id	sid	tid	token	lemma	pos
ejpam-5948	1	1	european	european	PROPN
ejpam-5948	1	2	journal	journal	PROPN
ejpam-5948	1	3	of	of	ADP
ejpam-5948	1	4	pure	pure	ADJ
ejpam-5948	1	5	and	and	CCONJ
ejpam-5948	1	6	applied	applied	ADJ
ejpam-5948	1	7	mathematics	mathematic	NOUN
ejpam-5948	1	8	2025	2025	NUM
ejpam-5948	1	9	,	,	PUNCT
ejpam-5948	1	10	vol	vol	NOUN
ejpam-5948	1	11	.	.	PROPN
ejpam-5948	1	12	18	18	NUM
ejpam-5948	1	13	,	,	PUNCT
ejpam-5948	1	14	issue	issue	NOUN
ejpam-5948	1	15	2	2	NUM
ejpam-5948	1	16	,	,	PUNCT
ejpam-5948	1	17	article	article	NOUN
ejpam-5948	1	18	number	number	NOUN
ejpam-5948	1	19	5948	5948	NUM
ejpam-5948	1	20	issn	issn	PROPN
ejpam-5948	1	21	1307	1307	NUM
ejpam-5948	1	22	-	-	SYM
ejpam-5948	1	23	5543	5543	NUM
ejpam-5948	1	24	–	–	PUNCT
ejpam-5948	1	25	ejpam.com	ejpam.com	X
ejpam-5948	1	26	published	publish	VERB
ejpam-5948	1	27	by	by	ADP
ejpam-5948	1	28	new	new	PROPN
ejpam-5948	1	29	york	york	PROPN
ejpam-5948	1	30	business	business	PROPN
ejpam-5948	1	31	global	global	ADJ
ejpam-5948	1	32	specific	specific	ADJ
ejpam-5948	1	33	identities	identity	NOUN
ejpam-5948	1	34	involving	involve	VERB
ejpam-5948	1	35	prime	prime	ADJ
ejpam-5948	1	36	ideals	ideal	NOUN
ejpam-5948	1	37	with	with	ADP
ejpam-5948	1	38	generalized	generalize	VERB
ejpam-5948	1	39	p	p	NOUN
ejpam-5948	1	40	-derivations	-derivation	NOUN
ejpam-5948	1	41	ali	ali	PROPN
ejpam-5948	1	42	yahya	yahya	PROPN
ejpam-5948	1	43	hummdi1	hummdi1	PROPN
ejpam-5948	1	44	,	,	PUNCT
ejpam-5948	1	45	radwan	radwan	PROPN
ejpam-5948	1	46	m.	m.	PROPN
ejpam-5948	1	47	al	al	PROPN
ejpam-5948	1	48	-	-	PUNCT
ejpam-5948	1	49	omary2,∗	omary2,∗	PROPN
ejpam-5948	1	50	,	,	PUNCT
ejpam-5948	1	51	zakia	zakia	PROPN
ejpam-5948	1	52	z.	z.	PROPN
ejpam-5948	1	53	al	al	PROPN
ejpam-5948	1	54	-	-	PUNCT
ejpam-5948	1	55	amery3	amery3	PROPN
ejpam-5948	1	56	1	1	NUM
ejpam-5948	1	57	department	department	NOUN
ejpam-5948	1	58	of	of	ADP
ejpam-5948	1	59	mathematics	mathematic	NOUN
ejpam-5948	1	60	,	,	PUNCT
ejpam-5948	1	61	college	college	NOUN
ejpam-5948	1	62	of	of	ADP
ejpam-5948	1	63	science	science	NOUN
ejpam-5948	1	64	,	,	PUNCT
ejpam-5948	1	65	king	king	PROPN
ejpam-5948	1	66	khalid	khalid	PROPN
ejpam-5948	1	67	university	university	PROPN
ejpam-5948	1	68	,	,	PUNCT
ejpam-5948	1	69	abha	abha	NOUN
ejpam-5948	1	70	61471	61471	NUM
ejpam-5948	1	71	,	,	PUNCT
ejpam-5948	1	72	saudi	saudi	PROPN
ejpam-5948	1	73	arabia	arabia	PROPN
ejpam-5948	1	74	2	2	NUM
ejpam-5948	1	75	department	department	NOUN
ejpam-5948	1	76	of	of	ADP
ejpam-5948	1	77	mathematics	mathematics	PROPN
ejpam-5948	1	78	,	,	PUNCT
ejpam-5948	1	79	ibb	ibb	PROPN
ejpam-5948	1	80	university	university	NOUN
ejpam-5948	1	81	,	,	PUNCT
ejpam-5948	1	82	ibb	ibb	NOUN
ejpam-5948	1	83	,	,	PUNCT
ejpam-5948	1	84	yemen	yemen	PROPN
ejpam-5948	1	85	3	3	NUM
ejpam-5948	1	86	department	department	NOUN
ejpam-5948	1	87	of	of	ADP
ejpam-5948	1	88	mathematics	mathematics	PROPN
ejpam-5948	1	89	,	,	PUNCT
ejpam-5948	1	90	aden	aden	PROPN
ejpam-5948	1	91	university	university	PROPN
ejpam-5948	1	92	,	,	PUNCT
ejpam-5948	1	93	aden	aden	PROPN
ejpam-5948	1	94	,	,	PUNCT
ejpam-5948	1	95	yemen	yemen	PROPN
ejpam-5948	1	96	abstract	abstract	NOUN
ejpam-5948	1	97	.	.	PUNCT
ejpam-5948	2	1	in	in	ADP
ejpam-5948	2	2	this	this	DET
ejpam-5948	2	3	article	article	NOUN
ejpam-5948	2	4	,	,	PUNCT
ejpam-5948	2	5	we	we	PRON
ejpam-5948	2	6	will	will	AUX
ejpam-5948	2	7	investigate	investigate	VERB
ejpam-5948	2	8	the	the	DET
ejpam-5948	2	9	commutativity	commutativity	NOUN
ejpam-5948	2	10	of	of	ADP
ejpam-5948	2	11	the	the	DET
ejpam-5948	2	12	factor	factor	NOUN
ejpam-5948	2	13	ring	ring	NOUN
ejpam-5948	2	14	ℜ/p	ℜ/p	PROPN
ejpam-5948	2	15	,	,	PUNCT
ejpam-5948	2	16	where	where	SCONJ
ejpam-5948	2	17	p	p	NOUN
ejpam-5948	2	18	is	be	AUX
ejpam-5948	2	19	a	a	DET
ejpam-5948	2	20	prime	prime	ADJ
ejpam-5948	2	21	ideal	ideal	NOUN
ejpam-5948	2	22	of	of	ADP
ejpam-5948	2	23	any	any	DET
ejpam-5948	2	24	ring	ring	NOUN
ejpam-5948	2	25	ℜ.	ℜ.	PROPN
ejpam-5948	3	1	this	this	DET
ejpam-5948	3	2	investigation	investigation	NOUN
ejpam-5948	3	3	will	will	AUX
ejpam-5948	3	4	be	be	AUX
ejpam-5948	3	5	carried	carry	VERB
ejpam-5948	3	6	out	out	ADP
ejpam-5948	3	7	using	use	VERB
ejpam-5948	3	8	generalized	generalized	ADJ
ejpam-5948	3	9	p	p	NOUN
ejpam-5948	3	10	-derivations	-derivation	NOUN
ejpam-5948	3	11	℧	℧	PROPN
ejpam-5948	3	12	and	and	CCONJ
ejpam-5948	3	13	⨿	⨿	NOUN
ejpam-5948	3	14	associated	associate	VERB
ejpam-5948	3	15	with	with	ADP
ejpam-5948	3	16	p	p	PROPN
ejpam-5948	3	17	-derivations	-derivation	NOUN
ejpam-5948	3	18	χ	χ	NOUN
ejpam-5948	3	19	and	and	CCONJ
ejpam-5948	3	20	∝	∝	PROPN
ejpam-5948	3	21	,	,	PUNCT
ejpam-5948	3	22	respectively	respectively	ADV
ejpam-5948	3	23	,	,	PUNCT
ejpam-5948	3	24	that	that	PRON
ejpam-5948	3	25	satisfy	satisfy	VERB
ejpam-5948	3	26	specific	specific	ADJ
ejpam-5948	3	27	functional	functional	ADJ
ejpam-5948	3	28	identities	identity	NOUN
ejpam-5948	3	29	linking	link	VERB
ejpam-5948	3	30	ℜ	ℜ	NOUN
ejpam-5948	3	31	to	to	ADP
ejpam-5948	3	32	p	p	PROPN
ejpam-5948	3	33	.	.	PUNCT
ejpam-5948	4	1	moreover	moreover	ADV
ejpam-5948	4	2	,	,	PUNCT
ejpam-5948	4	3	we	we	PRON
ejpam-5948	4	4	will	will	AUX
ejpam-5948	4	5	discuss	discuss	VERB
ejpam-5948	4	6	some	some	DET
ejpam-5948	4	7	related	relate	VERB
ejpam-5948	4	8	results	result	NOUN
ejpam-5948	4	9	.	.	PUNCT
ejpam-5948	5	1	finally	finally	ADV
ejpam-5948	5	2	,	,	PUNCT
ejpam-5948	5	3	to	to	PART
ejpam-5948	5	4	reinforce	reinforce	VERB
ejpam-5948	5	5	the	the	DET
ejpam-5948	5	6	importance	importance	NOUN
ejpam-5948	5	7	of	of	ADP
ejpam-5948	5	8	our	our	PRON
ejpam-5948	5	9	assumption	assumption	NOUN
ejpam-5948	5	10	regarding	regard	VERB
ejpam-5948	5	11	the	the	DET
ejpam-5948	5	12	primeness	primeness	NOUN
ejpam-5948	5	13	of	of	ADP
ejpam-5948	5	14	p	p	NOUN
ejpam-5948	5	15	,	,	PUNCT
ejpam-5948	5	16	we	we	PRON
ejpam-5948	5	17	will	will	AUX
ejpam-5948	5	18	provide	provide	VERB
ejpam-5948	5	19	some	some	DET
ejpam-5948	5	20	examples	example	NOUN
ejpam-5948	5	21	.	.	PUNCT
ejpam-5948	6	1	2020	2020	NUM
ejpam-5948	6	2	mathematics	mathematic	NOUN
ejpam-5948	6	3	subject	subject	NOUN
ejpam-5948	6	4	classifications	classification	NOUN
ejpam-5948	6	5	:	:	PUNCT
ejpam-5948	6	6	16w25	16w25	NUM
ejpam-5948	6	7	,	,	PUNCT
ejpam-5948	6	8	16n60	16n60	NUM
ejpam-5948	6	9	,	,	PUNCT
ejpam-5948	6	10	16u80	16u80	NUM
ejpam-5948	6	11	key	key	ADJ
ejpam-5948	6	12	words	word	NOUN
ejpam-5948	6	13	and	and	CCONJ
ejpam-5948	6	14	phrases	phrase	NOUN
ejpam-5948	6	15	:	:	PUNCT
ejpam-5948	6	16	generalized	generalize	VERB
ejpam-5948	6	17	p	p	NOUN
ejpam-5948	6	18	-derivation	-derivation	NOUN
ejpam-5948	6	19	,	,	PUNCT
ejpam-5948	6	20	integral	integral	ADJ
ejpam-5948	6	21	domain	domain	NOUN
ejpam-5948	6	22	,	,	PUNCT
ejpam-5948	6	23	prime	prime	ADJ
ejpam-5948	6	24	ideal	ideal	NOUN
ejpam-5948	6	25	,	,	PUNCT
ejpam-5948	6	26	factor	factor	NOUN
ejpam-5948	6	27	ring	ring	NOUN
ejpam-5948	6	28	1	1	NUM
ejpam-5948	6	29	.	.	PUNCT
ejpam-5948	6	30	introduction	introduction	NOUN
ejpam-5948	6	31	throughout	throughout	ADP
ejpam-5948	6	32	this	this	DET
ejpam-5948	6	33	article	article	NOUN
ejpam-5948	6	34	,	,	PUNCT
ejpam-5948	6	35	the	the	DET
ejpam-5948	6	36	symbol	symbol	NOUN
ejpam-5948	6	37	ℜ	ℜ	PROPN
ejpam-5948	6	38	denotes	denote	VERB
ejpam-5948	6	39	an	an	DET
ejpam-5948	6	40	associative	associative	ADJ
ejpam-5948	6	41	ring	ring	NOUN
ejpam-5948	6	42	with	with	ADP
ejpam-5948	6	43	center	center	NOUN
ejpam-5948	6	44	z(ℜ	z(ℜ	NUM
ejpam-5948	6	45	)	)	PUNCT
ejpam-5948	6	46	.	.	PUNCT
ejpam-5948	7	1	a	a	DET
ejpam-5948	7	2	ring	ring	NOUN
ejpam-5948	7	3	ℜ	ℜ	PROPN
ejpam-5948	7	4	is	be	AUX
ejpam-5948	7	5	said	say	VERB
ejpam-5948	7	6	to	to	PART
ejpam-5948	7	7	be	be	AUX
ejpam-5948	7	8	a	a	DET
ejpam-5948	7	9	prime	prime	ADJ
ejpam-5948	7	10	ring	ring	NOUN
ejpam-5948	7	11	if	if	SCONJ
ejpam-5948	7	12	for	for	ADP
ejpam-5948	7	13	any	any	DET
ejpam-5948	7	14	elements	element	NOUN
ejpam-5948	7	15	υ	υ	NOUN
ejpam-5948	7	16	,	,	PUNCT
ejpam-5948	7	17	℘	℘	NOUN
ejpam-5948	7	18	∈	∈	NOUN
ejpam-5948	7	19	ℜ	ℜ	PROPN
ejpam-5948	8	1	the	the	DET
ejpam-5948	8	2	condition	condition	NOUN
ejpam-5948	8	3	υℜ℘	υℜ℘	PROPN
ejpam-5948	8	4	=	=	PUNCT
ejpam-5948	8	5	{	{	PUNCT
ejpam-5948	8	6	0	0	NUM
ejpam-5948	8	7	}	}	PUNCT
ejpam-5948	8	8	implies	imply	VERB
ejpam-5948	8	9	that	that	SCONJ
ejpam-5948	8	10	at	at	ADV
ejpam-5948	8	11	least	least	ADJ
ejpam-5948	8	12	one	one	NUM
ejpam-5948	8	13	of	of	ADP
ejpam-5948	8	14	the	the	DET
ejpam-5948	8	15	elements	element	NOUN
ejpam-5948	8	16	υ	υ	NOUN
ejpam-5948	8	17	or	or	CCONJ
ejpam-5948	8	18	℘	℘	PROPN
ejpam-5948	8	19	must	must	AUX
ejpam-5948	8	20	be	be	AUX
ejpam-5948	8	21	zero	zero	NUM
ejpam-5948	8	22	.	.	PUNCT
ejpam-5948	9	1	a	a	DET
ejpam-5948	9	2	prime	prime	ADJ
ejpam-5948	9	3	ideal	ideal	NOUN
ejpam-5948	9	4	is	be	AUX
ejpam-5948	9	5	a	a	DET
ejpam-5948	9	6	proper	proper	ADJ
ejpam-5948	9	7	ideal	ideal	NOUN
ejpam-5948	9	8	p	p	NOUN
ejpam-5948	9	9	of	of	ADP
ejpam-5948	9	10	a	a	DET
ejpam-5948	9	11	ring	ring	NOUN
ejpam-5948	9	12	ℜ	ℜ	PROPN
ejpam-5948	9	13	such	such	ADJ
ejpam-5948	9	14	that	that	SCONJ
ejpam-5948	9	15	if	if	SCONJ
ejpam-5948	9	16	υℜ℘	υℜ℘	PROPN
ejpam-5948	9	17	⊆	⊆	NUM
ejpam-5948	9	18	p	p	NOUN
ejpam-5948	9	19	,	,	PUNCT
ejpam-5948	9	20	then	then	ADV
ejpam-5948	9	21	at	at	ADP
ejpam-5948	9	22	least	least	ADJ
ejpam-5948	9	23	one	one	NUM
ejpam-5948	9	24	of	of	ADP
ejpam-5948	9	25	the	the	DET
ejpam-5948	9	26	elements	element	NOUN
ejpam-5948	9	27	υ	υ	NOUN
ejpam-5948	9	28	or	or	CCONJ
ejpam-5948	9	29	℘	℘	PROPN
ejpam-5948	9	30	must	must	AUX
ejpam-5948	9	31	be	be	AUX
ejpam-5948	9	32	belong	belong	ADJ
ejpam-5948	9	33	to	to	ADP
ejpam-5948	9	34	p	p	PROPN
ejpam-5948	9	35	.	.	PUNCT
ejpam-5948	10	1	a	a	DET
ejpam-5948	10	2	ring	ring	NOUN
ejpam-5948	10	3	ℜ	ℜ	PROPN
ejpam-5948	10	4	is	be	AUX
ejpam-5948	10	5	said	say	VERB
ejpam-5948	10	6	to	to	PART
ejpam-5948	10	7	be	be	AUX
ejpam-5948	10	8	an	an	DET
ejpam-5948	10	9	integral	integral	ADJ
ejpam-5948	10	10	domain	domain	NOUN
ejpam-5948	10	11	if	if	SCONJ
ejpam-5948	10	12	it	it	PRON
ejpam-5948	10	13	is	be	AUX
ejpam-5948	10	14	a	a	DET
ejpam-5948	10	15	commutative	commutative	ADJ
ejpam-5948	10	16	ring	ring	NOUN
ejpam-5948	10	17	with	with	ADP
ejpam-5948	10	18	unity	unity	NOUN
ejpam-5948	10	19	and	and	CCONJ
ejpam-5948	10	20	has	have	VERB
ejpam-5948	10	21	no	no	DET
ejpam-5948	10	22	zero	zero	NUM
ejpam-5948	10	23	divisors	divisor	NOUN
ejpam-5948	10	24	.	.	PUNCT
ejpam-5948	11	1	every	every	DET
ejpam-5948	11	2	integral	integral	ADJ
ejpam-5948	11	3	domain	domain	NOUN
ejpam-5948	11	4	is	be	AUX
ejpam-5948	11	5	a	a	DET
ejpam-5948	11	6	prime	prime	ADJ
ejpam-5948	11	7	ring	ring	NOUN
ejpam-5948	11	8	,	,	PUNCT
ejpam-5948	11	9	but	but	CCONJ
ejpam-5948	11	10	the	the	DET
ejpam-5948	11	11	converse	converse	NOUN
ejpam-5948	11	12	is	be	AUX
ejpam-5948	11	13	not	not	PART
ejpam-5948	11	14	true	true	ADJ
ejpam-5948	11	15	in	in	ADP
ejpam-5948	11	16	general	general	ADJ
ejpam-5948	11	17	.	.	PUNCT
ejpam-5948	12	1	for	for	ADP
ejpam-5948	12	2	all	all	DET
ejpam-5948	12	3	υ	υ	NOUN
ejpam-5948	12	4	,	,	PUNCT
ejpam-5948	12	5	℘	℘	PROPN
ejpam-5948	12	6	∈	∈	PROPN
ejpam-5948	12	7	ℜ	ℜ	PROPN
ejpam-5948	12	8	,	,	PUNCT
ejpam-5948	12	9	the	the	DET
ejpam-5948	12	10	symbols	symbol	NOUN
ejpam-5948	12	11	[	[	X
ejpam-5948	12	12	υ	υ	NOUN
ejpam-5948	12	13	,	,	PUNCT
ejpam-5948	12	14	℘	℘	PROPN
ejpam-5948	12	15	]	]	X
ejpam-5948	12	16	=	=	SYM
ejpam-5948	12	17	υ℘	υ℘	ADP
ejpam-5948	12	18	−	−	NOUN
ejpam-5948	12	19	℘υ	℘υ	NOUN
ejpam-5948	12	20	and	and	CCONJ
ejpam-5948	12	21	(	(	PUNCT
ejpam-5948	12	22	υ	υ	NOUN
ejpam-5948	12	23	◦	◦	NOUN
ejpam-5948	12	24	℘	℘	NUM
ejpam-5948	12	25	)	)	PUNCT
ejpam-5948	12	26	=	=	PUNCT
ejpam-5948	12	27	υ℘	υ℘	VERB
ejpam-5948	12	28	+	+	NUM
ejpam-5948	12	29	℘υ	℘υ	NOUN
ejpam-5948	12	30	denote	denote	VERB
ejpam-5948	12	31	the	the	DET
ejpam-5948	12	32	commutator	commutator	NOUN
ejpam-5948	12	33	and	and	CCONJ
ejpam-5948	12	34	anticommutator	anticommutator	NOUN
ejpam-5948	12	35	,	,	PUNCT
ejpam-5948	12	36	respectively	respectively	ADV
ejpam-5948	12	37	.	.	PUNCT
ejpam-5948	13	1	for	for	ADP
ejpam-5948	13	2	a	a	DET
ejpam-5948	13	3	subset	subset	ADJ
ejpam-5948	13	4	θ	θ	NOUN
ejpam-5948	13	5	of	of	ADP
ejpam-5948	13	6	ℜ	ℜ	PROPN
ejpam-5948	13	7	,	,	PUNCT
ejpam-5948	13	8	a	a	DET
ejpam-5948	13	9	mapping	mapping	NOUN
ejpam-5948	13	10	χ	χ	X
ejpam-5948	13	11	:	:	PUNCT
ejpam-5948	13	12	θ	θ	NOUN
ejpam-5948	13	13	−→	−→	NOUN
ejpam-5948	13	14	ℜ	ℜ	PROPN
ejpam-5948	13	15	is	be	AUX
ejpam-5948	13	16	said	say	VERB
ejpam-5948	13	17	to	to	PART
ejpam-5948	13	18	be	be	AUX
ejpam-5948	13	19	centralizing	centralize	VERB
ejpam-5948	13	20	(	(	PUNCT
ejpam-5948	13	21	or	or	CCONJ
ejpam-5948	13	22	commuting	commuting	NOUN
ejpam-5948	13	23	)	)	PUNCT
ejpam-5948	13	24	on	on	ADP
ejpam-5948	13	25	θ	θ	PROPN
ejpam-5948	13	26	if	if	SCONJ
ejpam-5948	13	27	[	[	X
ejpam-5948	13	28	χ(υ	χ(υ	NOUN
ejpam-5948	13	29	)	)	PUNCT
ejpam-5948	13	30	,	,	PUNCT
ejpam-5948	13	31	υ	υ	X
ejpam-5948	13	32	]	]	X
ejpam-5948	13	33	∈	∈	PROPN
ejpam-5948	13	34	z(ℜ	z(ℜ	NUM
ejpam-5948	13	35	)	)	PUNCT
ejpam-5948	13	36	(	(	PUNCT
ejpam-5948	13	37	or	or	CCONJ
ejpam-5948	13	38	[	[	X
ejpam-5948	13	39	χ(υ	χ(υ	NOUN
ejpam-5948	13	40	)	)	PUNCT
ejpam-5948	13	41	,	,	PUNCT
ejpam-5948	13	42	υ	υ	X
ejpam-5948	13	43	]	]	X
ejpam-5948	13	44	=	=	SYM
ejpam-5948	13	45	0	0	NUM
ejpam-5948	13	46	)	)	PUNCT
ejpam-5948	13	47	for	for	ADP
ejpam-5948	13	48	all	all	PRON
ejpam-5948	13	49	υ	υ	DET
ejpam-5948	13	50	∈	∈	PROPN
ejpam-5948	13	51	θ	θ	PROPN
ejpam-5948	13	52	.	.	PUNCT
ejpam-5948	13	53	by	by	ADP
ejpam-5948	13	54	definition	definition	NOUN
ejpam-5948	13	55	,	,	PUNCT
ejpam-5948	13	56	a	a	DET
ejpam-5948	13	57	derivation	derivation	NOUN
ejpam-5948	13	58	is	be	AUX
ejpam-5948	13	59	an	an	DET
ejpam-5948	13	60	additive	additive	ADJ
ejpam-5948	13	61	mapping	mapping	NOUN
ejpam-5948	13	62	χ	χ	NOUN
ejpam-5948	13	63	from	from	ADP
ejpam-5948	13	64	ℜ	ℜ	PROPN
ejpam-5948	13	65	to	to	ADP
ejpam-5948	13	66	itself	itself	PRON
ejpam-5948	13	67	that	that	PRON
ejpam-5948	13	68	satisfies	satisfy	VERB
ejpam-5948	13	69	χ(υ℘	χ(υ℘	X
ejpam-5948	13	70	)	)	PUNCT
ejpam-5948	13	71	=	=	PUNCT
ejpam-5948	14	1	χ(υ)℘	χ(υ)℘	NOUN
ejpam-5948	14	2	+	+	CCONJ
ejpam-5948	14	3	υχ(℘	υχ(℘	PROPN
ejpam-5948	14	4	)	)	PUNCT
ejpam-5948	14	5	for	for	ADP
ejpam-5948	14	6	all	all	DET
ejpam-5948	14	7	υ	υ	NOUN
ejpam-5948	14	8	,	,	PUNCT
ejpam-5948	14	9	℘	℘	PROPN
ejpam-5948	14	10	∈	∈	NOUN
ejpam-5948	14	11	ℜ.	ℜ.	VERB
ejpam-5948	14	12	a	a	DET
ejpam-5948	14	13	generalized	generalized	ADJ
ejpam-5948	14	14	derivation	derivation	NOUN
ejpam-5948	14	15	,	,	PUNCT
ejpam-5948	14	16	on	on	ADP
ejpam-5948	14	17	the	the	DET
ejpam-5948	14	18	other	other	ADJ
ejpam-5948	14	19	hand	hand	NOUN
ejpam-5948	14	20	,	,	PUNCT
ejpam-5948	14	21	is	be	AUX
ejpam-5948	14	22	an	an	DET
ejpam-5948	14	23	additive	additive	ADJ
ejpam-5948	14	24	mapping	mapping	NOUN
ejpam-5948	14	25	℧	℧	VERB
ejpam-5948	14	26	from	from	ADP
ejpam-5948	14	27	ℜ	ℜ	PROPN
ejpam-5948	14	28	to	to	ADP
ejpam-5948	14	29	itself	itself	PRON
ejpam-5948	14	30	that	that	PRON
ejpam-5948	14	31	satisfies	satisfy	VERB
ejpam-5948	14	32	℧	℧	PROPN
ejpam-5948	14	33	(	(	PUNCT
ejpam-5948	14	34	υ℘	υ℘	NUM
ejpam-5948	14	35	)	)	PUNCT
ejpam-5948	15	1	=	=	SYM
ejpam-5948	15	2	℧	℧	PROPN
ejpam-5948	15	3	(	(	PUNCT
ejpam-5948	15	4	υ)℘	υ)℘	X
ejpam-5948	15	5	+	+	NOUN
ejpam-5948	15	6	υχ(℘	υχ(℘	PROPN
ejpam-5948	15	7	)	)	PUNCT
ejpam-5948	15	8	for	for	ADP
ejpam-5948	15	9	all	all	DET
ejpam-5948	15	10	∗corresponding	∗corresponde	VERB
ejpam-5948	15	11	author	author	NOUN
ejpam-5948	15	12	.	.	PUNCT
ejpam-5948	16	1	doi	doi	NOUN
ejpam-5948	16	2	:	:	PUNCT
ejpam-5948	16	3	https://doi.org/10.29020/nybg.ejpam.v18i2.5948	https://doi.org/10.29020/nybg.ejpam.v18i2.5948	ADJ
ejpam-5948	16	4	email	email	NOUN
ejpam-5948	16	5	addresses	address	NOUN
ejpam-5948	16	6	:	:	PUNCT
ejpam-5948	16	7	ahmdy@kku.edu.sa	ahmdy@kku.edu.sa	PROPN
ejpam-5948	16	8	(	(	PUNCT
ejpam-5948	16	9	a.y	a.y	PROPN
ejpam-5948	16	10	.	.	PROPN
ejpam-5948	16	11	hummdi	hummdi	PROPN
ejpam-5948	16	12	)	)	PUNCT
ejpam-5948	16	13	,	,	PUNCT
ejpam-5948	16	14	raradwan959@gmail.com	raradwan959@gmail.com	X
ejpam-5948	16	15	(	(	PUNCT
ejpam-5948	16	16	r.m	r.m	PROPN
ejpam-5948	16	17	.	.	PROPN
ejpam-5948	16	18	al	al	PROPN
ejpam-5948	16	19	-	-	PUNCT
ejpam-5948	16	20	omary	omary	NOUN
ejpam-5948	16	21	)	)	PUNCT
ejpam-5948	16	22	,	,	PUNCT
ejpam-5948	16	23	alameryzakia@gmail.com	alameryzakia@gmail.com	X
ejpam-5948	16	24	(	(	PUNCT
ejpam-5948	16	25	z.z	z.z	PROPN
ejpam-5948	16	26	.	.	PUNCT
ejpam-5948	17	1	al	al	PROPN
ejpam-5948	17	2	-	-	PUNCT
ejpam-5948	17	3	amery	amery	PROPN
ejpam-5948	17	4	)	)	PUNCT
ejpam-5948	17	5	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5948	18	1	1	1	NUM
ejpam-5948	18	2	copyright	copyright	NOUN
ejpam-5948	18	3	:	:	PUNCT
ejpam-5948	18	4	©	©	PROPN
ejpam-5948	18	5	2025	2025	NUM
ejpam-5948	18	6	the	the	DET
ejpam-5948	18	7	author(s	author(s	NOUN
ejpam-5948	18	8	)	)	PUNCT
ejpam-5948	18	9	.	.	PUNCT
ejpam-5948	19	1	(	(	PUNCT
ejpam-5948	19	2	cc	cc	NOUN
ejpam-5948	19	3	by	by	ADP
ejpam-5948	19	4	-	-	PUNCT
ejpam-5948	19	5	nc	nc	PROPN
ejpam-5948	19	6	4.0	4.0	NUM
ejpam-5948	19	7	)	)	PUNCT
ejpam-5948	19	8	a.y	a.y	PROPN
ejpam-5948	19	9	.	.	PROPN
ejpam-5948	19	10	hummdi	hummdi	PROPN
ejpam-5948	19	11	,	,	PUNCT
ejpam-5948	19	12	r.m	r.m	PROPN
ejpam-5948	19	13	.	.	PROPN
ejpam-5948	19	14	al	al	PROPN
ejpam-5948	19	15	-	-	PUNCT
ejpam-5948	19	16	omary	omary	PROPN
ejpam-5948	19	17	,	,	PUNCT
ejpam-5948	19	18	z.z	z.z	PROPN
ejpam-5948	19	19	.	.	PUNCT
ejpam-5948	19	20	al	al	PROPN
ejpam-5948	19	21	-	-	PUNCT
ejpam-5948	19	22	amery	amery	PROPN
ejpam-5948	19	23	/	/	SYM
ejpam-5948	19	24	eur	eur	PROPN
ejpam-5948	19	25	.	.	PUNCT
ejpam-5948	20	1	j.	j.	PROPN
ejpam-5948	20	2	pure	pure	PROPN
ejpam-5948	20	3	appl	appl	PROPN
ejpam-5948	20	4	.	.	PROPN
ejpam-5948	20	5	math	math	PROPN
ejpam-5948	20	6	,	,	PUNCT
ejpam-5948	20	7	18	18	NUM
ejpam-5948	20	8	(	(	PUNCT
ejpam-5948	20	9	2	2	NUM
ejpam-5948	20	10	)	)	PUNCT
ejpam-5948	20	11	(	(	PUNCT
ejpam-5948	20	12	2025	2025	NUM
ejpam-5948	20	13	)	)	PUNCT
ejpam-5948	20	14	,	,	PUNCT
ejpam-5948	20	15	5948	5948	NUM
ejpam-5948	20	16	2	2	NUM
ejpam-5948	20	17	of	of	ADP
ejpam-5948	20	18	12	12	NUM
ejpam-5948	20	19	υ	υ	NOUN
ejpam-5948	20	20	,	,	PUNCT
ejpam-5948	20	21	℘	℘	PROPN
ejpam-5948	20	22	∈	∈	PROPN
ejpam-5948	20	23	ℜ	ℜ	PROPN
ejpam-5948	20	24	,	,	PUNCT
ejpam-5948	20	25	where	where	SCONJ
ejpam-5948	20	26	χ	χ	NOUN
ejpam-5948	20	27	is	be	AUX
ejpam-5948	20	28	the	the	DET
ejpam-5948	20	29	associated	associated	ADJ
ejpam-5948	20	30	derivation	derivation	NOUN
ejpam-5948	20	31	with	with	ADP
ejpam-5948	20	32	℧	℧	PROPN
ejpam-5948	20	33	.	.	PUNCT
ejpam-5948	21	1	it	it	PRON
ejpam-5948	21	2	is	be	AUX
ejpam-5948	21	3	evident	evident	ADJ
ejpam-5948	21	4	that	that	SCONJ
ejpam-5948	21	5	every	every	DET
ejpam-5948	21	6	derivation	derivation	NOUN
ejpam-5948	21	7	is	be	AUX
ejpam-5948	21	8	a	a	DET
ejpam-5948	21	9	generalized	generalized	ADJ
ejpam-5948	21	10	derivation	derivation	NOUN
ejpam-5948	21	11	,	,	PUNCT
ejpam-5948	21	12	but	but	CCONJ
ejpam-5948	21	13	the	the	DET
ejpam-5948	21	14	converse	converse	NOUN
ejpam-5948	21	15	is	be	AUX
ejpam-5948	21	16	not	not	PART
ejpam-5948	21	17	true	true	ADJ
ejpam-5948	21	18	in	in	ADP
ejpam-5948	21	19	general	general	ADJ
ejpam-5948	21	20	.	.	PUNCT
ejpam-5948	22	1	another	another	DET
ejpam-5948	22	2	special	special	ADJ
ejpam-5948	22	3	case	case	NOUN
ejpam-5948	22	4	of	of	ADP
ejpam-5948	22	5	a	a	DET
ejpam-5948	22	6	generalized	generalized	ADJ
ejpam-5948	22	7	derivation	derivation	NOUN
ejpam-5948	22	8	occurs	occur	VERB
ejpam-5948	22	9	when	when	SCONJ
ejpam-5948	22	10	χ	χ	X
ejpam-5948	22	11	is	be	AUX
ejpam-5948	22	12	restricted	restrict	VERB
ejpam-5948	22	13	to	to	PART
ejpam-5948	22	14	be	be	AUX
ejpam-5948	22	15	zero	zero	NUM
ejpam-5948	22	16	.	.	PUNCT
ejpam-5948	23	1	this	this	PRON
ejpam-5948	23	2	is	be	AUX
ejpam-5948	23	3	called	call	VERB
ejpam-5948	23	4	a	a	DET
ejpam-5948	23	5	multiplier	multipli	ADJ
ejpam-5948	23	6	,	,	PUNCT
ejpam-5948	23	7	ξ	ξ	PROPN
ejpam-5948	23	8	,	,	PUNCT
ejpam-5948	23	9	defined	define	VERB
ejpam-5948	23	10	as	as	ADP
ejpam-5948	23	11	an	an	DET
ejpam-5948	23	12	additive	additive	ADJ
ejpam-5948	23	13	mapping	mapping	NOUN
ejpam-5948	23	14	ξ	ξ	NOUN
ejpam-5948	23	15	:	:	PUNCT
ejpam-5948	23	16	ℜ	ℜ	PROPN
ejpam-5948	23	17	−→	−→	NOUN
ejpam-5948	23	18	ℜ	ℜ	NOUN
ejpam-5948	23	19	by	by	ADP
ejpam-5948	23	20	the	the	DET
ejpam-5948	23	21	rules	rule	NOUN
ejpam-5948	23	22	ξ(υ℘	ξ(υ℘	NUM
ejpam-5948	23	23	)	)	PUNCT
ejpam-5948	24	1	=	=	SYM
ejpam-5948	24	2	ξ(υ)℘	ξ(υ)℘	PROPN
ejpam-5948	24	3	and	and	CCONJ
ejpam-5948	24	4	ξ(υ℘	ξ(υ℘	NUM
ejpam-5948	24	5	)	)	PUNCT
ejpam-5948	24	6	=	=	SYM
ejpam-5948	24	7	υξ(℘	υξ(℘	PROPN
ejpam-5948	24	8	)	)	PUNCT
ejpam-5948	24	9	for	for	ADP
ejpam-5948	24	10	all	all	DET
ejpam-5948	24	11	υ	υ	NOUN
ejpam-5948	24	12	,	,	PUNCT
ejpam-5948	24	13	℘	℘	PROPN
ejpam-5948	24	14	∈	∈	NOUN
ejpam-5948	24	15	ℜ.	ℜ.	ADJ
ejpam-5948	24	16	these	these	PRON
ejpam-5948	24	17	are	be	AUX
ejpam-5948	24	18	referred	refer	VERB
ejpam-5948	24	19	to	to	ADP
ejpam-5948	24	20	as	as	ADP
ejpam-5948	24	21	left	left	ADJ
ejpam-5948	24	22	and	and	CCONJ
ejpam-5948	24	23	right	right	ADJ
ejpam-5948	24	24	multipliers	multiplier	NOUN
ejpam-5948	24	25	,	,	PUNCT
ejpam-5948	24	26	respectively	respectively	ADV
ejpam-5948	24	27	.	.	PUNCT
ejpam-5948	25	1	if	if	SCONJ
ejpam-5948	25	2	ξ	ξ	PROPN
ejpam-5948	25	3	is	be	AUX
ejpam-5948	25	4	both	both	PRON
ejpam-5948	25	5	a	a	DET
ejpam-5948	25	6	right	right	NOUN
ejpam-5948	25	7	and	and	CCONJ
ejpam-5948	25	8	left	leave	VERB
ejpam-5948	25	9	multiplier	multiplier	ADV
ejpam-5948	25	10	,	,	PUNCT
ejpam-5948	25	11	it	it	PRON
ejpam-5948	25	12	is	be	AUX
ejpam-5948	25	13	simply	simply	ADV
ejpam-5948	25	14	called	call	VERB
ejpam-5948	25	15	a	a	DET
ejpam-5948	25	16	multiplier	multipli	ADJ
ejpam-5948	25	17	.	.	PUNCT
ejpam-5948	26	1	examples	example	NOUN
ejpam-5948	26	2	and	and	CCONJ
ejpam-5948	26	3	counterexamples	counterexample	NOUN
ejpam-5948	26	4	of	of	ADP
ejpam-5948	26	5	these	these	DET
ejpam-5948	26	6	concepts	concept	NOUN
ejpam-5948	26	7	can	can	AUX
ejpam-5948	26	8	be	be	AUX
ejpam-5948	26	9	found	find	VERB
ejpam-5948	26	10	in	in	ADP
ejpam-5948	26	11	the	the	DET
ejpam-5948	26	12	literature	literature	NOUN
ejpam-5948	26	13	.	.	PUNCT
ejpam-5948	27	1	the	the	DET
ejpam-5948	27	2	mapping	mapping	NOUN
ejpam-5948	27	3	χ	χ	X
ejpam-5948	27	4	:	:	PUNCT
ejpam-5948	27	5	ℜ	ℜ	ADJ
ejpam-5948	27	6	−→	−→	NOUN
ejpam-5948	27	7	ℜ	ℜ	PROPN
ejpam-5948	27	8	is	be	AUX
ejpam-5948	27	9	called	call	VERB
ejpam-5948	27	10	p	p	NOUN
ejpam-5948	27	11	-additive	-additive	NOUN
ejpam-5948	27	12	if	if	SCONJ
ejpam-5948	27	13	it	it	PRON
ejpam-5948	27	14	satisfies	satisfy	VERB
ejpam-5948	27	15	χ(υ+℘)−χ(υ)−χ(℘	χ(υ+℘)−χ(υ)−χ(℘	NOUN
ejpam-5948	27	16	)	)	PUNCT
ejpam-5948	27	17	∈	∈	PROPN
ejpam-5948	27	18	p	p	NOUN
ejpam-5948	27	19	for	for	ADP
ejpam-5948	27	20	all	all	DET
ejpam-5948	27	21	υ	υ	NOUN
ejpam-5948	27	22	,	,	PUNCT
ejpam-5948	27	23	℘	℘	PROPN
ejpam-5948	27	24	∈	∈	NOUN
ejpam-5948	27	25	ℜ.	ℜ.	VERB
ejpam-5948	27	26	a	a	DET
ejpam-5948	27	27	p	p	NOUN
ejpam-5948	27	28	-additive	-additive	ADJ
ejpam-5948	27	29	mapping	mapping	NOUN
ejpam-5948	27	30	χ	χ	NOUN
ejpam-5948	27	31	is	be	AUX
ejpam-5948	27	32	called	call	VERB
ejpam-5948	27	33	a	a	DET
ejpam-5948	27	34	p	p	NOUN
ejpam-5948	27	35	-derivation	-derivation	NOUN
ejpam-5948	27	36	if	if	SCONJ
ejpam-5948	27	37	it	it	PRON
ejpam-5948	27	38	satisfies	satisfy	VERB
ejpam-5948	27	39	the	the	DET
ejpam-5948	27	40	relation	relation	NOUN
ejpam-5948	27	41	χ(υ℘)−	χ(υ℘)−	PROPN
ejpam-5948	27	42	χ(υ)℘−	χ(υ)℘−	ADP
ejpam-5948	27	43	υχ(℘	υχ(℘	PROPN
ejpam-5948	27	44	)	)	PUNCT
ejpam-5948	27	45	∈	∈	PROPN
ejpam-5948	27	46	p	p	NOUN
ejpam-5948	27	47	for	for	ADP
ejpam-5948	27	48	all	all	DET
ejpam-5948	27	49	υ	υ	NOUN
ejpam-5948	27	50	,	,	PUNCT
ejpam-5948	27	51	℘	℘	PROPN
ejpam-5948	27	52	∈	∈	NOUN
ejpam-5948	27	53	ℜ.	ℜ.	VERB
ejpam-5948	27	54	a	a	DET
ejpam-5948	27	55	p	p	NOUN
ejpam-5948	27	56	-additive	-additive	ADJ
ejpam-5948	27	57	mapping	mapping	NOUN
ejpam-5948	27	58	℧	℧	NOUN
ejpam-5948	27	59	:	:	PUNCT
ejpam-5948	27	60	ℜ	ℜ	ADV
ejpam-5948	27	61	−→	−→	NOUN
ejpam-5948	27	62	ℜ	ℜ	PROPN
ejpam-5948	27	63	is	be	AUX
ejpam-5948	27	64	called	call	VERB
ejpam-5948	27	65	a	a	DET
ejpam-5948	27	66	generalized	generalize	VERB
ejpam-5948	27	67	p	p	NOUN
ejpam-5948	27	68	-derivation	-derivation	NOUN
ejpam-5948	27	69	associated	associate	VERB
ejpam-5948	27	70	with	with	ADP
ejpam-5948	27	71	a	a	DET
ejpam-5948	27	72	p	p	NOUN
ejpam-5948	27	73	-derivation	-derivation	NOUN
ejpam-5948	27	74	χ	χ	ADP
ejpam-5948	27	75	if	if	SCONJ
ejpam-5948	27	76	it	it	PRON
ejpam-5948	27	77	satisfies	satisfy	VERB
ejpam-5948	27	78	℧	℧	PROPN
ejpam-5948	27	79	(	(	PUNCT
ejpam-5948	27	80	υ℘)−	υ℘)−	NOUN
ejpam-5948	27	81	℧	℧	NOUN
ejpam-5948	27	82	(υ)℘−	(υ)℘−	NOUN
ejpam-5948	27	83	υχ(℘	υχ(℘	PROPN
ejpam-5948	27	84	)	)	PUNCT
ejpam-5948	27	85	∈	∈	PROPN
ejpam-5948	27	86	p	p	NOUN
ejpam-5948	27	87	for	for	ADP
ejpam-5948	27	88	all	all	DET
ejpam-5948	27	89	υ	υ	NOUN
ejpam-5948	27	90	,	,	PUNCT
ejpam-5948	27	91	℘	℘	PROPN
ejpam-5948	27	92	∈	∈	NOUN
ejpam-5948	27	93	ℜ.	ℜ.	PROPN
ejpam-5948	27	94	additionally	additionally	ADV
ejpam-5948	27	95	,	,	PUNCT
ejpam-5948	27	96	assuming	assume	VERB
ejpam-5948	27	97	χ	χ	PRON
ejpam-5948	27	98	is	be	AUX
ejpam-5948	27	99	a	a	DET
ejpam-5948	27	100	p	p	X
ejpam-5948	27	101	-trivial	-trivial	NOUN
ejpam-5948	27	102	(	(	PUNCT
ejpam-5948	27	103	i.e.	i.e.	X
ejpam-5948	27	104	,	,	PUNCT
ejpam-5948	27	105	χ(ℜ	χ(ℜ	NOUN
ejpam-5948	27	106	)	)	PUNCT
ejpam-5948	27	107	⊆	⊆	NUM
ejpam-5948	27	108	p	p	NOUN
ejpam-5948	27	109	)	)	PUNCT
ejpam-5948	27	110	in	in	ADP
ejpam-5948	27	111	the	the	DET
ejpam-5948	27	112	last	last	ADJ
ejpam-5948	27	113	relation	relation	NOUN
ejpam-5948	27	114	gives	give	VERB
ejpam-5948	27	115	us	we	PRON
ejpam-5948	27	116	a	a	DET
ejpam-5948	27	117	p	p	NOUN
ejpam-5948	27	118	-left	-left	NOUN
ejpam-5948	27	119	multiplier	multipli	ADJ
ejpam-5948	27	120	concept	concept	NOUN
ejpam-5948	27	121	,	,	PUNCT
ejpam-5948	27	122	defined	define	VERB
ejpam-5948	27	123	as	as	ADP
ejpam-5948	27	124	ξ(υ℘	ξ(υ℘	NUM
ejpam-5948	27	125	)	)	PUNCT
ejpam-5948	27	126	−	−	PROPN
ejpam-5948	28	1	ξ(υ)℘	ξ(υ)℘	NUM
ejpam-5948	28	2	∈	∈	PROPN
ejpam-5948	28	3	p	p	NOUN
ejpam-5948	28	4	for	for	ADP
ejpam-5948	28	5	all	all	DET
ejpam-5948	28	6	υ	υ	NOUN
ejpam-5948	28	7	,	,	PUNCT
ejpam-5948	28	8	℘	℘	PROPN
ejpam-5948	28	9	∈	∈	NOUN
ejpam-5948	28	10	ℜ.	ℜ.	VERB
ejpam-5948	28	11	the	the	DET
ejpam-5948	28	12	p	p	NOUN
ejpam-5948	28	13	-right	-right	NOUN
ejpam-5948	28	14	multiplier	multiplier	ADV
ejpam-5948	28	15	is	be	AUX
ejpam-5948	28	16	defined	define	VERB
ejpam-5948	28	17	as	as	ADP
ejpam-5948	28	18	ξ(υ℘	ξ(υ℘	NOUN
ejpam-5948	28	19	)	)	PUNCT
ejpam-5948	28	20	−	−	NUM
ejpam-5948	28	21	υξ(℘	υξ(℘	PROPN
ejpam-5948	28	22	)	)	PUNCT
ejpam-5948	28	23	∈	∈	PROPN
ejpam-5948	28	24	p	p	NOUN
ejpam-5948	28	25	for	for	ADP
ejpam-5948	28	26	all	all	DET
ejpam-5948	28	27	υ	υ	NOUN
ejpam-5948	28	28	,	,	PUNCT
ejpam-5948	28	29	℘	℘	PROPN
ejpam-5948	28	30	∈	∈	NOUN
ejpam-5948	28	31	ℜ.	ℜ.	PROPN
ejpam-5948	28	32	moreover	moreover	ADV
ejpam-5948	28	33	,	,	PUNCT
ejpam-5948	28	34	ξ	ξ	PROPN
ejpam-5948	28	35	is	be	AUX
ejpam-5948	28	36	considered	consider	VERB
ejpam-5948	28	37	a	a	DET
ejpam-5948	28	38	p	p	NOUN
ejpam-5948	28	39	-multiplier	-multiplier	NOUN
ejpam-5948	28	40	if	if	SCONJ
ejpam-5948	28	41	it	it	PRON
ejpam-5948	28	42	is	be	AUX
ejpam-5948	28	43	both	both	PRON
ejpam-5948	28	44	a	a	DET
ejpam-5948	28	45	p	p	NOUN
ejpam-5948	28	46	-left	-left	NOUN
ejpam-5948	28	47	and	and	CCONJ
ejpam-5948	28	48	p	p	NOUN
ejpam-5948	28	49	-right	-right	NOUN
ejpam-5948	28	50	multiplier	multiplier	ADV
ejpam-5948	28	51	.	.	PUNCT
ejpam-5948	29	1	it	it	PRON
ejpam-5948	29	2	is	be	AUX
ejpam-5948	29	3	clear	clear	ADJ
ejpam-5948	29	4	that	that	SCONJ
ejpam-5948	29	5	every	every	DET
ejpam-5948	29	6	generalized	generalized	ADJ
ejpam-5948	29	7	derivation	derivation	NOUN
ejpam-5948	29	8	is	be	AUX
ejpam-5948	29	9	a	a	DET
ejpam-5948	29	10	generalized	generalized	ADJ
ejpam-5948	29	11	p	p	NOUN
ejpam-5948	29	12	-derivation	-derivation	NOUN
ejpam-5948	29	13	,	,	PUNCT
ejpam-5948	29	14	and	and	CCONJ
ejpam-5948	29	15	that	that	SCONJ
ejpam-5948	29	16	every	every	DET
ejpam-5948	29	17	left	leave	VERB
ejpam-5948	29	18	multiplier	multiplier	ADV
ejpam-5948	29	19	is	be	AUX
ejpam-5948	29	20	also	also	ADV
ejpam-5948	29	21	a	a	DET
ejpam-5948	29	22	p	p	ADJ
ejpam-5948	29	23	-left	-left	NOUN
ejpam-5948	29	24	multiplier	multiplier	ADV
ejpam-5948	29	25	,	,	PUNCT
ejpam-5948	29	26	but	but	CCONJ
ejpam-5948	29	27	the	the	DET
ejpam-5948	29	28	converse	converse	NOUN
ejpam-5948	29	29	may	may	AUX
ejpam-5948	29	30	not	not	PART
ejpam-5948	29	31	be	be	AUX
ejpam-5948	29	32	true	true	ADJ
ejpam-5948	29	33	in	in	ADP
ejpam-5948	29	34	general	general	ADJ
ejpam-5948	29	35	.	.	PUNCT
ejpam-5948	30	1	for	for	ADP
ejpam-5948	30	2	examples	example	NOUN
ejpam-5948	30	3	and	and	CCONJ
ejpam-5948	30	4	counterexamples	counterexample	NOUN
ejpam-5948	30	5	regarding	regard	VERB
ejpam-5948	30	6	the	the	DET
ejpam-5948	30	7	existence	existence	NOUN
ejpam-5948	30	8	of	of	ADP
ejpam-5948	30	9	these	these	DET
ejpam-5948	30	10	concepts	concept	NOUN
ejpam-5948	30	11	,	,	PUNCT
ejpam-5948	30	12	refer	refer	VERB
ejpam-5948	30	13	to	to	ADP
ejpam-5948	30	14	[	[	X
ejpam-5948	30	15	1	1	NUM
ejpam-5948	30	16	]	]	PUNCT
ejpam-5948	30	17	.	.	PUNCT
ejpam-5948	31	1	derivations	derivation	NOUN
ejpam-5948	31	2	are	be	AUX
ejpam-5948	31	3	a	a	DET
ejpam-5948	31	4	crucial	crucial	ADJ
ejpam-5948	31	5	area	area	NOUN
ejpam-5948	31	6	of	of	ADP
ejpam-5948	31	7	study	study	NOUN
ejpam-5948	31	8	in	in	ADP
ejpam-5948	31	9	algebraic	algebraic	ADJ
ejpam-5948	31	10	structure	structure	NOUN
ejpam-5948	31	11	theory	theory	NOUN
ejpam-5948	31	12	.	.	PUNCT
ejpam-5948	32	1	they	they	PRON
ejpam-5948	32	2	have	have	VERB
ejpam-5948	32	3	their	their	PRON
ejpam-5948	32	4	origins	origin	NOUN
ejpam-5948	32	5	in	in	ADP
ejpam-5948	32	6	analytic	analytic	ADJ
ejpam-5948	32	7	theory	theory	NOUN
ejpam-5948	32	8	,	,	PUNCT
ejpam-5948	32	9	invariant	invariant	ADJ
ejpam-5948	32	10	theory	theory	NOUN
ejpam-5948	32	11	,	,	PUNCT
ejpam-5948	32	12	and	and	CCONJ
ejpam-5948	32	13	galois	galois	PROPN
ejpam-5948	32	14	theory	theory	NOUN
ejpam-5948	32	15	.	.	PUNCT
ejpam-5948	33	1	derivations	derivation	NOUN
ejpam-5948	33	2	play	play	VERB
ejpam-5948	33	3	a	a	DET
ejpam-5948	33	4	vital	vital	ADJ
ejpam-5948	33	5	role	role	NOUN
ejpam-5948	33	6	in	in	ADP
ejpam-5948	33	7	both	both	DET
ejpam-5948	33	8	physics	physics	NOUN
ejpam-5948	33	9	and	and	CCONJ
ejpam-5948	33	10	mathematics	mathematic	NOUN
ejpam-5948	33	11	.	.	PUNCT
ejpam-5948	34	1	many	many	ADJ
ejpam-5948	34	2	researchers	researcher	NOUN
ejpam-5948	34	3	have	have	AUX
ejpam-5948	34	4	presented	present	VERB
ejpam-5948	34	5	derivations	derivation	NOUN
ejpam-5948	34	6	of	of	ADP
ejpam-5948	34	7	various	various	ADJ
ejpam-5948	34	8	algebraic	algebraic	ADJ
ejpam-5948	34	9	structures	structure	NOUN
ejpam-5948	34	10	such	such	ADJ
ejpam-5948	34	11	as	as	ADP
ejpam-5948	34	12	rings	ring	NOUN
ejpam-5948	34	13	,	,	PUNCT
ejpam-5948	34	14	and	and	CCONJ
ejpam-5948	34	15	near	near	ADP
ejpam-5948	34	16	rings	ring	NOUN
ejpam-5948	34	17	.	.	PUNCT
ejpam-5948	35	1	based	base	VERB
ejpam-5948	35	2	on	on	ADP
ejpam-5948	35	3	derivations	derivation	NOUN
ejpam-5948	35	4	in	in	ADP
ejpam-5948	35	5	rings	ring	NOUN
ejpam-5948	35	6	,	,	PUNCT
ejpam-5948	35	7	posner	posner	NOUN
ejpam-5948	35	8	proved	prove	VERB
ejpam-5948	35	9	in	in	ADP
ejpam-5948	35	10	[	[	X
ejpam-5948	35	11	2	2	X
ejpam-5948	35	12	]	]	PUNCT
ejpam-5948	35	13	that	that	SCONJ
ejpam-5948	35	14	if	if	SCONJ
ejpam-5948	35	15	a	a	DET
ejpam-5948	35	16	non	non	ADJ
ejpam-5948	35	17	-	-	ADJ
ejpam-5948	35	18	zero	zero	ADJ
ejpam-5948	35	19	derivation	derivation	NOUN
ejpam-5948	35	20	χ	χ	ADP
ejpam-5948	35	21	centralizing	centralize	VERB
ejpam-5948	35	22	on	on	ADP
ejpam-5948	35	23	a	a	DET
ejpam-5948	35	24	prime	prime	ADJ
ejpam-5948	35	25	ring	ring	NOUN
