id	sid	tid	token	lemma	pos
iajs-828	1	1	ibn	ibn	PROPN
iajs-828	1	2	alhaitham	alhaitham	NOUN
iajs-828	1	3	j.	j.	PROPN
iajs-828	1	4	for	for	ADP
iajs-828	1	5	pure	pure	ADJ
iajs-828	1	6	&	&	CCONJ
iajs-828	1	7	appl	appl	PROPN
iajs-828	1	8	.	.	PUNCT
iajs-828	2	1	sci	sci	PROPN
iajs-828	2	2	.	.	PUNCT
iajs-828	2	3	vol.24	vol.24	NOUN
iajs-828	2	4	(	(	PUNCT
iajs-828	2	5	3	3	NUM
iajs-828	2	6	)	)	PUNCT
iajs-828	2	7	2011	2011	NUM
iajs-828	2	8	(	(	PUNCT
iajs-828	2	9	,)strongly	,)strongly	ADV
iajs-828	2	10	derivations	derivation	VERB
iajs-828	2	11	pairs	pair	NOUN
iajs-828	2	12	on	on	ADP
iajs-828	2	13	rings	ring	NOUN
iajs-828	2	14	i.	i.	PROPN
iajs-828	2	15	a.	a.	PROPN
iajs-828	2	16	saed	saed	PROPN
iajs-828	2	17	university	university	PROPN
iajs-828	2	18	of	of	ADP
iajs-828	2	19	technology	technology	NOUN
iajs-828	2	20	received	receive	VERB
iajs-828	2	21	in	in	ADP
iajs-828	2	22	:	:	PUNCT
iajs-828	2	23	30	30	NUM
iajs-828	2	24	november	november	NOUN
iajs-828	2	25	2010	2010	NUM
iajs-828	2	26	accepted	accept	VERB
iajs-828	2	27	in	in	ADP
iajs-828	2	28	:	:	PUNCT
iajs-828	2	29	27	27	NUM
iajs-828	2	30	february	february	NOUN
iajs-828	2	31	2011	2011	NUM
iajs-828	2	32	abstract	abstract	ADV
iajs-828	2	33	let	let	VERB
iajs-828	2	34	r	r	PRON
iajs-828	2	35	be	be	AUX
iajs-828	2	36	an	an	DET
iajs-828	2	37	associative	associative	ADJ
iajs-828	2	38	ring	ring	NOUN
iajs-828	2	39	.	.	PUNCT
iajs-828	3	1	in	in	ADP
iajs-828	3	2	this	this	DET
iajs-828	3	3	paper	paper	NOUN
iajs-828	3	4	we	we	PRON
iajs-828	3	5	present	present	VERB
iajs-828	3	6	the	the	DET
iajs-828	3	7	definition	definition	NOUN
iajs-828	3	8	of	of	ADP
iajs-828	3	9	(	(	PUNCT
iajs-828	3	10	,)strongly	,)strongly	ADV
iajs-828	3	11	derivation	derivation	NOUN
iajs-828	3	12	pair	pair	NOUN
iajs-828	3	13	and	and	CCONJ
iajs-828	3	14	jordan	jordan	PROPN
iajs-828	3	15	(	(	PUNCT
iajs-828	3	16	,)strongly	,)strongly	ADV
iajs-828	3	17	derivation	derivation	ADJ
iajs-828	3	18	pair	pair	NOUN
iajs-828	3	19	on	on	ADP
iajs-828	3	20	a	a	DET
iajs-828	3	21	ring	ring	NOUN
iajs-828	3	22	r	r	NOUN
iajs-828	3	23	,	,	PUNCT
iajs-828	3	24	and	and	CCONJ
iajs-828	3	25	study	study	VERB
iajs-828	3	26	the	the	DET
iajs-828	3	27	relation	relation	NOUN
iajs-828	3	28	between	between	ADP
iajs-828	3	29	them	they	PRON
iajs-828	3	30	.	.	PUNCT
iajs-828	4	1	also	also	ADV
iajs-828	4	2	,	,	PUNCT
iajs-828	4	3	we	we	PRON
iajs-828	4	4	study	study	VERB
iajs-828	4	5	prime	prime	ADJ
