	63	25	,	,	PUNCT
ejpam-5210	63	26	y	y	PROPN
ejpam-5210	63	27	∈	∈	PROPN
ejpam-5210	63	28	r.	r.	PROPN
ejpam-5210	63	29	from	from	ADP
ejpam-5210	63	30	[	[	X
ejpam-5210	63	31	b	b	X
ejpam-5210	63	32	,	,	PUNCT
ejpam-5210	63	33	xy	xy	PROPN
ejpam-5210	63	34	]	]	X
ejpam-5210	63	35	=	=	PUNCT
ejpam-5210	64	1	[	[	X
ejpam-5210	64	2	b	b	NOUN
ejpam-5210	64	3	,	,	PUNCT
ejpam-5210	64	4	x]y	x]y	PROPN
ejpam-5210	64	5	+	+	PROPN
ejpam-5210	65	1	x[b	x[b	PROPN
ejpam-5210	65	2	,	,	PUNCT
ejpam-5210	65	3	y	y	PROPN
ejpam-5210	65	4	]	]	X
ejpam-5210	65	5	,	,	PUNCT
ejpam-5210	65	6	b	b	PROPN
ejpam-5210	65	7	∈	∈	PROPN
ejpam-5210	65	8	xa	xa	PROPN
ejpam-5210	65	9	,	,	PUNCT
ejpam-5210	65	10	(	(	PUNCT
ejpam-5210	65	11	1	1	X
ejpam-5210	65	12	)	)	PUNCT
ejpam-5210	65	13	we	we	PRON
ejpam-5210	65	14	get	get	VERB
ejpam-5210	65	15	a[b	a[b	NOUN
ejpam-5210	65	16	,	,	PUNCT
ejpam-5210	65	17	xy	xy	X
ejpam-5210	65	18	]	]	X
ejpam-5210	65	19	=	=	SYM
ejpam-5210	65	20	0	0	NUM
ejpam-5210	65	21	,	,	PUNCT
ejpam-5210	65	22	then	then	ADV
ejpam-5210	65	23	ax[b	ax[b	PROPN
ejpam-5210	65	24	,	,	PUNCT
ejpam-5210	65	25	y	y	PROPN
ejpam-5210	65	26	]	]	X
ejpam-5210	65	27	=	=	PUNCT
ejpam-5210	66	1	0	0	X
ejpam-5210	66	2	.	.	PUNCT
ejpam-5210	67	1	hence	hence	ADV
ejpam-5210	67	2	ayx[b	ayx[b	PROPN
ejpam-5210	67	3	,	,	PUNCT
ejpam-5210	67	4	y	y	PROPN
ejpam-5210	67	5	]	]	X
ejpam-5210	67	6	=	=	SYM
ejpam-5210	67	7	0	0	NUM
ejpam-5210	67	8	and	and	CCONJ
ejpam-5210	67	9	yax[b	yax[b	PROPN
ejpam-5210	67	10	,	,	PUNCT
ejpam-5210	67	11	y	y	PROPN
ejpam-5210	67	12	]	]	X
ejpam-5210	67	13	=	=	SYM
ejpam-5210	67	14	0	0	X
ejpam-5210	67	15	.	.	PUNCT
ejpam-5210	68	1	thus	thus	ADV
ejpam-5210	68	2	,	,	PUNCT
ejpam-5210	68	3	we	we	PRON
ejpam-5210	68	4	deduce	deduce	VERB
ejpam-5210	68	5	that	that	SCONJ
ejpam-5210	68	6	(	(	PUNCT
ejpam-5210	68	7	r[a	r[a	NOUN
ejpam-5210	68	8	,	,	PUNCT
ejpam-5210	68	9	y]r)2	y]r)2	NOUN
ejpam-5210	68	10	=	=	SYM
ejpam-5210	68	11	0	0	NUM
ejpam-5210	68	12	,	,	PUNCT
ejpam-5210	68	13	a	a	DET
ejpam-5210	68	14	∈	∈	PROPN
ejpam-5210	68	15	r.	r.	NOUN
ejpam-5210	68	16	(	(	PUNCT
ejpam-5210	68	17	2	2	NUM
ejpam-5210	68	18	)	)	PUNCT
ejpam-5210	68	19	in	in	ADP
ejpam-5210	68	20	addition	addition	NOUN
ejpam-5210	68	21	0	0	NUM
ejpam-5210	68	22	=	=	SYM
ejpam-5210	68	23	d(a[b	d(a[b	NOUN
ejpam-5210	68	24	,	,	PUNCT
ejpam-5210	68	25	x	x	NOUN
ejpam-5210	68	26	]	]	X
ejpam-5210	68	27	)	)	PUNCT
ejpam-5210	69	1	=	=	SYM
ejpam-5210	69	2	d(a)[b	d(a)[b	PROPN
ejpam-5210	69	3	,	,	PUNCT
ejpam-5210	69	4	x	x	X
ejpam-5210	69	5	]	]	X
ejpam-5210	69	6	.	.	PUNCT
ejpam-5210	70	1	multiply	multiply	VERB
ejpam-5210	70	2	the	the	DET
ejpam-5210	70	3	identity	identity	NOUN
ejpam-5210	70	4	(	(	PUNCT
ejpam-5210	70	5	1	1	NUM
ejpam-5210	70	6	)	)	PUNCT
ejpam-5210	70	7	from	from	ADP
ejpam-5210	70	8	the	the	DET
ejpam-5210	70	9	left	left	NOUN
ejpam-5210	70	10	by	by	ADP
ejpam-5210	70	11	d(a	d(a	PROPN
ejpam-5210	70	12	)	)	PUNCT
ejpam-5210	70	13	,	,	PUNCT
ejpam-5210	70	14	then	then	ADV
ejpam-5210	70	15	we	we	PRON
ejpam-5210	70	16	get	get	VERB
ejpam-5210	70	17	d(a)x[b	d(a)x[b	ADJ
ejpam-5210	70	18	,	,	PUNCT
ejpam-5210	70	19	y	y	X
ejpam-5210	70	20	]	]	X
ejpam-5210	71	1	=	=	SYM
ejpam-5210	72	1	0	0	X
ejpam-5210	72	2	.	.	PUNCT
ejpam-5210	73	1	therefore	therefore	ADV
ejpam-5210	73	2	,	,	PUNCT
ejpam-5210	73	3	0	0	X
ejpam-5210	73	4	=	=	SYM
ejpam-5210	73	5	δ(ax[d(b	δ(ax[d(b	X
ejpam-5210	73	6	)	)	PUNCT
ejpam-5210	73	7	,	,	PUNCT
ejpam-5210	73	8	y	y	PROPN
ejpam-5210	73	9	]	]	X
ejpam-5210	73	10	=	=	SYM
ejpam-5210	73	11	δ(a)x[d(b	δ(a)x[d(b	PROPN
ejpam-5210	73	12	)	)	PUNCT
ejpam-5210	73	13	,	,	PUNCT
ejpam-5210	73	14	y	y	NOUN
ejpam-5210	73	15	]	]	X
ejpam-5210	73	16	,	,	PUNCT
ejpam-5210	73	17	and	and	CCONJ
ejpam-5210	73	18	by	by	ADP
ejpam-5210	73	19	the	the	DET
ejpam-5210	73	20	similar	similar	ADJ
ejpam-5210	73	21	argument	argument	NOUN
ejpam-5210	73	22	,	,	PUNCT
ejpam-5210	73	23	we	we	PRON
ejpam-5210	73	24	have	have	VERB
ejpam-5210	73	25	δm1	δm1	NOUN
ejpam-5210	73	26	1	1	NUM
ejpam-5210	73	27	·	·	PUNCT
ejpam-5210	73	28	·	·	PUNCT
ejpam-5210	73	29	·	·	PUNCT
ejpam-5210	74	1	δmk	δmk	PRON
ejpam-5210	75	1	k	k	X
ejpam-5210	75	2	(	(	PUNCT
ejpam-5210	75	3	a)x[δm1	a)x[δm1	PROPN
ejpam-5210	75	4	1	1	NUM
ejpam-5210	75	5	·	·	PUNCT
ejpam-5210	75	6	·	·	PUNCT
ejpam-5210	75	7	·	·	PUNCT
ejpam-5210	76	1	δmk	δmk	PRON
ejpam-5210	77	1	k	k	X
ejpam-5210	77	2	(	(	PUNCT
ejpam-5210	77	3	a	a	PROPN
ejpam-5210	77	4	)	)	PUNCT
ejpam-5210	77	5	,	,	PUNCT
ejpam-5210	77	6	y	y	PROPN
ejpam-5210	77	7	]	]	X
ejpam-5210	77	8	=	=	SYM
ejpam-5210	77	9	0	0	NUM
ejpam-5210	77	10	,	,	PUNCT
ejpam-5210	77	11	for	for	ADP
ejpam-5210	77	12	any	any	DET
ejpam-5210	77	13	integers	integer	NOUN
ejpam-5210	77	14	k	k	X
ejpam-5210	77	15	≥	≥	PROPN
ejpam-5210	77	16	1,mi	1,mi	NUM
ejpam-5210	77	17	≥	≥	NOUN
ejpam-5210	77	18	0	0	PUNCT
ejpam-5210	77	19	and	and	CCONJ
ejpam-5210	77	20	derivations	derivation	NOUN
ejpam-5210	77	21	δi	δi	PROPN
ejpam-5210	77	22	∈	∈	PROPN
ejpam-5210	77	23	�	�	PROPN
ejpam-5210	77	24	,	,	PUNCT
ejpam-5210	77	25	where	where	SCONJ
ejpam-5210	77	26	i	i	PRON
ejpam-5210	77	27	=	=	NOUN
ejpam-5210	77	28	1	1	NUM
ejpam-5210	77	29	,	,	PUNCT
ejpam-5210	77	30	...	...	PUNCT
ejpam-5210	77	31	,	,	PUNCT
ejpam-5210	77	32	k.	k.	PROPN
ejpam-5210	77	33	as	as	ADP
ejpam-5210	77	34	in	in	ADP
ejpam-5210	77	35	the	the	DET
ejpam-5210	77	36	proof	proof	NOUN
ejpam-5210	77	37	of	of	ADP
ejpam-5210	77	38	the	the	DET
ejpam-5210	77	39	condition	condition	NOUN
ejpam-5210	77	40	(	(	PUNCT
ejpam-5210	77	41	2	2	NUM
ejpam-5210	77	42	)	)	PUNCT
ejpam-5210	77	43	,	,	PUNCT
ejpam-5210	77	44	we	we	PRON
ejpam-5210	77	45	deduce	deduce	VERB
ejpam-5210	77	46	that	that	PRON
ejpam-5210	77	47	(	(	PUNCT
ejpam-5210	77	48	r[δm1	r[δm1	NOUN
ejpam-5210	77	49	1	1	NUM
ejpam-5210	77	50	·	·	PUNCT
ejpam-5210	77	51	·	·	PUNCT
ejpam-5210	77	52	·	·	PUNCT
ejpam-5210	78	1	δmk	δmk	PRON
ejpam-5210	78	2	k	k	X
ejpam-5210	78	3	(	(	PUNCT
ejpam-5210	78	4	a	a	NOUN
ejpam-5210	78	5	)	)	PUNCT
ejpam-5210	78	6	,	,	PUNCT
ejpam-5210	78	7	y]r)2	y]r)2	NOUN
ejpam-5210	78	8	=	=	SYM
ejpam-5210	78	9	0	0	X
ejpam-5210	78	10	.	.	PUNCT
ejpam-5210	79	1	then	then	ADV
ejpam-5210	79	2	,	,	PUNCT
ejpam-5210	79	3	i	i	PRON
ejpam-5210	79	4	=	=	PUNCT
ejpam-5210	80	1	∞∑	∞∑	NOUN
ejpam-5210	80	2	k=1	k=1	PUNCT
ejpam-5210	80	3	∑	∑	ADP
ejpam-5210	80	4	δi	δi	PROPN
ejpam-5210	80	5	∈	∈	PROPN
ejpam-5210	80	6	�	�	PROPN
ejpam-5210	80	7	r[δm1	r[δm1	PROPN
ejpam-5210	80	8	1	1	NUM
ejpam-5210	80	9	·	·	PUNCT
ejpam-5210	80	10	·	·	PUNCT
ejpam-5210	80	11	·	·	PUNCT
ejpam-5210	81	1	δmk	δmk	PRON
ejpam-5210	81	2	k	k	X
ejpam-5210	81	3	(	(	PUNCT
ejpam-5210	81	4	a	a	NOUN
ejpam-5210	81	5	)	)	PUNCT
ejpam-5210	81	6	,	,	PUNCT
ejpam-5210	81	7	y]r	y]r	NOUN
ejpam-5210	81	8	,	,	PUNCT
ejpam-5210	81	9	y	y	PROPN
ejpam-5210	81	10	∈	∈	PROPN
ejpam-5210	81	11	r	r	NOUN
ejpam-5210	81	12	is	be	AUX
ejpam-5210	81	13	a	a	DET
ejpam-5210	81	14	sum	sum	NOUN
ejpam-5210	81	15	of	of	ADP
ejpam-5210	81	16	nilpotent	nilpotent	ADJ
ejpam-5210	81	17	ideals	ideal	NOUN
ejpam-5210	81	18	,	,	PUNCT
ejpam-5210	81	19	therefore	therefore	ADV
ejpam-5210	81	20	it	it	PRON
ejpam-5210	81	21	will	will	AUX
ejpam-5210	81	22	be	be	AUX
ejpam-5210	81	23	a	a	DET
ejpam-5210	81	24	nil	nil	ADJ
ejpam-5210	81	25	ideal	ideal	NOUN
ejpam-5210	81	26	as	as	ADV
ejpam-5210	81	27	well	well	ADV
ejpam-5210	81	28	.	.	PUNCT
ejpam-5210	82	1	since	since	SCONJ
ejpam-5210	82	2	i	i	PRON
ejpam-5210	82	3	is	be	AUX
ejpam-5210	82	4	a	a	DET
ejpam-5210	82	5	�	�	NOUN
ejpam-5210	82	6	ideal	ideal	NOUN
ejpam-5210	82	7	,	,	PUNCT
ejpam-5210	82	8	we	we	PRON
ejpam-5210	82	9	get	get	VERB
ejpam-5210	82	10	i	i	PRON
ejpam-5210	82	11	=	=	ADJ
ejpam-5210	82	12	o	o	NOUN
ejpam-5210	82	13	,	,	PUNCT
ejpam-5210	82	14	hence	hence	ADV
ejpam-5210	82	15	a	a	DET
ejpam-5210	82	16	∈	∈	PROPN
ejpam-5210	82	17	z(r	z(r	NOUN
ejpam-5210	82	18	)	)	PUNCT
ejpam-5210	82	19	by	by	ADP
ejpam-5210	82	20	the	the	DET
ejpam-5210	82	21	same	same	ADJ
ejpam-5210	82	22	way	way	NOUN
ejpam-5210	82	23	,	,	PUNCT
ejpam-5210	82	24	we	we	PRON
ejpam-5210	82	25	prove	prove	VERB
ejpam-5210	82	26	the	the	DET
ejpam-5210	82	27	following	follow	VERB
ejpam-5210	82	28	lemma	lemma	PROPN
ejpam-5210	82	29	lemma	lemma	PROPN
ejpam-5210	82	30	6	6	NUM
ejpam-5210	82	31	.	.	PUNCT
ejpam-5210	83	1	let	let	VERB
ejpam-5210	83	2	r	r	PRON
ejpam-5210	83	3	be	be	AUX
ejpam-5210	83	4	a	a	DET
ejpam-5210	83	5	2	2	NUM
ejpam-5210	83	6	-	-	PUNCT
ejpam-5210	83	7	torsion	torsion	NOUN
ejpam-5210	83	8	-	-	PUNCT
ejpam-5210	83	9	free	free	ADJ
ejpam-5210	83	10	�	�	PROPN
ejpam-5210	83	11	-semiprime	-semiprime	PROPN
ejpam-5210	83	12	ring	ring	NOUN
ejpam-5210	83	13	,	,	PUNCT
ejpam-5210	83	14	u	u	NOUN
ejpam-5210	83	15	its	its	PRON
ejpam-5210	83	16	lie	lie	NOUN
ejpam-5210	83	17	�	�	NOUN
ejpam-5210	83	18	-ideal	-ideal	NOUN
ejpam-5210	83	19	.	.	PUNCT
ejpam-5210	84	1	if	if	SCONJ
ejpam-5210	84	2	a	a	DET
ejpam-5210	84	3	∈	∈	PROPN
ejpam-5210	84	4	cr([δ	cr([δ	PROPN
ejpam-5210	84	5	s1	s1	PROPN
ejpam-5210	84	6	1	1	NUM
ejpam-5210	84	7	·	·	PUNCT
ejpam-5210	84	8	·	·	PUNCT
ejpam-5210	84	9	·	·	PUNCT
ejpam-5210	84	10	δsll	δsll	PROPN
ejpam-5210	84	11	(	(	PUNCT
ejpam-5210	84	12	a	a	NOUN
ejpam-5210	84	13	)	)	PUNCT
ejpam-5210	84	14	,	,	PUNCT
ejpam-5210	84	15	u	u	NOUN
ejpam-5210	84	16	]	]	X
ejpam-5210	84	17	)	)	PUNCT
ejpam-5210	84	18	,	,	PUNCT
ejpam-5210	84	19	for	for	ADP
ejpam-5210	84	20	any	any	DET
ejpam-5210	84	21	integers	integer	NOUN
ejpam-5210	84	22	si	si	X
ejpam-5210	84	23	≥	≥	PROPN
ejpam-5210	84	24	0	0	NUM
ejpam-5210	84	25	,	,	PUNCT
ejpam-5210	84	26	l	l	PROPN
ejpam-5210	84	27	≥	≥	NOUN
ejpam-5210	84	28	1	1	NUM
ejpam-5210	84	29	and	and	CCONJ
ejpam-5210	84	30	derivations	derivation	NOUN
ejpam-5210	84	31	δi	δi	PROPN
ejpam-5210	84	32	∈	∈	PROPN
ejpam-5210	84	33	�	�	PROPN
ejpam-5210	84	34	,	,	PUNCT
ejpam-5210	84	35	where	where	SCONJ
ejpam-5210	84	36	i	i	PRON
ejpam-5210	84	37	=	=	NOUN
ejpam-5210	84	38	1	1	NUM
ejpam-5210	84	39	,	,	PUNCT
ejpam-5210	84	40	.	.	PUNCT
ejpam-5210	84	41	.	.	PUNCT
ejpam-5210	84	42	.	.	PUNCT
ejpam-5210	85	1	,	,	PUNCT
ejpam-5210	85	2	l.	l.	PROPN
ejpam-5210	85	3	then	then	ADV
ejpam-5210	85	4	a	a	DET
ejpam-5210	85	5	∈	∈	NOUN
ejpam-5210	85	6	cr(u	cr(u	NOUN
ejpam-5210	85	7	)	)	PUNCT
ejpam-5210	85	8	.	.	PUNCT
ejpam-5210	86	1	m.	m.	NOUN
ejpam-5210	86	2	alosaimi	alosaimi	PROPN
ejpam-5210	86	3	et	et	PROPN
ejpam-5210	86	4	al	al	PROPN
ejpam-5210	86	5	.	.	PUNCT
ejpam-5210	86	6	/	/	SYM
ejpam-5210	86	7	eur	eur	PROPN
ejpam-5210	86	8	.	.	PUNCT
ejpam-5210	87	1	j.	j.	PROPN
ejpam-5210	87	2	pure	pure	PROPN
ejpam-5210	87	3	appl	appl	PROPN
ejpam-5210	87	4	.	.	PROPN
ejpam-5210	87	5	math	math	PROPN
ejpam-5210	87	6	,	,	PUNCT
ejpam-5210	87	7	17	17	NUM
ejpam-5210	87	8	(	(	PUNCT
ejpam-5210	87	9	3	3	NUM
ejpam-5210	87	10	)	)	PUNCT
ejpam-5210	87	11	(	(	PUNCT
ejpam-5210	87	12	2024	2024	NUM
ejpam-5210	87	13	)	)	PUNCT
ejpam-5210	87	14	,	,	PUNCT
ejpam-5210	87	15	2264	2264	NUM
ejpam-5210	87	16	-	-	SYM
ejpam-5210	87	17	2275	2275	NUM
ejpam-5210	87	18	2267	2267	NUM
ejpam-5210	87	19	proof	proof	NOUN
ejpam-5210	87	20	.	.	PUNCT
ejpam-5210	88	1	let	let	VERB
ejpam-5210	88	2	u	u	NOUN
ejpam-5210	88	3	,	,	PUNCT
ejpam-5210	88	4	v	v	PROPN
ejpam-5210	88	5	∈	∈	PROPN
ejpam-5210	88	6	u	u	NOUN
ejpam-5210	88	7	,	,	PUNCT
ejpam-5210	88	8	si	si	X
ejpam-5210	88	9	≥	≥	PROPN
ejpam-5210	88	10	0	0	NUM
ejpam-5210	88	11	,	,	PUNCT
ejpam-5210	88	12	l	l	X
ejpam-5210	88	13	≥	≥	NUM
ejpam-5210	88	14	1	1	NUM
ejpam-5210	88	15	be	be	AUX
ejpam-5210	88	16	any	any	DET
ejpam-5210	88	17	integers	integer	NOUN
ejpam-5210	88	18	and	and	CCONJ
ejpam-5210	88	19	φ	φ	NOUN
ejpam-5210	88	20	,	,	PUNCT
ejpam-5210	88	21	δi	δi	PROPN
ejpam-5210	88	22	∈	∈	PROPN
ejpam-5210	88	23	�	�	PROPN
ejpam-5210	88	24	be	be	AUX
ejpam-5210	88	25	any	any	DET
ejpam-5210	88	26	derivation	derivation	NOUN
ejpam-5210	88	27	,	,	PUNCT
ejpam-5210	88	28	where	where	SCONJ
ejpam-5210	88	29	i	i	PRON
ejpam-5210	88	30	=	=	NOUN
ejpam-5210	88	31	1	1	NUM
ejpam-5210	88	32	,	,	PUNCT
ejpam-5210	88	33	.	.	PUNCT
ejpam-5210	88	34	.	.	PUNCT
ejpam-5210	89	1	.	.	PUNCT
ejpam-5210	90	1	,	,	PUNCT
ejpam-5210	90	2	l.	l.	PROPN
ejpam-5210	90	3	since	since	SCONJ
ejpam-5210	90	4	φ(δs11	φ(δs11	PROPN
ejpam-5210	90	5	·	·	PUNCT
ejpam-5210	90	6	·	·	PUNCT
ejpam-5210	90	7	·	·	PUNCT
ejpam-5210	91	1	δsll	δsll	PROPN
ejpam-5210	91	2	(	(	PUNCT
ejpam-5210	91	3	a)[a	a)[a	PROPN
ejpam-5210	91	4	,	,	PUNCT
ejpam-5210	91	5	x	x	NOUN
ejpam-5210	91	6	]	]	X
ejpam-5210	91	7	)	)	PUNCT
ejpam-5210	91	8	=	=	SYM
ejpam-5210	91	9	φ([a	φ([a	PROPN
ejpam-5210	91	10	,	,	PUNCT
ejpam-5210	91	11	x]δs11	x]δs11	PROPN
ejpam-5210	91	12	·	·	PUNCT
ejpam-5210	91	13	·	·	PUNCT
ejpam-5210	91	14	·	·	PUNCT
ejpam-5210	91	15	δsll	δsll	PROPN
ejpam-5210	91	16	(	(	PUNCT
ejpam-5210	91	17	a	a	NOUN
ejpam-5210	91	18	)	)	PUNCT
ejpam-5210	91	19	)	)	PUNCT
ejpam-5210	91	20	,	,	PUNCT
ejpam-5210	91	21	we	we	PRON
ejpam-5210	91	22	have	have	VERB
ejpam-5210	91	23	that	that	PRON
ejpam-5210	91	24	,	,	PUNCT
ejpam-5210	91	25	δs11	δs11	PROPN
ejpam-5210	91	26	·	·	PUNCT
ejpam-5210	91	27	·	·	PUNCT
ejpam-5210	91	28	·	·	PUNCT
ejpam-5210	91	29	δsll	δsll	PROPN
ejpam-5210	91	30	(	(	PUNCT
ejpam-5210	91	31	a	a	X
ejpam-5210	91	32	)	)	PUNCT
ejpam-5210	91	33	∈	∈	PROPN
ejpam-5210	91	34	cr([δ	cr([δ	PROPN
ejpam-5210	91	35	s1	s1	PROPN
ejpam-5210	91	36	1	1	NUM
ejpam-5210	91	37	·	·	PUNCT
ejpam-5210	91	38	·	·	PUNCT
ejpam-5210	91	39	·	·	PUNCT
ejpam-5210	91	40	δsll	δsll	PROPN
ejpam-5210	91	41	(	(	PUNCT
ejpam-5210	91	42	a	a	NOUN
ejpam-5210	91	43	)	)	PUNCT
ejpam-5210	91	44	,	,	PUNCT
ejpam-5210	91	45	x	x	NOUN
ejpam-5210	91	46	]	]	X
ejpam-5210	91	47	)	)	PUNCT
ejpam-5210	91	48	.	.	PUNCT
ejpam-5210	92	1	then	then	ADV
ejpam-5210	92	2	from	from	ADP
ejpam-5210	92	3	δs11	δs11	PROPN
ejpam-5210	92	4	·	·	PUNCT
ejpam-5210	92	5	·	·	PUNCT
ejpam-5210	92	6	·	·	PUNCT
ejpam-5210	92	7	δsll	δsll	PROPN
ejpam-5210	92	8	(	(	PUNCT
ejpam-5210	92	9	a)[δ	a)[δ	PROPN
ejpam-5210	92	10	s1	s1	PROPN
ejpam-5210	92	11	1	1	NUM
ejpam-5210	92	12	·	·	PUNCT
ejpam-5210	92	13	·	·	PUNCT
ejpam-5210	92	14	·	·	PUNCT
ejpam-5210	92	15	δsll	δsll	PROPN
ejpam-5210	92	16	(	(	PUNCT
ejpam-5210	92	17	a	a	NOUN
ejpam-5210	92	18	)	)	PUNCT
ejpam-5210	92	19	,	,	PUNCT
ejpam-5210	92	20	uv	uv	NOUN
ejpam-5210	92	21	]	]	X
ejpam-5210	92	22	=	=	SYM
ejpam-5210	93	1	[	[	PUNCT
ejpam-5210	93	2	δs11	δs11	PROPN
ejpam-5210	93	3	·	·	PUNCT
ejpam-5210	93	4	·	·	PUNCT
ejpam-5210	93	5	·	·	PUNCT
ejpam-5210	93	6	δsll	δsll	PROPN
ejpam-5210	93	7	(	(	PUNCT
ejpam-5210	93	8	a	a	NOUN
ejpam-5210	93	9	)	)	PUNCT
ejpam-5210	93	10	,	,	PUNCT
ejpam-5210	93	11	uv]δ	uv]δ	PROPN
ejpam-5210	93	12	s1	s1	PROPN
ejpam-5210	93	13	1	1	NUM
ejpam-5210	93	14	·	·	PUNCT
ejpam-5210	93	15	·	·	PUNCT
ejpam-5210	93	16	·	·	PUNCT
ejpam-5210	93	17	δsll	δsll	PROPN
ejpam-5210	93	18	(	(	PUNCT
ejpam-5210	93	19	a	a	X
ejpam-5210	93	20	)	)	PUNCT
ejpam-5210	93	21	it	it	PRON
ejpam-5210	93	22	holds	hold	VERB
ejpam-5210	93	23	that	that	SCONJ
ejpam-5210	93	24	[	[	X
ejpam-5210	93	25	δs11	δs11	PROPN
ejpam-5210	93	26	·	·	PUNCT
ejpam-5210	93	27	·	·	PUNCT
ejpam-5210	93	28	·	·	PUNCT
ejpam-5210	93	29	δsll	δsll	PROPN
ejpam-5210	93	30	(	(	PUNCT
ejpam-5210	93	31	a	a	NOUN
ejpam-5210	93	32	)	)	PUNCT
ejpam-5210	93	33	,	,	PUNCT
ejpam-5210	93	34	u][δ	u][δ	PROPN
ejpam-5210	93	35	s1	s1	PROPN
ejpam-5210	93	36	1	1	NUM
ejpam-5210	93	37	·	·	PUNCT
ejpam-5210	94	1	·	·	PUNCT
ejpam-5210	94	2	·	·	PUNCT
ejpam-5210	94	3	δsll	δsll	PROPN
ejpam-5210	94	4	(	(	PUNCT
ejpam-5210	94	5	a	a	NOUN
ejpam-5210	94	6	)	)	PUNCT
ejpam-5210	94	7	,	,	PUNCT
ejpam-5210	94	8	v	v	NOUN
ejpam-5210	94	9	]	]	X
ejpam-5210	94	10	=	=	SYM
ejpam-5210	94	11	0	0	NUM
ejpam-5210	94	12	,	,	PUNCT
ejpam-5210	94	13	what	what	DET
ejpam-5210	94	14	forces	force	VERB
ejpam-5210	94	15	[	[	X
ejpam-5210	94	16	δs11	δs11	PROPN
ejpam-5210	94	17	·	·	PUNCT
ejpam-5210	94	18	·	·	PUNCT
ejpam-5210	94	19	·	·	PUNCT
ejpam-5210	94	20	δsll	δsll	PROPN
ejpam-5210	94	21	(	(	PUNCT
ejpam-5210	94	22	a	a	NOUN
ejpam-5210	94	23	)	)	PUNCT
ejpam-5210	94	24	,	,	PUNCT
ejpam-5210	94	25	u]t[δ	u]t[δ	PRON
ejpam-5210	94	26	s1	s1	PROPN
ejpam-5210	94	27	1	1	NUM
ejpam-5210	94	28	·	·	PUNCT
ejpam-5210	94	29	·	·	PUNCT
ejpam-5210	94	30	·	·	PUNCT
ejpam-5210	94	31	δsll	δsll	PROPN
ejpam-5210	94	32	(	(	PUNCT
ejpam-5210	94	33	a	a	NOUN
ejpam-5210	94	34	)	)	PUNCT
ejpam-5210	94	35	,	,	PUNCT
ejpam-5210	94	36	v	v	NOUN
ejpam-5210	94	37	]	]	X
ejpam-5210	94	38	=	=	SYM
ejpam-5210	94	39	0	0	NUM
ejpam-5210	94	40	,	,	PUNCT
ejpam-5210	94	41	where	where	SCONJ
ejpam-5210	94	42	t	t	PROPN
ejpam-5210	94	43	∈	∈	PROPN
ejpam-5210	94	44	r.	r.	PROPN
ejpam-5210	94	45	thus	thus	ADV
ejpam-5210	94	46	,	,	PUNCT
ejpam-5210	94	47	the	the	DET
ejpam-5210	94	48	sum	sum	NOUN