ejpam-5948	35	26	ℜ	ℜ	PROPN
ejpam-5948	35	27	,	,	PUNCT
ejpam-5948	35	28	then	then	ADV
ejpam-5948	35	29	ℜ	ℜ	PROPN
ejpam-5948	35	30	becomes	become	VERB
ejpam-5948	35	31	commutative	commutative	ADJ
ejpam-5948	35	32	.	.	PUNCT
ejpam-5948	36	1	recently	recently	ADV
ejpam-5948	36	2	,	,	PUNCT
ejpam-5948	36	3	several	several	ADJ
ejpam-5948	36	4	authors	author	NOUN
ejpam-5948	36	5	have	have	AUX
ejpam-5948	36	6	proven	prove	VERB
ejpam-5948	36	7	the	the	DET
ejpam-5948	36	8	commutativity	commutativity	NOUN
ejpam-5948	36	9	of	of	ADP
ejpam-5948	36	10	semiprime	semiprime	NOUN
ejpam-5948	36	11	and	and	CCONJ
ejpam-5948	36	12	prime	prime	ADJ
ejpam-5948	36	13	rings	ring	NOUN
ejpam-5948	36	14	by	by	ADP
ejpam-5948	36	15	utilizing	utilize	VERB
ejpam-5948	36	16	appropriately	appropriately	ADV
ejpam-5948	36	17	restricted	restrict	VERB
ejpam-5948	36	18	additive	additive	ADJ
ejpam-5948	36	19	mappings	mapping	NOUN
ejpam-5948	36	20	that	that	PRON
ejpam-5948	36	21	act	act	VERB
ejpam-5948	36	22	on	on	ADP
ejpam-5948	36	23	these	these	DET
ejpam-5948	36	24	rings	ring	NOUN
ejpam-5948	36	25	or	or	CCONJ
ejpam-5948	36	26	on	on	ADP
ejpam-5948	36	27	suitable	suitable	ADJ
ejpam-5948	36	28	subsets	subset	NOUN
ejpam-5948	36	29	of	of	ADP
ejpam-5948	36	30	them	they	PRON
ejpam-5948	36	31	.	.	PUNCT
ejpam-5948	37	1	these	these	DET
ejpam-5948	37	2	mappings	mapping	NOUN
ejpam-5948	37	3	include	include	VERB
ejpam-5948	37	4	derivations	derivation	NOUN
ejpam-5948	37	5	,	,	PUNCT
ejpam-5948	37	6	automorphisms	automorphism	NOUN
ejpam-5948	37	7	,	,	PUNCT
ejpam-5948	37	8	generalized	generalized	ADJ
ejpam-5948	37	9	derivations	derivation	NOUN
ejpam-5948	37	10	,	,	PUNCT
ejpam-5948	37	11	multipliers	multiplier	NOUN
ejpam-5948	37	12	,	,	PUNCT
ejpam-5948	37	13	and	and	CCONJ
ejpam-5948	37	14	others	other	NOUN
ejpam-5948	37	15	.	.	PUNCT
ejpam-5948	38	1	for	for	ADP
ejpam-5948	38	2	more	more	ADJ
ejpam-5948	38	3	details	detail	NOUN
ejpam-5948	38	4	,	,	PUNCT
ejpam-5948	38	5	one	one	PRON
ejpam-5948	38	6	can	can	AUX
ejpam-5948	38	7	refer	refer	VERB
ejpam-5948	38	8	to	to	ADP
ejpam-5948	38	9	[	[	X
ejpam-5948	38	10	3	3	NUM
ejpam-5948	38	11	]	]	PUNCT
ejpam-5948	38	12	,	,	PUNCT
ejpam-5948	38	13	[	[	X
ejpam-5948	38	14	4	4	NUM
ejpam-5948	38	15	]	]	PUNCT
ejpam-5948	38	16	,	,	PUNCT
ejpam-5948	38	17	and	and	CCONJ
ejpam-5948	38	18	[	[	X
ejpam-5948	38	19	5	5	NUM
ejpam-5948	38	20	]	]	PUNCT
ejpam-5948	38	21	.	.	PUNCT
ejpam-5948	39	1	inspired	inspire	VERB
ejpam-5948	39	2	by	by	ADP
ejpam-5948	39	3	previous	previous	ADJ
ejpam-5948	39	4	studies	study	NOUN
ejpam-5948	39	5	,	,	PUNCT
ejpam-5948	39	6	the	the	DET
ejpam-5948	39	7	commutativity	commutativity	NOUN
ejpam-5948	39	8	of	of	ADP
ejpam-5948	39	9	rings	ring	NOUN
ejpam-5948	39	10	has	have	AUX
ejpam-5948	39	11	been	be	AUX
ejpam-5948	39	12	discussed	discuss	VERB
ejpam-5948	39	13	in	in	ADP
ejpam-5948	39	14	a	a	DET
ejpam-5948	39	15	more	more	ADV
ejpam-5948	39	16	expansive	expansive	ADJ
ejpam-5948	39	17	way	way	NOUN
ejpam-5948	39	18	.	.	PUNCT
ejpam-5948	40	1	for	for	ADP
ejpam-5948	40	2	instance	instance	NOUN
ejpam-5948	40	3	,	,	PUNCT
ejpam-5948	40	4	the	the	DET
ejpam-5948	40	5	consideration	consideration	NOUN
ejpam-5948	40	6	of	of	ADP
ejpam-5948	40	7	whether	whether	SCONJ
ejpam-5948	40	8	the	the	DET
ejpam-5948	40	9	ring	ring	NOUN
ejpam-5948	40	10	ℜ	ℜ	PROPN
ejpam-5948	40	11	is	be	AUX
ejpam-5948	40	12	prime	prime	ADJ
ejpam-5948	40	13	or	or	CCONJ
ejpam-5948	40	14	semiprime	semiprime	NOUN
ejpam-5948	40	15	has	have	AUX
ejpam-5948	40	16	been	be	AUX
ejpam-5948	40	17	omitted	omit	VERB
ejpam-5948	40	18	,	,	PUNCT
ejpam-5948	40	19	and	and	CCONJ
ejpam-5948	40	20	instead	instead	ADV
ejpam-5948	40	21	the	the	DET
ejpam-5948	40	22	focus	focus	NOUN
ejpam-5948	40	23	has	have	AUX
ejpam-5948	40	24	shifted	shift	VERB
ejpam-5948	40	25	to	to	ADP
ejpam-5948	40	26	analyzing	analyze	VERB
ejpam-5948	40	27	the	the	DET
ejpam-5948	40	28	behavior	behavior	NOUN
ejpam-5948	40	29	of	of	ADP
ejpam-5948	40	30	a	a	DET
ejpam-5948	40	31	factor	factor	NOUN
ejpam-5948	40	32	ring	ring	NOUN
ejpam-5948	40	33	ℜ/p	ℜ/p	PROPN
ejpam-5948	40	34	,	,	PUNCT
ejpam-5948	40	35	where	where	SCONJ
ejpam-5948	40	36	p	p	NOUN
ejpam-5948	40	37	is	be	AUX
ejpam-5948	40	38	a	a	DET
ejpam-5948	40	39	prime	prime	ADJ
ejpam-5948	40	40	ideal	ideal	NOUN
ejpam-5948	40	41	of	of	ADP
ejpam-5948	40	42	ℜ.	ℜ.	PROPN
ejpam-5948	40	43	these	these	DET
ejpam-5948	40	44	studies	study	NOUN
ejpam-5948	40	45	involve	involve	VERB
ejpam-5948	40	46	the	the	DET
ejpam-5948	40	47	utilization	utilization	NOUN
ejpam-5948	40	48	of	of	ADP
ejpam-5948	40	49	additive	additive	ADJ
ejpam-5948	40	50	mappings	mapping	NOUN
ejpam-5948	40	51	that	that	PRON
ejpam-5948	40	52	satisfy	satisfy	VERB
ejpam-5948	40	53	certain	certain	ADJ
ejpam-5948	40	54	identities	identity	NOUN
ejpam-5948	40	55	when	when	SCONJ
ejpam-5948	40	56	acting	act	VERB
ejpam-5948	40	57	on	on	ADP
ejpam-5948	40	58	appropriate	appropriate	ADJ
ejpam-5948	40	59	subsets	subset	NOUN
ejpam-5948	40	60	of	of	ADP
ejpam-5948	40	61	the	the	DET
ejpam-5948	40	62	ring	ring	NOUN
ejpam-5948	40	63	ℜ.	ℜ.	ADJ
ejpam-5948	40	64	for	for	ADP
ejpam-5948	40	65	further	further	ADJ
ejpam-5948	40	66	details	detail	NOUN
ejpam-5948	40	67	,	,	PUNCT
ejpam-5948	40	68	please	please	INTJ
ejpam-5948	40	69	refer	refer	VERB
ejpam-5948	40	70	to	to	ADP
ejpam-5948	40	71	references	reference	NOUN
ejpam-5948	40	72	[	[	X
ejpam-5948	40	73	6	6	NUM
ejpam-5948	40	74	]	]	PUNCT
ejpam-5948	40	75	,	,	PUNCT
ejpam-5948	40	76	[	[	X
ejpam-5948	40	77	7	7	NUM
ejpam-5948	40	78	]	]	PUNCT
ejpam-5948	40	79	,	,	PUNCT
ejpam-5948	40	80	[	[	X
ejpam-5948	40	81	8	8	NUM
ejpam-5948	40	82	]	]	PUNCT
ejpam-5948	40	83	,	,	PUNCT
ejpam-5948	41	1	[	[	X
ejpam-5948	41	2	9	9	NUM
ejpam-5948	41	3	]	]	PUNCT
ejpam-5948	41	4	,	,	PUNCT
ejpam-5948	41	5	and	and	CCONJ
ejpam-5948	41	6	[	[	X
ejpam-5948	41	7	10	10	NUM
ejpam-5948	41	8	]	]	PUNCT
ejpam-5948	41	9	.	.	PUNCT
ejpam-5948	42	1	in	in	ADP
ejpam-5948	42	2	[	[	X
ejpam-5948	42	3	10	10	NUM
ejpam-5948	42	4	]	]	PUNCT
ejpam-5948	42	5	,	,	PUNCT
ejpam-5948	42	6	mouhssine	mouhssine	PROPN
ejpam-5948	42	7	et	et	PROPN
ejpam-5948	42	8	al	al	PROPN
ejpam-5948	42	9	.	.	PROPN
ejpam-5948	42	10	discuss	discuss	VERB
ejpam-5948	42	11	the	the	DET
ejpam-5948	42	12	behavior	behavior	NOUN
ejpam-5948	42	13	of	of	ADP
ejpam-5948	42	14	a	a	DET
ejpam-5948	42	15	factor	factor	NOUN
ejpam-5948	42	16	near	near	ADP
ejpam-5948	42	17	ring	ring	NOUN
ejpam-5948	42	18	ℵ/p	ℵ/p	VERB
ejpam-5948	42	19	when	when	SCONJ
ejpam-5948	42	20	a	a	DET
ejpam-5948	42	21	near	near	ADJ
ejpam-5948	42	22	ring	ring	NOUN
ejpam-5948	42	23	ℵ	ℵ	PROPN
ejpam-5948	42	24	admits	admit	VERB
ejpam-5948	42	25	an	an	DET
ejpam-5948	42	26	(	(	PUNCT
ejpam-5948	42	27	α	α	NOUN
ejpam-5948	42	28	,	,	PUNCT
ejpam-5948	42	29	τ)-p	τ)-p	ADP
ejpam-5948	42	30	-derivation	-derivation	NOUN
ejpam-5948	42	31	χ	χ	PRON
ejpam-5948	42	32	that	that	PRON
ejpam-5948	42	33	satisfies	satisfy	VERB
ejpam-5948	42	34	certain	certain	ADJ
ejpam-5948	42	35	identities	identity	NOUN
ejpam-5948	42	36	,	,	PUNCT
ejpam-5948	42	37	where	where	SCONJ
ejpam-5948	42	38	p	p	NOUN
ejpam-5948	42	39	is	be	AUX
ejpam-5948	42	40	a	a	DET
ejpam-5948	42	41	prime	prime	ADJ
ejpam-5948	42	42	ideal	ideal	NOUN
ejpam-5948	42	43	of	of	ADP
ejpam-5948	42	44	ℵ.	ℵ.	PROPN
ejpam-5948	42	45	in	in	ADP
ejpam-5948	42	46	2023	2023	NUM
ejpam-5948	42	47	,	,	PUNCT
ejpam-5948	42	48	[	[	X
ejpam-5948	42	49	11	11	NUM
ejpam-5948	42	50	]	]	PUNCT
ejpam-5948	42	51	oukhtite	oukhtite	NOUN
ejpam-5948	42	52	et	et	PROPN
ejpam-5948	42	53	al	al	PROPN
ejpam-5948	42	54	.	.	PROPN
ejpam-5948	42	55	examined	examine	VERB
ejpam-5948	42	56	the	the	DET
ejpam-5948	42	57	effect	effect	NOUN
ejpam-5948	42	58	of	of	ADP
ejpam-5948	42	59	specific	specific	ADJ
ejpam-5948	42	60	differential	differential	ADJ
ejpam-5948	42	61	identities	identity	NOUN
ejpam-5948	42	62	involving	involve	VERB
ejpam-5948	42	63	p	p	NOUN
ejpam-5948	42	64	-multipliers	-multiplier	NOUN
ejpam-5948	42	65	on	on	ADP
ejpam-5948	42	66	a	a	DET
ejpam-5948	42	67	factor	factor	NOUN
ejpam-5948	42	68	ring	ring	NOUN
ejpam-5948	42	69	ℜ/p	ℜ/p	PROPN
ejpam-5948	42	70	,	,	PUNCT
ejpam-5948	42	71	where	where	SCONJ
ejpam-5948	42	72	p	p	NOUN
ejpam-5948	42	73	is	be	AUX
ejpam-5948	42	74	a	a	DET
ejpam-5948	42	75	prime	prime	ADJ
ejpam-5948	42	76	ideal	ideal	NOUN
ejpam-5948	42	77	of	of	ADP
ejpam-5948	42	78	any	any	DET
ejpam-5948	42	79	ring	ring	NOUN
ejpam-5948	42	80	ℜ.	ℜ.	ADJ
ejpam-5948	42	81	in	in	ADP
ejpam-5948	42	82	the	the	DET
ejpam-5948	42	83	same	same	ADJ
ejpam-5948	42	84	year	year	NOUN
ejpam-5948	42	85	,	,	PUNCT
ejpam-5948	42	86	sandhu	sandhu	PROPN
ejpam-5948	42	87	et	et	PROPN
ejpam-5948	42	88	al	al	PROPN
ejpam-5948	42	89	.	.	PUNCT
ejpam-5948	43	1	[	[	X
ejpam-5948	43	2	1	1	X
ejpam-5948	43	3	]	]	PUNCT
ejpam-5948	43	4	investigated	investigate	VERB
ejpam-5948	43	5	the	the	DET
ejpam-5948	43	6	commutativity	commutativity	NOUN
ejpam-5948	43	7	of	of	ADP
ejpam-5948	43	8	a	a	DET
ejpam-5948	43	9	factor	factor	NOUN
ejpam-5948	43	10	ring	ring	NOUN
ejpam-5948	43	11	ℜ/p	ℜ/p	PROPN
ejpam-5948	43	12	by	by	ADP
ejpam-5948	43	13	exploring	explore	VERB
ejpam-5948	43	14	certain	certain	ADJ
ejpam-5948	43	15	identities	identity	NOUN
ejpam-5948	43	16	involving	involve	VERB
ejpam-5948	43	17	a	a	DET
ejpam-5948	43	18	mixture	mixture	NOUN
ejpam-5948	43	19	of	of	ADP
ejpam-5948	43	20	a	a	DET
ejpam-5948	43	21	generalized	generalize	VERB
ejpam-5948	43	22	p	p	NOUN
ejpam-5948	43	23	-derivation	-derivation	NOUN
ejpam-5948	43	24	and	and	CCONJ
ejpam-5948	43	25	a	a	DET
ejpam-5948	43	26	p	p	NOUN
ejpam-5948	43	27	-multiplier	-multiplier	NOUN
ejpam-5948	43	28	,	,	PUNCT
ejpam-5948	43	29	where	where	SCONJ
ejpam-5948	43	30	p	p	NOUN
ejpam-5948	43	31	is	be	AUX
ejpam-5948	43	32	a	a	DET
ejpam-5948	43	33	prime	prime	ADJ
ejpam-5948	43	34	ideal	ideal	NOUN
ejpam-5948	43	35	in	in	ADP
ejpam-5948	43	36	ℜ.	ℜ.	PROPN
ejpam-5948	43	37	in	in	ADP
ejpam-5948	43	38	this	this	DET
ejpam-5948	43	39	article	article	NOUN
ejpam-5948	43	40	,	,	PUNCT
ejpam-5948	43	41	we	we	PRON
ejpam-5948	43	42	will	will	AUX
ejpam-5948	43	43	further	far	ADV
ejpam-5948	43	44	investigate	investigate	VERB
ejpam-5948	43	45	the	the	DET
ejpam-5948	43	46	commutativity	commutativity	NOUN
ejpam-5948	43	47	of	of	ADP
ejpam-5948	43	48	a	a	DET
ejpam-5948	43	49	factor	factor	NOUN
ejpam-5948	43	50	ring	ring	NOUN
ejpam-5948	43	51	ℜ/p	ℜ/p	PROPN
ejpam-5948	43	52	.	.	PUNCT
ejpam-5948	44	1	we	we	PRON
ejpam-5948	44	2	will	will	AUX
ejpam-5948	44	3	accomplish	accomplish	VERB
ejpam-5948	44	4	this	this	PRON
ejpam-5948	44	5	by	by	ADP
ejpam-5948	44	6	assuming	assume	VERB
ejpam-5948	44	7	that	that	SCONJ
ejpam-5948	44	8	the	the	DET
ejpam-5948	44	9	arbitrary	arbitrary	ADJ
ejpam-5948	44	10	ring	ring	NOUN
ejpam-5948	44	11	ℜ	ℜ	PROPN
ejpam-5948	44	12	admits	admit	VERB
ejpam-5948	44	13	generalized	generalize	VERB
ejpam-5948	44	14	p	p	PROPN
ejpam-5948	44	15	a.y	a.y	PROPN
ejpam-5948	44	16	.	.	PROPN
ejpam-5948	45	1	hummdi	hummdi	PROPN
ejpam-5948	45	2	,	,	PUNCT
ejpam-5948	45	3	r.m	r.m	PROPN
ejpam-5948	45	4	.	.	PROPN
ejpam-5948	45	5	al	al	PROPN
ejpam-5948	45	6	-	-	PUNCT
ejpam-5948	45	7	omary	omary	PROPN
ejpam-5948	45	8	,	,	PUNCT
ejpam-5948	45	9	z.z	z.z	PROPN
ejpam-5948	45	10	.	.	PUNCT
ejpam-5948	45	11	al	al	PROPN
ejpam-5948	45	12	-	-	PUNCT
ejpam-5948	45	13	amery	amery	PROPN
ejpam-5948	45	14	/	/	SYM
ejpam-5948	45	15	eur	eur	PROPN
ejpam-5948	45	16	.	.	PUNCT
ejpam-5948	46	1	j.	j.	PROPN
ejpam-5948	46	2	pure	pure	PROPN
ejpam-5948	46	3	appl	appl	PROPN
ejpam-5948	46	4	.	.	PROPN
ejpam-5948	46	5	math	math	PROPN
ejpam-5948	46	6	,	,	PUNCT
ejpam-5948	46	7	18	18	NUM
ejpam-5948	46	8	(	(	PUNCT
ejpam-5948	46	9	2	2	NUM
ejpam-5948	46	10	)	)	PUNCT
ejpam-5948	46	11	(	(	PUNCT
ejpam-5948	46	12	2025	2025	NUM
ejpam-5948	46	13	)	)	PUNCT
ejpam-5948	46	14	,	,	PUNCT
ejpam-5948	46	15	5948	5948	NUM
ejpam-5948	46	16	3	3	NUM
ejpam-5948	46	17	of	of	ADP
ejpam-5948	46	18	12	12	NUM
ejpam-5948	46	19	derivations	derivation	NOUN
ejpam-5948	46	20	(	(	PUNCT
ejpam-5948	46	21	℧	℧	PROPN
ejpam-5948	46	22	,	,	PUNCT
ejpam-5948	46	23	χ	χ	NOUN
ejpam-5948	46	24	)	)	PUNCT
ejpam-5948	46	25	and	and	CCONJ
ejpam-5948	46	26	(	(	PUNCT
ejpam-5948	46	27	⨿,∝	⨿,∝	X
ejpam-5948	46	28	)	)	PUNCT
ejpam-5948	46	29	that	that	PRON
ejpam-5948	46	30	satisfy	satisfy	VERB
ejpam-5948	46	31	any	any	PRON
ejpam-5948	46	32	of	of	ADP
ejpam-5948	46	33	the	the	DET
ejpam-5948	46	34	following	follow	VERB
ejpam-5948	46	35	identities	identity	NOUN
ejpam-5948	46	36	for	for	ADP
ejpam-5948	46	37	each	each	DET
ejpam-5948	46	38	υ	υ	NOUN
ejpam-5948	46	39	,	,	PUNCT
ejpam-5948	46	40	℘	℘	NOUN
ejpam-5948	46	41	∈	∈	PROPN
ejpam-5948	46	42	ℜ	ℜ	PROPN
ejpam-5948	46	43	:	:	PUNCT
ejpam-5948	46	44	(	(	PUNCT
ejpam-5948	46	45	i	i	NOUN
ejpam-5948	46	46	)	)	PUNCT
ejpam-5948	47	1	[	[	X
ejpam-5948	47	2	χ(υ	χ(υ	NOUN
ejpam-5948	47	3	)	)	PUNCT
ejpam-5948	47	4	,	,	PUNCT
ejpam-5948	47	5	χ(℘)]±[℘,⨿(υ	χ(℘)]±[℘,⨿(υ	PROPN
ejpam-5948	47	6	)	)	PUNCT
ejpam-5948	47	7	]	]	PUNCT
ejpam-5948	48	1	∈	∈	PROPN
ejpam-5948	48	2	p	p	X
ejpam-5948	48	3	,	,	PUNCT
ejpam-5948	48	4	(	(	PUNCT
ejpam-5948	48	5	ii	ii	NOUN
ejpam-5948	48	6	)	)	PUNCT
ejpam-5948	48	7	χ(υ)	χ(υ)	PROPN
ejpam-5948	48	8	◦	◦	NOUN
ejpam-5948	48	9	χ(℘)±[℘,⨿(υ	χ(℘)±[℘,⨿(υ	NUM
ejpam-5948	48	10	)	)	PUNCT
ejpam-5948	48	11	]	]	PUNCT
ejpam-5948	49	1	∈	∈	PROPN
ejpam-5948	49	2	p	p	X
ejpam-5948	49	3	,	,	PUNCT
ejpam-5948	49	4	(	(	PUNCT
ejpam-5948	49	5	iii	iii	NOUN
ejpam-5948	49	6	)	)	PUNCT
ejpam-5948	50	1	[	[	X
ejpam-5948	50	2	℧	℧	X
ejpam-5948	50	3	(	(	PUNCT
ejpam-5948	50	4	υ	υ	NOUN
ejpam-5948	50	5	)	)	PUNCT
ejpam-5948	50	6	,	,	PUNCT
ejpam-5948	50	7	χ(℘)]±℘	χ(℘)]±℘	PROPN
ejpam-5948	50	8	◦	◦	NOUN
ejpam-5948	50	9	⨿(υ	⨿(υ	NUM
ejpam-5948	50	10	)	)	PUNCT
ejpam-5948	51	1	∈	∈	PROPN
ejpam-5948	51	2	p	p	NOUN
ejpam-5948	51	3	,	,	PUNCT
ejpam-5948	51	4	(	(	PUNCT
ejpam-5948	51	5	iv	iv	X
ejpam-5948	51	6	)	)	PUNCT
ejpam-5948	52	1	[	[	X
ejpam-5948	52	2	υ,⨿(℘	υ,⨿(℘	NOUN
ejpam-5948	52	3	)	)	PUNCT
ejpam-5948	52	4	]	]	PUNCT
ejpam-5948	52	5	±	±	NUM
ejpam-5948	52	6	℧	℧	PROPN
ejpam-5948	52	7	(	(	PUNCT
ejpam-5948	52	8	[	[	X
ejpam-5948	52	9	υ	υ	INTJ
ejpam-5948	52	10	,	,	PUNCT
ejpam-5948	52	11	℘	℘	PROPN
ejpam-5948	52	12	]	]	PUNCT
ejpam-5948	52	13	)	)	PUNCT
ejpam-5948	53	1	∈	∈	PROPN
ejpam-5948	53	2	p	p	NOUN
ejpam-5948	53	3	,	,	PUNCT
ejpam-5948	53	4	(	(	PUNCT
ejpam-5948	53	5	v	v	NOUN
ejpam-5948	53	6	)	)	PUNCT
ejpam-5948	53	7	℧	℧	NOUN
ejpam-5948	53	8	(	(	PUNCT
ejpam-5948	53	9	[	[	X
ejpam-5948	53	10	υ	υ	INTJ
ejpam-5948	53	11	,	,	PUNCT
ejpam-5948	53	12	℘	℘	PROPN
ejpam-5948	53	13	]	]	SYM
ejpam-5948	53	14	)	)	PUNCT
ejpam-5948	53	15	±	±	NUM
ejpam-5948	53	16	℧	℧	PROPN
ejpam-5948	53	17	(	(	PUNCT
ejpam-5948	53	18	℘	℘	PROPN
ejpam-5948	53	19	)	)	PUNCT
ejpam-5948	53	20	⨿	⨿	NOUN
ejpam-5948	53	21	(	(	PUNCT
ejpam-5948	53	22	υ	υ	NOUN
ejpam-5948	53	23	)	)	PUNCT
ejpam-5948	53	24	∈	∈	PROPN
ejpam-5948	53	25	p	p	NOUN
ejpam-5948	53	26	,	,	PUNCT
ejpam-5948	53	27	(	(	PUNCT
ejpam-5948	53	28	vi	vi	NOUN
ejpam-5948	53	29	)	)	PUNCT
ejpam-5948	53	30	℧	℧	NOUN
ejpam-5948	53	31	(	(	PUNCT
ejpam-5948	53	32	[	[	X
ejpam-5948	53	33	υ	υ	INTJ
ejpam-5948	53	34	,	,	PUNCT
ejpam-5948	53	35	℘	℘	PROPN
ejpam-5948	53	36	]	]	SYM
ejpam-5948	53	37	)	)	PUNCT
ejpam-5948	53	38	±	±	NUM
ejpam-5948	53	39	℧	℧	PROPN
ejpam-5948	53	40	(	(	PUNCT
ejpam-5948	53	41	υ	υ	NOUN
ejpam-5948	53	42	)	)	PUNCT
ejpam-5948	53	43	⨿	⨿	NOUN
ejpam-5948	53	44	(	(	PUNCT
ejpam-5948	53	45	℘	℘	PROPN
ejpam-5948	53	46	)	)	PUNCT
ejpam-5948	53	47	∈	∈	PROPN
ejpam-5948	53	48	p	p	NOUN
ejpam-5948	53	49	,	,	PUNCT
ejpam-5948	53	50	(	(	PUNCT
ejpam-5948	53	51	vii	vii	PROPN
ejpam-5948	53	52	)	)	PUNCT
ejpam-5948	54	1	[	[	X
ejpam-5948	54	2	υ,	υ,	X
ejpam-5948	54	3	℧	℧	NOUN
ejpam-5948	54	4	(℘	(℘	NUM
ejpam-5948	54	5	)	)	PUNCT
ejpam-5948	54	6	]	]	PUNCT
ejpam-5948	54	7	±	±	NUM
ejpam-5948	54	8	℧	℧	PROPN
ejpam-5948	54	9	(	(	PUNCT
ejpam-5948	54	10	℘	℘	PROPN
ejpam-5948	54	11	)	)	PUNCT
ejpam-5948	54	12	⨿	⨿	NOUN
ejpam-5948	54	13	(	(	PUNCT
ejpam-5948	54	14	υ	υ	NOUN
ejpam-5948	54	15	)	)	PUNCT
ejpam-5948	54	16	∈	∈	PROPN
ejpam-5948	54	17	p	p	NOUN
ejpam-5948	54	18	.	.	PUNCT
ejpam-5948	55	1	furthermore	furthermore	ADV
ejpam-5948	55	2	,	,	PUNCT
ejpam-5948	55	3	we	we	PRON
ejpam-5948	55	4	will	will	AUX
ejpam-5948	55	5	present	present	VERB
ejpam-5948	55	6	several	several	ADJ
ejpam-5948	55	7	related	related	ADJ
ejpam-5948	55	8	consequences	consequence	NOUN
ejpam-5948	55	9	and	and	CCONJ
ejpam-5948	55	10	provide	provide	VERB
ejpam-5948	55	11	examples	example	NOUN
ejpam-5948	55	12	to	to	PART
ejpam-5948	55	13	illustrate	illustrate	VERB
ejpam-5948	55	14	the	the	DET
ejpam-5948	55	15	significance	significance	NOUN
ejpam-5948	55	16	of	of	ADP
ejpam-5948	55	17	the	the	DET
ejpam-5948	55	18	assumptions	assumption	NOUN
ejpam-5948	55	19	in	in	ADP
ejpam-5948	55	20	our	our	PRON
ejpam-5948	55	21	theorems	theorem	NOUN
ejpam-5948	55	22	.	.	PUNCT
ejpam-5948	56	1	2	2	X
ejpam-5948	56	2	.	.	NUM
ejpam-5948	56	3	preliminaries	preliminary	NOUN
ejpam-5948	56	4	in	in	ADP
ejpam-5948	56	5	this	this	DET
ejpam-5948	56	6	section	section	NOUN
ejpam-5948	56	7	,	,	PUNCT
ejpam-5948	56	8	we	we	PRON
ejpam-5948	56	9	will	will	AUX
ejpam-5948	56	10	exhibit	exhibit	VERB
ejpam-5948	56	11	some	some	DET
ejpam-5948	56	12	important	important	ADJ
ejpam-5948	56	13	preliminaries	preliminary	NOUN
ejpam-5948	56	14	that	that	PRON
ejpam-5948	56	15	will	will	AUX
ejpam-5948	56	16	be	be	AUX
ejpam-5948	56	17	used	use	VERB
ejpam-5948	56	18	repeatedly	repeatedly	ADV
ejpam-5948	56	19	to	to	PART
ejpam-5948	56	20	develop	develop	VERB
ejpam-5948	56	21	proofs	proof	NOUN
ejpam-5948	56	22	of	of	ADP
ejpam-5948	56	23	our	our	PRON
ejpam-5948	56	24	main	main	ADJ
ejpam-5948	56	25	theorems	theorem	NOUN
ejpam-5948	56	26	.	.	PUNCT
ejpam-5948	57	1	lemma	lemma	PROPN
ejpam-5948	57	2	1	1	NUM
ejpam-5948	57	3	.	.	PUNCT
ejpam-5948	58	1	[	[	X
ejpam-5948	58	2	12	12	NUM
ejpam-5948	58	3	,	,	PUNCT
ejpam-5948	58	4	lemma	lemma	PROPN
ejpam-5948	58	5	2.4	2.4	NUM
ejpam-5948	58	6	]	]	PUNCT
ejpam-5948	58	7	let	let	VERB
ejpam-5948	58	8	ℜ	ℜ	PROPN
ejpam-5948	58	9	be	be	AUX
ejpam-5948	58	10	a	a	DET
ejpam-5948	58	11	ring	ring	NOUN
ejpam-5948	58	12	with	with	ADP
ejpam-5948	58	13	a	a	DET
ejpam-5948	58	14	semi	semi	ADJ
ejpam-5948	58	15	-	-	ADJ
ejpam-5948	58	16	prime	prime	ADJ
ejpam-5948	58	17	ideal	ideal	NOUN
ejpam-5948	58	18	p	p	NOUN
ejpam-5948	58	19	,	,	PUNCT
ejpam-5948	58	20	and	and	CCONJ
ejpam-5948	58	21	let	let	VERB
ejpam-5948	58	22	ℜ/p	ℜ/p	PROPN
ejpam-5948	58	23	be	be	AUX
ejpam-5948	58	24	2	2	NUM
ejpam-5948	58	25	-	-	PUNCT
ejpam-5948	58	26	torsion	torsion	NOUN
ejpam-5948	58	27	free	free	ADJ
ejpam-5948	58	28	.	.	PUNCT
ejpam-5948	59	1	if	if	SCONJ
ejpam-5948	59	2	χ	χ	NOUN
ejpam-5948	59	3	is	be	AUX
ejpam-5948	59	4	a	a	DET
ejpam-5948	59	5	derivation	derivation	NOUN
ejpam-5948	59	6	on	on	ADP
ejpam-5948	59	7	ℜ	ℜ	PROPN
ejpam-5948	59	8	such	such	ADJ
ejpam-5948	59	9	that	that	SCONJ
ejpam-5948	59	10	[	[	X
ejpam-5948	59	11	χ2(υ	χ2(υ	NOUN
ejpam-5948	59	12	)	)	PUNCT
ejpam-5948	59	13	,	,	PUNCT
ejpam-5948	59	14	υ	υ	X
ejpam-5948	59	15	]	]	X
ejpam-5948	59	16	∈	∈	PROPN
ejpam-5948	59	17	p	p	NOUN
ejpam-5948	59	18	,	,	PUNCT
ejpam-5948	59	19	then	then	ADV
ejpam-5948	59	20	χ	χ	X
ejpam-5948	59	21	is	be	AUX
ejpam-5948	59	22	p	p	NOUN
ejpam-5948	59	23	-commuting	-commute	VERB
ejpam-5948	59	24	on	on	ADP
ejpam-5948	59	25	ℜ.	ℜ.	PROPN
ejpam-5948	59	26	lemma	lemma	PROPN
ejpam-5948	59	27	2	2	X
ejpam-5948	59	28	.	.	PUNCT
ejpam-5948	60	1	[	[	X
ejpam-5948	60	2	1	1	NUM
ejpam-5948	60	3	,	,	PUNCT
ejpam-5948	60	4	lemma	lemma	PROPN
ejpam-5948	60	5	10	10	NUM
ejpam-5948	60	6	]	]	PUNCT
ejpam-5948	60	7	let	let	VERB
ejpam-5948	60	8	ℜ	ℜ	PROPN
ejpam-5948	60	9	be	be	AUX
ejpam-5948	60	10	a	a	DET
ejpam-5948	60	11	ring	ring	NOUN
ejpam-5948	60	12	that	that	PRON
ejpam-5948	60	13	admits	admit	VERB
ejpam-5948	60	14	a	a	DET
ejpam-5948	60	15	generalized	generalized	ADJ
ejpam-5948	60	16	p	p	NOUN
ejpam-5948	60	17	-derivation	-derivation	NOUN
ejpam-5948	60	18	℧	℧	PROPN
ejpam-5948	60	19	associated	associate	VERB
ejpam-5948	60	20	with	with	ADP
ejpam-5948	60	21	a	a	DET
ejpam-5948	60	22	p	p	NOUN
ejpam-5948	60	23	-derivation	-derivation	NOUN
ejpam-5948	60	24	χ	χ	NOUN
ejpam-5948	60	25	,	,	PUNCT
ejpam-5948	60	26	where	where	SCONJ
ejpam-5948	60	27	p	p	NOUN
ejpam-5948	60	28	is	be	AUX
ejpam-5948	60	29	a	a	DET
ejpam-5948	60	30	prime	prime	ADJ
ejpam-5948	60	31	ideal	ideal	NOUN
ejpam-5948	60	32	of	of	ADP
ejpam-5948	60	33	ℜ.	ℜ.	PROPN
ejpam-5948	60	34	(	(	PUNCT
ejpam-5948	60	35	i	i	NOUN
ejpam-5948	60	36	)	)	PUNCT
ejpam-5948	60	37	if	if	SCONJ
ejpam-5948	60	38	[	[	X
ejpam-5948	60	39	υ,	υ,	X
ejpam-5948	60	40	℧	℧	NOUN
ejpam-5948	60	41	(℘	(℘	NUM
ejpam-5948	60	42	)	)	PUNCT
ejpam-5948	60	43	]	]	PUNCT
ejpam-5948	61	1	∈	∈	PROPN
ejpam-5948	61	2	p	p	NOUN
ejpam-5948	61	3	satisfies	satisfie	NOUN
ejpam-5948	61	4	for	for	ADP
ejpam-5948	61	5	every	every	DET
ejpam-5948	61	6	elements	element	NOUN
ejpam-5948	61	7	υ	υ	NOUN
ejpam-5948	61	8	,	,	PUNCT
ejpam-5948	61	9	℘	℘	PROPN
ejpam-5948	61	10	∈	∈	PROPN
ejpam-5948	61	11	ℜ	ℜ	PROPN
ejpam-5948	61	12	,	,	PUNCT
ejpam-5948	61	13	then	then	ADV
ejpam-5948	61	14	ℜ/p	ℜ/p	PROPN
ejpam-5948	61	15	is	be	AUX
ejpam-5948	61	16	an	an	DET
ejpam-5948	61	17	integral	integral	ADJ
ejpam-5948	61	18	domain	domain	NOUN
ejpam-5948	61	19	or	or	CCONJ
ejpam-5948	61	20	℧	℧	PROPN
ejpam-5948	61	21	(	(	PUNCT
ejpam-5948	61	22	ℜ	ℜ	PROPN
ejpam-5948	61	23	)	)	PUNCT
ejpam-5948	61	24	⊂	⊂	PROPN
ejpam-5948	61	25	p	p	X
ejpam-5948	61	26	.	.	PUNCT
ejpam-5948	62	1	(	(	PUNCT
ejpam-5948	62	2	ii	ii	NOUN
ejpam-5948	62	3	)	)	PUNCT
ejpam-5948	62	4	if	if	SCONJ
ejpam-5948	62	5	υ	υ	PRON
ejpam-5948	62	6	◦	◦	NOUN
ejpam-5948	62	7	℧	℧	PROPN
ejpam-5948	62	8	(	(	PUNCT
ejpam-5948	62	9	℘	℘	PROPN
ejpam-5948	62	10	)	)	PUNCT
ejpam-5948	62	11	∈	∈	NOUN
ejpam-5948	62	12	p	p	NOUN
ejpam-5948	62	13	satisfies	satisfie	NOUN
ejpam-5948	62	14	for	for	ADP
ejpam-5948	62	15	every	every	DET
ejpam-5948	62	16	elements	element	NOUN
ejpam-5948	62	17	υ	υ	NOUN
ejpam-5948	62	18	,	,	PUNCT
ejpam-5948	62	19	℘	℘	PROPN
ejpam-5948	62	20	∈	∈	PROPN
ejpam-5948	62	21	ℜ	ℜ	PROPN
ejpam-5948	62	22	,	,	PUNCT
ejpam-5948	62	23	then	then	ADV
ejpam-5948	62	24	ℜ/p	ℜ/p	PROPN
ejpam-5948	62	25	is	be	AUX
ejpam-5948	62	26	an	an	DET
ejpam-5948	62	27	integral	integral	ADJ
ejpam-5948	62	28	domain	domain	NOUN
ejpam-5948	62	29	with	with	ADP
ejpam-5948	62	30	char(ℜ/p	char(ℜ/p	PROPN
ejpam-5948	62	31	)	)	PUNCT
ejpam-5948	63	1	=	=	SYM
ejpam-5948	63	2	2	2	NUM
ejpam-5948	63	3	or	or	CCONJ
ejpam-5948	63	4	℧	℧	PROPN
ejpam-5948	63	5	(	(	PUNCT
ejpam-5948	63	6	ℜ	ℜ	PROPN
ejpam-5948	63	7	)	)	PUNCT
ejpam-5948	64	1	⊂	⊂	PROPN
ejpam-5948	64	2	p	p	X
ejpam-5948	64	3	.	.	PUNCT
ejpam-5948	65	1	lemma	lemma	PROPN
ejpam-5948	65	2	3	3	X
ejpam-5948	65	3	.	.	PUNCT
ejpam-5948	65	4	consider	consider	VERB
ejpam-5948	65	5	a	a	DET
ejpam-5948	65	6	prime	prime	ADJ
ejpam-5948	65	7	ideal	ideal	NOUN
ejpam-5948	65	8	p	p	NOUN
ejpam-5948	65	9	of	of	ADP
ejpam-5948	65	10	an	an	DET
ejpam-5948	65	11	arbitrary	arbitrary	ADJ
ejpam-5948	65	12	ring	ring	NOUN
ejpam-5948	65	13	ℜ.	ℜ.	PROPN
ejpam-5948	65	14	if	if	SCONJ
ejpam-5948	65	15	ℜ	ℜ	PROPN
ejpam-5948	65	16	admits	admit	VERB
ejpam-5948	65	17	a	a	DET
ejpam-5948	65	18	generalized	generalized	ADJ
ejpam-5948	65	19	p	p	NOUN
ejpam-5948	65	20	-derivation	-derivation	NOUN
ejpam-5948	65	21	℧	℧	PROPN
ejpam-5948	65	22	associated	associate	VERB
ejpam-5948	65	23	with	with	ADP
ejpam-5948	65	24	a	a	DET
ejpam-5948	65	25	p	p	NOUN
ejpam-5948	65	26	-derivation	-derivation	NOUN
ejpam-5948	65	27	χ	χ	NOUN
ejpam-5948	66	1	such	such	ADJ
ejpam-5948	66	2	that	that	SCONJ
ejpam-5948	66	3	[	[	X
ejpam-5948	66	4	υ,	υ,	X
ejpam-5948	66	5	℧	℧	NOUN
ejpam-5948	66	6	(υ	(υ	NUM
ejpam-5948	66	7	)	)	PUNCT
ejpam-5948	66	8	]	]	PUNCT
ejpam-5948	67	1	∈	∈	PROPN
ejpam-5948	67	2	p	p	NOUN
ejpam-5948	67	3	for	for	ADP
ejpam-5948	67	4	all	all	PRON
ejpam-5948	67	5	υ	υ	PRON
ejpam-5948	67	6	∈	∈	PROPN
ejpam-5948	67	7	ℜ	ℜ	PROPN
ejpam-5948	67	8	,	,	PUNCT
ejpam-5948	67	9	then	then	ADV
ejpam-5948	67	10	either	either	CCONJ
ejpam-5948	67	11	ℜ/p	ℜ/p	PROPN
ejpam-5948	67	12	is	be	AUX
ejpam-5948	67	13	an	an	DET
ejpam-5948	67	14	integral	integral	ADJ
ejpam-5948	67	15	domain	domain	NOUN
ejpam-5948	67	16	or	or	CCONJ
ejpam-5948	67	17	χ(ℜ	χ(ℜ	NOUN
ejpam-5948	67	18	)	)	PUNCT
ejpam-5948	67	19	⊆	⊆	NUM
ejpam-5948	67	20	p	p	NOUN
ejpam-5948	67	21	.	.	PUNCT
ejpam-5948	68	1	proof	proof	NOUN
ejpam-5948	68	2	.	.	PUNCT
ejpam-5948	69	1	the	the	DET
ejpam-5948	69	2	proof	proof	NOUN
ejpam-5948	69	3	can	can	AUX
ejpam-5948	69	4	be	be	AUX
ejpam-5948	69	5	easily	easily	ADV
ejpam-5948	69	6	derived	derive	VERB
ejpam-5948	69	7	from	from	ADP
ejpam-5948	69	8	lemma	lemma	PROPN
ejpam-5948	69	9	2	2	PROPN
ejpam-5948	69	10	(	(	PUNCT
ejpam-5948	69	11	i	i	NOUN
ejpam-5948	69	12	)	)	PUNCT
ejpam-5948	69	13	,	,	PUNCT
ejpam-5948	69	14	so	so	SCONJ
ejpam-5948	69	15	it	it	PRON
ejpam-5948	69	16	may	may	AUX
ejpam-5948	69	17	be	be	AUX
ejpam-5948	69	18	skipped	skip	VERB
ejpam-5948	69	19	as	as	SCONJ
ejpam-5948	69	20	it	it	PRON
ejpam-5948	69	21	would	would	AUX
ejpam-5948	69	22	not	not	PART
ejpam-5948	69	23	result	result	VERB
ejpam-5948	69	24	in	in	ADP
ejpam-5948	69	25	any	any	DET
ejpam-5948	69	26	significant	significant	ADJ
ejpam-5948	69	27	changes	change	NOUN
ejpam-5948	69	28	.	.	PUNCT
ejpam-5948	70	1	the	the	DET
ejpam-5948	70	2	following	follow	VERB
ejpam-5948	70	3	corollary	corollary	NOUN
ejpam-5948	70	4	is	be	AUX
ejpam-5948	70	5	a	a	DET
ejpam-5948	70	6	special	special	ADJ
ejpam-5948	70	7	case	case	NOUN
ejpam-5948	70	8	of	of	ADP
ejpam-5948	70	9	the	the	DET
ejpam-5948	70	10	previous	previous	ADJ
ejpam-5948	70	11	lemma	lemma	PROPN
ejpam-5948	70	12	when	when	SCONJ
ejpam-5948	70	13	℧	℧	PROPN
ejpam-5948	70	14	=	=	SYM
ejpam-5948	70	15	χ	χ	X
ejpam-5948	70	16	.	.	PUNCT
ejpam-5948	71	1	corollary	corollary	ADJ
ejpam-5948	71	2	1	1	NUM
ejpam-5948	71	3	.	.	PUNCT
ejpam-5948	72	1	[	[	X
ejpam-5948	72	2	1	1	NUM
ejpam-5948	72	3	,	,	PUNCT
ejpam-5948	72	4	lemma	lemma	PROPN
ejpam-5948	72	5	1	1	NUM
ejpam-5948	72	6	]	]	PUNCT
ejpam-5948	72	7	consider	consider	VERB
ejpam-5948	72	8	a	a	DET
ejpam-5948	72	9	prime	prime	ADJ
ejpam-5948	72	10	ideal	ideal	NOUN
ejpam-5948	72	11	p	p	NOUN
ejpam-5948	72	12	of	of	ADP
ejpam-5948	72	13	an	an	DET
ejpam-5948	72	14	arbitrary	arbitrary	ADJ
ejpam-5948	72	15	ring	ring	NOUN
ejpam-5948	72	16	ℜ.	ℜ.	PROPN
ejpam-5948	72	17	if	if	SCONJ
ejpam-5948	72	18	ℜ	ℜ	PROPN
ejpam-5948	72	19	admits	admit	VERB
ejpam-5948	72	20	a	a	DET
ejpam-5948	72	21	p	p	NOUN
ejpam-5948	72	22	-derivation	-derivation	NOUN
ejpam-5948	72	23	χ	χ	NOUN
ejpam-5948	72	24	such	such	ADJ
ejpam-5948	72	25	that	that	SCONJ
ejpam-5948	72	26	[	[	X
ejpam-5948	72	27	υ	υ	X
ejpam-5948	72	28	,	,	PUNCT
ejpam-5948	72	29	χ(υ	χ(υ	PROPN
ejpam-5948	72	30	)	)	PUNCT
ejpam-5948	72	31	]	]	PUNCT
ejpam-5948	73	1	∈	∈	PROPN
ejpam-5948	73	2	p	p	NOUN
ejpam-5948	73	3	for	for	ADP
ejpam-5948	73	4	all	all	PRON
ejpam-5948	73	5	υ	υ	PRON
ejpam-5948	73	6	∈	∈	PROPN
ejpam-5948	73	7	ℜ	ℜ	PROPN
ejpam-5948	73	8	,	,	PUNCT
ejpam-5948	73	9	then	then	ADV
ejpam-5948	73	10	either	either	CCONJ
ejpam-5948	73	11	ℜ/p	ℜ/p	PROPN
ejpam-5948	73	12	is	be	AUX
ejpam-5948	73	13	an	an	DET
ejpam-5948	73	14	integral	integral	ADJ
ejpam-5948	73	15	domain	domain	NOUN
ejpam-5948	73	16	or	or	CCONJ
ejpam-5948	73	17	χ(ℜ	χ(ℜ	NOUN
ejpam-5948	73	18	)	)	PUNCT
ejpam-5948	73	19	⊆	⊆	NUM
ejpam-5948	73	20	p	p	NOUN
ejpam-5948	73	21	.	.	PUNCT
ejpam-5948	74	1	3	3	X
ejpam-5948	74	2	.	.	X
ejpam-5948	74	3	main	main	ADJ
ejpam-5948	74	4	results	result	NOUN
ejpam-5948	74	5	for	for	ADP
ejpam-5948	74	6	brevity	brevity	NOUN
ejpam-5948	74	7	,	,	PUNCT
ejpam-5948	74	8	let	let	VERB
ejpam-5948	74	9	(	(	PUNCT
ejpam-5948	74	10	℧	℧	PROPN
ejpam-5948	74	11	,	,	PUNCT
ejpam-5948	74	12	χ	χ	NOUN
ejpam-5948	74	13	)	)	PUNCT
ejpam-5948	74	14	and	and	CCONJ
ejpam-5948	74	15	(	(	PUNCT
ejpam-5948	74	16	⨿,∝	⨿,∝	X
ejpam-5948	74	17	)	)	PUNCT
ejpam-5948	74	18	symbolize	symbolize	NOUN
ejpam-5948	74	19	two	two	NUM
ejpam-5948	74	20	generalized	generalized	ADJ
ejpam-5948	74	21	p	p	NOUN
ejpam-5948	74	22	-derivations	-derivation	NOUN
ejpam-5948	74	23	associated	associate	VERB
ejpam-5948	74	24	with	with	ADP
ejpam-5948	74	25	p	p	PROPN
ejpam-5948	74	26	-derivations	-derivation	NOUN
ejpam-5948	74	27	χ	χ	NOUN
ejpam-5948	74	28	and	and	CCONJ
ejpam-5948	74	29	∝	∝	PROPN
ejpam-5948	74	30	,	,	PUNCT
ejpam-5948	74	31	respectively	respectively	ADV
ejpam-5948	74	32	.	.	PUNCT
ejpam-5948	75	1	the	the	DET
ejpam-5948	75	2	symbol	symbol	NOUN
ejpam-5948	75	3	idℜ	idℜ	PROPN
ejpam-5948	75	4	designates	designate	VERB
ejpam-5948	75	5	the	the	DET
ejpam-5948	75	6	identity	identity	NOUN
ejpam-5948	75	7	map	map	NOUN
ejpam-5948	75	8	idℜ	idℜ	NOUN
ejpam-5948	75	9	:	:	PUNCT
ejpam-5948	75	10	ℜ	ℜ	ADJ
ejpam-5948	75	11	−→	−→	NOUN
ejpam-5948	75	12	ℜ	ℜ	PROPN
ejpam-5948	75	13	defined	define	VERB
ejpam-5948	75	14	by	by	ADP
ejpam-5948	75	15	idℜ(υ	idℜ(υ	NOUN
ejpam-5948	75	16	)	)	PUNCT
ejpam-5948	75	17	=	=	SYM
ejpam-5948	75	18	υ	υ	PROPN
ejpam-5948	75	19	for	for	ADP
ejpam-5948	75	20	all	all	PRON
ejpam-5948	75	21	υ	υ	DET
ejpam-5948	75	22	∈	∈	NOUN
ejpam-5948	75	23	ℜ.	ℜ.	PROPN
ejpam-5948	75	24	in	in	ADP
ejpam-5948	75	25	their	their	PRON
ejpam-5948	75	26	work	work	NOUN
ejpam-5948	75	27	,	,	PUNCT
ejpam-5948	75	28	sandhu	sandhu	PROPN
ejpam-5948	75	29	et	et	PROPN
ejpam-5948	75	30	al	al	PROPN
ejpam-5948	75	31	.	.	PUNCT
ejpam-5948	76	1	[	[	X
ejpam-5948	76	2	1	1	NUM
ejpam-5948	76	3	,	,	PUNCT
ejpam-5948	76	4	lemma	lemma	PROPN
ejpam-5948	76	5	10	10	NUM
ejpam-5948	76	6	]	]	PUNCT
ejpam-5948	76	7	discussed	discuss	VERB
ejpam-5948	76	8	the	the	DET
ejpam-5948	76	9	behavior	behavior	NOUN
ejpam-5948	76	10	of	of	ADP
ejpam-5948	76	11	a	a	DET
ejpam-5948	76	12	factor	factor	NOUN
ejpam-5948	76	13	ring	ring	NOUN
ejpam-5948	76	14	ℜ/p	ℜ/p	PROPN
ejpam-5948	76	15	under	under	ADP
ejpam-5948	76	16	the	the	DET
ejpam-5948	76	17	influence	influence	NOUN
ejpam-5948	76	18	of	of	ADP
ejpam-5948	76	19	a	a	DET
ejpam-5948	76	20	generalized	generalized	ADJ
ejpam-5948	76	21	p	p	NOUN
ejpam-5948	76	22	derivation	derivation	NOUN
ejpam-5948	76	23	satisfying	satisfy	VERB
ejpam-5948	76	24	certain	certain	ADJ
ejpam-5948	76	25	algebraic	algebraic	ADJ
ejpam-5948	76	26	identities	identity	NOUN
ejpam-5948	76	27	involving	involve	VERB
ejpam-5948	76	28	a	a	DET
ejpam-5948	76	29	prime	prime	ADJ
ejpam-5948	76	30	ideal	ideal	NOUN
ejpam-5948	76	31	p	p	NOUN
ejpam-5948	76	32	of	of	ADP
ejpam-5948	76	33	any	any	DET
ejpam-5948	76	34	ring	ring	NOUN
ejpam-5948	76	35	ℜ.	ℜ.	ADJ
ejpam-5948	76	36	in	in	ADP
ejpam-5948	76	37	the	the	DET
ejpam-5948	76	38	following	follow	VERB
ejpam-5948	76	39	three	three	NUM
ejpam-5948	76	40	theorems	theorem	NOUN
ejpam-5948	76	41	,	,	PUNCT
ejpam-5948	76	42	we	we	PRON
ejpam-5948	76	43	will	will	AUX
ejpam-5948	76	44	expand	expand	VERB
ejpam-5948	76	45	upon	upon	SCONJ
ejpam-5948	76	46	those	those	DET
ejpam-5948	76	47	results	result	NOUN
ejpam-5948	76	48	under	under	ADP
ejpam-5948	76	49	the	the	DET
ejpam-5948	76	50	influence	influence	NOUN
ejpam-5948	76	51	of	of	ADP
ejpam-5948	76	52	a	a	DET
ejpam-5948	76	53	pair	pair	NOUN
ejpam-5948	76	54	of	of	ADP
ejpam-5948	76	55	generalized	generalized	ADJ
ejpam-5948	76	56	p	p	NOUN
ejpam-5948	76	57	-derivations	-derivation	NOUN
ejpam-5948	76	58	alternating	alternate	VERB
ejpam-5948	76	59	between	between	ADP
ejpam-5948	76	60	commutator	commutator	NOUN
ejpam-5948	76	61	and	and	CCONJ
ejpam-5948	76	62	anticommutator	anticommutator	NOUN
ejpam-5948	76	63	.	.	PUNCT
ejpam-5948	77	1	a.y	a.y	PROPN
ejpam-5948	77	2	.	.	PROPN
ejpam-5948	77	3	hummdi	hummdi	PROPN
ejpam-5948	77	4	,	,	PUNCT
ejpam-5948	77	5	r.m	r.m	PROPN
ejpam-5948	77	6	.	.	PROPN
ejpam-5948	77	7	al	al	PROPN
ejpam-5948	77	8	-	-	PUNCT
ejpam-5948	77	9	omary	omary	PROPN
ejpam-5948	77	10	,	,	PUNCT
ejpam-5948	77	11	z.z	z.z	PROPN
ejpam-5948	77	12	.	.	PUNCT
ejpam-5948	77	13	al	al	PROPN
ejpam-5948	77	14	-	-	PUNCT
ejpam-5948	77	15	amery	amery	PROPN
ejpam-5948	77	16	/	/	SYM
ejpam-5948	77	17	eur	eur	PROPN
ejpam-5948	77	18	.	.	PUNCT
ejpam-5948	78	1	j.	j.	PROPN
ejpam-5948	78	2	pure	pure	PROPN