iajs-828	4	6	rings	ring	NOUN
iajs-828	4	7	,	,	PUNCT
iajs-828	4	8	semiprime	semiprime	NOUN
iajs-828	4	9	rings	ring	NOUN
iajs-828	4	10	,	,	PUNCT
iajs-828	4	11	and	and	CCONJ
iajs-828	4	12	rings	ring	NOUN
iajs-828	4	13	that	that	PRON
iajs-828	4	14	have	have	VERB
iajs-828	4	15	commutator	commutator	NOUN
iajs-828	4	16	left	leave	VERB
iajs-828	4	17	nonzero	nonzero	PROPN
iajs-828	4	18	divisior	divisior	PROPN
iajs-828	4	19	with	with	ADP
iajs-828	4	20	(	(	PUNCT
iajs-828	4	21	,)strongly	,)strongly	ADV
iajs-828	4	22	derivation	derivation	ADJ
iajs-828	4	23	pair	pair	NOUN
iajs-828	4	24	,	,	PUNCT
iajs-828	4	25	to	to	PART
iajs-828	4	26	obtain	obtain	VERB
iajs-828	4	27	a	a	DET
iajs-828	4	28	(	(	PUNCT
iajs-828	4	29	,)derivation	,)derivation	NOUN
iajs-828	4	30	.	.	PUNCT
iajs-828	5	1	where	where	SCONJ
iajs-828	5	2	,	,	NOUN
iajs-828	5	3	:	:	PUNCT
iajs-828	5	4	rr	rr	ADV
iajs-828	5	5	are	be	AUX
iajs-828	5	6	two	two	NUM
iajs-828	5	7	mappings	mapping	NOUN
iajs-828	5	8	of	of	ADP
iajs-828	5	9	r.	r.	PROPN
iajs-828	5	10	keywords	keyword	VERB
iajs-828	5	11	prime	prime	ADJ
iajs-828	5	12	ring	ring	NOUN
iajs-828	5	13	,	,	PUNCT
iajs-828	5	14	semiprime	semiprime	NOUN
iajs-828	5	15	ring	ring	NOUN
iajs-828	5	16	,	,	PUNCT
iajs-828	5	17	(	(	PUNCT
iajs-828	5	18	,)-derivation	,)-derivation	NOUN
iajs-828	5	19	,	,	PUNCT
iajs-828	5	20	(	(	PUNCT
iajs-828	5	21	,)-strongly	,)-strongly	ADV
iajs-828	5	22	derivation	derivation	NOUN
iajs-828	5	23	pair	pair	NOUN
iajs-828	5	24	,	,	PUNCT
iajs-828	5	25	jordan	jordan	PROPN
iajs-828	5	26	(	(	PUNCT
iajs-828	5	27	,)strongly	,)strongly	ADV
iajs-828	5	28	derivation	derivation	ADJ
iajs-828	5	29	pair	pair	NOUN
iajs-828	5	30	.	.	PUNCT
iajs-828	6	1	§	§	NOUN
iajs-828	6	2	1	1	NUM
iajs-828	6	3	basic	basic	ADJ
iajs-828	6	4	concepts	concept	NOUN
iajs-828	6	5	deinition	deinition	VERB
iajs-828	6	6	1.1	1.1	NUM
iajs-828	6	7	:	:	PUNCT
iajs-828	7	1	[	[	X
iajs-828	7	2	1	1	X
iajs-828	7	3	]	]	PUNCT
iajs-828	7	4	a	a	DET
iajs-828	7	5	nonempty	nonempty	ADV
iajs-828	7	6	set	set	VERB
iajs-828	7	7	r	r	NOUN
iajs-828	7	8	is	be	AUX
iajs-828	7	9	said	say	VERB
iajs-828	7	10	to	to	PART
iajs-828	7	11	be	be	AUX
iajs-828	7	12	associative	associative	ADJ
iajs-828	7	13	ring	ring	NOUN
iajs-828	7	14	if	if	SCONJ
iajs-828	7	15	in	in	ADP
iajs-828	7	16	r	r	NOUN
iajs-828	7	17	there	there	PRON
iajs-828	7	18	are	be	VERB
iajs-828	7	19	defined	define	VERB
iajs-828	7	20	two	two	NUM
iajs-828	7	21	operations	operation	NOUN