ejpam-5210	94	49	of	of	ADP
ejpam-5210	94	50	nilpotent	nilpotent	ADJ
ejpam-5210	94	51	idal	idal	NOUN
ejpam-5210	94	52	of	of	ADP
ejpam-5210	94	53	r	r	NOUN
ejpam-5210	94	54	is	be	AUX
ejpam-5210	94	55	�	�	NOUN
ejpam-5210	94	56	-ideal	-ideal	NOUN
ejpam-5210	94	57	.	.	PUNCT
ejpam-5210	95	1	then	then	ADV
ejpam-5210	95	2	a	a	DET
ejpam-5210	95	3	∈	∈	PROPN
ejpam-5210	95	4	c(r	c(r	NOUN
ejpam-5210	95	5	)	)	PUNCT
ejpam-5210	95	6	.	.	PUNCT
ejpam-5210	96	1	now	now	ADV
ejpam-5210	96	2	,	,	PUNCT
ejpam-5210	96	3	we	we	PRON
ejpam-5210	96	4	will	will	AUX
ejpam-5210	96	5	extend	extend	VERB
ejpam-5210	96	6	result	result	NOUN
ejpam-5210	96	7	given	give	VERB
ejpam-5210	96	8	by	by	ADP
ejpam-5210	96	9	[	[	X
ejpam-5210	96	10	8	8	NUM
ejpam-5210	96	11	,	,	PUNCT
ejpam-5210	96	12	theorem	theorem	VERB
ejpam-5210	96	13	3	3	NUM
ejpam-5210	96	14	]	]	PUNCT
ejpam-5210	96	15	.	.	PUNCT
ejpam-5210	97	1	proposition	proposition	NOUN
ejpam-5210	97	2	1	1	NUM
ejpam-5210	97	3	.	.	PUNCT
ejpam-5210	98	1	let	let	VERB
ejpam-5210	98	2	r	r	PRON
ejpam-5210	98	3	be	be	AUX
ejpam-5210	98	4	a	a	DET
ejpam-5210	98	5	2	2	NUM
ejpam-5210	98	6	-	-	PUNCT
ejpam-5210	98	7	torsion	torsion	NOUN
ejpam-5210	98	8	-	-	PUNCT
ejpam-5210	98	9	free	free	ADJ
ejpam-5210	98	10	�	�	PROPN
ejpam-5210	98	11	-semiprime	-semiprime	NOUN
ejpam-5210	98	12	ring	ring	NOUN
ejpam-5210	98	13	,	,	PUNCT
ejpam-5210	98	14	w	w	ADP
ejpam-5210	98	15	its	its	PRON
ejpam-5210	98	16	associative	associative	ADJ
ejpam-5210	98	17	�	�	PROPN
ejpam-5210	98	18	-subring	-subring	NOUN
ejpam-5210	98	19	and	and	CCONJ
ejpam-5210	98	20	u	u	PRON
ejpam-5210	98	21	its	its	PRON
ejpam-5210	98	22	lie	lie	NOUN
ejpam-5210	98	23	�	�	NOUN
ejpam-5210	98	24	-ideal	-ideal	NOUN
ejpam-5210	98	25	.	.	PUNCT
ejpam-5210	99	1	if	if	SCONJ
ejpam-5210	99	2	[	[	X
ejpam-5210	99	3	w	w	NOUN
ejpam-5210	99	4	,	,	PUNCT
ejpam-5210	99	5	u	u	NOUN
ejpam-5210	99	6	]	]	PUNCT
ejpam-5210	99	7	⊆	⊆	NUM
ejpam-5210	99	8	w	w	NOUN
ejpam-5210	99	9	,	,	PUNCT
ejpam-5210	99	10	then	then	ADV
ejpam-5210	99	11	[	[	X
ejpam-5210	99	12	w	w	X
ejpam-5210	99	13	,	,	PUNCT
ejpam-5210	99	14	u	u	NOUN
ejpam-5210	99	15	]	]	X
ejpam-5210	99	16	=	=	SYM
ejpam-5210	99	17	0	0	NUM
ejpam-5210	99	18	or	or	CCONJ
ejpam-5210	99	19	w	w	PROPN
ejpam-5210	99	20	contains	contain	VERB
ejpam-5210	99	21	a	a	DET
ejpam-5210	99	22	non	non	ADJ
ejpam-5210	99	23	-	-	ADJ
ejpam-5210	99	24	zero	zero	NUM
ejpam-5210	99	25	associative	associative	ADJ
ejpam-5210	99	26	�	�	NOUN
ejpam-5210	99	27	-ideal	-ideal	NOUN
ejpam-5210	99	28	of	of	ADP
ejpam-5210	99	29	r.	r.	NOUN
ejpam-5210	99	30	proof	proof	NOUN
ejpam-5210	99	31	.	.	PUNCT
ejpam-5210	100	1	let	let	VERB
ejpam-5210	100	2	x	x	PRON
ejpam-5210	100	3	,	,	PUNCT
ejpam-5210	100	4	y	y	PROPN
ejpam-5210	100	5	,	,	PUNCT
ejpam-5210	100	6	r	r	NOUN
ejpam-5210	100	7	∈	∈	PROPN
ejpam-5210	100	8	r	r	NOUN
ejpam-5210	100	9	,	,	PUNCT
ejpam-5210	100	10	t1	t1	PROPN
ejpam-5210	100	11	,	,	PUNCT
ejpam-5210	100	12	t	t	PROPN
ejpam-5210	100	13	∈	∈	PROPN
ejpam-5210	100	14	u	u	NOUN
ejpam-5210	100	15	∩w	∩w	NOUN
ejpam-5210	100	16	and	and	CCONJ
ejpam-5210	100	17	v	v	NOUN
ejpam-5210	100	18	,	,	PUNCT
ejpam-5210	100	19	w	w	PROPN
ejpam-5210	100	20	,	,	PUNCT
ejpam-5210	100	21	w1	w1	NOUN
ejpam-5210	100	22	,	,	PUNCT
ejpam-5210	100	23	s	s	PROPN
ejpam-5210	100	24	,	,	PUNCT
ejpam-5210	100	25	a	a	PRON
ejpam-5210	100	26	,	,	PUNCT
ejpam-5210	100	27	b	b	PROPN
ejpam-5210	100	28	∈	∈	PROPN
ejpam-5210	100	29	w	w	PROPN
ejpam-5210	100	30	.	.	PUNCT
ejpam-5210	101	1	assume	assume	VERB
ejpam-5210	101	2	that	that	SCONJ
ejpam-5210	102	1	[	[	X
ejpam-5210	102	2	w	w	NOUN
ejpam-5210	102	3	,	,	PUNCT
ejpam-5210	102	4	u	u	NOUN
ejpam-5210	102	5	]	]	PUNCT
ejpam-5210	102	6	̸=	̸=	PROPN
ejpam-5210	102	7	0	0	NUM
ejpam-5210	102	8	.	.	PUNCT
ejpam-5210	103	1	by	by	ADP
ejpam-5210	103	2	lemma	lemma	PROPN
ejpam-5210	103	3	2	2	NUM
ejpam-5210	103	4	,	,	PUNCT
ejpam-5210	103	5	[	[	X
ejpam-5210	103	6	u	u	NOUN
ejpam-5210	103	7	,	,	PUNCT
ejpam-5210	103	8	u	u	NOUN
ejpam-5210	103	9	]	]	PUNCT
ejpam-5210	103	10	̸=	̸=	PROPN
ejpam-5210	103	11	0	0	NUM
ejpam-5210	103	12	.	.	PUNCT
ejpam-5210	104	1	since	since	SCONJ
ejpam-5210	104	2	the	the	DET
ejpam-5210	104	3	subring	subre	VERB
ejpam-5210	104	4	u	u	NOUN
ejpam-5210	104	5	of	of	ADP
ejpam-5210	104	6	r	r	NOUN
ejpam-5210	104	7	generated	generate	VERB
ejpam-5210	104	8	by	by	ADP
ejpam-5210	104	9	u	u	NOUN
ejpam-5210	104	10	satisfies	satisfie	NOUN
ejpam-5210	104	11	that	that	SCONJ
ejpam-5210	104	12	δ(u	δ(u	PROPN
ejpam-5210	104	13	)	)	PUNCT
ejpam-5210	104	14	⊆	⊆	NUM
ejpam-5210	104	15	u	u	NOUN
ejpam-5210	104	16	,	,	PUNCT
ejpam-5210	104	17	for	for	ADP
ejpam-5210	104	18	all	all	DET
ejpam-5210	104	19	δ	δ	PROPN
ejpam-5210	104	20	∈	∈	PROPN
ejpam-5210	104	21	�	�	PROPN
ejpam-5210	104	22	,	,	PUNCT
ejpam-5210	104	23	then	then	ADV
ejpam-5210	104	24	,	,	PUNCT
ejpam-5210	104	25	as	as	ADP
ejpam-5210	104	26	in	in	ADP
ejpam-5210	104	27	the	the	DET
ejpam-5210	104	28	proof	proof	NOUN
ejpam-5210	104	29	of	of	ADP
ejpam-5210	104	30	[	[	X
ejpam-5210	104	31	8	8	NUM
ejpam-5210	104	32	,	,	PUNCT
ejpam-5210	104	33	theorem	theorem	VERB
ejpam-5210	104	34	3	3	NUM
ejpam-5210	104	35	]	]	PUNCT
ejpam-5210	104	36	,	,	PUNCT
ejpam-5210	104	37	we	we	PRON
ejpam-5210	104	38	can	can	AUX
ejpam-5210	104	39	obtain	obtain	VERB
ejpam-5210	104	40	that	that	DET
ejpam-5210	104	41	r[a	r[a	NOUN
ejpam-5210	104	42	,	,	PUNCT
ejpam-5210	104	43	b]rzr	b]rzr	VERB
ejpam-5210	104	44	⊆	⊆	NUM
ejpam-5210	104	45	uzr	uzr	NOUN
ejpam-5210	104	46	⊆	⊆	NUM
ejpam-5210	104	47	w	w	NOUN
ejpam-5210	104	48	,	,	PUNCT
ejpam-5210	104	49	where	where	SCONJ
ejpam-5210	104	50	z	z	NOUN
ejpam-5210	104	51	=	=	PUNCT
ejpam-5210	105	1	[	[	X
ejpam-5210	105	2	s	s	ADP
ejpam-5210	105	3	,	,	PUNCT
ejpam-5210	105	4	t][t	t][t	VERB
ejpam-5210	105	5	,	,	PUNCT
ejpam-5210	105	6	w	w	NOUN
ejpam-5210	105	7	]	]	X
ejpam-5210	105	8	.	.	PUNCT
ejpam-5210	106	1	thus	thus	ADV
ejpam-5210	106	2	,	,	PUNCT
ejpam-5210	106	3	the	the	DET
ejpam-5210	106	4	sum	sum	NOUN
ejpam-5210	106	5	of	of	ADP
ejpam-5210	106	6	nilpotent	nilpotent	ADJ
ejpam-5210	106	7	idal	idal	NOUN
ejpam-5210	106	8	of	of	ADP
ejpam-5210	106	9	r	r	NOUN
ejpam-5210	106	10	is	be	AUX
ejpam-5210	106	11	�	�	NOUN
ejpam-5210	106	12	-ideal	-ideal	NOUN
ejpam-5210	106	13	of	of	ADP
ejpam-5210	106	14	r	r	NOUN
ejpam-5210	106	15	contained	contain	VERB
ejpam-5210	106	16	in	in	ADP
ejpam-5210	106	17	w	w	PROPN
ejpam-5210	106	18	.	.	PUNCT
ejpam-5210	107	1	otherwise	otherwise	ADV
ejpam-5210	107	2	[	[	X
ejpam-5210	107	3	a	a	PRON
ejpam-5210	107	4	,	,	PUNCT
ejpam-5210	107	5	b]rzr	b]rzr	ADJ
ejpam-5210	107	6	=	=	SYM
ejpam-5210	107	7	0	0	NUM
ejpam-5210	107	8	,	,	PUNCT
ejpam-5210	107	9	and	and	CCONJ
ejpam-5210	107	10	consequently	consequently	ADV
ejpam-5210	107	11	a	a	DET
ejpam-5210	107	12	=	=	SYM
ejpam-5210	107	13	∑	∑	PROPN
ejpam-5210	107	14	s	s	PROPN
ejpam-5210	107	15	,	,	PUNCT
ejpam-5210	107	16	w	w	PROPN
ejpam-5210	107	17	∈	∈	PROPN
ejpam-5210	107	18	w	w	PROPN
ejpam-5210	107	19	t	t	PROPN
ejpam-5210	107	20	∈	∈	PROPN
ejpam-5210	107	21	u	u	PROPN
ejpam-5210	107	22	∩w	∩w	ADJ
ejpam-5210	107	23	rzr	rzr	PROPN
ejpam-5210	107	24	is	be	AUX
ejpam-5210	107	25	a	a	DET
ejpam-5210	107	26	�	�	NOUN
ejpam-5210	107	27	-ideal	-ideal	NOUN
ejpam-5210	107	28	such	such	ADJ
ejpam-5210	107	29	that	that	SCONJ
ejpam-5210	107	30	[	[	X
ejpam-5210	107	31	a	a	X
ejpam-5210	107	32	,	,	PUNCT
ejpam-5210	107	33	b	b	NOUN
ejpam-5210	107	34	]	]	X
ejpam-5210	107	35	∈	∈	NOUN
ejpam-5210	107	36	annla	annla	NOUN
ejpam-5210	107	37	.	.	PUNCT
ejpam-5210	108	1	but	but	CCONJ
ejpam-5210	108	2	a	a	PRON
ejpam-5210	108	3	is	be	AUX
ejpam-5210	108	4	non	non	ADJ
ejpam-5210	108	5	-	-	ADJ
ejpam-5210	108	6	zero	zero	NUM
ejpam-5210	108	7	and	and	CCONJ
ejpam-5210	108	8	a	a	DET
ejpam-5210	108	9	∩	∩	ADJ
ejpam-5210	108	10	annla	annla	NOUN
ejpam-5210	108	11	=	=	SYM
ejpam-5210	108	12	0	0	NUM
ejpam-5210	108	13	,	,	PUNCT
ejpam-5210	108	14	implies	imply	VERB
ejpam-5210	108	15	that	that	SCONJ
ejpam-5210	108	16	[	[	X
ejpam-5210	108	17	a	a	DET
ejpam-5210	108	18	,	,	PUNCT
ejpam-5210	108	19	b	b	NOUN
ejpam-5210	108	20	]	]	X
ejpam-5210	108	21	=	=	SYM
ejpam-5210	108	22	0	0	X
ejpam-5210	108	23	.	.	PUNCT
ejpam-5210	108	24	m.	m.	NOUN
ejpam-5210	109	1	alosaimi	alosaimi	PROPN
ejpam-5210	109	2	et	et	PROPN
ejpam-5210	109	3	al	al	PROPN
ejpam-5210	109	4	.	.	PUNCT
ejpam-5210	109	5	/	/	SYM
ejpam-5210	109	6	eur	eur	PROPN
ejpam-5210	109	7	.	.	PUNCT
ejpam-5210	110	1	j.	j.	PROPN
ejpam-5210	110	2	pure	pure	PROPN
ejpam-5210	110	3	appl	appl	PROPN
ejpam-5210	110	4	.	.	PROPN
ejpam-5210	110	5	math	math	PROPN
ejpam-5210	110	6	,	,	PUNCT
ejpam-5210	110	7	17	17	NUM
ejpam-5210	110	8	(	(	PUNCT
ejpam-5210	110	9	3	3	NUM
ejpam-5210	110	10	)	)	PUNCT
ejpam-5210	110	11	(	(	PUNCT
ejpam-5210	110	12	2024	2024	NUM
ejpam-5210	110	13	)	)	PUNCT
ejpam-5210	110	14	,	,	PUNCT
ejpam-5210	110	15	2264	2264	NUM
ejpam-5210	110	16	-	-	SYM
ejpam-5210	110	17	2275	2275	NUM
ejpam-5210	110	18	2268	2268	NUM
ejpam-5210	110	19	inasmuch	inasmuch	PROPN
ejpam-5210	110	20	b	b	NOUN
ejpam-5210	110	21	=	=	SYM
ejpam-5210	110	22	z	z	PROPN
ejpam-5210	110	23	∈	∈	PROPN
ejpam-5210	110	24	u	u	NOUN
ejpam-5210	110	25	,	,	PUNCT
ejpam-5210	110	26	we	we	PRON
ejpam-5210	110	27	have	have	VERB
ejpam-5210	110	28	[	[	X
ejpam-5210	110	29	z	z	X
ejpam-5210	110	30	,	,	PUNCT
ejpam-5210	110	31	r	r	NOUN
ejpam-5210	110	32	]	]	X
ejpam-5210	110	33	⊆	⊆	NUM
ejpam-5210	110	34	u	u	NOUN
ejpam-5210	110	35	and	and	CCONJ
ejpam-5210	110	36	δm1	δm1	VERB
ejpam-5210	110	37	1	1	NUM
ejpam-5210	110	38	·	·	PUNCT
ejpam-5210	110	39	·	·	PUNCT
ejpam-5210	110	40	·	·	PUNCT
ejpam-5210	111	1	δmk	δmk	PRON
ejpam-5210	112	1	k	k	X
ejpam-5210	112	2	(	(	PUNCT
ejpam-5210	112	3	z	z	X
ejpam-5210	112	4	)	)	PUNCT
ejpam-5210	112	5	∈	∈	PROPN
ejpam-5210	112	6	u	u	NOUN
ejpam-5210	112	7	,	,	PUNCT
ejpam-5210	112	8	for	for	ADP
ejpam-5210	112	9	any	any	DET
ejpam-5210	112	10	integers	integer	NOUN
ejpam-5210	112	11	k	k	X
ejpam-5210	112	12	≥	≥	NUM
ejpam-5210	112	13	1	1	NUM
ejpam-5210	112	14	,	,	PUNCT
ejpam-5210	112	15	mi	mi	PROPN
ejpam-5210	112	16	≥	≥	PROPN
ejpam-5210	112	17	0	0	NUM
ejpam-5210	112	18	and	and	CCONJ
ejpam-5210	112	19	derivations	derivation	NOUN
ejpam-5210	112	20	δi	δi	PROPN
ejpam-5210	112	21	∈	∈	PROPN
ejpam-5210	112	22	�	�	PROPN
ejpam-5210	112	23	(	(	PUNCT
ejpam-5210	112	24	i	i	NOUN
ejpam-5210	112	25	=	=	NOUN
ejpam-5210	112	26	1	1	NUM
ejpam-5210	112	27	,	,	PUNCT
ejpam-5210	112	28	.	.	PUNCT
ejpam-5210	112	29	.	.	PUNCT
ejpam-5210	112	30	.	.	PUNCT
ejpam-5210	113	1	,	,	PUNCT
ejpam-5210	113	2	k	k	X
ejpam-5210	113	3	)	)	PUNCT
ejpam-5210	113	4	,	,	PUNCT
ejpam-5210	113	5	we	we	PRON
ejpam-5210	113	6	conclude	conclude	VERB
ejpam-5210	113	7	that	that	SCONJ
ejpam-5210	113	8	[	[	X
ejpam-5210	113	9	δm1	δm1	NOUN
ejpam-5210	113	10	1	1	NUM
ejpam-5210	113	11	·	·	PUNCT
ejpam-5210	113	12	·	·	PUNCT
ejpam-5210	113	13	·	·	PUNCT
ejpam-5210	114	1	δmk	δmk	PRON
ejpam-5210	115	1	k	k	X
ejpam-5210	115	2	(	(	PUNCT
ejpam-5210	115	3	z	z	NOUN
ejpam-5210	115	4	)	)	PUNCT
ejpam-5210	115	5	,	,	PUNCT
ejpam-5210	115	6	r	r	X
ejpam-5210	115	7	]	]	X
ejpam-5210	115	8	⊆	⊆	NUM
ejpam-5210	115	9	u	u	NOUN
ejpam-5210	115	10	what	what	PRON
ejpam-5210	115	11	gives	give	VERB
ejpam-5210	115	12	that	that	DET
ejpam-5210	115	13	z	z	PROPN
ejpam-5210	115	14	∈	∈	PROPN
ejpam-5210	115	15	cr([δ	cr([δ	PROPN
ejpam-5210	115	16	m1	m1	PROPN
ejpam-5210	115	17	1	1	NUM
ejpam-5210	115	18	·	·	PUNCT
ejpam-5210	115	19	·	·	PUNCT
ejpam-5210	115	20	·	·	PUNCT
ejpam-5210	116	1	δmk	δmk	PRON
ejpam-5210	117	1	k	k	X
ejpam-5210	117	2	(	(	PUNCT
ejpam-5210	117	3	z	z	NOUN
ejpam-5210	117	4	)	)	PUNCT
ejpam-5210	117	5	,	,	PUNCT
ejpam-5210	117	6	r	r	NOUN
ejpam-5210	117	7	]	]	PUNCT
ejpam-5210	117	8	)	)	PUNCT
ejpam-5210	117	9	.	.	PUNCT
ejpam-5210	118	1	by	by	ADP
ejpam-5210	118	2	lemma	lemma	PROPN
ejpam-5210	118	3	5	5	NUM
ejpam-5210	118	4	,	,	PUNCT
ejpam-5210	118	5	z	z	PROPN
ejpam-5210	118	6	∈	∈	PROPN
ejpam-5210	118	7	z(r	z(r	PROPN
ejpam-5210	118	8	)	)	PUNCT
ejpam-5210	118	9	.	.	PUNCT
ejpam-5210	119	1	then	then	ADV
ejpam-5210	119	2	b	b	X
ejpam-5210	119	3	=	=	SYM
ejpam-5210	119	4	∑	∑	PROPN
ejpam-5210	119	5	s	s	PROPN
ejpam-5210	119	6	,	,	PUNCT
ejpam-5210	119	7	w	w	PROPN
ejpam-5210	119	8	∈	∈	PROPN
ejpam-5210	119	9	w	w	PROPN
ejpam-5210	119	10	t	t	PROPN
ejpam-5210	119	11	∈	∈	PROPN
ejpam-5210	119	12	u	u	NOUN
ejpam-5210	119	13	∩w	∩w	NOUN
ejpam-5210	120	1	[	[	X
ejpam-5210	120	2	s	s	NOUN
ejpam-5210	120	3	,	,	PUNCT
ejpam-5210	120	4	t][t	t][t	VERB
ejpam-5210	120	5	,	,	PUNCT
ejpam-5210	120	6	w]r	w]r	VERB
ejpam-5210	120	7	⊆	⊆	NUM
ejpam-5210	120	8	w	w	NOUN
ejpam-5210	120	9	is	be	AUX
ejpam-5210	120	10	a	a	DET
ejpam-5210	120	11	�	�	NOUN
ejpam-5210	120	12	-ideal	-ideal	NOUN
ejpam-5210	120	13	of	of	ADP
ejpam-5210	120	14	r.	r.	PROPN
ejpam-5210	120	15	therefore	therefore	ADV
ejpam-5210	120	16	,	,	PUNCT
ejpam-5210	120	17	b	b	X
ejpam-5210	120	18	=	=	SYM
ejpam-5210	120	19	0	0	PROPN
ejpam-5210	120	20	and	and	CCONJ
ejpam-5210	120	21	,	,	PUNCT
ejpam-5210	120	22	as	as	ADP
ejpam-5210	120	23	a	a	DET
ejpam-5210	120	24	consequence	consequence	NOUN
ejpam-5210	120	25	,	,	PUNCT
ejpam-5210	120	26	z	z	NOUN
ejpam-5210	120	27	=	=	SYM
ejpam-5210	120	28	0	0	X
ejpam-5210	120	29	.	.	PUNCT
ejpam-5210	121	1	this	this	PRON
ejpam-5210	121	2	means	mean	VERB
ejpam-5210	121	3	that	that	SCONJ
ejpam-5210	122	1	[	[	X
ejpam-5210	122	2	s	s	X
ejpam-5210	122	3	,	,	PUNCT
ejpam-5210	122	4	t][t	t][t	VERB
ejpam-5210	122	5	,	,	PUNCT
ejpam-5210	122	6	w	w	NOUN
ejpam-5210	122	7	]	]	X
ejpam-5210	122	8	=	=	SYM
ejpam-5210	122	9	0	0	X
ejpam-5210	122	10	.	.	PUNCT
ejpam-5210	122	11	(	(	PUNCT
ejpam-5210	122	12	3	3	X
ejpam-5210	122	13	)	)	PUNCT
ejpam-5210	122	14	replace	replace	NOUN
ejpam-5210	122	15	w	w	NOUN
ejpam-5210	122	16	by	by	ADP
ejpam-5210	122	17	vw	vw	PROPN
ejpam-5210	122	18	in	in	ADP
ejpam-5210	122	19	the	the	DET
ejpam-5210	122	20	identity	identity	NOUN
ejpam-5210	122	21	(	(	PUNCT
ejpam-5210	122	22	3	3	NUM
ejpam-5210	122	23	)	)	PUNCT
ejpam-5210	122	24	.	.	PUNCT
ejpam-5210	123	1	then	then	ADV
ejpam-5210	123	2	[	[	X
ejpam-5210	123	3	s	s	X
ejpam-5210	123	4	,	,	PUNCT
ejpam-5210	123	5	t]v[t	t]v[t	X
ejpam-5210	123	6	,	,	PUNCT
ejpam-5210	123	7	w	w	NOUN
ejpam-5210	123	8	]	]	X
ejpam-5210	123	9	=	=	SYM
ejpam-5210	123	10	0	0	PUNCT
ejpam-5210	123	11	and	and	CCONJ
ejpam-5210	123	12	consequently	consequently	ADV
ejpam-5210	123	13	[	[	X
ejpam-5210	123	14	s	s	X
ejpam-5210	123	15	,	,	PUNCT
ejpam-5210	123	16	t]w	t]w	VERB
ejpam-5210	123	17	[	[	X
ejpam-5210	123	18	t	t	X
ejpam-5210	123	19	,	,	PUNCT
ejpam-5210	123	20	w	w	NOUN
ejpam-5210	123	21	]	]	X
ejpam-5210	123	22	=	=	SYM
ejpam-5210	123	23	0	0	X
ejpam-5210	123	24	.	.	PUNCT
ejpam-5210	124	1	(	(	PUNCT
ejpam-5210	124	2	4	4	X
ejpam-5210	124	3	)	)	PUNCT
ejpam-5210	124	4	linearize	linearize	VERB
ejpam-5210	124	5	the	the	DET
ejpam-5210	124	6	identity	identity	NOUN
ejpam-5210	124	7	(	(	PUNCT
ejpam-5210	124	8	3	3	NUM
ejpam-5210	124	9	)	)	PUNCT
ejpam-5210	124	10	on	on	ADP
ejpam-5210	124	11	t	t	PROPN
ejpam-5210	124	12	and	and	CCONJ
ejpam-5210	124	13	put	put	VERB
ejpam-5210	124	14	s	s	NOUN
ejpam-5210	124	15	=	=	X
ejpam-5210	124	16	w	w	PROPN
ejpam-5210	124	17	=	=	SYM
ejpam-5210	124	18	a	a	NOUN
ejpam-5210	124	19	;	;	PUNCT
ejpam-5210	124	20	then	then	ADV
ejpam-5210	124	21	[	[	X
ejpam-5210	124	22	a	a	DET
ejpam-5210	124	23	,	,	PUNCT
ejpam-5210	124	24	t1][a	t1][a	NOUN
ejpam-5210	124	25	,	,	PUNCT
ejpam-5210	124	26	t	t	PROPN
ejpam-5210	124	27	]	]	PUNCT
ejpam-5210	125	1	+	+	CCONJ
ejpam-5210	125	2	[	[	X
ejpam-5210	125	3	a	a	DET
ejpam-5210	125	4	,	,	PUNCT
ejpam-5210	125	5	t][a	t][a	NOUN
ejpam-5210	125	6	,	,	PUNCT
ejpam-5210	125	7	t1	t1	NOUN
ejpam-5210	125	8	]	]	X
ejpam-5210	125	9	=	=	SYM
ejpam-5210	125	10	0	0	X
ejpam-5210	125	11	.	.	PUNCT
ejpam-5210	126	1	(	(	PUNCT
ejpam-5210	126	2	5	5	NUM
ejpam-5210	126	3	)	)	PUNCT
ejpam-5210	126	4	since	since	SCONJ
ejpam-5210	126	5	x	x	PRON
ejpam-5210	126	6	:	:	PUNCT
ejpam-5210	126	7	=	=	PUNCT
ejpam-5210	127	1	[	[	X
ejpam-5210	127	2	[	[	X
ejpam-5210	127	3	a	a	X
ejpam-5210	127	4	,	,	PUNCT
ejpam-5210	127	5	t1	t1	NOUN
ejpam-5210	127	6	]	]	X
ejpam-5210	127	7	,	,	PUNCT
ejpam-5210	127	8	r	r	X
ejpam-5210	127	9	]	]	X
ejpam-5210	127	10	∈	∈	PROPN
ejpam-5210	127	11	u	u	NOUN
ejpam-5210	127	12	and	and	CCONJ
ejpam-5210	127	13	2[a	2[a	NUM
ejpam-5210	127	14	,	,	PUNCT
ejpam-5210	127	15	t1]r[a	t1]r[a	NOUN
ejpam-5210	127	16	,	,	PUNCT
ejpam-5210	127	17	t1	t1	NOUN
ejpam-5210	127	18	]	]	X
ejpam-5210	127	19	=	=	PUNCT
ejpam-5210	128	1	[	[	X
ejpam-5210	128	2	x	x	X
ejpam-5210	128	3	,	,	PUNCT
ejpam-5210	128	4	[	[	X
ejpam-5210	128	5	a	a	DET
ejpam-5210	128	6	,	,	PUNCT
ejpam-5210	128	7	t1	t1	NOUN
ejpam-5210	128	8	]	]	X