ejpam-5948	78	3	appl	appl	PROPN
ejpam-5948	78	4	.	.	PROPN
ejpam-5948	78	5	math	math	PROPN
ejpam-5948	78	6	,	,	PUNCT
ejpam-5948	78	7	18	18	NUM
ejpam-5948	78	8	(	(	PUNCT
ejpam-5948	78	9	2	2	NUM
ejpam-5948	78	10	)	)	PUNCT
ejpam-5948	78	11	(	(	PUNCT
ejpam-5948	78	12	2025	2025	NUM
ejpam-5948	78	13	)	)	PUNCT
ejpam-5948	78	14	,	,	PUNCT
ejpam-5948	78	15	5948	5948	NUM
ejpam-5948	78	16	4	4	NUM
ejpam-5948	78	17	of	of	ADP
ejpam-5948	78	18	12	12	NUM
ejpam-5948	78	19	theorem	theorem	NOUN
ejpam-5948	78	20	1	1	NUM
ejpam-5948	78	21	.	.	PUNCT
ejpam-5948	79	1	let	let	VERB
ejpam-5948	79	2	ℜ	ℜ	PROPN
ejpam-5948	79	3	be	be	AUX
ejpam-5948	79	4	a	a	DET
ejpam-5948	79	5	ring	ring	NOUN
ejpam-5948	79	6	equipped	equip	VERB
ejpam-5948	79	7	with	with	ADP
ejpam-5948	79	8	a	a	DET
ejpam-5948	79	9	p	p	NOUN
ejpam-5948	79	10	-derivation	-derivation	NOUN
ejpam-5948	79	11	χ	χ	NOUN
ejpam-5948	79	12	and	and	CCONJ
ejpam-5948	79	13	a	a	DET
ejpam-5948	79	14	generalized	generalized	ADJ
ejpam-5948	79	15	p	p	NOUN
ejpam-5948	79	16	-derivation	-derivation	NOUN
ejpam-5948	79	17	(	(	PUNCT
ejpam-5948	79	18	⨿,∝	⨿,∝	NOUN
ejpam-5948	79	19	)	)	PUNCT
ejpam-5948	79	20	such	such	ADJ
ejpam-5948	79	21	that	that	DET
ejpam-5948	79	22	char(ℜ/p	char(ℜ/p	NOUN
ejpam-5948	79	23	)	)	PUNCT
ejpam-5948	80	1	̸=	̸=	PROPN
ejpam-5948	80	2	2	2	NUM
ejpam-5948	80	3	.	.	PUNCT
ejpam-5948	81	1	then	then	ADV
ejpam-5948	81	2	,	,	PUNCT
ejpam-5948	81	3	[	[	X
ejpam-5948	81	4	χ(υ	χ(υ	X
ejpam-5948	81	5	)	)	PUNCT
ejpam-5948	81	6	,	,	PUNCT
ejpam-5948	81	7	χ(℘	χ(℘	PROPN
ejpam-5948	81	8	)	)	PUNCT
ejpam-5948	81	9	]	]	PUNCT
ejpam-5948	81	10	±	±	NOUN
ejpam-5948	82	1	[	[	X
ejpam-5948	82	2	℘,⨿(υ	℘,⨿(υ	X
ejpam-5948	82	3	)	)	PUNCT
ejpam-5948	82	4	]	]	PUNCT
ejpam-5948	83	1	∈	∈	PROPN
ejpam-5948	83	2	p	p	NOUN
ejpam-5948	83	3	for	for	ADP
ejpam-5948	83	4	all	all	DET
ejpam-5948	83	5	υ	υ	NOUN
ejpam-5948	83	6	,	,	PUNCT
ejpam-5948	83	7	℘	℘	NOUN
ejpam-5948	83	8	∈	∈	NOUN
ejpam-5948	83	9	ℜ	ℜ	PROPN
ejpam-5948	83	10	if	if	SCONJ
ejpam-5948	83	11	and	and	CCONJ
ejpam-5948	83	12	only	only	ADV
ejpam-5948	83	13	if	if	SCONJ
ejpam-5948	83	14	one	one	NUM
ejpam-5948	83	15	of	of	ADP
ejpam-5948	83	16	the	the	DET
ejpam-5948	83	17	following	follow	VERB
ejpam-5948	83	18	is	be	AUX
ejpam-5948	83	19	true	true	ADJ
ejpam-5948	83	20	:	:	PUNCT
ejpam-5948	83	21	(	(	PUNCT
ejpam-5948	83	22	i	i	NOUN
ejpam-5948	83	23	)	)	PUNCT
ejpam-5948	84	1	ℜ/p	ℜ/p	PROPN
ejpam-5948	84	2	is	be	AUX
ejpam-5948	84	3	an	an	DET
ejpam-5948	84	4	integral	integral	ADJ
ejpam-5948	84	5	domain	domain	NOUN
ejpam-5948	84	6	;	;	PUNCT
ejpam-5948	84	7	(	(	PUNCT
ejpam-5948	84	8	ii	ii	NOUN
ejpam-5948	84	9	)	)	PUNCT
ejpam-5948	84	10	χ(ℜ	χ(ℜ	NOUN
ejpam-5948	84	11	)	)	PUNCT
ejpam-5948	84	12	⊆	⊆	NUM
ejpam-5948	84	13	p	p	NOUN
ejpam-5948	84	14	,	,	PUNCT
ejpam-5948	84	15	and	and	CCONJ
ejpam-5948	84	16	⨿(ℜ	⨿(ℜ	NOUN
ejpam-5948	84	17	)	)	PUNCT
ejpam-5948	84	18	⊆	⊆	NUM
ejpam-5948	84	19	p	p	NOUN
ejpam-5948	84	20	.	.	PUNCT
ejpam-5948	85	1	proof	proof	NOUN
ejpam-5948	85	2	.	.	PUNCT
ejpam-5948	86	1	according	accord	VERB
ejpam-5948	86	2	to	to	ADP
ejpam-5948	86	3	the	the	DET
ejpam-5948	86	4	given	give	VERB
ejpam-5948	86	5	hypothesis	hypothesis	NOUN
ejpam-5948	86	6	,	,	PUNCT
ejpam-5948	86	7	for	for	ADP
ejpam-5948	86	8	every	every	DET
ejpam-5948	86	9	υ	υ	NOUN
ejpam-5948	86	10	,	,	PUNCT
ejpam-5948	86	11	℘	℘	PROPN
ejpam-5948	86	12	∈	∈	PROPN
ejpam-5948	86	13	ℜ	ℜ	PROPN
ejpam-5948	86	14	,	,	PUNCT
ejpam-5948	86	15	we	we	PRON
ejpam-5948	86	16	have	have	VERB
ejpam-5948	86	17	[	[	X
ejpam-5948	86	18	χ(υ	χ(υ	NOUN
ejpam-5948	86	19	)	)	PUNCT
ejpam-5948	86	20	,	,	PUNCT
ejpam-5948	86	21	χ(℘)]±	χ(℘)]±	VERB
ejpam-5948	86	22	[	[	X
ejpam-5948	86	23	℘,⨿(υ	℘,⨿(υ	X
ejpam-5948	86	24	)	)	PUNCT
ejpam-5948	86	25	]	]	PUNCT
ejpam-5948	87	1	∈	∈	PROPN
ejpam-5948	87	2	p.	p.	NOUN
ejpam-5948	87	3	(	(	PUNCT
ejpam-5948	87	4	1	1	X
ejpam-5948	87	5	)	)	PUNCT
ejpam-5948	87	6	replacing	replace	VERB
ejpam-5948	87	7	℘	℘	NOUN
ejpam-5948	87	8	by	by	ADP
ejpam-5948	87	9	℘ℏ	℘ℏ	NOUN
ejpam-5948	87	10	in	in	ADP
ejpam-5948	87	11	equation	equation	NOUN
ejpam-5948	87	12	(	(	PUNCT
ejpam-5948	87	13	1	1	NUM
ejpam-5948	87	14	)	)	PUNCT
ejpam-5948	87	15	and	and	CCONJ
ejpam-5948	87	16	applying	apply	VERB
ejpam-5948	87	17	it	it	PRON
ejpam-5948	87	18	,	,	PUNCT
ejpam-5948	87	19	we	we	PRON
ejpam-5948	87	20	obtain	obtain	VERB
ejpam-5948	87	21	χ(℘)[χ(υ	χ(℘)[χ(υ	NOUN
ejpam-5948	87	22	)	)	PUNCT
ejpam-5948	87	23	,	,	PUNCT
ejpam-5948	87	24	ℏ	ℏ	X
ejpam-5948	87	25	]	]	PUNCT
ejpam-5948	88	1	+	+	CCONJ
ejpam-5948	89	1	[	[	X
ejpam-5948	89	2	χ(υ	χ(υ	NOUN
ejpam-5948	89	3	)	)	PUNCT
ejpam-5948	89	4	,	,	PUNCT
ejpam-5948	89	5	℘]χ(ℏ	℘]χ(ℏ	PROPN
ejpam-5948	89	6	)	)	PUNCT
ejpam-5948	89	7	∈	∈	PROPN
ejpam-5948	89	8	p	p	NOUN
ejpam-5948	89	9	for	for	ADP
ejpam-5948	89	10	all	all	DET
ejpam-5948	89	11	υ	υ	NOUN
ejpam-5948	89	12	,	,	PUNCT
ejpam-5948	89	13	℘	℘	PROPN
ejpam-5948	89	14	,	,	PUNCT
ejpam-5948	89	15	ℏ	ℏ	PROPN
ejpam-5948	89	16	∈	∈	NOUN
ejpam-5948	89	17	ℜ.	ℜ.	PROPN
ejpam-5948	89	18	(	(	PUNCT
ejpam-5948	89	19	2	2	X
ejpam-5948	89	20	)	)	PUNCT
ejpam-5948	89	21	taking	take	VERB
ejpam-5948	89	22	υ	υ	NOUN
ejpam-5948	89	23	=	=	PUNCT
ejpam-5948	89	24	℘	℘	X
ejpam-5948	89	25	=	=	SYM
ejpam-5948	89	26	ℏ	ℏ	PROPN
ejpam-5948	89	27	in	in	ADP
ejpam-5948	89	28	equation	equation	NOUN
ejpam-5948	89	29	(	(	PUNCT
ejpam-5948	89	30	2	2	NUM
ejpam-5948	89	31	)	)	PUNCT
ejpam-5948	89	32	,	,	PUNCT
ejpam-5948	89	33	we	we	PRON
ejpam-5948	89	34	obtain	obtain	VERB
ejpam-5948	89	35	χ(υ)[χ(υ	χ(υ)[χ(υ	NOUN
ejpam-5948	89	36	)	)	PUNCT
ejpam-5948	89	37	,	,	PUNCT
ejpam-5948	89	38	υ]+[χ(υ	υ]+[χ(υ	NUM
ejpam-5948	89	39	)	)	PUNCT
ejpam-5948	89	40	,	,	PUNCT
ejpam-5948	89	41	υ]χ(υ	υ]χ(υ	PROPN
ejpam-5948	89	42	)	)	PUNCT
ejpam-5948	89	43	=	=	SYM
ejpam-5948	89	44	χ2(υ)υ−χ(υ)υχ(υ)+χ(υ)υχ(υ)−υχ2(υ	χ2(υ)υ−χ(υ)υχ(υ)+χ(υ)υχ(υ)−υχ2(υ	NUM
ejpam-5948	89	45	)	)	PUNCT
ejpam-5948	89	46	∈	∈	PROPN
ejpam-5948	89	47	p	p	NOUN
ejpam-5948	89	48	for	for	ADP
ejpam-5948	89	49	all	all	PRON
ejpam-5948	89	50	υ	υ	DET
ejpam-5948	89	51	∈	∈	PROPN
ejpam-5948	89	52	ℜ.	ℜ.	PROPN
ejpam-5948	89	53	(	(	PUNCT
ejpam-5948	89	54	3	3	NUM
ejpam-5948	89	55	)	)	PUNCT
ejpam-5948	89	56	from	from	ADP
ejpam-5948	89	57	equation	equation	NOUN
ejpam-5948	89	58	(	(	PUNCT
ejpam-5948	89	59	3	3	NUM
ejpam-5948	89	60	)	)	PUNCT
ejpam-5948	89	61	,	,	PUNCT
ejpam-5948	89	62	we	we	PRON
ejpam-5948	89	63	conclude	conclude	VERB
ejpam-5948	89	64	[	[	X
ejpam-5948	89	65	χ2(υ	χ2(υ	NOUN
ejpam-5948	89	66	)	)	PUNCT
ejpam-5948	89	67	,	,	PUNCT
ejpam-5948	89	68	υ	υ	X
ejpam-5948	89	69	]	]	X
ejpam-5948	89	70	∈	∈	PROPN
ejpam-5948	89	71	p	p	NOUN
ejpam-5948	89	72	for	for	ADP
ejpam-5948	89	73	all	all	PRON
ejpam-5948	89	74	υ	υ	DET
ejpam-5948	89	75	∈	∈	NOUN
ejpam-5948	89	76	ℜ.	ℜ.	VERB
ejpam-5948	89	77	by	by	ADP
ejpam-5948	89	78	using	use	VERB
ejpam-5948	89	79	lemma	lemma	PROPN
ejpam-5948	89	80	1	1	NUM
ejpam-5948	89	81	,	,	PUNCT
ejpam-5948	89	82	we	we	PRON
ejpam-5948	89	83	obtain	obtain	VERB
ejpam-5948	89	84	[	[	X
ejpam-5948	89	85	χ(υ	χ(υ	NOUN
ejpam-5948	89	86	)	)	PUNCT
ejpam-5948	89	87	,	,	PUNCT
ejpam-5948	89	88	υ	υ	X
ejpam-5948	89	89	]	]	X
ejpam-5948	89	90	∈	∈	PROPN
ejpam-5948	89	91	p	p	NOUN
ejpam-5948	89	92	for	for	ADP
ejpam-5948	89	93	all	all	PRON
ejpam-5948	89	94	υ	υ	DET
ejpam-5948	89	95	∈	∈	PROPN
ejpam-5948	89	96	ℜ.	ℜ.	PROPN
ejpam-5948	89	97	therefore	therefore	ADV
ejpam-5948	89	98	,	,	PUNCT
ejpam-5948	89	99	corollary	corollary	ADJ
ejpam-5948	89	100	1	1	NUM
ejpam-5948	89	101	implies	imply	VERB
ejpam-5948	89	102	that	that	SCONJ
ejpam-5948	89	103	either	either	CCONJ
ejpam-5948	89	104	ℜ/p	ℜ/p	PROPN
ejpam-5948	89	105	is	be	AUX
ejpam-5948	89	106	an	an	DET
ejpam-5948	89	107	integral	integral	ADJ
ejpam-5948	89	108	domain	domain	NOUN
ejpam-5948	89	109	or	or	CCONJ
ejpam-5948	89	110	χ(ℜ	χ(ℜ	NOUN
ejpam-5948	89	111	)	)	PUNCT
ejpam-5948	89	112	⊆	⊆	NUM
ejpam-5948	89	113	p	p	NOUN
ejpam-5948	89	114	.	.	PUNCT
ejpam-5948	90	1	assuming	assume	VERB
ejpam-5948	90	2	that	that	SCONJ
ejpam-5948	90	3	χ(ℜ	χ(ℜ	NOUN
ejpam-5948	90	4	)	)	PUNCT
ejpam-5948	90	5	⊆	⊆	NUM
ejpam-5948	90	6	p	p	NOUN
ejpam-5948	90	7	,	,	PUNCT
ejpam-5948	90	8	then	then	ADV
ejpam-5948	90	9	equation	equation	NOUN
ejpam-5948	90	10	(	(	PUNCT
ejpam-5948	90	11	1	1	X
ejpam-5948	90	12	)	)	PUNCT
ejpam-5948	90	13	reduces	reduce	VERB
ejpam-5948	90	14	to	to	ADP
ejpam-5948	90	15	[	[	X
ejpam-5948	90	16	℘,⨿(υ	℘,⨿(υ	X
ejpam-5948	90	17	)	)	PUNCT
ejpam-5948	90	18	]	]	PUNCT
ejpam-5948	91	1	∈	∈	PROPN
ejpam-5948	91	2	p	p	NOUN
ejpam-5948	91	3	for	for	ADP
ejpam-5948	91	4	every	every	DET
ejpam-5948	91	5	υ	υ	NOUN
ejpam-5948	91	6	,	,	PUNCT
ejpam-5948	91	7	℘	℘	NOUN
ejpam-5948	91	8	∈	∈	NOUN
ejpam-5948	91	9	ℜ.	ℜ.	VERB
ejpam-5948	91	10	by	by	ADP
ejpam-5948	91	11	using	use	VERB
ejpam-5948	91	12	lemma	lemma	PROPN
ejpam-5948	91	13	2	2	PROPN
ejpam-5948	91	14	(	(	PUNCT
ejpam-5948	91	15	i	i	NOUN
ejpam-5948	91	16	)	)	PUNCT
ejpam-5948	91	17	,	,	PUNCT
ejpam-5948	91	18	we	we	PRON
ejpam-5948	91	19	can	can	AUX
ejpam-5948	91	20	conclude	conclude	VERB
ejpam-5948	91	21	that	that	SCONJ
ejpam-5948	91	22	either	either	CCONJ
ejpam-5948	91	23	ℜ/p	ℜ/p	PROPN
ejpam-5948	91	24	is	be	AUX
ejpam-5948	91	25	an	an	DET
ejpam-5948	91	26	integral	integral	ADJ
ejpam-5948	91	27	domain	domain	NOUN
ejpam-5948	91	28	or	or	CCONJ
ejpam-5948	91	29	⨿(ℜ	⨿(ℜ	NOUN
ejpam-5948	91	30	)	)	PUNCT
ejpam-5948	92	1	⊆	⊆	NUM
ejpam-5948	92	2	p	p	NOUN
ejpam-5948	92	3	.	.	PUNCT
ejpam-5948	93	1	the	the	DET
ejpam-5948	93	2	following	follow	VERB
ejpam-5948	93	3	corollary	corollary	NOUN
ejpam-5948	93	4	directly	directly	ADV
ejpam-5948	93	5	results	result	NOUN
ejpam-5948	93	6	from	from	ADP
ejpam-5948	93	7	replacing	replace	VERB
ejpam-5948	93	8	⨿	⨿	NOUN
ejpam-5948	93	9	by	by	ADP
ejpam-5948	93	10	idℜ	idℜ	NOUN
ejpam-5948	93	11	in	in	ADP
ejpam-5948	93	12	the	the	DET
ejpam-5948	93	13	previous	previous	ADJ
ejpam-5948	93	14	theorem	theorem	NOUN
ejpam-5948	93	15	and	and	CCONJ
ejpam-5948	93	16	following	follow	VERB
ejpam-5948	93	17	arguments	argument	NOUN
ejpam-5948	93	18	similar	similar	ADJ
ejpam-5948	93	19	to	to	ADP
ejpam-5948	93	20	those	those	PRON
ejpam-5948	93	21	used	use	VERB
ejpam-5948	93	22	.	.	PUNCT
ejpam-5948	94	1	corollary	corollary	ADJ
ejpam-5948	94	2	2	2	NUM
ejpam-5948	94	3	.	.	PUNCT
ejpam-5948	95	1	let	let	VERB
ejpam-5948	95	2	ℜ	ℜ	PROPN
ejpam-5948	95	3	be	be	AUX
ejpam-5948	95	4	a	a	DET
ejpam-5948	95	5	ring	ring	NOUN
ejpam-5948	95	6	equipped	equip	VERB
ejpam-5948	95	7	with	with	ADP
ejpam-5948	95	8	a	a	DET
ejpam-5948	95	9	p	p	NOUN
ejpam-5948	95	10	-derivation	-derivation	NOUN
ejpam-5948	95	11	χ	χ	ADP
ejpam-5948	95	12	such	such	ADJ
ejpam-5948	95	13	that	that	DET
ejpam-5948	95	14	char(ℜ/p	char(ℜ/p	NOUN
ejpam-5948	95	15	)	)	PUNCT
ejpam-5948	95	16	̸=	̸=	PROPN
ejpam-5948	95	17	2	2	NUM
ejpam-5948	95	18	.	.	PUNCT
ejpam-5948	96	1	then	then	ADV
ejpam-5948	96	2	,	,	PUNCT
ejpam-5948	96	3	[	[	X
ejpam-5948	96	4	χ(υ	χ(υ	NOUN
ejpam-5948	96	5	)	)	PUNCT
ejpam-5948	96	6	,	,	PUNCT
ejpam-5948	96	7	χ(℘)]±	χ(℘)]±	VERB
ejpam-5948	96	8	[	[	SYM
ejpam-5948	96	9	℘	℘	NOUN
ejpam-5948	96	10	,	,	PUNCT
ejpam-5948	96	11	υ	υ	NOUN
ejpam-5948	96	12	]	]	X
ejpam-5948	96	13	∈	∈	PROPN
ejpam-5948	96	14	p	p	NOUN
ejpam-5948	96	15	for	for	ADP
ejpam-5948	96	16	all	all	DET
ejpam-5948	96	17	υ	υ	NOUN
ejpam-5948	96	18	,	,	PUNCT
ejpam-5948	96	19	℘	℘	NOUN
ejpam-5948	96	20	∈	∈	NOUN
ejpam-5948	96	21	ℜ	ℜ	PROPN
ejpam-5948	96	22	if	if	SCONJ
ejpam-5948	96	23	and	and	CCONJ
ejpam-5948	96	24	only	only	ADV
ejpam-5948	96	25	if	if	SCONJ
ejpam-5948	96	26	ℜ/p	ℜ/p	PROPN
ejpam-5948	96	27	is	be	AUX
ejpam-5948	96	28	an	an	DET
ejpam-5948	96	29	integral	integral	ADJ
ejpam-5948	96	30	domain	domain	NOUN
ejpam-5948	96	31	.	.	PUNCT
ejpam-5948	97	1	theorem	theorem	NOUN
ejpam-5948	97	2	2	2	NUM
ejpam-5948	97	3	.	.	PUNCT
ejpam-5948	98	1	let	let	VERB
ejpam-5948	98	2	ℜ	ℜ	PROPN
ejpam-5948	98	3	be	be	AUX
ejpam-5948	98	4	a	a	DET
ejpam-5948	98	5	ring	ring	NOUN
ejpam-5948	98	6	equipped	equip	VERB
ejpam-5948	98	7	with	with	ADP
ejpam-5948	98	8	a	a	DET
ejpam-5948	98	9	p	p	NOUN
ejpam-5948	98	10	-derivation	-derivation	NOUN
ejpam-5948	98	11	χ	χ	NOUN
ejpam-5948	98	12	and	and	CCONJ
ejpam-5948	98	13	a	a	DET
ejpam-5948	98	14	generalized	generalized	ADJ
ejpam-5948	98	15	p	p	NOUN
ejpam-5948	98	16	-derivation	-derivation	NOUN
ejpam-5948	98	17	(	(	PUNCT
ejpam-5948	98	18	⨿,∝	⨿,∝	NOUN
ejpam-5948	98	19	)	)	PUNCT
ejpam-5948	98	20	such	such	ADJ
ejpam-5948	98	21	that	that	DET
ejpam-5948	98	22	char(ℜ/p	char(ℜ/p	NOUN
ejpam-5948	98	23	)	)	PUNCT
ejpam-5948	99	1	̸=	̸=	PROPN
ejpam-5948	99	2	2	2	NUM
ejpam-5948	99	3	.	.	PUNCT
ejpam-5948	100	1	then	then	ADV
ejpam-5948	100	2	,	,	PUNCT
ejpam-5948	100	3	χ(υ	χ(υ	PROPN
ejpam-5948	100	4	)	)	PUNCT
ejpam-5948	100	5	◦	◦	NOUN
ejpam-5948	100	6	χ(℘)±	χ(℘)±	ADJ
ejpam-5948	101	1	[	[	X
ejpam-5948	101	2	℘,⨿(υ	℘,⨿(υ	X
ejpam-5948	101	3	)	)	PUNCT
ejpam-5948	101	4	]	]	PUNCT
ejpam-5948	102	1	∈	∈	PROPN
ejpam-5948	102	2	p	p	NOUN
ejpam-5948	102	3	for	for	ADP
ejpam-5948	102	4	all	all	DET
ejpam-5948	102	5	υ	υ	NOUN
ejpam-5948	102	6	,	,	PUNCT
ejpam-5948	102	7	℘	℘	NOUN
ejpam-5948	102	8	∈	∈	NOUN
ejpam-5948	102	9	ℜ	ℜ	PROPN
ejpam-5948	102	10	if	if	SCONJ
ejpam-5948	102	11	and	and	CCONJ
ejpam-5948	102	12	only	only	ADV
ejpam-5948	102	13	if	if	SCONJ
ejpam-5948	102	14	one	one	NUM
ejpam-5948	102	15	of	of	ADP
ejpam-5948	102	16	the	the	DET
ejpam-5948	102	17	following	follow	VERB
ejpam-5948	102	18	is	be	AUX
ejpam-5948	102	19	true	true	ADJ
ejpam-5948	102	20	:	:	PUNCT
ejpam-5948	102	21	(	(	PUNCT
ejpam-5948	102	22	i	i	NOUN
ejpam-5948	102	23	)	)	PUNCT
ejpam-5948	103	1	ℜ/p	ℜ/p	PROPN
ejpam-5948	103	2	is	be	AUX
ejpam-5948	103	3	an	an	DET
ejpam-5948	103	4	integral	integral	ADJ
ejpam-5948	103	5	domain	domain	NOUN
ejpam-5948	103	6	and	and	CCONJ
ejpam-5948	103	7	χ(ℜ	χ(ℜ	NOUN
ejpam-5948	103	8	)	)	PUNCT
ejpam-5948	103	9	⊆	⊆	NUM
ejpam-5948	103	10	p	p	NOUN
ejpam-5948	103	11	.	.	PUNCT
ejpam-5948	104	1	(	(	PUNCT
ejpam-5948	104	2	ii	ii	NOUN
ejpam-5948	104	3	)	)	PUNCT
ejpam-5948	104	4	χ(ℜ	χ(ℜ	NOUN
ejpam-5948	104	5	)	)	PUNCT
ejpam-5948	105	1	⊆	⊆	NUM
ejpam-5948	105	2	p	p	NOUN
ejpam-5948	105	3	and	and	CCONJ
ejpam-5948	105	4	⨿(ℜ	⨿(ℜ	NOUN
ejpam-5948	105	5	)	)	PUNCT
ejpam-5948	105	6	⊆	⊆	NUM
ejpam-5948	105	7	p	p	NOUN
ejpam-5948	105	8	.	.	PUNCT
ejpam-5948	106	1	proof	proof	NOUN
ejpam-5948	106	2	.	.	PUNCT
ejpam-5948	107	1	for	for	ADP
ejpam-5948	107	2	each	each	DET
ejpam-5948	107	3	υ	υ	NOUN
ejpam-5948	107	4	,	,	PUNCT
ejpam-5948	107	5	℘	℘	PROPN
ejpam-5948	107	6	∈	∈	PROPN
ejpam-5948	107	7	ℜ	ℜ	PROPN
ejpam-5948	107	8	,	,	PUNCT
ejpam-5948	107	9	given	give	VERB
ejpam-5948	107	10	the	the	DET
ejpam-5948	107	11	hypothesis	hypothesis	NOUN
ejpam-5948	107	12	χ(υ	χ(υ	NOUN
ejpam-5948	107	13	)	)	PUNCT
ejpam-5948	107	14	◦	◦	NOUN
ejpam-5948	107	15	χ(℘	χ(℘	NUM
ejpam-5948	107	16	)	)	PUNCT
ejpam-5948	107	17	+	+	CCONJ
ejpam-5948	107	18	[	[	X
ejpam-5948	107	19	℘,⨿(υ	℘,⨿(υ	X
ejpam-5948	107	20	)	)	PUNCT
ejpam-5948	107	21	]	]	PUNCT
ejpam-5948	108	1	∈	∈	PROPN
ejpam-5948	108	2	p.	p.	NOUN
ejpam-5948	108	3	(	(	PUNCT
ejpam-5948	108	4	4	4	NUM
ejpam-5948	108	5	)	)	PUNCT
ejpam-5948	108	6	by	by	ADP
ejpam-5948	108	7	substituting	substitute	VERB
ejpam-5948	108	8	℘	℘	NOUN
ejpam-5948	108	9	with	with	ADP
ejpam-5948	108	10	℘ℏ	℘ℏ	NOUN
ejpam-5948	108	11	in	in	ADP
ejpam-5948	108	12	equation	equation	NOUN
ejpam-5948	108	13	(	(	PUNCT
ejpam-5948	108	14	4	4	NUM
ejpam-5948	108	15	)	)	PUNCT
ejpam-5948	108	16	and	and	CCONJ
ejpam-5948	108	17	applying	apply	VERB
ejpam-5948	108	18	it	it	PRON
ejpam-5948	108	19	,	,	PUNCT
ejpam-5948	108	20	we	we	PRON
ejpam-5948	108	21	obtain	obtain	VERB
ejpam-5948	108	22	[	[	X
ejpam-5948	108	23	χ(υ	χ(υ	NOUN
ejpam-5948	108	24	)	)	PUNCT
ejpam-5948	108	25	,	,	PUNCT
ejpam-5948	108	26	℘]χ(ℏ)−	℘]χ(ℏ)−	PROPN
ejpam-5948	108	27	χ(℘)[χ(υ	χ(℘)[χ(υ	NOUN
ejpam-5948	108	28	)	)	PUNCT
ejpam-5948	108	29	,	,	PUNCT
ejpam-5948	108	30	ℏ	ℏ	X
ejpam-5948	108	31	]	]	X
ejpam-5948	108	32	∈	∈	PROPN
ejpam-5948	108	33	p	p	NOUN
ejpam-5948	108	34	for	for	ADP
ejpam-5948	108	35	all	all	DET
ejpam-5948	108	36	υ	υ	NOUN
ejpam-5948	108	37	,	,	PUNCT
ejpam-5948	108	38	℘	℘	PROPN
ejpam-5948	108	39	,	,	PUNCT
ejpam-5948	109	1	ℏ	ℏ	PROPN
ejpam-5948	109	2	∈	∈	NOUN
ejpam-5948	109	3	ℜ.	ℜ.	PROPN
ejpam-5948	109	4	(	(	PUNCT
ejpam-5948	109	5	5	5	X
ejpam-5948	109	6	)	)	PUNCT
ejpam-5948	109	7	a.y	a.y	PROPN
ejpam-5948	109	8	.	.	PROPN
ejpam-5948	109	9	hummdi	hummdi	PROPN
ejpam-5948	109	10	,	,	PUNCT
ejpam-5948	109	11	r.m	r.m	PROPN
ejpam-5948	109	12	.	.	PROPN
ejpam-5948	109	13	al	al	PROPN
ejpam-5948	109	14	-	-	PUNCT
ejpam-5948	109	15	omary	omary	PROPN
ejpam-5948	109	16	,	,	PUNCT
ejpam-5948	109	17	z.z	z.z	PROPN
ejpam-5948	109	18	.	.	PUNCT
ejpam-5948	109	19	al	al	PROPN
ejpam-5948	109	20	-	-	PUNCT
ejpam-5948	109	21	amery	amery	PROPN
ejpam-5948	109	22	/	/	SYM
ejpam-5948	109	23	eur	eur	PROPN
ejpam-5948	109	24	.	.	PUNCT
ejpam-5948	110	1	j.	j.	PROPN
ejpam-5948	110	2	pure	pure	PROPN
ejpam-5948	110	3	appl	appl	PROPN
ejpam-5948	110	4	.	.	PROPN
ejpam-5948	110	5	math	math	PROPN
ejpam-5948	110	6	,	,	PUNCT
ejpam-5948	110	7	18	18	NUM
ejpam-5948	110	8	(	(	PUNCT
ejpam-5948	110	9	2	2	NUM
ejpam-5948	110	10	)	)	PUNCT
ejpam-5948	110	11	(	(	PUNCT
ejpam-5948	110	12	2025	2025	NUM
ejpam-5948	110	13	)	)	PUNCT
ejpam-5948	110	14	,	,	PUNCT
ejpam-5948	110	15	5948	5948	NUM
ejpam-5948	110	16	5	5	NUM
ejpam-5948	110	17	of	of	ADP
ejpam-5948	110	18	12	12	NUM
ejpam-5948	110	19	replacing	replace	VERB
ejpam-5948	110	20	℘	℘	PROPN
ejpam-5948	110	21	by	by	ADP
ejpam-5948	110	22	κ℘	κ℘	NOUN
ejpam-5948	110	23	in	in	ADP
ejpam-5948	110	24	equation	equation	NOUN
ejpam-5948	110	25	(	(	PUNCT
ejpam-5948	110	26	5	5	NUM
ejpam-5948	110	27	)	)	PUNCT
ejpam-5948	110	28	and	and	CCONJ
ejpam-5948	110	29	applying	apply	VERB
ejpam-5948	110	30	it	it	PRON
ejpam-5948	110	31	,	,	PUNCT
ejpam-5948	110	32	we	we	PRON
ejpam-5948	110	33	get	get	VERB
ejpam-5948	110	34	[	[	X
ejpam-5948	110	35	χ(υ	χ(υ	NOUN
ejpam-5948	110	36	)	)	PUNCT
ejpam-5948	110	37	,	,	PUNCT
ejpam-5948	110	38	κ]℘χ(ℏ)−	κ]℘χ(ℏ)−	PROPN
ejpam-5948	110	39	χ(κ)℘[χ(υ	χ(κ)℘[χ(υ	PROPN
ejpam-5948	110	40	)	)	PUNCT
ejpam-5948	110	41	,	,	PUNCT
ejpam-5948	110	42	ℏ	ℏ	X
ejpam-5948	110	43	]	]	X
ejpam-5948	110	44	∈	∈	PROPN
ejpam-5948	110	45	p	p	NOUN
ejpam-5948	110	46	for	for	ADP
ejpam-5948	110	47	all	all	DET
ejpam-5948	110	48	υ	υ	NOUN
ejpam-5948	110	49	,	,	PUNCT
ejpam-5948	110	50	℘	℘	PROPN
ejpam-5948	110	51	,	,	PUNCT
ejpam-5948	110	52	ℏ	ℏ	PROPN
ejpam-5948	110	53	,	,	PUNCT
ejpam-5948	110	54	κ	κ	PROPN
ejpam-5948	110	55	∈	∈	PROPN
ejpam-5948	110	56	ℜ.	ℜ.	PROPN
ejpam-5948	110	57	(	(	PUNCT
ejpam-5948	110	58	6	6	X
ejpam-5948	110	59	)	)	PUNCT
ejpam-5948	110	60	setting	set	VERB
ejpam-5948	110	61	κ	κ	X
ejpam-5948	110	62	=	=	PUNCT
ejpam-5948	110	63	χ(υ	χ(υ	NOUN
ejpam-5948	110	64	)	)	PUNCT
ejpam-5948	110	65	in	in	ADP
ejpam-5948	110	66	equation	equation	NOUN
ejpam-5948	110	67	(	(	PUNCT
ejpam-5948	110	68	6	6	NUM
ejpam-5948	110	69	)	)	PUNCT
ejpam-5948	110	70	,	,	PUNCT
ejpam-5948	110	71	we	we	PRON
ejpam-5948	110	72	get	get	VERB
ejpam-5948	110	73	χ2(υ)℘[χ(υ	χ2(υ)℘[χ(υ	NOUN
ejpam-5948	110	74	)	)	PUNCT
ejpam-5948	110	75	,	,	PUNCT
ejpam-5948	110	76	ℏ	ℏ	X
ejpam-5948	110	77	]	]	X
ejpam-5948	110	78	∈	∈	PROPN
ejpam-5948	110	79	p	p	NOUN
ejpam-5948	110	80	for	for	ADP
ejpam-5948	110	81	all	all	DET
ejpam-5948	110	82	υ	υ	NOUN
ejpam-5948	110	83	,	,	PUNCT
ejpam-5948	110	84	℘	℘	PROPN
ejpam-5948	110	85	,	,	PUNCT
ejpam-5948	110	86	ℏ	ℏ	PROPN
ejpam-5948	110	87	∈	∈	NOUN
ejpam-5948	110	88	ℜ.	ℜ.	PROPN
ejpam-5948	110	89	(	(	PUNCT
ejpam-5948	110	90	7	7	X
ejpam-5948	110	91	)	)	PUNCT
ejpam-5948	110	92	replacing	replace	VERB
ejpam-5948	110	93	℘	℘	PROPN
ejpam-5948	110	94	by	by	ADP
ejpam-5948	110	95	ℏ℘	ℏ℘	NOUN
ejpam-5948	110	96	in	in	ADP
ejpam-5948	110	97	equation	equation	NOUN
ejpam-5948	110	98	(	(	PUNCT
ejpam-5948	110	99	7	7	NUM
ejpam-5948	110	100	)	)	PUNCT
ejpam-5948	110	101	and	and	CCONJ
ejpam-5948	110	102	comparing	compare	VERB
ejpam-5948	110	103	the	the	DET
ejpam-5948	110	104	result	result	NOUN
ejpam-5948	110	105	with	with	ADP
ejpam-5948	110	106	equation	equation	NOUN
ejpam-5948	110	107	(	(	PUNCT
ejpam-5948	110	108	7	7	NUM
ejpam-5948	110	109	)	)	PUNCT
ejpam-5948	110	110	,	,	PUNCT
ejpam-5948	110	111	we	we	PRON
ejpam-5948	110	112	obtain	obtain	VERB
ejpam-5948	110	113	[	[	X
ejpam-5948	110	114	χ2(υ	χ2(υ	NOUN
ejpam-5948	110	115	)	)	PUNCT
ejpam-5948	110	116	,	,	PUNCT
ejpam-5948	110	117	ℏ]℘[χ(υ	ℏ]℘[χ(υ	PROPN
ejpam-5948	110	118	)	)	PUNCT
ejpam-5948	110	119	,	,	PUNCT
ejpam-5948	110	120	ℏ	ℏ	X
ejpam-5948	110	121	]	]	X
ejpam-5948	110	122	∈	∈	PROPN
ejpam-5948	110	123	p	p	NOUN
ejpam-5948	110	124	for	for	ADP
ejpam-5948	110	125	all	all	DET
ejpam-5948	110	126	υ	υ	NOUN
ejpam-5948	110	127	,	,	PUNCT
ejpam-5948	110	128	℘	℘	PROPN
ejpam-5948	110	129	,	,	PUNCT
ejpam-5948	110	130	ℏ	ℏ	PROPN
ejpam-5948	110	131	∈	∈	NOUN
ejpam-5948	110	132	ℜ.	ℜ.	PROPN
ejpam-5948	110	133	(	(	PUNCT
ejpam-5948	110	134	8)	8)	NUM
ejpam-5948	110	135	this	this	PRON
ejpam-5948	110	136	implies	imply	VERB
ejpam-5948	110	137	that	that	SCONJ
ejpam-5948	110	138	[	[	X
ejpam-5948	110	139	χ2(υ	χ2(υ	NUM
ejpam-5948	110	140	)	)	PUNCT
ejpam-5948	110	141	,	,	PUNCT
ejpam-5948	110	142	ℏ]ℜ[χ(υ	ℏ]ℜ[χ(υ	PROPN
ejpam-5948	110	143	)	)	PUNCT
ejpam-5948	110	144	,	,	PUNCT
ejpam-5948	110	145	ℏ	ℏ	X
ejpam-5948	110	146	]	]	X
ejpam-5948	110	147	⊆	⊆	NUM
ejpam-5948	110	148	p	p	NOUN
ejpam-5948	110	149	for	for	ADP
ejpam-5948	110	150	all	all	DET
ejpam-5948	110	151	υ	υ	NOUN
ejpam-5948	110	152	,	,	PUNCT
ejpam-5948	110	153	ℏ	ℏ	PROPN
ejpam-5948	110	154	∈	∈	NOUN
ejpam-5948	110	155	ℜ.	ℜ.	VERB
ejpam-5948	110	156	the	the	DET
ejpam-5948	110	157	primeness	primeness	NOUN
ejpam-5948	110	158	of	of	ADP
ejpam-5948	110	159	p	p	NOUN
ejpam-5948	110	160	yields	yield	NOUN
ejpam-5948	111	1	either	either	CCONJ
ejpam-5948	111	2	[	[	X
ejpam-5948	111	3	χ2(υ	χ2(υ	NOUN
ejpam-5948	111	4	)	)	PUNCT
ejpam-5948	111	5	,	,	PUNCT
ejpam-5948	111	6	ℏ	ℏ	X
ejpam-5948	111	7	]	]	X
ejpam-5948	111	8	∈	∈	PROPN
ejpam-5948	111	9	p	p	NOUN
ejpam-5948	111	10	or	or	CCONJ
ejpam-5948	111	11	[	[	X
ejpam-5948	111	12	χ(υ	χ(υ	NOUN
ejpam-5948	111	13	)	)	PUNCT
ejpam-5948	111	14	,	,	PUNCT
ejpam-5948	111	15	ℏ	ℏ	X
ejpam-5948	111	16	]	]	X
ejpam-5948	111	17	⊆	⊆	NUM
ejpam-5948	111	18	p	p	NOUN
ejpam-5948	111	19	for	for	ADP
ejpam-5948	111	20	all	all	DET
ejpam-5948	111	21	υ	υ	NOUN
ejpam-5948	111	22	,	,	PUNCT
ejpam-5948	111	23	ℏ	ℏ	PROPN
ejpam-5948	111	24	∈	∈	NOUN
ejpam-5948	111	25	ℜ.	ℜ.	PROPN
ejpam-5948	111	26	suppose	suppose	VERB
ejpam-5948	111	27	[	[	X
ejpam-5948	111	28	χ2(υ	χ2(υ	NOUN
ejpam-5948	111	29	)	)	PUNCT
ejpam-5948	111	30	,	,	PUNCT
ejpam-5948	111	31	ℏ	ℏ	X
ejpam-5948	111	32	]	]	X
ejpam-5948	111	33	∈	∈	PROPN
ejpam-5948	111	34	p	p	NOUN
ejpam-5948	111	35	for	for	ADP
ejpam-5948	111	36	all	all	DET
ejpam-5948	111	37	υ	υ	NOUN
ejpam-5948	111	38	,	,	PUNCT
ejpam-5948	111	39	ℏ	ℏ	PROPN
ejpam-5948	111	40	∈	∈	NOUN
ejpam-5948	111	41	ℜ.	ℜ.	PROPN
ejpam-5948	111	42	in	in	ADP
ejpam-5948	111	43	particular	particular	ADJ
ejpam-5948	111	44	,	,	PUNCT
ejpam-5948	111	45	we	we	PRON
ejpam-5948	111	46	have	have	VERB
ejpam-5948	111	47	[	[	X
ejpam-5948	111	48	χ2(υ	χ2(υ	NOUN
ejpam-5948	111	49	)	)	PUNCT
ejpam-5948	111	50	,	,	PUNCT
ejpam-5948	111	51	υ	υ	X
ejpam-5948	111	52	]	]	X
ejpam-5948	111	53	∈	∈	PROPN
ejpam-5948	111	54	p	p	NOUN
ejpam-5948	111	55	for	for	ADP
ejpam-5948	111	56	all	all	PRON
ejpam-5948	111	57	υ	υ	PRON
ejpam-5948	111	58	∈	∈	PROPN
ejpam-5948	111	59	ℜ.	ℜ.	PROPN
ejpam-5948	111	60	using	use	VERB
ejpam-5948	111	61	lemma	lemma	PROPN
ejpam-5948	111	62	1	1	NUM
ejpam-5948	111	63	,	,	PUNCT
ejpam-5948	111	64	we	we	PRON
ejpam-5948	111	65	find	find	VERB
ejpam-5948	111	66	that	that	SCONJ
ejpam-5948	111	67	[	[	X
ejpam-5948	111	68	χ(υ	χ(υ	NOUN
ejpam-5948	111	69	)	)	PUNCT
ejpam-5948	111	70	,	,	PUNCT
ejpam-5948	111	71	υ	υ	X
ejpam-5948	111	72	]	]	X
ejpam-5948	111	73	∈	∈	PROPN
ejpam-5948	111	74	p	p	NOUN
ejpam-5948	111	75	for	for	ADP
ejpam-5948	111	76	all	all	PRON
ejpam-5948	111	77	υ	υ	DET
ejpam-5948	111	78	∈	∈	PROPN
ejpam-5948	111	79	ℜ.	ℜ.	PROPN
ejpam-5948	111	80	this	this	PRON
ejpam-5948	111	81	leads	lead	VERB
ejpam-5948	111	82	to	to	ADP
ejpam-5948	111	83	either	either	CCONJ
ejpam-5948	111	84	ℜ/p	ℜ/p	PROPN
ejpam-5948	111	85	is	be	AUX
ejpam-5948	111	86	an	an	DET
ejpam-5948	111	87	integral	integral	ADJ
ejpam-5948	111	88	domain	domain	NOUN
ejpam-5948	111	89	or	or	CCONJ
ejpam-5948	111	90	χ(ℜ	χ(ℜ	NOUN
ejpam-5948	111	91	)	)	PUNCT
ejpam-5948	111	92	⊆	⊆	NUM
ejpam-5948	111	93	p	p	NOUN
ejpam-5948	111	94	,	,	PUNCT
ejpam-5948	111	95	by	by	ADP
ejpam-5948	111	96	using	use	VERB
ejpam-5948	111	97	corollary	corollary	ADJ
ejpam-5948	111	98	1	1	NUM
ejpam-5948	111	99	.	.	PUNCT
ejpam-5948	112	1	if	if	SCONJ
ejpam-5948	112	2	χ(ℜ	χ(ℜ	NOUN
ejpam-5948	112	3	)	)	PUNCT
ejpam-5948	112	4	⊆	⊆	NUM
ejpam-5948	112	5	p	p	NOUN
ejpam-5948	112	6	,	,	PUNCT
ejpam-5948	112	7	equation	equation	NOUN
ejpam-5948	112	8	(	(	PUNCT
ejpam-5948	112	9	4	4	NUM
ejpam-5948	112	10	)	)	PUNCT
ejpam-5948	112	11	reduces	reduce	VERB
ejpam-5948	112	12	to	to	ADP
ejpam-5948	112	13	[	[	X
ejpam-5948	112	14	℘,⨿(υ	℘,⨿(υ	X
ejpam-5948	112	15	)	)	PUNCT
ejpam-5948	112	16	]	]	PUNCT
ejpam-5948	113	1	∈	∈	PROPN
ejpam-5948	113	2	p	p	NOUN
ejpam-5948	113	3	for	for	ADP
ejpam-5948	113	4	all	all	DET
ejpam-5948	113	5	υ	υ	NOUN
ejpam-5948	113	6	,	,	PUNCT
ejpam-5948	113	7	℘	℘	PROPN
ejpam-5948	113	8	∈	∈	NOUN
ejpam-5948	113	9	ℜ.	ℜ.	PROPN
ejpam-5948	113	10	thus	thus	ADV
ejpam-5948	113	11	,	,	PUNCT
ejpam-5948	113	12	lemma	lemma	PROPN
ejpam-5948	113	13	2	2	NUM
ejpam-5948	113	14	(	(	PUNCT
ejpam-5948	113	15	i	i	NOUN
ejpam-5948	113	16	)	)	PUNCT
ejpam-5948	113	17	implies	imply	VERB
ejpam-5948	113	18	that	that	SCONJ
ejpam-5948	113	19	either	either	CCONJ
ejpam-5948	113	20	ℜ/p	ℜ/p	PROPN
ejpam-5948	113	21	is	be	AUX
ejpam-5948	113	22	an	an	DET
ejpam-5948	113	23	integral	integral	ADJ
ejpam-5948	113	24	domain	domain	NOUN
ejpam-5948	113	25	or	or	CCONJ
ejpam-5948	113	26	⨿(ℜ	⨿(ℜ	NOUN
ejpam-5948	113	27	)	)	PUNCT
ejpam-5948	113	28	⊆	⊆	NUM
ejpam-5948	113	29	p	p	NOUN
ejpam-5948	113	30	.	.	PUNCT
ejpam-5948	114	1	assuming	assume	VERB
ejpam-5948	114	2	ℜ/p	ℜ/p	PROPN
ejpam-5948	114	3	is	be	AUX
ejpam-5948	114	4	an	an	DET
ejpam-5948	114	5	integral	integral	ADJ
ejpam-5948	114	6	domain	domain	NOUN
ejpam-5948	114	7	,	,	PUNCT
ejpam-5948	114	8	equation	equation	NOUN
ejpam-5948	114	9	(	(	PUNCT
ejpam-5948	114	10	4	4	X
ejpam-5948	114	11	)	)	PUNCT
ejpam-5948	114	12	becomes	become	VERB
ejpam-5948	114	13	2χ(υ)χ(℘	2χ(υ)χ(℘	NUM
ejpam-5948	114	14	)	)	PUNCT
ejpam-5948	114	15	∈	∈	PROPN
ejpam-5948	114	16	p	p	NOUN
ejpam-5948	114	17	for	for	ADP
ejpam-5948	114	18	all	all	DET
ejpam-5948	114	19	υ	υ	NOUN
ejpam-5948	114	20	,	,	PUNCT
ejpam-5948	114	21	℘	℘	NOUN
ejpam-5948	114	22	∈	∈	NOUN
ejpam-5948	114	23	ℜ.	ℜ.	PROPN
ejpam-5948	114	24	since	since	SCONJ
ejpam-5948	114	25	char(ℜ/p	char(ℜ/p	NOUN
ejpam-5948	114	26	)	)	PUNCT
ejpam-5948	115	1	̸=	̸=	PROPN
ejpam-5948	115	2	2	2	NUM
ejpam-5948	115	3	,	,	PUNCT
ejpam-5948	115	4	then	then	ADV
ejpam-5948	115	5	χ(υ)χ(℘	χ(υ)χ(℘	PROPN
ejpam-5948	115	6	)	)	PUNCT
ejpam-5948	115	7	∈	∈	PROPN
ejpam-5948	115	8	p	p	NOUN
ejpam-5948	115	9	for	for	ADP
ejpam-5948	115	10	all	all	DET
ejpam-5948	115	11	υ	υ	NOUN
ejpam-5948	115	12	,	,	PUNCT
ejpam-5948	115	13	℘	℘	NOUN
ejpam-5948	115	14	∈	∈	NOUN
ejpam-5948	115	15	ℜ.	ℜ.	VERB
ejpam-5948	115	16	by	by	ADP
ejpam-5948	115	17	replacing	replace	VERB
ejpam-5948	115	18	℘	℘	PROPN
ejpam-5948	115	19	with	with	ADP
ejpam-5948	115	20	℘ℏ	℘ℏ	NOUN
ejpam-5948	115	21	in	in	ADP
ejpam-5948	115	22	the	the	DET
ejpam-5948	115	23	previous	previous	ADJ
ejpam-5948	115	24	equation	equation	NOUN
ejpam-5948	115	25	and	and	CCONJ
ejpam-5948	115	26	using	use	VERB
ejpam-5948	115	27	it	it	PRON
ejpam-5948	115	28	,	,	PUNCT
ejpam-5948	115	29	we	we	PRON
ejpam-5948	115	30	arrive	arrive	VERB
ejpam-5948	115	31	at	at	ADP
ejpam-5948	115	32	χ(υ)ℜχ(ℏ	χ(υ)ℜχ(ℏ	PROPN
ejpam-5948	115	33	)	)	PUNCT
ejpam-5948	115	34	⊆	⊆	NUM
ejpam-5948	115	35	p	p	NOUN
ejpam-5948	115	36	for	for	ADP
ejpam-5948	115	37	all	all	DET
ejpam-5948	115	38	υ	υ	NOUN
ejpam-5948	115	39	,	,	PUNCT
ejpam-5948	115	40	ℏ	ℏ	PROPN
ejpam-5948	115	41	∈	∈	NOUN
ejpam-5948	115	42	ℜ.	ℜ.	VERB
ejpam-5948	115	43	by	by	ADP
ejpam-5948	115	44	utilizing	utilize	VERB
ejpam-5948	115	45	the	the	DET
ejpam-5948	115	46	primeness	primeness	NOUN
ejpam-5948	115	47	of	of	ADP
ejpam-5948	115	48	p	p	NOUN
ejpam-5948	115	49	,	,	PUNCT
ejpam-5948	115	50	we	we	PRON
ejpam-5948	115	51	can	can	AUX
ejpam-5948	115	52	conclude	conclude	VERB
ejpam-5948	115	53	that	that	DET
ejpam-5948	115	54	χ(ℜ	χ(ℜ	NOUN
ejpam-5948	115	55	)	)	PUNCT
ejpam-5948	115	56	⊆	⊆	NUM
ejpam-5948	115	57	p	p	NOUN
ejpam-5948	115	58	.	.	PUNCT
ejpam-5948	116	1	the	the	DET
ejpam-5948	116	2	proof	proof	NOUN
ejpam-5948	116	3	is	be	AUX
ejpam-5948	116	4	complete	complete	ADJ
ejpam-5948	116	5	.	.	PUNCT
ejpam-5948	117	1	to	to	PART
ejpam-5948	117	2	prove	prove	VERB
ejpam-5948	117	3	the	the	DET
ejpam-5948	117	4	theorem	theorem	NOUN
ejpam-5948	117	5	for	for	ADP
ejpam-5948	117	6	the	the	DET
ejpam-5948	117	7	identity	identity	NOUN
ejpam-5948	117	8	χ(υ	χ(υ	NOUN
ejpam-5948	117	9	)	)	PUNCT
ejpam-5948	117	10	◦	◦	NOUN
ejpam-5948	117	11	χ(℘	χ(℘	NUM
ejpam-5948	117	12	)	)	PUNCT
ejpam-5948	117	13	−	−	PROPN
ejpam-5948	118	1	[	[	X
ejpam-5948	118	2	℘,⨿(υ	℘,⨿(υ	X
ejpam-5948	118	3	)	)	PUNCT
ejpam-5948	118	4	]	]	PUNCT
ejpam-5948	119	1	∈	∈	PROPN
ejpam-5948	119	2	p	p	NOUN
ejpam-5948	119	3	for	for	ADP
ejpam-5948	119	4	all	all	DET
ejpam-5948	119	5	υ	υ	NOUN
ejpam-5948	119	6	,	,	PUNCT
ejpam-5948	119	7	℘	℘	PROPN
ejpam-5948	119	8	∈	∈	PROPN
ejpam-5948	119	9	ℜ	ℜ	PROPN
ejpam-5948	119	10	,	,	PUNCT
ejpam-5948	119	11	simply	simply	ADV
ejpam-5948	119	12	repeat	repeat	VERB
ejpam-5948	119	13	the	the	DET
ejpam-5948	119	14	previous	previous	ADJ
ejpam-5948	119	15	arguments	argument	NOUN
ejpam-5948	119	16	to	to	PART
ejpam-5948	119	17	obtain	obtain	VERB
ejpam-5948	119	18	the	the	DET
ejpam-5948	119	19	desired	desire	VERB
ejpam-5948	119	20	result	result	NOUN
ejpam-5948	119	21	.	.	PUNCT
ejpam-5948	120	1	the	the	DET
ejpam-5948	120	2	following	follow	VERB
ejpam-5948	120	3	corollary	corollary	NOUN
ejpam-5948	120	4	can	can	AUX
ejpam-5948	120	5	be	be	AUX
ejpam-5948	120	6	derived	derive	VERB
ejpam-5948	120	7	immediately	immediately	ADV
ejpam-5948	120	8	from	from	ADP
ejpam-5948	120	9	the	the	DET
ejpam-5948	120	10	previous	previous	ADJ
ejpam-5948	120	11	theorem	theorem	NOUN
ejpam-5948	120	12	by	by	ADP
ejpam-5948	120	13	replacing	replace	VERB
ejpam-5948	120	14	⨿	⨿	NOUN
ejpam-5948	120	15	by	by	ADP
ejpam-5948	120	16	idℜ.	idℜ.	PROPN
ejpam-5948	120	17	corollary	corollary	ADJ
ejpam-5948	120	18	3	3	X
ejpam-5948	120	19	.	.	PUNCT
ejpam-5948	121	1	let	let	VERB
ejpam-5948	121	2	ℜ	ℜ	PROPN
ejpam-5948	121	3	be	be	AUX
ejpam-5948	121	4	a	a	DET
ejpam-5948	121	5	ring	ring	NOUN
ejpam-5948	121	6	equipped	equip	VERB
ejpam-5948	121	7	with	with	ADP
ejpam-5948	121	8	a	a	DET
ejpam-5948	121	9	p	p	NOUN
ejpam-5948	121	10	-derivation	-derivation	NOUN
ejpam-5948	121	11	χ	χ	ADP
ejpam-5948	121	12	such	such	ADJ
ejpam-5948	121	13	that	that	DET
ejpam-5948	121	14	char(ℜ/p	char(ℜ/p	NOUN
ejpam-5948	121	15	)	)	PUNCT
ejpam-5948	122	1	̸=	̸=	PROPN
ejpam-5948	122	2	2	2	NUM
ejpam-5948	122	3	.	.	PUNCT
ejpam-5948	123	1	then	then	ADV
ejpam-5948	123	2	,	,	PUNCT
ejpam-5948	123	3	χ(υ	χ(υ	PROPN
ejpam-5948	123	4	)	)	PUNCT
ejpam-5948	123	5	◦	◦	NOUN
ejpam-5948	123	6	χ(℘	χ(℘	NUM
ejpam-5948	123	7	)	)	PUNCT
ejpam-5948	123	8	±	±	NOUN
ejpam-5948	124	1	[	[	X
ejpam-5948	124	2	℘	℘	NOUN
ejpam-5948	124	3	,	,	PUNCT
ejpam-5948	124	4	υ	υ	NOUN
ejpam-5948	124	5	]	]	X
ejpam-5948	124	6	∈	∈	PROPN
ejpam-5948	124	7	p	p	NOUN
ejpam-5948	124	8	for	for	ADP
ejpam-5948	124	9	all	all	DET
ejpam-5948	124	10	υ	υ	NOUN
ejpam-5948	124	11	,	,	PUNCT
ejpam-5948	124	12	℘	℘	NOUN
ejpam-5948	124	13	∈	∈	NOUN
ejpam-5948	124	14	ℜ	ℜ	PROPN
ejpam-5948	124	15	if	if	SCONJ
ejpam-5948	124	16	and	and	CCONJ
ejpam-5948	124	17	only	only	ADV
ejpam-5948	124	18	if	if	SCONJ
ejpam-5948	124	19	ℜ/p	ℜ/p	PROPN
ejpam-5948	124	20	is	be	AUX
ejpam-5948	124	21	an	an	DET
ejpam-5948	124	22	integral	integral	ADJ
ejpam-5948	124	23	domain	domain	NOUN
ejpam-5948	124	24	and	and	CCONJ