iajs-828	7	22	,	,	PUNCT
iajs-828	7	23	denoted	denote	VERB
iajs-828	7	24	by	by	ADP
iajs-828	7	25	+	+	CCONJ
iajs-828	7	26	and	and	CCONJ
iajs-828	7	27	.	.	PUNCT
iajs-828	8	1	respectively	respectively	ADV
iajs-828	8	2	,	,	PUNCT
iajs-828	8	3	such	such	ADJ
iajs-828	8	4	that	that	PRON
iajs-828	8	5	for	for	ADP
iajs-828	8	6	all	all	DET
iajs-828	8	7	a	a	DET
iajs-828	8	8	,	,	PUNCT
iajs-828	8	9	b	b	NOUN
iajs-828	8	10	,	,	PUNCT
iajs-828	8	11	c	c	NOUN
iajs-828	8	12	in	in	ADP
iajs-828	8	13	r	r	NOUN
iajs-828	8	14	:	:	PUNCT
iajs-828	8	15	1a	1a	NOUN
iajs-828	8	16	+	+	CCONJ
iajs-828	8	17	b	b	X
iajs-828	8	18	is	be	AUX
iajs-828	8	19	in	in	ADP
iajs-828	8	20	r	r	NOUN
iajs-828	8	21	2a	2a	NUM
iajs-828	8	22	+	+	CCONJ
iajs-828	8	23	b	b	X
iajs-828	8	24	=	=	SYM
iajs-828	8	25	b	b	PROPN
iajs-828	8	26	+	+	CCONJ
iajs-828	8	27	a	a	DET
iajs-828	8	28	3(a+b	3(a+b	NUM
iajs-828	8	29	)	)	PUNCT
iajs-828	9	1	+	+	NUM
iajs-828	10	1	c	c	NOUN
iajs-828	10	2	=	=	PUNCT
iajs-828	10	3	a	a	PRON
iajs-828	10	4	+	+	X
iajs-828	10	5	(	(	PUNCT
iajs-828	10	6	b+c	b+c	NUM
iajs-828	10	7	)	)	PUNCT
iajs-828	10	8	4there	4there	PRON
iajs-828	10	9	is	be	AUX
iajs-828	10	10	an	an	DET
iajs-828	10	11	element	element	NOUN
iajs-828	10	12	0	0	NUM
iajs-828	10	13	in	in	ADP
iajs-828	10	14	r	r	NOUN
iajs-828	10	15	such	such	ADJ
iajs-828	10	16	that	that	SCONJ
iajs-828	10	17	a+0	a+0	ADV
iajs-828	10	18	=	=	SYM
iajs-828	10	19	a	a	PRON
iajs-828	10	20	(	(	PUNCT
iajs-828	10	21	for	for	ADP
iajs-828	10	22	every	every	DET
iajs-828	10	23	a	a	PRON
iajs-828	10	24	in	in	ADP
iajs-828	10	25	r	r	NOUN
iajs-828	10	26	)	)	PUNCT
iajs-828	10	27	5there	5there	NUM
iajs-828	10	28	exists	exist	VERB
iajs-828	10	29	an	an	DET
iajs-828	10	30	element	element	NOUN
iajs-828	10	31	–	–	PUNCT
iajs-828	10	32	a	a	PRON
iajs-828	10	33	in	in	ADP
iajs-828	10	34	r	r	NOUN
iajs-828	10	35	such	such	DET
iajs-828	10	36	that	that	SCONJ
iajs-828	10	37	a	a	DET
iajs-828	10	38	+	+	X
iajs-828	10	39	(	(	PUNCT
iajs-828	10	40	-a)=0	-a)=0	NUM
iajs-828	10	41	.	.	PUNCT
iajs-828	10	42	6a	6a	NOUN
iajs-828	10	43	.	.	PUNCT
iajs-828	11	1	b	b	NOUN
iajs-828	11	2	is	be	AUX
iajs-828	11	3	in	in	ADP
iajs-828	11	4	r.	r.	PROPN
iajs-828	11	5	7a	7a	NUM
iajs-828	11	6	.	.	PUNCT
iajs-828	12	1	(	(	PUNCT
iajs-828	12	2	b.c	b.c	PROPN
iajs-828	12	3	)	)	PUNCT
iajs-828	12	4	=	=	PRON
iajs-828	13	1	(	(	PUNCT
iajs-828	13	2	a.b).c	a.b).c	PROPN
iajs-828	13	3	8a	8a	NUM
iajs-828	13	4	.	.	PUNCT
iajs-828	14	1	(	(	PUNCT