ejpam-5210	128	9	]	]	X
ejpam-5210	128	10	∈	∈	PROPN
ejpam-5210	128	11	w	w	PROPN
ejpam-5210	128	12	,	,	PUNCT
ejpam-5210	128	13	we	we	PRON
ejpam-5210	128	14	see	see	VERB
ejpam-5210	128	15	that	that	PRON
ejpam-5210	128	16	,	,	PUNCT
ejpam-5210	128	17	using	use	VERB
ejpam-5210	128	18	(	(	PUNCT
ejpam-5210	128	19	5	5	NUM
ejpam-5210	128	20	)	)	PUNCT
ejpam-5210	128	21	2[a	2[a	NUM
ejpam-5210	128	22	,	,	PUNCT
ejpam-5210	128	23	t1]r[a	t1]r[a	NOUN
ejpam-5210	128	24	,	,	PUNCT
ejpam-5210	128	25	t1	t1	NOUN
ejpam-5210	128	26	]	]	X
ejpam-5210	128	27	⊆	⊆	NUM
ejpam-5210	128	28	w	w	NOUN
ejpam-5210	128	29	,	,	PUNCT
ejpam-5210	128	30	and	and	CCONJ
ejpam-5210	128	31	,	,	PUNCT
ejpam-5210	128	32	in	in	ADP
ejpam-5210	128	33	view	view	NOUN
ejpam-5210	128	34	of	of	ADP
ejpam-5210	128	35	the	the	DET
ejpam-5210	128	36	identity	identity	NOUN
ejpam-5210	128	37	(	(	PUNCT
ejpam-5210	128	38	4	4	NUM
ejpam-5210	128	39	)	)	PUNCT
ejpam-5210	128	40	,	,	PUNCT
ejpam-5210	128	41	[	[	X
ejpam-5210	128	42	s	s	X
ejpam-5210	128	43	,	,	PUNCT
ejpam-5210	128	44	t][a	t][a	NOUN
ejpam-5210	128	45	,	,	PUNCT
ejpam-5210	128	46	t1]r[a	t1]r[a	NOUN
ejpam-5210	128	47	,	,	PUNCT
ejpam-5210	128	48	t1][w	t1][w	NOUN
ejpam-5210	128	49	,	,	PUNCT
ejpam-5210	128	50	t	t	X
ejpam-5210	128	51	]	]	X
ejpam-5210	128	52	=	=	SYM
ejpam-5210	128	53	0	0	X
ejpam-5210	128	54	.	.	PUNCT
ejpam-5210	128	55	(	(	PUNCT
ejpam-5210	128	56	6	6	NUM
ejpam-5210	128	57	)	)	PUNCT
ejpam-5210	128	58	in	in	ADP
ejpam-5210	128	59	the	the	DET
ejpam-5210	128	60	identity	identity	NOUN
ejpam-5210	128	61	(	(	PUNCT
ejpam-5210	128	62	6	6	NUM
ejpam-5210	128	63	)	)	PUNCT
ejpam-5210	128	64	,	,	PUNCT
ejpam-5210	128	65	put	put	VERB
ejpam-5210	128	66	s	s	PRON
ejpam-5210	128	67	=	=	NOUN
ejpam-5210	128	68	a	a	PROPN
ejpam-5210	128	69	=	=	SYM
ejpam-5210	128	70	w	w	NOUN
ejpam-5210	128	71	;	;	PUNCT
ejpam-5210	128	72	we	we	PRON
ejpam-5210	128	73	get	get	VERB
ejpam-5210	128	74	[	[	X
ejpam-5210	128	75	a	a	DET
ejpam-5210	128	76	,	,	PUNCT
ejpam-5210	128	77	t][a	t][a	NOUN
ejpam-5210	128	78	,	,	PUNCT
ejpam-5210	128	79	t1]r[a	t1]r[a	NOUN
ejpam-5210	128	80	,	,	PUNCT
ejpam-5210	128	81	t][a	t][a	NOUN
ejpam-5210	128	82	,	,	PUNCT
ejpam-5210	128	83	t1	t1	NOUN
ejpam-5210	128	84	]	]	X
ejpam-5210	128	85	=	=	SYM
ejpam-5210	129	1	0	0	X
ejpam-5210	129	2	.	.	PUNCT
ejpam-5210	130	1	this	this	PRON
ejpam-5210	130	2	means	mean	VERB
ejpam-5210	130	3	that	that	SCONJ
ejpam-5210	130	4	(	(	PUNCT
ejpam-5210	130	5	r[a	r[a	NOUN
ejpam-5210	130	6	,	,	PUNCT
ejpam-5210	130	7	t][a	t][a	NOUN
ejpam-5210	130	8	,	,	PUNCT
ejpam-5210	130	9	t1]r)2	t1]r)2	PROPN
ejpam-5210	130	10	=	=	SYM
ejpam-5210	130	11	0	0	X
ejpam-5210	130	12	.	.	PUNCT
ejpam-5210	131	1	since	since	SCONJ
ejpam-5210	131	2	c	c	X
ejpam-5210	131	3	=	=	PUNCT
ejpam-5210	131	4	∑	∑	PUNCT
ejpam-5210	131	5	a	a	DET
ejpam-5210	131	6	∈	∈	PROPN
ejpam-5210	131	7	w	w	PROPN
ejpam-5210	131	8	t	t	PROPN
ejpam-5210	131	9	,	,	PUNCT
ejpam-5210	131	10	t1	t1	PROPN
ejpam-5210	131	11	∈	∈	PROPN
ejpam-5210	131	12	u	u	NOUN
ejpam-5210	131	13	∩w	∩w	NOUN
ejpam-5210	131	14	r[a	r[a	PROPN
ejpam-5210	131	15	,	,	PUNCT
ejpam-5210	131	16	t][a	t][a	NOUN
ejpam-5210	131	17	,	,	PUNCT
ejpam-5210	131	18	t1]r	t1]r	X
ejpam-5210	131	19	m.	m.	PROPN
ejpam-5210	131	20	alosaimi	alosaimi	PROPN
ejpam-5210	131	21	et	et	PROPN
ejpam-5210	131	22	al	al	PROPN
ejpam-5210	131	23	.	.	PUNCT
ejpam-5210	131	24	/	/	SYM
ejpam-5210	131	25	eur	eur	PROPN
ejpam-5210	131	26	.	.	PUNCT
ejpam-5210	132	1	j.	j.	PROPN
ejpam-5210	132	2	pure	pure	PROPN
ejpam-5210	132	3	appl	appl	PROPN
ejpam-5210	132	4	.	.	PROPN
ejpam-5210	132	5	math	math	PROPN
ejpam-5210	132	6	,	,	PUNCT
ejpam-5210	132	7	17	17	NUM
ejpam-5210	132	8	(	(	PUNCT
ejpam-5210	132	9	3	3	NUM
ejpam-5210	132	10	)	)	PUNCT
ejpam-5210	132	11	(	(	PUNCT
ejpam-5210	132	12	2024	2024	NUM
ejpam-5210	132	13	)	)	PUNCT
ejpam-5210	132	14	,	,	PUNCT
ejpam-5210	132	15	2264	2264	NUM
ejpam-5210	132	16	-	-	SYM
ejpam-5210	132	17	2275	2275	NUM
ejpam-5210	132	18	2269	2269	NUM
ejpam-5210	132	19	is	be	AUX
ejpam-5210	132	20	a	a	DET
ejpam-5210	132	21	�	�	NOUN
ejpam-5210	132	22	-ideal	-ideal	NOUN
ejpam-5210	132	23	,	,	PUNCT
ejpam-5210	132	24	which	which	PRON
ejpam-5210	132	25	is	be	AUX
ejpam-5210	132	26	a	a	DET
ejpam-5210	132	27	sum	sum	NOUN
ejpam-5210	132	28	of	of	ADP
ejpam-5210	132	29	nilpotent	nilpotent	ADJ
ejpam-5210	132	30	ideals	ideal	NOUN
ejpam-5210	132	31	,	,	PUNCT
ejpam-5210	132	32	we	we	PRON
ejpam-5210	132	33	deduce	deduce	VERB
ejpam-5210	132	34	that	that	PRON
ejpam-5210	132	35	c	c	AUX
ejpam-5210	132	36	=	=	SYM
ejpam-5210	132	37	0	0	NUM
ejpam-5210	133	1	and	and	CCONJ
ejpam-5210	133	2	so	so	ADV
ejpam-5210	133	3	[	[	X
ejpam-5210	133	4	a	a	DET
ejpam-5210	133	5	,	,	PUNCT
ejpam-5210	133	6	t][a	t][a	NOUN
ejpam-5210	133	7	,	,	PUNCT
ejpam-5210	133	8	t1	t1	NOUN
ejpam-5210	133	9	]	]	X
ejpam-5210	134	1	=	=	SYM
ejpam-5210	134	2	0	0	X
ejpam-5210	134	3	.	.	PUNCT
ejpam-5210	135	1	(	(	PUNCT
ejpam-5210	135	2	7	7	X
ejpam-5210	135	3	)	)	PUNCT
ejpam-5210	135	4	we	we	PRON
ejpam-5210	135	5	linearize	linearize	VERB
ejpam-5210	135	6	the	the	DET
ejpam-5210	135	7	identity	identity	NOUN
ejpam-5210	135	8	(	(	PUNCT
ejpam-5210	135	9	7	7	NUM
ejpam-5210	135	10	)	)	PUNCT
ejpam-5210	135	11	on	on	ADP
ejpam-5210	135	12	a	a	PRON
ejpam-5210	135	13	to	to	PART
ejpam-5210	135	14	get	get	VERB
ejpam-5210	135	15	[	[	X
ejpam-5210	135	16	a	a	X
ejpam-5210	135	17	,	,	PUNCT
ejpam-5210	135	18	t][b	t][b	PROPN
ejpam-5210	135	19	,	,	PUNCT
ejpam-5210	135	20	t1	t1	PROPN
ejpam-5210	135	21	]	]	X
ejpam-5210	136	1	+	+	CCONJ
ejpam-5210	136	2	[	[	X
ejpam-5210	136	3	b	b	NOUN
ejpam-5210	136	4	,	,	PUNCT
ejpam-5210	136	5	t][a	t][a	NOUN
ejpam-5210	136	6	,	,	PUNCT
ejpam-5210	136	7	t1	t1	NOUN
ejpam-5210	136	8	]	]	X
ejpam-5210	136	9	=	=	SYM
ejpam-5210	137	1	0	0	X
ejpam-5210	137	2	.	.	PUNCT
ejpam-5210	138	1	using	use	VERB
ejpam-5210	138	2	the	the	DET
ejpam-5210	138	3	previous	previous	ADJ
ejpam-5210	138	4	relation	relation	NOUN
ejpam-5210	138	5	in	in	ADP
ejpam-5210	138	6	the	the	DET
ejpam-5210	138	7	identity	identity	NOUN
ejpam-5210	138	8	(	(	PUNCT
ejpam-5210	138	9	6	6	NUM
ejpam-5210	138	10	)	)	PUNCT
ejpam-5210	138	11	with	with	ADP
ejpam-5210	138	12	w	w	PROPN
ejpam-5210	138	13	=	=	SYM
ejpam-5210	138	14	b	b	PROPN
ejpam-5210	138	15	,	,	PUNCT
ejpam-5210	138	16	we	we	PRON
ejpam-5210	138	17	obtain	obtain	VERB
ejpam-5210	138	18	[	[	X
ejpam-5210	138	19	s	s	X
ejpam-5210	138	20	,	,	PUNCT
ejpam-5210	138	21	t][a	t][a	NOUN
ejpam-5210	138	22	,	,	PUNCT
ejpam-5210	138	23	t1]r[b	t1]r[b	ADJ
ejpam-5210	138	24	,	,	PUNCT
ejpam-5210	138	25	t1][a	t1][a	NOUN
ejpam-5210	138	26	,	,	PUNCT
ejpam-5210	138	27	t	t	PROPN
ejpam-5210	138	28	]	]	X
ejpam-5210	138	29	=	=	SYM
ejpam-5210	139	1	0	0	X
ejpam-5210	139	2	.	.	PUNCT
ejpam-5210	140	1	(	(	PUNCT
ejpam-5210	140	2	8)	8)	NUM
ejpam-5210	140	3	by	by	ADP
ejpam-5210	140	4	linearization	linearization	NOUN
ejpam-5210	140	5	the	the	DET
ejpam-5210	140	6	identity	identity	NOUN
ejpam-5210	140	7	(	(	PUNCT
ejpam-5210	140	8	3	3	NUM
ejpam-5210	140	9	)	)	PUNCT
ejpam-5210	140	10	for	for	ADP
ejpam-5210	140	11	t	t	PROPN
ejpam-5210	140	12	,	,	PUNCT
ejpam-5210	140	13	we	we	PRON
ejpam-5210	140	14	have	have	VERB
ejpam-5210	140	15	[	[	X
ejpam-5210	140	16	s	s	X
ejpam-5210	140	17	,	,	PUNCT
ejpam-5210	140	18	t][t1	t][t1	PROPN
ejpam-5210	140	19	,	,	PUNCT
ejpam-5210	140	20	w	w	NOUN
ejpam-5210	140	21	]	]	X
ejpam-5210	141	1	+	+	CCONJ
ejpam-5210	141	2	[	[	X
ejpam-5210	141	3	s	s	NOUN
ejpam-5210	141	4	,	,	PUNCT
ejpam-5210	141	5	t1][t	t1][t	NOUN
ejpam-5210	141	6	,	,	PUNCT
ejpam-5210	141	7	w	w	NOUN
ejpam-5210	141	8	]	]	X
ejpam-5210	141	9	=	=	SYM
ejpam-5210	141	10	0	0	X
ejpam-5210	141	11	.	.	PUNCT
ejpam-5210	142	1	in	in	ADP
ejpam-5210	142	2	view	view	NOUN
ejpam-5210	142	3	of	of	ADP
ejpam-5210	142	4	it	it	PRON
ejpam-5210	142	5	,	,	PUNCT
ejpam-5210	142	6	from	from	ADP
ejpam-5210	142	7	the	the	DET
ejpam-5210	142	8	identity	identity	NOUN
ejpam-5210	142	9	(	(	PUNCT
ejpam-5210	142	10	8)	8)	NUM
ejpam-5210	142	11	,	,	PUNCT
ejpam-5210	142	12	by	by	ADP
ejpam-5210	142	13	replacing	replace	VERB
ejpam-5210	142	14	b	b	NOUN
ejpam-5210	142	15	instead	instead	ADV
ejpam-5210	142	16	of	of	ADP
ejpam-5210	142	17	s	s	PRON
ejpam-5210	142	18	and	and	CCONJ
ejpam-5210	142	19	w	w	NOUN
ejpam-5210	142	20	by	by	ADP
ejpam-5210	142	21	a	a	X
ejpam-5210	142	22	,	,	PUNCT
ejpam-5210	142	23	it	it	PRON
ejpam-5210	142	24	follows	follow	VERB
ejpam-5210	142	25	that	that	SCONJ
ejpam-5210	142	26	[	[	X
ejpam-5210	142	27	s	s	NOUN
ejpam-5210	142	28	,	,	PUNCT
ejpam-5210	142	29	t][a	t][a	NOUN
ejpam-5210	142	30	,	,	PUNCT
ejpam-5210	142	31	t1]r[s	t1]r[s	PROPN
ejpam-5210	142	32	,	,	PUNCT
ejpam-5210	142	33	t][a	t][a	NOUN
ejpam-5210	142	34	,	,	PUNCT
ejpam-5210	142	35	t1	t1	NOUN
ejpam-5210	142	36	]	]	X
ejpam-5210	142	37	=	=	SYM
ejpam-5210	143	1	0	0	X
ejpam-5210	143	2	.	.	PUNCT
ejpam-5210	144	1	then	then	ADV
ejpam-5210	144	2	d	d	X
ejpam-5210	144	3	=	=	PUNCT
ejpam-5210	144	4	∑	∑	PROPN
ejpam-5210	144	5	a	a	PRON
ejpam-5210	144	6	,	,	PUNCT
ejpam-5210	144	7	s	s	NOUN
ejpam-5210	144	8	∈	∈	PROPN
ejpam-5210	144	9	w	w	PROPN
ejpam-5210	144	10	t	t	PROPN
ejpam-5210	144	11	,	,	PUNCT
ejpam-5210	144	12	t1	t1	PROPN
ejpam-5210	144	13	∈	∈	PROPN
ejpam-5210	144	14	u	u	NOUN
ejpam-5210	144	15	∩w	∩w	ADJ
ejpam-5210	144	16	r[s	r[s	PROPN
ejpam-5210	144	17	,	,	PUNCT
ejpam-5210	144	18	t][a	t][a	NOUN
ejpam-5210	144	19	,	,	PUNCT
ejpam-5210	144	20	t1]r	t1]r	ADV
ejpam-5210	144	21	is	be	AUX
ejpam-5210	144	22	a	a	DET
ejpam-5210	144	23	�	�	NOUN
ejpam-5210	144	24	-ideal	-ideal	NOUN
ejpam-5210	144	25	.	.	PUNCT
ejpam-5210	145	1	then	then	ADV
ejpam-5210	145	2	d	d	X
ejpam-5210	145	3	=	=	SYM
ejpam-5210	145	4	0	0	NUM
ejpam-5210	145	5	and	and	CCONJ
ejpam-5210	145	6	[	[	X
ejpam-5210	145	7	s	s	X
ejpam-5210	145	8	,	,	PUNCT
ejpam-5210	145	9	t][a	t][a	NOUN
ejpam-5210	145	10	,	,	PUNCT
ejpam-5210	145	11	t1	t1	NOUN
ejpam-5210	145	12	]	]	X
ejpam-5210	146	1	=	=	SYM
ejpam-5210	146	2	0	0	X
ejpam-5210	146	3	.	.	PUNCT
ejpam-5210	146	4	denote	denote	VERB
ejpam-5210	147	1	[	[	X
ejpam-5210	147	2	w	w	X
ejpam-5210	147	3	,	,	PUNCT
ejpam-5210	147	4	[	[	X
ejpam-5210	147	5	u	u	NOUN
ejpam-5210	147	6	,	,	PUNCT
ejpam-5210	147	7	w	w	PROPN
ejpam-5210	147	8	]	]	X
ejpam-5210	147	9	]	]	PUNCT
ejpam-5210	147	10	by	by	ADP
ejpam-5210	147	11	w1	w1	PROPN
ejpam-5210	147	12	.	.	PUNCT
ejpam-5210	148	1	then	then	ADV
ejpam-5210	148	2	w1	w1	PROPN
ejpam-5210	148	3	is	be	AUX
ejpam-5210	148	4	a	a	DET
ejpam-5210	148	5	lie	lie	NOUN
ejpam-5210	148	6	�	�	NOUN
ejpam-5210	148	7	-ideal	-ideal	NOUN
ejpam-5210	148	8	of	of	ADP
ejpam-5210	148	9	r	r	NOUN
ejpam-5210	148	10	and	and	CCONJ
ejpam-5210	148	11	[	[	X
ejpam-5210	148	12	s	s	X
ejpam-5210	148	13	,	,	PUNCT
ejpam-5210	148	14	t]w1	t]w1	NOUN
ejpam-5210	148	15	=	=	SYM
ejpam-5210	148	16	0	0	X
ejpam-5210	148	17	.	.	PUNCT
ejpam-5210	149	1	furthermore	furthermore	ADV
ejpam-5210	149	2	,	,	PUNCT
ejpam-5210	149	3	[	[	X
ejpam-5210	149	4	u	u	NOUN
ejpam-5210	149	5	,	,	PUNCT
ejpam-5210	149	6	w1	w1	NOUN
ejpam-5210	149	7	]	]	PUNCT
ejpam-5210	149	8	⊆	⊆	NUM
ejpam-5210	149	9	w1	w1	NOUN
ejpam-5210	149	10	,	,	PUNCT
ejpam-5210	149	11	[	[	X
ejpam-5210	149	12	s	s	X
ejpam-5210	149	13	,	,	PUNCT
ejpam-5210	149	14	t]uw1	t]uw1	X
ejpam-5210	149	15	=	=	SYM
ejpam-5210	149	16	0	0	NUM
ejpam-5210	149	17	and	and	CCONJ
ejpam-5210	149	18	[	[	X
ejpam-5210	149	19	s	s	X
ejpam-5210	149	20	,	,	PUNCT
ejpam-5210	149	21	t]uw1	t]uw1	X
ejpam-5210	149	22	=	=	SYM
ejpam-5210	149	23	0	0	NUM
ejpam-5210	149	24	.	.	PUNCT
ejpam-5210	150	1	from	from	ADP
ejpam-5210	150	2	the	the	DET
ejpam-5210	150	3	equation	equation	NOUN
ejpam-5210	150	4	r[a	r[a	NOUN
ejpam-5210	150	5	,	,	PUNCT
ejpam-5210	150	6	b]r	b]r	VERB
ejpam-5210	150	7	⊆	⊆	NUM
ejpam-5210	150	8	u	u	NOUN
ejpam-5210	150	9	,	,	PUNCT
ejpam-5210	150	10	we	we	PRON
ejpam-5210	150	11	deduce	deduce	VERB
ejpam-5210	150	12	that	that	SCONJ
ejpam-5210	150	13	[	[	X
ejpam-5210	150	14	s	s	X
ejpam-5210	150	15	,	,	PUNCT
ejpam-5210	150	16	t]r[a	t]r[a	NOUN
ejpam-5210	150	17	,	,	PUNCT
ejpam-5210	150	18	b]rw1	b]rw1	NOUN
ejpam-5210	150	19	=	=	SYM
ejpam-5210	150	20	0	0	X
ejpam-5210	150	21	.	.	PUNCT
ejpam-5210	150	22	assume	assume	VERB
ejpam-5210	150	23	that	that	SCONJ
ejpam-5210	150	24	p	p	X
ejpam-5210	150	25	,	,	PUNCT
ejpam-5210	150	26	q	q	PROPN
ejpam-5210	150	27	∈	∈	PROPN
ejpam-5210	150	28	u	u	NOUN
ejpam-5210	150	29	∩w	∩w	NOUN
ejpam-5210	150	30	,	,	PUNCT
ejpam-5210	150	31	then	then	ADV
ejpam-5210	150	32	we	we	PRON
ejpam-5210	150	33	get	get	VERB
ejpam-5210	150	34	[	[	X
ejpam-5210	150	35	p	p	X
ejpam-5210	150	36	,	,	PUNCT
ejpam-5210	150	37	q]r[p	q]r[p	NOUN
ejpam-5210	150	38	,	,	PUNCT
ejpam-5210	150	39	q]rw1	q]rw1	NOUN
ejpam-5210	150	40	=	=	SYM
ejpam-5210	150	41	0	0	NUM
ejpam-5210	150	42	.	.	PUNCT
ejpam-5210	151	1	therefore	therefore	ADV
ejpam-5210	151	2	,	,	PUNCT
ejpam-5210	151	3	(	(	PUNCT
ejpam-5210	151	4	r[s	r[s	PROPN
ejpam-5210	151	5	,	,	PUNCT
ejpam-5210	151	6	t]r)3	t]r)3	ADP
ejpam-5210	151	7	=	=	NOUN
ejpam-5210	151	8	0	0	X
ejpam-5210	151	9	.	.	PUNCT
ejpam-5210	152	1	then	then	ADV
ejpam-5210	152	2	e	e	X
ejpam-5210	152	3	=	=	PUNCT
ejpam-5210	152	4	∑	∑	PROPN
ejpam-5210	152	5	p	p	X
ejpam-5210	152	6	,	,	PUNCT
ejpam-5210	152	7	q	q	PROPN
ejpam-5210	152	8	∈	∈	PROPN
ejpam-5210	152	9	u	u	NOUN
ejpam-5210	152	10	∩w	∩w	PROPN
ejpam-5210	152	11	r[p	r[p	PROPN
ejpam-5210	152	12	,	,	PUNCT
ejpam-5210	152	13	q]r	q]r	NOUN
ejpam-5210	152	14	is	be	AUX
ejpam-5210	152	15	a	a	DET
ejpam-5210	152	16	nil	nil	ADJ
ejpam-5210	152	17	�	�	NOUN
ejpam-5210	152	18	-ideal	-ideal	NOUN
ejpam-5210	152	19	of	of	ADP
ejpam-5210	152	20	r	r	NOUN
ejpam-5210	152	21	,	,	PUNCT
ejpam-5210	152	22	hence	hence	ADV
ejpam-5210	152	23	[	[	X
ejpam-5210	152	24	p	p	X
ejpam-5210	152	25	,	,	PUNCT
ejpam-5210	152	26	q	q	X
ejpam-5210	152	27	]	]	X
ejpam-5210	152	28	=	=	SYM
ejpam-5210	152	29	0	0	X
ejpam-5210	152	30	.	.	PUNCT
ejpam-5210	153	1	as	as	ADP
ejpam-5210	153	2	a	a	DET
ejpam-5210	153	3	consequence	consequence	NOUN
ejpam-5210	153	4	,	,	PUNCT
ejpam-5210	153	5	t	t	PROPN
ejpam-5210	153	6	∈	∈	PROPN
ejpam-5210	154	1	[	[	X
ejpam-5210	154	2	u	u	NOUN
ejpam-5210	154	3	,	,	PUNCT
ejpam-5210	154	4	w	w	PROPN
ejpam-5210	154	5	]	]	PUNCT
ejpam-5210	154	6	is	be	AUX
ejpam-5210	154	7	commuting	commute	VERB
ejpam-5210	154	8	with	with	ADP
ejpam-5210	154	9	[	[	X
ejpam-5210	154	10	u	u	NOUN
ejpam-5210	154	11	,	,	PUNCT
ejpam-5210	154	12	[	[	X
ejpam-5210	154	13	u	u	NOUN
ejpam-5210	154	14	,	,	PUNCT
ejpam-5210	154	15	w	w	PROPN
ejpam-5210	154	16	]	]	X
ejpam-5210	154	17	]	]	PUNCT
ejpam-5210	154	18	.	.	PUNCT
ejpam-5210	155	1	by	by	ADP
ejpam-5210	155	2	lemma	lemma	PROPN
ejpam-5210	155	3	3	3	NUM
ejpam-5210	155	4	,	,	PUNCT
ejpam-5210	155	5	t	t	PROPN
ejpam-5210	155	6	∈	∈	PROPN
ejpam-5210	155	7	cr(u	cr(u	NOUN
ejpam-5210	155	8	)	)	PUNCT
ejpam-5210	155	9	.	.	PUNCT
ejpam-5210	156	1	then	then	ADV
ejpam-5210	156	2	t	t	PROPN
ejpam-5210	156	3	∈	∈	PROPN
ejpam-5210	156	4	cr([δ	cr([δ	PROPN
ejpam-5210	156	5	m1	m1	PROPN
ejpam-5210	156	6	1	1	NUM
ejpam-5210	156	7	·	·	PUNCT
ejpam-5210	156	8	·	·	PUNCT
ejpam-5210	156	9	·	·	PUNCT
ejpam-5210	157	1	δmk	δmk	PRON
ejpam-5210	157	2	k	k	X
ejpam-5210	157	3	(	(	PUNCT
ejpam-5210	157	4	t	t	PROPN
ejpam-5210	157	5	)	)	PUNCT
ejpam-5210	157	6	,	,	PUNCT
ejpam-5210	157	7	u	u	NOUN
ejpam-5210	157	8	]	]	X
ejpam-5210	157	9	)	)	PUNCT
ejpam-5210	157	10	,	,	PUNCT
ejpam-5210	157	11	m.	m.	NOUN
ejpam-5210	157	12	alosaimi	alosaimi	PROPN
ejpam-5210	157	13	et	et	PROPN
ejpam-5210	157	14	al	al	PROPN
ejpam-5210	157	15	.	.	PUNCT
ejpam-5210	157	16	/	/	SYM
ejpam-5210	157	17	eur	eur	PROPN
ejpam-5210	157	18	.	.	PUNCT
ejpam-5210	158	1	j.	j.	PROPN
ejpam-5210	158	2	pure	pure	PROPN
ejpam-5210	158	3	appl	appl	PROPN
ejpam-5210	158	4	.	.	PROPN
ejpam-5210	158	5	math	math	PROPN
ejpam-5210	158	6	,	,	PUNCT
ejpam-5210	158	7	17	17	NUM
ejpam-5210	158	8	(	(	PUNCT
ejpam-5210	158	9	3	3	NUM
ejpam-5210	158	10	)	)	PUNCT
ejpam-5210	158	11	(	(	PUNCT
ejpam-5210	158	12	2024	2024	NUM
ejpam-5210	158	13	)	)	PUNCT
ejpam-5210	158	14	,	,	PUNCT
ejpam-5210	158	15	2264	2264	NUM
ejpam-5210	158	16	-	-	SYM
ejpam-5210	158	17	2275	2275	NUM
ejpam-5210	158	18	2270	2270	NUM
ejpam-5210	158	19	for	for	ADP
ejpam-5210	158	20	any	any	DET
ejpam-5210	158	21	integers	integer	NOUN
ejpam-5210	158	22	k	k	X
ejpam-5210	158	23	≥	≥	NUM
ejpam-5210	158	24	1	1	NUM
ejpam-5210	158	25	,	,	PUNCT
ejpam-5210	158	26	mi	mi	PROPN
ejpam-5210	158	27	≥	≥	PROPN
ejpam-5210	158	28	0	0	NUM
ejpam-5210	158	29	and	and	CCONJ
ejpam-5210	158	30	derivations	derivation	NOUN
ejpam-5210	158	31	δi	δi	PROPN
ejpam-5210	158	32	∈	∈	PROPN
ejpam-5210	158	33	�	�	PROPN
ejpam-5210	158	34	,	,	PUNCT