ejpam-5948	124	25	χ(ℜ	χ(ℜ	NOUN
ejpam-5948	124	26	)	)	PUNCT
ejpam-5948	124	27	⊆	⊆	NUM
ejpam-5948	124	28	p	p	NOUN
ejpam-5948	124	29	.	.	PUNCT
ejpam-5948	125	1	theorem	theorem	NOUN
ejpam-5948	125	2	3	3	X
ejpam-5948	125	3	.	.	PUNCT
ejpam-5948	126	1	let	let	VERB
ejpam-5948	126	2	ℜ	ℜ	PROPN
ejpam-5948	126	3	be	be	AUX
ejpam-5948	126	4	a	a	DET
ejpam-5948	126	5	ring	ring	NOUN
ejpam-5948	126	6	equipped	equip	VERB
ejpam-5948	126	7	with	with	ADP
ejpam-5948	126	8	generalized	generalize	VERB
ejpam-5948	126	9	p	p	NOUN
ejpam-5948	126	10	-derivations	-derivation	NOUN
ejpam-5948	126	11	(	(	PUNCT
ejpam-5948	126	12	℧	℧	PROPN
ejpam-5948	126	13	,	,	PUNCT
ejpam-5948	126	14	χ	χ	NOUN
ejpam-5948	126	15	)	)	PUNCT
ejpam-5948	126	16	and	and	CCONJ
ejpam-5948	126	17	(	(	PUNCT
ejpam-5948	126	18	⨿,∝	⨿,∝	X
ejpam-5948	126	19	)	)	PUNCT
ejpam-5948	126	20	such	such	ADJ
ejpam-5948	126	21	that	that	DET
ejpam-5948	126	22	char(ℜ/p	char(ℜ/p	NOUN
ejpam-5948	126	23	)	)	PUNCT
ejpam-5948	127	1	̸=	̸=	PROPN
ejpam-5948	127	2	2	2	NUM
ejpam-5948	127	3	.	.	PUNCT
ejpam-5948	128	1	then	then	ADV
ejpam-5948	128	2	[	[	X
ejpam-5948	128	3	℧	℧	PROPN
ejpam-5948	128	4	(	(	PUNCT
ejpam-5948	128	5	υ	υ	NOUN
ejpam-5948	128	6	)	)	PUNCT
ejpam-5948	128	7	,	,	PUNCT
ejpam-5948	128	8	χ(℘	χ(℘	PROPN
ejpam-5948	128	9	)	)	PUNCT
ejpam-5948	128	10	]	]	PUNCT
ejpam-5948	128	11	±	±	NUM
ejpam-5948	128	12	℘	℘	PROPN
ejpam-5948	128	13	◦	◦	NOUN
ejpam-5948	128	14	⨿(υ	⨿(υ	NUM
ejpam-5948	128	15	)	)	PUNCT
ejpam-5948	129	1	∈	∈	PROPN
ejpam-5948	129	2	p	p	NOUN
ejpam-5948	129	3	for	for	ADP
ejpam-5948	129	4	all	all	DET
ejpam-5948	129	5	υ	υ	NOUN
ejpam-5948	129	6	,	,	PUNCT
ejpam-5948	129	7	℘	℘	NOUN
ejpam-5948	129	8	∈	∈	NOUN
ejpam-5948	129	9	ℜ	ℜ	PROPN
ejpam-5948	129	10	if	if	SCONJ
ejpam-5948	129	11	and	and	CCONJ
ejpam-5948	129	12	only	only	ADV
ejpam-5948	129	13	if	if	SCONJ
ejpam-5948	129	14	one	one	NUM
ejpam-5948	129	15	of	of	ADP
ejpam-5948	129	16	the	the	DET
ejpam-5948	129	17	following	follow	VERB
ejpam-5948	129	18	satisfies	satisfie	NOUN
ejpam-5948	129	19	(	(	PUNCT
ejpam-5948	129	20	i	i	NOUN
ejpam-5948	129	21	)	)	PUNCT
ejpam-5948	130	1	ℜ/p	ℜ/p	PROPN
ejpam-5948	130	2	is	be	AUX
ejpam-5948	130	3	an	an	DET
ejpam-5948	130	4	integral	integral	ADJ
ejpam-5948	130	5	domain	domain	NOUN
ejpam-5948	130	6	and	and	CCONJ
ejpam-5948	130	7	⨿(ℜ	⨿(ℜ	NOUN
ejpam-5948	130	8	)	)	PUNCT
ejpam-5948	130	9	⊆	⊆	NUM
ejpam-5948	130	10	p	p	NOUN
ejpam-5948	130	11	;	;	PUNCT
ejpam-5948	130	12	(	(	PUNCT
ejpam-5948	130	13	ii	ii	NOUN
ejpam-5948	130	14	)	)	PUNCT
ejpam-5948	130	15	℧	℧	PROPN
ejpam-5948	130	16	(	(	PUNCT
ejpam-5948	130	17	ℜ	ℜ	PROPN
ejpam-5948	130	18	)	)	PUNCT
ejpam-5948	130	19	⊆	⊆	NUM
ejpam-5948	130	20	p	p	NOUN
ejpam-5948	130	21	and	and	CCONJ
ejpam-5948	130	22	⨿(ℜ	⨿(ℜ	NOUN
ejpam-5948	130	23	)	)	PUNCT
ejpam-5948	130	24	⊆	⊆	NUM
ejpam-5948	130	25	p	p	NOUN
ejpam-5948	130	26	;	;	PUNCT
ejpam-5948	130	27	(	(	PUNCT
ejpam-5948	130	28	iii	iii	NOUN
ejpam-5948	130	29	)	)	PUNCT
ejpam-5948	130	30	χ(ℜ	χ(ℜ	NOUN
ejpam-5948	130	31	)	)	PUNCT
ejpam-5948	130	32	⊆	⊆	NUM
ejpam-5948	130	33	p	p	NOUN
ejpam-5948	130	34	,	,	PUNCT
ejpam-5948	130	35	and	and	CCONJ
ejpam-5948	130	36	⨿(ℜ	⨿(ℜ	NOUN
ejpam-5948	130	37	)	)	PUNCT
ejpam-5948	130	38	⊆	⊆	NUM
ejpam-5948	130	39	p	p	NOUN
ejpam-5948	130	40	.	.	PUNCT
ejpam-5948	131	1	proof	proof	NOUN
ejpam-5948	131	2	.	.	PUNCT
ejpam-5948	132	1	from	from	ADP
ejpam-5948	132	2	the	the	DET
ejpam-5948	132	3	given	give	VERB
ejpam-5948	132	4	hypothesis	hypothesis	NOUN
ejpam-5948	132	5	for	for	ADP
ejpam-5948	132	6	each	each	DET
ejpam-5948	132	7	υ	υ	NOUN
ejpam-5948	132	8	,	,	PUNCT
ejpam-5948	132	9	℘	℘	PROPN
ejpam-5948	132	10	∈	∈	PROPN
ejpam-5948	132	11	ℜ	ℜ	PROPN
ejpam-5948	132	12	,	,	PUNCT
ejpam-5948	132	13	we	we	PRON
ejpam-5948	132	14	have	have	VERB
ejpam-5948	132	15	[	[	X
ejpam-5948	132	16	℧	℧	X
ejpam-5948	132	17	(	(	PUNCT
ejpam-5948	132	18	υ	υ	NOUN
ejpam-5948	132	19	)	)	PUNCT
ejpam-5948	132	20	,	,	PUNCT
ejpam-5948	132	21	χ(℘	χ(℘	PROPN
ejpam-5948	132	22	)	)	PUNCT
ejpam-5948	132	23	]	]	PUNCT
ejpam-5948	133	1	+	+	CCONJ
ejpam-5948	133	2	℘	℘	PROPN
ejpam-5948	133	3	◦	◦	NOUN
ejpam-5948	133	4	⨿(υ	⨿(υ	NUM
ejpam-5948	133	5	)	)	PUNCT
ejpam-5948	134	1	∈	∈	PROPN
ejpam-5948	134	2	p	p	NOUN
ejpam-5948	134	3	,	,	PUNCT
ejpam-5948	134	4	for	for	ADP
ejpam-5948	134	5	all	all	DET
ejpam-5948	134	6	υ	υ	NOUN
ejpam-5948	134	7	,	,	PUNCT
ejpam-5948	134	8	℘	℘	PROPN
ejpam-5948	134	9	∈	∈	NOUN
ejpam-5948	134	10	ℜ.	ℜ.	PROPN
ejpam-5948	134	11	(	(	PUNCT
ejpam-5948	134	12	9	9	NUM
ejpam-5948	134	13	)	)	PUNCT
ejpam-5948	134	14	replacing	replace	VERB
ejpam-5948	134	15	℘	℘	NOUN
ejpam-5948	134	16	by	by	ADP
ejpam-5948	134	17	℘ℏ	℘ℏ	NOUN
ejpam-5948	134	18	in	in	ADP
ejpam-5948	134	19	equation	equation	NOUN
ejpam-5948	134	20	(	(	PUNCT
ejpam-5948	134	21	9	9	NUM
ejpam-5948	134	22	)	)	PUNCT
ejpam-5948	134	23	and	and	CCONJ
ejpam-5948	134	24	applying	apply	VERB
ejpam-5948	134	25	it	it	PRON
ejpam-5948	134	26	,	,	PUNCT
ejpam-5948	134	27	we	we	PRON
ejpam-5948	134	28	get	get	VERB
ejpam-5948	134	29	χ(℘)[	χ(℘)[	NOUN
ejpam-5948	134	30	℧	℧	NOUN
ejpam-5948	134	31	(υ	(υ	NOUN
ejpam-5948	134	32	)	)	PUNCT
ejpam-5948	134	33	,	,	PUNCT
ejpam-5948	134	34	ℏ	ℏ	X
ejpam-5948	134	35	]	]	X
ejpam-5948	134	36	+	+	CCONJ
ejpam-5948	134	37	℘[	℘[	PROPN
ejpam-5948	134	38	℧	℧	NOUN
ejpam-5948	134	39	(υ	(υ	NUM
ejpam-5948	134	40	)	)	PUNCT
ejpam-5948	134	41	,	,	PUNCT
ejpam-5948	134	42	χ(ℏ	χ(ℏ	PROPN
ejpam-5948	134	43	)	)	PUNCT
ejpam-5948	134	44	]	]	PUNCT
ejpam-5948	135	1	+	+	CCONJ
ejpam-5948	135	2	[	[	X
ejpam-5948	135	3	℧	℧	X
ejpam-5948	135	4	(	(	PUNCT
ejpam-5948	135	5	υ	υ	NOUN
ejpam-5948	135	6	)	)	PUNCT
ejpam-5948	135	7	,	,	PUNCT
ejpam-5948	135	8	℘]χ(ℏ	℘]χ(ℏ	PROPN
ejpam-5948	135	9	)	)	PUNCT
ejpam-5948	135	10	+	+	NUM
ejpam-5948	135	11	℘[ℏ,⨿(υ	℘[ℏ,⨿(υ	NOUN
ejpam-5948	135	12	)	)	PUNCT
ejpam-5948	135	13	]	]	PUNCT
ejpam-5948	136	1	∈	∈	PROPN
ejpam-5948	136	2	p	p	X
ejpam-5948	136	3	,	,	PUNCT
ejpam-5948	136	4	for	for	ADP
ejpam-5948	136	5	all	all	DET
ejpam-5948	136	6	υ	υ	NOUN
ejpam-5948	136	7	,	,	PUNCT
ejpam-5948	136	8	℘	℘	PROPN
ejpam-5948	136	9	,	,	PUNCT
ejpam-5948	136	10	ℏ	ℏ	PROPN
ejpam-5948	136	11	∈	∈	NOUN
ejpam-5948	136	12	ℜ.	ℜ.	PROPN
ejpam-5948	136	13	(	(	PUNCT
ejpam-5948	136	14	10	10	NUM
ejpam-5948	136	15	)	)	PUNCT
ejpam-5948	136	16	a.y	a.y	PROPN
ejpam-5948	136	17	.	.	PROPN
ejpam-5948	136	18	hummdi	hummdi	PROPN
ejpam-5948	136	19	,	,	PUNCT
ejpam-5948	136	20	r.m	r.m	PROPN
ejpam-5948	136	21	.	.	PROPN
ejpam-5948	136	22	al	al	PROPN
ejpam-5948	136	23	-	-	PUNCT
ejpam-5948	136	24	omary	omary	PROPN
ejpam-5948	136	25	,	,	PUNCT
ejpam-5948	136	26	z.z	z.z	PROPN
ejpam-5948	136	27	.	.	PUNCT
ejpam-5948	136	28	al	al	PROPN
ejpam-5948	136	29	-	-	PUNCT
ejpam-5948	136	30	amery	amery	PROPN
ejpam-5948	136	31	/	/	SYM
ejpam-5948	136	32	eur	eur	PROPN
ejpam-5948	136	33	.	.	PUNCT
ejpam-5948	137	1	j.	j.	PROPN
ejpam-5948	137	2	pure	pure	PROPN
ejpam-5948	137	3	appl	appl	PROPN
ejpam-5948	137	4	.	.	PROPN
ejpam-5948	137	5	math	math	PROPN
ejpam-5948	137	6	,	,	PUNCT
ejpam-5948	137	7	18	18	NUM
ejpam-5948	137	8	(	(	PUNCT
ejpam-5948	137	9	2	2	NUM
ejpam-5948	137	10	)	)	PUNCT
ejpam-5948	137	11	(	(	PUNCT
ejpam-5948	137	12	2025	2025	NUM
ejpam-5948	137	13	)	)	PUNCT
ejpam-5948	137	14	,	,	PUNCT
ejpam-5948	137	15	5948	5948	NUM
ejpam-5948	137	16	6	6	NUM
ejpam-5948	137	17	of	of	ADP
ejpam-5948	137	18	12	12	NUM
ejpam-5948	137	19	taking	take	VERB
ejpam-5948	137	20	℘	℘	PROPN
ejpam-5948	137	21	=	=	SYM
ejpam-5948	137	22	κ℘	κ℘	NOUN
ejpam-5948	137	23	in	in	ADP
ejpam-5948	137	24	equation	equation	NOUN
ejpam-5948	137	25	(	(	PUNCT
ejpam-5948	137	26	10	10	NUM
ejpam-5948	137	27	)	)	PUNCT
ejpam-5948	137	28	and	and	CCONJ
ejpam-5948	137	29	using	use	VERB
ejpam-5948	137	30	it	it	PRON
ejpam-5948	137	31	,	,	PUNCT
ejpam-5948	137	32	we	we	PRON
ejpam-5948	137	33	get	get	VERB
ejpam-5948	137	34	χ(κ)℘[	χ(κ)℘[	ADJ
ejpam-5948	137	35	℧	℧	NOUN
ejpam-5948	137	36	(υ	(υ	NUM
ejpam-5948	137	37	)	)	PUNCT
ejpam-5948	137	38	,	,	PUNCT
ejpam-5948	137	39	ℏ	ℏ	PROPN
ejpam-5948	137	40	]	]	PUNCT
ejpam-5948	138	1	+	+	CCONJ
ejpam-5948	138	2	[	[	X
ejpam-5948	138	3	℧	℧	X
ejpam-5948	138	4	(	(	PUNCT
ejpam-5948	138	5	υ	υ	NOUN
ejpam-5948	138	6	)	)	PUNCT
ejpam-5948	138	7	,	,	PUNCT
ejpam-5948	138	8	κ]℘χ(ℏ	κ]℘χ(ℏ	X
ejpam-5948	138	9	)	)	PUNCT
ejpam-5948	138	10	∈	∈	PROPN
ejpam-5948	138	11	p	p	NOUN
ejpam-5948	138	12	for	for	ADP
ejpam-5948	138	13	all	all	DET
ejpam-5948	138	14	υ	υ	NOUN
ejpam-5948	138	15	,	,	PUNCT
ejpam-5948	138	16	℘	℘	PROPN
ejpam-5948	138	17	,	,	PUNCT
ejpam-5948	138	18	ℏ	ℏ	PROPN
ejpam-5948	138	19	,	,	PUNCT
ejpam-5948	138	20	κ	κ	PROPN
ejpam-5948	138	21	∈	∈	PROPN
ejpam-5948	138	22	ℜ.	ℜ.	PROPN
ejpam-5948	138	23	letting	let	VERB
ejpam-5948	138	24	κ	κ	X
ejpam-5948	138	25	=	=	SYM
ejpam-5948	138	26	ℏ	ℏ	PROPN
ejpam-5948	138	27	,	,	PUNCT
ejpam-5948	138	28	we	we	PRON
ejpam-5948	138	29	obtain	obtain	VERB
ejpam-5948	138	30	χ(ℏ)℘[	χ(ℏ)℘[	NOUN
ejpam-5948	138	31	℧	℧	NOUN
ejpam-5948	138	32	(υ	(υ	NUM
ejpam-5948	138	33	)	)	PUNCT
ejpam-5948	138	34	,	,	PUNCT
ejpam-5948	138	35	ℏ	ℏ	PROPN
ejpam-5948	138	36	]	]	PUNCT
ejpam-5948	139	1	+	+	CCONJ
ejpam-5948	139	2	[	[	X
ejpam-5948	139	3	℧	℧	X
ejpam-5948	139	4	(	(	PUNCT
ejpam-5948	139	5	υ	υ	NOUN
ejpam-5948	139	6	)	)	PUNCT
ejpam-5948	139	7	,	,	PUNCT
ejpam-5948	139	8	ℏ]℘χ(ℏ	ℏ]℘χ(ℏ	NOUN
ejpam-5948	139	9	)	)	PUNCT
ejpam-5948	139	10	∈	∈	PROPN
ejpam-5948	139	11	p	p	NOUN
ejpam-5948	139	12	for	for	ADP
ejpam-5948	139	13	all	all	DET
ejpam-5948	139	14	υ	υ	NOUN
ejpam-5948	139	15	,	,	PUNCT
ejpam-5948	139	16	℘	℘	PROPN
ejpam-5948	139	17	,	,	PUNCT
ejpam-5948	139	18	ℏ	ℏ	PROPN
ejpam-5948	139	19	∈	∈	NOUN
ejpam-5948	139	20	ℜ.	ℜ.	PROPN
ejpam-5948	139	21	therefore	therefore	ADV
ejpam-5948	139	22	,	,	PUNCT
ejpam-5948	139	23	we	we	PRON
ejpam-5948	139	24	can	can	AUX
ejpam-5948	139	25	deduce	deduce	VERB
ejpam-5948	139	26	that	that	DET
ejpam-5948	139	27	χ(ℏ)℘[	χ(ℏ)℘[	NOUN
ejpam-5948	139	28	℧	℧	NOUN
ejpam-5948	139	29	(υ	(υ	NUM
ejpam-5948	139	30	)	)	PUNCT
ejpam-5948	139	31	,	,	PUNCT
ejpam-5948	140	1	ℏ	ℏ	X
ejpam-5948	140	2	]	]	X
ejpam-5948	140	3	∈	∈	PROPN
ejpam-5948	140	4	p	p	NOUN
ejpam-5948	140	5	and	and	CCONJ
ejpam-5948	140	6	[	[	X
ejpam-5948	140	7	℧	℧	PROPN
ejpam-5948	140	8	(	(	PUNCT
ejpam-5948	140	9	υ	υ	NOUN
ejpam-5948	140	10	)	)	PUNCT
ejpam-5948	140	11	,	,	PUNCT
ejpam-5948	140	12	ℏ]℘χ(ℏ	ℏ]℘χ(ℏ	NOUN
ejpam-5948	140	13	)	)	PUNCT
ejpam-5948	140	14	∈	∈	PROPN
ejpam-5948	140	15	p	p	NOUN
ejpam-5948	140	16	for	for	ADP
ejpam-5948	140	17	all	all	DET
ejpam-5948	140	18	υ	υ	NOUN
ejpam-5948	140	19	,	,	PUNCT
ejpam-5948	140	20	℘	℘	PROPN
ejpam-5948	140	21	,	,	PUNCT
ejpam-5948	140	22	ℏ	ℏ	PROPN
ejpam-5948	140	23	∈	∈	NOUN
ejpam-5948	140	24	ℜ.	ℜ.	PROPN
ejpam-5948	140	25	that	that	PRON
ejpam-5948	140	26	is	be	AUX
ejpam-5948	140	27	,	,	PUNCT
ejpam-5948	140	28	χ(ℏ)ℜ[	χ(ℏ)ℜ[	VERB
ejpam-5948	140	29	℧	℧	NOUN
ejpam-5948	140	30	(υ	(υ	NUM
ejpam-5948	140	31	)	)	PUNCT
ejpam-5948	140	32	,	,	PUNCT
ejpam-5948	140	33	ℏ	ℏ	X
ejpam-5948	140	34	]	]	X
ejpam-5948	140	35	⊆	⊆	NUM
ejpam-5948	140	36	p	p	NOUN
ejpam-5948	140	37	for	for	ADP
ejpam-5948	140	38	all	all	DET
ejpam-5948	140	39	υ	υ	NOUN
ejpam-5948	140	40	,	,	PUNCT
ejpam-5948	140	41	ℏ	ℏ	PROPN
ejpam-5948	140	42	∈	∈	NOUN
ejpam-5948	140	43	ℜ.	ℜ.	VERB
ejpam-5948	140	44	the	the	DET
ejpam-5948	140	45	primeness	primeness	NOUN
ejpam-5948	140	46	of	of	ADP
ejpam-5948	140	47	p	p	PROPN
ejpam-5948	140	48	implies	imply	VERB
ejpam-5948	140	49	either	either	CCONJ
ejpam-5948	140	50	χ(ℏ	χ(ℏ	PROPN
ejpam-5948	140	51	)	)	PUNCT
ejpam-5948	140	52	∈	∈	PROPN
ejpam-5948	140	53	p	p	NOUN
ejpam-5948	140	54	for	for	ADP
ejpam-5948	140	55	all	all	DET
ejpam-5948	140	56	ℏ	ℏ	PRON
ejpam-5948	140	57	∈	∈	PROPN
ejpam-5948	140	58	ℜ	ℜ	PROPN
ejpam-5948	140	59	or	or	CCONJ
ejpam-5948	140	60	[	[	X
ejpam-5948	140	61	℧	℧	PROPN
ejpam-5948	140	62	(	(	PUNCT
ejpam-5948	140	63	υ	υ	NOUN
ejpam-5948	140	64	)	)	PUNCT
ejpam-5948	140	65	,	,	PUNCT
ejpam-5948	140	66	ℏ	ℏ	X
ejpam-5948	140	67	]	]	X
ejpam-5948	140	68	∈	∈	PROPN
ejpam-5948	140	69	p	p	NOUN
ejpam-5948	140	70	for	for	ADP
ejpam-5948	140	71	all	all	DET
ejpam-5948	140	72	υ	υ	NOUN
ejpam-5948	140	73	,	,	PUNCT
ejpam-5948	140	74	ℏ	ℏ	PROPN
ejpam-5948	140	75	∈	∈	NOUN
ejpam-5948	140	76	ℜ.	ℜ.	PROPN
ejpam-5948	140	77	in	in	ADP
ejpam-5948	140	78	the	the	DET
ejpam-5948	140	79	second	second	ADJ
ejpam-5948	140	80	case	case	NOUN
ejpam-5948	140	81	,	,	PUNCT
ejpam-5948	140	82	lemma	lemma	PROPN
ejpam-5948	140	83	2	2	NUM
ejpam-5948	140	84	(	(	PUNCT
ejpam-5948	140	85	i	i	NOUN
ejpam-5948	140	86	)	)	PUNCT
ejpam-5948	140	87	forces	force	NOUN
ejpam-5948	140	88	either	either	CCONJ
ejpam-5948	140	89	ℜ/p	ℜ/p	PROPN
ejpam-5948	140	90	is	be	AUX
ejpam-5948	140	91	an	an	DET
ejpam-5948	140	92	integral	integral	ADJ
ejpam-5948	140	93	domain	domain	NOUN
ejpam-5948	140	94	or	or	CCONJ
ejpam-5948	140	95	℧	℧	PROPN
ejpam-5948	140	96	(	(	PUNCT
ejpam-5948	140	97	ℜ	ℜ	PROPN
ejpam-5948	140	98	)	)	PUNCT
ejpam-5948	140	99	⊆	⊆	NUM
ejpam-5948	140	100	p	p	NOUN
ejpam-5948	140	101	.	.	PUNCT
ejpam-5948	141	1	let	let	VERB
ejpam-5948	141	2	’s	’s	PRON
ejpam-5948	141	3	examine	examine	VERB
ejpam-5948	141	4	the	the	DET
ejpam-5948	141	5	case	case	NOUN
ejpam-5948	141	6	when	when	SCONJ
ejpam-5948	141	7	℧	℧	PROPN
ejpam-5948	141	8	(	(	PUNCT
ejpam-5948	141	9	ℜ	ℜ	NOUN
ejpam-5948	141	10	)	)	PUNCT
ejpam-5948	141	11	⊆	⊆	NUM
ejpam-5948	141	12	p	p	NOUN
ejpam-5948	141	13	.	.	PUNCT
ejpam-5948	142	1	this	this	PRON
ejpam-5948	142	2	reduces	reduce	VERB
ejpam-5948	142	3	equation	equation	NOUN
ejpam-5948	142	4	(	(	PUNCT
ejpam-5948	142	5	9	9	NUM
ejpam-5948	142	6	)	)	PUNCT
ejpam-5948	142	7	to	to	ADP
ejpam-5948	142	8	℘	℘	VERB
ejpam-5948	142	9	◦	◦	NOUN
ejpam-5948	142	10	⨿(υ	⨿(υ	NUM
ejpam-5948	142	11	)	)	PUNCT
ejpam-5948	143	1	∈	∈	PROPN
ejpam-5948	143	2	p	p	NOUN
ejpam-5948	143	3	for	for	ADP
ejpam-5948	143	4	all	all	DET
ejpam-5948	143	5	υ	υ	NOUN
ejpam-5948	143	6	,	,	PUNCT
ejpam-5948	143	7	℘	℘	NOUN
ejpam-5948	143	8	∈	∈	NOUN
ejpam-5948	143	9	ℜ.	ℜ.	ADJ
ejpam-5948	143	10	using	use	VERB
ejpam-5948	143	11	the	the	DET
ejpam-5948	143	12	hypothesis	hypothesis	NOUN
ejpam-5948	143	13	that	that	PRON
ejpam-5948	143	14	char(ℜ/p	char(ℜ/p	NOUN
ejpam-5948	143	15	)	)	PUNCT
ejpam-5948	144	1	̸=	̸=	PROPN
ejpam-5948	144	2	2	2	NUM
ejpam-5948	144	3	together	together	ADV
ejpam-5948	144	4	with	with	ADP
ejpam-5948	144	5	lemma	lemma	PROPN
ejpam-5948	144	6	2	2	NUM
ejpam-5948	144	7	(	(	PUNCT
ejpam-5948	144	8	ii	ii	NOUN
ejpam-5948	144	9	)	)	PUNCT
ejpam-5948	144	10	,	,	PUNCT
ejpam-5948	144	11	we	we	PRON
ejpam-5948	144	12	find	find	VERB
ejpam-5948	144	13	either	either	CCONJ
ejpam-5948	144	14	ℜ/p	ℜ/p	PROPN
ejpam-5948	144	15	is	be	AUX
ejpam-5948	144	16	an	an	DET
ejpam-5948	144	17	integral	integral	ADJ
ejpam-5948	144	18	domain	domain	NOUN
ejpam-5948	144	19	or	or	CCONJ
ejpam-5948	144	20	⨿(ℜ	⨿(ℜ	NOUN
ejpam-5948	144	21	)	)	PUNCT
ejpam-5948	144	22	⊆	⊆	NUM
ejpam-5948	144	23	p	p	NOUN
ejpam-5948	144	24	.	.	PUNCT
ejpam-5948	145	1	now	now	ADV
ejpam-5948	145	2	,	,	PUNCT
ejpam-5948	145	3	assuming	assume	VERB
ejpam-5948	145	4	ℜ/p	ℜ/p	PROPN
ejpam-5948	145	5	is	be	AUX
ejpam-5948	145	6	an	an	DET
ejpam-5948	145	7	integral	integral	ADJ
ejpam-5948	145	8	domain	domain	NOUN
ejpam-5948	145	9	,	,	PUNCT
ejpam-5948	145	10	equation	equation	NOUN
ejpam-5948	145	11	(	(	PUNCT
ejpam-5948	145	12	9	9	NUM
ejpam-5948	145	13	)	)	PUNCT
ejpam-5948	145	14	reduces	reduce	VERB
ejpam-5948	145	15	to	to	ADP
ejpam-5948	145	16	2℘ℜ⨿	2℘ℜ⨿	PROPN
ejpam-5948	145	17	(	(	PUNCT
ejpam-5948	145	18	υ	υ	NOUN
ejpam-5948	145	19	)	)	PUNCT
ejpam-5948	145	20	⊆	⊆	NUM
ejpam-5948	145	21	p	p	NOUN
ejpam-5948	145	22	for	for	ADP
ejpam-5948	145	23	all	all	DET
ejpam-5948	145	24	υ	υ	NOUN
ejpam-5948	145	25	,	,	PUNCT
ejpam-5948	145	26	℘	℘	NOUN
ejpam-5948	145	27	∈	∈	NOUN
ejpam-5948	145	28	ℜ.	ℜ.	VERB
ejpam-5948	145	29	by	by	ADP
ejpam-5948	145	30	using	use	VERB
ejpam-5948	145	31	the	the	DET
ejpam-5948	145	32	primeness	primeness	NOUN
ejpam-5948	145	33	of	of	ADP
ejpam-5948	145	34	p	p	NOUN
ejpam-5948	145	35	and	and	CCONJ
ejpam-5948	145	36	the	the	DET
ejpam-5948	145	37	fact	fact	NOUN
ejpam-5948	145	38	that	that	SCONJ
ejpam-5948	145	39	char(ℜ/p	char(ℜ/p	NOUN
ejpam-5948	145	40	)	)	PUNCT
ejpam-5948	146	1	̸=	̸=	PROPN
ejpam-5948	146	2	2	2	NUM
ejpam-5948	146	3	,	,	PUNCT
ejpam-5948	146	4	we	we	PRON
ejpam-5948	146	5	can	can	AUX
ejpam-5948	146	6	conclude	conclude	VERB
ejpam-5948	146	7	that	that	DET
ejpam-5948	146	8	⨿(ℜ	⨿(ℜ	NOUN
ejpam-5948	146	9	)	)	PUNCT
ejpam-5948	146	10	⊆	⊆	NUM
ejpam-5948	146	11	p	p	NOUN
ejpam-5948	146	12	.	.	PUNCT
ejpam-5948	147	1	on	on	ADP
ejpam-5948	147	2	the	the	DET
ejpam-5948	147	3	other	other	ADJ
ejpam-5948	147	4	hand	hand	NOUN
ejpam-5948	147	5	,	,	PUNCT
ejpam-5948	147	6	if	if	SCONJ
ejpam-5948	147	7	χ(ℏ	χ(ℏ	NOUN
ejpam-5948	147	8	)	)	PUNCT
ejpam-5948	147	9	∈	∈	PROPN
ejpam-5948	147	10	p	p	NOUN
ejpam-5948	147	11	for	for	ADP
ejpam-5948	147	12	all	all	DET
ejpam-5948	147	13	ℏ	ℏ	PRON
ejpam-5948	147	14	∈	∈	PROPN
ejpam-5948	147	15	ℜ	ℜ	PROPN
ejpam-5948	147	16	,	,	PUNCT
ejpam-5948	147	17	then	then	ADV
ejpam-5948	147	18	equation	equation	NOUN
ejpam-5948	147	19	(	(	PUNCT
ejpam-5948	147	20	9	9	X
ejpam-5948	147	21	)	)	PUNCT
ejpam-5948	147	22	becomes	become	VERB
ejpam-5948	147	23	℘	℘	PROPN
ejpam-5948	147	24	◦	◦	NOUN
ejpam-5948	147	25	⨿(υ	⨿(υ	NUM
ejpam-5948	147	26	)	)	PUNCT
ejpam-5948	148	1	∈	∈	PROPN
ejpam-5948	148	2	p	p	NOUN
ejpam-5948	148	3	for	for	ADP
ejpam-5948	148	4	all	all	DET
ejpam-5948	148	5	υ	υ	NOUN
ejpam-5948	148	6	,	,	PUNCT
ejpam-5948	148	7	℘	℘	NOUN
ejpam-5948	148	8	∈	∈	NOUN
ejpam-5948	148	9	ℜ.	ℜ.	VERB
ejpam-5948	148	10	by	by	ADP
ejpam-5948	148	11	repeating	repeat	VERB
ejpam-5948	148	12	the	the	DET
ejpam-5948	148	13	previous	previous	ADJ
ejpam-5948	148	14	discussion	discussion	NOUN
ejpam-5948	148	15	,	,	PUNCT
ejpam-5948	148	16	we	we	PRON
ejpam-5948	148	17	arrive	arrive	VERB
ejpam-5948	148	18	at	at	ADP
ejpam-5948	148	19	the	the	DET
ejpam-5948	148	20	required	required	ADJ
ejpam-5948	148	21	conclusion	conclusion	NOUN
ejpam-5948	148	22	.	.	PUNCT
ejpam-5948	149	1	to	to	PART
ejpam-5948	149	2	prove	prove	VERB
ejpam-5948	149	3	the	the	DET
ejpam-5948	149	4	theorem	theorem	NOUN
ejpam-5948	149	5	for	for	ADP
ejpam-5948	149	6	the	the	DET
ejpam-5948	149	7	identity	identity	NOUN
ejpam-5948	149	8	[	[	X
ejpam-5948	149	9	℧	℧	PROPN
ejpam-5948	149	10	(	(	PUNCT
ejpam-5948	149	11	υ	υ	NOUN
ejpam-5948	149	12	)	)	PUNCT
ejpam-5948	149	13	,	,	PUNCT
ejpam-5948	149	14	χ(℘	χ(℘	PROPN
ejpam-5948	149	15	)	)	PUNCT
ejpam-5948	149	16	]	]	PUNCT
ejpam-5948	150	1	−	−	PROPN
ejpam-5948	150	2	℘	℘	PROPN
ejpam-5948	150	3	◦	◦	NOUN
ejpam-5948	150	4	⨿(υ	⨿(υ	NUM
ejpam-5948	150	5	)	)	PUNCT
ejpam-5948	151	1	∈	∈	PROPN
ejpam-5948	151	2	p	p	NOUN
ejpam-5948	151	3	for	for	ADP
ejpam-5948	151	4	all	all	DET
ejpam-5948	151	5	υ	υ	NOUN
ejpam-5948	151	6	,	,	PUNCT
ejpam-5948	151	7	℘	℘	PROPN
ejpam-5948	151	8	∈	∈	PROPN
ejpam-5948	151	9	ℜ	ℜ	PROPN
ejpam-5948	151	10	,	,	PUNCT
ejpam-5948	151	11	simply	simply	ADV
ejpam-5948	151	12	repeat	repeat	VERB
ejpam-5948	151	13	the	the	DET
ejpam-5948	151	14	previous	previous	ADJ
ejpam-5948	151	15	arguments	argument	NOUN
ejpam-5948	151	16	to	to	PART
ejpam-5948	151	17	obtain	obtain	VERB
ejpam-5948	151	18	the	the	DET
ejpam-5948	151	19	desired	desire	VERB
ejpam-5948	151	20	result	result	NOUN
ejpam-5948	151	21	.	.	PUNCT
ejpam-5948	152	1	in	in	ADP
ejpam-5948	152	2	[	[	X
ejpam-5948	152	3	13	13	NUM
ejpam-5948	152	4	]	]	PUNCT
ejpam-5948	152	5	,	,	PUNCT
ejpam-5948	152	6	quadri	quadri	PROPN
ejpam-5948	152	7	et	et	PROPN
ejpam-5948	152	8	al	al	PROPN
ejpam-5948	152	9	.	.	PROPN
ejpam-5948	152	10	discussed	discuss	VERB
ejpam-5948	152	11	the	the	DET
ejpam-5948	152	12	behavior	behavior	NOUN
ejpam-5948	152	13	of	of	ADP
ejpam-5948	152	14	a	a	DET
ejpam-5948	152	15	prime	prime	ADJ
ejpam-5948	152	16	ring	ring	NOUN
ejpam-5948	152	17	ℜ	ℜ	PROPN
ejpam-5948	152	18	that	that	PRON
ejpam-5948	152	19	admits	admit	VERB
ejpam-5948	152	20	a	a	DET
ejpam-5948	152	21	generalized	generalized	ADJ
ejpam-5948	152	22	derivation	derivation	NOUN
ejpam-5948	152	23	(	(	PUNCT
ejpam-5948	152	24	℧	℧	PROPN
ejpam-5948	152	25	,	,	PUNCT
ejpam-5948	152	26	χ	χ	NOUN
ejpam-5948	152	27	)	)	PUNCT
ejpam-5948	152	28	satisfying	satisfy	VERB
ejpam-5948	152	29	℧	℧	PROPN
ejpam-5948	152	30	[	[	X
ejpam-5948	152	31	υ	υ	NOUN
ejpam-5948	152	32	,	,	PUNCT
ejpam-5948	152	33	℘]−	℘]−	NOUN
ejpam-5948	153	1	[	[	X
ejpam-5948	153	2	υ	υ	NOUN
ejpam-5948	153	3	,	,	PUNCT
ejpam-5948	153	4	℘	℘	PROPN
ejpam-5948	153	5	]	]	PUNCT
ejpam-5948	153	6	=	=	SYM
ejpam-5948	153	7	0	0	NUM
ejpam-5948	153	8	for	for	ADP
ejpam-5948	153	9	all	all	DET
ejpam-5948	153	10	υ	υ	NOUN
ejpam-5948	153	11	,	,	PUNCT
ejpam-5948	153	12	℘	℘	PROPN
ejpam-5948	153	13	∈	∈	PROPN
ejpam-5948	153	14	υ	υ	NOUN
ejpam-5948	153	15	,	,	PUNCT
ejpam-5948	153	16	where	where	SCONJ
ejpam-5948	153	17	υ	υ	NOUN
ejpam-5948	153	18	is	be	AUX
ejpam-5948	153	19	a	a	DET
ejpam-5948	153	20	nonzero	nonzero	ADJ
ejpam-5948	153	21	ideal	ideal	NOUN
ejpam-5948	153	22	of	of	ADP
ejpam-5948	153	23	ℜ.	ℜ.	PROPN
ejpam-5948	153	24	in	in	ADP
ejpam-5948	153	25	the	the	DET
ejpam-5948	153	26	context	context	NOUN
ejpam-5948	153	27	of	of	ADP
ejpam-5948	153	28	two	two	NUM
ejpam-5948	153	29	generalized	generalized	ADJ
ejpam-5948	153	30	derivations	derivation	NOUN
ejpam-5948	153	31	,	,	PUNCT
ejpam-5948	153	32	rehman	rehman	NOUN
ejpam-5948	153	33	et	et	PROPN
ejpam-5948	153	34	al	al	PROPN
ejpam-5948	153	35	.	.	PUNCT
ejpam-5948	154	1	[	[	X
ejpam-5948	154	2	14	14	NUM
ejpam-5948	154	3	]	]	PUNCT
ejpam-5948	154	4	discussed	discuss	VERB
ejpam-5948	154	5	the	the	DET
ejpam-5948	154	6	behavior	behavior	NOUN
ejpam-5948	154	7	of	of	ADP
ejpam-5948	154	8	a	a	DET
ejpam-5948	154	9	2	2	NUM
ejpam-5948	154	10	-	-	PUNCT
ejpam-5948	154	11	torsion	torsion	NOUN
ejpam-5948	154	12	free	free	ADJ
ejpam-5948	154	13	∗-prime	∗-prime	PROPN
ejpam-5948	154	14	ring	ring	NOUN
ejpam-5948	154	15	with	with	ADP
ejpam-5948	154	16	the	the	DET
ejpam-5948	154	17	identity	identity	NOUN
ejpam-5948	154	18	[	[	X
ejpam-5948	154	19	υ,⨿(℘)]−	υ,⨿(℘)]−	PROPN
ejpam-5948	154	20	℧	℧	PROPN
ejpam-5948	154	21	(	(	PUNCT
ejpam-5948	154	22	[	[	X
ejpam-5948	154	23	υ	υ	INTJ
ejpam-5948	154	24	,	,	PUNCT
ejpam-5948	154	25	℘	℘	PROPN
ejpam-5948	154	26	]	]	PUNCT
ejpam-5948	154	27	)	)	PUNCT
ejpam-5948	154	28	=	=	SYM
ejpam-5948	154	29	0	0	NUM
ejpam-5948	154	30	for	for	ADP
ejpam-5948	154	31	all	all	DET
ejpam-5948	154	32	υ	υ	NOUN
ejpam-5948	154	33	,	,	PUNCT
ejpam-5948	154	34	℘	℘	PROPN
ejpam-5948	154	35	∈	∈	PROPN
ejpam-5948	154	36	λ	λ	NOUN
ejpam-5948	154	37	,	,	PUNCT
ejpam-5948	154	38	where	where	SCONJ
ejpam-5948	154	39	λ	λ	PROPN
ejpam-5948	154	40	is	be	AUX
ejpam-5948	154	41	a	a	DET
ejpam-5948	154	42	nonzero	nonzero	PROPN
ejpam-5948	154	43	square	square	PROPN
ejpam-5948	154	44	closed	close	VERB
ejpam-5948	154	45	∗-lie	∗-lie	PROPN
ejpam-5948	154	46	ideal	ideal	NOUN
ejpam-5948	154	47	of	of	ADP
ejpam-5948	154	48	ℜ.	ℜ.	PROPN
ejpam-5948	154	49	bouchannafa	bouchannafa	PROPN
ejpam-5948	154	50	et	et	PROPN
ejpam-5948	154	51	al	al	PROPN
ejpam-5948	154	52	.	.	PUNCT
ejpam-5948	155	1	[	[	X
ejpam-5948	155	2	15	15	NUM
ejpam-5948	155	3	]	]	PUNCT
ejpam-5948	155	4	studied	study	VERB
ejpam-5948	155	5	the	the	DET
ejpam-5948	155	6	relationship	relationship	NOUN
ejpam-5948	155	7	between	between	ADP
ejpam-5948	155	8	a	a	DET
ejpam-5948	155	9	factor	factor	NOUN
ejpam-5948	155	10	ring	ring	NOUN
ejpam-5948	155	11	ℜ/p	ℜ/p	PROPN
ejpam-5948	155	12	and	and	CCONJ
ejpam-5948	155	13	a	a	DET
ejpam-5948	155	14	generalized	generalized	ADJ
ejpam-5948	155	15	derivation	derivation	NOUN
ejpam-5948	155	16	(	(	PUNCT
ejpam-5948	155	17	℧	℧	PROPN
ejpam-5948	155	18	,	,	PUNCT
ejpam-5948	155	19	χ	χ	NOUN
ejpam-5948	155	20	)	)	PUNCT
ejpam-5948	155	21	satisfying	satisfy	VERB
ejpam-5948	155	22	℧	℧	PROPN
ejpam-5948	156	1	[	[	X
ejpam-5948	156	2	υ	υ	NOUN
ejpam-5948	156	3	,	,	PUNCT
ejpam-5948	156	4	℘]−	℘]−	X
ejpam-5948	157	1	[	[	X
ejpam-5948	157	2	℧	℧	PROPN
ejpam-5948	157	3	(	(	PUNCT
ejpam-5948	157	4	υ	υ	NOUN
ejpam-5948	157	5	)	)	PUNCT
ejpam-5948	157	6	,	,	PUNCT
ejpam-5948	157	7	℘	℘	PROPN
ejpam-5948	157	8	]	]	PUNCT
ejpam-5948	157	9	∈	∈	PROPN
ejpam-5948	157	10	z(ℜ/p	z(ℜ/p	NUM
ejpam-5948	157	11	)	)	PUNCT
ejpam-5948	157	12	for	for	ADP
ejpam-5948	157	13	all	all	DET
ejpam-5948	157	14	υ	υ	NOUN
ejpam-5948	157	15	,	,	PUNCT
ejpam-5948	157	16	℘	℘	PROPN
ejpam-5948	157	17	∈	∈	PROPN
ejpam-5948	157	18	ℜ	ℜ	PROPN
ejpam-5948	157	19	,	,	PUNCT
ejpam-5948	157	20	without	without	ADP
ejpam-5948	157	21	imposing	impose	VERB
ejpam-5948	157	22	primeness	primeness	NOUN
ejpam-5948	157	23	on	on	ADP
ejpam-5948	157	24	a	a	DET
ejpam-5948	157	25	ring	ring	NOUN
ejpam-5948	157	26	or	or	CCONJ
ejpam-5948	157	27	char(ℜ/p	char(ℜ/p	NOUN
ejpam-5948	157	28	)	)	PUNCT
ejpam-5948	158	1	̸=	̸=	PROPN
ejpam-5948	158	2	2	2	NUM
ejpam-5948	158	3	,	,	PUNCT
ejpam-5948	158	4	where	where	SCONJ
ejpam-5948	158	5	p	p	NOUN
ejpam-5948	158	6	is	be	AUX
ejpam-5948	158	7	a	a	DET
ejpam-5948	158	8	prime	prime	ADJ
ejpam-5948	158	9	ideal	ideal	NOUN
ejpam-5948	158	10	of	of	ADP
ejpam-5948	158	11	ℜ.	ℜ.	PROPN
ejpam-5948	158	12	building	building	NOUN
ejpam-5948	158	13	on	on	ADP
ejpam-5948	158	14	these	these	DET
ejpam-5948	158	15	previous	previous	ADJ
ejpam-5948	158	16	findings	finding	NOUN
ejpam-5948	158	17	,	,	PUNCT
ejpam-5948	158	18	it	it	PRON
ejpam-5948	158	19	is	be	AUX
ejpam-5948	158	20	natural	natural	ADJ
ejpam-5948	158	21	to	to	PART
ejpam-5948	158	22	inquire	inquire	VERB
ejpam-5948	158	23	about	about	ADP
ejpam-5948	158	24	the	the	DET
ejpam-5948	158	25	situation	situation	NOUN
ejpam-5948	158	26	of	of	ADP
ejpam-5948	158	27	a	a	DET
ejpam-5948	158	28	factor	factor	NOUN
ejpam-5948	158	29	ring	ring	NOUN
ejpam-5948	158	30	ℜ/p	ℜ/p	PROPN
ejpam-5948	158	31	when	when	SCONJ
ejpam-5948	158	32	ℜ	ℜ	PROPN
ejpam-5948	158	33	admits	admit	VERB
ejpam-5948	158	34	generalized	generalize	VERB
ejpam-5948	158	35	p	p	NOUN
ejpam-5948	158	36	-derivations	-derivation	NOUN
ejpam-5948	158	37	(	(	PUNCT
ejpam-5948	158	38	℧	℧	PROPN
ejpam-5948	158	39	,	,	PUNCT
ejpam-5948	158	40	χ	χ	NOUN
ejpam-5948	158	41	)	)	PUNCT
ejpam-5948	158	42	and	and	CCONJ
ejpam-5948	158	43	(	(	PUNCT
ejpam-5948	158	44	⨿,∝	⨿,∝	X
ejpam-5948	158	45	)	)	PUNCT
ejpam-5948	158	46	that	that	PRON
ejpam-5948	158	47	satisfy	satisfy	VERB
ejpam-5948	158	48	the	the	DET
ejpam-5948	158	49	identity	identity	NOUN
ejpam-5948	158	50	[	[	X
ejpam-5948	158	51	υ,⨿(℘	υ,⨿(℘	NOUN
ejpam-5948	158	52	)	)	PUNCT
ejpam-5948	158	53	]	]	PUNCT
ejpam-5948	158	54	±	±	NUM
ejpam-5948	158	55	℧	℧	PROPN
ejpam-5948	158	56	(	(	PUNCT
ejpam-5948	158	57	[	[	X
ejpam-5948	158	58	υ	υ	INTJ
ejpam-5948	158	59	,	,	PUNCT
ejpam-5948	158	60	℘	℘	PROPN
ejpam-5948	158	61	]	]	PUNCT
ejpam-5948	158	62	)	)	PUNCT
ejpam-5948	158	63	∈	∈	PROPN
ejpam-5948	158	64	p	p	NOUN
ejpam-5948	158	65	for	for	ADP
ejpam-5948	158	66	all	all	DET
ejpam-5948	158	67	υ	υ	NOUN
ejpam-5948	158	68	,	,	PUNCT
ejpam-5948	158	69	℘	℘	PROPN
ejpam-5948	158	70	∈	∈	NOUN
ejpam-5948	158	71	ℜ.	ℜ.	ADJ
ejpam-5948	158	72	to	to	PART
ejpam-5948	158	73	address	address	VERB
ejpam-5948	158	74	this	this	DET
ejpam-5948	158	75	question	question	NOUN
ejpam-5948	158	76	,	,	PUNCT
ejpam-5948	158	77	we	we	PRON
ejpam-5948	158	78	will	will	AUX
ejpam-5948	158	79	now	now	ADV
ejpam-5948	158	80	present	present	VERB
ejpam-5948	158	81	the	the	DET
ejpam-5948	158	82	following	follow	VERB
ejpam-5948	158	83	theorem	theorem	NOUN
ejpam-5948	158	84	.	.	PUNCT
ejpam-5948	158	85	theorem	theorem	NOUN
ejpam-5948	158	86	4	4	NUM
ejpam-5948	158	87	.	.	PUNCT
ejpam-5948	158	88	let	let	VERB
ejpam-5948	158	89	ℜ	ℜ	PROPN
ejpam-5948	158	90	be	be	AUX
ejpam-5948	158	91	a	a	DET
ejpam-5948	158	92	ring	ring	NOUN
ejpam-5948	158	93	equipped	equip	VERB
ejpam-5948	158	94	with	with	ADP
ejpam-5948	158	95	generalized	generalize	VERB
ejpam-5948	158	96	p	p	NOUN
ejpam-5948	158	97	-derivations	-derivation	NOUN
ejpam-5948	158	98	(	(	PUNCT
ejpam-5948	158	99	℧	℧	PROPN
ejpam-5948	158	100	,	,	PUNCT
ejpam-5948	158	101	χ	χ	NOUN
ejpam-5948	158	102	)	)	PUNCT
ejpam-5948	158	103	and	and	CCONJ
ejpam-5948	158	104	(	(	PUNCT
ejpam-5948	158	105	⨿,∝	⨿,∝	X
ejpam-5948	158	106	)	)	PUNCT
ejpam-5948	158	107	such	such	ADJ
ejpam-5948	158	108	that	that	SCONJ
ejpam-5948	158	109	[	[	X
ejpam-5948	158	110	υ,⨿(℘	υ,⨿(℘	NOUN
ejpam-5948	158	111	)	)	PUNCT
ejpam-5948	158	112	]	]	PUNCT
ejpam-5948	158	113	±	±	NUM
ejpam-5948	158	114	℧	℧	PROPN
ejpam-5948	158	115	(	(	PUNCT
ejpam-5948	158	116	[	[	X
ejpam-5948	158	117	υ	υ	INTJ
ejpam-5948	158	118	,	,	PUNCT
ejpam-5948	158	119	℘	℘	PROPN
ejpam-5948	158	120	]	]	PUNCT
ejpam-5948	158	121	)	)	PUNCT
ejpam-5948	158	122	∈	∈	PROPN
ejpam-5948	158	123	p	p	NOUN
ejpam-5948	158	124	for	for	ADP
ejpam-5948	158	125	all	all	DET
ejpam-5948	158	126	υ	υ	NOUN
ejpam-5948	158	127	,	,	PUNCT
ejpam-5948	158	128	℘	℘	PROPN
ejpam-5948	158	129	∈	∈	NOUN
ejpam-5948	158	130	ℜ.	ℜ.	PROPN
ejpam-5948	158	131	then	then	ADV
ejpam-5948	158	132	,	,	PUNCT
ejpam-5948	158	133	ℜ/p	ℜ/p	PROPN
ejpam-5948	158	134	is	be	AUX
ejpam-5948	158	135	an	an	DET
ejpam-5948	158	136	integral	integral	ADJ
ejpam-5948	158	137	domain	domain	NOUN
ejpam-5948	158	138	or	or	CCONJ
ejpam-5948	158	139	(	(	PUNCT
ejpam-5948	158	140	⨿±	⨿±	VERB
ejpam-5948	158	141	℧	℧	NOUN
ejpam-5948	158	142	)	)	PUNCT
ejpam-5948	158	143	(	(	PUNCT
ejpam-5948	158	144	ℜ	ℜ	PROPN
ejpam-5948	158	145	)	)	PUNCT
ejpam-5948	158	146	⊆	⊆	NUM
ejpam-5948	158	147	p	p	NOUN
ejpam-5948	158	148	.	.	PUNCT
ejpam-5948	159	1	proof	proof	NOUN
ejpam-5948	159	2	.	.	PUNCT
ejpam-5948	160	1	our	our	PRON
ejpam-5948	160	2	initial	initial	ADJ
ejpam-5948	160	3	hypothesis	hypothesis	NOUN
ejpam-5948	160	4	states	state	NOUN
ejpam-5948	160	5	:	:	PUNCT
ejpam-5948	161	1	[	[	X
ejpam-5948	161	2	υ,⨿(℘)]±	υ,⨿(℘)]±	NOUN
ejpam-5948	161	3	℧	℧	PROPN
ejpam-5948	161	4	(	(	PUNCT
ejpam-5948	161	5	[	[	X
ejpam-5948	161	6	υ	υ	INTJ
ejpam-5948	161	7	,	,	PUNCT
ejpam-5948	161	8	℘	℘	PROPN
ejpam-5948	161	9	]	]	PUNCT
ejpam-5948	161	10	)	)	PUNCT
ejpam-5948	161	11	∈	∈	PROPN
ejpam-5948	161	12	p	p	X
ejpam-5948	161	13	,	,	PUNCT
ejpam-5948	161	14	for	for	ADP
ejpam-5948	161	15	all	all	DET
ejpam-5948	161	16	υ	υ	NOUN
ejpam-5948	161	17	,	,	PUNCT
ejpam-5948	161	18	℘	℘	PROPN
ejpam-5948	161	19	∈	∈	NOUN
ejpam-5948	161	20	ℜ.	ℜ.	PROPN
ejpam-5948	161	21	(	(	PUNCT
ejpam-5948	161	22	11	11	NUM
ejpam-5948	161	23	)	)	PUNCT
ejpam-5948	161	24	substituting	substitute	VERB
ejpam-5948	161	25	℘	℘	NOUN
ejpam-5948	161	26	with	with	ADP
ejpam-5948	161	27	℘ℏ	℘ℏ	NOUN
ejpam-5948	161	28	in	in	ADP
ejpam-5948	161	29	equation	equation	NOUN
ejpam-5948	161	30	(	(	PUNCT
ejpam-5948	161	31	11	11	NUM
ejpam-5948	161	32	)	)	PUNCT
ejpam-5948	161	33	and	and	CCONJ
ejpam-5948	161	34	applying	apply	VERB
ejpam-5948	161	35	it	it	PRON
ejpam-5948	161	36	,	,	PUNCT
ejpam-5948	161	37	we	we	PRON
ejpam-5948	161	38	get	get	VERB
ejpam-5948	161	39	⨿(℘)[υ	⨿(℘)[υ	NOUN
ejpam-5948	161	40	,	,	PUNCT
ejpam-5948	161	41	ℏ]+℘[υ,∝	ℏ]+℘[υ,∝	ADP
ejpam-5948	161	42	(	(	PUNCT
ejpam-5948	161	43	ℏ)]+[υ	ℏ)]+[υ	NOUN
ejpam-5948	161	44	,	,	PUNCT
ejpam-5948	161	45	℘	℘	PROPN
ejpam-5948	161	46	]	]	PUNCT
ejpam-5948	161	47	∝	∝	PROPN
ejpam-5948	161	48	(	(	PUNCT
ejpam-5948	161	49	ℏ)±	ℏ)±	PROPN
ejpam-5948	161	50	℧	℧	PROPN
ejpam-5948	161	51	(℘)[υ	(℘)[υ	PROPN
ejpam-5948	161	52	,	,	PUNCT
ejpam-5948	161	53	ℏ]±℘χ([υ	ℏ]±℘χ([υ	PROPN
ejpam-5948	161	54	,	,	PUNCT
ejpam-5948	161	55	ℏ])±[υ	ℏ])±[υ	PROPN
ejpam-5948	161	56	,	,	PUNCT
ejpam-5948	161	57	℘]χ(ℏ	℘]χ(ℏ	PROPN
ejpam-5948	161	58	)	)	PUNCT
ejpam-5948	161	59	∈	∈	PROPN
ejpam-5948	161	60	p	p	NOUN
ejpam-5948	161	61	for	for	ADP
ejpam-5948	161	62	all	all	DET
ejpam-5948	161	63	υ	υ	NOUN
ejpam-5948	161	64	,	,	PUNCT
ejpam-5948	161	65	℘	℘	PROPN
ejpam-5948	161	66	,	,	PUNCT
ejpam-5948	161	67	ℏ	ℏ	PROPN
ejpam-5948	161	68	∈	∈	NOUN
ejpam-5948	161	69	ℜ.	ℜ.	PROPN
ejpam-5948	161	70	(	(	PUNCT
ejpam-5948	161	71	12	12	NUM
ejpam-5948	161	72	)	)	PUNCT
ejpam-5948	161	73	taking	take	VERB
ejpam-5948	161	74	υ	υ	NOUN
ejpam-5948	161	75	=	=	X
ejpam-5948	161	76	ℏ	ℏ	PROPN
ejpam-5948	161	77	in	in	ADP
ejpam-5948	161	78	equation	equation	NOUN
ejpam-5948	161	79	(	(	PUNCT
ejpam-5948	161	80	12	12	NUM
ejpam-5948	161	81	)	)	PUNCT
ejpam-5948	161	82	,	,	PUNCT
ejpam-5948	161	83	we	we	PRON
ejpam-5948	161	84	get	get	VERB