iajs-828	14	2	b+c	b+c	NUM
iajs-828	14	3	)	)	PUNCT
iajs-828	14	4	=	=	SYM
iajs-828	14	5	a.b	a.b	PROPN
iajs-828	14	6	+	+	SYM
iajs-828	14	7	a.c	a.c	PROPN
iajs-828	14	8	and	and	CCONJ
iajs-828	14	9	(	(	PUNCT
iajs-828	14	10	b+c).a	b+c).a	PROPN
iajs-828	14	11	=	=	SYM
iajs-828	14	12	b.a	b.a	PROPN
iajs-828	14	13	+	+	PROPN
iajs-828	14	14	c.a	c.a	PROPN
iajs-828	14	15	deinition	deinition	NOUN
iajs-828	14	16	1.2	1.2	NUM
iajs-828	14	17	:	:	PUNCT
iajs-828	15	1	[	[	X
iajs-828	15	2	1	1	X
iajs-828	15	3	]	]	PUNCT
iajs-828	15	4	a	a	DET
iajs-828	15	5	ring	ring	NOUN
iajs-828	15	6	r	r	NOUN
iajs-828	15	7	is	be	AUX
iajs-828	15	8	called	call	VERB
iajs-828	15	9	prime	prime	ADJ
iajs-828	15	10	ring	ring	NOUN
iajs-828	15	11	if	if	SCONJ
iajs-828	15	12	for	for	ADP
iajs-828	15	13	any	any	DET
iajs-828	15	14	a	a	PRON
iajs-828	15	15	,	,	PUNCT
iajs-828	15	16	br	br	PROPN
iajs-828	15	17	,	,	PUNCT
iajs-828	15	18	a	a	DET
iajs-828	15	19	r	r	NOUN
iajs-828	15	20	b	b	NOUN
iajs-828	15	21	=	=	PUNCT
iajs-828	15	22	{	{	PUNCT
iajs-828	15	23	0	0	NUM
iajs-828	15	24	}	}	PUNCT
iajs-828	15	25	,	,	PUNCT
iajs-828	15	26	implies	imply	VERB
iajs-828	15	27	that	that	SCONJ
iajs-828	15	28	either	either	CCONJ
iajs-828	15	29	a=0	a=0	PROPN
iajs-828	15	30	or	or	CCONJ
iajs-828	15	31	b=0	b=0	PROPN
iajs-828	15	32	.	.	PUNCT
iajs-828	16	1	definition	definition	NOUN
iajs-828	16	2	1.3:[1	1.3:[1	NUM
iajs-828	16	3	]	]	X
iajs-828	16	4	a	a	DET
iajs-828	16	5	ring	ring	NOUN
iajs-828	16	6	r	r	NOUN
iajs-828	16	7	is	be	AUX
iajs-828	16	8	called	call	VERB
iajs-828	16	9	semiprime	semiprime	NOUN
iajs-828	16	10	ring	ring	NOUN
iajs-828	16	11	if	if	SCONJ
iajs-828	16	12	for	for	ADP
iajs-828	16	13	any	any	DET
iajs-828	16	14	ar	ar	NOUN
iajs-828	16	15	,	,	PUNCT
iajs-828	16	16	ara	ara	NOUN
iajs-828	16	17	=	=	PUNCT
iajs-828	16	18	{	{	PUNCT
iajs-828	16	19	0	0	NUM
iajs-828	16	20	}	}	PUNCT
iajs-828	16	21	,	,	PUNCT
iajs-828	16	22	implies	imply	VERB
iajs-828	16	23	that	that	SCONJ
iajs-828	16	24	a=0	a=0	PROPN
iajs-828	16	25	.	.	PUNCT
iajs-828	16	26	remark	remark	PROPN
iajs-828	16	27	1.4:[1	1.4:[1	NUM
iajs-828	16	28	]	]	X
iajs-828	16	29	every	every	DET
iajs-828	16	30	prime	prime	ADJ
iajs-828	16	31	ring	ring	NOUN
iajs-828	16	32	is	be	AUX
iajs-828	16	33	semiprime	semiprime	NOUN
iajs-828	16	34	ring	ring	NOUN
iajs-828	16	35	,	,	PUNCT
iajs-828	16	36	but	but	CCONJ
iajs-828	16	37	the	the	DET
iajs-828	16	38	converse	converse	NOUN
iajs-828	16	39	in	in	ADP
iajs-828	16	40	general	general	ADJ