ejpam-5210	158	35	where	where	SCONJ
ejpam-5210	158	36	i	i	PRON
ejpam-5210	158	37	=	=	NOUN
ejpam-5210	158	38	1	1	NUM
ejpam-5210	158	39	,	,	PUNCT
ejpam-5210	158	40	.	.	PUNCT
ejpam-5210	158	41	.	.	PUNCT
ejpam-5210	159	1	.	.	PUNCT
ejpam-5210	160	1	,	,	PUNCT
ejpam-5210	160	2	k	k	PROPN
ejpam-5210	160	3	and	and	CCONJ
ejpam-5210	160	4	t	t	PROPN
ejpam-5210	160	5	∈	∈	PROPN
ejpam-5210	160	6	z(r	z(r	PROPN
ejpam-5210	160	7	)	)	PUNCT
ejpam-5210	160	8	.	.	PUNCT
ejpam-5210	161	1	this	this	PRON
ejpam-5210	161	2	means	mean	VERB
ejpam-5210	161	3	that	that	SCONJ
ejpam-5210	161	4	[	[	X
ejpam-5210	161	5	u	u	NOUN
ejpam-5210	161	6	,	,	PUNCT
ejpam-5210	161	7	w	w	PROPN
ejpam-5210	161	8	]	]	PUNCT
ejpam-5210	161	9	⊆	⊆	NUM
ejpam-5210	161	10	z(r	z(r	NOUN
ejpam-5210	161	11	)	)	PUNCT
ejpam-5210	161	12	.	.	PUNCT
ejpam-5210	162	1	since	since	SCONJ
ejpam-5210	162	2	u	u	PROPN
ejpam-5210	162	3	∈	∈	PROPN
ejpam-5210	162	4	u	u	NOUN
ejpam-5210	162	5	is	be	AUX
ejpam-5210	162	6	commuting	commute	VERB
ejpam-5210	162	7	with	with	ADP
ejpam-5210	162	8	[	[	X
ejpam-5210	162	9	δm1	δm1	NOUN
ejpam-5210	162	10	1	1	NUM
ejpam-5210	162	11	·	·	PUNCT
ejpam-5210	162	12	·	·	PUNCT
ejpam-5210	163	1	·	·	PUNCT
ejpam-5210	164	1	δmk	δmk	PRON
ejpam-5210	165	1	k	k	X
ejpam-5210	165	2	(	(	PUNCT
ejpam-5210	165	3	u	u	NOUN
ejpam-5210	165	4	)	)	PUNCT
ejpam-5210	165	5	,	,	PUNCT
ejpam-5210	165	6	w	w	PROPN
ejpam-5210	165	7	]	]	X
ejpam-5210	165	8	,	,	PUNCT
ejpam-5210	165	9	for	for	ADP
ejpam-5210	165	10	any	any	DET
ejpam-5210	165	11	w	w	PROPN
ejpam-5210	165	12	∈	∈	PROPN
ejpam-5210	165	13	w	w	NOUN
ejpam-5210	165	14	,	,	PUNCT
ejpam-5210	165	15	we	we	PRON
ejpam-5210	165	16	deduce	deduce	VERB
ejpam-5210	165	17	that	that	SCONJ
ejpam-5210	165	18	[	[	X
ejpam-5210	165	19	u	u	NOUN
ejpam-5210	165	20	,	,	PUNCT
ejpam-5210	165	21	w	w	NOUN
ejpam-5210	165	22	]	]	X
ejpam-5210	165	23	=	=	SYM
ejpam-5210	165	24	0	0	NUM
ejpam-5210	165	25	,	,	PUNCT
ejpam-5210	165	26	is	be	AUX
ejpam-5210	165	27	a	a	DET
ejpam-5210	165	28	contradiction	contradiction	NOUN
ejpam-5210	165	29	.	.	PUNCT
ejpam-5210	166	1	lemma	lemma	PROPN
ejpam-5210	166	2	7	7	X
ejpam-5210	166	3	.	.	PUNCT
ejpam-5210	167	1	let	let	VERB
ejpam-5210	167	2	r	r	PRON
ejpam-5210	167	3	be	be	AUX
ejpam-5210	167	4	a	a	DET
ejpam-5210	167	5	2	2	NUM
ejpam-5210	167	6	-	-	PUNCT
ejpam-5210	167	7	torsion	torsion	NOUN
ejpam-5210	167	8	-	-	PUNCT
ejpam-5210	167	9	free	free	ADJ
ejpam-5210	167	10	�	�	PROPN
ejpam-5210	167	11	-semiprime	-semiprime	PROPN
ejpam-5210	167	12	ring	ring	NOUN
ejpam-5210	167	13	,	,	PUNCT
ejpam-5210	167	14	u	u	NOUN
ejpam-5210	167	15	its	its	PRON
ejpam-5210	167	16	lie	lie	NOUN
ejpam-5210	167	17	�	�	NOUN
ejpam-5210	167	18	-ideal	-ideal	NOUN
ejpam-5210	167	19	.	.	PUNCT
ejpam-5210	168	1	if	if	SCONJ
ejpam-5210	168	2	a	a	DET
ejpam-5210	168	3	⊆	⊆	NUM
ejpam-5210	168	4	u	u	NOUN
ejpam-5210	168	5	and	and	CCONJ
ejpam-5210	168	6	satisfies	satisfie	NOUN
ejpam-5210	168	7	that	that	SCONJ
ejpam-5210	168	8	δ(a	δ(a	PROPN
ejpam-5210	168	9	)	)	PUNCT
ejpam-5210	168	10	⊆	⊆	NUM
ejpam-5210	168	11	a	a	PRON
ejpam-5210	168	12	for	for	ADP
ejpam-5210	168	13	all	all	DET
ejpam-5210	168	14	δ	δ	PROPN
ejpam-5210	168	15	∈	∈	PROPN
ejpam-5210	168	16	�	�	PROPN
ejpam-5210	168	17	and	and	CCONJ
ejpam-5210	168	18	it	it	PRON
ejpam-5210	168	19	is	be	AUX
ejpam-5210	168	20	an	an	DET
ejpam-5210	168	21	additive	additive	ADJ
ejpam-5210	168	22	subgroup	subgroup	NOUN
ejpam-5210	168	23	such	such	ADJ
ejpam-5210	168	24	that	that	SCONJ
ejpam-5210	168	25	[	[	X
ejpam-5210	168	26	u	u	NOUN
ejpam-5210	168	27	,	,	PUNCT
ejpam-5210	168	28	a	a	X
ejpam-5210	168	29	]	]	X
ejpam-5210	168	30	⊆	⊆	NUM
ejpam-5210	168	31	a	a	PRON
ejpam-5210	168	32	and	and	CCONJ
ejpam-5210	168	33	[	[	X
ejpam-5210	168	34	a	a	X
ejpam-5210	168	35	,	,	PUNCT
ejpam-5210	168	36	a	a	PRON
ejpam-5210	168	37	]	]	X
ejpam-5210	168	38	⊆	⊆	NUM
ejpam-5210	168	39	z(r	z(r	NOUN
ejpam-5210	168	40	)	)	PUNCT
ejpam-5210	168	41	,	,	PUNCT
ejpam-5210	168	42	then	then	ADV
ejpam-5210	168	43	[	[	X
ejpam-5210	168	44	a	a	X
ejpam-5210	168	45	,	,	PUNCT
ejpam-5210	168	46	u	u	NOUN
ejpam-5210	168	47	]	]	X
ejpam-5210	168	48	=	=	SYM
ejpam-5210	168	49	0	0	X
ejpam-5210	168	50	.	.	PUNCT
ejpam-5210	169	1	proof	proof	NOUN
ejpam-5210	169	2	.	.	PUNCT
ejpam-5210	170	1	let	let	VERB
ejpam-5210	170	2	u	u	PRON
ejpam-5210	170	3	∈	∈	PROPN
ejpam-5210	170	4	u	u	NOUN
ejpam-5210	170	5	and	and	CCONJ
ejpam-5210	170	6	x	x	PROPN
ejpam-5210	170	7	∈	∈	PROPN
ejpam-5210	170	8	r.	r.	NOUN
ejpam-5210	170	9	if	if	SCONJ
ejpam-5210	170	10	[	[	X
ejpam-5210	170	11	a	a	X
ejpam-5210	170	12	,	,	PUNCT
ejpam-5210	170	13	a	a	X
ejpam-5210	170	14	]	]	X
ejpam-5210	170	15	=	=	SYM
ejpam-5210	170	16	0	0	NUM
ejpam-5210	170	17	,	,	PUNCT
ejpam-5210	170	18	then	then	ADV
ejpam-5210	170	19	[	[	X
ejpam-5210	170	20	a	a	X
ejpam-5210	170	21	,	,	PUNCT
ejpam-5210	170	22	u	u	NOUN
ejpam-5210	170	23	]	]	X
ejpam-5210	170	24	∈	∈	PROPN
ejpam-5210	170	25	a	a	DET
ejpam-5210	170	26	∩	∩	NOUN
ejpam-5210	170	27	cr(a	cr(a	NUM
ejpam-5210	170	28	)	)	PUNCT
ejpam-5210	170	29	.	.	PUNCT
ejpam-5210	171	1	by	by	ADP
ejpam-5210	171	2	lemma	lemma	PROPN
ejpam-5210	171	3	5	5	NUM
ejpam-5210	171	4	,	,	PUNCT
ejpam-5210	171	5	[	[	X
ejpam-5210	171	6	a	a	X
ejpam-5210	171	7	,	,	PUNCT
ejpam-5210	171	8	u	u	NOUN
ejpam-5210	171	9	]	]	X
ejpam-5210	171	10	=	=	SYM
ejpam-5210	171	11	0	0	X
ejpam-5210	171	12	.	.	PUNCT
ejpam-5210	172	1	therefore	therefore	ADV
ejpam-5210	172	2	,	,	PUNCT
ejpam-5210	172	3	we	we	PRON
ejpam-5210	172	4	assume	assume	VERB
ejpam-5210	172	5	that	that	SCONJ
ejpam-5210	172	6	0	0	NUM
ejpam-5210	173	1	̸=	̸=	PROPN
ejpam-5210	173	2	[	[	X
ejpam-5210	173	3	a	a	X
ejpam-5210	173	4	,	,	PUNCT
ejpam-5210	173	5	b	b	X
ejpam-5210	173	6	]	]	X
ejpam-5210	173	7	∈	∈	PROPN
ejpam-5210	173	8	z(r	z(r	PROPN
ejpam-5210	173	9	)	)	PUNCT
ejpam-5210	173	10	for	for	ADP
ejpam-5210	173	11	some	some	DET
ejpam-5210	173	12	a	a	PRON
ejpam-5210	173	13	,	,	PUNCT
ejpam-5210	173	14	b	b	X
ejpam-5210	173	15	∈	∈	NOUN
ejpam-5210	173	16	a.	a.	NOUN
ejpam-5210	173	17	as	as	ADP
ejpam-5210	173	18	in	in	ADP
ejpam-5210	173	19	the	the	DET
ejpam-5210	173	20	proof	proof	NOUN
ejpam-5210	173	21	of	of	ADP
ejpam-5210	173	22	[	[	X
ejpam-5210	173	23	8	8	NUM
ejpam-5210	173	24	,	,	PUNCT
ejpam-5210	173	25	lemma	lemma	PROPN
ejpam-5210	173	26	4	4	NUM
ejpam-5210	173	27	]	]	PUNCT
ejpam-5210	173	28	,	,	PUNCT
ejpam-5210	173	29	we	we	PRON
ejpam-5210	173	30	can	can	AUX
ejpam-5210	173	31	obtain	obtain	VERB
ejpam-5210	173	32	that	that	PRON
ejpam-5210	173	33	[	[	X
ejpam-5210	173	34	a	a	DET
ejpam-5210	173	35	,	,	PUNCT
ejpam-5210	173	36	b]4	b]4	NOUN
ejpam-5210	173	37	=	=	NOUN
ejpam-5210	173	38	0	0	X
ejpam-5210	173	39	.	.	PUNCT
ejpam-5210	174	1	since	since	SCONJ
ejpam-5210	174	2	[	[	X
ejpam-5210	174	3	a	a	X
ejpam-5210	174	4	,	,	PUNCT
ejpam-5210	174	5	a	a	PRON
ejpam-5210	174	6	]	]	X
ejpam-5210	174	7	⊆	⊆	NUM
ejpam-5210	174	8	z(r	z(r	NOUN
ejpam-5210	174	9	)	)	PUNCT
ejpam-5210	174	10	,	,	PUNCT
ejpam-5210	174	11	we	we	PRON
ejpam-5210	174	12	deduce	deduce	VERB
ejpam-5210	174	13	that	that	SCONJ
ejpam-5210	174	14	i	i	PRON
ejpam-5210	174	15	=	=	PUNCT
ejpam-5210	174	16	∑	∑	PUNCT
ejpam-5210	174	17	a	a	DET
ejpam-5210	174	18	,	,	PUNCT
ejpam-5210	174	19	b∈a	b∈a	NOUN
ejpam-5210	174	20	[	[	X
ejpam-5210	174	21	a	a	PRON
ejpam-5210	174	22	,	,	PUNCT
ejpam-5210	174	23	b]r	b]r	NOUN
ejpam-5210	174	24	is	be	AUX
ejpam-5210	174	25	a	a	DET
ejpam-5210	174	26	nil	nil	ADJ
ejpam-5210	174	27	�	�	NOUN
ejpam-5210	174	28	-ideal	-ideal	NOUN
ejpam-5210	174	29	.	.	PUNCT
ejpam-5210	175	1	then	then	ADV
ejpam-5210	175	2	i	i	PRON
ejpam-5210	175	3	=	=	NOUN
ejpam-5210	175	4	0	0	PROPN
ejpam-5210	175	5	,	,	PUNCT
ejpam-5210	175	6	which	which	PRON
ejpam-5210	175	7	is	be	AUX
ejpam-5210	175	8	a	a	DET
ejpam-5210	175	9	contradiction	contradiction	NOUN
ejpam-5210	175	10	.	.	PUNCT
ejpam-5210	176	1	corollary	corollary	ADJ
ejpam-5210	176	2	1	1	NUM
ejpam-5210	176	3	.	.	PUNCT
ejpam-5210	177	1	let	let	VERB
ejpam-5210	177	2	r	r	PRON
ejpam-5210	177	3	be	be	AUX
ejpam-5210	177	4	a	a	DET
ejpam-5210	177	5	2	2	NUM
ejpam-5210	177	6	-	-	PUNCT
ejpam-5210	177	7	torsion	torsion	NOUN
ejpam-5210	177	8	-	-	PUNCT
ejpam-5210	177	9	free	free	ADJ
ejpam-5210	177	10	�	�	PROPN
ejpam-5210	177	11	-semiprime	-semiprime	PROPN
ejpam-5210	177	12	ring	ring	NOUN
ejpam-5210	177	13	,	,	PUNCT
ejpam-5210	177	14	u	u	NOUN
ejpam-5210	177	15	its	its	PRON
ejpam-5210	177	16	lie	lie	NOUN
ejpam-5210	177	17	�	�	NOUN
ejpam-5210	177	18	-ideal	-ideal	NOUN
ejpam-5210	177	19	and	and	CCONJ
ejpam-5210	177	20	v	v	ADJ
ejpam-5210	177	21	satisfies	satisfie	NOUN
ejpam-5210	177	22	that	that	PRON
ejpam-5210	177	23	δ(v	δ(v	PROPN
ejpam-5210	177	24	)	)	PUNCT
ejpam-5210	178	1	⊆	⊆	NUM
ejpam-5210	178	2	v	v	NOUN
ejpam-5210	178	3	for	for	ADP
ejpam-5210	178	4	all	all	DET
ejpam-5210	178	5	δ	δ	PROPN
ejpam-5210	178	6	∈	∈	PROPN
ejpam-5210	178	7	�	�	PROPN
ejpam-5210	178	8	and	and	CCONJ
ejpam-5210	178	9	it	it	PRON
ejpam-5210	178	10	is	be	AUX
ejpam-5210	178	11	an	an	DET
ejpam-5210	178	12	additive	additive	ADJ
ejpam-5210	178	13	subgroup	subgroup	NOUN
ejpam-5210	178	14	of	of	ADP
ejpam-5210	178	15	r	r	NOUN
ejpam-5210	178	16	such	such	ADJ
ejpam-5210	178	17	that	that	SCONJ
ejpam-5210	179	1	[	[	X
ejpam-5210	179	2	v	v	NOUN
ejpam-5210	179	3	,	,	PUNCT
ejpam-5210	179	4	u	u	NOUN
ejpam-5210	179	5	]	]	PUNCT
ejpam-5210	179	6	⊆	⊆	NUM
ejpam-5210	179	7	v	v	NOUN
ejpam-5210	179	8	.	.	PUNCT
ejpam-5210	180	1	then	then	ADV
ejpam-5210	180	2	either	either	CCONJ
ejpam-5210	180	3	[	[	X
ejpam-5210	180	4	v	v	NOUN
ejpam-5210	180	5	,	,	PUNCT
ejpam-5210	180	6	u	u	NOUN
ejpam-5210	180	7	]	]	X
ejpam-5210	180	8	=	=	SYM
ejpam-5210	180	9	0	0	PUNCT
ejpam-5210	180	10	or	or	CCONJ
ejpam-5210	180	11	there	there	PRON
ejpam-5210	180	12	exists	exist	VERB
ejpam-5210	180	13	a	a	DET
ejpam-5210	180	14	�	�	NOUN
ejpam-5210	180	15	-ideal	-ideal	NOUN
ejpam-5210	180	16	m	m	NOUN
ejpam-5210	180	17	of	of	ADP
ejpam-5210	180	18	r	r	NOUN
ejpam-5210	181	1	such	such	ADJ
ejpam-5210	181	2	that	that	DET
ejpam-5210	181	3	0	0	NUM
ejpam-5210	181	4	̸=	̸=	PROPN
ejpam-5210	181	5	[	[	X
ejpam-5210	181	6	m	m	X
ejpam-5210	181	7	,	,	PUNCT
ejpam-5210	181	8	r	r	X
ejpam-5210	181	9	]	]	X
ejpam-5210	181	10	⊆	⊆	NUM
ejpam-5210	181	11	v	v	NOUN
ejpam-5210	181	12	(	(	PUNCT
ejpam-5210	181	13	in	in	ADP
ejpam-5210	181	14	particular	particular	ADJ
ejpam-5210	181	15	,	,	PUNCT
ejpam-5210	181	16	in	in	ADP
ejpam-5210	181	17	the	the	DET
ejpam-5210	181	18	second	second	ADJ
ejpam-5210	181	19	case	case	NOUN
ejpam-5210	181	20	,	,	PUNCT
ejpam-5210	181	21	v	v	NOUN
ejpam-5210	181	22	contains	contain	VERB
ejpam-5210	181	23	a	a	DET
ejpam-5210	181	24	non	non	ADJ
ejpam-5210	181	25	-	-	ADJ
ejpam-5210	181	26	zero	zero	NUM
ejpam-5210	181	27	lie	lie	NOUN
ejpam-5210	181	28	�	�	NOUN
ejpam-5210	181	29	-ideal	-ideal	NOUN
ejpam-5210	181	30	of	of	ADP
ejpam-5210	181	31	r	r	NOUN
ejpam-5210	181	32	)	)	PUNCT
ejpam-5210	181	33	.	.	PUNCT
ejpam-5210	182	1	proof	proof	NOUN
ejpam-5210	182	2	.	.	PUNCT
ejpam-5210	183	1	clearly	clearly	ADV
ejpam-5210	183	2	that	that	SCONJ
ejpam-5210	183	3	a	a	PRON
ejpam-5210	184	1	=	=	X
ejpam-5210	185	1	[	[	X
ejpam-5210	185	2	v	v	NOUN
ejpam-5210	185	3	,	,	PUNCT
ejpam-5210	185	4	u	u	NOUN
ejpam-5210	185	5	]	]	PUNCT
ejpam-5210	185	6	⊆	⊆	NUM
ejpam-5210	185	7	v	v	ADP
ejpam-5210	185	8	∩	∩	ADJ
ejpam-5210	185	9	u	u	NOUN
ejpam-5210	185	10	,	,	PUNCT
ejpam-5210	185	11	where	where	SCONJ
ejpam-5210	185	12	δ(a	δ(a	PROPN
ejpam-5210	185	13	)	)	PUNCT
ejpam-5210	185	14	⊆	⊆	NUM
ejpam-5210	185	15	a	a	PRON
ejpam-5210	185	16	for	for	ADP
ejpam-5210	185	17	all	all	DET
ejpam-5210	185	18	δ	δ	PROPN
ejpam-5210	185	19	∈	∈	PROPN
ejpam-5210	185	20	�	�	PROPN
ejpam-5210	185	21	and	and	CCONJ
ejpam-5210	185	22	[	[	X
ejpam-5210	185	23	a	a	X
ejpam-5210	185	24	,	,	PUNCT
ejpam-5210	185	25	u	u	NOUN
ejpam-5210	185	26	]	]	PUNCT
ejpam-5210	185	27	⊆	⊆	NUM
ejpam-5210	185	28	[	[	X
ejpam-5210	185	29	v	v	NOUN
ejpam-5210	185	30	,	,	PUNCT
ejpam-5210	185	31	u	u	NOUN
ejpam-5210	185	32	]	]	PUNCT
ejpam-5210	185	33	.	.	PUNCT
ejpam-5210	186	1	then	then	ADV
ejpam-5210	186	2	t	t	PROPN
ejpam-5210	186	3	=	=	PUNCT
ejpam-5210	186	4	{	{	PUNCT
ejpam-5210	186	5	x	x	PUNCT
ejpam-5210	186	6	∈	∈	PROPN
ejpam-5210	186	7	r	r	NOUN
ejpam-5210	187	1	|	|	NOUN
ejpam-5210	187	2	[	[	X
ejpam-5210	187	3	x	x	X
ejpam-5210	187	4	,	,	PUNCT
ejpam-5210	187	5	r	r	NOUN
ejpam-5210	187	6	]	]	X
ejpam-5210	187	7	⊆	⊆	NUM
ejpam-5210	187	8	u	u	NOUN
ejpam-5210	187	9	}	}	PUNCT
ejpam-5210	187	10	is	be	AUX
ejpam-5210	187	11	a	a	DET
ejpam-5210	187	12	�	�	PROPN
ejpam-5210	187	13	-subring	-subring	NOUN
ejpam-5210	187	14	of	of	ADP
ejpam-5210	187	15	r.	r.	PROPN
ejpam-5210	187	16	let	let	VERB
ejpam-5210	187	17	t0	t0	PROPN
ejpam-5210	187	18	be	be	AUX
ejpam-5210	187	19	a	a	DET
ejpam-5210	187	20	subring	subring	NOUN
ejpam-5210	187	21	of	of	ADP
ejpam-5210	187	22	t	t	PROPN
ejpam-5210	187	23	generated	generate	VERB
ejpam-5210	187	24	by	by	ADP
ejpam-5210	187	25	[	[	X
ejpam-5210	187	26	a	a	X
ejpam-5210	187	27	,	,	PUNCT
ejpam-5210	187	28	a	a	PRON
ejpam-5210	187	29	]	]	X
ejpam-5210	187	30	.	.	PUNCT
ejpam-5210	188	1	then	then	ADV
ejpam-5210	188	2	t0	t0	PROPN
ejpam-5210	188	3	satisfies	satisfy	VERB
ejpam-5210	188	4	that	that	SCONJ
ejpam-5210	188	5	δ(t	δ(t	NOUN
ejpam-5210	188	6	)	)	PUNCT
ejpam-5210	188	7	⊆	⊆	NUM
ejpam-5210	188	8	t	t	NOUN
ejpam-5210	188	9	for	for	ADP
ejpam-5210	188	10	all	all	DET
ejpam-5210	188	11	δ	δ	PROPN
ejpam-5210	188	12	∈	∈	PROPN
ejpam-5210	188	13	�	�	PROPN
ejpam-5210	188	14	.	.	PUNCT
ejpam-5210	189	1	inasmuch	inasmuch	ADJ
ejpam-5210	190	1	[	[	X
ejpam-5210	190	2	[	[	X
ejpam-5210	190	3	a	a	X
ejpam-5210	190	4	,	,	PUNCT
ejpam-5210	190	5	a	a	PRON
ejpam-5210	190	6	]	]	X
ejpam-5210	190	7	,	,	PUNCT
ejpam-5210	190	8	u	u	X
ejpam-5210	190	9	]	]	PUNCT
ejpam-5210	190	10	⊆	⊆	NUM
ejpam-5210	190	11	[	[	X
ejpam-5210	190	12	a	a	X
ejpam-5210	190	13	,	,	PUNCT
ejpam-5210	190	14	a	a	PRON
ejpam-5210	190	15	]	]	X
ejpam-5210	190	16	,	,	PUNCT
ejpam-5210	190	17	we	we	PRON
ejpam-5210	190	18	have	have	VERB
ejpam-5210	190	19	[	[	X
ejpam-5210	190	20	t0	t0	NOUN
ejpam-5210	190	21	,	,	PUNCT
ejpam-5210	190	22	u	u	X
ejpam-5210	190	23	]	]	PUNCT
ejpam-5210	190	24	⊆	⊆	NUM
ejpam-5210	190	25	t0	t0	NOUN
ejpam-5210	190	26	.	.	PUNCT
ejpam-5210	191	1	by	by	ADP
ejpam-5210	191	2	lemma	lemma	PROPN
ejpam-5210	191	3	3	3	NUM
ejpam-5210	191	4	,	,	PUNCT
ejpam-5210	191	5	[	[	X
ejpam-5210	191	6	t0	t0	X
ejpam-5210	191	7	,	,	PUNCT
ejpam-5210	191	8	u	u	NOUN
ejpam-5210	191	9	]	]	X
ejpam-5210	191	10	=	=	SYM
ejpam-5210	191	11	0	0	NUM
ejpam-5210	191	12	or	or	CCONJ
ejpam-5210	191	13	t0	t0	PROPN
ejpam-5210	191	14	contains	contain	VERB
ejpam-5210	191	15	a	a	DET
ejpam-5210	191	16	non	non	ADJ
ejpam-5210	191	17	-	-	ADJ
ejpam-5210	191	18	zero	zero	NUM
ejpam-5210	191	19	�	�	NOUN
ejpam-5210	191	20	-ideal	-ideal	NOUN
ejpam-5210	191	21	of	of	ADP
ejpam-5210	191	22	r.	r.	PROPN
ejpam-5210	191	23	a	a	X
ejpam-5210	191	24	)	)	PUNCT
ejpam-5210	191	25	if	if	SCONJ
ejpam-5210	191	26	[	[	X
ejpam-5210	191	27	t0	t0	NOUN
ejpam-5210	191	28	,	,	PUNCT
ejpam-5210	191	29	u	u	NOUN
ejpam-5210	191	30	]	]	X
ejpam-5210	191	31	=	=	SYM
ejpam-5210	191	32	0	0	NUM
ejpam-5210	191	33	,	,	PUNCT
ejpam-5210	191	34	then	then	ADV
ejpam-5210	191	35	using	use	VERB
ejpam-5210	191	36	the	the	DET
ejpam-5210	191	37	fact	fact	NOUN
ejpam-5210	191	38	that	that	SCONJ
ejpam-5210	191	39	[	[	X
ejpam-5210	191	40	a	a	X
ejpam-5210	191	41	,	,	PUNCT
ejpam-5210	191	42	a	a	DET
ejpam-5210	191	43	]	]	X
ejpam-5210	191	44	⊆	⊆	NUM
ejpam-5210	191	45	t0	t0	NOUN
ejpam-5210	191	46	we	we	PRON
ejpam-5210	191	47	have	have	AUX
ejpam-5210	191	48	[	[	X
ejpam-5210	191	49	[	[	X
ejpam-5210	191	50	a	a	X
ejpam-5210	191	51	,	,	PUNCT
ejpam-5210	191	52	a	a	PRON
ejpam-5210	191	53	]	]	X
ejpam-5210	191	54	,	,	PUNCT
ejpam-5210	191	55	u	u	NOUN
ejpam-5210	191	56	]	]	X
ejpam-5210	191	57	=	=	SYM
ejpam-5210	191	58	0	0	X
ejpam-5210	191	59	.	.	PUNCT
ejpam-5210	191	60	m.	m.	NOUN
ejpam-5210	192	1	alosaimi	alosaimi	PROPN
ejpam-5210	192	2	et	et	PROPN
ejpam-5210	192	3	al	al	PROPN
ejpam-5210	192	4	.	.	PUNCT
ejpam-5210	192	5	/	/	SYM
ejpam-5210	192	6	eur	eur	PROPN
ejpam-5210	192	7	.	.	PUNCT
ejpam-5210	193	1	j.	j.	PROPN
ejpam-5210	193	2	pure	pure	PROPN
ejpam-5210	193	3	appl	appl	PROPN
ejpam-5210	193	4	.	.	PROPN
ejpam-5210	193	5	math	math	PROPN
ejpam-5210	193	6	,	,	PUNCT
ejpam-5210	193	7	17	17	NUM
ejpam-5210	193	8	(	(	PUNCT
ejpam-5210	193	9	3	3	NUM
ejpam-5210	193	10	)	)	PUNCT
ejpam-5210	193	11	(	(	PUNCT
ejpam-5210	193	12	2024	2024	NUM
ejpam-5210	193	13	)	)	PUNCT
ejpam-5210	193	14	,	,	PUNCT
ejpam-5210	193	15	2264	2264	NUM
ejpam-5210	193	16	-	-	SYM
ejpam-5210	193	17	2275	2275	NUM
ejpam-5210	193	18	2271	2271	NUM