ejpam-5948	161	85	℘[υ,∝	℘[υ,∝	NOUN
ejpam-5948	161	86	(	(	PUNCT
ejpam-5948	161	87	υ	υ	NOUN
ejpam-5948	161	88	)	)	PUNCT
ejpam-5948	161	89	]	]	PUNCT
ejpam-5948	162	1	+	+	CCONJ
ejpam-5948	162	2	[	[	X
ejpam-5948	162	3	υ	υ	INTJ
ejpam-5948	162	4	,	,	PUNCT
ejpam-5948	162	5	℘	℘	PROPN
ejpam-5948	162	6	]	]	PUNCT
ejpam-5948	162	7	∝	∝	PROPN
ejpam-5948	162	8	(	(	PUNCT
ejpam-5948	162	9	υ)±	υ)±	X
ejpam-5948	162	10	[	[	X
ejpam-5948	162	11	υ	υ	NOUN
ejpam-5948	162	12	,	,	PUNCT
ejpam-5948	162	13	℘]χ(υ	℘]χ(υ	ADJ
ejpam-5948	162	14	)	)	PUNCT
ejpam-5948	162	15	∈	∈	PROPN
ejpam-5948	162	16	p	p	NOUN
ejpam-5948	162	17	for	for	ADP
ejpam-5948	162	18	all	all	DET
ejpam-5948	162	19	υ	υ	NOUN
ejpam-5948	162	20	,	,	PUNCT
ejpam-5948	162	21	℘	℘	PROPN
ejpam-5948	162	22	∈	∈	NOUN
ejpam-5948	162	23	ℜ.	ℜ.	PROPN
ejpam-5948	162	24	(	(	PUNCT
ejpam-5948	162	25	13	13	NUM
ejpam-5948	162	26	)	)	PUNCT
ejpam-5948	162	27	a.y	a.y	PROPN
ejpam-5948	162	28	.	.	PROPN
ejpam-5948	162	29	hummdi	hummdi	PROPN
ejpam-5948	162	30	,	,	PUNCT
ejpam-5948	162	31	r.m	r.m	PROPN
ejpam-5948	162	32	.	.	PROPN
ejpam-5948	162	33	al	al	PROPN
ejpam-5948	162	34	-	-	PUNCT
ejpam-5948	162	35	omary	omary	PROPN
ejpam-5948	162	36	,	,	PUNCT
ejpam-5948	162	37	z.z	z.z	PROPN
ejpam-5948	162	38	.	.	PUNCT
ejpam-5948	162	39	al	al	PROPN
ejpam-5948	162	40	-	-	PUNCT
ejpam-5948	162	41	amery	amery	PROPN
ejpam-5948	162	42	/	/	SYM
ejpam-5948	162	43	eur	eur	PROPN
ejpam-5948	162	44	.	.	PUNCT
ejpam-5948	163	1	j.	j.	PROPN
ejpam-5948	163	2	pure	pure	PROPN
ejpam-5948	163	3	appl	appl	PROPN
ejpam-5948	163	4	.	.	PROPN
ejpam-5948	163	5	math	math	PROPN
ejpam-5948	163	6	,	,	PUNCT
ejpam-5948	163	7	18	18	NUM
ejpam-5948	163	8	(	(	PUNCT
ejpam-5948	163	9	2	2	NUM
ejpam-5948	163	10	)	)	PUNCT
ejpam-5948	163	11	(	(	PUNCT
ejpam-5948	163	12	2025	2025	NUM
ejpam-5948	163	13	)	)	PUNCT
ejpam-5948	163	14	,	,	PUNCT
ejpam-5948	163	15	5948	5948	NUM
ejpam-5948	163	16	7	7	NUM
ejpam-5948	163	17	of	of	ADP
ejpam-5948	163	18	12	12	NUM
ejpam-5948	163	19	by	by	ADP
ejpam-5948	163	20	replacing	replace	VERB
ejpam-5948	163	21	℘	℘	PROPN
ejpam-5948	163	22	with	with	ADP
ejpam-5948	163	23	κ℘	κ℘	NOUN
ejpam-5948	163	24	in	in	ADP
ejpam-5948	163	25	equation	equation	NOUN
ejpam-5948	163	26	(	(	PUNCT
ejpam-5948	163	27	13	13	NUM
ejpam-5948	163	28	)	)	PUNCT
ejpam-5948	163	29	and	and	CCONJ
ejpam-5948	163	30	comparing	compare	VERB
ejpam-5948	163	31	it	it	PRON
ejpam-5948	163	32	with	with	ADP
ejpam-5948	163	33	(	(	PUNCT
ejpam-5948	163	34	13	13	NUM
ejpam-5948	163	35	)	)	PUNCT
ejpam-5948	163	36	,	,	PUNCT
ejpam-5948	163	37	we	we	PRON
ejpam-5948	163	38	obtain	obtain	VERB
ejpam-5948	163	39	[	[	X
ejpam-5948	163	40	υ	υ	NOUN
ejpam-5948	163	41	,	,	PUNCT
ejpam-5948	163	42	κ]ℜ(∝	κ]ℜ(∝	PROPN
ejpam-5948	163	43	(	(	PUNCT
ejpam-5948	163	44	υ)±	υ)±	NOUN
ejpam-5948	163	45	χ(υ	χ(υ	NOUN
ejpam-5948	163	46	)	)	PUNCT
ejpam-5948	163	47	)	)	PUNCT
ejpam-5948	164	1	⊆	⊆	NUM
ejpam-5948	164	2	p	p	NOUN
ejpam-5948	164	3	for	for	ADP
ejpam-5948	164	4	all	all	DET
ejpam-5948	164	5	υ	υ	PROPN
ejpam-5948	164	6	,	,	PUNCT
ejpam-5948	164	7	κ	κ	PROPN
ejpam-5948	164	8	∈	∈	PROPN
ejpam-5948	164	9	ℜ.	ℜ.	PROPN
ejpam-5948	164	10	utilizing	utilize	VERB
ejpam-5948	164	11	the	the	DET
ejpam-5948	164	12	primeness	primeness	NOUN
ejpam-5948	164	13	of	of	ADP
ejpam-5948	164	14	p	p	NOUN
ejpam-5948	164	15	,	,	PUNCT
ejpam-5948	164	16	we	we	PRON
ejpam-5948	164	17	can	can	AUX
ejpam-5948	164	18	conclude	conclude	VERB
ejpam-5948	164	19	that	that	SCONJ
ejpam-5948	164	20	either	either	CCONJ
ejpam-5948	164	21	ℜ/p	ℜ/p	PROPN
ejpam-5948	164	22	is	be	AUX
ejpam-5948	164	23	an	an	DET
ejpam-5948	164	24	integral	integral	ADJ
ejpam-5948	164	25	domain	domain	NOUN
ejpam-5948	164	26	or	or	CCONJ
ejpam-5948	164	27	∝	∝	PROPN
ejpam-5948	164	28	(	(	PUNCT
ejpam-5948	164	29	υ)±	υ)±	NOUN
ejpam-5948	164	30	χ(υ	χ(υ	NOUN
ejpam-5948	164	31	)	)	PUNCT
ejpam-5948	164	32	∈	∈	PROPN
ejpam-5948	164	33	p	p	NOUN
ejpam-5948	164	34	for	for	ADP
ejpam-5948	164	35	all	all	PRON
ejpam-5948	164	36	υ	υ	DET
ejpam-5948	164	37	∈	∈	PROPN
ejpam-5948	164	38	ℜ.	ℜ.	PROPN
ejpam-5948	164	39	let	let	VERB
ejpam-5948	164	40	’s	’s	NOUN
ejpam-5948	164	41	examine	examine	VERB
ejpam-5948	164	42	the	the	DET
ejpam-5948	164	43	case	case	NOUN
ejpam-5948	164	44	when	when	SCONJ
ejpam-5948	164	45	∝	∝	PROPN
ejpam-5948	164	46	(	(	PUNCT
ejpam-5948	164	47	υ)±	υ)±	NOUN
ejpam-5948	164	48	χ(υ	χ(υ	PROPN
ejpam-5948	164	49	)	)	PUNCT
ejpam-5948	164	50	∈	∈	PROPN
ejpam-5948	164	51	p	p	NOUN
ejpam-5948	164	52	for	for	ADP
ejpam-5948	164	53	all	all	PRON
ejpam-5948	164	54	υ	υ	DET
ejpam-5948	164	55	∈	∈	PROPN
ejpam-5948	164	56	ℜ.	ℜ.	PROPN
ejpam-5948	164	57	(	(	PUNCT
ejpam-5948	164	58	14	14	NUM
ejpam-5948	164	59	)	)	PUNCT
ejpam-5948	164	60	this	this	PRON
ejpam-5948	164	61	reduces	reduce	VERB
ejpam-5948	164	62	equation	equation	NOUN
ejpam-5948	164	63	(	(	PUNCT
ejpam-5948	164	64	13	13	NUM
ejpam-5948	164	65	)	)	PUNCT
ejpam-5948	164	66	to	to	PART
ejpam-5948	164	67	℘[υ,∝	℘[υ,∝	NOUN
ejpam-5948	164	68	(	(	PUNCT
ejpam-5948	164	69	υ	υ	NOUN
ejpam-5948	164	70	)	)	PUNCT
ejpam-5948	164	71	]	]	PUNCT
ejpam-5948	165	1	∈	∈	PROPN
ejpam-5948	165	2	p	p	NOUN
ejpam-5948	165	3	for	for	ADP
ejpam-5948	165	4	all	all	DET
ejpam-5948	165	5	υ	υ	NOUN
ejpam-5948	165	6	,	,	PUNCT
ejpam-5948	165	7	℘	℘	PROPN
ejpam-5948	165	8	∈	∈	NOUN
ejpam-5948	165	9	ℜ.	ℜ.	VERB
ejpam-5948	165	10	the	the	DET
ejpam-5948	165	11	primeness	primeness	NOUN
ejpam-5948	165	12	of	of	ADP
ejpam-5948	165	13	p	p	NOUN
ejpam-5948	165	14	,	,	PUNCT
ejpam-5948	165	15	along	along	ADP
ejpam-5948	165	16	with	with	ADP
ejpam-5948	165	17	corollary	corollary	ADJ
ejpam-5948	165	18	1	1	NUM
ejpam-5948	165	19	,	,	PUNCT
ejpam-5948	165	20	forces	force	NOUN
ejpam-5948	165	21	either	either	CCONJ
ejpam-5948	165	22	ℜ/p	ℜ/p	PROPN
ejpam-5948	165	23	is	be	AUX
ejpam-5948	165	24	an	an	DET
ejpam-5948	165	25	integral	integral	ADJ
ejpam-5948	165	26	domain	domain	NOUN
ejpam-5948	165	27	or∝	or∝	NOUN
ejpam-5948	165	28	(	(	PUNCT
ejpam-5948	165	29	ℜ	ℜ	PROPN
ejpam-5948	165	30	)	)	PUNCT
ejpam-5948	165	31	⊆	⊆	NUM
ejpam-5948	165	32	p	p	NOUN
ejpam-5948	165	33	.	.	PUNCT
ejpam-5948	166	1	if∝	if∝	NOUN
ejpam-5948	166	2	(	(	PUNCT
ejpam-5948	166	3	ℜ	ℜ	PROPN
ejpam-5948	166	4	)	)	PUNCT
ejpam-5948	166	5	⊆	⊆	NUM
ejpam-5948	166	6	p	p	NOUN
ejpam-5948	166	7	for	for	ADP
ejpam-5948	166	8	all	all	DET
ejpam-5948	166	9	υ	υ	PRON
ejpam-5948	166	10	∈	∈	PROPN
ejpam-5948	166	11	ℜ	ℜ	PROPN
ejpam-5948	166	12	,	,	PUNCT
ejpam-5948	166	13	then	then	ADV
ejpam-5948	166	14	equation	equation	NOUN
ejpam-5948	166	15	(	(	PUNCT
ejpam-5948	166	16	14	14	NUM
ejpam-5948	166	17	)	)	PUNCT
ejpam-5948	166	18	becomes	become	VERB
ejpam-5948	166	19	χ(υ	χ(υ	NOUN
ejpam-5948	166	20	)	)	PUNCT
ejpam-5948	166	21	∈	∈	PROPN
ejpam-5948	166	22	p	p	NOUN
ejpam-5948	166	23	for	for	ADP
ejpam-5948	166	24	all	all	PRON
ejpam-5948	166	25	υ	υ	PRON
ejpam-5948	166	26	∈	∈	NOUN
ejpam-5948	166	27	ℜ.	ℜ.	PROPN
ejpam-5948	166	28	hence	hence	ADV
ejpam-5948	166	29	,	,	PUNCT
ejpam-5948	166	30	equation	equation	NOUN
ejpam-5948	166	31	(	(	PUNCT
ejpam-5948	166	32	12	12	NUM
ejpam-5948	166	33	)	)	PUNCT
ejpam-5948	166	34	reduces	reduce	VERB
ejpam-5948	166	35	to	to	PART
ejpam-5948	166	36	(	(	PUNCT
ejpam-5948	166	37	⨿(℘	⨿(℘	NOUN
ejpam-5948	166	38	)	)	PUNCT
ejpam-5948	166	39	±	±	NUM
ejpam-5948	166	40	℧	℧	PROPN
ejpam-5948	166	41	(	(	PUNCT
ejpam-5948	166	42	℘))[υ	℘))[υ	PROPN
ejpam-5948	166	43	,	,	PUNCT
ejpam-5948	166	44	ℏ	ℏ	X
ejpam-5948	166	45	]	]	X
ejpam-5948	166	46	∈	∈	PROPN
ejpam-5948	166	47	p	p	NOUN
ejpam-5948	166	48	for	for	ADP
ejpam-5948	166	49	all	all	DET
ejpam-5948	166	50	υ	υ	NOUN
ejpam-5948	166	51	,	,	PUNCT
ejpam-5948	166	52	℘	℘	PROPN
ejpam-5948	166	53	,	,	PUNCT
ejpam-5948	166	54	ℏ	ℏ	PROPN
ejpam-5948	166	55	∈	∈	NOUN
ejpam-5948	166	56	ℜ.	ℜ.	PROPN
ejpam-5948	166	57	for	for	ADP
ejpam-5948	166	58	any	any	DET
ejpam-5948	166	59	τ	τ	PROPN
ejpam-5948	166	60	∈	∈	PROPN
ejpam-5948	166	61	ℜ	ℜ	PROPN
ejpam-5948	166	62	,	,	PUNCT
ejpam-5948	166	63	replacing	replace	VERB
ejpam-5948	166	64	υ	υ	PRON
ejpam-5948	166	65	by	by	ADP
ejpam-5948	166	66	υτ	υτ	NOUN
ejpam-5948	166	67	in	in	ADP
ejpam-5948	166	68	the	the	DET
ejpam-5948	166	69	last	last	ADJ
ejpam-5948	166	70	equation	equation	NOUN
ejpam-5948	166	71	and	and	CCONJ
ejpam-5948	166	72	using	use	VERB
ejpam-5948	166	73	it	it	PRON
ejpam-5948	166	74	,	,	PUNCT
ejpam-5948	166	75	we	we	PRON
ejpam-5948	166	76	get	get	AUX
ejpam-5948	166	77	(	(	PUNCT
ejpam-5948	166	78	⨿(℘)±	⨿(℘)±	ADJ
ejpam-5948	166	79	℧	℧	PROPN
ejpam-5948	166	80	(℘))ℜ[τ	(℘))ℜ[τ	PROPN
ejpam-5948	166	81	,	,	PUNCT
ejpam-5948	166	82	ℏ	ℏ	X
ejpam-5948	166	83	]	]	X
ejpam-5948	166	84	⊆	⊆	NUM
ejpam-5948	166	85	p	p	NOUN
ejpam-5948	166	86	for	for	ADP
ejpam-5948	166	87	all	all	DET
ejpam-5948	166	88	τ	τ	PROPN
ejpam-5948	166	89	,	,	PUNCT
ejpam-5948	166	90	℘	℘	PROPN
ejpam-5948	166	91	,	,	PUNCT
ejpam-5948	166	92	ℏ	ℏ	PROPN
ejpam-5948	166	93	∈	∈	NOUN
ejpam-5948	166	94	ℜ.	ℜ.	PROPN
ejpam-5948	166	95	again	again	ADV
ejpam-5948	166	96	,	,	PUNCT
ejpam-5948	166	97	the	the	DET
ejpam-5948	166	98	primeness	primeness	NOUN
ejpam-5948	166	99	of	of	ADP
ejpam-5948	166	100	p	p	PROPN
ejpam-5948	166	101	gives	give	VERB
ejpam-5948	166	102	either	either	CCONJ
ejpam-5948	166	103	ℜ/p	ℜ/p	PROPN
ejpam-5948	166	104	is	be	AUX
ejpam-5948	166	105	an	an	DET
ejpam-5948	166	106	integral	integral	ADJ
ejpam-5948	166	107	domain	domain	NOUN
ejpam-5948	166	108	or	or	CCONJ
ejpam-5948	166	109	⨿(℘)±	⨿(℘)±	NUM
ejpam-5948	166	110	℧	℧	NOUN
ejpam-5948	166	111	(℘	(℘	NUM
ejpam-5948	166	112	)	)	PUNCT
ejpam-5948	166	113	∈	∈	PROPN
ejpam-5948	166	114	p	p	NOUN
ejpam-5948	166	115	for	for	ADP
ejpam-5948	166	116	all	all	DET
ejpam-5948	166	117	℘	℘	PROPN
ejpam-5948	166	118	∈	∈	NOUN
ejpam-5948	166	119	ℜ.	ℜ.	PROPN
ejpam-5948	166	120	therefore	therefore	ADV
ejpam-5948	166	121	,	,	PUNCT
ejpam-5948	166	122	(	(	PUNCT
ejpam-5948	166	123	⨿±	⨿±	VERB
ejpam-5948	166	124	℧	℧	NOUN
ejpam-5948	166	125	)	)	PUNCT
ejpam-5948	166	126	(	(	PUNCT
ejpam-5948	166	127	ℜ	ℜ	PROPN
ejpam-5948	166	128	)	)	PUNCT
ejpam-5948	166	129	⊆	⊆	NUM
ejpam-5948	166	130	p	p	NOUN
ejpam-5948	166	131	.	.	PUNCT
ejpam-5948	167	1	now	now	ADV
ejpam-5948	167	2	we	we	PRON
ejpam-5948	167	3	are	be	AUX
ejpam-5948	167	4	prepared	prepared	ADJ
ejpam-5948	167	5	to	to	PART
ejpam-5948	167	6	gather	gather	VERB
ejpam-5948	167	7	several	several	ADJ
ejpam-5948	167	8	corollaries	corollary	NOUN
ejpam-5948	167	9	as	as	ADP
ejpam-5948	167	10	applications	application	NOUN
ejpam-5948	167	11	of	of	ADP
ejpam-5948	167	12	theorem	theorem	NOUN
ejpam-5948	167	13	4	4	NUM
ejpam-5948	167	14	as	as	SCONJ
ejpam-5948	167	15	follows	follow	VERB
ejpam-5948	167	16	:	:	PUNCT
ejpam-5948	167	17	corollary	corollary	ADJ
ejpam-5948	167	18	4	4	X
ejpam-5948	167	19	.	.	PUNCT
ejpam-5948	168	1	let	let	VERB
ejpam-5948	168	2	ℜ	ℜ	PROPN
ejpam-5948	168	3	be	be	AUX
ejpam-5948	168	4	a	a	DET
ejpam-5948	168	5	ring	ring	NOUN
ejpam-5948	168	6	equipped	equip	VERB
ejpam-5948	168	7	with	with	ADP
ejpam-5948	168	8	p	p	PROPN
ejpam-5948	168	9	-derivations	-derivation	NOUN
ejpam-5948	168	10	χ	χ	NOUN
ejpam-5948	168	11	and	and	CCONJ
ejpam-5948	168	12	∝	∝	PRON
ejpam-5948	168	13	such	such	ADJ
ejpam-5948	168	14	that	that	SCONJ
ejpam-5948	168	15	[	[	X
ejpam-5948	168	16	υ,∝	υ,∝	INTJ
ejpam-5948	168	17	(	(	PUNCT
ejpam-5948	168	18	℘)]±	℘)]±	NOUN
ejpam-5948	168	19	χ([υ	χ([υ	PROPN
ejpam-5948	168	20	,	,	PUNCT
ejpam-5948	168	21	℘	℘	PROPN
ejpam-5948	168	22	]	]	PUNCT
ejpam-5948	168	23	)	)	PUNCT
ejpam-5948	168	24	∈	∈	PROPN
ejpam-5948	168	25	p	p	NOUN
ejpam-5948	168	26	for	for	ADP
ejpam-5948	168	27	all	all	DET
ejpam-5948	168	28	υ	υ	NOUN
ejpam-5948	168	29	,	,	PUNCT
ejpam-5948	168	30	℘	℘	PROPN
ejpam-5948	168	31	∈	∈	NOUN
ejpam-5948	168	32	ℜ.	ℜ.	PROPN
ejpam-5948	168	33	then	then	ADV
ejpam-5948	168	34	ℜ/p	ℜ/p	PROPN
ejpam-5948	168	35	is	be	AUX
ejpam-5948	168	36	an	an	DET
ejpam-5948	168	37	integral	integral	ADJ
ejpam-5948	168	38	domain	domain	NOUN
ejpam-5948	168	39	or	or	CCONJ
ejpam-5948	168	40	(	(	PUNCT
ejpam-5948	168	41	∝	∝	PROPN
ejpam-5948	168	42	±χ)(ℜ	±χ)(ℜ	PROPN
ejpam-5948	168	43	)	)	PUNCT
ejpam-5948	168	44	⊆	⊆	NUM
ejpam-5948	168	45	p	p	NOUN
ejpam-5948	168	46	.	.	PUNCT
ejpam-5948	169	1	corollary	corollary	ADJ
ejpam-5948	169	2	5	5	NUM
ejpam-5948	169	3	.	.	PUNCT
ejpam-5948	170	1	let	let	VERB
ejpam-5948	170	2	ℜ	ℜ	PROPN
ejpam-5948	170	3	be	be	AUX
ejpam-5948	170	4	a	a	DET
ejpam-5948	170	5	ring	ring	NOUN
ejpam-5948	170	6	with	with	ADP
ejpam-5948	170	7	char(ℜ/p	char(ℜ/p	PROPN
ejpam-5948	170	8	)	)	PUNCT
ejpam-5948	170	9	̸=	̸=	PROPN
ejpam-5948	170	10	2	2	NUM
ejpam-5948	170	11	.	.	PUNCT
ejpam-5948	171	1	if	if	SCONJ
ejpam-5948	171	2	ℜ	ℜ	PROPN
ejpam-5948	171	3	is	be	AUX
ejpam-5948	171	4	equipped	equip	VERB
ejpam-5948	171	5	with	with	ADP
ejpam-5948	171	6	a	a	DET
ejpam-5948	171	7	generalized	generalized	ADJ
ejpam-5948	171	8	p	p	NOUN
ejpam-5948	171	9	-derivation	-derivation	NOUN
ejpam-5948	171	10	(	(	PUNCT
ejpam-5948	171	11	℧	℧	PROPN
ejpam-5948	171	12	,	,	PUNCT
ejpam-5948	171	13	χ	χ	NOUN
ejpam-5948	171	14	)	)	PUNCT
ejpam-5948	171	15	such	such	ADJ
ejpam-5948	171	16	that	that	SCONJ
ejpam-5948	172	1	[	[	X
ejpam-5948	172	2	υ,	υ,	X
ejpam-5948	172	3	℧	℧	NOUN
ejpam-5948	172	4	(℘	(℘	NUM
ejpam-5948	172	5	)	)	PUNCT
ejpam-5948	172	6	]	]	PUNCT
ejpam-5948	173	1	+	+	CCONJ
ejpam-5948	173	2	℧	℧	PROPN
ejpam-5948	173	3	(	(	PUNCT
ejpam-5948	173	4	[	[	X
ejpam-5948	173	5	υ	υ	INTJ
ejpam-5948	173	6	,	,	PUNCT
ejpam-5948	173	7	℘	℘	PROPN
ejpam-5948	173	8	]	]	PUNCT
ejpam-5948	173	9	)	)	PUNCT
ejpam-5948	173	10	∈	∈	PROPN
ejpam-5948	173	11	p	p	NOUN
ejpam-5948	173	12	for	for	ADP
ejpam-5948	173	13	all	all	DET
ejpam-5948	173	14	υ	υ	NOUN
ejpam-5948	173	15	,	,	PUNCT
ejpam-5948	173	16	℘	℘	PROPN
ejpam-5948	173	17	∈	∈	PROPN
ejpam-5948	173	18	ℜ	ℜ	PROPN
ejpam-5948	173	19	,	,	PUNCT
ejpam-5948	173	20	then	then	ADV
ejpam-5948	173	21	ℜ/p	ℜ/p	PROPN
ejpam-5948	173	22	is	be	AUX
ejpam-5948	173	23	an	an	DET
ejpam-5948	173	24	integral	integral	ADJ
ejpam-5948	173	25	domain	domain	NOUN
ejpam-5948	173	26	or	or	CCONJ
ejpam-5948	173	27	℧	℧	PROPN
ejpam-5948	173	28	(	(	PUNCT
ejpam-5948	173	29	ℜ	ℜ	PROPN
ejpam-5948	173	30	)	)	PUNCT
ejpam-5948	173	31	⊆	⊆	NUM
ejpam-5948	173	32	p	p	NOUN
ejpam-5948	173	33	.	.	PUNCT
ejpam-5948	174	1	by	by	ADP
ejpam-5948	174	2	setting	set	VERB
ejpam-5948	174	3	⨿	⨿	NOUN
ejpam-5948	174	4	=	=	SYM
ejpam-5948	174	5	idℜ	idℜ	PROPN
ejpam-5948	174	6	,	,	PUNCT
ejpam-5948	174	7	we	we	PRON
ejpam-5948	174	8	obtain	obtain	VERB
ejpam-5948	174	9	a	a	DET
ejpam-5948	174	10	generalization	generalization	NOUN
ejpam-5948	174	11	of	of	ADP
ejpam-5948	174	12	[	[	X
ejpam-5948	174	13	16	16	NUM
ejpam-5948	174	14	,	,	PUNCT
ejpam-5948	174	15	theorem	theorem	VERB
ejpam-5948	174	16	1	1	NUM
ejpam-5948	174	17	]	]	PUNCT
ejpam-5948	174	18	as	as	SCONJ
ejpam-5948	174	19	shown	show	VERB
ejpam-5948	174	20	in	in	ADP
ejpam-5948	174	21	the	the	DET
ejpam-5948	174	22	following	follow	VERB
ejpam-5948	174	23	corollary	corollary	NOUN
ejpam-5948	174	24	:	:	PUNCT
ejpam-5948	174	25	corollary	corollary	ADJ
ejpam-5948	174	26	6	6	NUM
ejpam-5948	174	27	.	.	PUNCT
ejpam-5948	175	1	let	let	VERB
ejpam-5948	175	2	ℜ	ℜ	PROPN
ejpam-5948	175	3	be	be	AUX
ejpam-5948	175	4	a	a	DET
ejpam-5948	175	5	ring	ring	NOUN
ejpam-5948	175	6	equipped	equip	VERB
ejpam-5948	175	7	with	with	ADP
ejpam-5948	175	8	a	a	DET
ejpam-5948	175	9	generalized	generalized	ADJ
ejpam-5948	175	10	p	p	NOUN
ejpam-5948	175	11	-derivation	-derivation	NOUN
ejpam-5948	175	12	(	(	PUNCT
ejpam-5948	175	13	℧	℧	PROPN
ejpam-5948	175	14	,	,	PUNCT
ejpam-5948	175	15	χ	χ	NOUN
ejpam-5948	175	16	)	)	PUNCT
ejpam-5948	176	1	such	such	ADJ
ejpam-5948	176	2	that	that	SCONJ
ejpam-5948	176	3	℧	℧	PROPN
ejpam-5948	176	4	(	(	PUNCT
ejpam-5948	176	5	[	[	X
ejpam-5948	176	6	υ	υ	INTJ
ejpam-5948	176	7	,	,	PUNCT
ejpam-5948	176	8	℘])±([υ	℘])±([υ	PROPN
ejpam-5948	176	9	,	,	PUNCT
ejpam-5948	176	10	℘	℘	PROPN
ejpam-5948	176	11	]	]	PUNCT
ejpam-5948	176	12	)	)	PUNCT
ejpam-5948	176	13	∈	∈	PROPN
ejpam-5948	176	14	p	p	NOUN
ejpam-5948	176	15	for	for	ADP
ejpam-5948	176	16	all	all	DET
ejpam-5948	176	17	υ	υ	NOUN
ejpam-5948	176	18	,	,	PUNCT
ejpam-5948	176	19	℘	℘	PROPN
ejpam-5948	176	20	∈	∈	NOUN
ejpam-5948	176	21	ℜ.	ℜ.	PROPN
ejpam-5948	176	22	then	then	ADV
ejpam-5948	176	23	ℜ/p	ℜ/p	PROPN
ejpam-5948	176	24	is	be	AUX
ejpam-5948	176	25	an	an	DET
ejpam-5948	176	26	integral	integral	ADJ
ejpam-5948	176	27	domain	domain	NOUN
ejpam-5948	176	28	or	or	CCONJ
ejpam-5948	176	29	(	(	PUNCT
ejpam-5948	176	30	℧	℧	NOUN
ejpam-5948	176	31	±idℜ)(ℜ	±idℜ)(ℜ	NOUN
ejpam-5948	176	32	)	)	PUNCT
ejpam-5948	176	33	⊆	⊆	NUM
ejpam-5948	176	34	p	p	NOUN
ejpam-5948	176	35	.	.	PUNCT
ejpam-5948	177	1	by	by	ADP
ejpam-5948	177	2	setting	set	VERB
ejpam-5948	177	3	℧	℧	PROPN
ejpam-5948	177	4	=	=	SYM
ejpam-5948	177	5	χ	χ	NOUN
ejpam-5948	177	6	,	,	PUNCT
ejpam-5948	177	7	we	we	PRON
ejpam-5948	177	8	get	get	VERB
ejpam-5948	177	9	a	a	DET
ejpam-5948	177	10	generalization	generalization	NOUN
ejpam-5948	177	11	of	of	ADP
ejpam-5948	177	12	[	[	X
ejpam-5948	177	13	17	17	NUM
ejpam-5948	177	14	,	,	PUNCT
ejpam-5948	177	15	theorem	theorem	VERB
ejpam-5948	177	16	3	3	NUM
ejpam-5948	177	17	]	]	PUNCT
ejpam-5948	177	18	as	as	SCONJ
ejpam-5948	177	19	shown	show	VERB
ejpam-5948	177	20	in	in	ADP
ejpam-5948	177	21	the	the	DET
ejpam-5948	177	22	following	follow	VERB
ejpam-5948	177	23	corollary	corollary	NOUN
ejpam-5948	177	24	:	:	PUNCT
ejpam-5948	177	25	corollary	corollary	ADJ
ejpam-5948	177	26	7	7	NUM
ejpam-5948	177	27	.	.	PUNCT
ejpam-5948	178	1	let	let	VERB
ejpam-5948	178	2	ℜ	ℜ	PROPN
ejpam-5948	178	3	be	be	AUX
ejpam-5948	178	4	a	a	DET
ejpam-5948	178	5	ring	ring	NOUN
ejpam-5948	178	6	equipped	equip	VERB
ejpam-5948	178	7	with	with	ADP
ejpam-5948	178	8	a	a	DET
ejpam-5948	178	9	p	p	NOUN
ejpam-5948	178	10	-derivation	-derivation	NOUN
ejpam-5948	178	11	χ	χ	ADP
ejpam-5948	178	12	such	such	ADJ
ejpam-5948	178	13	that	that	DET
ejpam-5948	178	14	χ([υ	χ([υ	PROPN
ejpam-5948	178	15	,	,	PUNCT
ejpam-5948	178	16	℘])±[υ	℘])±[υ	NOUN
ejpam-5948	178	17	,	,	PUNCT
ejpam-5948	178	18	℘	℘	PROPN
ejpam-5948	178	19	]	]	PUNCT
ejpam-5948	178	20	∈	∈	PROPN
ejpam-5948	178	21	p	p	NOUN
ejpam-5948	178	22	for	for	ADP
ejpam-5948	178	23	all	all	DET
ejpam-5948	178	24	υ	υ	NOUN
ejpam-5948	178	25	,	,	PUNCT
ejpam-5948	178	26	℘	℘	PROPN
ejpam-5948	178	27	∈	∈	NOUN
ejpam-5948	178	28	ℜ.	ℜ.	PROPN
ejpam-5948	178	29	then	then	ADV
ejpam-5948	178	30	ℜ/p	ℜ/p	PROPN
ejpam-5948	178	31	is	be	AUX
ejpam-5948	178	32	an	an	DET
ejpam-5948	178	33	integral	integral	ADJ
ejpam-5948	178	34	domain	domain	NOUN
ejpam-5948	178	35	.	.	PUNCT
ejpam-5948	179	1	the	the	DET
ejpam-5948	179	2	following	following	ADJ
ejpam-5948	179	3	example	example	NOUN
ejpam-5948	179	4	is	be	AUX
ejpam-5948	179	5	devoted	devote	VERB
ejpam-5948	179	6	to	to	ADP
ejpam-5948	179	7	explaining	explain	VERB
ejpam-5948	179	8	the	the	DET
ejpam-5948	179	9	significance	significance	NOUN
ejpam-5948	179	10	of	of	ADP
ejpam-5948	179	11	the	the	DET
ejpam-5948	179	12	primeness	primeness	NOUN
ejpam-5948	179	13	hypothesis	hypothesis	NOUN
ejpam-5948	179	14	of	of	ADP
ejpam-5948	179	15	p	p	NOUN
ejpam-5948	179	16	in	in	ADP
ejpam-5948	179	17	the	the	DET
ejpam-5948	179	18	previous	previous	ADJ
ejpam-5948	179	19	theorems	theorem	NOUN
ejpam-5948	179	20	.	.	PUNCT
ejpam-5948	179	21	example	example	NOUN
ejpam-5948	180	1	1	1	NUM
ejpam-5948	180	2	.	.	PUNCT
ejpam-5948	180	3	let	let	VERB
ejpam-5948	180	4	ℜ	ℜ	PROPN
ejpam-5948	180	5	=	=	SYM
ejpam-5948	180	6	c[υ]×m2(ψ	c[υ]×m2(ψ	PROPN
ejpam-5948	180	7	)	)	PUNCT
ejpam-5948	180	8	,	,	PUNCT
ejpam-5948	180	9	where	where	SCONJ
ejpam-5948	180	10	c[υ	c[υ	NOUN
ejpam-5948	180	11	]	]	X
ejpam-5948	180	12	is	be	AUX
ejpam-5948	180	13	the	the	DET
ejpam-5948	180	14	polynomial	polynomial	ADJ
ejpam-5948	180	15	ring	ring	NOUN
ejpam-5948	180	16	of	of	ADP
ejpam-5948	180	17	complex	complex	ADJ
ejpam-5948	180	18	numbers	number	NOUN
ejpam-5948	180	19	with	with	ADP
ejpam-5948	180	20	determinant	determinant	ADJ
ejpam-5948	180	21	υ	υ	NOUN
ejpam-5948	180	22	,	,	PUNCT
ejpam-5948	180	23	and	and	CCONJ
ejpam-5948	180	24	let	let	VERB
ejpam-5948	180	25	ψ	ψ	PART
ejpam-5948	180	26	be	be	AUX
ejpam-5948	180	27	any	any	DET
ejpam-5948	180	28	ring	ring	NOUN
ejpam-5948	180	29	.	.	PUNCT
ejpam-5948	181	1	let	let	VERB
ejpam-5948	181	2	p	p	NOUN
ejpam-5948	181	3	=	=	PRON
ejpam-5948	181	4	{	{	PUNCT
ejpam-5948	181	5	(	(	PUNCT
ejpam-5948	181	6	0	0	NUM
ejpam-5948	181	7	,	,	PUNCT
ejpam-5948	181	8	0	0	NUM
ejpam-5948	181	9	)	)	PUNCT
ejpam-5948	181	10	}	}	PUNCT
ejpam-5948	181	11	.	.	PUNCT
ejpam-5948	182	1	define	define	VERB
ejpam-5948	182	2	(	(	PUNCT
ejpam-5948	182	3	℧	℧	PROPN
ejpam-5948	182	4	,	,	PUNCT
ejpam-5948	182	5	χ	χ	NOUN
ejpam-5948	182	6	)	)	PUNCT
ejpam-5948	182	7	,	,	PUNCT
ejpam-5948	182	8	(	(	PUNCT
ejpam-5948	182	9	⨿,∝	⨿,∝	X
ejpam-5948	182	10	)	)	PUNCT
ejpam-5948	182	11	:	:	PUNCT
ejpam-5948	182	12	ℜ	ℜ	ADV
ejpam-5948	182	13	−→	−→	NOUN
ejpam-5948	182	14	ℜ	ℜ	ADJ
ejpam-5948	182	15	by	by	ADP
ejpam-5948	182	16	℧	℧	PROPN
ejpam-5948	182	17	(	(	PUNCT
ejpam-5948	182	18	t(υ	t(υ	NOUN
ejpam-5948	182	19	)	)	PUNCT
ejpam-5948	182	20	,	,	PUNCT
ejpam-5948	182	21	ψ	ψ	X
ejpam-5948	182	22	)	)	PUNCT
ejpam-5948	182	23	=	=	SYM
ejpam-5948	182	24	χ(t(υ	χ(t(υ	NOUN
ejpam-5948	182	25	)	)	PUNCT
ejpam-5948	182	26	,	,	PUNCT
ejpam-5948	182	27	ψ	ψ	X
ejpam-5948	182	28	)	)	PUNCT
ejpam-5948	182	29	=	=	SYM
ejpam-5948	182	30	(	(	PUNCT
ejpam-5948	182	31	t′(υ	t′(υ	PROPN
ejpam-5948	182	32	)	)	PUNCT
ejpam-5948	182	33	<	<	X
ejpam-5948	182	34	υ2	υ2	PROPN
ejpam-5948	182	35	>	>	X
ejpam-5948	182	36	,	,	PUNCT
ejpam-5948	182	37	0	0	NUM
ejpam-5948	182	38	)	)	PUNCT
ejpam-5948	182	39	,	,	PUNCT
ejpam-5948	182	40	a.y	a.y	PROPN
ejpam-5948	182	41	.	.	PROPN
ejpam-5948	182	42	hummdi	hummdi	PROPN
ejpam-5948	182	43	,	,	PUNCT
ejpam-5948	182	44	r.m	r.m	PROPN
ejpam-5948	182	45	.	.	PROPN
ejpam-5948	182	46	al	al	PROPN
ejpam-5948	182	47	-	-	PUNCT
ejpam-5948	182	48	omary	omary	PROPN
ejpam-5948	182	49	,	,	PUNCT
ejpam-5948	182	50	z.z	z.z	PROPN
ejpam-5948	182	51	.	.	PUNCT
ejpam-5948	182	52	al	al	PROPN
ejpam-5948	182	53	-	-	PUNCT
ejpam-5948	182	54	amery	amery	PROPN
ejpam-5948	182	55	/	/	SYM
ejpam-5948	182	56	eur	eur	PROPN
ejpam-5948	182	57	.	.	PUNCT
ejpam-5948	183	1	j.	j.	PROPN
ejpam-5948	183	2	pure	pure	PROPN
ejpam-5948	183	3	appl	appl	PROPN
ejpam-5948	183	4	.	.	PROPN
ejpam-5948	183	5	math	math	PROPN
ejpam-5948	183	6	,	,	PUNCT
ejpam-5948	183	7	18	18	NUM
ejpam-5948	183	8	(	(	PUNCT
ejpam-5948	183	9	2	2	NUM
ejpam-5948	183	10	)	)	PUNCT
ejpam-5948	183	11	(	(	PUNCT
ejpam-5948	183	12	2025	2025	NUM
ejpam-5948	183	13	)	)	PUNCT
ejpam-5948	183	14	,	,	PUNCT
ejpam-5948	183	15	5948	5948	NUM
ejpam-5948	183	16	8	8	NUM
ejpam-5948	183	17	of	of	ADP
ejpam-5948	183	18	12	12	NUM
ejpam-5948	183	19	and	and	CCONJ
ejpam-5948	183	20	⨿(t(υ	⨿(t(υ	NOUN
ejpam-5948	183	21	)	)	PUNCT
ejpam-5948	183	22	,	,	PUNCT
ejpam-5948	183	23	ψ	ψ	X
ejpam-5948	183	24	)	)	PUNCT
ejpam-5948	184	1	=	=	NOUN
ejpam-5948	184	2	∝	∝	X
ejpam-5948	184	3	(	(	PUNCT
ejpam-5948	184	4	t(υ	t(υ	NOUN
ejpam-5948	184	5	)	)	PUNCT
ejpam-5948	184	6	,	,	PUNCT
ejpam-5948	184	7	ψ	ψ	X
ejpam-5948	184	8	)	)	PUNCT
ejpam-5948	184	9	=	=	SYM
ejpam-5948	184	10	(	(	PUNCT
ejpam-5948	184	11	t′(υ	t′(υ	PROPN
ejpam-5948	184	12	)	)	PUNCT
ejpam-5948	184	13	<	<	X
ejpam-5948	184	14	υ3	υ3	PROPN
ejpam-5948	184	15	>	>	X
ejpam-5948	184	16	,	,	PUNCT
ejpam-5948	184	17	0	0	NUM
ejpam-5948	184	18	)	)	PUNCT
ejpam-5948	184	19	.	.	PUNCT
ejpam-5948	185	1	it	it	PRON
ejpam-5948	185	2	is	be	AUX
ejpam-5948	185	3	easy	easy	ADJ
ejpam-5948	185	4	to	to	PART
ejpam-5948	185	5	verify	verify	VERB
ejpam-5948	185	6	that	that	PRON
ejpam-5948	185	7	℧	℧	PROPN
ejpam-5948	185	8	and	and	CCONJ
ejpam-5948	185	9	⨿	⨿	NOUN
ejpam-5948	185	10	are	be	AUX
ejpam-5948	185	11	generalized	generalized	ADJ
ejpam-5948	185	12	derivations	derivation	NOUN
ejpam-5948	185	13	of	of	ADP
ejpam-5948	185	14	ℜ	ℜ	PROPN
ejpam-5948	185	15	associated	associate	VERB
ejpam-5948	185	16	with	with	ADP
ejpam-5948	185	17	derivations	derivation	NOUN
ejpam-5948	185	18	χ	χ	NOUN
ejpam-5948	185	19	and	and	CCONJ
ejpam-5948	185	20	∝	∝	PROPN
ejpam-5948	185	21	,	,	PUNCT
ejpam-5948	185	22	respectively	respectively	ADV
ejpam-5948	185	23	.	.	PUNCT
ejpam-5948	186	1	it	it	PRON
ejpam-5948	186	2	can	can	AUX
ejpam-5948	186	3	also	also	ADV
ejpam-5948	186	4	be	be	AUX
ejpam-5948	186	5	verified	verify	VERB
ejpam-5948	186	6	that	that	SCONJ
ejpam-5948	186	7	ℜ	ℜ	PROPN
ejpam-5948	186	8	satisfies	satisfy	VERB
ejpam-5948	186	9	the	the	DET
ejpam-5948	186	10	identities	identity	NOUN
ejpam-5948	186	11	in	in	ADP
ejpam-5948	186	12	theorems	theorem	NOUN
ejpam-5948	186	13	1	1	NUM
ejpam-5948	186	14	,	,	PUNCT
ejpam-5948	186	15	3	3	NUM
ejpam-5948	186	16	(	(	PUNCT
ejpam-5948	186	17	i	i	NOUN
ejpam-5948	186	18	)	)	PUNCT
ejpam-5948	186	19	,	,	PUNCT
ejpam-5948	186	20	and	and	CCONJ
ejpam-5948	187	1	4	4	X
ejpam-5948	187	2	.	.	PUNCT
ejpam-5948	188	1	however	however	ADV
ejpam-5948	188	2	,	,	PUNCT
ejpam-5948	188	3	neither	neither	CCONJ
ejpam-5948	188	4	ℜ/p	ℜ/p	PROPN
ejpam-5948	188	5	is	be	AUX
ejpam-5948	188	6	an	an	DET
ejpam-5948	188	7	integral	integral	ADJ
ejpam-5948	188	8	domain	domain	NOUN
ejpam-5948	188	9	nor	nor	CCONJ
ejpam-5948	188	10	χ	χ	NOUN
ejpam-5948	188	11	,	,	PUNCT
ejpam-5948	188	12	℧	℧	PROPN
ejpam-5948	188	13	,	,	PUNCT
ejpam-5948	188	14	⨿	⨿	NOUN
ejpam-5948	188	15	and	and	CCONJ
ejpam-5948	188	16	℧	℧	PROPN
ejpam-5948	188	17	±	±	NOUN
ejpam-5948	188	18	⨿	⨿	PRON
ejpam-5948	188	19	map	map	VERB
ejpam-5948	188	20	ℜ	ℜ	PROPN
ejpam-5948	188	21	to	to	ADP
ejpam-5948	188	22	p	p	NOUN
ejpam-5948	188	23	.	.	PUNCT
ejpam-5948	189	1	it	it	PRON
ejpam-5948	189	2	is	be	AUX
ejpam-5948	189	3	important	important	ADJ
ejpam-5948	189	4	to	to	PART
ejpam-5948	189	5	note	note	VERB
ejpam-5948	189	6	that	that	SCONJ
ejpam-5948	189	7	p	p	NOUN
ejpam-5948	189	8	is	be	AUX
ejpam-5948	189	9	not	not	PART
ejpam-5948	189	10	a	a	DET
ejpam-5948	189	11	prime	prime	ADJ
ejpam-5948	189	12	ideal	ideal	NOUN
ejpam-5948	189	13	of	of	ADP
ejpam-5948	189	14	ℜ	ℜ	PROPN
ejpam-5948	189	15	,	,	PUNCT
ejpam-5948	189	16	since	since	SCONJ
ejpam-5948	189	17	(	(	PUNCT
ejpam-5948	189	18	0	0	NUM
ejpam-5948	189	19	,	,	PUNCT
ejpam-5948	189	20	(	(	PUNCT
ejpam-5948	189	21	υ	υ	NOUN
ejpam-5948	189	22	0	0	NUM
ejpam-5948	189	23	0	0	NUM
ejpam-5948	189	24	0	0	NUM
ejpam-5948	189	25	)	)	PUNCT
ejpam-5948	189	26	)	)	PUNCT
ejpam-5948	190	1	(	(	PUNCT
ejpam-5948	190	2	0	0	NUM
ejpam-5948	190	3	,	,	PUNCT
ejpam-5948	190	4	(	(	PUNCT
ejpam-5948	190	5	0	0	NUM
ejpam-5948	190	6	0	0	NUM
ejpam-5948	190	7	0	0	NUM
ejpam-5948	190	8	℘	℘	PROPN
ejpam-5948	190	9	)	)	PUNCT
ejpam-5948	190	10	)	)	PUNCT
ejpam-5948	191	1	∈	∈	PROPN
ejpam-5948	191	2	p	p	NOUN
ejpam-5948	191	3	,	,	PUNCT
ejpam-5948	191	4	but	but	CCONJ
ejpam-5948	191	5	neither	neither	CCONJ
ejpam-5948	191	6	(	(	PUNCT
ejpam-5948	191	7	0	0	NUM
ejpam-5948	191	8	,	,	PUNCT
ejpam-5948	191	9	(	(	PUNCT
ejpam-5948	191	10	υ	υ	NOUN
ejpam-5948	191	11	0	0	NUM
ejpam-5948	191	12	0	0	NUM
ejpam-5948	191	13	0	0	NUM
ejpam-5948	191	14	)	)	PUNCT
ejpam-5948	191	15	)	)	PUNCT
ejpam-5948	192	1	∈	∈	PROPN
ejpam-5948	192	2	p	p	NOUN
ejpam-5948	192	3	nor	nor	CCONJ
ejpam-5948	192	4	(	(	PUNCT
ejpam-5948	192	5	0	0	NUM
ejpam-5948	192	6	,	,	PUNCT
ejpam-5948	192	7	(	(	PUNCT
ejpam-5948	192	8	0	0	NUM
ejpam-5948	192	9	0	0	NUM
ejpam-5948	192	10	0	0	NUM
ejpam-5948	192	11	℘	℘	PROPN
ejpam-5948	192	12	)	)	PUNCT
ejpam-5948	192	13	)	)	PUNCT
ejpam-5948	193	1	∈	∈	PROPN
ejpam-5948	193	2	p	p	NOUN
ejpam-5948	193	3	.	.	PUNCT
ejpam-5948	194	1	therefore	therefore	ADV
ejpam-5948	194	2	,	,	PUNCT
ejpam-5948	194	3	the	the	DET
ejpam-5948	194	4	primeness	primeness	NOUN
ejpam-5948	194	5	condition	condition	NOUN
ejpam-5948	194	6	in	in	ADP
ejpam-5948	194	7	theorems	theorem	NOUN
ejpam-5948	194	8	1	1	NUM
ejpam-5948	194	9	,	,	PUNCT
ejpam-5948	194	10	3	3	NUM
ejpam-5948	194	11	(	(	PUNCT
ejpam-5948	194	12	i	i	NOUN
ejpam-5948	194	13	)	)	PUNCT
ejpam-5948	194	14	,	,	PUNCT
ejpam-5948	194	15	and	and	CCONJ
ejpam-5948	194	16	4	4	NUM
ejpam-5948	194	17	is	be	AUX
ejpam-5948	194	18	essential	essential	ADJ
ejpam-5948	194	19	.	.	PUNCT
ejpam-5948	195	1	rehman	rehman	NOUN
ejpam-5948	195	2	et	et	PROPN
ejpam-5948	195	3	al	al	PROPN
ejpam-5948	195	4	.	.	PUNCT
ejpam-5948	196	1	[	[	X
ejpam-5948	196	2	12	12	NUM
ejpam-5948	196	3	]	]	PUNCT
ejpam-5948	196	4	found	find	VERB
ejpam-5948	196	5	that	that	SCONJ
ejpam-5948	196	6	either	either	DET
ejpam-5948	196	7	χ(ℜ	χ(ℜ	NOUN
ejpam-5948	196	8	)	)	PUNCT
ejpam-5948	196	9	⊆	⊆	NUM
ejpam-5948	196	10	p	p	NOUN
ejpam-5948	196	11	or	or	CCONJ
ejpam-5948	196	12	ℜ/p	ℜ/p	PROPN
ejpam-5948	196	13	is	be	AUX
ejpam-5948	196	14	an	an	DET
ejpam-5948	196	15	integral	integral	ADJ
ejpam-5948	196	16	domain	domain	NOUN
ejpam-5948	196	17	when	when	SCONJ
ejpam-5948	196	18	ℜ	ℜ	PROPN
ejpam-5948	196	19	admits	admit	VERB
ejpam-5948	196	20	generalized	generalized	ADJ
ejpam-5948	196	21	derivations	derivation	NOUN
ejpam-5948	196	22	(	(	PUNCT
ejpam-5948	196	23	℧	℧	PROPN
ejpam-5948	196	24	,	,	PUNCT
ejpam-5948	196	25	χ	χ	NOUN
ejpam-5948	196	26	)	)	PUNCT
ejpam-5948	196	27	and	and	CCONJ
ejpam-5948	196	28	(	(	PUNCT
ejpam-5948	196	29	⨿,∝	⨿,∝	X
ejpam-5948	196	30	)	)	PUNCT
ejpam-5948	196	31	that	that	PRON
ejpam-5948	196	32	satisfy	satisfy	VERB
ejpam-5948	196	33	the	the	DET
ejpam-5948	196	34	identities	identity	NOUN
ejpam-5948	196	35	℧	℧	PROPN
ejpam-5948	196	36	(	(	PUNCT
ejpam-5948	196	37	υ℘)+	υ℘)+	NOUN
ejpam-5948	196	38	℧	℧	PROPN
ejpam-5948	196	39	(υ)⨿	(υ)⨿	PUNCT
ejpam-5948	196	40	(	(	PUNCT
ejpam-5948	196	41	℘	℘	PROPN
ejpam-5948	196	42	)	)	PUNCT
ejpam-5948	196	43	∈	∈	PROPN
ejpam-5948	196	44	p	p	NOUN
ejpam-5948	196	45	or	or	CCONJ
ejpam-5948	196	46	℧	℧	PROPN
ejpam-5948	196	47	(	(	PUNCT
ejpam-5948	196	48	υ)⨿	υ)⨿	X
ejpam-5948	196	49	(	(	PUNCT
ejpam-5948	196	50	℘	℘	PROPN
ejpam-5948	196	51	)	)	PUNCT
ejpam-5948	196	52	+	+	CCONJ
ejpam-5948	197	1	[	[	X
ejpam-5948	197	2	υ	υ	INTJ
ejpam-5948	197	3	,	,	PUNCT
ejpam-5948	197	4	℘	℘	PROPN
ejpam-5948	197	5	]	]	X
ejpam-5948	197	6	∈	∈	PROPN
ejpam-5948	197	7	p	p	NOUN
ejpam-5948	197	8	for	for	ADP
ejpam-5948	197	9	all	all	DET
ejpam-5948	197	10	υ	υ	NOUN
ejpam-5948	197	11	,	,	PUNCT
ejpam-5948	197	12	℘	℘	PROPN
ejpam-5948	197	13	∈	∈	PROPN
ejpam-5948	197	14	ℜ	ℜ	PROPN
ejpam-5948	197	15	,	,	PUNCT
ejpam-5948	197	16	where	where	SCONJ
ejpam-5948	197	17	p	p	NOUN
ejpam-5948	197	18	is	be	AUX
ejpam-5948	197	19	a	a	DET
ejpam-5948	197	20	prime	prime	ADJ
ejpam-5948	197	21	ideal	ideal	NOUN
ejpam-5948	197	22	of	of	ADP
ejpam-5948	197	23	ℜ.	ℜ.	PROPN
ejpam-5948	197	24	in	in	ADP
ejpam-5948	197	25	the	the	DET
ejpam-5948	197	26	upcoming	upcoming	NOUN
ejpam-5948	197	27	theorem	theorem	NOUN
ejpam-5948	197	28	,	,	PUNCT
ejpam-5948	197	29	we	we	PRON
ejpam-5948	197	30	will	will	AUX
ejpam-5948	197	31	explore	explore	VERB
ejpam-5948	197	32	how	how	SCONJ
ejpam-5948	197	33	the	the	DET
ejpam-5948	197	34	identities	identity	NOUN
ejpam-5948	197	35	(	(	PUNCT
ejpam-5948	197	36	i	i	NOUN
ejpam-5948	197	37	)	)	PUNCT
ejpam-5948	197	38	℧	℧	PROPN
ejpam-5948	197	39	(	(	PUNCT
ejpam-5948	197	40	[	[	X
ejpam-5948	197	41	υ	υ	INTJ
ejpam-5948	197	42	,	,	PUNCT
ejpam-5948	197	43	℘])±	℘])±	ADP
ejpam-5948	197	44	℧	℧	PROPN
ejpam-5948	197	45	(	(	PUNCT
ejpam-5948	197	46	℘	℘	PROPN
ejpam-5948	197	47	)	)	PUNCT
ejpam-5948	197	48	⨿	⨿	NOUN
ejpam-5948	197	49	(	(	PUNCT
ejpam-5948	197	50	υ	υ	NOUN
ejpam-5948	197	51	)	)	PUNCT
ejpam-5948	197	52	∈	∈	PROPN
ejpam-5948	197	53	p	p	NOUN
ejpam-5948	197	54	and	and	CCONJ
ejpam-5948	197	55	(	(	PUNCT
ejpam-5948	197	56	ii	ii	NOUN
ejpam-5948	197	57	)	)	PUNCT
ejpam-5948	197	58	℧	℧	PROPN
ejpam-5948	197	59	(	(	PUNCT
ejpam-5948	197	60	[	[	X
ejpam-5948	197	61	υ	υ	INTJ
ejpam-5948	197	62	,	,	PUNCT
ejpam-5948	197	63	℘	℘	PROPN
ejpam-5948	197	64	]	]	SYM
ejpam-5948	197	65	)	)	PUNCT
ejpam-5948	197	66	±	±	NUM
ejpam-5948	197	67	℧	℧	PROPN
ejpam-5948	197	68	(	(	PUNCT
ejpam-5948	197	69	υ	υ	NOUN
ejpam-5948	197	70	)	)	PUNCT
ejpam-5948	197	71	⨿	⨿	NOUN
ejpam-5948	197	72	(	(	PUNCT
ejpam-5948	197	73	℘	℘	PROPN
ejpam-5948	197	74	)	)	PUNCT
ejpam-5948	197	75	∈	∈	PROPN
ejpam-5948	197	76	p	p	NOUN
ejpam-5948	197	77	for	for	ADP
ejpam-5948	197	78	all	all	DET
ejpam-5948	197	79	υ	υ	NOUN
ejpam-5948	197	80	,	,	PUNCT