iajs-828	16	41	is	be	AUX
iajs-828	16	42	not	not	PART
iajs-828	16	43	true	true	ADJ
iajs-828	16	44	.	.	PUNCT
iajs-828	17	1	the	the	DET
iajs-828	17	2	following	follow	VERB
iajs-828	17	3	example	example	NOUN
iajs-828	17	4	justifies	justify	VERB
iajs-828	17	5	this	this	DET
iajs-828	17	6	remark	remark	NOUN
iajs-828	17	7	.	.	PUNCT
iajs-828	18	1	ibn	ibn	PROPN
iajs-828	18	2	alhaitham	alhaitham	PROPN
iajs-828	18	3	j.	j.	PROPN
iajs-828	18	4	for	for	ADP
iajs-828	18	5	pure	pure	ADJ
iajs-828	18	6	&	&	CCONJ
iajs-828	18	7	appl	appl	PROPN
iajs-828	18	8	.	.	PUNCT
iajs-828	19	1	sci	sci	PROPN
iajs-828	19	2	.	.	PUNCT
iajs-828	19	3	vol.24	vol.24	NOUN
iajs-828	19	4	(	(	PUNCT
iajs-828	19	5	3	3	NUM
iajs-828	19	6	)	)	PUNCT
iajs-828	19	7	2011	2011	NUM
iajs-828	19	8	example	example	NOUN
iajs-828	19	9	1.5	1.5	NUM
iajs-828	19	10	:	:	PUNCT
iajs-828	20	1	[	[	X
iajs-828	20	2	1	1	NUM
iajs-828	20	3	]	]	X
iajs-828	20	4	r	r	NOUN
iajs-828	20	5	=	=	SYM
iajs-828	20	6	z6	z6	PROPN
iajs-828	20	7	is	be	AUX
iajs-828	20	8	a	a	DET
iajs-828	20	9	semiprime	semiprime	NOUN
iajs-828	20	10	ring	ring	NOUN
iajs-828	20	11	but	but	CCONJ
iajs-828	20	12	is	be	AUX
iajs-828	20	13	not	not	PART
iajs-828	20	14	prime	prime	ADJ
iajs-828	20	15	.	.	PUNCT
iajs-828	21	1	let	let	VERB
iajs-828	21	2	a	a	DET
iajs-828	21	3			NOUN
iajs-828	21	4	r	r	NOUN
iajs-828	21	5	such	such	ADJ
iajs-828	21	6	that	that	DET
iajs-828	21	7	ara	ara	NOUN
iajs-828	21	8	=	=	X
iajs-828	21	9	{	{	PUNCT
iajs-828	21	10	0	0	NUM
iajs-828	21	11	}	}	PUNCT
iajs-828	21	12	,	,	PUNCT
iajs-828	21	13	implies	imply	VERB
iajs-828	21	14	that	that	SCONJ
iajs-828	21	15	a	a	DET
iajs-828	21	16	2	2	NUM
iajs-828	21	17	=	=	SYM
iajs-828	21	18	0	0	NUM
iajs-828	21	19	,	,	PUNCT
iajs-828	21	20	hence	hence	ADV
iajs-828	21	21	a=0	a=0	ADV
iajs-828	21	22	,	,	PUNCT
iajs-828	21	23	therefore	therefore	ADV
iajs-828	21	24	r	r	NOUN
iajs-828	21	25	is	be	AUX
iajs-828	21	26	a	a	DET
iajs-828	21	27	semiprime	semiprime	NOUN
iajs-828	21	28	ring	ring	NOUN
iajs-828	21	29	.	.	PUNCT
iajs-828	22	1	but	but	CCONJ
iajs-828	22	2	r	r	NOUN
iajs-828	22	3	is	be	AUX
iajs-828	22	4	not	not	PART
iajs-828	22	5	prime	prime	ADJ
iajs-828	22	6	,	,	PUNCT
iajs-828	22	7	since	since	SCONJ
iajs-828	22	8	20	20	NUM
iajs-828	22	9	and	and	CCONJ
iajs-828	22	10	30	30	NUM
iajs-828	22	11	implies	imply	VERB
iajs-828	22	12	that	that	SCONJ
iajs-828	22	13	2r3	2r3	NUM
iajs-828	22	14	=	=	SYM
iajs-828	22	15	{	{	PUNCT