ejpam-5210	193	19	since	since	SCONJ
ejpam-5210	193	20	[	[	X
ejpam-5210	193	21	a	a	X
ejpam-5210	193	22	,	,	PUNCT
ejpam-5210	193	23	a	a	DET
ejpam-5210	193	24	]	]	X
ejpam-5210	193	25	satisfies	satisfie	NOUN
ejpam-5210	193	26	δ([a	δ([a	PROPN
ejpam-5210	193	27	,	,	PUNCT
ejpam-5210	193	28	a	a	DET
ejpam-5210	193	29	]	]	X
ejpam-5210	193	30	)	)	PUNCT
ejpam-5210	193	31	⊆	⊆	NUM
ejpam-5210	194	1	[	[	X
ejpam-5210	194	2	a	a	X
ejpam-5210	194	3	,	,	PUNCT
ejpam-5210	194	4	a	a	X
ejpam-5210	194	5	]	]	X
ejpam-5210	194	6	for	for	ADP
ejpam-5210	194	7	all	all	DET
ejpam-5210	194	8	δ	δ	PROPN
ejpam-5210	194	9	∈	∈	PROPN
ejpam-5210	194	10	�	�	PROPN
ejpam-5210	194	11	,	,	PUNCT
ejpam-5210	194	12	we	we	PRON
ejpam-5210	194	13	conclude	conclude	VERB
ejpam-5210	194	14	that	that	SCONJ
ejpam-5210	194	15	for	for	ADP
ejpam-5210	194	16	a	a	DET
ejpam-5210	194	17	∈	∈	PROPN
ejpam-5210	194	18	[	[	X
ejpam-5210	194	19	a	a	X
ejpam-5210	194	20	,	,	PUNCT
ejpam-5210	194	21	a	a	PRON
ejpam-5210	194	22	]	]	X
ejpam-5210	194	23	we	we	PRON
ejpam-5210	194	24	have	have	VERB
ejpam-5210	194	25	[	[	X
ejpam-5210	194	26	δm1	δm1	NOUN
ejpam-5210	194	27	1	1	NUM
ejpam-5210	194	28	·	·	PUNCT
ejpam-5210	194	29	·	·	PUNCT
ejpam-5210	194	30	·	·	PUNCT
ejpam-5210	195	1	δmk	δmk	PRON
ejpam-5210	196	1	k	k	X
ejpam-5210	196	2	(	(	PUNCT
ejpam-5210	196	3	a	a	NOUN
ejpam-5210	196	4	)	)	PUNCT
ejpam-5210	196	5	,	,	PUNCT
ejpam-5210	196	6	[	[	X
ejpam-5210	196	7	a	a	X
ejpam-5210	196	8	,	,	PUNCT
ejpam-5210	196	9	r	r	NOUN
ejpam-5210	196	10	]	]	X
ejpam-5210	196	11	]	]	X
ejpam-5210	196	12	=	=	SYM
ejpam-5210	196	13	0	0	X
ejpam-5210	196	14	.	.	PUNCT
ejpam-5210	197	1	by	by	ADP
ejpam-5210	197	2	lemma	lemma	PROPN
ejpam-5210	197	3	5	5	NUM
ejpam-5210	197	4	,	,	PUNCT
ejpam-5210	197	5	a	a	PRON
ejpam-5210	197	6	will	will	AUX
ejpam-5210	197	7	be	be	AUX
ejpam-5210	197	8	from	from	ADP
ejpam-5210	197	9	z(r	z(r	NOUN
ejpam-5210	197	10	)	)	PUNCT
ejpam-5210	197	11	.	.	PUNCT
ejpam-5210	198	1	this	this	PRON
ejpam-5210	198	2	means	mean	VERB
ejpam-5210	198	3	that	that	SCONJ
ejpam-5210	198	4	[	[	X
ejpam-5210	198	5	a	a	X
ejpam-5210	198	6	,	,	PUNCT
ejpam-5210	198	7	a	a	PRON
ejpam-5210	198	8	]	]	X
ejpam-5210	198	9	⊆	⊆	NUM
ejpam-5210	198	10	z(r	z(r	NUM
ejpam-5210	198	11	)	)	PUNCT
ejpam-5210	198	12	.	.	PUNCT
ejpam-5210	199	1	by	by	ADP
ejpam-5210	199	2	lemma	lemma	PROPN
ejpam-5210	199	3	7	7	NUM
ejpam-5210	199	4	,	,	PUNCT
ejpam-5210	199	5	[	[	X
ejpam-5210	199	6	a	a	X
ejpam-5210	199	7	,	,	PUNCT
ejpam-5210	199	8	u	u	NOUN
ejpam-5210	199	9	]	]	X
ejpam-5210	199	10	=	=	SYM
ejpam-5210	200	1	0	0	X
ejpam-5210	200	2	.	.	PUNCT
ejpam-5210	200	3	hence	hence	ADV
ejpam-5210	200	4	by	by	ADP
ejpam-5210	200	5	lemma	lemma	PROPN
ejpam-5210	200	6	3	3	NUM
ejpam-5210	200	7	a	a	DET
ejpam-5210	200	8	⊆	⊆	NUM
ejpam-5210	200	9	z(r	z(r	NOUN
ejpam-5210	200	10	)	)	PUNCT
ejpam-5210	200	11	and	and	CCONJ
ejpam-5210	200	12	[	[	X
ejpam-5210	200	13	v	v	NOUN
ejpam-5210	200	14	,	,	PUNCT
ejpam-5210	200	15	u	u	NOUN
ejpam-5210	200	16	]	]	X
ejpam-5210	200	17	∈	∈	PROPN
ejpam-5210	200	18	a	a	PRON
ejpam-5210	200	19	for	for	ADP
ejpam-5210	200	20	any	any	DET
ejpam-5210	200	21	v	v	NUM
ejpam-5210	200	22	∈	∈	PROPN
ejpam-5210	200	23	v	v	NOUN
ejpam-5210	200	24	and	and	CCONJ
ejpam-5210	200	25	u	u	NOUN
ejpam-5210	200	26	∈	∈	PROPN
ejpam-5210	200	27	u	u	NOUN
ejpam-5210	200	28	.	.	PUNCT
ejpam-5210	201	1	then	then	ADV
ejpam-5210	201	2	[	[	X
ejpam-5210	201	3	δm1	δm1	NOUN
ejpam-5210	201	4	1	1	NUM
ejpam-5210	201	5	·	·	PUNCT
ejpam-5210	201	6	·	·	PUNCT
ejpam-5210	201	7	·	·	PUNCT
ejpam-5210	202	1	δmk	δmk	PRON
ejpam-5210	203	1	k	k	X
ejpam-5210	203	2	(	(	PUNCT
ejpam-5210	203	3	v	v	NOUN
ejpam-5210	203	4	)	)	PUNCT
ejpam-5210	203	5	,	,	PUNCT
ejpam-5210	203	6	[	[	X
ejpam-5210	203	7	v	v	NOUN
ejpam-5210	203	8	,	,	PUNCT
ejpam-5210	203	9	u	u	NOUN
ejpam-5210	203	10	]	]	X
ejpam-5210	203	11	]	]	X
ejpam-5210	203	12	=	=	SYM
ejpam-5210	203	13	0	0	NUM
ejpam-5210	203	14	for	for	ADP
ejpam-5210	203	15	any	any	DET
ejpam-5210	203	16	integers	integer	NOUN
ejpam-5210	203	17	k	k	X
ejpam-5210	203	18	≥	≥	NUM
ejpam-5210	203	19	1	1	NUM
ejpam-5210	203	20	,	,	PUNCT
ejpam-5210	203	21	mi	mi	PROPN
ejpam-5210	203	22	≥	≥	PROPN
ejpam-5210	203	23	0	0	NUM
ejpam-5210	203	24	and	and	CCONJ
ejpam-5210	203	25	derivations	derivation	NOUN
ejpam-5210	203	26	δi	δi	PROPN
ejpam-5210	203	27	∈	∈	PROPN
ejpam-5210	203	28	�	�	PROPN
ejpam-5210	203	29	where	where	SCONJ
ejpam-5210	203	30	i	i	PRON
ejpam-5210	203	31	=	=	NOUN
ejpam-5210	203	32	1	1	NUM
ejpam-5210	203	33	,	,	PUNCT
ejpam-5210	203	34	.	.	PUNCT
ejpam-5210	203	35	.	.	PUNCT
ejpam-5210	203	36	.	.	PUNCT
ejpam-5210	204	1	,	,	PUNCT
ejpam-5210	205	1	k	k	NOUN
ejpam-5210	205	2	,	,	PUNCT
ejpam-5210	205	3	and	and	CCONJ
ejpam-5210	205	4	by	by	ADP
ejpam-5210	205	5	lemma	lemma	PROPN
ejpam-5210	205	6	5	5	NUM
ejpam-5210	205	7	,	,	PUNCT
ejpam-5210	205	8	v	v	NOUN
ejpam-5210	205	9	∈	∈	NOUN
ejpam-5210	205	10	cr(u	cr(u	NOUN
ejpam-5210	205	11	)	)	PUNCT
ejpam-5210	205	12	what	what	PRON
ejpam-5210	205	13	forces	force	VERB
ejpam-5210	205	14	that	that	SCONJ
ejpam-5210	206	1	[	[	X
ejpam-5210	206	2	v	v	NOUN
ejpam-5210	206	3	,	,	PUNCT
ejpam-5210	206	4	u	u	NOUN
ejpam-5210	206	5	]	]	X
ejpam-5210	206	6	=	=	SYM
ejpam-5210	206	7	0	0	NUM
ejpam-5210	206	8	.	.	PUNCT
ejpam-5210	206	9	b	b	X
ejpam-5210	206	10	)	)	PUNCT
ejpam-5210	206	11	assume	assume	VERB
ejpam-5210	206	12	that	that	SCONJ
ejpam-5210	206	13	t0	t0	PROPN
ejpam-5210	206	14	contains	contain	VERB
ejpam-5210	206	15	a	a	DET
ejpam-5210	206	16	non	non	ADJ
ejpam-5210	206	17	-	-	ADJ
ejpam-5210	206	18	zero	zero	NUM
ejpam-5210	206	19	�	�	NOUN
ejpam-5210	206	20	-ideal	-ideal	NOUN
ejpam-5210	206	21	m	m	NOUN
ejpam-5210	206	22	of	of	ADP
ejpam-5210	206	23	r.	r.	PROPN
ejpam-5210	206	24	then	then	ADV
ejpam-5210	207	1	[	[	X
ejpam-5210	207	2	m	m	X
ejpam-5210	207	3	,	,	PUNCT
ejpam-5210	207	4	r	r	NOUN
ejpam-5210	207	5	]	]	X
ejpam-5210	207	6	̸=	̸=	PROPN
ejpam-5210	207	7	0	0	NUM
ejpam-5210	207	8	or	or	CCONJ
ejpam-5210	207	9	[	[	X
ejpam-5210	207	10	m	m	X
ejpam-5210	207	11	,	,	PUNCT
ejpam-5210	207	12	r	r	X
ejpam-5210	207	13	]	]	X
ejpam-5210	207	14	=	=	SYM
ejpam-5210	207	15	0	0	X
ejpam-5210	207	16	.	.	PUNCT
ejpam-5210	208	1	in	in	ADP
ejpam-5210	208	2	the	the	DET
ejpam-5210	208	3	last	last	ADJ
ejpam-5210	208	4	case	case	NOUN
ejpam-5210	208	5	mc(r	mc(r	PUNCT
ejpam-5210	208	6	)	)	PUNCT
ejpam-5210	208	7	=	=	SYM
ejpam-5210	208	8	0	0	NUM
ejpam-5210	208	9	and	and	CCONJ
ejpam-5210	208	10	mt0	mt0	VERB
ejpam-5210	208	11	=	=	NOUN
ejpam-5210	208	12	0	0	X
ejpam-5210	208	13	.	.	PUNCT
ejpam-5210	209	1	as	as	ADP
ejpam-5210	209	2	a	a	DET
ejpam-5210	209	3	consequence	consequence	NOUN
ejpam-5210	209	4	,	,	PUNCT
ejpam-5210	209	5	m2	m2	PROPN
ejpam-5210	209	6	=	=	PROPN
ejpam-5210	209	7	0	0	PROPN
ejpam-5210	209	8	.	.	PUNCT
ejpam-5210	209	9	by	by	ADP
ejpam-5210	209	10	the	the	DET
ejpam-5210	209	11	�	�	PROPN
ejpam-5210	209	12	-semiprimeness	-semiprimeness	NOUN
ejpam-5210	209	13	of	of	ADP
ejpam-5210	209	14	r	r	NOUN
ejpam-5210	209	15	,	,	PUNCT
ejpam-5210	209	16	m	m	VERB
ejpam-5210	209	17	=	=	NOUN
ejpam-5210	209	18	0	0	NUM
ejpam-5210	209	19	,	,	PUNCT
ejpam-5210	209	20	which	which	PRON
ejpam-5210	209	21	is	be	AUX
ejpam-5210	209	22	a	a	DET
ejpam-5210	209	23	contradiction	contradiction	NOUN
ejpam-5210	209	24	.	.	PUNCT
ejpam-5210	210	1	now	now	ADV
ejpam-5210	210	2	we	we	PRON
ejpam-5210	210	3	extended	extend	VERB
ejpam-5210	210	4	[	[	X
ejpam-5210	210	5	5	5	NUM
ejpam-5210	210	6	,	,	PUNCT
ejpam-5210	210	7	theorem	theorem	VERB
ejpam-5210	210	8	1	1	NUM
ejpam-5210	210	9	]	]	PUNCT
ejpam-5210	210	10	in	in	ADP
ejpam-5210	210	11	the	the	DET
ejpam-5210	210	12	next	next	ADJ
ejpam-5210	210	13	proposition	proposition	NOUN
ejpam-5210	210	14	proposition	proposition	NOUN
ejpam-5210	210	15	2	2	NUM
ejpam-5210	210	16	.	.	PUNCT
ejpam-5210	211	1	let	let	VERB
ejpam-5210	211	2	r	r	PRON
ejpam-5210	211	3	be	be	AUX
ejpam-5210	211	4	a	a	DET
ejpam-5210	211	5	2	2	NUM
ejpam-5210	211	6	-	-	PUNCT
ejpam-5210	211	7	torsion	torsion	NOUN
ejpam-5210	211	8	-	-	PUNCT
ejpam-5210	211	9	free	free	ADJ
ejpam-5210	211	10	δ	δ	PROPN
ejpam-5210	211	11	-	-	PUNCT
ejpam-5210	211	12	semiprime	semiprime	NOUN
ejpam-5210	211	13	ring	ring	NOUN
ejpam-5210	211	14	,	,	PUNCT
ejpam-5210	211	15	u	u	PROPN
ejpam-5210	211	16	its	its	PRON
ejpam-5210	211	17	δ	δ	NOUN
ejpam-5210	211	18	-	-	PUNCT
ejpam-5210	211	19	ideal	ideal	ADJ
ejpam-5210	211	20	,	,	PUNCT
ejpam-5210	211	21	where	where	SCONJ
ejpam-5210	211	22	0	0	NUM
ejpam-5210	211	23	̸=	̸=	PROPN
ejpam-5210	211	24	δ	δ	PROPN
ejpam-5210	211	25	∈	∈	PROPN
ejpam-5210	211	26	d.	d.	NOUN
ejpam-5210	211	27	if	if	SCONJ
ejpam-5210	211	28	δ2(u	δ2(u	PROPN
ejpam-5210	211	29	)	)	PUNCT
ejpam-5210	211	30	=	=	SYM
ejpam-5210	211	31	0	0	NUM
ejpam-5210	211	32	,	,	PUNCT
ejpam-5210	211	33	then	then	ADV
ejpam-5210	211	34	δ(u	δ(u	NOUN
ejpam-5210	211	35	)	)	PUNCT
ejpam-5210	211	36	⊆	⊆	NUM
ejpam-5210	211	37	z(r	z(r	NUM
ejpam-5210	211	38	)	)	PUNCT
ejpam-5210	211	39	.	.	PUNCT
ejpam-5210	212	1	proof	proof	NOUN
ejpam-5210	212	2	.	.	PUNCT
ejpam-5210	213	1	let	let	VERB
ejpam-5210	213	2	a	a	DET
ejpam-5210	213	3	,	,	PUNCT
ejpam-5210	213	4	b	b	NOUN
ejpam-5210	213	5	,	,	PUNCT
ejpam-5210	213	6	u	u	NOUN
ejpam-5210	213	7	,	,	PUNCT
ejpam-5210	213	8	v	v	NOUN
ejpam-5210	213	9	∈	∈	PROPN
ejpam-5210	213	10	u	u	NOUN
ejpam-5210	213	11	and	and	CCONJ
ejpam-5210	213	12	x	x	NOUN
ejpam-5210	213	13	,	,	PUNCT
ejpam-5210	213	14	r	r	PROPN
ejpam-5210	213	15	∈	∈	PROPN
ejpam-5210	213	16	r.	r.	NOUN
ejpam-5210	213	17	from	from	ADP
ejpam-5210	213	18	0	0	PROPN
ejpam-5210	213	19	=	=	SYM
ejpam-5210	213	20	δ2([u	δ2([u	PROPN
ejpam-5210	213	21	,	,	PUNCT
ejpam-5210	213	22	v	v	NOUN
ejpam-5210	213	23	]	]	X
ejpam-5210	213	24	)	)	PUNCT
ejpam-5210	213	25	=	=	SYM
ejpam-5210	214	1	2[δ(u	2[δ(u	NUM
ejpam-5210	214	2	)	)	PUNCT
ejpam-5210	214	3	,	,	PUNCT
ejpam-5210	214	4	δ(v	δ(v	PROPN
ejpam-5210	214	5	)	)	PUNCT
ejpam-5210	214	6	]	]	PUNCT
ejpam-5210	214	7	,	,	PUNCT
ejpam-5210	214	8	we	we	PRON
ejpam-5210	214	9	deduced	deduce	VERB
ejpam-5210	214	10	u	u	NOUN
ejpam-5210	214	11	is	be	AUX
ejpam-5210	214	12	commutative	commutative	ADJ
ejpam-5210	214	13	.	.	PUNCT
ejpam-5210	215	1	since	since	SCONJ
ejpam-5210	215	2	u[u	u[u	NOUN
ejpam-5210	215	3	,	,	PUNCT
ejpam-5210	215	4	r	r	NOUN
ejpam-5210	215	5	]	]	X
ejpam-5210	215	6	=	=	SYM
ejpam-5210	215	7	u(ur	u(ur	PROPN
ejpam-5210	215	8	−	−	PROPN
ejpam-5210	215	9	ru	ru	PROPN
ejpam-5210	215	10	)	)	PUNCT
ejpam-5210	215	11	=	=	X
ejpam-5210	215	12	u(ur)−	u(ur)−	NOUN
ejpam-5210	215	13	(	(	PUNCT
ejpam-5210	215	14	ur)u	ur)u	PROPN
ejpam-5210	215	15	=	=	PUNCT
ejpam-5210	216	1	[	[	X
ejpam-5210	216	2	u	u	NOUN
ejpam-5210	216	3	,	,	PUNCT
ejpam-5210	216	4	ur	ur	INTJ
ejpam-5210	216	5	]	]	X
ejpam-5210	216	6	∈	∈	PROPN
ejpam-5210	216	7	u	u	NOUN
ejpam-5210	216	8	it	it	PRON
ejpam-5210	216	9	follows	follow	VERB
ejpam-5210	216	10	that	that	SCONJ
ejpam-5210	216	11	0	0	NUM
ejpam-5210	216	12	=	=	SYM
ejpam-5210	216	13	δ2(u[u	δ2(u[u	NOUN
ejpam-5210	216	14	,	,	PUNCT
ejpam-5210	216	15	r	r	NOUN
ejpam-5210	216	16	]	]	X
ejpam-5210	216	17	)	)	PUNCT
ejpam-5210	216	18	=	=	SYM
ejpam-5210	217	1	2δ(u)δ([u	2δ(u)δ([u	NUM
ejpam-5210	217	2	,	,	PUNCT
ejpam-5210	217	3	r	r	NOUN
ejpam-5210	217	4	]	]	PUNCT
ejpam-5210	217	5	)	)	PUNCT
ejpam-5210	217	6	and	and	CCONJ
ejpam-5210	217	7	therefore	therefore	ADV
ejpam-5210	217	8	,	,	PUNCT
ejpam-5210	217	9	δ(u)δ([u	δ(u)δ([u	NUM
ejpam-5210	217	10	,	,	PUNCT
ejpam-5210	217	11	r	r	NOUN
ejpam-5210	217	12	]	]	X
ejpam-5210	217	13	)	)	PUNCT
ejpam-5210	217	14	=	=	SYM
ejpam-5210	218	1	0	0	X
ejpam-5210	218	2	.	.	X
ejpam-5210	218	3	multiplying	multiply	VERB
ejpam-5210	218	4	[	[	X
ejpam-5210	218	5	δ(u	δ(u	NOUN
ejpam-5210	218	6	)	)	PUNCT
ejpam-5210	218	7	,	,	PUNCT
ejpam-5210	218	8	rx	rx	VERB
ejpam-5210	218	9	]	]	PUNCT
ejpam-5210	218	10	=	=	PUNCT
ejpam-5210	219	1	[	[	X
ejpam-5210	219	2	δ(u	δ(u	NUM
ejpam-5210	219	3	)	)	PUNCT
ejpam-5210	219	4	,	,	PUNCT
ejpam-5210	219	5	r]x+	r]x+	VERB
ejpam-5210	219	6	r[δ(u	r[δ(u	NOUN
ejpam-5210	219	7	)	)	PUNCT
ejpam-5210	219	8	,	,	PUNCT
ejpam-5210	219	9	x	x	X
ejpam-5210	219	10	]	]	X
ejpam-5210	219	11	,	,	PUNCT
ejpam-5210	219	12	by	by	ADP
ejpam-5210	219	13	δ(u	δ(u	NOUN
ejpam-5210	219	14	)	)	PUNCT
ejpam-5210	219	15	on	on	ADP
ejpam-5210	219	16	left	left	ADV
ejpam-5210	219	17	we	we	PRON
ejpam-5210	219	18	get	get	VERB
ejpam-5210	219	19	δ(u)r[δ(u	δ(u)r[δ(u	NOUN
ejpam-5210	219	20	)	)	PUNCT
ejpam-5210	219	21	,	,	PUNCT
ejpam-5210	219	22	x	x	X
ejpam-5210	219	23	]	]	X
ejpam-5210	219	24	=	=	SYM
ejpam-5210	219	25	0	0	X
ejpam-5210	219	26	.	.	PUNCT
ejpam-5210	220	1	since	since	SCONJ
ejpam-5210	220	2	δ(u)xr[δ(u	δ(u)xr[δ(u	ADJ
ejpam-5210	220	3	)	)	PUNCT
ejpam-5210	220	4	,	,	PUNCT
ejpam-5210	220	5	x	x	X
ejpam-5210	220	6	]	]	X
ejpam-5210	220	7	=	=	SYM
ejpam-5210	220	8	0	0	NUM
ejpam-5210	220	9	,	,	PUNCT
ejpam-5210	220	10	xδ(u)r[δ(u	xδ(u)r[δ(u	PROPN
ejpam-5210	220	11	)	)	PUNCT
ejpam-5210	220	12	,	,	PUNCT
ejpam-5210	220	13	x	x	X
ejpam-5210	220	14	]	]	X
ejpam-5210	220	15	=	=	SYM
ejpam-5210	220	16	0	0	NUM
ejpam-5210	220	17	,	,	PUNCT
ejpam-5210	220	18	m.	m.	NOUN
ejpam-5210	220	19	alosaimi	alosaimi	PROPN
ejpam-5210	220	20	et	et	PROPN
ejpam-5210	220	21	al	al	PROPN
ejpam-5210	220	22	.	.	PUNCT
ejpam-5210	220	23	/	/	SYM
ejpam-5210	220	24	eur	eur	PROPN
ejpam-5210	220	25	.	.	PUNCT
ejpam-5210	221	1	j.	j.	PROPN
ejpam-5210	221	2	pure	pure	PROPN
ejpam-5210	221	3	appl	appl	PROPN
ejpam-5210	221	4	.	.	PROPN
ejpam-5210	221	5	math	math	PROPN
ejpam-5210	221	6	,	,	PUNCT
ejpam-5210	221	7	17	17	NUM
ejpam-5210	221	8	(	(	PUNCT
ejpam-5210	221	9	3	3	NUM
ejpam-5210	221	10	)	)	PUNCT
ejpam-5210	221	11	(	(	PUNCT
ejpam-5210	221	12	2024	2024	NUM
ejpam-5210	221	13	)	)	PUNCT
ejpam-5210	221	14	,	,	PUNCT
ejpam-5210	221	15	2264	2264	NUM
ejpam-5210	221	16	-	-	SYM
ejpam-5210	221	17	2275	2275	NUM
ejpam-5210	221	18	2272	2272	NUM
ejpam-5210	221	19	we	we	PRON
ejpam-5210	221	20	obtain	obtain	VERB
ejpam-5210	221	21	that	that	SCONJ
ejpam-5210	222	1	[	[	X
ejpam-5210	222	2	δ(u	δ(u	NOUN
ejpam-5210	222	3	)	)	PUNCT
ejpam-5210	222	4	,	,	PUNCT
ejpam-5210	222	5	x]r[δ(u	x]r[δ(u	PROPN
ejpam-5210	222	6	)	)	PUNCT
ejpam-5210	222	7	,	,	PUNCT
ejpam-5210	222	8	x	x	X
ejpam-5210	222	9	]	]	X
ejpam-5210	222	10	=	=	SYM
ejpam-5210	222	11	0	0	X
ejpam-5210	222	12	.	.	PUNCT
ejpam-5210	223	1	this	this	PRON
ejpam-5210	223	2	means	mean	VERB
ejpam-5210	223	3	that	that	SCONJ
ejpam-5210	223	4	iux	iux	ADV
ejpam-5210	223	5	=	=	PUNCT
ejpam-5210	223	6	r[δ(u	r[δ(u	NOUN
ejpam-5210	223	7	)	)	PUNCT
ejpam-5210	223	8	,	,	PUNCT
ejpam-5210	223	9	x]r	x]r	VERB
ejpam-5210	223	10	is	be	AUX
ejpam-5210	223	11	a	a	DET
ejpam-5210	223	12	nilpotent	nilpotent	ADJ
ejpam-5210	223	13	ideal	ideal	NOUN
ejpam-5210	223	14	.	.	PUNCT
ejpam-5210	224	1	inasmuch	inasmuch	ADJ
ejpam-5210	224	2	i	i	PRON
ejpam-5210	224	3	=	=	SYM
ejpam-5210	224	4	∑	∑	PUNCT
ejpam-5210	224	5	u∈u	u∈u	ADJ
ejpam-5210	224	6	,	,	PUNCT
ejpam-5210	224	7	x∈r	x∈r	PROPN
ejpam-5210	224	8	iux	iux	PROPN
ejpam-5210	224	9	is	be	AUX
ejpam-5210	224	10	a	a	DET
ejpam-5210	224	11	nil	nil	ADJ
ejpam-5210	224	12	δ	δ	NOUN
ejpam-5210	224	13	-	-	PUNCT
ejpam-5210	224	14	ideal	ideal	ADJ
ejpam-5210	224	15	,	,	PUNCT
ejpam-5210	224	16	hence	hence	ADV
ejpam-5210	224	17	we	we	PRON
ejpam-5210	224	18	conclude	conclude	VERB
ejpam-5210	224	19	that	that	SCONJ
ejpam-5210	224	20	i	i	PRON
ejpam-5210	224	21	=	=	NOUN
ejpam-5210	224	22	0	0	X
ejpam-5210	224	23	.	.	PUNCT
ejpam-5210	225	1	this	this	PRON
ejpam-5210	225	2	means	mean	VERB
ejpam-5210	225	3	that	that	SCONJ
ejpam-5210	225	4	δ(u	δ(u	NOUN
ejpam-5210	225	5	)	)	PUNCT
ejpam-5210	225	6	⊆	⊆	NUM
ejpam-5210	225	7	z(r	z(r	NOUN
ejpam-5210	225	8	)	)	PUNCT
ejpam-5210	225	9	.	.	PUNCT
ejpam-5210	226	1	now	now	ADV
ejpam-5210	226	2	we	we	PRON
ejpam-5210	226	3	will	will	AUX
ejpam-5210	226	4	investigate	investigate	VERB
ejpam-5210	226	5	the	the	DET
ejpam-5210	226	6	inverse	inverse	NOUN
ejpam-5210	226	7	problem	problem	NOUN
ejpam-5210	226	8	and	and	CCONJ
ejpam-5210	226	9	prove	prove	VERB
ejpam-5210	226	10	the	the	DET
ejpam-5210	226	11	main	main	ADJ
ejpam-5210	226	12	result	result	NOUN
ejpam-5210	226	13	.	.	PUNCT
ejpam-5210	227	1	theorem	theorem	NOUN
ejpam-5210	227	2	1	1	NUM
ejpam-5210	227	3	.	.	PUNCT
ejpam-5210	228	1	let	let	VERB
ejpam-5210	228	2	r	r	PRON
ejpam-5210	228	3	be	be	AUX
ejpam-5210	228	4	a	a	DET
ejpam-5210	228	5	2	2	NUM
ejpam-5210	228	6	-	-	PUNCT
ejpam-5210	228	7	torsion	torsion	NOUN
ejpam-5210	228	8	-	-	PUNCT
ejpam-5210	228	9	free	free	ADJ
ejpam-5210	228	10	ring	ring	NOUN
ejpam-5210	228	11	.	.	PUNCT
ejpam-5210	229	1	if	if	SCONJ
ejpam-5210	229	2	r	r	NOUN
ejpam-5210	229	3	is	be	AUX