ejpam-5948	197	81	℘	℘	PROPN
ejpam-5948	197	82	∈	∈	PROPN
ejpam-5948	197	83	ℜ	ℜ	PROPN
ejpam-5948	197	84	,	,	PUNCT
ejpam-5948	197	85	affect	affect	VERB
ejpam-5948	197	86	the	the	DET
ejpam-5948	197	87	relationship	relationship	NOUN
ejpam-5948	197	88	between	between	ADP
ejpam-5948	197	89	a	a	DET
ejpam-5948	197	90	factor	factor	NOUN
ejpam-5948	197	91	ring	ring	NOUN
ejpam-5948	197	92	ℜ/p	ℜ/p	PROPN
ejpam-5948	197	93	and	and	CCONJ
ejpam-5948	197	94	generalized	generalize	VERB
ejpam-5948	197	95	p	p	ADJ
ejpam-5948	197	96	-derivations	-derivation	NOUN
ejpam-5948	197	97	.	.	PUNCT
ejpam-5948	198	1	theorem	theorem	NOUN
ejpam-5948	198	2	5	5	NUM
ejpam-5948	198	3	.	.	PUNCT
ejpam-5948	199	1	let	let	VERB
ejpam-5948	199	2	ℜ	ℜ	PROPN
ejpam-5948	199	3	be	be	AUX
ejpam-5948	199	4	a	a	DET
ejpam-5948	199	5	ring	ring	NOUN
ejpam-5948	199	6	equipped	equip	VERB
ejpam-5948	199	7	with	with	ADP
ejpam-5948	199	8	generalized	generalize	VERB
ejpam-5948	199	9	p	p	NOUN
ejpam-5948	199	10	-derivations	-derivation	NOUN
ejpam-5948	199	11	(	(	PUNCT
ejpam-5948	199	12	℧	℧	PROPN
ejpam-5948	199	13	,	,	PUNCT
ejpam-5948	199	14	χ	χ	NOUN
ejpam-5948	199	15	)	)	PUNCT
ejpam-5948	199	16	and	and	CCONJ
ejpam-5948	199	17	(	(	PUNCT
ejpam-5948	199	18	⨿,∝	⨿,∝	X
ejpam-5948	199	19	)	)	PUNCT
ejpam-5948	199	20	such	such	ADJ
ejpam-5948	199	21	that	that	SCONJ
ejpam-5948	199	22	one	one	NUM
ejpam-5948	199	23	of	of	ADP
ejpam-5948	199	24	the	the	DET
ejpam-5948	199	25	following	follow	VERB
ejpam-5948	199	26	identities	identity	NOUN
ejpam-5948	199	27	holds	hold	VERB
ejpam-5948	199	28	for	for	ADP
ejpam-5948	199	29	every	every	DET
ejpam-5948	199	30	υ	υ	NOUN
ejpam-5948	199	31	,	,	PUNCT
ejpam-5948	199	32	℘	℘	NOUN
ejpam-5948	199	33	∈	∈	PROPN
ejpam-5948	199	34	ℜ	ℜ	PROPN
ejpam-5948	199	35	:	:	PUNCT
ejpam-5948	199	36	(	(	PUNCT
ejpam-5948	199	37	i	i	NOUN
ejpam-5948	199	38	)	)	PUNCT
ejpam-5948	199	39	℧	℧	PROPN
ejpam-5948	199	40	(	(	PUNCT
ejpam-5948	199	41	[	[	X
ejpam-5948	199	42	υ	υ	INTJ
ejpam-5948	199	43	,	,	PUNCT
ejpam-5948	199	44	℘])±	℘])±	ADP
ejpam-5948	199	45	℧	℧	PROPN
ejpam-5948	199	46	(	(	PUNCT
ejpam-5948	199	47	℘)⨿	℘)⨿	NOUN
ejpam-5948	199	48	(	(	PUNCT
ejpam-5948	199	49	υ	υ	NOUN
ejpam-5948	199	50	)	)	PUNCT
ejpam-5948	199	51	∈	∈	PROPN
ejpam-5948	199	52	p	p	NOUN
ejpam-5948	199	53	,	,	PUNCT
ejpam-5948	199	54	(	(	PUNCT
ejpam-5948	199	55	ii	ii	NOUN
ejpam-5948	199	56	)	)	PUNCT
ejpam-5948	199	57	℧	℧	PROPN
ejpam-5948	199	58	(	(	PUNCT
ejpam-5948	199	59	[	[	X
ejpam-5948	199	60	υ	υ	INTJ
ejpam-5948	199	61	,	,	PUNCT
ejpam-5948	199	62	℘])±	℘])±	ADJ
ejpam-5948	199	63	℧	℧	NOUN
ejpam-5948	199	64	(υ)⨿(℘	(υ)⨿(℘	NOUN
ejpam-5948	199	65	)	)	PUNCT
ejpam-5948	199	66	∈	∈	PROPN
ejpam-5948	199	67	p	p	NOUN
ejpam-5948	199	68	.	.	PUNCT
ejpam-5948	200	1	then	then	ADV
ejpam-5948	200	2	either	either	CCONJ
ejpam-5948	200	3	∝	∝	PROPN
ejpam-5948	200	4	(	(	PUNCT
ejpam-5948	200	5	ℜ	ℜ	PROPN
ejpam-5948	200	6	)	)	PUNCT
ejpam-5948	200	7	⊆	⊆	NUM
ejpam-5948	200	8	p	p	NOUN
ejpam-5948	200	9	and	and	CCONJ
ejpam-5948	200	10	ℜ/p	ℜ/p	PROPN
ejpam-5948	200	11	is	be	AUX
ejpam-5948	200	12	an	an	DET
ejpam-5948	200	13	integral	integral	ADJ
ejpam-5948	200	14	domain	domain	NOUN
ejpam-5948	200	15	,	,	PUNCT
ejpam-5948	200	16	or	or	CCONJ
ejpam-5948	200	17	℧	℧	PROPN
ejpam-5948	200	18	(	(	PUNCT
ejpam-5948	200	19	ℜ	ℜ	PROPN
ejpam-5948	200	20	)	)	PUNCT
ejpam-5948	200	21	is	be	AUX
ejpam-5948	200	22	a	a	DET
ejpam-5948	200	23	subset	subset	NOUN
ejpam-5948	200	24	of	of	ADP
ejpam-5948	200	25	p	p	NOUN
ejpam-5948	200	26	.	.	PUNCT
ejpam-5948	201	1	proof	proof	NOUN
ejpam-5948	201	2	.	.	PUNCT
ejpam-5948	202	1	(	(	PUNCT
ejpam-5948	202	2	i	i	NOUN
ejpam-5948	202	3	)	)	PUNCT
ejpam-5948	202	4	according	accord	VERB
ejpam-5948	202	5	to	to	ADP
ejpam-5948	202	6	the	the	DET
ejpam-5948	202	7	basic	basic	ADJ
ejpam-5948	202	8	assumption	assumption	NOUN
ejpam-5948	202	9	,	,	PUNCT
ejpam-5948	202	10	we	we	PRON
ejpam-5948	202	11	have	have	VERB
ejpam-5948	202	12	℧	℧	PROPN
ejpam-5948	202	13	(	(	PUNCT
ejpam-5948	202	14	[	[	X
ejpam-5948	202	15	υ	υ	INTJ
ejpam-5948	202	16	,	,	PUNCT
ejpam-5948	202	17	℘])±	℘])±	ADP
ejpam-5948	202	18	℧	℧	PROPN
ejpam-5948	202	19	(	(	PUNCT
ejpam-5948	202	20	℘)⨿	℘)⨿	NOUN
ejpam-5948	202	21	(	(	PUNCT
ejpam-5948	202	22	υ	υ	NOUN
ejpam-5948	202	23	)	)	PUNCT
ejpam-5948	202	24	∈	∈	PROPN
ejpam-5948	202	25	p	p	NOUN
ejpam-5948	202	26	for	for	ADP
ejpam-5948	202	27	all	all	DET
ejpam-5948	202	28	υ	υ	NOUN
ejpam-5948	202	29	,	,	PUNCT
ejpam-5948	202	30	℘	℘	PROPN
ejpam-5948	202	31	∈	∈	NOUN
ejpam-5948	202	32	ℜ.	ℜ.	PROPN
ejpam-5948	202	33	(	(	PUNCT
ejpam-5948	202	34	15	15	NUM
ejpam-5948	202	35	)	)	PUNCT
ejpam-5948	202	36	setting	set	VERB
ejpam-5948	202	37	υ	υ	NOUN
ejpam-5948	202	38	=	=	PUNCT
ejpam-5948	202	39	υℏ	υℏ	NOUN
ejpam-5948	202	40	in	in	ADP
ejpam-5948	202	41	equation	equation	NOUN
ejpam-5948	202	42	(	(	PUNCT
ejpam-5948	202	43	15	15	NUM
ejpam-5948	202	44	)	)	PUNCT
ejpam-5948	202	45	and	and	CCONJ
ejpam-5948	202	46	using	use	VERB
ejpam-5948	202	47	it	it	PRON
ejpam-5948	202	48	,	,	PUNCT
ejpam-5948	202	49	we	we	PRON
ejpam-5948	202	50	get	get	VERB
ejpam-5948	202	51	℧	℧	PROPN
ejpam-5948	202	52	(	(	PUNCT
ejpam-5948	202	53	υ)[ℏ	υ)[ℏ	PROPN
ejpam-5948	202	54	,	,	PUNCT
ejpam-5948	202	55	℘	℘	PROPN
ejpam-5948	202	56	]	]	PUNCT
ejpam-5948	203	1	+	+	CCONJ
ejpam-5948	203	2	υχ([ℏ	υχ([ℏ	PROPN
ejpam-5948	203	3	,	,	PUNCT
ejpam-5948	203	4	℘	℘	PROPN
ejpam-5948	203	5	]	]	PUNCT
ejpam-5948	203	6	)	)	PUNCT
ejpam-5948	204	1	+	+	CCONJ
ejpam-5948	205	1	[	[	X
ejpam-5948	205	2	υ	υ	X
ejpam-5948	205	3	,	,	PUNCT
ejpam-5948	205	4	℘]χ(ℏ)±	℘]χ(ℏ)±	NUM
ejpam-5948	205	5	℧	℧	PROPN
ejpam-5948	205	6	(	(	PUNCT
ejpam-5948	205	7	℘)υ	℘)υ	PROPN
ejpam-5948	205	8	∝	∝	PROPN
ejpam-5948	205	9	(	(	PUNCT
ejpam-5948	205	10	ℏ	ℏ	PROPN
ejpam-5948	205	11	)	)	PUNCT
ejpam-5948	205	12	∈	∈	PROPN
ejpam-5948	205	13	p	p	NOUN
ejpam-5948	205	14	for	for	ADP
ejpam-5948	205	15	all	all	DET
ejpam-5948	205	16	υ	υ	NOUN
ejpam-5948	205	17	,	,	PUNCT
ejpam-5948	205	18	℘	℘	PROPN
ejpam-5948	205	19	,	,	PUNCT
ejpam-5948	205	20	ℏ	ℏ	PROPN
ejpam-5948	205	21	∈	∈	NOUN
ejpam-5948	205	22	ℜ.	ℜ.	PROPN
ejpam-5948	205	23	(	(	PUNCT
ejpam-5948	205	24	16	16	NUM
ejpam-5948	205	25	)	)	PUNCT
ejpam-5948	205	26	by	by	ADP
ejpam-5948	205	27	setting	set	VERB
ejpam-5948	205	28	ℏ	ℏ	NOUN
ejpam-5948	205	29	=	=	PUNCT
ejpam-5948	205	30	℘	℘	PROPN
ejpam-5948	205	31	,	,	PUNCT
ejpam-5948	205	32	we	we	PRON
ejpam-5948	205	33	have	have	VERB
ejpam-5948	205	34	[	[	X
ejpam-5948	205	35	υ	υ	X
ejpam-5948	205	36	,	,	PUNCT
ejpam-5948	205	37	℘]χ(℘)±	℘]χ(℘)±	VERB
ejpam-5948	205	38	℧	℧	PROPN
ejpam-5948	205	39	(	(	PUNCT
ejpam-5948	205	40	℘)υ	℘)υ	PROPN
ejpam-5948	205	41	∝	∝	PROPN
ejpam-5948	205	42	(	(	PUNCT
ejpam-5948	205	43	℘	℘	PROPN
ejpam-5948	205	44	)	)	PUNCT
ejpam-5948	205	45	∈	∈	PROPN
ejpam-5948	205	46	p	p	NOUN
ejpam-5948	205	47	for	for	ADP
ejpam-5948	205	48	all	all	DET
ejpam-5948	205	49	υ	υ	NOUN
ejpam-5948	205	50	,	,	PUNCT
ejpam-5948	205	51	℘	℘	PROPN
ejpam-5948	205	52	∈	∈	NOUN
ejpam-5948	205	53	ℜ.	ℜ.	PROPN
ejpam-5948	205	54	(	(	PUNCT
ejpam-5948	205	55	17	17	NUM
ejpam-5948	205	56	)	)	PUNCT
ejpam-5948	205	57	replacing	replace	VERB
ejpam-5948	205	58	υ	υ	NOUN
ejpam-5948	205	59	by	by	ADP
ejpam-5948	205	60	℘υ	℘υ	NOUN
ejpam-5948	205	61	in	in	ADP
ejpam-5948	205	62	equation	equation	NOUN
ejpam-5948	205	63	(	(	PUNCT
ejpam-5948	205	64	17	17	NUM
ejpam-5948	205	65	)	)	PUNCT
ejpam-5948	205	66	and	and	CCONJ
ejpam-5948	205	67	utilizing	utilize	VERB
ejpam-5948	205	68	it	it	PRON
ejpam-5948	205	69	,	,	PUNCT
ejpam-5948	205	70	we	we	PRON
ejpam-5948	205	71	get	get	VERB
ejpam-5948	205	72	℧	℧	PROPN
ejpam-5948	205	73	(	(	PUNCT
ejpam-5948	205	74	℘)℘υ	℘)℘υ	PROPN
ejpam-5948	205	75	∝	∝	PROPN
ejpam-5948	205	76	(	(	PUNCT
ejpam-5948	205	77	℘	℘	PROPN
ejpam-5948	205	78	)	)	PUNCT
ejpam-5948	205	79	−	−	ADP
ejpam-5948	205	80	℘	℘	PROPN
ejpam-5948	205	81	℧	℧	PROPN
ejpam-5948	205	82	(℘)υ	(℘)υ	PUNCT
ejpam-5948	205	83	∝	∝	PROPN
ejpam-5948	205	84	(	(	PUNCT
ejpam-5948	205	85	℘	℘	PROPN
ejpam-5948	205	86	)	)	PUNCT
ejpam-5948	205	87	=	=	PUNCT
ejpam-5948	206	1	[	[	X
ejpam-5948	206	2	℧	℧	X
ejpam-5948	206	3	(	(	PUNCT
ejpam-5948	206	4	℘	℘	PROPN
ejpam-5948	206	5	)	)	PUNCT
ejpam-5948	206	6	,	,	PUNCT
ejpam-5948	206	7	℘]υ	℘]υ	PROPN
ejpam-5948	206	8	∝	∝	PROPN
ejpam-5948	206	9	(	(	PUNCT
ejpam-5948	206	10	℘	℘	PROPN
ejpam-5948	206	11	)	)	PUNCT
ejpam-5948	206	12	∈	∈	PROPN
ejpam-5948	206	13	p	p	NOUN
ejpam-5948	206	14	for	for	ADP
ejpam-5948	206	15	all	all	DET
ejpam-5948	206	16	υ	υ	NOUN
ejpam-5948	206	17	,	,	PUNCT
ejpam-5948	206	18	℘	℘	PROPN
ejpam-5948	206	19	∈	∈	NOUN
ejpam-5948	206	20	ℜ.	ℜ.	PROPN
ejpam-5948	206	21	this	this	PRON
ejpam-5948	206	22	implies	imply	VERB
ejpam-5948	206	23	that	that	SCONJ
ejpam-5948	206	24	[	[	X
ejpam-5948	206	25	℧	℧	PROPN
ejpam-5948	206	26	(	(	PUNCT
ejpam-5948	206	27	℘	℘	PROPN
ejpam-5948	206	28	)	)	PUNCT
ejpam-5948	206	29	,	,	PUNCT
ejpam-5948	206	30	℘]ℜ	℘]ℜ	VERB
ejpam-5948	206	31	∝	∝	PROPN
ejpam-5948	206	32	(	(	PUNCT
ejpam-5948	206	33	℘	℘	PROPN
ejpam-5948	206	34	)	)	PUNCT
ejpam-5948	206	35	⊆	⊆	NUM
ejpam-5948	206	36	p	p	NOUN
ejpam-5948	206	37	for	for	ADP
ejpam-5948	206	38	all	all	DET
ejpam-5948	206	39	℘	℘	PROPN
ejpam-5948	206	40	∈	∈	PRON
ejpam-5948	206	41	ℜ.	ℜ.	VERB
ejpam-5948	206	42	the	the	DET
ejpam-5948	206	43	primeness	primeness	NOUN
ejpam-5948	206	44	of	of	ADP
ejpam-5948	206	45	p	p	PROPN
ejpam-5948	206	46	implies	imply	VERB
ejpam-5948	206	47	either	either	CCONJ
ejpam-5948	206	48	[	[	X
ejpam-5948	206	49	℧	℧	PROPN
ejpam-5948	206	50	(	(	PUNCT
ejpam-5948	206	51	℘	℘	PROPN
ejpam-5948	206	52	)	)	PUNCT
ejpam-5948	206	53	,	,	PUNCT
ejpam-5948	206	54	℘	℘	PROPN
ejpam-5948	206	55	]	]	PUNCT
ejpam-5948	206	56	∈	∈	PROPN
ejpam-5948	206	57	p	p	NOUN
ejpam-5948	206	58	or	or	CCONJ
ejpam-5948	206	59	∝	∝	PROPN
ejpam-5948	206	60	(	(	PUNCT
ejpam-5948	206	61	℘	℘	PROPN
ejpam-5948	206	62	)	)	PUNCT
ejpam-5948	206	63	∈	∈	PROPN
ejpam-5948	206	64	p	p	NOUN
ejpam-5948	206	65	for	for	ADP
ejpam-5948	206	66	all	all	DET
ejpam-5948	206	67	℘	℘	PROPN
ejpam-5948	206	68	∈	∈	NOUN
ejpam-5948	206	69	ℜ.	ℜ.	ADJ
ejpam-5948	206	70	to	to	PART
ejpam-5948	206	71	complete	complete	VERB
ejpam-5948	206	72	the	the	DET
ejpam-5948	206	73	proof	proof	NOUN
ejpam-5948	206	74	,	,	PUNCT
ejpam-5948	206	75	we	we	PRON
ejpam-5948	206	76	will	will	AUX
ejpam-5948	206	77	discuss	discuss	VERB
ejpam-5948	206	78	the	the	DET
ejpam-5948	206	79	following	follow	VERB
ejpam-5948	206	80	two	two	NUM
ejpam-5948	206	81	cases	case	NOUN
ejpam-5948	206	82	:	:	PUNCT
ejpam-5948	206	83	case	case	NOUN
ejpam-5948	206	84	(	(	PUNCT
ejpam-5948	206	85	a	a	X
ejpam-5948	206	86	):	):	PUNCT
ejpam-5948	206	87	if	if	SCONJ
ejpam-5948	206	88	[	[	X
ejpam-5948	206	89	℧	℧	X
ejpam-5948	206	90	(	(	PUNCT
ejpam-5948	206	91	℘	℘	PROPN
ejpam-5948	206	92	)	)	PUNCT
ejpam-5948	206	93	,	,	PUNCT
ejpam-5948	206	94	℘	℘	PROPN
ejpam-5948	206	95	]	]	PUNCT
ejpam-5948	206	96	∈	∈	PROPN
ejpam-5948	206	97	p	p	NOUN
ejpam-5948	206	98	for	for	ADP
ejpam-5948	206	99	all	all	DET
ejpam-5948	206	100	℘	℘	PROPN
ejpam-5948	206	101	∈	∈	PROPN
ejpam-5948	206	102	ℜ	ℜ	PROPN
ejpam-5948	206	103	,	,	PUNCT
ejpam-5948	206	104	this	this	PRON
ejpam-5948	206	105	implies	imply	VERB
ejpam-5948	206	106	that	that	SCONJ
ejpam-5948	206	107	ℜ/p	ℜ/p	PROPN
ejpam-5948	206	108	is	be	AUX
ejpam-5948	206	109	an	an	DET
ejpam-5948	206	110	integral	integral	ADJ
ejpam-5948	206	111	domain	domain	NOUN
ejpam-5948	206	112	or	or	CCONJ
ejpam-5948	206	113	χ(ℜ	χ(ℜ	NOUN
ejpam-5948	206	114	)	)	PUNCT
ejpam-5948	206	115	⊆	⊆	NUM
ejpam-5948	206	116	p	p	NOUN
ejpam-5948	206	117	by	by	ADP
ejpam-5948	206	118	using	use	VERB
ejpam-5948	206	119	lemma	lemma	PROPN
ejpam-5948	206	120	3	3	NUM
ejpam-5948	206	121	.	.	PUNCT
ejpam-5948	207	1	if	if	SCONJ
ejpam-5948	207	2	ℜ/p	ℜ/p	PROPN
ejpam-5948	207	3	is	be	AUX
ejpam-5948	207	4	an	an	DET
ejpam-5948	207	5	integral	integral	ADJ
ejpam-5948	207	6	domain	domain	NOUN
ejpam-5948	207	7	,	,	PUNCT
ejpam-5948	207	8	equation	equation	NOUN
ejpam-5948	207	9	(	(	PUNCT
ejpam-5948	207	10	17	17	NUM
ejpam-5948	207	11	)	)	PUNCT
ejpam-5948	207	12	yields	yield	NOUN
ejpam-5948	207	13	℧	℧	PROPN
ejpam-5948	207	14	(	(	PUNCT
ejpam-5948	207	15	℘)ℜ	℘)ℜ	PUNCT
ejpam-5948	207	16	∝	∝	PROPN
ejpam-5948	207	17	(	(	PUNCT
ejpam-5948	207	18	℘	℘	PROPN
ejpam-5948	207	19	)	)	PUNCT
ejpam-5948	207	20	⊆	⊆	NUM
ejpam-5948	207	21	p	p	NOUN
ejpam-5948	207	22	for	for	ADP
ejpam-5948	207	23	all	all	DET
ejpam-5948	207	24	℘	℘	PROPN
ejpam-5948	207	25	∈	∈	PRON
ejpam-5948	207	26	ℜ.	ℜ.	VERB
ejpam-5948	207	27	the	the	DET
ejpam-5948	207	28	primeness	primeness	NOUN
ejpam-5948	207	29	of	of	ADP
ejpam-5948	207	30	p	p	NOUN
ejpam-5948	207	31	leads	lead	VERB
ejpam-5948	207	32	to	to	ADP
ejpam-5948	207	33	either	either	CCONJ
ejpam-5948	207	34	℧	℧	PROPN
ejpam-5948	207	35	(	(	PUNCT
ejpam-5948	207	36	ℜ	ℜ	PROPN
ejpam-5948	207	37	)	)	PUNCT
ejpam-5948	207	38	⊆	⊆	NUM
ejpam-5948	207	39	p	p	NOUN
ejpam-5948	207	40	or	or	CCONJ
ejpam-5948	207	41	∝	∝	PROPN
ejpam-5948	207	42	(	(	PUNCT
ejpam-5948	207	43	ℜ	ℜ	PROPN
ejpam-5948	207	44	)	)	PUNCT
ejpam-5948	207	45	⊆	⊆	NUM
ejpam-5948	207	46	p	p	NOUN
ejpam-5948	207	47	.	.	PUNCT
ejpam-5948	208	1	on	on	ADP
ejpam-5948	208	2	the	the	DET
ejpam-5948	208	3	other	other	ADJ
ejpam-5948	208	4	hand	hand	NOUN
ejpam-5948	208	5	,	,	PUNCT
ejpam-5948	208	6	if	if	SCONJ
ejpam-5948	208	7	χ(ℜ	χ(ℜ	AUX
ejpam-5948	208	8	)	)	PUNCT
ejpam-5948	208	9	⊆	⊆	NUM
ejpam-5948	208	10	p	p	NOUN
ejpam-5948	208	11	,	,	PUNCT
ejpam-5948	208	12	then	then	ADV
ejpam-5948	208	13	equation	equation	NOUN
ejpam-5948	208	14	(	(	PUNCT
ejpam-5948	208	15	17	17	NUM
ejpam-5948	208	16	)	)	PUNCT
ejpam-5948	208	17	gives	give	VERB
ejpam-5948	208	18	℧	℧	PROPN
ejpam-5948	208	19	(	(	PUNCT
ejpam-5948	208	20	℘)ℜ	℘)ℜ	PUNCT
ejpam-5948	208	21	∝	∝	PROPN
ejpam-5948	208	22	(	(	PUNCT
ejpam-5948	208	23	℘	℘	PROPN
ejpam-5948	208	24	)	)	PUNCT
ejpam-5948	208	25	⊆	⊆	NUM
ejpam-5948	208	26	p	p	NOUN
ejpam-5948	208	27	for	for	ADP
ejpam-5948	208	28	all	all	DET
ejpam-5948	208	29	℘	℘	PROPN
ejpam-5948	208	30	∈	∈	NOUN
ejpam-5948	208	31	ℜ.	ℜ.	VERB
ejpam-5948	208	32	again	again	ADV
ejpam-5948	208	33	,	,	PUNCT
ejpam-5948	208	34	the	the	DET
ejpam-5948	208	35	primeness	primeness	NOUN
ejpam-5948	208	36	of	of	ADP
ejpam-5948	208	37	p	p	NOUN
ejpam-5948	208	38	leads	lead	VERB
ejpam-5948	208	39	to	to	ADP
ejpam-5948	208	40	either	either	CCONJ
ejpam-5948	208	41	℧	℧	PROPN
ejpam-5948	208	42	(	(	PUNCT
ejpam-5948	208	43	ℜ	ℜ	PROPN
ejpam-5948	208	44	)	)	PUNCT
ejpam-5948	208	45	⊆	⊆	NUM
ejpam-5948	208	46	p	p	NOUN
ejpam-5948	208	47	or	or	CCONJ
ejpam-5948	208	48	∝	∝	PROPN
ejpam-5948	208	49	(	(	PUNCT
ejpam-5948	208	50	ℜ	ℜ	PROPN
ejpam-5948	208	51	)	)	PUNCT
ejpam-5948	208	52	⊆	⊆	NUM
ejpam-5948	208	53	p	p	NOUN
ejpam-5948	208	54	.	.	PUNCT
ejpam-5948	209	1	in	in	ADP
ejpam-5948	209	2	the	the	DET
ejpam-5948	209	3	scenario	scenario	NOUN
ejpam-5948	209	4	where	where	SCONJ
ejpam-5948	209	5	∝	∝	PROPN
ejpam-5948	209	6	(	(	PUNCT
ejpam-5948	209	7	ℜ	ℜ	PROPN
ejpam-5948	209	8	)	)	PUNCT
ejpam-5948	209	9	⊆	⊆	NUM
ejpam-5948	209	10	p	p	NOUN
ejpam-5948	209	11	,	,	PUNCT
ejpam-5948	209	12	equation	equation	NOUN
ejpam-5948	209	13	(	(	PUNCT
ejpam-5948	209	14	16	16	NUM
ejpam-5948	209	15	)	)	PUNCT
ejpam-5948	209	16	becomes	become	VERB
ejpam-5948	209	17	℧	℧	PROPN
ejpam-5948	209	18	(	(	PUNCT
ejpam-5948	209	19	υ)[ℏ	υ)[ℏ	PROPN
ejpam-5948	209	20	,	,	PUNCT
ejpam-5948	209	21	℘	℘	PROPN
ejpam-5948	209	22	]	]	X
ejpam-5948	209	23	∈	∈	PROPN
ejpam-5948	209	24	p	p	NOUN
ejpam-5948	209	25	for	for	ADP
ejpam-5948	209	26	all	all	DET
ejpam-5948	209	27	υ	υ	NOUN
ejpam-5948	209	28	,	,	PUNCT
ejpam-5948	209	29	℘	℘	PROPN
ejpam-5948	209	30	,	,	PUNCT
ejpam-5948	209	31	ℏ	ℏ	PROPN
ejpam-5948	209	32	∈	∈	PROPN
ejpam-5948	209	33	ℜ.	ℜ.	PROPN
ejpam-5948	209	34	a.y	a.y	PROPN
ejpam-5948	209	35	.	.	PROPN
ejpam-5948	209	36	hummdi	hummdi	PROPN
ejpam-5948	209	37	,	,	PUNCT
ejpam-5948	209	38	r.m	r.m	PROPN
ejpam-5948	209	39	.	.	PROPN
ejpam-5948	209	40	al	al	PROPN
ejpam-5948	209	41	-	-	PUNCT
ejpam-5948	209	42	omary	omary	PROPN
ejpam-5948	209	43	,	,	PUNCT
ejpam-5948	209	44	z.z	z.z	PROPN
ejpam-5948	209	45	.	.	PUNCT
ejpam-5948	209	46	al	al	PROPN
ejpam-5948	209	47	-	-	PUNCT
ejpam-5948	209	48	amery	amery	PROPN
ejpam-5948	209	49	/	/	SYM
ejpam-5948	209	50	eur	eur	PROPN
ejpam-5948	209	51	.	.	PUNCT
ejpam-5948	210	1	j.	j.	PROPN
ejpam-5948	210	2	pure	pure	PROPN
ejpam-5948	210	3	appl	appl	PROPN
ejpam-5948	210	4	.	.	PROPN
ejpam-5948	210	5	math	math	PROPN
ejpam-5948	210	6	,	,	PUNCT
ejpam-5948	210	7	18	18	NUM
ejpam-5948	210	8	(	(	PUNCT
ejpam-5948	210	9	2	2	NUM
ejpam-5948	210	10	)	)	PUNCT
ejpam-5948	210	11	(	(	PUNCT
ejpam-5948	210	12	2025	2025	NUM
ejpam-5948	210	13	)	)	PUNCT
ejpam-5948	210	14	,	,	PUNCT
ejpam-5948	210	15	5948	5948	NUM
ejpam-5948	210	16	9	9	NUM
ejpam-5948	210	17	of	of	ADP
ejpam-5948	210	18	12	12	NUM
ejpam-5948	210	19	replacing	replace	VERB
ejpam-5948	210	20	υ	υ	NOUN
ejpam-5948	210	21	by	by	ADP
ejpam-5948	210	22	υκ	υκ	PROPN
ejpam-5948	210	23	in	in	ADP
ejpam-5948	210	24	the	the	DET
ejpam-5948	210	25	last	last	ADJ
ejpam-5948	210	26	equation	equation	NOUN
ejpam-5948	210	27	yields	yield	VERB
ejpam-5948	210	28	℧	℧	PROPN
ejpam-5948	210	29	(	(	PUNCT
ejpam-5948	210	30	υ)ℜ[ℏ	υ)ℜ[ℏ	PROPN
ejpam-5948	210	31	,	,	PUNCT
ejpam-5948	210	32	℘	℘	NOUN
ejpam-5948	210	33	]	]	PUNCT
ejpam-5948	210	34	⊆	⊆	NUM
ejpam-5948	210	35	p	p	NOUN
ejpam-5948	210	36	for	for	ADP
ejpam-5948	210	37	all	all	DET
ejpam-5948	210	38	υ	υ	NOUN
ejpam-5948	210	39	,	,	PUNCT
ejpam-5948	210	40	℘	℘	PROPN
ejpam-5948	210	41	,	,	PUNCT
ejpam-5948	210	42	ℏ	ℏ	PROPN
ejpam-5948	210	43	∈	∈	NOUN
ejpam-5948	210	44	ℜ.	ℜ.	PROPN
ejpam-5948	210	45	since	since	SCONJ
ejpam-5948	210	46	p	p	NOUN
ejpam-5948	210	47	is	be	AUX
ejpam-5948	210	48	prime	prime	ADJ
ejpam-5948	210	49	,	,	PUNCT
ejpam-5948	210	50	either	either	CCONJ
ejpam-5948	210	51	℧	℧	PROPN
ejpam-5948	210	52	(	(	PUNCT
ejpam-5948	210	53	ℜ	ℜ	PROPN
ejpam-5948	210	54	)	)	PUNCT
ejpam-5948	210	55	⊆	⊆	NUM
ejpam-5948	210	56	p	p	NOUN
ejpam-5948	210	57	or	or	CCONJ
ejpam-5948	210	58	ℜ/p	ℜ/p	PROPN
ejpam-5948	210	59	is	be	AUX
ejpam-5948	210	60	an	an	DET
ejpam-5948	210	61	integral	integral	ADJ
ejpam-5948	210	62	domain	domain	NOUN
ejpam-5948	210	63	.	.	PUNCT
ejpam-5948	211	1	case	case	NOUN
ejpam-5948	211	2	(	(	PUNCT
ejpam-5948	211	3	b	b	X
ejpam-5948	211	4	):	):	PUNCT
ejpam-5948	211	5	if	if	SCONJ
ejpam-5948	211	6	∝	∝	PROPN
ejpam-5948	211	7	(	(	PUNCT
ejpam-5948	211	8	ℜ	ℜ	PROPN
ejpam-5948	211	9	)	)	PUNCT
ejpam-5948	211	10	⊆	⊆	NUM
ejpam-5948	211	11	p	p	NOUN
ejpam-5948	211	12	,	,	PUNCT
ejpam-5948	211	13	then	then	ADV
ejpam-5948	211	14	equation	equation	NOUN
ejpam-5948	211	15	(	(	PUNCT
ejpam-5948	211	16	17	17	NUM
ejpam-5948	211	17	)	)	PUNCT
ejpam-5948	211	18	becomes	become	VERB
ejpam-5948	211	19	[	[	X
ejpam-5948	211	20	υ	υ	NOUN
ejpam-5948	211	21	,	,	PUNCT
ejpam-5948	211	22	℘]χ(℘	℘]χ(℘	NOUN
ejpam-5948	211	23	)	)	PUNCT
ejpam-5948	211	24	∈	∈	PROPN
ejpam-5948	211	25	p	p	NOUN
ejpam-5948	211	26	for	for	ADP
ejpam-5948	211	27	all	all	DET
ejpam-5948	211	28	υ	υ	NOUN
ejpam-5948	211	29	,	,	PUNCT
ejpam-5948	211	30	℘	℘	PROPN
ejpam-5948	211	31	∈	∈	NOUN
ejpam-5948	211	32	ℜ.	ℜ.	PROPN
ejpam-5948	211	33	substituting	substitute	VERB
ejpam-5948	211	34	υ	υ	NOUN
ejpam-5948	211	35	with	with	ADP
ejpam-5948	211	36	ℏυ	ℏυ	ADV
ejpam-5948	211	37	in	in	ADP
ejpam-5948	211	38	the	the	DET
ejpam-5948	211	39	last	last	ADJ
ejpam-5948	211	40	expression	expression	NOUN
ejpam-5948	211	41	and	and	CCONJ
ejpam-5948	211	42	using	use	VERB
ejpam-5948	211	43	it	it	PRON
ejpam-5948	211	44	,	,	PUNCT
ejpam-5948	211	45	we	we	PRON
ejpam-5948	211	46	obtain	obtain	VERB
ejpam-5948	211	47	[	[	X
ejpam-5948	211	48	ℏ	ℏ	NOUN
ejpam-5948	211	49	,	,	PUNCT
ejpam-5948	211	50	℘]ℜχ(℘	℘]ℜχ(℘	PROPN
ejpam-5948	211	51	)	)	PUNCT
ejpam-5948	211	52	⊆	⊆	NUM
ejpam-5948	211	53	p	p	NOUN
ejpam-5948	211	54	for	for	ADP
ejpam-5948	211	55	all	all	DET
ejpam-5948	211	56	ℏ	ℏ	NOUN
ejpam-5948	211	57	,	,	PUNCT
ejpam-5948	211	58	℘	℘	X
ejpam-5948	211	59	∈	∈	NOUN
ejpam-5948	211	60	ℜ.	ℜ.	VERB
ejpam-5948	211	61	the	the	DET
ejpam-5948	211	62	primeness	primeness	NOUN
ejpam-5948	211	63	of	of	ADP
ejpam-5948	211	64	p	p	PROPN
ejpam-5948	211	65	implies	imply	VERB
ejpam-5948	211	66	either	either	CCONJ
ejpam-5948	211	67	ℜ/p	ℜ/p	PROPN
ejpam-5948	211	68	is	be	AUX
ejpam-5948	211	69	an	an	DET
ejpam-5948	211	70	integral	integral	ADJ
ejpam-5948	211	71	domain	domain	NOUN
ejpam-5948	211	72	or	or	CCONJ
ejpam-5948	211	73	χ(ℜ	χ(ℜ	NOUN
ejpam-5948	211	74	)	)	PUNCT
ejpam-5948	211	75	⊆	⊆	NUM
ejpam-5948	211	76	p	p	NOUN
ejpam-5948	211	77	.	.	PUNCT
ejpam-5948	212	1	in	in	ADP
ejpam-5948	212	2	the	the	DET
ejpam-5948	212	3	second	second	ADJ
ejpam-5948	212	4	case	case	NOUN
ejpam-5948	212	5	,	,	PUNCT
ejpam-5948	212	6	as	as	SCONJ
ejpam-5948	212	7	discussed	discuss	VERB
ejpam-5948	212	8	in	in	ADP
ejpam-5948	212	9	case	case	NOUN
ejpam-5948	212	10	(	(	PUNCT
ejpam-5948	212	11	a	a	X
ejpam-5948	212	12	)	)	PUNCT
ejpam-5948	212	13	,	,	PUNCT
ejpam-5948	212	14	we	we	PRON
ejpam-5948	212	15	conclude	conclude	VERB
ejpam-5948	212	16	that	that	SCONJ
ejpam-5948	212	17	either	either	CCONJ
ejpam-5948	212	18	ℜ/p	ℜ/p	PROPN
ejpam-5948	212	19	is	be	AUX
ejpam-5948	212	20	an	an	DET
ejpam-5948	212	21	integral	integral	ADJ
ejpam-5948	212	22	domain	domain	NOUN
ejpam-5948	212	23	or	or	CCONJ
ejpam-5948	212	24	℧	℧	PROPN
ejpam-5948	212	25	(	(	PUNCT
ejpam-5948	212	26	ℜ	ℜ	PROPN
ejpam-5948	212	27	)	)	PUNCT
ejpam-5948	212	28	⊆	⊆	NUM
ejpam-5948	212	29	p	p	NOUN
ejpam-5948	212	30	.	.	PUNCT
ejpam-5948	213	1	(	(	PUNCT
ejpam-5948	213	2	ii	ii	NOUN
ejpam-5948	213	3	)	)	PUNCT
ejpam-5948	213	4	to	to	PART
ejpam-5948	213	5	prove	prove	VERB
ejpam-5948	213	6	this	this	DET
ejpam-5948	213	7	part	part	NOUN
ejpam-5948	213	8	,	,	PUNCT
ejpam-5948	213	9	simply	simply	ADV
ejpam-5948	213	10	reiterate	reiterate	VERB
ejpam-5948	213	11	the	the	DET
ejpam-5948	213	12	arguments	argument	NOUN
ejpam-5948	213	13	used	use	VERB
ejpam-5948	213	14	in	in	ADP
ejpam-5948	213	15	the	the	DET
ejpam-5948	213	16	proof	proof	NOUN
ejpam-5948	213	17	of	of	ADP
ejpam-5948	213	18	part	part	NOUN
ejpam-5948	213	19	(	(	PUNCT
ejpam-5948	213	20	i	i	NOUN
ejpam-5948	213	21	)	)	PUNCT
ejpam-5948	213	22	with	with	ADP
ejpam-5948	213	23	some	some	DET
ejpam-5948	213	24	minor	minor	ADJ
ejpam-5948	213	25	modifications	modification	NOUN
ejpam-5948	213	26	to	to	PART
ejpam-5948	213	27	achieve	achieve	VERB
ejpam-5948	213	28	the	the	DET
ejpam-5948	213	29	desired	desire	VERB
ejpam-5948	213	30	result	result	NOUN
ejpam-5948	213	31	.	.	PUNCT
ejpam-5948	214	1	corollary	corollary	ADJ
ejpam-5948	214	2	8	8	NUM
ejpam-5948	214	3	.	.	PUNCT
ejpam-5948	215	1	let	let	VERB
ejpam-5948	215	2	ℜ	ℜ	PROPN
ejpam-5948	215	3	be	be	AUX
ejpam-5948	215	4	a	a	DET
ejpam-5948	215	5	ring	ring	NOUN
ejpam-5948	215	6	equipped	equip	VERB
ejpam-5948	215	7	with	with	ADP
ejpam-5948	215	8	a	a	DET
ejpam-5948	215	9	generalized	generalized	ADJ
ejpam-5948	215	10	p	p	NOUN
ejpam-5948	215	11	-derivation	-derivation	NOUN
ejpam-5948	215	12	(	(	PUNCT
ejpam-5948	215	13	℧	℧	PROPN
ejpam-5948	215	14	,	,	PUNCT
ejpam-5948	215	15	χ	χ	NOUN
ejpam-5948	215	16	)	)	PUNCT
ejpam-5948	215	17	such	such	ADJ
ejpam-5948	215	18	that	that	SCONJ
ejpam-5948	215	19	℧	℧	PROPN
ejpam-5948	215	20	(	(	PUNCT
ejpam-5948	215	21	[	[	X
ejpam-5948	215	22	υ	υ	INTJ
ejpam-5948	215	23	,	,	PUNCT
ejpam-5948	215	24	℘])±	℘])±	VERB
ejpam-5948	215	25	℧	℧	NOUN
ejpam-5948	215	26	(℘)	(℘)	SYM
ejpam-5948	215	27	℧	℧	NOUN
ejpam-5948	215	28	(υ	(υ	NOUN
ejpam-5948	215	29	)	)	PUNCT
ejpam-5948	215	30	∈	∈	PROPN
ejpam-5948	215	31	p	p	NOUN
ejpam-5948	215	32	for	for	ADP
ejpam-5948	215	33	all	all	DET
ejpam-5948	215	34	υ	υ	NOUN
ejpam-5948	215	35	,	,	PUNCT
ejpam-5948	215	36	℘	℘	PROPN
ejpam-5948	215	37	∈	∈	NOUN
ejpam-5948	215	38	ℜ.	ℜ.	PROPN
ejpam-5948	215	39	then	then	ADV
ejpam-5948	215	40	χ(ℜ	χ(ℜ	NOUN
ejpam-5948	215	41	)	)	PUNCT
ejpam-5948	215	42	⊆	⊆	NUM
ejpam-5948	215	43	p	p	NOUN
ejpam-5948	215	44	and	and	CCONJ
ejpam-5948	215	45	ℜ/p	ℜ/p	PROPN
ejpam-5948	215	46	is	be	AUX
ejpam-5948	215	47	an	an	DET
ejpam-5948	215	48	integral	integral	ADJ
ejpam-5948	215	49	domain	domain	NOUN
ejpam-5948	215	50	,	,	PUNCT
ejpam-5948	215	51	or	or	CCONJ
ejpam-5948	215	52	℧	℧	PROPN
ejpam-5948	215	53	(	(	PUNCT
ejpam-5948	215	54	ℜ	ℜ	PROPN
ejpam-5948	215	55	)	)	PUNCT
ejpam-5948	215	56	is	be	AUX
ejpam-5948	215	57	a	a	DET
ejpam-5948	215	58	subset	subset	NOUN
ejpam-5948	215	59	of	of	ADP
ejpam-5948	215	60	p	p	PROPN
ejpam-5948	215	61	.	.	PUNCT
ejpam-5948	216	1	corollary	corollary	ADJ
ejpam-5948	216	2	9	9	NUM
ejpam-5948	216	3	.	.	PUNCT
ejpam-5948	217	1	let	let	VERB
ejpam-5948	217	2	ℜ	ℜ	PROPN
ejpam-5948	217	3	be	be	AUX
ejpam-5948	217	4	a	a	DET
ejpam-5948	217	5	ring	ring	NOUN
ejpam-5948	217	6	equipped	equip	VERB
ejpam-5948	217	7	with	with	ADP
ejpam-5948	217	8	p	p	PROPN
ejpam-5948	217	9	-derivations	-derivation	NOUN
ejpam-5948	217	10	χ	χ	NOUN
ejpam-5948	217	11	and	and	CCONJ
ejpam-5948	217	12	∝	∝	PRON
ejpam-5948	217	13	such	such	ADJ
ejpam-5948	217	14	that	that	DET
ejpam-5948	217	15	χ([υ	χ([υ	PROPN
ejpam-5948	217	16	,	,	PUNCT
ejpam-5948	217	17	℘])±	℘])±	X
ejpam-5948	217	18	χ(℘	χ(℘	PROPN
ejpam-5948	217	19	)	)	PUNCT
ejpam-5948	217	20	∝	∝	PROPN
ejpam-5948	217	21	(	(	PUNCT
ejpam-5948	217	22	υ	υ	NOUN
ejpam-5948	217	23	)	)	PUNCT
ejpam-5948	217	24	∈	∈	PROPN
ejpam-5948	217	25	p	p	NOUN
ejpam-5948	217	26	for	for	ADP
ejpam-5948	217	27	all	all	DET
ejpam-5948	217	28	υ	υ	NOUN
ejpam-5948	217	29	,	,	PUNCT
ejpam-5948	217	30	℘	℘	PROPN
ejpam-5948	217	31	∈	∈	NOUN
ejpam-5948	217	32	ℜ.	ℜ.	PROPN
ejpam-5948	217	33	then	then	ADV
ejpam-5948	217	34	∝	∝	PROPN
ejpam-5948	217	35	(	(	PUNCT
ejpam-5948	217	36	ℜ	ℜ	PROPN
ejpam-5948	217	37	)	)	PUNCT
ejpam-5948	217	38	⊆	⊆	NUM
ejpam-5948	217	39	p	p	NOUN
ejpam-5948	217	40	and	and	CCONJ
ejpam-5948	217	41	ℜ/p	ℜ/p	PROPN
ejpam-5948	217	42	is	be	AUX
ejpam-5948	217	43	an	an	DET
ejpam-5948	217	44	integral	integral	ADJ
ejpam-5948	217	45	domain	domain	NOUN
ejpam-5948	217	46	,	,	PUNCT
ejpam-5948	217	47	or	or	CCONJ
ejpam-5948	217	48	χ(ℜ	χ(ℜ	NOUN
ejpam-5948	217	49	)	)	PUNCT
ejpam-5948	217	50	is	be	AUX
ejpam-5948	217	51	a	a	DET
ejpam-5948	217	52	subset	subset	NOUN
ejpam-5948	217	53	of	of	ADP
ejpam-5948	217	54	p	p	PROPN
ejpam-5948	217	55	.	.	PUNCT
ejpam-5948	218	1	example	example	NOUN
ejpam-5948	219	1	2	2	NUM
ejpam-5948	219	2	.	.	PUNCT
ejpam-5948	219	3	let	let	VERB
ejpam-5948	219	4	ℜ	ℜ	NOUN
ejpam-5948	219	5	=	=	SYM
ejpam-5948	219	6	k22	k22	NOUN
ejpam-5948	219	7	as	as	ADP
ejpam-5948	219	8	in	in	ADP
ejpam-5948	219	9	[	[	X
ejpam-5948	219	10	18	18	NUM
ejpam-5948	219	11	,	,	PUNCT
ejpam-5948	219	12	example	example	NOUN
ejpam-5948	219	13	2.1	2.1	NUM
ejpam-5948	219	14	]	]	PUNCT
ejpam-5948	219	15	,	,	PUNCT
ejpam-5948	219	16	and	and	CCONJ
ejpam-5948	219	17	let	let	VERB
ejpam-5948	219	18	p	p	NOUN
ejpam-5948	219	19	=	=	X
ejpam-5948	219	20	{	{	PUNCT
ejpam-5948	219	21	0	0	NUM
ejpam-5948	219	22	}	}	PUNCT
ejpam-5948	219	23	.	.	PUNCT
ejpam-5948	220	1	define	define	VERB
ejpam-5948	220	2	(	(	PUNCT
ejpam-5948	220	3	℧	℧	PROPN
ejpam-5948	220	4	,	,	PUNCT
ejpam-5948	220	5	χ	χ	NOUN
ejpam-5948	220	6	)	)	PUNCT
ejpam-5948	220	7	:	:	PUNCT
ejpam-5948	220	8	ℜ	ℜ	ADV
ejpam-5948	220	9	−→	−→	NOUN
ejpam-5948	220	10	ℜ	ℜ	ADJ
ejpam-5948	220	11	by	by	ADP
ejpam-5948	220	12	℧	℧	PROPN
ejpam-5948	220	13	(	(	PUNCT
ejpam-5948	220	14	υ	υ	NOUN
ejpam-5948	220	15	)	)	PUNCT
ejpam-5948	220	16	=	=	SYM
ejpam-5948	220	17	χ(υ	χ(υ	PROPN
ejpam-5948	220	18	)	)	PUNCT
ejpam-5948	220	19	=	=	PRON
ejpam-5948	220	20	{	{	PUNCT
ejpam-5948	220	21	0	0	NUM
ejpam-5948	220	22	if	if	SCONJ
ejpam-5948	220	23	υ	υ	NOUN
ejpam-5948	220	24	=	=	NOUN
ejpam-5948	220	25	0	0	NUM
ejpam-5948	220	26	,	,	PUNCT
ejpam-5948	220	27	c	c	X
ejpam-5948	220	28	;	;	PUNCT
ejpam-5948	220	29	c	c	X
ejpam-5948	220	30	if	if	SCONJ
ejpam-5948	220	31	υ	υ	PROPN
ejpam-5948	220	32	=	=	SYM
ejpam-5948	220	33	a	a	PROPN
ejpam-5948	220	34	,	,	PUNCT
ejpam-5948	220	35	b.	b.	PROPN
ejpam-5948	221	1	it	it	PRON
ejpam-5948	221	2	is	be	AUX
ejpam-5948	221	3	easy	easy	ADJ
ejpam-5948	221	4	to	to	PART
ejpam-5948	221	5	verify	verify	VERB
ejpam-5948	221	6	that	that	SCONJ
ejpam-5948	221	7	℧	℧	PROPN
ejpam-5948	221	8	is	be	AUX
ejpam-5948	221	9	a	a	DET
ejpam-5948	221	10	generalized	generalized	ADJ
ejpam-5948	221	11	derivation	derivation	NOUN
ejpam-5948	221	12	of	of	ADP
ejpam-5948	221	13	ℜ	ℜ	PROPN
ejpam-5948	221	14	associated	associate	VERB
ejpam-5948	221	15	with	with	ADP
ejpam-5948	221	16	derivation	derivation	NOUN
ejpam-5948	221	17	χ	χ	NOUN
ejpam-5948	221	18	.	.	PUNCT
ejpam-5948	222	1	it	it	PRON
ejpam-5948	222	2	can	can	AUX
ejpam-5948	222	3	also	also	ADV
ejpam-5948	222	4	be	be	AUX
ejpam-5948	222	5	verified	verify	VERB
ejpam-5948	222	6	that	that	SCONJ
ejpam-5948	222	7	ℜ	ℜ	PROPN
ejpam-5948	222	8	satisfies	satisfy	VERB
ejpam-5948	222	9	the	the	DET
ejpam-5948	222	10	identity	identity	NOUN
ejpam-5948	222	11	in	in	ADP
ejpam-5948	222	12	corollary	corollary	ADJ
ejpam-5948	222	13	8	8	NUM
ejpam-5948	222	14	.	.	PUNCT
ejpam-5948	223	1	however	however	ADV
ejpam-5948	223	2	,	,	PUNCT
ejpam-5948	223	3	neither	neither	CCONJ
ejpam-5948	223	4	ℜ/p	ℜ/p	PROPN
ejpam-5948	223	5	is	be	AUX
ejpam-5948	223	6	an	an	DET
ejpam-5948	223	7	integral	integral	ADJ
ejpam-5948	223	8	domain	domain	NOUN
ejpam-5948	223	9	nor	nor	CCONJ
ejpam-5948	223	10	℧	℧	PROPN
ejpam-5948	223	11	(	(	PUNCT
ejpam-5948	223	12	ℜ	ℜ	PROPN
ejpam-5948	223	13	)	)	PUNCT
ejpam-5948	223	14	⊆	⊆	NUM
ejpam-5948	223	15	p	p	NOUN
ejpam-5948	223	16	.	.	PUNCT
ejpam-5948	224	1	it	it	PRON
ejpam-5948	224	2	is	be	AUX
ejpam-5948	224	3	important	important	ADJ
ejpam-5948	224	4	to	to	PART
ejpam-5948	224	5	note	note	VERB
ejpam-5948	224	6	that	that	SCONJ
ejpam-5948	224	7	p	p	NOUN
ejpam-5948	224	8	is	be	AUX
ejpam-5948	224	9	not	not	PART
ejpam-5948	224	10	a	a	DET
ejpam-5948	224	11	prime	prime	ADJ
ejpam-5948	224	12	ideal	ideal	NOUN
ejpam-5948	224	13	of	of	ADP
ejpam-5948	224	14	ℜ	ℜ	PROPN
ejpam-5948	224	15	,	,	PUNCT
ejpam-5948	224	16	since	since	SCONJ
ejpam-5948	224	17	aℜc	aℜc	ADP
ejpam-5948	224	18	⊆	⊆	NUM
ejpam-5948	224	19	p	p	NOUN
ejpam-5948	224	20	,	,	PUNCT
ejpam-5948	224	21	but	but	CCONJ
ejpam-5948	224	22	neither	neither	CCONJ
ejpam-5948	224	23	a	a	DET
ejpam-5948	224	24	∈	∈	PROPN
ejpam-5948	224	25	p	p	NOUN
ejpam-5948	224	26	nor	nor	CCONJ
ejpam-5948	224	27	c	c	NOUN
ejpam-5948	224	28	∈	∈	PROPN
ejpam-5948	224	29	p	p	NOUN
ejpam-5948	224	30	.	.	PUNCT
ejpam-5948	225	1	therefore	therefore	ADV
ejpam-5948	225	2	,	,	PUNCT
ejpam-5948	225	3	the	the	DET
ejpam-5948	225	4	primeness	primeness	NOUN
ejpam-5948	225	5	condition	condition	NOUN
ejpam-5948	225	6	in	in	ADP
ejpam-5948	225	7	corollary	corollary	ADJ
ejpam-5948	225	8	8	8	NUM
ejpam-5948	225	9	is	be	AUX
ejpam-5948	225	10	essential	essential	ADJ
ejpam-5948	225	11	.	.	PUNCT
ejpam-5948	226	1	bouchannafa	bouchannafa	PROPN
ejpam-5948	226	2	et	et	PROPN
ejpam-5948	226	3	al	al	PROPN
ejpam-5948	226	4	.	.	PUNCT
ejpam-5948	227	1	[	[	X
ejpam-5948	227	2	9	9	NUM
ejpam-5948	227	3	,	,	PUNCT
ejpam-5948	227	4	theorem	theorem	VERB
ejpam-5948	227	5	4	4	NUM
ejpam-5948	227	6	]	]	PUNCT
ejpam-5948	227	7	investigated	investigate	VERB
ejpam-5948	227	8	that	that	SCONJ
ejpam-5948	227	9	the	the	DET
ejpam-5948	227	10	derivations	derivation	NOUN
ejpam-5948	227	11	χ	χ	NOUN
ejpam-5948	227	12	and	and	CCONJ
ejpam-5948	227	13	∝	∝	PROPN
ejpam-5948	227	14	are	be	AUX
ejpam-5948	227	15	subsets	subset	NOUN
ejpam-5948	227	16	of	of	ADP
ejpam-5948	227	17	a	a	DET
ejpam-5948	227	18	prime	prime	ADJ
ejpam-5948	227	19	ideal	ideal	NOUN
ejpam-5948	227	20	p	p	NOUN
ejpam-5948	227	21	,	,	PUNCT
ejpam-5948	227	22	or	or	CCONJ
ejpam-5948	227	23	the	the	DET
ejpam-5948	227	24	factor	factor	NOUN
ejpam-5948	227	25	ring	ring	NOUN
ejpam-5948	227	26	ℜ/p	ℜ/p	PROPN
ejpam-5948	227	27	is	be	AUX
ejpam-5948	227	28	an	an	DET
ejpam-5948	227	29	integral	integral	ADJ
ejpam-5948	227	30	domain	domain	NOUN
ejpam-5948	227	31	.	.	PUNCT
ejpam-5948	228	1	this	this	PRON
ejpam-5948	228	2	occurs	occur	VERB
ejpam-5948	228	3	when	when	SCONJ
ejpam-5948	228	4	one	one	NUM
ejpam-5948	228	5	of	of	ADP
ejpam-5948	228	6	the	the	DET
ejpam-5948	228	7	following	follow	VERB
ejpam-5948	228	8	identities	identity	NOUN