ejpam-5210	229	4	a	a	DET
ejpam-5210	229	5	d	d	ADJ
ejpam-5210	229	6	-	-	PUNCT
ejpam-5210	229	7	semiprime	semiprime	NOUN
ejpam-5210	229	8	ring	ring	NOUN
ejpam-5210	229	9	then	then	ADV
ejpam-5210	229	10	one	one	NUM
ejpam-5210	229	11	of	of	ADP
ejpam-5210	229	12	the	the	DET
ejpam-5210	229	13	following	follow	VERB
ejpam-5210	229	14	holds	hold	VERB
ejpam-5210	229	15	:	:	PUNCT
ejpam-5210	229	16	(	(	PUNCT
ejpam-5210	229	17	1	1	X
ejpam-5210	229	18	)	)	PUNCT
ejpam-5210	229	19	r	r	NOUN
ejpam-5210	229	20	is	be	AUX
ejpam-5210	229	21	a	a	DET
ejpam-5210	229	22	commutative	commutative	ADJ
ejpam-5210	229	23	ring	ring	NOUN
ejpam-5210	229	24	,	,	PUNCT
ejpam-5210	229	25	(	(	PUNCT
ejpam-5210	229	26	2	2	X
ejpam-5210	229	27	)	)	PUNCT
ejpam-5210	229	28	d	d	NOUN
ejpam-5210	229	29	is	be	AUX
ejpam-5210	229	30	a	a	DET
ejpam-5210	229	31	semiprime	semiprime	NOUN
ejpam-5210	229	32	ring	ring	NOUN
ejpam-5210	229	33	.	.	PUNCT
ejpam-5210	230	1	proof	proof	NOUN
ejpam-5210	230	2	.	.	PUNCT
ejpam-5210	231	1	assume	assume	VERB
ejpam-5210	231	2	that	that	SCONJ
ejpam-5210	231	3	r	r	NOUN
ejpam-5210	231	4	is	be	AUX
ejpam-5210	231	5	not	not	PART
ejpam-5210	231	6	commutative	commutative	ADJ
ejpam-5210	231	7	,	,	PUNCT
ejpam-5210	231	8	then	then	ADV
ejpam-5210	231	9	c(r	c(r	NOUN
ejpam-5210	231	10	)	)	PUNCT
ejpam-5210	231	11	̸=	̸=	PROPN
ejpam-5210	231	12	0	0	NUM
ejpam-5210	231	13	.	.	PUNCT
ejpam-5210	232	1	by	by	ADP
ejpam-5210	232	2	the	the	DET
ejpam-5210	232	3	d	d	NOUN
ejpam-5210	232	4	-	-	NOUN
ejpam-5210	232	5	semiprimeness	semiprimeness	NOUN
ejpam-5210	232	6	of	of	ADP
ejpam-5210	232	7	r	r	NOUN
ejpam-5210	232	8	,	,	PUNCT
ejpam-5210	232	9	c(r)2	c(r)2	NOUN
ejpam-5210	232	10	̸=	̸=	PROPN
ejpam-5210	232	11	0	0	NUM
ejpam-5210	232	12	.	.	PUNCT
ejpam-5210	232	13	suppose	suppose	VERB
ejpam-5210	232	14	that	that	SCONJ
ejpam-5210	232	15	b	b	PROPN
ejpam-5210	232	16	is	be	AUX
ejpam-5210	232	17	a	a	DET
ejpam-5210	232	18	non	non	ADJ
ejpam-5210	232	19	-	-	ADJ
ejpam-5210	232	20	zero	zero	NUM
ejpam-5210	232	21	ideal	ideal	NOUN
ejpam-5210	232	22	of	of	ADP
ejpam-5210	232	23	d	d	PROPN
ejpam-5210	232	24	,	,	PUNCT
ejpam-5210	232	25	where	where	SCONJ
ejpam-5210	232	26	[	[	X
ejpam-5210	232	27	b	b	NOUN
ejpam-5210	232	28	,	,	PUNCT
ejpam-5210	232	29	b	b	NOUN
ejpam-5210	232	30	]	]	X
ejpam-5210	232	31	=	=	SYM
ejpam-5210	232	32	0	0	X
ejpam-5210	232	33	.	.	PUNCT
ejpam-5210	233	1	let	let	VERB
ejpam-5210	233	2	j	j	PROPN
ejpam-5210	233	3	=	=	SYM
ejpam-5210	233	4	b	b	PROPN
ejpam-5210	233	5	∩	∩	X
ejpam-5210	233	6	i	i	PROPN
ejpam-5210	233	7	d	d	PROPN
ejpam-5210	233	8	and	and	CCONJ
ejpam-5210	233	9	x	x	PROPN
ejpam-5210	233	10	,	,	PUNCT
ejpam-5210	233	11	y	y	PROPN
ejpam-5210	233	12	,	,	PUNCT
ejpam-5210	233	13	r	r	PROPN
ejpam-5210	233	14	,	,	PUNCT
ejpam-5210	233	15	t	t	PROPN
ejpam-5210	233	16	∈	∈	PROPN
ejpam-5210	233	17	r.	r.	PROPN
ejpam-5210	233	18	(	(	PUNCT
ejpam-5210	233	19	a	a	X
ejpam-5210	233	20	)	)	PUNCT
ejpam-5210	233	21	if	if	SCONJ
ejpam-5210	233	22	j	j	PROPN
ejpam-5210	233	23	=	=	SYM
ejpam-5210	233	24	0	0	PROPN
ejpam-5210	233	25	,	,	PUNCT
ejpam-5210	233	26	then	then	ADV
ejpam-5210	233	27	,	,	PUNCT
ejpam-5210	233	28	for	for	ADP
ejpam-5210	233	29	any	any	DET
ejpam-5210	233	30	d	d	PROPN
ejpam-5210	233	31	∈	∈	PROPN
ejpam-5210	233	32	b	b	PROPN
ejpam-5210	233	33	,	,	PUNCT
ejpam-5210	233	34	∂d(x	∂d(x	ADJ
ejpam-5210	233	35	)	)	PUNCT
ejpam-5210	233	36	=	=	PUNCT
ejpam-5210	234	1	[	[	X
ejpam-5210	234	2	d	d	NOUN
ejpam-5210	234	3	,	,	PUNCT
ejpam-5210	234	4	∂x	∂x	PROPN
ejpam-5210	234	5	]	]	X
ejpam-5210	234	6	=	=	SYM
ejpam-5210	234	7	0	0	NUM
ejpam-5210	234	8	that	that	PRON
ejpam-5210	234	9	is	be	AUX
ejpam-5210	234	10	d(x	d(x	NOUN
ejpam-5210	234	11	)	)	PUNCT
ejpam-5210	234	12	∈	∈	PROPN
ejpam-5210	234	13	z(r	z(r	PROPN
ejpam-5210	234	14	)	)	PUNCT
ejpam-5210	234	15	.	.	PUNCT
ejpam-5210	235	1	then	then	ADV
ejpam-5210	235	2	,	,	PUNCT
ejpam-5210	235	3	for	for	ADP
ejpam-5210	235	4	any	any	DET
ejpam-5210	235	5	z	z	PROPN
ejpam-5210	235	6	∈	∈	PROPN
ejpam-5210	235	7	cr(x	cr(x	X
ejpam-5210	235	8	)	)	PUNCT
ejpam-5210	235	9	,	,	PUNCT
ejpam-5210	235	10	we	we	PRON
ejpam-5210	235	11	obtain	obtain	VERB
ejpam-5210	235	12	that	that	DET
ejpam-5210	235	13	d([x	d([x	PROPN
ejpam-5210	235	14	,	,	PUNCT
ejpam-5210	235	15	y	y	NOUN
ejpam-5210	235	16	]	]	X
ejpam-5210	235	17	)	)	PUNCT
ejpam-5210	235	18	=	=	PUNCT
ejpam-5210	236	1	[	[	X
ejpam-5210	236	2	d(x	d(x	NOUN
ejpam-5210	236	3	)	)	PUNCT
ejpam-5210	236	4	,	,	PUNCT
ejpam-5210	236	5	y	y	X
ejpam-5210	236	6	]	]	PUNCT
ejpam-5210	237	1	+	+	CCONJ
ejpam-5210	237	2	[	[	X
ejpam-5210	237	3	x	x	X
ejpam-5210	237	4	,	,	PUNCT
ejpam-5210	237	5	d(y	d(y	PROPN
ejpam-5210	237	6	)	)	PUNCT
ejpam-5210	237	7	]	]	PUNCT
ejpam-5210	238	1	=	=	SYM
ejpam-5210	238	2	0	0	NUM
ejpam-5210	238	3	,	,	PUNCT
ejpam-5210	238	4	d(z)[x	d(z)[x	PROPN
ejpam-5210	238	5	,	,	PUNCT
ejpam-5210	238	6	y	y	NOUN
ejpam-5210	238	7	]	]	X
ejpam-5210	238	8	=	=	SYM
ejpam-5210	238	9	d(z[x	d(z[x	PROPN
ejpam-5210	238	10	,	,	PUNCT
ejpam-5210	238	11	y	y	NOUN
ejpam-5210	238	12	]	]	X
ejpam-5210	238	13	)	)	PUNCT
ejpam-5210	238	14	=	=	SYM
ejpam-5210	238	15	d([x	d([x	PROPN
ejpam-5210	238	16	,	,	PUNCT
ejpam-5210	238	17	zy	zy	NOUN
ejpam-5210	238	18	]	]	X
ejpam-5210	238	19	)	)	PUNCT
ejpam-5210	238	20	=	=	SYM
ejpam-5210	238	21	0	0	PUNCT
ejpam-5210	238	22	=	=	SYM
ejpam-5210	238	23	d([x	d([x	PROPN
ejpam-5210	238	24	,	,	PUNCT
ejpam-5210	238	25	y]z	y]z	NOUN
ejpam-5210	238	26	)	)	PUNCT
ejpam-5210	238	27	=	=	PUNCT
ejpam-5210	239	1	[	[	X
ejpam-5210	239	2	x	x	X
ejpam-5210	239	3	,	,	PUNCT
ejpam-5210	239	4	y]d(z	y]d(z	PROPN
ejpam-5210	239	5	)	)	PUNCT
ejpam-5210	239	6	and	and	CCONJ
ejpam-5210	239	7	rd(z)t	rd(z)t	X
ejpam-5210	239	8	=	=	SYM
ejpam-5210	239	9	rtd(z	rtd(z	PROPN
ejpam-5210	239	10	)	)	PUNCT
ejpam-5210	239	11	+	+	NUM
ejpam-5210	239	12	r[d(z	r[d(z	NOUN
ejpam-5210	239	13	)	)	PUNCT
ejpam-5210	239	14	,	,	PUNCT
ejpam-5210	239	15	t	t	X
ejpam-5210	239	16	]	]	X
ejpam-5210	239	17	=	=	PUNCT
ejpam-5210	239	18	rtd(z	rtd(z	PROPN
ejpam-5210	239	19	)	)	PUNCT
ejpam-5210	239	20	.	.	PUNCT
ejpam-5210	240	1	assume	assume	VERB
ejpam-5210	240	2	that	that	SCONJ
ejpam-5210	240	3	x	x	X
ejpam-5210	240	4	/∈	/∈	PUNCT
ejpam-5210	240	5	z(r	z(r	NOUN
ejpam-5210	240	6	)	)	PUNCT
ejpam-5210	240	7	.	.	PUNCT
ejpam-5210	241	1	the	the	DET
ejpam-5210	241	2	ideal	ideal	ADJ
ejpam-5210	241	3	ax	ax	NOUN
ejpam-5210	241	4	generated	generate	VERB
ejpam-5210	241	5	by	by	ADP
ejpam-5210	241	6	all	all	DET
ejpam-5210	241	7	d(z	d(z	NOUN
ejpam-5210	241	8	)	)	PUNCT
ejpam-5210	241	9	,	,	PUNCT
ejpam-5210	241	10	where	where	SCONJ
ejpam-5210	241	11	d	d	PROPN
ejpam-5210	241	12	∈	∈	PROPN
ejpam-5210	241	13	d	d	NOUN
ejpam-5210	241	14	and	and	CCONJ
ejpam-5210	241	15	z	z	PROPN
ejpam-5210	241	16	∈	∈	PROPN
ejpam-5210	241	17	cr(x	cr(x	X
ejpam-5210	241	18	)	)	PUNCT
ejpam-5210	241	19	,	,	PUNCT
ejpam-5210	241	20	is	be	AUX
ejpam-5210	241	21	a	a	DET
ejpam-5210	241	22	d	d	NOUN
ejpam-5210	241	23	-	-	NOUN
ejpam-5210	241	24	ideal	ideal	NOUN
ejpam-5210	241	25	of	of	ADP
ejpam-5210	241	26	r.	r.	PROPN
ejpam-5210	241	27	if	if	SCONJ
ejpam-5210	241	28	ax	ax	NOUN
ejpam-5210	241	29	̸=	̸=	PROPN
ejpam-5210	241	30	0	0	NUM
ejpam-5210	241	31	,	,	PUNCT
ejpam-5210	241	32	then	then	ADV
ejpam-5210	241	33	,	,	PUNCT
ejpam-5210	241	34	using	use	VERB
ejpam-5210	241	35	the	the	DET
ejpam-5210	241	36	non	non	ADJ
ejpam-5210	241	37	commutativity	commutativity	NOUN
ejpam-5210	241	38	of	of	ADP
ejpam-5210	241	39	a	a	DET
ejpam-5210	241	40	ring	ring	NOUN
ejpam-5210	241	41	r	r	NOUN
ejpam-5210	241	42	and	and	CCONJ
ejpam-5210	241	43	the	the	DET
ejpam-5210	241	44	definition	definition	NOUN
ejpam-5210	241	45	of	of	ADP
ejpam-5210	241	46	the	the	DET
ejpam-5210	241	47	annihilator	annihilator	NOUN
ejpam-5210	241	48	,	,	PUNCT
ejpam-5210	241	49	we	we	PRON
ejpam-5210	241	50	see	see	VERB
ejpam-5210	241	51	that	that	SCONJ
ejpam-5210	241	52	annax	annax	PROPN
ejpam-5210	241	53	is	be	AUX
ejpam-5210	241	54	a	a	DET
ejpam-5210	241	55	non	non	ADJ
ejpam-5210	241	56	-	-	ADJ
ejpam-5210	241	57	zero	zero	ADJ
ejpam-5210	241	58	d	d	NOUN
ejpam-5210	241	59	-	-	PUNCT
ejpam-5210	241	60	ideal	ideal	ADJ
ejpam-5210	241	61	,	,	PUNCT
ejpam-5210	241	62	which	which	PRON
ejpam-5210	241	63	is	be	AUX
ejpam-5210	241	64	a	a	DET
ejpam-5210	241	65	contradiction	contradiction	NOUN
ejpam-5210	241	66	.	.	PUNCT
ejpam-5210	242	1	hence	hence	ADV
ejpam-5210	242	2	,	,	PUNCT
ejpam-5210	242	3	d(cr(x	d(cr(x	NOUN
ejpam-5210	242	4	)	)	PUNCT
ejpam-5210	242	5	)	)	PUNCT
ejpam-5210	243	1	=	=	PUNCT
ejpam-5210	243	2	0	0	X
ejpam-5210	243	3	.	.	PUNCT
ejpam-5210	244	1	if	if	SCONJ
ejpam-5210	244	2	d(z(r	d(z(r	PROPN
ejpam-5210	244	3	)	)	PUNCT
ejpam-5210	244	4	)	)	PUNCT
ejpam-5210	245	1	=	=	SYM
ejpam-5210	245	2	0	0	NUM
ejpam-5210	245	3	,	,	PUNCT
ejpam-5210	245	4	then	then	ADV
ejpam-5210	245	5	d(r	d(r	NOUN
ejpam-5210	245	6	)	)	PUNCT
ejpam-5210	245	7	=	=	SYM
ejpam-5210	245	8	0	0	NUM
ejpam-5210	246	1	and	and	CCONJ
ejpam-5210	246	2	so	so	ADV
ejpam-5210	246	3	d	d	NOUN
ejpam-5210	246	4	=	=	SYM
ejpam-5210	246	5	0	0	NUM
ejpam-5210	246	6	.	.	PUNCT
ejpam-5210	247	1	therefore	therefore	ADV
ejpam-5210	247	2	,	,	PUNCT
ejpam-5210	247	3	we	we	PRON
ejpam-5210	247	4	assume	assume	VERB
ejpam-5210	247	5	that	that	SCONJ
ejpam-5210	247	6	d(z(r	d(z(r	PROPN
ejpam-5210	247	7	)	)	PUNCT
ejpam-5210	247	8	)	)	PUNCT
ejpam-5210	248	1	̸=	̸=	PROPN
ejpam-5210	248	2	0	0	NUM
ejpam-5210	248	3	.	.	PUNCT
ejpam-5210	249	1	if	if	SCONJ
ejpam-5210	249	2	a	a	DET
ejpam-5210	249	3	∈	∈	PROPN
ejpam-5210	249	4	z(r	z(r	NOUN
ejpam-5210	249	5	)	)	PUNCT
ejpam-5210	249	6	,	,	PUNCT
ejpam-5210	249	7	then	then	ADV
ejpam-5210	249	8	acr(x	acr(x	X
ejpam-5210	249	9	)	)	PUNCT
ejpam-5210	249	10	⊆	⊆	NUM
ejpam-5210	249	11	cr(x	cr(x	X
ejpam-5210	249	12	)	)	PUNCT
ejpam-5210	249	13	and	and	CCONJ
ejpam-5210	249	14	then	then	ADV
ejpam-5210	249	15	d(cr(x)a	d(cr(x)a	NUM
ejpam-5210	249	16	)	)	PUNCT
ejpam-5210	249	17	=	=	SYM
ejpam-5210	249	18	cr(x)d(a	cr(x)d(a	PROPN
ejpam-5210	249	19	)	)	PUNCT
ejpam-5210	249	20	=	=	SYM
ejpam-5210	249	21	0	0	X
ejpam-5210	249	22	=	=	SYM
ejpam-5210	249	23	d(acr(x	d(acr(x	NOUN
ejpam-5210	249	24	)	)	PUNCT
ejpam-5210	249	25	)	)	PUNCT
ejpam-5210	250	1	=	=	PUNCT
ejpam-5210	250	2	d(a)cr(x	d(a)cr(x	X
ejpam-5210	250	3	)	)	PUNCT
ejpam-5210	250	4	and	and	CCONJ
ejpam-5210	250	5	consequently	consequently	ADV
ejpam-5210	250	6	d(r	d(r	NOUN
ejpam-5210	250	7	)	)	PUNCT
ejpam-5210	250	8	⊆	⊆	NUM
ejpam-5210	250	9	anncr(x	anncr(x	PROPN
ejpam-5210	250	10	)	)	PUNCT
ejpam-5210	250	11	.	.	PUNCT
ejpam-5210	251	1	m.	m.	NOUN
ejpam-5210	251	2	alosaimi	alosaimi	PROPN
ejpam-5210	251	3	et	et	PROPN
ejpam-5210	251	4	al	al	PROPN
ejpam-5210	251	5	.	.	PUNCT
ejpam-5210	251	6	/	/	SYM
ejpam-5210	251	7	eur	eur	PROPN
ejpam-5210	251	8	.	.	PUNCT
ejpam-5210	252	1	j.	j.	PROPN
ejpam-5210	252	2	pure	pure	PROPN
ejpam-5210	252	3	appl	appl	PROPN
ejpam-5210	252	4	.	.	PROPN
ejpam-5210	252	5	math	math	PROPN
ejpam-5210	252	6	,	,	PUNCT
ejpam-5210	252	7	17	17	NUM
ejpam-5210	252	8	(	(	PUNCT
ejpam-5210	252	9	3	3	NUM
ejpam-5210	252	10	)	)	PUNCT
ejpam-5210	252	11	(	(	PUNCT
ejpam-5210	252	12	2024	2024	NUM
ejpam-5210	252	13	)	)	PUNCT
ejpam-5210	252	14	,	,	PUNCT
ejpam-5210	252	15	2264	2264	NUM
ejpam-5210	252	16	-	-	SYM
ejpam-5210	252	17	2275	2275	NUM
ejpam-5210	252	18	2273	2273	NUM
ejpam-5210	252	19	in	in	ADP
ejpam-5210	252	20	view	view	NOUN
ejpam-5210	252	21	of	of	ADP
ejpam-5210	252	22	lemma	lemma	PROPN
ejpam-5210	252	23	4	4	NUM
ejpam-5210	252	24	,	,	PUNCT
ejpam-5210	252	25	d(r	d(r	NOUN
ejpam-5210	252	26	)	)	PUNCT
ejpam-5210	252	27	⊆	⊆	NUM
ejpam-5210	252	28	z(r	z(r	NOUN
ejpam-5210	252	29	)	)	PUNCT
ejpam-5210	252	30	for	for	ADP
ejpam-5210	252	31	d	d	PROPN
ejpam-5210	252	32	∈	∈	PROPN
ejpam-5210	252	33	b.	b.	PROPN
ejpam-5210	252	34	let	let	VERB
ejpam-5210	252	35	db(r	db(r	NOUN
ejpam-5210	252	36	)	)	PUNCT
ejpam-5210	252	37	by	by	ADP
ejpam-5210	252	38	the	the	DET
ejpam-5210	252	39	ideal	ideal	NOUN
ejpam-5210	252	40	of	of	ADP
ejpam-5210	252	41	r	r	NOUN
ejpam-5210	252	42	generated	generate	VERB
ejpam-5210	252	43	by	by	ADP
ejpam-5210	252	44	all	all	DET
ejpam-5210	252	45	d(r	d(r	NOUN
ejpam-5210	252	46	)	)	PUNCT
ejpam-5210	252	47	,	,	PUNCT
ejpam-5210	252	48	where	where	SCONJ
ejpam-5210	252	49	d	d	PROPN
ejpam-5210	252	50	∈	∈	PROPN
ejpam-5210	252	51	b.	b.	PROPN
ejpam-5210	252	52	then	then	ADV
ejpam-5210	252	53	u	u	X
ejpam-5210	252	54	∈	∈	PROPN
ejpam-5210	252	55	cr(u	cr(u	NOUN
ejpam-5210	252	56	)	)	PUNCT
ejpam-5210	252	57	⊆	⊆	NUM
ejpam-5210	252	58	anndb(r	anndb(r	NOUN
ejpam-5210	252	59	)	)	PUNCT
ejpam-5210	252	60	for	for	ADP
ejpam-5210	252	61	any	any	DET
ejpam-5210	252	62	u	u	PROPN
ejpam-5210	252	63	∈	∈	PROPN
ejpam-5210	252	64	db(r	db(r	NOUN
ejpam-5210	252	65	)	)	PUNCT
ejpam-5210	252	66	and	and	CCONJ
ejpam-5210	252	67	so	so	ADV
ejpam-5210	252	68	u	u	PROPN
ejpam-5210	252	69	∈	∈	PROPN
ejpam-5210	252	70	db(r)∩anndb(r	db(r)∩anndb(r	PROPN
ejpam-5210	252	71	)	)	PUNCT
ejpam-5210	252	72	=	=	SYM
ejpam-5210	253	1	0	0	X
ejpam-5210	253	2	.	.	PUNCT
ejpam-5210	254	1	this	this	PRON
ejpam-5210	254	2	means	mean	VERB
ejpam-5210	254	3	that	that	SCONJ
ejpam-5210	254	4	db(r	db(r	NOUN
ejpam-5210	254	5	)	)	PUNCT
ejpam-5210	254	6	=	=	SYM
ejpam-5210	254	7	0	0	NUM
ejpam-5210	254	8	,	,	PUNCT
ejpam-5210	254	9	which	which	PRON
ejpam-5210	254	10	leads	lead	VERB
ejpam-5210	254	11	to	to	ADP
ejpam-5210	254	12	a	a	DET
ejpam-5210	254	13	contradiction	contradiction	NOUN
ejpam-5210	254	14	.	.	PUNCT
ejpam-5210	255	1	(	(	PUNCT
ejpam-5210	255	2	b	b	X
ejpam-5210	255	3	)	)	PUNCT
ejpam-5210	255	4	assume	assume	VERB
ejpam-5210	255	5	that	that	SCONJ
ejpam-5210	255	6	j	j	PROPN
ejpam-5210	255	7	̸=	̸=	PROPN
ejpam-5210	255	8	0	0	NUM
ejpam-5210	255	9	.	.	PUNCT
ejpam-5210	256	1	then	then	ADV
ejpam-5210	256	2	i	i	PRON
ejpam-5210	256	3	=	=	PUNCT
ejpam-5210	256	4	{	{	PUNCT
ejpam-5210	256	5	t	t	NOUN
ejpam-5210	256	6	∈	∈	PROPN
ejpam-5210	256	7	r	r	NOUN
ejpam-5210	257	1	|	|	ADV
ejpam-5210	257	2	∂t	∂t	PROPN
ejpam-5210	257	3	∈	∈	PROPN
ejpam-5210	257	4	j	j	PROPN
ejpam-5210	257	5	}	}	PUNCT
ejpam-5210	257	6	is	be	AUX
ejpam-5210	257	7	a	a	DET
ejpam-5210	257	8	non	non	ADJ
ejpam-5210	257	9	-	-	ADJ
ejpam-5210	257	10	zero	zero	ADJ
ejpam-5210	257	11	d	d	NOUN
ejpam-5210	257	12	-	-	NOUN
ejpam-5210	257	13	ideal	ideal	NOUN
ejpam-5210	257	14	of	of	ADP
ejpam-5210	257	15	r	r	NOUN
ejpam-5210	257	16	and	and	CCONJ
ejpam-5210	257	17	∂[t1,t2	∂[t1,t2	PROPN
ejpam-5210	257	18	]	]	PUNCT
ejpam-5210	258	1	=	=	PUNCT
ejpam-5210	259	1	[	[	X
ejpam-5210	259	2	∂t1	∂t1	X
ejpam-5210	259	3	,	,	PUNCT
ejpam-5210	259	4	∂t2	∂t2	NOUN
ejpam-5210	259	5	]	]	X
ejpam-5210	259	6	∈	∈	PROPN
ejpam-5210	259	7	[	[	X
ejpam-5210	259	8	j	j	PROPN
ejpam-5210	259	9	,	,	PUNCT
ejpam-5210	259	10	j	j	PROPN
ejpam-5210	259	11	]	]	X
ejpam-5210	259	12	=	=	PUNCT
ejpam-5210	259	13	0	0	NUM
ejpam-5210	259	14	for	for	ADP
ejpam-5210	259	15	any	any	DET
ejpam-5210	259	16	ti	ti	PROPN
ejpam-5210	259	17	∈	∈	PROPN
ejpam-5210	259	18	j	j	PROPN
ejpam-5210	259	19	and	and	CCONJ
ejpam-5210	259	20	,	,	PUNCT
ejpam-5210	259	21	as	as	ADP
ejpam-5210	259	22	a	a	DET
ejpam-5210	259	23	consequence	consequence	NOUN
ejpam-5210	259	24	,	,	PUNCT
ejpam-5210	259	25	[	[	X
ejpam-5210	259	26	i	i	X
ejpam-5210	259	27	,	,	PUNCT
ejpam-5210	259	28	i	i	PRON
ejpam-5210	259	29	]	]	PUNCT
ejpam-5210	259	30	⊆	⊆	NUM
ejpam-5210	259	31	z(r	z(r	NUM
ejpam-5210	259	32	)	)	PUNCT
ejpam-5210	259	33	.	.	PUNCT
ejpam-5210	260	1	(	(	PUNCT
ejpam-5210	260	2	9	9	X
ejpam-5210	260	3	)	)	PUNCT
ejpam-5210	260	4	let	let	VERB
ejpam-5210	260	5	t	t	PROPN
ejpam-5210	260	6	(	(	PUNCT
ejpam-5210	260	7	i	i	NOUN
ejpam-5210	260	8	)	)	PUNCT
ejpam-5210	261	1	=	=	PRON
ejpam-5210	261	2	{	{	PUNCT
ejpam-5210	261	3	w	w	NOUN
ejpam-5210	261	4	∈	∈	PROPN
ejpam-5210	261	5	r	r	NOUN
ejpam-5210	262	1	|	|	NOUN
ejpam-5210	263	1	[	[	X
ejpam-5210	263	2	w	w	NOUN
ejpam-5210	263	3	,	,	PUNCT
ejpam-5210	263	4	r	r	NOUN
ejpam-5210	263	5	]	]	PUNCT
ejpam-5210	263	6	⊆	⊆	NUM
ejpam-5210	263	7	i	i	PROPN
ejpam-5210	263	8	}	}	PUNCT
ejpam-5210	263	9	.	.	PUNCT
ejpam-5210	264	1	then	then	ADV
ejpam-5210	264	2	i	i	PRON
ejpam-5210	264	3	⊆	⊆	NUM
ejpam-5210	264	4	t	t	PROPN
ejpam-5210	264	5	(	(	PUNCT
ejpam-5210	264	6	i	i	NOUN
ejpam-5210	264	7	)	)	PUNCT
ejpam-5210	264	8	and	and	CCONJ
ejpam-5210	264	9	t	t	PROPN
ejpam-5210	264	10	(	(	PUNCT
ejpam-5210	264	11	b	b	NOUN
ejpam-5210	264	12	)	)	PUNCT
ejpam-5210	264	13	is	be	AUX
ejpam-5210	264	14	an	an	DET
ejpam-5210	264	15	associative	associative	ADJ
ejpam-5210	264	16	subring	subring	NOUN
ejpam-5210	264	17	and	and	CCONJ
ejpam-5210	264	18	a	a	DET
ejpam-5210	264	19	lie	lie	NOUN
ejpam-5210	264	20	ideal	ideal	NOUN