ejpam-5948	228	9	holds	hold	VERB
ejpam-5948	228	10	:	:	PUNCT
ejpam-5948	228	11	℧	℧	PROPN
ejpam-5948	228	12	(	(	PUNCT
ejpam-5948	228	13	υ)	υ)	NOUN
ejpam-5948	228	14	℧	℧	NOUN
ejpam-5948	228	15	(℘)±⨿(υ℘	(℘)±⨿(υ℘	NOUN
ejpam-5948	228	16	)	)	PUNCT
ejpam-5948	228	17	∈	∈	PROPN
ejpam-5948	228	18	z(ℜ/p	z(ℜ/p	NUM
ejpam-5948	228	19	)	)	PUNCT
ejpam-5948	228	20	,	,	PUNCT
ejpam-5948	228	21	or	or	CCONJ
ejpam-5948	228	22	[	[	X
ejpam-5948	228	23	℧	℧	PROPN
ejpam-5948	228	24	(	(	PUNCT
ejpam-5948	228	25	υ	υ	NOUN
ejpam-5948	228	26	)	)	PUNCT
ejpam-5948	228	27	,	,	PUNCT
ejpam-5948	228	28	℘]±⨿(υ℘	℘]±⨿(υ℘	NUM
ejpam-5948	229	1	)	)	PUNCT
ejpam-5948	229	2	∈	∈	PROPN
ejpam-5948	229	3	z(ℜ/p	z(ℜ/p	NUM
ejpam-5948	229	4	)	)	PUNCT
ejpam-5948	229	5	for	for	ADP
ejpam-5948	229	6	all	all	DET
ejpam-5948	229	7	υ	υ	NOUN
ejpam-5948	229	8	,	,	PUNCT
ejpam-5948	229	9	℘	℘	PROPN
ejpam-5948	229	10	∈	∈	PROPN
ejpam-5948	229	11	υ	υ	NOUN
ejpam-5948	229	12	,	,	PUNCT
ejpam-5948	229	13	where	where	SCONJ
ejpam-5948	229	14	υ	υ	NOUN
ejpam-5948	229	15	is	be	AUX
ejpam-5948	229	16	a	a	DET
ejpam-5948	229	17	non	non	ADJ
ejpam-5948	229	18	-	-	ADJ
ejpam-5948	229	19	zero	zero	NUM
ejpam-5948	229	20	ideal	ideal	NOUN
ejpam-5948	229	21	of	of	ADP
ejpam-5948	229	22	ℜ	ℜ	PROPN
ejpam-5948	229	23	,	,	PUNCT
ejpam-5948	229	24	(	(	PUNCT
ejpam-5948	229	25	℧	℧	PROPN
ejpam-5948	229	26	,	,	PUNCT
ejpam-5948	229	27	χ	χ	NOUN
ejpam-5948	229	28	)	)	PUNCT
ejpam-5948	229	29	and	and	CCONJ
ejpam-5948	229	30	(	(	PUNCT
ejpam-5948	229	31	⨿,∝	⨿,∝	X
ejpam-5948	229	32	)	)	PUNCT
ejpam-5948	229	33	are	be	AUX
ejpam-5948	229	34	generalized	generalized	ADJ
ejpam-5948	229	35	derivations	derivation	NOUN
ejpam-5948	229	36	in	in	ADP
ejpam-5948	229	37	ℜ.	ℜ.	PROPN
ejpam-5948	229	38	in	in	ADP
ejpam-5948	229	39	the	the	DET
ejpam-5948	229	40	following	following	NOUN
ejpam-5948	229	41	theorem	theorem	NOUN
ejpam-5948	229	42	,	,	PUNCT
ejpam-5948	229	43	we	we	PRON
ejpam-5948	229	44	will	will	AUX
ejpam-5948	229	45	examine	examine	VERB
ejpam-5948	229	46	the	the	DET
ejpam-5948	229	47	structure	structure	NOUN
ejpam-5948	229	48	of	of	ADP
ejpam-5948	229	49	a	a	DET
ejpam-5948	229	50	factor	factor	NOUN
ejpam-5948	229	51	ring	ring	NOUN
ejpam-5948	229	52	ℜ/p	ℜ/p	PROPN
ejpam-5948	229	53	under	under	ADP
ejpam-5948	229	54	the	the	DET
ejpam-5948	229	55	influence	influence	NOUN
ejpam-5948	229	56	of	of	ADP
ejpam-5948	229	57	a	a	DET
ejpam-5948	229	58	pair	pair	NOUN
ejpam-5948	229	59	of	of	ADP
ejpam-5948	229	60	generalized	generalized	ADJ
ejpam-5948	229	61	p	p	NOUN
ejpam-5948	229	62	-derivations	-derivation	NOUN
ejpam-5948	229	63	that	that	PRON
ejpam-5948	229	64	satisfy	satisfy	VERB
ejpam-5948	229	65	any	any	PRON
ejpam-5948	229	66	of	of	ADP
ejpam-5948	229	67	the	the	DET
ejpam-5948	229	68	following	follow	VERB
ejpam-5948	229	69	algebraic	algebraic	ADJ
ejpam-5948	229	70	identities	identity	NOUN
ejpam-5948	229	71	for	for	ADP
ejpam-5948	229	72	every	every	DET
ejpam-5948	229	73	υ	υ	NOUN
ejpam-5948	229	74	,	,	PUNCT
ejpam-5948	229	75	℘	℘	NOUN
ejpam-5948	229	76	∈	∈	PROPN
ejpam-5948	229	77	ℜ	ℜ	PROPN
ejpam-5948	229	78	:	:	PUNCT
ejpam-5948	229	79	[	[	X
ejpam-5948	229	80	υ,	υ,	X
ejpam-5948	229	81	℧	℧	NOUN
ejpam-5948	229	82	(℘)]±	(℘)]±	PUNCT
ejpam-5948	229	83	℧	℧	NOUN
ejpam-5948	229	84	(	(	PUNCT
ejpam-5948	229	85	℘)⨿	℘)⨿	NOUN
ejpam-5948	229	86	(	(	PUNCT
ejpam-5948	229	87	υ	υ	NOUN
ejpam-5948	229	88	)	)	PUNCT
ejpam-5948	229	89	∈	∈	PROPN
ejpam-5948	229	90	p	p	NOUN
ejpam-5948	229	91	.	.	PUNCT
ejpam-5948	230	1	theorem	theorem	ADJ
ejpam-5948	230	2	6	6	NUM
ejpam-5948	230	3	.	.	PUNCT
ejpam-5948	231	1	let	let	VERB
ejpam-5948	231	2	ℜ	ℜ	PROPN
ejpam-5948	231	3	be	be	AUX
ejpam-5948	231	4	a	a	DET
ejpam-5948	231	5	ring	ring	NOUN
ejpam-5948	231	6	equipped	equip	VERB
ejpam-5948	231	7	with	with	ADP
ejpam-5948	231	8	generalized	generalize	VERB
ejpam-5948	231	9	p	p	NOUN
ejpam-5948	231	10	-derivations	-derivation	NOUN
ejpam-5948	231	11	(	(	PUNCT
ejpam-5948	231	12	℧	℧	PROPN
ejpam-5948	231	13	,	,	PUNCT
ejpam-5948	231	14	χ	χ	NOUN
ejpam-5948	231	15	)	)	PUNCT
ejpam-5948	231	16	and	and	CCONJ
ejpam-5948	231	17	(	(	PUNCT
ejpam-5948	231	18	⨿,∝	⨿,∝	X
ejpam-5948	231	19	)	)	PUNCT
ejpam-5948	231	20	such	such	ADJ
ejpam-5948	231	21	that	that	SCONJ
ejpam-5948	231	22	[	[	X
ejpam-5948	231	23	υ,	υ,	X
ejpam-5948	231	24	℧	℧	NOUN
ejpam-5948	231	25	(℘)]±	(℘)]±	PUNCT
ejpam-5948	231	26	℧	℧	NOUN
ejpam-5948	231	27	(	(	PUNCT
ejpam-5948	231	28	℘)⨿	℘)⨿	NOUN
ejpam-5948	231	29	(	(	PUNCT
ejpam-5948	231	30	υ	υ	NOUN
ejpam-5948	231	31	)	)	PUNCT
ejpam-5948	231	32	∈	∈	PROPN
ejpam-5948	231	33	p	p	NOUN
ejpam-5948	231	34	for	for	ADP
ejpam-5948	231	35	all	all	DET
ejpam-5948	231	36	υ	υ	NOUN
ejpam-5948	231	37	,	,	PUNCT
ejpam-5948	231	38	℘	℘	PROPN
ejpam-5948	231	39	∈	∈	NOUN
ejpam-5948	231	40	ℜ.	ℜ.	PROPN
ejpam-5948	231	41	then	then	ADV
ejpam-5948	231	42	one	one	NUM
ejpam-5948	231	43	of	of	ADP
ejpam-5948	231	44	the	the	DET
ejpam-5948	231	45	following	follow	VERB
ejpam-5948	231	46	is	be	AUX
ejpam-5948	231	47	true	true	ADJ
ejpam-5948	231	48	:	:	PUNCT
ejpam-5948	231	49	(	(	PUNCT
ejpam-5948	231	50	i	i	NOUN
ejpam-5948	231	51	)	)	PUNCT
ejpam-5948	231	52	℧	℧	PROPN
ejpam-5948	231	53	(	(	PUNCT
ejpam-5948	231	54	ℜ	ℜ	PROPN
ejpam-5948	231	55	)	)	PUNCT
ejpam-5948	231	56	⊆	⊆	NUM
ejpam-5948	231	57	p	p	NOUN
ejpam-5948	231	58	;	;	PUNCT
ejpam-5948	231	59	(	(	PUNCT
ejpam-5948	231	60	ii	ii	NOUN
ejpam-5948	231	61	)	)	PUNCT
ejpam-5948	231	62	χ(ℜ	χ(ℜ	NOUN
ejpam-5948	231	63	)	)	PUNCT
ejpam-5948	231	64	⊆	⊆	NUM
ejpam-5948	231	65	p	p	NOUN
ejpam-5948	231	66	and	and	CCONJ
ejpam-5948	231	67	∝	∝	PROPN
ejpam-5948	231	68	(	(	PUNCT
ejpam-5948	231	69	ℜ	ℜ	PROPN
ejpam-5948	231	70	)	)	PUNCT
ejpam-5948	231	71	⊆	⊆	NUM
ejpam-5948	231	72	p	p	NOUN
ejpam-5948	231	73	;	;	PUNCT
ejpam-5948	231	74	(	(	PUNCT
ejpam-5948	231	75	iii	iii	X
ejpam-5948	231	76	)	)	PUNCT
ejpam-5948	231	77	∝	∝	PROPN
ejpam-5948	231	78	(	(	PUNCT
ejpam-5948	231	79	ℜ	ℜ	PROPN
ejpam-5948	231	80	)	)	PUNCT
ejpam-5948	231	81	⊆	⊆	NUM
ejpam-5948	231	82	p	p	NOUN
ejpam-5948	231	83	and	and	CCONJ
ejpam-5948	231	84	ℜ/p	ℜ/p	PROPN
ejpam-5948	231	85	is	be	AUX
ejpam-5948	231	86	an	an	DET
ejpam-5948	231	87	integral	integral	ADJ
ejpam-5948	231	88	domain	domain	NOUN
ejpam-5948	231	89	.	.	PUNCT
ejpam-5948	232	1	a.y	a.y	PROPN
ejpam-5948	232	2	.	.	PROPN
ejpam-5948	232	3	hummdi	hummdi	PROPN
ejpam-5948	232	4	,	,	PUNCT
ejpam-5948	232	5	r.m	r.m	PROPN
ejpam-5948	232	6	.	.	PROPN
ejpam-5948	232	7	al	al	PROPN
ejpam-5948	232	8	-	-	PUNCT
ejpam-5948	232	9	omary	omary	PROPN
ejpam-5948	232	10	,	,	PUNCT
ejpam-5948	232	11	z.z	z.z	PROPN
ejpam-5948	232	12	.	.	PUNCT
ejpam-5948	232	13	al	al	PROPN
ejpam-5948	232	14	-	-	PUNCT
ejpam-5948	232	15	amery	amery	PROPN
ejpam-5948	232	16	/	/	SYM
ejpam-5948	232	17	eur	eur	PROPN
ejpam-5948	232	18	.	.	PUNCT
ejpam-5948	233	1	j.	j.	PROPN
ejpam-5948	233	2	pure	pure	PROPN
ejpam-5948	233	3	appl	appl	PROPN
ejpam-5948	233	4	.	.	PROPN
ejpam-5948	233	5	math	math	PROPN
ejpam-5948	233	6	,	,	PUNCT
ejpam-5948	233	7	18	18	NUM
ejpam-5948	233	8	(	(	PUNCT
ejpam-5948	233	9	2	2	NUM
ejpam-5948	233	10	)	)	PUNCT
ejpam-5948	233	11	(	(	PUNCT
ejpam-5948	233	12	2025	2025	NUM
ejpam-5948	233	13	)	)	PUNCT
ejpam-5948	233	14	,	,	PUNCT
ejpam-5948	233	15	5948	5948	NUM
ejpam-5948	233	16	10	10	NUM
ejpam-5948	233	17	of	of	ADP
ejpam-5948	233	18	12	12	NUM
ejpam-5948	233	19	proof	proof	NOUN
ejpam-5948	233	20	.	.	PUNCT
ejpam-5948	234	1	the	the	DET
ejpam-5948	234	2	initial	initial	ADJ
ejpam-5948	234	3	hypothesis	hypothesis	NOUN
ejpam-5948	234	4	states	state	NOUN
ejpam-5948	234	5	:	:	PUNCT
ejpam-5948	235	1	[	[	X
ejpam-5948	235	2	υ,	υ,	X
ejpam-5948	235	3	℧	℧	NOUN
ejpam-5948	235	4	(℘)]±	(℘)]±	PUNCT
ejpam-5948	235	5	℧	℧	NOUN
ejpam-5948	235	6	(	(	PUNCT
ejpam-5948	235	7	℘)⨿	℘)⨿	NOUN
ejpam-5948	235	8	(	(	PUNCT
ejpam-5948	235	9	υ	υ	NOUN
ejpam-5948	235	10	)	)	PUNCT
ejpam-5948	235	11	∈	∈	PROPN
ejpam-5948	235	12	p	p	NOUN
ejpam-5948	235	13	for	for	ADP
ejpam-5948	235	14	all	all	DET
ejpam-5948	235	15	υ	υ	NOUN
ejpam-5948	235	16	,	,	PUNCT
ejpam-5948	235	17	℘	℘	PROPN
ejpam-5948	235	18	∈	∈	NOUN
ejpam-5948	235	19	ℜ.	ℜ.	PROPN
ejpam-5948	235	20	(	(	PUNCT
ejpam-5948	235	21	18	18	NUM
ejpam-5948	235	22	)	)	PUNCT
ejpam-5948	235	23	substituting	substitute	VERB
ejpam-5948	235	24	υ	υ	NOUN
ejpam-5948	235	25	by	by	ADP
ejpam-5948	235	26	υℏ	υℏ	NOUN
ejpam-5948	235	27	in	in	ADP
ejpam-5948	235	28	equation	equation	NOUN
ejpam-5948	235	29	(	(	PUNCT
ejpam-5948	235	30	18	18	NUM
ejpam-5948	235	31	)	)	PUNCT
ejpam-5948	235	32	and	and	CCONJ
ejpam-5948	235	33	using	use	VERB
ejpam-5948	235	34	it	it	PRON
ejpam-5948	235	35	,	,	PUNCT
ejpam-5948	235	36	we	we	PRON
ejpam-5948	235	37	get	get	VERB
ejpam-5948	235	38	υ[ℏ,	υ[ℏ,	NOUN
ejpam-5948	235	39	℧	℧	NOUN
ejpam-5948	235	40	(℘)]±	(℘)]±	SYM
ejpam-5948	235	41	℧	℧	NOUN
ejpam-5948	235	42	(	(	PUNCT
ejpam-5948	235	43	℘)υ	℘)υ	PROPN
ejpam-5948	235	44	∝	∝	PROPN
ejpam-5948	235	45	(	(	PUNCT
ejpam-5948	235	46	ℏ	ℏ	PROPN
ejpam-5948	235	47	)	)	PUNCT
ejpam-5948	235	48	∈	∈	PROPN
ejpam-5948	235	49	p	p	NOUN
ejpam-5948	235	50	for	for	ADP
ejpam-5948	235	51	all	all	DET
ejpam-5948	235	52	υ	υ	NOUN
ejpam-5948	235	53	,	,	PUNCT
ejpam-5948	235	54	℘	℘	PROPN
ejpam-5948	235	55	,	,	PUNCT
ejpam-5948	235	56	ℏ	ℏ	PROPN
ejpam-5948	235	57	∈	∈	NOUN
ejpam-5948	235	58	ℜ.	ℜ.	PROPN
ejpam-5948	235	59	(	(	PUNCT
ejpam-5948	235	60	19	19	NUM
ejpam-5948	235	61	)	)	PUNCT
ejpam-5948	235	62	replacing	replace	VERB
ejpam-5948	235	63	υ	υ	NOUN
ejpam-5948	235	64	by	by	ADP
ejpam-5948	235	65	℘υ	℘υ	NOUN
ejpam-5948	235	66	in	in	ADP
ejpam-5948	235	67	equation	equation	NOUN
ejpam-5948	235	68	(	(	PUNCT
ejpam-5948	235	69	19	19	NUM
ejpam-5948	235	70	)	)	PUNCT
ejpam-5948	235	71	and	and	CCONJ
ejpam-5948	235	72	comparing	compare	VERB
ejpam-5948	235	73	it	it	PRON
ejpam-5948	235	74	with	with	ADP
ejpam-5948	235	75	(	(	PUNCT
ejpam-5948	235	76	19	19	NUM
ejpam-5948	235	77	)	)	PUNCT
ejpam-5948	235	78	,	,	PUNCT
ejpam-5948	235	79	we	we	PRON
ejpam-5948	235	80	obtain	obtain	VERB
ejpam-5948	235	81	[	[	PUNCT
ejpam-5948	235	82	℧	℧	NOUN
ejpam-5948	235	83	(	(	PUNCT
ejpam-5948	235	84	℘	℘	PROPN
ejpam-5948	235	85	)	)	PUNCT
ejpam-5948	235	86	,	,	PUNCT
ejpam-5948	235	87	℘]ℜ	℘]ℜ	VERB
ejpam-5948	235	88	∝	∝	PROPN
ejpam-5948	235	89	(	(	PUNCT
ejpam-5948	235	90	ℏ	ℏ	PROPN
ejpam-5948	235	91	)	)	PUNCT
ejpam-5948	235	92	⊆	⊆	NUM
ejpam-5948	235	93	p	p	NOUN
ejpam-5948	235	94	for	for	ADP
ejpam-5948	235	95	all	all	DET
ejpam-5948	235	96	℘	℘	NOUN
ejpam-5948	235	97	,	,	PUNCT
ejpam-5948	235	98	ℏ	ℏ	PROPN
ejpam-5948	235	99	∈	∈	NOUN
ejpam-5948	235	100	ℜ.	ℜ.	PROPN
ejpam-5948	235	101	primeness	primeness	NOUN
ejpam-5948	235	102	of	of	ADP
ejpam-5948	235	103	p	p	NOUN
ejpam-5948	235	104	forces	force	NOUN
ejpam-5948	235	105	that	that	SCONJ
ejpam-5948	235	106	[	[	X
ejpam-5948	235	107	℧	℧	PROPN
ejpam-5948	235	108	(	(	PUNCT
ejpam-5948	235	109	℘	℘	PROPN
ejpam-5948	235	110	)	)	PUNCT
ejpam-5948	235	111	,	,	PUNCT
ejpam-5948	235	112	℘	℘	PROPN
ejpam-5948	235	113	]	]	PUNCT
ejpam-5948	235	114	∈	∈	PROPN
ejpam-5948	235	115	p	p	NOUN
ejpam-5948	235	116	or	or	CCONJ
ejpam-5948	235	117	∝	∝	PROPN
ejpam-5948	235	118	(	(	PUNCT
ejpam-5948	235	119	ℏ	ℏ	NOUN
ejpam-5948	235	120	)	)	PUNCT
ejpam-5948	235	121	∈	∈	PROPN
ejpam-5948	235	122	p	p	NOUN
ejpam-5948	235	123	for	for	ADP
ejpam-5948	235	124	all	all	DET
ejpam-5948	235	125	℘	℘	NOUN
ejpam-5948	235	126	,	,	PUNCT
ejpam-5948	235	127	ℏ	ℏ	PROPN
ejpam-5948	235	128	∈	∈	NOUN
ejpam-5948	235	129	ℜ.	ℜ.	PROPN
ejpam-5948	235	130	to	to	PART
ejpam-5948	235	131	complete	complete	VERB
ejpam-5948	235	132	the	the	DET
ejpam-5948	235	133	proof	proof	NOUN
ejpam-5948	235	134	,	,	PUNCT
ejpam-5948	235	135	let	let	VERB
ejpam-5948	235	136	’s	’s	PRON
ejpam-5948	235	137	discuss	discuss	VERB
ejpam-5948	235	138	the	the	DET
ejpam-5948	235	139	following	follow	VERB
ejpam-5948	235	140	two	two	NUM
ejpam-5948	235	141	cases	case	NOUN
ejpam-5948	235	142	:	:	PUNCT
ejpam-5948	235	143	case	case	NOUN
ejpam-5948	235	144	(	(	PUNCT
ejpam-5948	235	145	a	a	X
ejpam-5948	235	146	):	):	PUNCT
ejpam-5948	235	147	if	if	SCONJ
ejpam-5948	235	148	[	[	X
ejpam-5948	235	149	℧	℧	X
ejpam-5948	235	150	(	(	PUNCT
ejpam-5948	235	151	℘	℘	PROPN
ejpam-5948	235	152	)	)	PUNCT
ejpam-5948	235	153	,	,	PUNCT
ejpam-5948	235	154	℘	℘	PROPN
ejpam-5948	235	155	]	]	PUNCT
ejpam-5948	235	156	∈	∈	PROPN
ejpam-5948	235	157	p	p	NOUN
ejpam-5948	235	158	for	for	ADP
ejpam-5948	235	159	all	all	DET
ejpam-5948	235	160	℘	℘	PROPN
ejpam-5948	235	161	∈	∈	PROPN
ejpam-5948	235	162	ℜ	ℜ	PROPN
ejpam-5948	235	163	,	,	PUNCT
ejpam-5948	235	164	lemma	lemma	PROPN
ejpam-5948	235	165	3	3	NUM
ejpam-5948	235	166	gives	give	VERB
ejpam-5948	235	167	χ(ℜ	χ(ℜ	NOUN
ejpam-5948	235	168	)	)	PUNCT
ejpam-5948	235	169	⊆	⊆	NUM
ejpam-5948	235	170	p	p	NOUN
ejpam-5948	235	171	or	or	CCONJ
ejpam-5948	235	172	ℜ/p	ℜ/p	PROPN
ejpam-5948	235	173	is	be	AUX
ejpam-5948	235	174	an	an	DET
ejpam-5948	235	175	integral	integral	ADJ
ejpam-5948	235	176	domain	domain	NOUN
ejpam-5948	235	177	.	.	PUNCT
ejpam-5948	236	1	let	let	VERB
ejpam-5948	236	2	’s	’s	PRON
ejpam-5948	236	3	examine	examine	VERB
ejpam-5948	236	4	χ(ℜ	χ(ℜ	NOUN
ejpam-5948	236	5	)	)	PUNCT
ejpam-5948	236	6	⊆	⊆	NUM
ejpam-5948	236	7	p	p	NOUN
ejpam-5948	236	8	.	.	PUNCT
ejpam-5948	237	1	then	then	ADV
ejpam-5948	237	2	substituting	substitute	VERB
ejpam-5948	237	3	℘	℘	PROPN
ejpam-5948	237	4	with	with	ADP
ejpam-5948	237	5	℘κ	℘κ	NOUN
ejpam-5948	237	6	in	in	ADP
ejpam-5948	237	7	equation	equation	NOUN
ejpam-5948	237	8	(	(	PUNCT
ejpam-5948	237	9	18	18	NUM
ejpam-5948	237	10	)	)	PUNCT
ejpam-5948	237	11	,	,	PUNCT
ejpam-5948	237	12	we	we	PRON
ejpam-5948	237	13	deduce	deduce	VERB
ejpam-5948	237	14	that	that	SCONJ
ejpam-5948	237	15	[	[	X
ejpam-5948	237	16	υ,	υ,	X
ejpam-5948	237	17	℧	℧	X
ejpam-5948	237	18	(℘)]κ	(℘)]κ	PROPN
ejpam-5948	237	19	+	+	NUM
ejpam-5948	237	20	℧	℧	PROPN
ejpam-5948	237	21	(	(	PUNCT
ejpam-5948	237	22	℘)[υ	℘)[υ	PROPN
ejpam-5948	237	23	,	,	PUNCT
ejpam-5948	237	24	κ	κ	X
ejpam-5948	237	25	]	]	X
ejpam-5948	237	26	±	±	NUM
ejpam-5948	237	27	℧	℧	PROPN
ejpam-5948	237	28	(	(	PUNCT
ejpam-5948	237	29	℘)κ	℘)κ	PROPN
ejpam-5948	237	30	⨿	⨿	X
ejpam-5948	237	31	(	(	PUNCT
ejpam-5948	237	32	υ	υ	NOUN
ejpam-5948	237	33	)	)	PUNCT
ejpam-5948	237	34	∈	∈	PROPN
ejpam-5948	237	35	p	p	NOUN
ejpam-5948	237	36	for	for	ADP
ejpam-5948	237	37	all	all	DET
ejpam-5948	237	38	υ	υ	NOUN
ejpam-5948	237	39	,	,	PUNCT
ejpam-5948	237	40	℘	℘	PROPN
ejpam-5948	237	41	,	,	PUNCT
ejpam-5948	237	42	κ	κ	PROPN
ejpam-5948	237	43	∈	∈	PROPN
ejpam-5948	237	44	ℜ.	ℜ.	PROPN
ejpam-5948	237	45	right	right	ADJ
ejpam-5948	237	46	multiplying	multiplying	NOUN
ejpam-5948	237	47	of	of	ADP
ejpam-5948	237	48	equation	equation	NOUN
ejpam-5948	237	49	(	(	PUNCT
ejpam-5948	237	50	18	18	NUM
ejpam-5948	237	51	)	)	PUNCT
ejpam-5948	237	52	by	by	ADP
ejpam-5948	237	53	κ	κ	NOUN
ejpam-5948	237	54	and	and	CCONJ
ejpam-5948	237	55	comparing	compare	VERB
ejpam-5948	237	56	it	it	PRON
ejpam-5948	237	57	with	with	ADP
ejpam-5948	237	58	the	the	DET
ejpam-5948	237	59	last	last	ADJ
ejpam-5948	237	60	equation	equation	NOUN
ejpam-5948	237	61	,	,	PUNCT
ejpam-5948	237	62	we	we	PRON
ejpam-5948	237	63	get	get	VERB
ejpam-5948	237	64	℧	℧	PROPN
ejpam-5948	237	65	(	(	PUNCT
ejpam-5948	237	66	℘)[υ	℘)[υ	PROPN
ejpam-5948	237	67	,	,	PUNCT
ejpam-5948	237	68	κ]±	κ]±	NUM
ejpam-5948	237	69	℧	℧	PROPN
ejpam-5948	237	70	(	(	PUNCT
ejpam-5948	237	71	℘)[κ,⨿(υ	℘)[κ,⨿(υ	NOUN
ejpam-5948	237	72	)	)	PUNCT
ejpam-5948	237	73	]	]	PUNCT
ejpam-5948	238	1	∈	∈	PROPN
ejpam-5948	238	2	p	p	NOUN
ejpam-5948	238	3	.	.	PUNCT
ejpam-5948	239	1	that	that	PRON
ejpam-5948	239	2	is	be	AUX
ejpam-5948	239	3	,	,	PUNCT
ejpam-5948	239	4	℧	℧	PROPN
ejpam-5948	239	5	(	(	PUNCT
ejpam-5948	239	6	℘)ℜ([υ	℘)ℜ([υ	NOUN
ejpam-5948	239	7	,	,	PUNCT
ejpam-5948	239	8	κ]∓	κ]∓	NOUN
ejpam-5948	240	1	[	[	X
ejpam-5948	240	2	⨿(υ	⨿(υ	NUM
ejpam-5948	240	3	)	)	PUNCT
ejpam-5948	240	4	,	,	PUNCT
ejpam-5948	240	5	κ	κ	NOUN
ejpam-5948	240	6	]	]	X
ejpam-5948	240	7	)	)	PUNCT
ejpam-5948	240	8	⊆	⊆	NUM
ejpam-5948	240	9	p	p	NOUN
ejpam-5948	240	10	for	for	ADP
ejpam-5948	240	11	all	all	DET
ejpam-5948	240	12	υ	υ	NOUN
ejpam-5948	240	13	,	,	PUNCT
ejpam-5948	240	14	℘	℘	PROPN
ejpam-5948	240	15	,	,	PUNCT
ejpam-5948	240	16	κ	κ	PROPN
ejpam-5948	240	17	∈	∈	PROPN
ejpam-5948	240	18	ℜ.	ℜ.	PROPN
ejpam-5948	240	19	primeness	primeness	NOUN
ejpam-5948	240	20	of	of	ADP
ejpam-5948	240	21	p	p	PROPN
ejpam-5948	240	22	implies	imply	VERB
ejpam-5948	240	23	that	that	SCONJ
ejpam-5948	240	24	℧	℧	PROPN
ejpam-5948	240	25	(	(	PUNCT
ejpam-5948	240	26	℘	℘	PROPN
ejpam-5948	240	27	)	)	PUNCT
ejpam-5948	240	28	∈	∈	PROPN
ejpam-5948	240	29	p	p	NOUN
ejpam-5948	240	30	or	or	CCONJ
ejpam-5948	240	31	[	[	X
ejpam-5948	240	32	υ	υ	X
ejpam-5948	240	33	,	,	PUNCT
ejpam-5948	240	34	κ	κ	X
ejpam-5948	240	35	]	]	X
ejpam-5948	240	36	∓	∓	PROPN
ejpam-5948	241	1	[	[	X
ejpam-5948	241	2	⨿(υ	⨿(υ	NUM
ejpam-5948	241	3	)	)	PUNCT
ejpam-5948	241	4	,	,	PUNCT
ejpam-5948	241	5	κ	κ	X
ejpam-5948	241	6	]	]	X
ejpam-5948	241	7	∈	∈	PROPN
ejpam-5948	241	8	p	p	NOUN
ejpam-5948	241	9	for	for	ADP
ejpam-5948	241	10	all	all	DET
ejpam-5948	241	11	υ	υ	NOUN
ejpam-5948	241	12	,	,	PUNCT
ejpam-5948	241	13	℘	℘	PROPN
ejpam-5948	241	14	,	,	PUNCT
ejpam-5948	241	15	κ	κ	NOUN
ejpam-5948	241	16	∈	∈	PROPN
ejpam-5948	241	17	ℜ.	ℜ.	PROPN
ejpam-5948	241	18	from	from	ADP
ejpam-5948	241	19	the	the	DET
ejpam-5948	241	20	first	first	ADJ
ejpam-5948	241	21	scenario	scenario	NOUN
ejpam-5948	241	22	,	,	PUNCT
ejpam-5948	241	23	we	we	PRON
ejpam-5948	241	24	conclude	conclude	VERB
ejpam-5948	241	25	℧	℧	PROPN
ejpam-5948	241	26	(	(	PUNCT
ejpam-5948	241	27	ℜ	ℜ	PROPN
ejpam-5948	241	28	)	)	PUNCT
ejpam-5948	241	29	⊆	⊆	NUM
ejpam-5948	241	30	p	p	NOUN
ejpam-5948	241	31	.	.	PUNCT
ejpam-5948	242	1	the	the	DET
ejpam-5948	242	2	second	second	ADJ
ejpam-5948	242	3	scenario	scenario	NOUN
ejpam-5948	242	4	with	with	ADP
ejpam-5948	242	5	substitution	substitution	NOUN
ejpam-5948	242	6	κ	κ	NOUN
ejpam-5948	242	7	=	=	PUNCT
ejpam-5948	242	8	κυ	κυ	ADP
ejpam-5948	242	9	,	,	PUNCT
ejpam-5948	242	10	yields	yield	NOUN
ejpam-5948	242	11	[	[	X
ejpam-5948	242	12	⨿(υ	⨿(υ	NUM
ejpam-5948	242	13	)	)	PUNCT
ejpam-5948	242	14	,	,	PUNCT
ejpam-5948	242	15	υ]κ	υ]κ	VERB
ejpam-5948	242	16	∈	∈	PROPN
ejpam-5948	242	17	p	p	PROPN
ejpam-5948	242	18	for	for	ADP
ejpam-5948	242	19	all	all	DET
ejpam-5948	242	20	υ	υ	PROPN
ejpam-5948	242	21	,	,	PUNCT
ejpam-5948	242	22	κ	κ	PROPN
ejpam-5948	242	23	∈	∈	PROPN
ejpam-5948	242	24	ℜ.	ℜ.	PROPN
ejpam-5948	242	25	again	again	ADV
ejpam-5948	242	26	,	,	PUNCT
ejpam-5948	242	27	primeness	primeness	NOUN
ejpam-5948	242	28	of	of	ADP
ejpam-5948	242	29	p	p	NOUN
ejpam-5948	242	30	with	with	ADP
ejpam-5948	242	31	utilizing	utilize	VERB
ejpam-5948	242	32	lemma	lemma	PROPN
ejpam-5948	242	33	3	3	NUM
ejpam-5948	242	34	,	,	PUNCT
ejpam-5948	242	35	we	we	PRON
ejpam-5948	242	36	conclude	conclude	VERB
ejpam-5948	242	37	ℜ/p	ℜ/p	PROPN
ejpam-5948	242	38	an	an	DET
ejpam-5948	242	39	integral	integral	ADJ
ejpam-5948	242	40	domain	domain	NOUN
ejpam-5948	242	41	or	or	CCONJ
ejpam-5948	242	42	∝	∝	PROPN
ejpam-5948	242	43	(	(	PUNCT
ejpam-5948	242	44	ℜ	ℜ	PROPN
ejpam-5948	242	45	)	)	PUNCT
ejpam-5948	242	46	⊆	⊆	NUM
ejpam-5948	242	47	p	p	NOUN
ejpam-5948	242	48	.	.	PUNCT
ejpam-5948	243	1	now	now	ADV
ejpam-5948	243	2	,	,	PUNCT
ejpam-5948	243	3	if	if	SCONJ
ejpam-5948	243	4	we	we	PRON
ejpam-5948	243	5	consider	consider	VERB
ejpam-5948	243	6	the	the	DET
ejpam-5948	243	7	scenario	scenario	NOUN
ejpam-5948	243	8	when	when	SCONJ
ejpam-5948	243	9	ℜ/p	ℜ/p	PROPN
ejpam-5948	243	10	is	be	AUX
ejpam-5948	243	11	an	an	DET
ejpam-5948	243	12	integral	integral	ADJ
ejpam-5948	243	13	domain	domain	NOUN
ejpam-5948	243	14	,	,	PUNCT
ejpam-5948	243	15	then	then	ADV
ejpam-5948	243	16	equation	equation	NOUN
ejpam-5948	243	17	(	(	PUNCT
ejpam-5948	243	18	19	19	NUM
ejpam-5948	243	19	)	)	PUNCT
ejpam-5948	243	20	simplifies	simplifie	NOUN
ejpam-5948	243	21	to	to	ADP
ejpam-5948	243	22	℧	℧	PROPN
ejpam-5948	243	23	(	(	PUNCT
ejpam-5948	243	24	℘)υ	℘)υ	PROPN
ejpam-5948	243	25	∝	∝	PROPN
ejpam-5948	243	26	(	(	PUNCT
ejpam-5948	243	27	ℏ	ℏ	PROPN
ejpam-5948	243	28	)	)	PUNCT
ejpam-5948	243	29	∈	∈	PROPN
ejpam-5948	243	30	p	p	NOUN
ejpam-5948	243	31	for	for	ADP
ejpam-5948	243	32	all	all	DET
ejpam-5948	243	33	υ	υ	NOUN
ejpam-5948	243	34	,	,	PUNCT
ejpam-5948	243	35	℘	℘	PROPN
ejpam-5948	243	36	,	,	PUNCT
ejpam-5948	243	37	ℏ	ℏ	PROPN
ejpam-5948	243	38	∈	∈	PROPN
ejpam-5948	243	39	ℜ	ℜ	PROPN
ejpam-5948	243	40	,	,	PUNCT
ejpam-5948	243	41	which	which	PRON
ejpam-5948	243	42	implies	imply	VERB
ejpam-5948	243	43	℧	℧	PROPN
ejpam-5948	243	44	(	(	PUNCT
ejpam-5948	243	45	℘)ℜ	℘)ℜ	PUNCT
ejpam-5948	243	46	∝	∝	PROPN
ejpam-5948	243	47	(	(	PUNCT
ejpam-5948	243	48	ℏ	ℏ	PROPN
ejpam-5948	243	49	)	)	PUNCT
ejpam-5948	243	50	⊆	⊆	NUM
ejpam-5948	243	51	p	p	NOUN
ejpam-5948	243	52	for	for	ADP
ejpam-5948	243	53	all	all	DET
ejpam-5948	243	54	℘	℘	NOUN
ejpam-5948	243	55	,	,	PUNCT
ejpam-5948	243	56	ℏ	ℏ	PROPN
ejpam-5948	243	57	∈	∈	NOUN
ejpam-5948	243	58	ℜ.	ℜ.	PROPN
ejpam-5948	243	59	primeness	primeness	NOUN
ejpam-5948	243	60	of	of	ADP
ejpam-5948	243	61	p	p	NOUN
ejpam-5948	243	62	implies	imply	VERB
ejpam-5948	243	63	℧	℧	PROPN
ejpam-5948	243	64	(	(	PUNCT
ejpam-5948	243	65	ℜ	ℜ	PROPN
ejpam-5948	243	66	)	)	PUNCT
ejpam-5948	243	67	⊆	⊆	NUM
ejpam-5948	243	68	p	p	NOUN
ejpam-5948	243	69	or	or	CCONJ
ejpam-5948	243	70	∝	∝	PROPN
ejpam-5948	243	71	(	(	PUNCT
ejpam-5948	243	72	ℜ	ℜ	PROPN
ejpam-5948	243	73	)	)	PUNCT
ejpam-5948	243	74	⊆	⊆	NUM
ejpam-5948	243	75	p	p	NOUN
ejpam-5948	243	76	.	.	PUNCT
ejpam-5948	244	1	case	case	NOUN
ejpam-5948	244	2	(	(	PUNCT
ejpam-5948	244	3	b	b	X
ejpam-5948	244	4	):	):	PUNCT
ejpam-5948	244	5	if	if	SCONJ
ejpam-5948	244	6	∝	∝	PROPN
ejpam-5948	244	7	(	(	PUNCT
ejpam-5948	244	8	ℏ	ℏ	PROPN
ejpam-5948	244	9	)	)	PUNCT
ejpam-5948	244	10	∈	∈	PROPN
ejpam-5948	244	11	p	p	NOUN
ejpam-5948	244	12	for	for	ADP
ejpam-5948	244	13	all	all	DET
ejpam-5948	244	14	ℏ	ℏ	PRON
ejpam-5948	244	15	∈	∈	PROPN
ejpam-5948	244	16	ℜ	ℜ	PROPN
ejpam-5948	244	17	,	,	PUNCT
ejpam-5948	244	18	equation	equation	NOUN
ejpam-5948	244	19	(	(	PUNCT
ejpam-5948	244	20	19	19	NUM
ejpam-5948	244	21	)	)	PUNCT
ejpam-5948	244	22	reduces	reduce	VERB
ejpam-5948	244	23	to	to	ADP
ejpam-5948	244	24	υ[ℏ,	υ[ℏ,	NOUN
ejpam-5948	244	25	℧	℧	NOUN
ejpam-5948	244	26	(℘	(℘	NOUN
ejpam-5948	244	27	)	)	PUNCT
ejpam-5948	244	28	]	]	PUNCT
ejpam-5948	245	1	∈	∈	PROPN
ejpam-5948	245	2	p	p	NOUN
ejpam-5948	245	3	for	for	ADP
ejpam-5948	245	4	all	all	DET
ejpam-5948	245	5	υ	υ	NOUN
ejpam-5948	245	6	,	,	PUNCT
ejpam-5948	245	7	℘	℘	PROPN
ejpam-5948	245	8	,	,	PUNCT
ejpam-5948	245	9	ℏ	ℏ	PROPN
ejpam-5948	245	10	∈	∈	NOUN
ejpam-5948	245	11	ℜ.	ℜ.	PROPN
ejpam-5948	245	12	primeness	primeness	NOUN
ejpam-5948	245	13	of	of	ADP
ejpam-5948	245	14	p	p	NOUN
ejpam-5948	245	15	forces	force	NOUN
ejpam-5948	245	16	that	that	SCONJ
ejpam-5948	246	1	[	[	X
ejpam-5948	246	2	ℏ,	ℏ,	NOUN
ejpam-5948	246	3	℧	℧	NOUN
ejpam-5948	246	4	(℘	(℘	NUM
ejpam-5948	246	5	)	)	PUNCT
ejpam-5948	246	6	]	]	PUNCT
ejpam-5948	247	1	∈	∈	PROPN
ejpam-5948	247	2	p	p	NOUN
ejpam-5948	247	3	for	for	ADP
ejpam-5948	247	4	all	all	DET
ejpam-5948	247	5	℘	℘	NOUN
ejpam-5948	247	6	,	,	PUNCT
ejpam-5948	247	7	ℏ	ℏ	PROPN
ejpam-5948	247	8	∈	∈	NOUN
ejpam-5948	247	9	ℜ.	ℜ.	VERB
ejpam-5948	247	10	by	by	ADP
ejpam-5948	247	11	using	use	VERB
ejpam-5948	247	12	lemma	lemma	PROPN
ejpam-5948	247	13	2	2	PROPN
ejpam-5948	247	14	(	(	PUNCT
ejpam-5948	247	15	i	i	NOUN
ejpam-5948	247	16	)	)	PUNCT
ejpam-5948	247	17	,	,	PUNCT
ejpam-5948	247	18	we	we	PRON
ejpam-5948	247	19	conclude	conclude	VERB
ejpam-5948	247	20	ℜ/p	ℜ/p	PROPN
ejpam-5948	247	21	is	be	AUX
ejpam-5948	247	22	an	an	DET
ejpam-5948	247	23	integral	integral	ADJ
ejpam-5948	247	24	domain	domain	NOUN
ejpam-5948	247	25	or	or	CCONJ
ejpam-5948	247	26	℧	℧	PROPN
ejpam-5948	247	27	(	(	PUNCT
ejpam-5948	247	28	ℜ	ℜ	PROPN
ejpam-5948	247	29	)	)	PUNCT
ejpam-5948	247	30	⊆	⊆	NUM
ejpam-5948	247	31	p	p	NOUN
ejpam-5948	247	32	.	.	PUNCT
ejpam-5948	248	1	if	if	SCONJ
ejpam-5948	248	2	ℜ/p	ℜ/p	PROPN
ejpam-5948	248	3	is	be	AUX
ejpam-5948	248	4	an	an	DET
ejpam-5948	248	5	integral	integral	ADJ
ejpam-5948	248	6	domain	domain	NOUN
ejpam-5948	248	7	,	,	PUNCT
ejpam-5948	248	8	then	then	ADV
ejpam-5948	248	9	as	as	SCONJ
ejpam-5948	248	10	discussed	discuss	VERB
ejpam-5948	248	11	above	above	ADP
ejpam-5948	248	12	the	the	DET
ejpam-5948	248	13	desired	desire	VERB
ejpam-5948	248	14	result	result	NOUN
ejpam-5948	248	15	can	can	AUX
ejpam-5948	248	16	be	be	AUX
ejpam-5948	248	17	obtained	obtain	VERB
ejpam-5948	248	18	.	.	PUNCT
ejpam-5948	249	1	the	the	DET
ejpam-5948	249	2	following	follow	VERB
ejpam-5948	249	3	examples	example	NOUN
ejpam-5948	249	4	aim	aim	VERB
ejpam-5948	249	5	to	to	PART
ejpam-5948	249	6	emphasize	emphasize	VERB
ejpam-5948	249	7	the	the	DET
ejpam-5948	249	8	necessity	necessity	NOUN
ejpam-5948	249	9	of	of	ADP
ejpam-5948	249	10	the	the	DET
ejpam-5948	249	11	primeness	primeness	NOUN
ejpam-5948	249	12	condition	condition	NOUN
ejpam-5948	249	13	of	of	ADP
ejpam-5948	249	14	p	p	NOUN
ejpam-5948	249	15	in	in	ADP
ejpam-5948	249	16	theorems	theorem	NOUN
ejpam-5948	249	17	[	[	X
ejpam-5948	249	18	1–6	1–6	NUM
ejpam-5948	249	19	]	]	PUNCT
ejpam-5948	249	20	.	.	PUNCT
ejpam-5948	250	1	example	example	NOUN
ejpam-5948	251	1	3	3	X
ejpam-5948	251	2	.	.	PUNCT
ejpam-5948	251	3	let	let	VERB
ejpam-5948	251	4	ℜ	ℜ	NOUN
ejpam-5948	251	5	=	=	PUNCT
ejpam-5948	251	6	{	{	PUNCT
ejpam-5948	251	7	0	0	ADV
ejpam-5948	251	8	υ	υ	PRON
ejpam-5948	251	9	℘	℘	PROPN
ejpam-5948	251	10	0	0	NUM
ejpam-5948	251	11	0	0	NUM
ejpam-5948	251	12	4ℏ	4ℏ	NUM
ejpam-5948	251	13	0	0	NUM
ejpam-5948	251	14	0	0	NUM
ejpam-5948	251	15	0	0	NUM
ejpam-5948	252	1			PROPN
ejpam-5948	252	2	|	|	ADV
ejpam-5948	252	3	υ	υ	NOUN
ejpam-5948	252	4	,	,	PUNCT
ejpam-5948	252	5	℘	℘	PROPN
ejpam-5948	252	6	,	,	PUNCT
ejpam-5948	252	7	ℏ	ℏ	PROPN
ejpam-5948	252	8	∈	∈	PROPN
ejpam-5948	252	9	z8	z8	NOUN
ejpam-5948	252	10	}	}	PUNCT
ejpam-5948	252	11	,	,	PUNCT
ejpam-5948	252	12	and	and	CCONJ
ejpam-5948	252	13	let	let	VERB
ejpam-5948	252	14	p	p	NOUN
ejpam-5948	252	15	=	=	X
ejpam-5948	252	16	{	{	PUNCT
ejpam-5948	252	17	0	0	ADV
ejpam-5948	252	18	0	0	NUM
ejpam-5948	252	19	0	0	NUM
ejpam-5948	252	20	0	0	NUM
ejpam-5948	252	21	0	0	NUM
ejpam-5948	252	22	0	0	NUM
ejpam-5948	252	23	0	0	NUM
ejpam-5948	252	24	0	0	NUM
ejpam-5948	252	25	0	0	NUM
ejpam-5948	252	26			PROPN
ejpam-5948	252	27	}	}	PUNCT
ejpam-5948	252	28	.	.	PUNCT
ejpam-5948	253	1	defined	define	VERB
ejpam-5948	253	2	(	(	PUNCT
ejpam-5948	253	3	℧	℧	PROPN
ejpam-5948	253	4	,	,	PUNCT
ejpam-5948	253	5	χ	χ	NOUN
ejpam-5948	253	6	)	)	PUNCT
ejpam-5948	253	7	;	;	PUNCT
ejpam-5948	253	8	(	(	PUNCT
ejpam-5948	253	9	⨿,∝	⨿,∝	X
ejpam-5948	253	10	)	)	PUNCT
ejpam-5948	253	11	:	:	PUNCT
ejpam-5948	253	12	ℜ	ℜ	ADV
ejpam-5948	253	13	−→	−→	NOUN
ejpam-5948	253	14	ℜ	ℜ	ADJ
ejpam-5948	253	15	by	by	ADP
ejpam-5948	253	16	℧	℧	PROPN
ejpam-5948	253	17	0	0	ADP
ejpam-5948	253	18	υ	υ	DET
ejpam-5948	253	19	℘	℘	PROPN
ejpam-5948	253	20	0	0	NUM
ejpam-5948	253	21	0	0	NUM
ejpam-5948	253	22	4ℏ	4ℏ	NUM
ejpam-5948	253	23	0	0	NUM
ejpam-5948	253	24	0	0	NUM
ejpam-5948	253	25	0	0	NUM
ejpam-5948	254	1			PROPN
ejpam-5948	254	2	=	=	PUNCT
ejpam-5948	254	3	0	0	ADP
ejpam-5948	254	4	2υ	2υ	NUM
ejpam-5948	254	5	0	0	SYM
ejpam-5948	254	6	0	0	NUM
ejpam-5948	254	7	0	0	NUM
ejpam-5948	254	8	0	0	NUM
ejpam-5948	254	9	0	0	NUM
ejpam-5948	254	10	0	0	NUM
ejpam-5948	254	11	0	0	NUM
ejpam-5948	255	1			PROPN
ejpam-5948	255	2	with	with	ADP
ejpam-5948	255	3	χ	χ	PROPN
ejpam-5948	255	4	0	0	ADP
ejpam-5948	255	5	υ	υ	DET
ejpam-5948	255	6	℘	℘	PROPN
ejpam-5948	255	7	0	0	NUM
ejpam-5948	255	8	0	0	NUM
ejpam-5948	255	9	4ℏ	4ℏ	NUM
ejpam-5948	255	10	0	0	NUM
ejpam-5948	255	11	0	0	NUM
ejpam-5948	255	12	0	0	NUM
ejpam-5948	256	1			PROPN
ejpam-5948	256	2	=	=	SYM
ejpam-5948	256	3	0	0	ADP
ejpam-5948	256	4	0	0	NUM
ejpam-5948	256	5	ℏ	ℏ	NOUN
ejpam-5948	256	6	0	0	NUM
ejpam-5948	256	7	0	0	NUM
ejpam-5948	256	8	0	0	NUM
ejpam-5948	256	9	0	0	NUM
ejpam-5948	256	10	0	0	NUM
ejpam-5948	256	11	0	0	NUM
ejpam-5948	257	1			PROPN
ejpam-5948	257	2	and	and	CCONJ
ejpam-5948	257	3	⨿	⨿	PRON
ejpam-5948	257	4	0	0	ADP
ejpam-5948	257	5	υ	υ	DET
ejpam-5948	257	6	℘	℘	PROPN
ejpam-5948	257	7	0	0	NUM
ejpam-5948	257	8	0	0	NUM
ejpam-5948	257	9	4ℏ	4ℏ	NUM
ejpam-5948	257	10	0	0	NUM
ejpam-5948	257	11	0	0	NUM
ejpam-5948	257	12	0	0	NUM
ejpam-5948	258	1			PROPN
ejpam-5948	258	2	=	=	PUNCT
ejpam-5948	258	3	0	0	ADP
ejpam-5948	258	4	4℘	4℘	NUM
ejpam-5948	258	5	0	0	NUM
ejpam-5948	258	6	0	0	NUM
ejpam-5948	258	7	0	0	NUM
ejpam-5948	258	8	0	0	NUM
ejpam-5948	258	9	0	0	NUM
ejpam-5948	258	10	0	0	NUM
ejpam-5948	258	11	0	0	NUM
ejpam-5948	259	1			PROPN
ejpam-5948	259	2	with	with	ADP
ejpam-5948	259	3	∝	∝	PRON
ejpam-5948	259	4	0	0	ADP
ejpam-5948	259	5	υ	υ	DET
ejpam-5948	259	6	℘	℘	PROPN
ejpam-5948	259	7	0	0	NUM
ejpam-5948	259	8	0	0	NUM
ejpam-5948	259	9	4ℏ	4ℏ	NUM
ejpam-5948	259	10	0	0	NUM
ejpam-5948	259	11	0	0	NUM
ejpam-5948	259	12	0	0	NUM
ejpam-5948	260	1			PROPN
ejpam-5948	260	2	=	=	PUNCT
ejpam-5948	260	3	0	0	ADP
ejpam-5948	260	4	0	0	NUM
ejpam-5948	260	5	υ	υ	NOUN
ejpam-5948	260	6	0	0	NUM
ejpam-5948	260	7	0	0	NUM
ejpam-5948	260	8	0	0	NUM
ejpam-5948	260	9	0	0	NUM
ejpam-5948	260	10	0	0	NUM
ejpam-5948	260	11	0	0	NUM
ejpam-5948	261	1			PROPN
ejpam-5948	261	2	.	.	PUNCT
ejpam-5948	262	1	a.y	a.y	PROPN
ejpam-5948	262	2	.	.	PROPN
ejpam-5948	263	1	hummdi	hummdi	PROPN
ejpam-5948	263	2	,	,	PUNCT
ejpam-5948	263	3	r.m	r.m	PROPN
ejpam-5948	263	4	.	.	PROPN
ejpam-5948	263	5	al	al	PROPN
ejpam-5948	263	6	-	-	PUNCT
ejpam-5948	263	7	omary	omary	PROPN
ejpam-5948	263	8	,	,	PUNCT
ejpam-5948	263	9	z.z	z.z	PROPN
ejpam-5948	263	10	.	.	PUNCT
ejpam-5948	263	11	al	al	PROPN
ejpam-5948	263	12	-	-	PUNCT
ejpam-5948	263	13	amery	amery	PROPN
ejpam-5948	263	14	/	/	SYM
ejpam-5948	263	15	eur	eur	PROPN
ejpam-5948	263	16	.	.	PUNCT
ejpam-5948	264	1	j.	j.	PROPN
ejpam-5948	264	2	pure	pure	PROPN
ejpam-5948	264	3	appl	appl	PROPN
ejpam-5948	264	4	.	.	PROPN
ejpam-5948	264	5	math	math	PROPN
ejpam-5948	264	6	,	,	PUNCT
ejpam-5948	264	7	18	18	NUM
ejpam-5948	264	8	(	(	PUNCT
ejpam-5948	264	9	2	2	NUM
ejpam-5948	264	10	)	)	PUNCT
ejpam-5948	264	11	(	(	PUNCT
ejpam-5948	264	12	2025	2025	NUM
ejpam-5948	264	13	)	)	PUNCT
ejpam-5948	264	14	,	,	PUNCT
ejpam-5948	264	15	5948	5948	NUM
ejpam-5948	264	16	11	11	NUM
ejpam-5948	264	17	of	of	ADP
ejpam-5948	264	18	12	12	NUM
ejpam-5948	264	19	it	it	PRON
ejpam-5948	264	20	is	be	AUX
ejpam-5948	264	21	easy	easy	ADJ
ejpam-5948	264	22	to	to	PART
ejpam-5948	264	23	verify	verify	VERB
ejpam-5948	264	24	that	that	PRON
ejpam-5948	264	25	℧	℧	PROPN
ejpam-5948	264	26	and	and	CCONJ
ejpam-5948	264	27	⨿	⨿	NOUN
ejpam-5948	264	28	are	be	AUX
ejpam-5948	264	29	generalized	generalized	ADJ
ejpam-5948	264	30	derivations	derivation	NOUN
ejpam-5948	264	31	of	of	ADP
ejpam-5948	264	32	ℜ	ℜ	PROPN
ejpam-5948	264	33	associated	associate	VERB
ejpam-5948	264	34	with	with	ADP
ejpam-5948	264	35	derivations	derivation	NOUN
ejpam-5948	264	36	χ	χ	NOUN
ejpam-5948	264	37	and	and	CCONJ
ejpam-5948	264	38	∝	∝	PROPN
ejpam-5948	264	39	,	,	PUNCT
ejpam-5948	264	40	respectively	respectively	ADV
ejpam-5948	264	41	.	.	PUNCT
ejpam-5948	265	1	it	it	PRON
ejpam-5948	265	2	can	can	AUX
ejpam-5948	265	3	also	also	ADV
ejpam-5948	265	4	be	be	AUX
ejpam-5948	265	5	verified	verify	VERB
ejpam-5948	265	6	that	that	SCONJ
ejpam-5948	265	7	ℜ	ℜ	PROPN
ejpam-5948	265	8	satisfies	satisfy	VERB
ejpam-5948	265	9	the	the	DET
ejpam-5948	265	10	identities	identity	NOUN
ejpam-5948	265	11	in	in	ADP
ejpam-5948	265	12	theorems	theorem	NOUN
ejpam-5948	266	1	[	[	X
ejpam-5948	266	2	1	1	NUM
ejpam-5948	266	3	–	–	PUNCT
ejpam-5948	266	4	6	6	NUM
ejpam-5948	266	5	]	]	PUNCT
ejpam-5948	266	6	.	.	PUNCT
ejpam-5948	267	1	however	however	ADV
ejpam-5948	267	2	,	,	PUNCT
ejpam-5948	267	3	neither	neither	CCONJ
ejpam-5948	267	4	ℜ/p	ℜ/p	PROPN
ejpam-5948	267	5	integral	integral	ADJ
ejpam-5948	267	6	domain	domain	NOUN
ejpam-5948	267	7	nor	nor	CCONJ
ejpam-5948	267	8	χ	χ	NOUN
ejpam-5948	267	9	,	,	PUNCT
ejpam-5948	267	10	∝	∝	PROPN
ejpam-5948	267	11	,	,	PUNCT
ejpam-5948	267	12	℧	℧	PROPN
ejpam-5948	267	13	,	,	PUNCT
ejpam-5948	267	14	⨿	⨿	NOUN
ejpam-5948	267	15	and	and	CCONJ