ejpam-5210	264	21	of	of	ADP
ejpam-5210	264	22	r	r	NOUN
ejpam-5210	264	23	(	(	PUNCT
ejpam-5210	264	24	see	see	VERB
ejpam-5210	264	25	[	[	X
ejpam-5210	264	26	7	7	NUM
ejpam-5210	264	27	,	,	PUNCT
ejpam-5210	264	28	lemma	lemma	PROPN
ejpam-5210	264	29	3	3	NUM
ejpam-5210	264	30	]	]	PUNCT
ejpam-5210	264	31	)	)	PUNCT
ejpam-5210	264	32	.	.	PUNCT
ejpam-5210	265	1	since	since	SCONJ
ejpam-5210	265	2	[	[	X
ejpam-5210	265	3	i	i	PRON
ejpam-5210	265	4	,	,	PUNCT
ejpam-5210	265	5	t	t	PROPN
ejpam-5210	265	6	(	(	PUNCT
ejpam-5210	265	7	i	i	NOUN
ejpam-5210	265	8	)	)	PUNCT
ejpam-5210	265	9	]	]	PUNCT
ejpam-5210	266	1	⊆	⊆	NUM
ejpam-5210	266	2	i	i	PRON
ejpam-5210	266	3	,	,	PUNCT
ejpam-5210	266	4	we	we	PRON
ejpam-5210	266	5	deduce	deduce	VERB
ejpam-5210	266	6	that	that	PRON
ejpam-5210	266	7	0	0	NUM
ejpam-5210	266	8	=	=	SYM
ejpam-5210	266	9	∂t1([t1	∂t1([t1	PROPN
ejpam-5210	266	10	,	,	PUNCT
ejpam-5210	266	11	t	t	PROPN
ejpam-5210	266	12	2	2	NUM
ejpam-5210	266	13	2	2	NUM
ejpam-5210	266	14	]	]	PUNCT
ejpam-5210	266	15	)	)	PUNCT
ejpam-5210	266	16	=	=	SYM
ejpam-5210	266	17	2∂t1(t2	2∂t1(t2	NUM
ejpam-5210	266	18	)	)	PUNCT
ejpam-5210	266	19	2	2	NUM
ejpam-5210	266	20	.	.	X
ejpam-5210	266	21	from	from	ADP
ejpam-5210	266	22	this	this	PRON
ejpam-5210	266	23	and	and	CCONJ
ejpam-5210	266	24	the	the	DET
ejpam-5210	266	25	condition	condition	NOUN
ejpam-5210	266	26	in	in	ADP
ejpam-5210	266	27	the	the	DET
ejpam-5210	266	28	equation	equation	NOUN
ejpam-5210	266	29	(	(	PUNCT
ejpam-5210	266	30	9	9	X
ejpam-5210	266	31	)	)	PUNCT
ejpam-5210	266	32	it	it	PRON
ejpam-5210	266	33	holds	hold	VERB
ejpam-5210	266	34	that	that	DET
ejpam-5210	266	35	∂t1(t2	∂t1(t2	NOUN
ejpam-5210	266	36	)	)	PUNCT
ejpam-5210	266	37	∈	∈	PROPN
ejpam-5210	266	38	p(r	p(r	PROPN
ejpam-5210	266	39	)	)	PUNCT
ejpam-5210	266	40	∩	∩	NOUN
ejpam-5210	266	41	z(r	z(r	NOUN
ejpam-5210	266	42	)	)	PUNCT
ejpam-5210	266	43	.	.	PUNCT
ejpam-5210	267	1	then	then	ADV
ejpam-5210	267	2	∑	∑	PROPN
ejpam-5210	267	3	t1,t2∈i	t1,t2∈i	PROPN
ejpam-5210	268	1	[	[	X
ejpam-5210	268	2	t1	t1	NOUN
ejpam-5210	268	3	,	,	PUNCT
ejpam-5210	268	4	t2]r	t2]r	PRON
ejpam-5210	268	5	is	be	AUX
ejpam-5210	268	6	a	a	DET
ejpam-5210	268	7	nil	nil	ADJ
ejpam-5210	268	8	d	d	NOUN
ejpam-5210	268	9	-	-	NOUN
ejpam-5210	268	10	ideal	ideal	NOUN
ejpam-5210	268	11	of	of	ADP
ejpam-5210	268	12	r	r	NOUN
ejpam-5210	268	13	,	,	PUNCT
ejpam-5210	268	14	which	which	PRON
ejpam-5210	268	15	is	be	AUX
ejpam-5210	268	16	a	a	DET
ejpam-5210	268	17	contradiction	contradiction	NOUN
ejpam-5210	268	18	.	.	PUNCT
ejpam-5210	269	1	4	4	X
ejpam-5210	269	2	.	.	X
ejpam-5210	269	3	conclusion	conclusion	NOUN
ejpam-5210	269	4	through	through	ADP
ejpam-5210	269	5	this	this	DET
ejpam-5210	269	6	work	work	NOUN
ejpam-5210	269	7	,	,	PUNCT
ejpam-5210	269	8	firstly	firstly	ADV
ejpam-5210	269	9	,	,	PUNCT
ejpam-5210	269	10	we	we	PRON
ejpam-5210	269	11	found	find	VERB
ejpam-5210	269	12	some	some	DET
ejpam-5210	269	13	properties	property	NOUN
ejpam-5210	269	14	of	of	ADP
ejpam-5210	269	15	lie	lie	NOUN
ejpam-5210	269	16	ideals	ideal	NOUN
ejpam-5210	269	17	in	in	ADP
ejpam-5210	269	18	d	d	ADJ
ejpam-5210	269	19	-	-	ADJ
ejpam-5210	269	20	semiprime	semiprime	ADJ
ejpam-5210	269	21	ring	ring	NOUN
ejpam-5210	269	22	,	,	PUNCT
ejpam-5210	269	23	then	then	ADV
ejpam-5210	269	24	we	we	PRON
ejpam-5210	269	25	demonstrated	demonstrate	VERB
ejpam-5210	269	26	when	when	SCONJ
ejpam-5210	269	27	a	a	DET
ejpam-5210	269	28	commutator	commutator	NOUN
ejpam-5210	269	29	of	of	ADP
ejpam-5210	269	30	a	a	DET
ejpam-5210	269	31	composite	composite	ADJ
ejpam-5210	269	32	derivation	derivation	NOUN
ejpam-5210	269	33	for	for	ADP
ejpam-5210	269	34	an	an	DET
ejpam-5210	269	35	element	element	NOUN
ejpam-5210	269	36	and	and	CCONJ
ejpam-5210	269	37	lie	lie	VERB
ejpam-5210	269	38	ideal	ideal	NOUN
ejpam-5210	269	39	in	in	ADP
ejpam-5210	269	40	a	a	DET
ejpam-5210	269	41	d	d	ADJ
ejpam-5210	269	42	-	-	PUNCT
ejpam-5210	269	43	semiprime	semiprime	ADJ
ejpam-5210	269	44	ring	ring	NOUN
ejpam-5210	269	45	equal	equal	ADJ
ejpam-5210	269	46	to	to	ADP
ejpam-5210	269	47	zero	zero	NUM
ejpam-5210	269	48	,	,	PUNCT
ejpam-5210	269	49	implies	imply	VERB
ejpam-5210	269	50	the	the	DET
ejpam-5210	269	51	element	element	NOUN
ejpam-5210	269	52	belongs	belong	VERB
ejpam-5210	269	53	to	to	ADP
ejpam-5210	269	54	center	center	NOUN
ejpam-5210	269	55	of	of	ADP
ejpam-5210	269	56	this	this	DET
ejpam-5210	269	57	ring	ring	NOUN
ejpam-5210	269	58	.	.	PUNCT
ejpam-5210	270	1	also	also	ADV
ejpam-5210	270	2	,	,	PUNCT
ejpam-5210	270	3	we	we	PRON
ejpam-5210	270	4	investigated	investigate	VERB
ejpam-5210	270	5	the	the	DET
ejpam-5210	270	6	relationship	relationship	NOUN
ejpam-5210	270	7	between	between	ADP
ejpam-5210	270	8	an	an	DET
ejpam-5210	270	9	element	element	NOUN
ejpam-5210	270	10	of	of	ADP
ejpam-5210	270	11	lie	lie	NOUN
ejpam-5210	270	12	�	�	PROPN
ejpam-5210	270	13	ideal	ideal	NOUN
ejpam-5210	270	14	and	and	CCONJ
ejpam-5210	270	15	the	the	DET
ejpam-5210	270	16	center	center	NOUN
ejpam-5210	270	17	(	(	PUNCT
ejpam-5210	270	18	the	the	DET
ejpam-5210	270	19	commutator	commutator	NOUN
ejpam-5210	270	20	ideal	ideal	PROPN
ejpam-5210	270	21	)	)	PUNCT
ejpam-5210	270	22	of	of	ADP
ejpam-5210	270	23	a	a	DET
ejpam-5210	270	24	ring	ring	NOUN
ejpam-5210	270	25	.	.	PUNCT
ejpam-5210	271	1	after	after	ADP
ejpam-5210	271	2	that	that	PRON
ejpam-5210	271	3	,	,	PUNCT
ejpam-5210	271	4	we	we	PRON
ejpam-5210	271	5	showed	show	VERB
ejpam-5210	271	6	that	that	SCONJ
ejpam-5210	271	7	for	for	ADP
ejpam-5210	271	8	any	any	DET
ejpam-5210	271	9	an	an	DET
ejpam-5210	271	10	ideal	ideal	NOUN
ejpam-5210	271	11	contained	contain	VERB
ejpam-5210	271	12	in	in	ADP
ejpam-5210	271	13	lie	lie	NOUN
ejpam-5210	271	14	�	�	NOUN
ejpam-5210	271	15	-ideal	-ideal	NOUN
ejpam-5210	271	16	of	of	ADP
ejpam-5210	271	17	a	a	DET
ejpam-5210	271	18	�	�	PROPN
ejpam-5210	271	19	-semiprime	-semiprime	NOUN
ejpam-5210	271	20	ring	ring	NOUN
ejpam-5210	271	21	,	,	PUNCT
ejpam-5210	271	22	their	their	PRON
ejpam-5210	271	23	commutator	commutator	NOUN
ejpam-5210	271	24	must	must	AUX
ejpam-5210	271	25	be	be	AUX
ejpam-5210	271	26	contained	contain	VERB
ejpam-5210	271	27	in	in	ADP
ejpam-5210	271	28	the	the	DET
ejpam-5210	271	29	ideal	ideal	NOUN
ejpam-5210	271	30	itself	itself	PRON
ejpam-5210	271	31	.	.	PUNCT
ejpam-5210	272	1	furthermore	furthermore	ADV
ejpam-5210	272	2	,	,	PUNCT
ejpam-5210	272	3	we	we	PRON
ejpam-5210	272	4	related	relate	VERB
ejpam-5210	272	5	between	between	ADP
ejpam-5210	272	6	the	the	DET
ejpam-5210	272	7	commutator	commutator	NOUN
ejpam-5210	272	8	of	of	ADP
ejpam-5210	272	9	�	�	PROPN
ejpam-5210	272	10	ideal	ideal	NOUN
ejpam-5210	272	11	and	and	CCONJ
ejpam-5210	272	12	every	every	DET
ejpam-5210	272	13	associative	associative	ADJ
ejpam-5210	272	14	subgroup	subgroup	NOUN
ejpam-5210	272	15	of	of	ADP
ejpam-5210	272	16	it	it	PRON
ejpam-5210	272	17	under	under	ADP
ejpam-5210	272	18	specific	specific	ADJ
ejpam-5210	272	19	conditions	condition	NOUN
ejpam-5210	272	20	.	.	PUNCT
ejpam-5210	273	1	in	in	ADP
ejpam-5210	273	2	addition	addition	NOUN
ejpam-5210	273	3	,	,	PUNCT
ejpam-5210	273	4	if	if	SCONJ
ejpam-5210	273	5	any	any	DET
ejpam-5210	273	6	δ	δ	NOUN
ejpam-5210	273	7	-	-	PUNCT
ejpam-5210	273	8	ideal	ideal	NOUN
ejpam-5210	273	9	of	of	ADP
ejpam-5210	273	10	a	a	DET
ejpam-5210	273	11	δsemiprime	δsemiprime	NOUN
ejpam-5210	273	12	ring	ring	NOUN
ejpam-5210	273	13	satisfies	satisfie	NOUN
ejpam-5210	273	14	δ2(u	δ2(u	NUM
ejpam-5210	273	15	)	)	PUNCT
ejpam-5210	273	16	=	=	SYM
ejpam-5210	273	17	0	0	NUM
ejpam-5210	273	18	,	,	PUNCT
ejpam-5210	273	19	this	this	PRON
ejpam-5210	273	20	gives	give	VERB
ejpam-5210	273	21	δ(u	δ(u	PROPN
ejpam-5210	273	22	)	)	PUNCT
ejpam-5210	273	23	is	be	AUX
ejpam-5210	273	24	contained	contain	VERB
ejpam-5210	273	25	in	in	ADP
ejpam-5210	273	26	the	the	DET
ejpam-5210	273	27	center	center	NOUN
ejpam-5210	273	28	of	of	ADP
ejpam-5210	273	29	ring	ring	NOUN
ejpam-5210	273	30	.	.	PUNCT
ejpam-5210	274	1	finally	finally	ADV
ejpam-5210	274	2	,	,	PUNCT
ejpam-5210	274	3	we	we	PRON
ejpam-5210	274	4	proved	prove	VERB
ejpam-5210	274	5	that	that	SCONJ
ejpam-5210	274	6	every	every	DET
ejpam-5210	274	7	a	a	DET
ejpam-5210	274	8	2	2	NUM
ejpam-5210	274	9	-	-	PUNCT
ejpam-5210	274	10	torsion	torsion	NOUN
ejpam-5210	274	11	-	-	PUNCT
ejpam-5210	274	12	free	free	ADJ
ejpam-5210	274	13	d	d	NOUN
ejpam-5210	274	14	-	-	NOUN
ejpam-5210	274	15	semiprime	semiprime	NOUN
ejpam-5210	274	16	with	with	ADP
ejpam-5210	274	17	an	an	DET
ejpam-5210	274	18	identity	identity	NOUN
ejpam-5210	274	19	ring	ring	NOUN
ejpam-5210	274	20	is	be	AUX
ejpam-5210	274	21	either	either	CCONJ
ejpam-5210	274	22	commutative	commutative	ADJ
ejpam-5210	274	23	or	or	CCONJ
ejpam-5210	274	24	a	a	DET
ejpam-5210	274	25	d	d	NOUN
ejpam-5210	274	26	is	be	AUX
ejpam-5210	274	27	a	a	DET
ejpam-5210	274	28	semiprime	semiprime	NOUN
ejpam-5210	274	29	ring	ring	NOUN
ejpam-5210	274	30	.	.	PUNCT
ejpam-5210	275	1	references	reference	NOUN
ejpam-5210	275	2	2274	2274	NUM
ejpam-5210	275	3	references	reference	NOUN
ejpam-5210	275	4	[	[	X
ejpam-5210	275	5	1	1	NUM
ejpam-5210	275	6	]	]	X
ejpam-5210	275	7	o	o	X
ejpam-5210	275	8	artemovych	artemovych	NOUN
ejpam-5210	275	9	and	and	CCONJ
ejpam-5210	275	10	m	m	PROPN
ejpam-5210	275	11	lukashenko	lukashenko	PROPN
ejpam-5210	275	12	.	.	PUNCT
ejpam-5210	276	1	lie	lie	NOUN
ejpam-5210	276	2	and	and	CCONJ
ejpam-5210	276	3	jordan	jordan	PROPN
ejpam-5210	276	4	structures	structure	NOUN
ejpam-5210	276	5	of	of	ADP
ejpam-5210	276	6	differentially	differentially	ADV
ejpam-5210	276	7	semiprime	semiprime	NOUN
ejpam-5210	276	8	rings	ring	NOUN
ejpam-5210	276	9	.	.	PUNCT
ejpam-5210	277	1	algebra	algebra	NOUN
ejpam-5210	277	2	and	and	CCONJ
ejpam-5210	277	3	discrete	discrete	ADJ
ejpam-5210	277	4	mathematics	mathematic	NOUN
ejpam-5210	277	5	,	,	PUNCT
ejpam-5210	277	6	20(1	20(1	NUM
ejpam-5210	277	7	)	)	PUNCT
ejpam-5210	277	8	,	,	PUNCT
ejpam-5210	277	9	2015	2015	NUM
ejpam-5210	277	10	.	.	PUNCT
ejpam-5210	278	1	[	[	X
ejpam-5210	278	2	2	2	NUM
ejpam-5210	278	3	]	]	X
ejpam-5210	278	4	k	k	PROPN
ejpam-5210	278	5	beidar	beidar	NOUN
ejpam-5210	278	6	and	and	CCONJ
ejpam-5210	278	7	a	a	DET
ejpam-5210	278	8	mikhalev	mikhalev	NOUN
ejpam-5210	278	9	.	.	PUNCT
ejpam-5210	279	1	orthogonal	orthogonal	ADJ
ejpam-5210	279	2	completeness	completeness	NOUN
ejpam-5210	279	3	and	and	CCONJ
ejpam-5210	279	4	minimal	minimal	ADJ
ejpam-5210	279	5	prime	prime	ADJ
ejpam-5210	279	6	ideals	ideal	NOUN
ejpam-5210	279	7	.	.	PUNCT
ejpam-5210	280	1	journal	journal	NOUN
ejpam-5210	280	2	of	of	ADP
ejpam-5210	280	3	soviet	soviet	PROPN
ejpam-5210	280	4	mathematics	mathematic	NOUN
ejpam-5210	280	5	,	,	PUNCT
ejpam-5210	280	6	35:2876–2882	35:2876–2882	PROPN
ejpam-5210	280	7	,	,	PUNCT
ejpam-5210	280	8	1986	1986	NUM
ejpam-5210	280	9	.	.	PUNCT
ejpam-5210	281	1	[	[	X
ejpam-5210	281	2	3	3	X
ejpam-5210	281	3	]	]	X
ejpam-5210	281	4	j	j	PROPN
ejpam-5210	281	5	bergen	bergen	PROPN
ejpam-5210	281	6	,	,	PUNCT
ejpam-5210	281	7	i	i	PRON
ejpam-5210	281	8	herstein	herstein	NOUN
ejpam-5210	281	9	,	,	PUNCT
ejpam-5210	281	10	and	and	CCONJ
ejpam-5210	281	11	w	w	PROPN
ejpam-5210	281	12	jeanne	jeanne	PROPN
ejpam-5210	281	13	.	.	PROPN
ejpam-5210	281	14	lie	lie	PROPN
ejpam-5210	281	15	ideals	ideal	NOUN
ejpam-5210	281	16	and	and	CCONJ
ejpam-5210	281	17	derivations	derivation	NOUN
ejpam-5210	281	18	of	of	ADP
ejpam-5210	281	19	prime	prime	ADJ
ejpam-5210	281	20	rings	ring	NOUN
ejpam-5210	281	21	.	.	PUNCT
ejpam-5210	282	1	journal	journal	PROPN
ejpam-5210	282	2	of	of	ADP
ejpam-5210	282	3	algebra	algebra	PROPN
ejpam-5210	282	4	,	,	PUNCT
ejpam-5210	282	5	71(1):259–267	71(1):259–267	PROPN
ejpam-5210	282	6	,	,	PUNCT
ejpam-5210	282	7	1981	1981	NUM
ejpam-5210	282	8	.	.	PUNCT
ejpam-5210	283	1	[	[	X
ejpam-5210	283	2	4	4	X
ejpam-5210	283	3	]	]	SYM
ejpam-5210	283	4	m	m	VERB
ejpam-5210	283	5	brešar	brešar	ADJ
ejpam-5210	283	6	,	,	PUNCT
ejpam-5210	283	7	m	m	NOUN
ejpam-5210	283	8	chebotar	chebotar	ADJ
ejpam-5210	283	9	,	,	PUNCT
ejpam-5210	283	10	and	and	CCONJ
ejpam-5210	283	11	w	w	PROPN
ejpam-5210	283	12	martindale	martindale	PROPN
ejpam-5210	283	13	.	.	PUNCT
ejpam-5210	284	1	functional	functional	ADJ
ejpam-5210	284	2	identities	identity	NOUN
ejpam-5210	284	3	.	.	PUNCT
ejpam-5210	285	1	springer	springer	NOUN
ejpam-5210	285	2	science	science	PROPN
ejpam-5210	285	3	&	&	CCONJ
ejpam-5210	285	4	business	business	NOUN
ejpam-5210	285	5	media	medium	NOUN
ejpam-5210	285	6	,	,	PUNCT
ejpam-5210	285	7	2007	2007	NUM
ejpam-5210	285	8	.	.	PUNCT
ejpam-5210	286	1	[	[	X
ejpam-5210	286	2	5	5	NUM
ejpam-5210	286	3	]	]	PUNCT
ejpam-5210	286	4	m	m	AUX
ejpam-5210	286	5	chebotar	chebotar	ADJ
ejpam-5210	286	6	and	and	CCONJ
ejpam-5210	286	7	a	a	DET
ejpam-5210	286	8	giambruno	giambruno	NOUN
ejpam-5210	286	9	.	.	PUNCT
ejpam-5210	287	1	lie	lie	NOUN
ejpam-5210	287	2	ideals	ideal	NOUN
ejpam-5210	287	3	and	and	CCONJ
ejpam-5210	287	4	nil	nil	NOUN
ejpam-5210	287	5	derivations	derivation	NOUN
ejpam-5210	287	6	.	.	PUNCT
ejpam-5210	288	1	bollettino	bollettino	PROPN
ejpam-5210	288	2	dell’unione	dell’unione	PROPN
ejpam-5210	288	3	mathmatica	mathmatica	PROPN
ejpam-5210	288	4	italiana	italiana	PROPN
ejpam-5210	288	5	.	.	PROPN
ejpam-5210	288	6	,	,	PUNCT
ejpam-5210	288	7	4(3):497–503	4(3):497–503	NUM
ejpam-5210	288	8	,	,	PUNCT
ejpam-5210	288	9	1985	1985	NUM
ejpam-5210	288	10	.	.	PUNCT
ejpam-5210	289	1	[	[	X
ejpam-5210	289	2	6	6	NUM
ejpam-5210	289	3	]	]	PUNCT
ejpam-5210	289	4	m	m	AUX
ejpam-5210	289	5	chebotar	chebotar	ADJ
ejpam-5210	289	6	and	and	CCONJ
ejpam-5210	289	7	p	p	PROPN
ejpam-5210	289	8	lee	lee	PROPN
ejpam-5210	289	9	.	.	PUNCT
ejpam-5210	290	1	prime	prime	PROPN
ejpam-5210	290	2	lie	lie	NOUN
ejpam-5210	290	3	rings	ring	NOUN
ejpam-5210	290	4	of	of	ADP
ejpam-5210	290	5	derivations	derivation	NOUN
ejpam-5210	290	6	of	of	ADP
ejpam-5210	290	7	commutative	commutative	ADJ
ejpam-5210	290	8	rings	ring	NOUN
ejpam-5210	290	9	.	.	PUNCT
ejpam-5210	291	1	communications	communication	NOUN
ejpam-5210	291	2	in	in	ADP
ejpam-5210	291	3	algebraregistered	algebraregistere	VERB
ejpam-5210	291	4	,	,	PUNCT
ejpam-5210	291	5	34(12):4339–4344	34(12):4339–4344	NUM
ejpam-5210	291	6	,	,	PUNCT
ejpam-5210	291	7	2006	2006	NUM
ejpam-5210	291	8	.	.	PUNCT
ejpam-5210	292	1	[	[	X
ejpam-5210	292	2	7	7	X
ejpam-5210	292	3	]	]	X
ejpam-5210	292	4	i	i	PRON
ejpam-5210	292	5	herstein	herstein	NOUN
ejpam-5210	292	6	.	.	PUNCT
ejpam-5210	293	1	on	on	ADP
ejpam-5210	293	2	the	the	DET
ejpam-5210	293	3	lie	lie	NOUN
ejpam-5210	293	4	and	and	CCONJ
ejpam-5210	293	5	jordan	jordan	PROPN
ejpam-5210	293	6	rings	ring	NOUN
ejpam-5210	293	7	of	of	ADP
ejpam-5210	293	8	a	a	DET
ejpam-5210	293	9	simple	simple	ADJ
ejpam-5210	293	10	associative	associative	ADJ
ejpam-5210	293	11	ring	ring	NOUN
ejpam-5210	293	12	.	.	PUNCT
ejpam-5210	294	1	american	american	PROPN
ejpam-5210	294	2	journal	journal	PROPN
ejpam-5210	294	3	of	of	ADP
ejpam-5210	294	4	mathematics	mathematic	NOUN
ejpam-5210	294	5	,	,	PUNCT
ejpam-5210	294	6	77(2):279–285	77(2):279–285	PROPN
ejpam-5210	294	7	,	,	PUNCT
ejpam-5210	294	8	1955	1955	NUM
ejpam-5210	294	9	.	.	PUNCT
ejpam-5210	295	1	[	[	X
ejpam-5210	295	2	8	8	NUM
ejpam-5210	295	3	]	]	X
ejpam-5210	295	4	i	i	PRON
ejpam-5210	295	5	herstein	herstein	NOUN
ejpam-5210	295	6	.	.	PUNCT
ejpam-5210	296	1	on	on	ADP
ejpam-5210	296	2	the	the	DET
ejpam-5210	296	3	lie	lie	NOUN
ejpam-5210	296	4	structure	structure	NOUN
ejpam-5210	296	5	of	of	ADP
ejpam-5210	296	6	an	an	DET
ejpam-5210	296	7	associative	associative	ADJ
ejpam-5210	296	8	ring	ring	NOUN
ejpam-5210	296	9	.	.	PUNCT
ejpam-5210	297	1	journal	journal	PROPN
ejpam-5210	297	2	of	of	ADP
ejpam-5210	297	3	algebra	algebra	PROPN
ejpam-5210	297	4	,	,	PUNCT
ejpam-5210	297	5	14(4):561	14(4):561	NUM
ejpam-5210	297	6	–	–	PUNCT
ejpam-5210	297	7	571	571	NUM
ejpam-5210	297	8	,	,	PUNCT
ejpam-5210	297	9	1970	1970	NUM
ejpam-5210	297	10	.	.	PUNCT
ejpam-5210	298	1	[	[	X
ejpam-5210	298	2	9	9	NUM
ejpam-5210	298	3	]	]	X
ejpam-5210	298	4	i	i	PROPN
ejpam-5210	298	5	herstein	herstein	NOUN
ejpam-5210	298	6	.	.	PUNCT
ejpam-5210	299	1	noncommutative	noncommutative	ADJ
ejpam-5210	299	2	rings	ring	NOUN
ejpam-5210	299	3	.	.	PUNCT
ejpam-5210	299	4	,	,	PUNCT
ejpam-5210	299	5	volume	volume	NOUN
ejpam-5210	299	6	15	15	NUM
ejpam-5210	299	7	.	.	PUNCT
ejpam-5210	300	1	american	american	PROPN
ejpam-5210	300	2	mathematical	mathematical	PROPN
ejpam-5210	300	3	soc	soc	PROPN
ejpam-5210	300	4	.	.	PUNCT
ejpam-5210	300	5	,	,	PUNCT
ejpam-5210	300	6	1994	1994	NUM
ejpam-5210	300	7	.	.	PUNCT
ejpam-5210	301	1	[	[	X
ejpam-5210	301	2	10	10	NUM
ejpam-5210	301	3	]	]	X