ejpam-5948	267	16	℧	℧	PROPN
ejpam-5948	267	17	±	±	X
ejpam-5948	267	18	⨿	⨿	NOUN
ejpam-5948	267	19	mapping	map	VERB
ejpam-5948	267	20	ℜ	ℜ	PROPN
ejpam-5948	267	21	to	to	ADP
ejpam-5948	267	22	p	p	PRON
ejpam-5948	267	23	.	.	PUNCT
ejpam-5948	268	1	it	it	PRON
ejpam-5948	268	2	is	be	AUX
ejpam-5948	268	3	important	important	ADJ
ejpam-5948	268	4	to	to	PART
ejpam-5948	268	5	note	note	VERB
ejpam-5948	268	6	that	that	SCONJ
ejpam-5948	268	7	p	p	NOUN
ejpam-5948	268	8	is	be	AUX
ejpam-5948	268	9	not	not	PART
ejpam-5948	268	10	a	a	DET
ejpam-5948	268	11	prime	prime	ADJ
ejpam-5948	268	12	ideal	ideal	NOUN
ejpam-5948	268	13	of	of	ADP
ejpam-5948	268	14	ℜ	ℜ	PROPN
ejpam-5948	268	15	,	,	PUNCT
ejpam-5948	268	16	since	since	SCONJ
ejpam-5948	268	17	0	0	ADP
ejpam-5948	268	18	υ	υ	PROPN
ejpam-5948	268	19	0	0	NUM
ejpam-5948	268	20	0	0	NUM
ejpam-5948	268	21	0	0	NUM
ejpam-5948	268	22	0	0	NUM
ejpam-5948	268	23	0	0	NUM
ejpam-5948	268	24	0	0	NUM
ejpam-5948	268	25	0	0	NUM
ejpam-5948	268	26	2	2	PROPN
ejpam-5948	268	27	∈	∈	PROPN
ejpam-5948	268	28	p	p	NOUN
ejpam-5948	268	29	,	,	PUNCT
ejpam-5948	268	30	but0	but0	NOUN
ejpam-5948	268	31	υ	υ	NOUN
ejpam-5948	268	32	0	0	NUM
ejpam-5948	268	33	0	0	NUM
ejpam-5948	268	34	0	0	NUM
ejpam-5948	268	35	0	0	NUM
ejpam-5948	268	36	0	0	NUM
ejpam-5948	268	37	0	0	NUM
ejpam-5948	268	38	0	0	NUM
ejpam-5948	269	1			PROPN
ejpam-5948	269	2	/∈	/∈	PUNCT
ejpam-5948	270	1	p	p	X
ejpam-5948	270	2	.	.	PUNCT
ejpam-5948	271	1	therefore	therefore	ADV
ejpam-5948	271	2	,	,	PUNCT
ejpam-5948	271	3	the	the	DET
ejpam-5948	271	4	primeness	primeness	NOUN
ejpam-5948	271	5	condition	condition	NOUN
ejpam-5948	271	6	in	in	ADP
ejpam-5948	271	7	theorems	theorem	NOUN
ejpam-5948	271	8	[	[	X
ejpam-5948	271	9	1–6	1–6	NUM
ejpam-5948	271	10	]	]	X
ejpam-5948	271	11	is	be	AUX
ejpam-5948	271	12	essential	essential	ADJ
ejpam-5948	271	13	.	.	PUNCT
ejpam-5948	272	1	example	example	NOUN
ejpam-5948	273	1	4	4	X
ejpam-5948	273	2	.	.	PUNCT
ejpam-5948	273	3	let	let	VERB
ejpam-5948	273	4	ℜ	ℜ	NOUN
ejpam-5948	273	5	=	=	SYM
ejpam-5948	273	6	γ×h[x	γ×h[x	PROPN
ejpam-5948	273	7	]	]	PUNCT
ejpam-5948	273	8	,	,	PUNCT
ejpam-5948	273	9	where	where	SCONJ
ejpam-5948	273	10	γ	γ	X
ejpam-5948	273	11	=	=	SYM
ejpam-5948	273	12	{	{	PUNCT
ejpam-5948	273	13	γ	γ	X
ejpam-5948	273	14	=	=	SYM
ejpam-5948	273	15	υe21	υe21	PROPN
ejpam-5948	273	16	+	+	CCONJ
ejpam-5948	273	17	℘e31	℘e31	PROPN
ejpam-5948	273	18	+	+	NOUN
ejpam-5948	273	19	2ℏe32	2ℏe32	NUM
ejpam-5948	273	20	|	|	ADV
ejpam-5948	273	21	υ	υ	NOUN
ejpam-5948	273	22	,	,	PUNCT
ejpam-5948	273	23	℘	℘	PROPN
ejpam-5948	273	24	,	,	PUNCT
ejpam-5948	273	25	ℏ	ℏ	PROPN
ejpam-5948	273	26	∈	∈	NOUN
ejpam-5948	273	27	z4	z4	X
ejpam-5948	273	28	}	}	PUNCT
ejpam-5948	273	29	and	and	CCONJ
ejpam-5948	273	30	z[υ	z[υ	NUM
ejpam-5948	273	31	]	]	X
ejpam-5948	273	32	is	be	AUX
ejpam-5948	273	33	the	the	DET
ejpam-5948	273	34	polynomial	polynomial	ADJ
ejpam-5948	273	35	ring	ring	NOUN
ejpam-5948	273	36	of	of	ADP
ejpam-5948	273	37	quaternions	quaternion	NOUN
ejpam-5948	273	38	in	in	ADP
ejpam-5948	273	39	determinate	determinate	ADJ
ejpam-5948	273	40	υ	υ	NOUN
ejpam-5948	273	41	,	,	PUNCT
ejpam-5948	273	42	and	and	CCONJ
ejpam-5948	273	43	let	let	VERB
ejpam-5948	273	44	p	p	NOUN
ejpam-5948	273	45	=	=	PRON
ejpam-5948	273	46	{	{	PUNCT
ejpam-5948	273	47	(	(	PUNCT
ejpam-5948	273	48	0	0	NUM
ejpam-5948	273	49	,	,	PUNCT
ejpam-5948	273	50	0	0	NUM
ejpam-5948	273	51	)	)	PUNCT
ejpam-5948	273	52	}	}	PUNCT
ejpam-5948	273	53	.	.	PUNCT
ejpam-5948	274	1	define	define	VERB
ejpam-5948	274	2	(	(	PUNCT
ejpam-5948	274	3	℧	℧	PROPN
ejpam-5948	274	4	,	,	PUNCT
ejpam-5948	274	5	χ	χ	NOUN
ejpam-5948	274	6	)	)	PUNCT
ejpam-5948	274	7	,	,	PUNCT
ejpam-5948	274	8	(	(	PUNCT
ejpam-5948	274	9	⨿,∝	⨿,∝	X
ejpam-5948	274	10	)	)	PUNCT
ejpam-5948	274	11	:	:	PUNCT
ejpam-5948	274	12	ℜ	ℜ	ADV
ejpam-5948	274	13	−→	−→	NOUN
ejpam-5948	274	14	ℜ	ℜ	ADJ
ejpam-5948	274	15	by	by	ADP
ejpam-5948	274	16	℧	℧	PROPN
ejpam-5948	274	17	(	(	PUNCT
ejpam-5948	274	18	γ	γ	PROPN
ejpam-5948	274	19	,	,	PUNCT
ejpam-5948	274	20	t(υ	t(υ	NOUN
ejpam-5948	274	21	)	)	PUNCT
ejpam-5948	274	22	)	)	PUNCT
ejpam-5948	275	1	=	=	SYM
ejpam-5948	275	2	(	(	PUNCT
ejpam-5948	275	3	2υe21	2υe21	NUM
ejpam-5948	275	4	,	,	PUNCT
ejpam-5948	275	5	0	0	NUM
ejpam-5948	275	6	)	)	PUNCT
ejpam-5948	275	7	with	with	ADP
ejpam-5948	275	8	χ(γ	χ(γ	NOUN
ejpam-5948	275	9	,	,	PUNCT
ejpam-5948	275	10	t(xυ	t(xυ	NOUN
ejpam-5948	275	11	)	)	PUNCT
ejpam-5948	275	12	)	)	PUNCT
ejpam-5948	276	1	=	=	SYM
ejpam-5948	276	2	(	(	PUNCT
ejpam-5948	276	3	−ℏe31	−ℏe31	PROPN
ejpam-5948	276	4	,	,	PUNCT
ejpam-5948	276	5	0	0	NUM
ejpam-5948	276	6	)	)	PUNCT
ejpam-5948	276	7	,	,	PUNCT
ejpam-5948	276	8	and	and	CCONJ
ejpam-5948	276	9	⨿(γ	⨿(γ	PROPN
ejpam-5948	276	10	,	,	PUNCT
ejpam-5948	276	11	t(υ	t(υ	NOUN
ejpam-5948	276	12	)	)	PUNCT
ejpam-5948	276	13	)	)	PUNCT
ejpam-5948	277	1	=	=	PRON
ejpam-5948	277	2	(	(	PUNCT
ejpam-5948	277	3	2℘e21	2℘e21	NUM
ejpam-5948	277	4	,	,	PUNCT
ejpam-5948	277	5	0	0	NUM
ejpam-5948	277	6	)	)	PUNCT
ejpam-5948	277	7	with	with	ADP
ejpam-5948	277	8	∝	∝	PROPN
ejpam-5948	277	9	(	(	PUNCT
ejpam-5948	277	10	γ	γ	PROPN
ejpam-5948	277	11	,	,	PUNCT
ejpam-5948	277	12	t(υ	t(υ	NOUN
ejpam-5948	277	13	)	)	PUNCT
ejpam-5948	277	14	)	)	PUNCT
ejpam-5948	278	1	=	=	SYM
ejpam-5948	278	2	(	(	PUNCT
ejpam-5948	278	3	−υe31	−υe31	PROPN
ejpam-5948	278	4	,	,	PUNCT
ejpam-5948	278	5	0	0	NUM
ejpam-5948	278	6	)	)	PUNCT
ejpam-5948	278	7	.	.	PUNCT
ejpam-5948	279	1	it	it	PRON
ejpam-5948	279	2	is	be	AUX
ejpam-5948	279	3	easy	easy	ADJ
ejpam-5948	279	4	to	to	PART
ejpam-5948	279	5	verify	verify	VERB
ejpam-5948	279	6	that	that	PRON
ejpam-5948	279	7	℧	℧	PROPN
ejpam-5948	279	8	and	and	CCONJ
ejpam-5948	279	9	⨿	⨿	NOUN
ejpam-5948	279	10	are	be	AUX
ejpam-5948	279	11	generalized	generalized	ADJ
ejpam-5948	279	12	derivations	derivation	NOUN
ejpam-5948	279	13	of	of	ADP
ejpam-5948	279	14	ℜ	ℜ	PROPN
ejpam-5948	279	15	associated	associate	VERB
ejpam-5948	279	16	with	with	ADP
ejpam-5948	279	17	derivations	derivation	NOUN
ejpam-5948	279	18	χ	χ	NOUN
ejpam-5948	279	19	and	and	CCONJ
ejpam-5948	279	20	∝	∝	PROPN
ejpam-5948	279	21	,	,	PUNCT
ejpam-5948	279	22	respectively	respectively	ADV
ejpam-5948	279	23	.	.	PUNCT
ejpam-5948	280	1	it	it	PRON
ejpam-5948	280	2	can	can	AUX
ejpam-5948	280	3	also	also	ADV
ejpam-5948	280	4	be	be	AUX
ejpam-5948	280	5	verified	verify	VERB
ejpam-5948	280	6	that	that	SCONJ
ejpam-5948	280	7	ℜ	ℜ	PROPN
ejpam-5948	280	8	satisfies	satisfy	VERB
ejpam-5948	280	9	the	the	DET
ejpam-5948	280	10	identities	identity	NOUN
ejpam-5948	280	11	in	in	ADP
ejpam-5948	280	12	theorems	theorem	NOUN
ejpam-5948	281	1	[	[	X
ejpam-5948	281	2	1	1	NUM
ejpam-5948	281	3	–	–	PUNCT
ejpam-5948	281	4	6	6	NUM
ejpam-5948	281	5	]	]	PUNCT
ejpam-5948	281	6	.	.	PUNCT
ejpam-5948	282	1	however	however	ADV
ejpam-5948	282	2	,	,	PUNCT
ejpam-5948	282	3	neither	neither	CCONJ
ejpam-5948	282	4	ℜ/p	ℜ/p	PROPN
ejpam-5948	282	5	integral	integral	ADJ
ejpam-5948	282	6	domain	domain	NOUN
ejpam-5948	282	7	nor	nor	CCONJ
ejpam-5948	282	8	χ	χ	NOUN
ejpam-5948	282	9	,	,	PUNCT
ejpam-5948	282	10	∝	∝	PROPN
ejpam-5948	282	11	,	,	PUNCT
ejpam-5948	282	12	℧	℧	PROPN
ejpam-5948	282	13	,	,	PUNCT
ejpam-5948	282	14	⨿	⨿	NOUN
ejpam-5948	282	15	and	and	CCONJ
ejpam-5948	282	16	℧	℧	AUX
ejpam-5948	282	17	±⨿	±⨿	AUX
ejpam-5948	282	18	mapping	map	VERB
ejpam-5948	282	19	ℜ	ℜ	PROPN
ejpam-5948	282	20	to	to	ADP
ejpam-5948	282	21	p	p	PRON
ejpam-5948	282	22	.	.	PUNCT
ejpam-5948	283	1	it	it	PRON
ejpam-5948	283	2	is	be	AUX
ejpam-5948	283	3	important	important	ADJ
ejpam-5948	283	4	to	to	PART
ejpam-5948	283	5	note	note	VERB
ejpam-5948	283	6	that	that	SCONJ
ejpam-5948	283	7	p	p	NOUN
ejpam-5948	283	8	is	be	AUX
ejpam-5948	283	9	not	not	PART
ejpam-5948	283	10	a	a	DET
ejpam-5948	283	11	prime	prime	ADJ
ejpam-5948	283	12	ideal	ideal	NOUN
ejpam-5948	283	13	of	of	ADP
ejpam-5948	283	14	ℜ	ℜ	PROPN
ejpam-5948	283	15	,	,	PUNCT
ejpam-5948	283	16	since	since	SCONJ
ejpam-5948	283	17	(	(	PUNCT
ejpam-5948	283	18	υe21	υe21	PROPN
ejpam-5948	283	19	,	,	PUNCT
ejpam-5948	283	20	0)ℜ(2℘e32	0)ℜ(2℘e32	NUM
ejpam-5948	283	21	,	,	PUNCT
ejpam-5948	283	22	0	0	NUM
ejpam-5948	283	23	)	)	PUNCT
ejpam-5948	283	24	∈	∈	PROPN
ejpam-5948	283	25	p	p	NOUN
ejpam-5948	283	26	,	,	PUNCT
ejpam-5948	283	27	but	but	CCONJ
ejpam-5948	284	1	neither	neither	PRON
ejpam-5948	284	2	(	(	PUNCT
ejpam-5948	284	3	υe21	υe21	PROPN
ejpam-5948	284	4	,	,	PUNCT
ejpam-5948	284	5	0	0	NUM
ejpam-5948	284	6	)	)	PUNCT
ejpam-5948	284	7	∈	∈	PROPN
ejpam-5948	284	8	p	p	NOUN
ejpam-5948	284	9	nor	nor	CCONJ
ejpam-5948	284	10	(	(	PUNCT
ejpam-5948	284	11	2℘e32	2℘e32	NOUN
ejpam-5948	284	12	,	,	PUNCT
ejpam-5948	284	13	0	0	NUM
ejpam-5948	284	14	)	)	PUNCT
ejpam-5948	284	15	∈	∈	PROPN
ejpam-5948	284	16	p	p	NOUN
ejpam-5948	284	17	.	.	PUNCT
ejpam-5948	285	1	hence	hence	ADV
ejpam-5948	285	2	,	,	PUNCT
ejpam-5948	285	3	the	the	DET
ejpam-5948	285	4	primeness	primeness	NOUN
ejpam-5948	285	5	condition	condition	NOUN
ejpam-5948	285	6	in	in	ADP
ejpam-5948	285	7	theorems	theorem	NOUN
ejpam-5948	285	8	[	[	X
ejpam-5948	285	9	1	1	NUM
ejpam-5948	285	10	–	–	PUNCT
ejpam-5948	285	11	6	6	NUM
ejpam-5948	285	12	]	]	PUNCT
ejpam-5948	285	13	is	be	AUX
ejpam-5948	285	14	essential	essential	ADJ
ejpam-5948	285	15	.	.	PUNCT
ejpam-5948	286	1	funding	fund	VERB
ejpam-5948	286	2	this	this	DET
ejpam-5948	286	3	study	study	NOUN
ejpam-5948	286	4	was	be	AUX
ejpam-5948	286	5	carried	carry	VERB
ejpam-5948	286	6	out	out	ADP
ejpam-5948	286	7	with	with	ADP
ejpam-5948	286	8	financial	financial	ADJ
ejpam-5948	286	9	support	support	NOUN
ejpam-5948	286	10	from	from	ADP
ejpam-5948	286	11	the	the	DET
ejpam-5948	286	12	deanship	deanship	NOUN
ejpam-5948	286	13	of	of	ADP
ejpam-5948	286	14	scientific	scientific	ADJ
ejpam-5948	286	15	research	research	NOUN
ejpam-5948	286	16	at	at	ADP
ejpam-5948	286	17	king	king	PROPN
ejpam-5948	286	18	khalid	khalid	PROPN
ejpam-5948	286	19	university	university	PROPN
ejpam-5948	286	20	(	(	PUNCT
ejpam-5948	286	21	kku	kku	PROPN
ejpam-5948	286	22	)	)	PUNCT
ejpam-5948	286	23	,	,	PUNCT
ejpam-5948	286	24	abha	abha	NOUN
ejpam-5948	286	25	,	,	PUNCT
ejpam-5948	286	26	saudi	saudi	PROPN
ejpam-5948	286	27	arabia	arabia	PROPN
ejpam-5948	286	28	through	through	ADP
ejpam-5948	286	29	a	a	DET
ejpam-5948	286	30	large	large	ADJ
ejpam-5948	286	31	group	group	NOUN
ejpam-5948	286	32	research	research	NOUN
ejpam-5948	286	33	project	project	NOUN
ejpam-5948	286	34	under	under	ADP
ejpam-5948	286	35	grant	grant	PROPN
ejpam-5948	286	36	number	number	PROPN
ejpam-5948	286	37	rgp	rgp	PROPN
ejpam-5948	286	38	.	.	PUNCT
ejpam-5948	287	1	2/340/46	2/340/46	PROPN
ejpam-5948	287	2	references	reference	NOUN
ejpam-5948	287	3	[	[	X
ejpam-5948	287	4	1	1	NUM
ejpam-5948	287	5	]	]	PUNCT
ejpam-5948	287	6	g.	g.	PROPN
ejpam-5948	287	7	s.	s.	PROPN
ejpam-5948	287	8	sandhu	sandhu	PROPN
ejpam-5948	287	9	,	,	PUNCT
ejpam-5948	287	10	a.	a.	NOUN
ejpam-5948	287	11	boua	boua	NOUN
ejpam-5948	287	12	,	,	PUNCT
ejpam-5948	287	13	and	and	CCONJ
ejpam-5948	287	14	n.	n.	NOUN
ejpam-5948	287	15	ur	ur	PROPN
ejpam-5948	287	16	rehman	rehman	PROPN
ejpam-5948	287	17	.	.	PUNCT
ejpam-5948	288	1	some	some	DET
ejpam-5948	288	2	results	result	NOUN
ejpam-5948	288	3	involving	involve	VERB
ejpam-5948	288	4	p	p	NOUN
ejpam-5948	288	5	-derivations	-derivation	NOUN
ejpam-5948	288	6	and	and	CCONJ
ejpam-5948	288	7	prime	prime	ADJ
ejpam-5948	288	8	ideals	ideal	NOUN
ejpam-5948	288	9	in	in	ADP
ejpam-5948	288	10	rings	ring	NOUN
ejpam-5948	288	11	.	.	PUNCT
ejpam-5948	289	1	annali	annali	PROPN
ejpam-5948	289	2	dell’università	dell’università	PROPN
ejpam-5948	289	3	di	di	PROPN
ejpam-5948	289	4	ferrara	ferrara	NOUN
ejpam-5948	289	5	,	,	PUNCT
ejpam-5948	289	6	69:587–604	69:587–604	NUM
ejpam-5948	289	7	,	,	PUNCT
ejpam-5948	289	8	2023	2023	NUM
ejpam-5948	289	9	.	.	PUNCT
ejpam-5948	290	1	[	[	X
ejpam-5948	290	2	2	2	X
ejpam-5948	290	3	]	]	PUNCT
ejpam-5948	290	4	e.	e.	PROPN
ejpam-5948	290	5	posner	posner	PROPN
ejpam-5948	290	6	.	.	PUNCT
ejpam-5948	291	1	derivations	derivation	NOUN
ejpam-5948	291	2	in	in	ADP
ejpam-5948	291	3	prime	prime	ADJ
ejpam-5948	291	4	rings	ring	NOUN
ejpam-5948	291	5	.	.	PUNCT
ejpam-5948	292	1	proceedings	proceeding	NOUN
ejpam-5948	292	2	of	of	ADP
ejpam-5948	292	3	the	the	DET
ejpam-5948	292	4	american	american	PROPN
ejpam-5948	292	5	mathematical	mathematical	PROPN
ejpam-5948	292	6	society	society	NOUN
ejpam-5948	292	7	,	,	PUNCT
ejpam-5948	292	8	8:1093–1100	8:1093–1100	NUM
ejpam-5948	292	9	,	,	PUNCT
ejpam-5948	292	10	1957	1957	NUM
ejpam-5948	292	11	.	.	PUNCT
ejpam-5948	293	1	[	[	X
ejpam-5948	293	2	3	3	X
ejpam-5948	293	3	]	]	PUNCT
ejpam-5948	293	4	m.	m.	NOUN
ejpam-5948	293	5	bera	bera	NOUN
ejpam-5948	293	6	,	,	PUNCT
ejpam-5948	293	7	b.	b.	PROPN
ejpam-5948	293	8	dhara	dhara	PROPN
ejpam-5948	293	9	,	,	PUNCT
ejpam-5948	293	10	and	and	CCONJ
ejpam-5948	293	11	s.	s.	PROPN
ejpam-5948	293	12	kar	kar	PROPN
ejpam-5948	293	13	.	.	PUNCT
ejpam-5948	294	1	some	some	DET
ejpam-5948	294	2	identities	identity	NOUN
ejpam-5948	294	3	involving	involve	VERB
ejpam-5948	294	4	generalized	generalized	ADJ
ejpam-5948	294	5	(	(	PUNCT
ejpam-5948	294	6	α	α	NOUN
ejpam-5948	294	7	,	,	PUNCT
ejpam-5948	294	8	β)derivations	β)derivation	NOUN
ejpam-5948	294	9	in	in	ADP
ejpam-5948	294	10	prime	prime	ADJ
ejpam-5948	294	11	and	and	CCONJ
ejpam-5948	294	12	semiprime	semiprime	NOUN
ejpam-5948	294	13	rings	ring	NOUN
ejpam-5948	294	14	.	.	PUNCT
ejpam-5948	295	1	asian	asian	ADJ
ejpam-5948	295	2	-	-	PUNCT
ejpam-5948	295	3	european	european	ADJ
ejpam-5948	295	4	journal	journal	NOUN
ejpam-5948	295	5	of	of	ADP
ejpam-5948	295	6	mathematics	mathematic	NOUN
ejpam-5948	295	7	,	,	PUNCT
ejpam-5948	295	8	16:1–14	16:1–14	NUM
ejpam-5948	295	9	,	,	PUNCT
ejpam-5948	295	10	2023	2023	NUM
ejpam-5948	295	11	.	.	PUNCT
ejpam-5948	296	1	[	[	X
ejpam-5948	296	2	4	4	NUM
ejpam-5948	296	3	]	]	PUNCT
ejpam-5948	296	4	a.	a.	NOUN
ejpam-5948	296	5	y.	y.	PROPN
ejpam-5948	296	6	hummdi	hummdi	PROPN
ejpam-5948	296	7	,	,	PUNCT
ejpam-5948	296	8	z.	z.	PROPN
ejpam-5948	296	9	z.	z.	PROPN
ejpam-5948	296	10	al	al	PROPN
ejpam-5948	296	11	-	-	PUNCT
ejpam-5948	296	12	amery	amery	PROPN
ejpam-5948	296	13	,	,	PUNCT
ejpam-5948	296	14	and	and	CCONJ
ejpam-5948	296	15	r.	r.	PROPN
ejpam-5948	296	16	m.	m.	PROPN
ejpam-5948	296	17	al	al	PROPN
ejpam-5948	296	18	-	-	PUNCT
ejpam-5948	296	19	omary	omary	NOUN
ejpam-5948	296	20	.	.	PUNCT
ejpam-5948	297	1	factor	factor	NOUN
ejpam-5948	297	2	rings	ring	NOUN
ejpam-5948	297	3	with	with	ADP
ejpam-5948	297	4	algebraic	algebraic	ADJ
ejpam-5948	297	5	identities	identity	NOUN
ejpam-5948	297	6	via	via	ADP
ejpam-5948	297	7	generalized	generalized	ADJ
ejpam-5948	297	8	derivations	derivation	NOUN
ejpam-5948	297	9	.	.	PUNCT
ejpam-5948	298	1	axioms	axiom	NOUN
ejpam-5948	298	2	,	,	PUNCT
ejpam-5948	298	3	14(1):15	14(1):15	NUM
ejpam-5948	298	4	,	,	PUNCT
ejpam-5948	298	5	2025	2025	NUM
ejpam-5948	298	6	.	.	PUNCT
ejpam-5948	299	1	[	[	X
ejpam-5948	299	2	5	5	X
ejpam-5948	299	3	]	]	PUNCT
ejpam-5948	299	4	s.	s.	PROPN
ejpam-5948	299	5	k.	k.	PROPN
ejpam-5948	299	6	tiwari	tiwari	PROPN
ejpam-5948	299	7	,	,	PUNCT
ejpam-5948	299	8	r.	r.	PROPN
ejpam-5948	299	9	k.	k.	PROPN
ejpam-5948	299	10	sharma	sharma	PROPN
ejpam-5948	299	11	,	,	PUNCT
ejpam-5948	299	12	and	and	CCONJ
ejpam-5948	299	13	b.	b.	PROPN
ejpam-5948	299	14	dhara	dhara	PROPN
ejpam-5948	299	15	.	.	PUNCT
ejpam-5948	300	1	identities	identity	NOUN
ejpam-5948	300	2	related	relate	VERB
ejpam-5948	300	3	to	to	ADP
ejpam-5948	300	4	generalized	generalized	ADJ
ejpam-5948	300	5	derivations	derivation	NOUN
ejpam-5948	300	6	on	on	ADP
ejpam-5948	300	7	ideal	ideal	NOUN
ejpam-5948	300	8	in	in	ADP
ejpam-5948	300	9	prime	prime	ADJ
ejpam-5948	300	10	rings	ring	NOUN
ejpam-5948	300	11	.	.	PUNCT
ejpam-5948	301	1	beiträge	beiträge	NOUN
ejpam-5948	301	2	zur	zur	PROPN
ejpam-5948	301	3	algebra	algebra	PROPN
ejpam-5948	301	4	und	und	NOUN
ejpam-5948	301	5	geometrie	geometrie	NOUN
ejpam-5948	301	6	,	,	PUNCT
ejpam-5948	301	7	57(4):809–821	57(4):809–821	PROPN
ejpam-5948	301	8	,	,	PUNCT
ejpam-5948	301	9	2016	2016	NUM
ejpam-5948	301	10	.	.	PUNCT
ejpam-5948	302	1	a.y	a.y	PROPN
ejpam-5948	302	2	.	.	PROPN
ejpam-5948	302	3	hummdi	hummdi	PROPN
ejpam-5948	302	4	,	,	PUNCT
ejpam-5948	302	5	r.m	r.m	PROPN
ejpam-5948	302	6	.	.	PROPN
ejpam-5948	302	7	al	al	PROPN
ejpam-5948	302	8	-	-	PUNCT
ejpam-5948	302	9	omary	omary	PROPN
ejpam-5948	302	10	,	,	PUNCT
ejpam-5948	302	11	z.z	z.z	PROPN
ejpam-5948	302	12	.	.	PUNCT
ejpam-5948	302	13	al	al	PROPN
ejpam-5948	302	14	-	-	PUNCT
ejpam-5948	302	15	amery	amery	PROPN
ejpam-5948	302	16	/	/	SYM
ejpam-5948	302	17	eur	eur	PROPN
ejpam-5948	302	18	.	.	PUNCT
ejpam-5948	303	1	j.	j.	PROPN
ejpam-5948	303	2	pure	pure	PROPN
ejpam-5948	303	3	appl	appl	PROPN
ejpam-5948	303	4	.	.	PROPN
ejpam-5948	303	5	math	math	PROPN
ejpam-5948	303	6	,	,	PUNCT
ejpam-5948	303	7	18	18	NUM
ejpam-5948	303	8	(	(	PUNCT
ejpam-5948	303	9	2	2	NUM
ejpam-5948	303	10	)	)	PUNCT
ejpam-5948	303	11	(	(	PUNCT
ejpam-5948	303	12	2025	2025	NUM
ejpam-5948	303	13	)	)	PUNCT
ejpam-5948	303	14	,	,	PUNCT
ejpam-5948	303	15	5948	5948	NUM
ejpam-5948	303	16	12	12	NUM
ejpam-5948	303	17	of	of	ADP
ejpam-5948	303	18	12	12	NUM
ejpam-5948	303	19	[	[	SYM
ejpam-5948	303	20	6	6	NUM
ejpam-5948	303	21	]	]	PUNCT
ejpam-5948	303	22	f.	f.	PROPN
ejpam-5948	303	23	almahdi	almahdi	PROPN
ejpam-5948	303	24	,	,	PUNCT
ejpam-5948	303	25	a.	a.	NOUN
ejpam-5948	303	26	mamouni	mamouni	PROPN
ejpam-5948	303	27	,	,	PUNCT
ejpam-5948	303	28	and	and	CCONJ
ejpam-5948	303	29	m.	m.	NOUN
ejpam-5948	303	30	tamekkante	tamekkante	PROPN
ejpam-5948	303	31	.	.	PUNCT
ejpam-5948	304	1	a	a	DET
ejpam-5948	304	2	generalization	generalization	NOUN
ejpam-5948	304	3	of	of	ADP
ejpam-5948	304	4	posner	posner	NOUN
ejpam-5948	304	5	’s	’s	PART
ejpam-5948	304	6	theorem	theorem	NOUN
ejpam-5948	304	7	on	on	ADP
ejpam-5948	304	8	derivations	derivation	NOUN
ejpam-5948	304	9	in	in	ADP
ejpam-5948	304	10	ring	ring	NOUN
ejpam-5948	304	11	.	.	PUNCT
ejpam-5948	305	1	indian	indian	PROPN
ejpam-5948	305	2	journal	journal	PROPN
ejpam-5948	305	3	of	of	ADP
ejpam-5948	305	4	pure	pure	ADJ
ejpam-5948	305	5	and	and	CCONJ
ejpam-5948	305	6	applied	applied	ADJ
ejpam-5948	305	7	mathematics	mathematic	NOUN
ejpam-5948	305	8	,	,	PUNCT
ejpam-5948	305	9	51:187–194	51:187–194	NUM
ejpam-5948	305	10	,	,	PUNCT
ejpam-5948	305	11	2020	2020	NUM
ejpam-5948	305	12	.	.	PUNCT
ejpam-5948	306	1	[	[	X
ejpam-5948	306	2	7	7	X
ejpam-5948	306	3	]	]	X
ejpam-5948	306	4	n.	n.	PROPN
ejpam-5948	306	5	alsowait	alsowait	PROPN
ejpam-5948	306	6	,	,	PUNCT
ejpam-5948	306	7	r.	r.	PROPN
ejpam-5948	306	8	m.	m.	PROPN
ejpam-5948	306	9	al	al	PROPN
ejpam-5948	306	10	-	-	PUNCT
ejpam-5948	306	11	omary	omary	PROPN
ejpam-5948	306	12	,	,	PUNCT
ejpam-5948	306	13	z.	z.	PROPN
ejpam-5948	306	14	z.	z.	PROPN
ejpam-5948	306	15	al	al	PROPN
ejpam-5948	306	16	-	-	PUNCT
ejpam-5948	306	17	amery	amery	PROPN
ejpam-5948	306	18	,	,	PUNCT
ejpam-5948	306	19	and	and	CCONJ
ejpam-5948	306	20	m.	m.	PROPN
ejpam-5948	306	21	al	al	PROPN
ejpam-5948	306	22	-	-	PUNCT
ejpam-5948	306	23	shomrani	shomrani	PROPN
ejpam-5948	306	24	.	.	PUNCT
ejpam-5948	307	1	exploring	explore	VERB
ejpam-5948	307	2	commutativity	commutativity	NOUN
ejpam-5948	307	3	via	via	ADP
ejpam-5948	307	4	generalized	generalized	ADJ
ejpam-5948	307	5	(	(	PUNCT
ejpam-5948	307	6	α	α	NOUN
ejpam-5948	307	7	,	,	PUNCT
ejpam-5948	307	8	β)-derivations	β)-derivation	NOUN
ejpam-5948	307	9	involving	involve	VERB
ejpam-5948	307	10	prime	prime	ADJ
ejpam-5948	307	11	ideals	ideal	NOUN
ejpam-5948	307	12	.	.	PUNCT
ejpam-5948	308	1	mathematics	mathematic	NOUN
ejpam-5948	308	2	,	,	PUNCT
ejpam-5948	308	3	12(15):2325	12(15):2325	NUM
ejpam-5948	308	4	,	,	PUNCT
ejpam-5948	308	5	2024	2024	NUM
ejpam-5948	308	6	.	.	PUNCT
ejpam-5948	309	1	[	[	X
ejpam-5948	309	2	8	8	NUM
ejpam-5948	309	3	]	]	PUNCT
ejpam-5948	309	4	a.	a.	NOUN
ejpam-5948	309	5	boua	boua	NOUN
ejpam-5948	309	6	and	and	CCONJ
ejpam-5948	309	7	g.	g.	PROPN
ejpam-5948	309	8	s.	s.	PROPN
ejpam-5948	309	9	sandhu	sandhu	PROPN
ejpam-5948	309	10	.	.	PUNCT
ejpam-5948	310	1	results	result	NOUN
ejpam-5948	310	2	on	on	ADP
ejpam-5948	310	3	various	various	ADJ
ejpam-5948	310	4	derivations	derivation	NOUN
ejpam-5948	310	5	and	and	CCONJ
ejpam-5948	310	6	posner	posner	NOUN
ejpam-5948	310	7	’s	’s	PART
ejpam-5948	310	8	theorem	theorem	NOUN
ejpam-5948	310	9	in	in	ADP
ejpam-5948	310	10	prime	prime	ADJ
ejpam-5948	310	11	ideals	ideal	NOUN
ejpam-5948	310	12	of	of	ADP
ejpam-5948	310	13	rings	ring	NOUN
ejpam-5948	310	14	.	.	PUNCT
ejpam-5948	311	1	boletim	boletim	PROPN
ejpam-5948	311	2	da	da	PROPN
ejpam-5948	311	3	sociedade	sociedade	PROPN
ejpam-5948	311	4	paranaense	paranaense	PROPN
ejpam-5948	311	5	de	de	PROPN
ejpam-5948	311	6	matemática	matemática	PROPN
ejpam-5948	311	7	,	,	PUNCT
ejpam-5948	311	8	41:1–13	41:1–13	NUM
ejpam-5948	311	9	,	,	PUNCT
ejpam-5948	311	10	2023	2023	NUM
ejpam-5948	311	11	.	.	PUNCT
ejpam-5948	312	1	[	[	X
ejpam-5948	312	2	9	9	NUM
ejpam-5948	312	3	]	]	PUNCT
ejpam-5948	312	4	k.	k.	PROPN
ejpam-5948	312	5	bouchannafa	bouchannafa	PROPN
ejpam-5948	312	6	,	,	PUNCT
ejpam-5948	312	7	a.	a.	NOUN
ejpam-5948	312	8	mamouni	mamouni	PROPN
ejpam-5948	312	9	,	,	PUNCT
ejpam-5948	312	10	and	and	CCONJ
ejpam-5948	312	11	l.	l.	PROPN
ejpam-5948	312	12	oukhtite	oukhtite	PROPN
ejpam-5948	312	13	.	.	PUNCT
ejpam-5948	313	1	structure	structure	NOUN
ejpam-5948	313	2	of	of	ADP
ejpam-5948	313	3	a	a	DET
ejpam-5948	313	4	quotient	quotient	NOUN
ejpam-5948	313	5	ring	ring	NOUN
ejpam-5948	313	6	r	r	NOUN
ejpam-5948	313	7	/	/	SYM
ejpam-5948	313	8	p	p	NOUN
ejpam-5948	313	9	and	and	CCONJ
ejpam-5948	313	10	its	its	PRON
ejpam-5948	313	11	relation	relation	NOUN
ejpam-5948	313	12	with	with	ADP
ejpam-5948	313	13	generalized	generalized	ADJ
ejpam-5948	313	14	derivations	derivation	NOUN
ejpam-5948	313	15	of	of	ADP
ejpam-5948	313	16	r.	r.	PROPN
ejpam-5948	313	17	proyecciones	proyecciones	PROPN
ejpam-5948	313	18	,	,	PUNCT
ejpam-5948	313	19	41(3):623–642	41(3):623–642	PROPN
ejpam-5948	313	20	,	,	PUNCT
ejpam-5948	313	21	2022	2022	NUM
ejpam-5948	313	22	.	.	PUNCT
ejpam-5948	314	1	[	[	X
ejpam-5948	314	2	10	10	NUM
ejpam-5948	314	3	]	]	X
ejpam-5948	314	4	s.	s.	PROPN
ejpam-5948	314	5	mouhssine	mouhssine	PROPN
ejpam-5948	314	6	and	and	CCONJ
ejpam-5948	314	7	a.	a.	NOUN
ejpam-5948	314	8	boua	boua	NOUN
ejpam-5948	314	9	.	.	PUNCT
ejpam-5948	315	1	(	(	PUNCT
ejpam-5948	315	2	α	α	X
ejpam-5948	315	3	,	,	PUNCT
ejpam-5948	315	4	τ)-p	τ)-p	ADP
ejpam-5948	315	5	-derivations	-derivation	NOUN
ejpam-5948	315	6	on	on	ADP
ejpam-5948	315	7	left	left	ADJ
ejpam-5948	315	8	near	near	NOUN
ejpam-5948	315	9	-	-	PUNCT
ejpam-5948	315	10	rings	ring	NOUN
ejpam-5948	315	11	.	.	PUNCT
ejpam-5948	316	1	note	note	VERB
ejpam-5948	316	2	di	di	PROPN
ejpam-5948	316	3	matematica	matematica	PROPN
ejpam-5948	316	4	,	,	PUNCT
ejpam-5948	316	5	42(2):93–107	42(2):93–107	NUM
ejpam-5948	316	6	,	,	PUNCT
ejpam-5948	316	7	2022	2022	NUM
ejpam-5948	316	8	.	.	PUNCT
ejpam-5948	317	1	[	[	X
ejpam-5948	317	2	11	11	NUM
ejpam-5948	317	3	]	]	X
ejpam-5948	317	4	l.	l.	PROPN
ejpam-5948	317	5	oukhtite	oukhtite	PROPN
ejpam-5948	317	6	and	and	CCONJ
ejpam-5948	317	7	k.	k.	PROPN
ejpam-5948	317	8	bouchannafa	bouchannafa	PROPN
ejpam-5948	317	9	.	.	PUNCT
ejpam-5948	318	1	commutativity	commutativity	NOUN
ejpam-5948	318	2	criteria	criterion	NOUN
ejpam-5948	318	3	for	for	ADP
ejpam-5948	318	4	a	a	DET
ejpam-5948	318	5	factor	factor	NOUN
ejpam-5948	318	6	ring	ring	NOUN
ejpam-5948	318	7	r	r	NOUN
ejpam-5948	318	8	/	/	SYM
ejpam-5948	318	9	p	p	NOUN
ejpam-5948	318	10	arising	arise	VERB
ejpam-5948	318	11	from	from	ADP
ejpam-5948	318	12	p	p	PROPN
ejpam-5948	318	13	-centralizers	-centralizer	NOUN
ejpam-5948	318	14	.	.	PUNCT
ejpam-5948	319	1	kyungpook	kyungpook	PROPN
ejpam-5948	319	2	mathematical	mathematical	PROPN
ejpam-5948	319	3	journal	journal	PROPN
ejpam-5948	319	4	,	,	PUNCT
ejpam-5948	319	5	63(4):551–560	63(4):551–560	PROPN
ejpam-5948	319	6	,	,	PUNCT
ejpam-5948	319	7	2023	2023	NUM
ejpam-5948	319	8	.	.	PUNCT
ejpam-5948	320	1	[	[	X
ejpam-5948	320	2	12	12	NUM
ejpam-5948	320	3	]	]	X
ejpam-5948	320	4	n.	n.	PROPN
ejpam-5948	320	5	rehman	rehman	PROPN
ejpam-5948	320	6	and	and	CCONJ
ejpam-5948	320	7	h.	h.	PROPN
ejpam-5948	320	8	m.	m.	PROPN
ejpam-5948	320	9	alnoghashi	alnoghashi	PROPN
ejpam-5948	320	10	.	.	PUNCT
ejpam-5948	321	1	t	t	PROPN
ejpam-5948	321	2	-commuting	-commute	VERB
ejpam-5948	321	3	generalized	generalized	ADJ
ejpam-5948	321	4	derivations	derivation	NOUN
ejpam-5948	321	5	on	on	ADP
ejpam-5948	321	6	ideals	ideal	NOUN
ejpam-5948	321	7	and	and	CCONJ
ejpam-5948	321	8	semi	semi	ADJ
ejpam-5948	321	9	-	-	ADJ
ejpam-5948	321	10	prime	prime	ADJ
ejpam-5948	321	11	ideals	ideal	NOUN
ejpam-5948	321	12	.	.	PUNCT
ejpam-5948	322	1	boletim	boletim	PROPN
ejpam-5948	322	2	da	da	PROPN
ejpam-5948	322	3	sociedade	sociedade	PROPN
ejpam-5948	322	4	paranaense	paranaense	PROPN
ejpam-5948	322	5	de	de	PROPN
ejpam-5948	322	6	matemática	matemática	PROPN
ejpam-5948	322	7	,	,	PUNCT
ejpam-5948	322	8	42:1–15	42:1–15	NUM
ejpam-5948	322	9	,	,	PUNCT
ejpam-5948	322	10	2024	2024	NUM
ejpam-5948	322	11	.	.	PUNCT
ejpam-5948	323	1	[	[	X
ejpam-5948	323	2	13	13	NUM
ejpam-5948	323	3	]	]	PUNCT
ejpam-5948	323	4	m.	m.	NOUN
ejpam-5948	323	5	a.	a.	PROPN
ejpam-5948	323	6	quadri	quadri	PROPN
ejpam-5948	323	7	,	,	PUNCT
ejpam-5948	323	8	m.	m.	NOUN
ejpam-5948	323	9	s.	s.	PROPN
ejpam-5948	323	10	khan	khan	PROPN
ejpam-5948	323	11	,	,	PUNCT
ejpam-5948	323	12	and	and	CCONJ
ejpam-5948	323	13	n.	n.	PROPN
ejpam-5948	323	14	rehman	rehman	PROPN
ejpam-5948	323	15	.	.	PUNCT
ejpam-5948	324	1	generalized	generalized	ADJ
ejpam-5948	324	2	derivations	derivation	NOUN
ejpam-5948	324	3	and	and	CCONJ
ejpam-5948	324	4	commutativity	commutativity	NOUN
ejpam-5948	324	5	of	of	ADP
ejpam-5948	324	6	prime	prime	ADJ
ejpam-5948	324	7	rings	ring	NOUN
ejpam-5948	324	8	.	.	PUNCT
ejpam-5948	325	1	indian	indian	PROPN
ejpam-5948	325	2	journal	journal	PROPN
ejpam-5948	325	3	of	of	ADP
ejpam-5948	325	4	pure	pure	ADJ
ejpam-5948	325	5	and	and	CCONJ
ejpam-5948	325	6	applied	applied	ADJ
ejpam-5948	325	7	mathematics	mathematic	NOUN
ejpam-5948	325	8	,	,	PUNCT
ejpam-5948	325	9	34:1393–1396	34:1393–1396	NUM
ejpam-5948	325	10	,	,	PUNCT
ejpam-5948	325	11	2003	2003	NUM
ejpam-5948	325	12	.	.	PUNCT
ejpam-5948	326	1	[	[	X
ejpam-5948	326	2	14	14	NUM
ejpam-5948	326	3	]	]	X
ejpam-5948	326	4	n.	n.	PROPN
ejpam-5948	326	5	rehman	rehman	PROPN
ejpam-5948	326	6	,	,	PUNCT
ejpam-5948	326	7	r.	r.	PROPN
ejpam-5948	326	8	m.	m.	PROPN
ejpam-5948	326	9	al	al	PROPN
ejpam-5948	326	10	-	-	PUNCT
ejpam-5948	326	11	omary	omary	NOUN
ejpam-5948	326	12	,	,	PUNCT
ejpam-5948	326	13	and	and	CCONJ
ejpam-5948	326	14	a.	a.	NOUN
ejpam-5948	326	15	z.	z.	PROPN
ejpam-5948	326	16	ansari	ansari	PROPN
ejpam-5948	326	17	.	.	PUNCT
ejpam-5948	327	1	on	on	ADP
ejpam-5948	327	2	lie	lie	NOUN
ejpam-5948	327	3	ideals	ideal	NOUN
ejpam-5948	327	4	of	of	ADP
ejpam-5948	327	5	prime	prime	ADJ
ejpam-5948	327	6	rings	ring	NOUN
ejpam-5948	327	7	with	with	ADP
ejpam-5948	327	8	generalized	generalized	ADJ
ejpam-5948	327	9	derivations	derivation	NOUN
ejpam-5948	327	10	.	.	PUNCT
ejpam-5948	328	1	bolet́ın	bolet́ın	X
ejpam-5948	328	2	de	de	X
ejpam-5948	328	3	la	la	PROPN
ejpam-5948	328	4	sociedad	sociedad	PROPN
ejpam-5948	328	5	matemática	matemática	PROPN
ejpam-5948	328	6	mexicana	mexicana	PROPN
ejpam-5948	328	7	,	,	PUNCT
ejpam-5948	328	8	21(1):19–26	21(1):19–26	NUM
ejpam-5948	328	9	,	,	PUNCT
ejpam-5948	328	10	2015	2015	NUM
ejpam-5948	328	11	.	.	PUNCT
ejpam-5948	329	1	[	[	X
ejpam-5948	329	2	15	15	NUM
ejpam-5948	329	3	]	]	X
ejpam-5948	329	4	k.	k.	PROPN
ejpam-5948	329	5	bouchannafa	bouchannafa	PROPN
ejpam-5948	329	6	,	,	PUNCT
ejpam-5948	329	7	m.	m.	NOUN
ejpam-5948	329	8	a.	a.	PROPN
ejpam-5948	329	9	idrissi	idrissi	PROPN
ejpam-5948	329	10	,	,	PUNCT
ejpam-5948	329	11	and	and	CCONJ
ejpam-5948	329	12	l.	l.	PROPN
ejpam-5948	329	13	oukhtite	oukhtite	PROPN
ejpam-5948	329	14	.	.	PUNCT
ejpam-5948	330	1	relationship	relationship	NOUN
ejpam-5948	330	2	between	between	ADP
ejpam-5948	330	3	the	the	DET
ejpam-5948	330	4	structure	structure	NOUN
ejpam-5948	330	5	of	of	ADP
ejpam-5948	330	6	a	a	DET
ejpam-5948	330	7	quotient	quotient	NOUN
ejpam-5948	330	8	ring	ring	NOUN
ejpam-5948	330	9	and	and	CCONJ
ejpam-5948	330	10	the	the	DET
ejpam-5948	330	11	behavior	behavior	NOUN
ejpam-5948	330	12	of	of	ADP
ejpam-5948	330	13	certain	certain	ADJ
ejpam-5948	330	14	additive	additive	ADJ
ejpam-5948	330	15	mapping	mapping	NOUN
ejpam-5948	330	16	.	.	PUNCT
ejpam-5948	331	1	communications	communication	NOUN
ejpam-5948	331	2	of	of	ADP
ejpam-5948	331	3	the	the	DET
ejpam-5948	331	4	korean	korean	ADJ
ejpam-5948	331	5	mathematical	mathematical	ADJ
ejpam-5948	331	6	society	society	NOUN
ejpam-5948	331	7	,	,	PUNCT
ejpam-5948	331	8	37(2):359–370	37(2):359–370	NUM
ejpam-5948	331	9	,	,	PUNCT
ejpam-5948	331	10	2022	2022	NUM
ejpam-5948	331	11	.	.	PUNCT
ejpam-5948	332	1	[	[	X
ejpam-5948	332	2	16	16	NUM
ejpam-5948	332	3	]	]	PUNCT
ejpam-5948	332	4	m.	m.	NOUN
ejpam-5948	332	5	ashraf	ashraf	PROPN
ejpam-5948	332	6	,	,	PUNCT
ejpam-5948	332	7	a.	a.	PROPN
ejpam-5948	332	8	ali	ali	PROPN
ejpam-5948	332	9	,	,	PUNCT
ejpam-5948	332	10	and	and	CCONJ
ejpam-5948	332	11	r.	r.	PROPN
ejpam-5948	332	12	rani	rani	PROPN
ejpam-5948	332	13	.	.	PUNCT
ejpam-5948	333	1	on	on	ADP
ejpam-5948	333	2	generalized	generalized	ADJ
ejpam-5948	333	3	derivations	derivation	NOUN
ejpam-5948	333	4	of	of	ADP
ejpam-5948	333	5	prime	prime	ADJ
ejpam-5948	333	6	rings	ring	NOUN
ejpam-5948	333	7	.	.	PUNCT
ejpam-5948	334	1	southeast	southeast	ADJ
ejpam-5948	334	2	asian	asian	ADJ
ejpam-5948	334	3	bulletin	bulletin	NOUN
ejpam-5948	334	4	of	of	ADP
ejpam-5948	334	5	mathematics	mathematic	NOUN
ejpam-5948	334	6	,	,	PUNCT
ejpam-5948	334	7	29(4):669–675	29(4):669–675	NUM
ejpam-5948	334	8	,	,	PUNCT
ejpam-5948	334	9	2005	2005	NUM
ejpam-5948	334	10	.	.	PUNCT
ejpam-5948	335	1	[	[	X
ejpam-5948	335	2	17	17	NUM
ejpam-5948	335	3	]	]	PUNCT
ejpam-5948	335	4	m.	m.	NOUN
ejpam-5948	335	5	n.	n.	PROPN
ejpam-5948	335	6	daif	daif	PROPN
ejpam-5948	335	7	and	and	CCONJ
ejpam-5948	335	8	h.	h.	PROPN
ejpam-5948	335	9	e.	e.	PROPN
ejpam-5948	335	10	bell	bell	PROPN
ejpam-5948	335	11	.	.	PUNCT
ejpam-5948	336	1	remarks	remark	NOUN
ejpam-5948	336	2	on	on	ADP
ejpam-5948	336	3	derivations	derivation	NOUN
ejpam-5948	336	4	on	on	ADP
ejpam-5948	336	5	semiprime	semiprime	NOUN
ejpam-5948	336	6	rings	ring	NOUN
ejpam-5948	336	7	.	.	PUNCT
ejpam-5948	337	1	international	international	ADJ
ejpam-5948	337	2	journal	journal	PROPN
ejpam-5948	337	3	of	of	ADP
ejpam-5948	337	4	mathematics	mathematics	PROPN
ejpam-5948	337	5	and	and	CCONJ
ejpam-5948	337	6	mathematical	mathematical	ADJ
ejpam-5948	337	7	sciences	science	NOUN
ejpam-5948	337	8	,	,	PUNCT
ejpam-5948	337	9	15:205–206	15:205–206	NUM
ejpam-5948	337	10	,	,	PUNCT
ejpam-5948	337	11	1992	1992	NUM
ejpam-5948	337	12	.	.	PUNCT
ejpam-5948	338	1	[	[	X
ejpam-5948	338	2	18	18	NUM
ejpam-5948	338	3	]	]	X
ejpam-5948	338	4	r.	r.	PROPN
ejpam-5948	338	5	m.	m.	PROPN
ejpam-5948	338	6	al	al	PROPN
ejpam-5948	338	7	-	-	PUNCT
ejpam-5948	338	8	omary	omary	PROPN
ejpam-5948	338	9	and	and	CCONJ
ejpam-5948	338	10	s.	s.	PROPN
ejpam-5948	338	11	k.	k.	PROPN
ejpam-5948	338	12	nauman	nauman	PROPN
ejpam-5948	338	13	.	.	PUNCT
ejpam-5948	339	1	generalized	generalized	ADJ
ejpam-5948	339	2	derivations	derivation	NOUN
ejpam-5948	339	3	on	on	ADP
ejpam-5948	339	4	prime	prime	ADJ
ejpam-5948	339	5	rings	ring	NOUN
ejpam-5948	339	6	satisfying	satisfy	VERB
ejpam-5948	339	7	certain	certain	ADJ
ejpam-5948	339	8	identities	identity	NOUN
ejpam-5948	339	9	.	.	PUNCT
ejpam-5948	340	1	communications	communication	NOUN
ejpam-5948	340	2	of	of	ADP
ejpam-5948	340	3	the	the	DET
ejpam-5948	340	4	korean	korean	ADJ
ejpam-5948	340	5	mathematical	mathematical	ADJ
ejpam-5948	340	6	society	society	NOUN
ejpam-5948	340	7	,	,	PUNCT
ejpam-5948	340	8	36(2):229	36(2):229	NUM
ejpam-5948	340	9	–	–	PUNCT
ejpam-5948	340	10	238	238	NUM
ejpam-5948	340	11	,	,	PUNCT
ejpam-5948	340	12	2021	2021	NUM
ejpam-5948	340	13	.	.	PUNCT