ejpam-5210	301	4	i	i	PRON
ejpam-5210	301	5	herstein	herstein	NOUN
ejpam-5210	301	6	.	.	PUNCT
ejpam-5210	302	1	topics	topic	NOUN
ejpam-5210	302	2	in	in	ADP
ejpam-5210	302	3	ring	ring	NOUN
ejpam-5210	302	4	theory	theory	NOUN
ejpam-5210	302	5	.	.	PUNCT
ejpam-5210	303	1	springer	springer	NOUN
ejpam-5210	303	2	,	,	PUNCT
ejpam-5210	303	3	2011	2011	NUM
ejpam-5210	303	4	.	.	PUNCT
ejpam-5210	304	1	[	[	X
ejpam-5210	304	2	11	11	NUM
ejpam-5210	304	3	]	]	X
ejpam-5210	304	4	y	y	PROPN
ejpam-5210	304	5	hirano	hirano	PROPN
ejpam-5210	304	6	,	,	PUNCT
ejpam-5210	304	7	h	h	NOUN
ejpam-5210	304	8	tominaga	tominaga	NOUN
ejpam-5210	304	9	,	,	PUNCT
ejpam-5210	304	10	and	and	CCONJ
ejpam-5210	304	11	a	a	DET
ejpam-5210	304	12	trzepizur	trzepizur	NOUN
ejpam-5210	304	13	.	.	PUNCT
ejpam-5210	305	1	on	on	ADP
ejpam-5210	305	2	a	a	DET
ejpam-5210	305	3	theorem	theorem	NOUN
ejpam-5210	305	4	of	of	ADP
ejpam-5210	305	5	posner	posner	NOUN
ejpam-5210	305	6	.	.	PUNCT
ejpam-5210	306	1	math	math	NOUN
ejpam-5210	306	2	.	.	PUNCT
ejpam-5210	307	1	j.	j.	PROPN
ejpam-5210	307	2	okayama	okayama	PROPN
ejpam-5210	307	3	univ	univ	PROPN
ejpam-5210	307	4	,	,	PUNCT
ejpam-5210	307	5	27:25–32	27:25–32	NUM
ejpam-5210	307	6	,	,	PUNCT
ejpam-5210	307	7	1985	1985	NUM
ejpam-5210	307	8	.	.	PUNCT
ejpam-5210	308	1	[	[	X
ejpam-5210	308	2	12	12	NUM
ejpam-5210	308	3	]	]	X
ejpam-5210	308	4	m	m	VERB
ejpam-5210	308	5	hongan	hongan	ADJ
ejpam-5210	308	6	and	and	CCONJ
ejpam-5210	308	7	a	a	DET
ejpam-5210	308	8	trzepizur	trzepizur	NOUN
ejpam-5210	308	9	.	.	PUNCT
ejpam-5210	309	1	on	on	ADP
ejpam-5210	309	2	generalization	generalization	NOUN
ejpam-5210	309	3	of	of	ADP
ejpam-5210	309	4	a	a	DET
ejpam-5210	309	5	theorem	theorem	NOUN
ejpam-5210	309	6	of	of	ADP
ejpam-5210	309	7	posner	posner	NOUN
ejpam-5210	309	8	.	.	PUNCT
ejpam-5210	310	1	mathematical	mathematical	ADJ
ejpam-5210	310	2	journal	journal	PROPN
ejpam-5210	310	3	of	of	ADP
ejpam-5210	310	4	okayama	okayama	PROPN
ejpam-5210	310	5	university	university	PROPN
ejpam-5210	310	6	,	,	PUNCT
ejpam-5210	310	7	27(1):19–23	27(1):19–23	NUM
ejpam-5210	310	8	,	,	PUNCT
ejpam-5210	310	9	1985	1985	NUM
ejpam-5210	310	10	.	.	PUNCT
ejpam-5210	311	1	[	[	X
ejpam-5210	311	2	13	13	NUM
ejpam-5210	311	3	]	]	PUNCT
ejpam-5210	311	4	n	n	PRON
ejpam-5210	311	5	jacobson	jacobson	PROPN
ejpam-5210	311	6	.	.	PUNCT
ejpam-5210	312	1	abstract	abstract	ADJ
ejpam-5210	312	2	derivation	derivation	NOUN
ejpam-5210	312	3	and	and	CCONJ
ejpam-5210	312	4	lie	lie	NOUN
ejpam-5210	312	5	algebras	algebra	NOUN
ejpam-5210	312	6	.	.	PUNCT
ejpam-5210	313	1	transactions	transaction	NOUN
ejpam-5210	313	2	of	of	ADP
ejpam-5210	313	3	the	the	DET
ejpam-5210	313	4	american	american	PROPN
ejpam-5210	313	5	mathematical	mathematical	PROPN
ejpam-5210	313	6	society	society	NOUN
ejpam-5210	313	7	,	,	PUNCT
ejpam-5210	313	8	42(2):206–224	42(2):206–224	PROPN
ejpam-5210	313	9	,	,	PUNCT
ejpam-5210	313	10	1937	1937	NUM
ejpam-5210	313	11	.	.	PUNCT
ejpam-5210	314	1	[	[	X
ejpam-5210	314	2	14	14	NUM
ejpam-5210	314	3	]	]	X
ejpam-5210	314	4	n	n	PRON
ejpam-5210	314	5	jacobson	jacobson	PROPN
ejpam-5210	314	6	.	.	PROPN
ejpam-5210	315	1	lie	lie	PROPN
ejpam-5210	315	2	algebras	algebras	PROPN
ejpam-5210	315	3	.	.	PUNCT
ejpam-5210	316	1	number	number	NOUN
ejpam-5210	316	2	10	10	NUM
ejpam-5210	316	3	.	.	PUNCT
ejpam-5210	317	1	courier	courier	NOUN
ejpam-5210	317	2	corporation	corporation	NOUN
ejpam-5210	317	3	.	.	PUNCT
ejpam-5210	317	4	,	,	PUNCT
ejpam-5210	317	5	1979	1979	NUM
ejpam-5210	317	6	.	.	PUNCT
ejpam-5210	318	1	[	[	X
ejpam-5210	318	2	15	15	NUM
ejpam-5210	318	3	]	]	X
ejpam-5210	318	4	a	a	DET
ejpam-5210	318	5	jordan	jordan	PROPN
ejpam-5210	318	6	.	.	PUNCT
ejpam-5210	319	1	noetherian	noetherian	ADJ
ejpam-5210	319	2	ore	ore	NOUN
ejpam-5210	319	3	extensions	extension	NOUN
ejpam-5210	319	4	and	and	CCONJ
ejpam-5210	319	5	jacobson	jacobson	PROPN
ejpam-5210	319	6	rings	ring	NOUN
ejpam-5210	319	7	.	.	PUNCT
ejpam-5210	320	1	journal	journal	PROPN
ejpam-5210	320	2	of	of	ADP
ejpam-5210	320	3	the	the	DET
ejpam-5210	320	4	london	london	PROPN
ejpam-5210	320	5	mathematical	mathematical	ADJ
ejpam-5210	320	6	society	society	NOUN
ejpam-5210	320	7	,	,	PUNCT
ejpam-5210	320	8	2(3):281–291	2(3):281–291	NUM
ejpam-5210	320	9	,	,	PUNCT
ejpam-5210	320	10	1975	1975	NUM
ejpam-5210	320	11	.	.	PUNCT
ejpam-5210	321	1	[	[	X
ejpam-5210	321	2	16	16	NUM
ejpam-5210	321	3	]	]	X
ejpam-5210	321	4	c	c	PROPN
ejpam-5210	321	5	jordan	jordan	PROPN
ejpam-5210	321	6	and	and	CCONJ
ejpam-5210	321	7	d	d	PROPN
ejpam-5210	321	8	jordan	jordan	PROPN
ejpam-5210	321	9	.	.	PUNCT
ejpam-5210	322	1	lie	lie	PROPN
ejpam-5210	322	2	rings	ring	NOUN
ejpam-5210	322	3	of	of	ADP
ejpam-5210	322	4	derivations	derivation	NOUN
ejpam-5210	322	5	of	of	ADP
ejpam-5210	322	6	associative	associative	ADJ
ejpam-5210	322	7	rings	ring	NOUN
ejpam-5210	322	8	.	.	PUNCT
ejpam-5210	323	1	journal	journal	PROPN
ejpam-5210	323	2	of	of	ADP
ejpam-5210	323	3	the	the	DET
ejpam-5210	323	4	london	london	PROPN
ejpam-5210	323	5	mathematical	mathematical	ADJ
ejpam-5210	323	6	society	society	NOUN
ejpam-5210	323	7	,	,	PUNCT
ejpam-5210	323	8	2(1):33–41	2(1):33–41	NUM
ejpam-5210	323	9	,	,	PUNCT
ejpam-5210	323	10	1978	1978	NUM
ejpam-5210	323	11	.	.	PUNCT
ejpam-5210	324	1	[	[	X
ejpam-5210	324	2	17	17	NUM
ejpam-5210	324	3	]	]	X
ejpam-5210	324	4	c	c	PROPN
ejpam-5210	324	5	jordan	jordan	PROPN
ejpam-5210	324	6	and	and	CCONJ
ejpam-5210	324	7	d	d	PROPN
ejpam-5210	324	8	jordan	jordan	PROPN
ejpam-5210	324	9	.	.	PUNCT
ejpam-5210	325	1	the	the	DET
ejpam-5210	325	2	lie	lie	NOUN
ejpam-5210	325	3	structure	structure	NOUN
ejpam-5210	325	4	of	of	ADP
ejpam-5210	325	5	a	a	DET
ejpam-5210	325	6	commutative	commutative	ADJ
ejpam-5210	325	7	ring	ring	NOUN
ejpam-5210	325	8	with	with	ADP
ejpam-5210	325	9	derivation	derivation	NOUN
ejpam-5210	325	10	.	.	PUNCT
ejpam-5210	326	1	journal	journal	NOUN
ejpam-5210	326	2	of	of	ADP
ejpam-5210	326	3	the	the	DET
ejpam-5210	326	4	london	london	PROPN
ejpam-5210	326	5	mathematical	mathematical	ADJ
ejpam-5210	326	6	society	society	NOUN
ejpam-5210	326	7	,	,	PUNCT
ejpam-5210	326	8	2(18):39–49	2(18):39–49	NUM
ejpam-5210	326	9	,	,	PUNCT
ejpam-5210	326	10	1978	1978	NUM
ejpam-5210	326	11	.	.	PUNCT
ejpam-5210	327	1	references	reference	NOUN
ejpam-5210	327	2	2275	2275	NUM
ejpam-5210	327	3	[	[	X
ejpam-5210	327	4	18	18	NUM
ejpam-5210	327	5	]	]	PUNCT
ejpam-5210	327	6	a	a	DET
ejpam-5210	327	7	al	al	PROPN
ejpam-5210	327	8	khalaf	khalaf	PROPN
ejpam-5210	327	9	,	,	PUNCT
ejpam-5210	327	10	o	o	PROPN
ejpam-5210	327	11	artemovych	artemovych	NOUN
ejpam-5210	327	12	,	,	PUNCT
ejpam-5210	327	13	and	and	CCONJ
ejpam-5210	327	14	i	i	PRON
ejpam-5210	327	15	taha	taha	PROPN
ejpam-5210	327	16	.	.	PUNCT
ejpam-5210	328	1	derivations	derivation	NOUN
ejpam-5210	328	2	in	in	ADP
ejpam-5210	328	3	differentially	differentially	ADV
ejpam-5210	328	4	prime	prime	ADJ
ejpam-5210	328	5	rings	ring	NOUN
ejpam-5210	328	6	.	.	PUNCT
ejpam-5210	329	1	journal	journal	PROPN
ejpam-5210	329	2	of	of	ADP
ejpam-5210	329	3	algebra	algebra	PROPN
ejpam-5210	329	4	and	and	CCONJ
ejpam-5210	329	5	its	its	PRON
ejpam-5210	329	6	applications	application	NOUN
ejpam-5210	329	7	,	,	PUNCT
ejpam-5210	329	8	17(7):1850129	17(7):1850129	NUM
ejpam-5210	329	9	,	,	PUNCT
ejpam-5210	329	10	2018	2018	NUM
ejpam-5210	329	11	.	.	PUNCT
ejpam-5210	330	1	[	[	X
ejpam-5210	330	2	19	19	NUM
ejpam-5210	330	3	]	]	PUNCT
ejpam-5210	330	4	a	a	DET
ejpam-5210	330	5	al	al	PROPN
ejpam-5210	330	6	khalaf	khalaf	PROPN
ejpam-5210	330	7	,	,	PUNCT
ejpam-5210	330	8	o	o	PROPN
ejpam-5210	330	9	artemovych	artemovych	NOUN
ejpam-5210	330	10	,	,	PUNCT
ejpam-5210	330	11	and	and	CCONJ
ejpam-5210	330	12	i	i	PRON
ejpam-5210	330	13	taha	taha	PROPN
ejpam-5210	330	14	.	.	PUNCT
ejpam-5210	330	15	rings	ring	NOUN
ejpam-5210	330	16	with	with	ADP
ejpam-5210	330	17	simple	simple	ADJ
ejpam-5210	330	18	lie	lie	NOUN
ejpam-5210	330	19	rings	ring	NOUN
ejpam-5210	330	20	of	of	ADP
ejpam-5210	330	21	lie	lie	NOUN
ejpam-5210	330	22	and	and	CCONJ
ejpam-5210	330	23	jordan	jordan	PROPN
ejpam-5210	330	24	derivations	derivations	PROPN
ejpam-5210	330	25	.	.	PUNCT
ejpam-5210	331	1	journal	journal	PROPN
ejpam-5210	331	2	of	of	ADP
ejpam-5210	331	3	algebra	algebra	PROPN
ejpam-5210	331	4	and	and	CCONJ
ejpam-5210	331	5	its	its	PRON
ejpam-5210	331	6	applications	application	NOUN
ejpam-5210	331	7	,	,	PUNCT
ejpam-5210	331	8	17(4):1850078	17(4):1850078	NUM
ejpam-5210	331	9	,	,	PUNCT
ejpam-5210	331	10	2018	2018	NUM
ejpam-5210	331	11	.	.	PUNCT
ejpam-5210	332	1	[	[	X
ejpam-5210	332	2	20	20	NUM
ejpam-5210	332	3	]	]	PUNCT
ejpam-5210	332	4	a	a	DET
ejpam-5210	332	5	al	al	PROPN
ejpam-5210	332	6	khalaf	khalaf	PROPN
ejpam-5210	332	7	,	,	PUNCT
ejpam-5210	332	8	o	o	PROPN
ejpam-5210	332	9	artemovych	artemovych	NOUN
ejpam-5210	332	10	,	,	PUNCT
ejpam-5210	332	11	and	and	CCONJ
ejpam-5210	332	12	i	i	PRON
ejpam-5210	332	13	taha	taha	PROPN
ejpam-5210	332	14	.	.	PUNCT
ejpam-5210	333	1	derivations	derivation	NOUN
ejpam-5210	333	2	of	of	ADP
ejpam-5210	333	3	differentially	differentially	ADV
ejpam-5210	333	4	semiprime	semiprime	NOUN
ejpam-5210	333	5	rings	ring	NOUN
ejpam-5210	333	6	.	.	PUNCT
ejpam-5210	334	1	asian	asian	ADJ
ejpam-5210	334	2	-	-	PUNCT
ejpam-5210	334	3	european	european	ADJ
ejpam-5210	334	4	journal	journal	NOUN
ejpam-5210	334	5	of	of	ADP
ejpam-5210	334	6	mathematics	mathematic	NOUN
ejpam-5210	334	7	,	,	PUNCT
ejpam-5210	334	8	12(5):1950079	12(5):1950079	NUM
ejpam-5210	334	9	,	,	PUNCT
ejpam-5210	334	10	2019	2019	NUM
ejpam-5210	334	11	.	.	PUNCT
ejpam-5210	335	1	[	[	X
ejpam-5210	335	2	21	21	NUM
ejpam-5210	335	3	]	]	PUNCT
ejpam-5210	335	4	a	a	DET
ejpam-5210	335	5	al	al	PROPN
ejpam-5210	335	6	khalaf	khalaf	PROPN
ejpam-5210	335	7	,	,	PUNCT
ejpam-5210	335	8	o	o	PROPN
ejpam-5210	335	9	artemovych	artemovych	NOUN
ejpam-5210	335	10	,	,	PUNCT
ejpam-5210	335	11	and	and	CCONJ
ejpam-5210	335	12	i	i	PRON
ejpam-5210	335	13	taha	taha	PROPN
ejpam-5210	335	14	.	.	PUNCT
ejpam-5210	336	1	commutators	commutator	NOUN
ejpam-5210	336	2	in	in	ADP
ejpam-5210	336	3	semiprime	semiprime	NOUN
ejpam-5210	336	4	gamma	gamma	PROPN
ejpam-5210	336	5	rings	ring	NOUN
ejpam-5210	336	6	.	.	PUNCT
ejpam-5210	337	1	asian	asian	ADJ
ejpam-5210	337	2	-	-	PUNCT
ejpam-5210	337	3	european	european	ADJ
ejpam-5210	337	4	journal	journal	NOUN
ejpam-5210	337	5	of	of	ADP
ejpam-5210	337	6	mathematics	mathematic	NOUN
ejpam-5210	337	7	,	,	PUNCT
ejpam-5210	337	8	13(4):2050078	13(4):2050078	NUM
ejpam-5210	337	9	,	,	PUNCT
ejpam-5210	337	10	2020	2020	NUM
ejpam-5210	337	11	.	.	PUNCT
ejpam-5210	338	1	[	[	X
ejpam-5210	338	2	22	22	NUM
ejpam-5210	338	3	]	]	X
ejpam-5210	338	4	j	j	PROPN
ejpam-5210	338	5	lambek	lambek	PROPN
ejpam-5210	338	6	.	.	PUNCT
ejpam-5210	339	1	lectures	lecture	NOUN
ejpam-5210	339	2	on	on	ADP
ejpam-5210	339	3	rings	ring	NOUN
ejpam-5210	339	4	and	and	CCONJ
ejpam-5210	339	5	modules	module	NOUN
ejpam-5210	339	6	.	.	PUNCT
ejpam-5210	339	7	,	,	PUNCT
ejpam-5210	339	8	volume	volume	NOUN
ejpam-5210	339	9	28	28	NUM
ejpam-5210	339	10	.	.	PUNCT
ejpam-5210	340	1	american	american	PROPN
ejpam-5210	340	2	mathematical	mathematical	PROPN
ejpam-5210	340	3	soc	soc	PROPN
ejpam-5210	340	4	.	.	PUNCT
ejpam-5210	340	5	,	,	PUNCT
ejpam-5210	340	6	2009	2009	NUM
ejpam-5210	340	7	.	.	PUNCT
ejpam-5210	341	1	[	[	X
ejpam-5210	341	2	23	23	NUM
ejpam-5210	341	3	]	]	X
ejpam-5210	341	4	p	p	X
ejpam-5210	341	5	lee	lee	PROPN
ejpam-5210	341	6	and	and	CCONJ
ejpam-5210	341	7	c	c	PROPN
ejpam-5210	341	8	liu	liu	PROPN
ejpam-5210	341	9	.	.	PUNCT
ejpam-5210	342	1	prime	prime	PROPN
ejpam-5210	342	2	lie	lie	NOUN
ejpam-5210	342	3	rings	ring	NOUN
ejpam-5210	342	4	of	of	ADP
ejpam-5210	342	5	derivations	derivation	NOUN
ejpam-5210	342	6	of	of	ADP
ejpam-5210	342	7	commutative	commutative	PROPN
ejpam-5210	342	8	rings	ring	NOUN
ejpam-5210	342	9	ii	ii	PROPN
ejpam-5210	342	10	.	.	PUNCT
ejpam-5210	343	1	communications	communication	NOUN
ejpam-5210	343	2	in	in	ADP
ejpam-5210	343	3	algebra	algebra	PROPN
ejpam-5210	343	4	®	®	NOUN
ejpam-5210	343	5	,	,	PUNCT
ejpam-5210	343	6	35(4):39–49	35(4):39–49	NUM
ejpam-5210	343	7	,	,	PUNCT
ejpam-5210	343	8	2007	2007	NUM
ejpam-5210	343	9	.	.	PUNCT
ejpam-5210	344	1	[	[	X
ejpam-5210	344	2	24	24	NUM
ejpam-5210	344	3	]	]	X
ejpam-5210	345	1	c	c	PROPN
ejpam-5210	345	2	liu	liu	PROPN
ejpam-5210	345	3	.	.	PROPN
ejpam-5210	345	4	semiprime	semiprime	PROPN
ejpam-5210	345	5	lie	lie	PROPN
ejpam-5210	345	6	rings	ring	NOUN
ejpam-5210	345	7	of	of	ADP
ejpam-5210	345	8	derivations	derivation	NOUN
ejpam-5210	345	9	of	of	ADP
ejpam-5210	345	10	commutative	commutative	ADJ
ejpam-5210	345	11	rings	ring	NOUN
ejpam-5210	345	12	.	.	PUNCT
ejpam-5210	346	1	contemporary	contemporary	ADJ
ejpam-5210	346	2	mathematics	mathematics	PROPN
ejpam-5210	346	3	,	,	PUNCT
ejpam-5210	346	4	420:259–268	420:259–268	NUM
ejpam-5210	346	5	,	,	PUNCT
ejpam-5210	346	6	2007	2007	NUM
ejpam-5210	346	7	.	.	PUNCT
ejpam-5210	347	1	[	[	X
ejpam-5210	347	2	25	25	NUM
ejpam-5210	347	3	]	]	PUNCT
ejpam-5210	347	4	a	a	DET
ejpam-5210	347	5	nowicki	nowicki	NOUN
ejpam-5210	347	6	.	.	PUNCT
ejpam-5210	348	1	the	the	DET
ejpam-5210	348	2	lie	lie	NOUN
ejpam-5210	348	3	structure	structure	NOUN
ejpam-5210	348	4	of	of	ADP
ejpam-5210	348	5	a	a	DET
ejpam-5210	348	6	commutative	commutative	ADJ
ejpam-5210	348	7	ring	ring	NOUN
ejpam-5210	348	8	with	with	ADP
ejpam-5210	348	9	a	a	DET
ejpam-5210	348	10	derivation	derivation	NOUN
ejpam-5210	348	11	.	.	PUNCT
ejpam-5210	349	1	archiv	archiv	PROPN
ejpam-5210	349	2	der	der	PROPN
ejpam-5210	349	3	mathematik	mathematik	PROPN
ejpam-5210	349	4	,	,	PUNCT
ejpam-5210	349	5	45:328–335	45:328–335	PROPN
ejpam-5210	349	6	,	,	PUNCT
ejpam-5210	349	7	1985	1985	NUM
ejpam-5210	349	8	.	.	PUNCT
ejpam-5210	350	1	[	[	X
ejpam-5210	350	2	26	26	NUM
ejpam-5210	350	3	]	]	X
ejpam-5210	350	4	d	d	X
ejpam-5210	350	5	passman	passman	NOUN
ejpam-5210	350	6	.	.	PUNCT
ejpam-5210	351	1	simple	simple	ADJ
ejpam-5210	351	2	lie	lie	NOUN
ejpam-5210	351	3	algebras	algebra	NOUN
ejpam-5210	351	4	of	of	ADP
ejpam-5210	351	5	witt	witt	PROPN
ejpam-5210	351	6	type	type	NOUN
ejpam-5210	351	7	.	.	PUNCT
ejpam-5210	352	1	journal	journal	PROPN
ejpam-5210	352	2	of	of	ADP
ejpam-5210	352	3	algebra	algebra	PROPN
ejpam-5210	352	4	,	,	PUNCT
ejpam-5210	352	5	206(2):682–692	206(2):682–692	NUM
ejpam-5210	352	6	,	,	PUNCT
ejpam-5210	352	7	1998	1998	NUM
ejpam-5210	352	8	.	.	PUNCT
ejpam-5210	353	1	[	[	X
ejpam-5210	353	2	27	27	NUM
ejpam-5210	353	3	]	]	X
ejpam-5210	353	4	i	i	PRON
ejpam-5210	353	5	taha	taha	PROPN
ejpam-5210	353	6	,	,	PUNCT
ejpam-5210	353	7	r	r	PROPN
ejpam-5210	353	8	masri	masri	PROPN
ejpam-5210	353	9	,	,	PUNCT
ejpam-5210	353	10	and	and	CCONJ
ejpam-5210	353	11	a	a	DET
ejpam-5210	353	12	al	al	PROPN
ejpam-5210	353	13	khalaf	khalaf	PROPN
ejpam-5210	353	14	.	.	PUNCT
ejpam-5210	354	1	derivations	derivation	NOUN
ejpam-5210	354	2	in	in	ADP
ejpam-5210	354	3	differentially	differentially	ADV
ejpam-5210	354	4	δ	δ	NOUN
ejpam-5210	354	5	-	-	ADJ
ejpam-5210	354	6	prime	prime	NOUN
ejpam-5210	354	7	rings	ring	NOUN
ejpam-5210	354	8	.	.	PUNCT
ejpam-5210	355	1	european	european	PROPN
ejpam-5210	355	2	journal	journal	PROPN
ejpam-5210	355	3	of	of	ADP
ejpam-5210	355	4	pure	pure	ADJ
ejpam-5210	355	5	and	and	CCONJ
ejpam-5210	355	6	applied	applied	ADJ
ejpam-5210	355	7	mathematics	mathematic	NOUN
ejpam-5210	355	8	,	,	PUNCT
ejpam-5210	355	9	15(2):454–466	15(2):454–466	PROPN
ejpam-5210	355	10	,	,	PUNCT
ejpam-5210	355	11	2022	2022	NUM
ejpam-5210	355	12	.	.	PUNCT
ejpam-5210	356	1	[	[	X
ejpam-5210	356	2	28	28	NUM
ejpam-5210	356	3	]	]	X
ejpam-5210	356	4	i	i	PRON
ejpam-5210	356	5	taha	taha	PROPN
ejpam-5210	356	6	,	,	PUNCT
ejpam-5210	356	7	r	r	PROPN
ejpam-5210	356	8	masri	masri	PROPN
ejpam-5210	356	9	,	,	PUNCT
ejpam-5210	356	10	and	and	CCONJ
ejpam-5210	356	11	a	a	DET
ejpam-5210	356	12	al	al	PROPN
ejpam-5210	356	13	khalaf	khalaf	PROPN
ejpam-5210	356	14	.	.	PUNCT
ejpam-5210	357	1	reverse	reverse	ADJ
ejpam-5210	357	2	derivations	derivation	NOUN
ejpam-5210	357	3	on	on	ADP
ejpam-5210	357	4	δ	δ	NOUN
ejpam-5210	357	5	-	-	PUNCT
ejpam-5210	357	6	prime	prime	NOUN
ejpam-5210	357	7	rings	ring	NOUN
ejpam-5210	357	8	.	.	PUNCT
ejpam-5210	358	1	european	european	PROPN
ejpam-5210	358	2	journal	journal	PROPN
ejpam-5210	358	3	of	of	ADP
ejpam-5210	358	4	pure	pure	ADJ
ejpam-5210	358	5	and	and	CCONJ
ejpam-5210	358	6	applied	applied	ADJ
ejpam-5210	358	7	mathematics	mathematic	NOUN
ejpam-5210	358	8	,	,	PUNCT
ejpam-5210	358	9	15(4):2032–2042	15(4):2032–2042	NUM
ejpam-5210	358	10	,	,	PUNCT
ejpam-5210	358	11	2022	2022	NUM
ejpam-5210	358	12	.	.	PUNCT