iajs-828	22	16	0	0	NUM
iajs-828	22	17	}	}	PUNCT
iajs-828	22	18	.	.	PUNCT
iajs-828	23	1	definition	definition	NOUN
iajs-828	23	2	1.6:[2	1.6:[2	NUM
iajs-828	23	3	]	]	PUNCT
iajs-828	23	4	a	a	DET
iajs-828	23	5	ring	ring	NOUN
iajs-828	23	6	r	r	NOUN
iajs-828	23	7	is	be	AUX
iajs-828	23	8	said	say	VERB
iajs-828	23	9	to	to	PART
iajs-828	23	10	be	be	AUX
iajs-828	23	11	n	n	CCONJ
iajs-828	23	12	-	-	PUNCT
iajs-828	23	13	torsion	torsion	NOUN
iajs-828	23	14	free	free	ADJ
iajs-828	23	15	,	,	PUNCT
iajs-828	23	16	where	where	SCONJ
iajs-828	23	17	n≠0	n≠0	PROPN
iajs-828	23	18	is	be	AUX
iajs-828	23	19	an	an	DET
iajs-828	23	20	integer	integer	NOUN
iajs-828	23	21	if	if	SCONJ
iajs-828	23	22	whenever	whenever	SCONJ
iajs-828	23	23	n	n	PRON
iajs-828	23	24	a=0	a=0	X
iajs-828	23	25	,	,	PUNCT
iajs-828	23	26	with	with	ADP
iajs-828	23	27	ar	ar	NOUN
iajs-828	23	28	,	,	PUNCT
iajs-828	23	29	then	then	ADV
iajs-828	23	30	a=0	a=0	PROPN
iajs-828	23	31	.	.	PUNCT
iajs-828	24	1	definition	definition	NOUN
iajs-828	24	2	1.7:[2	1.7:[2	NUM
iajs-828	24	3	]	]	PUNCT
iajs-828	24	4	let	let	VERB
iajs-828	24	5	r	r	PRON
iajs-828	24	6	be	be	AUX
iajs-828	24	7	a	a	DET
iajs-828	24	8	ring	ring	NOUN
iajs-828	24	9	.	.	PUNCT
iajs-828	25	1	a	a	DET
iajs-828	25	2	lie	lie	NOUN
iajs-828	25	3	product	product	NOUN
iajs-828	25	4	[	[	X
iajs-828	25	5	,	,	PUNCT
iajs-828	25	6	]	]	PUNCT
iajs-828	25	7	on	on	ADP
iajs-828	25	8	r	r	NOUN
iajs-828	25	9	is	be	AUX
iajs-828	25	10	defined	define	VERB
iajs-828	25	11	as[x	as[x	ADJ
iajs-828	25	12	,	,	PUNCT
iajs-828	25	13	y]=xy	y]=xy	PROPN
iajs-828	25	14	–	–	PUNCT
iajs-828	25	15	yx	yx	NOUN
iajs-828	25	16	,	,	PUNCT
iajs-828	25	17	for	for	ADP
iajs-828	25	18	all	all	DET
iajs-828	25	19	x	x	NOUN
iajs-828	25	20	,	,	PUNCT
iajs-828	25	21	y	y	PROPN
iajs-828	25	22			PROPN
iajs-828	25	23	r.	r.	PROPN
iajs-828	25	24	definition	definition	NOUN
iajs-828	25	25	1.8:[2	1.8:[2	NUM
iajs-828	25	26	]	]	PUNCT
iajs-828	25	27	let	let	VERB
iajs-828	25	28	r	r	PRON
iajs-828	25	29	be	be	AUX
iajs-828	25	30	a	a	DET
iajs-828	25	31	ring	ring	NOUN
iajs-828	25	32	.	.	PUNCT
iajs-828	26	1	an	an	DET
iajs-828	26	2	additive	additive	ADJ
iajs-828	26	3	mapping	mapping	NOUN
iajs-828	26	4	d	d	NOUN
iajs-828	26	5	:	:	PUNCT
iajs-828	26	6	rr	rr	ADV
iajs-828	26	7	is	be	AUX
iajs-828	26	8	called	call	VERB
iajs-828	26	9	a	a	DET
iajs-828	26	10	derivation	derivation	NOUN
iajs-828	26	11	if	if	SCONJ
iajs-828	26	12	d(xy)=	d(xy)=	PROPN
iajs-828	26	13	d(x)y	d(x)y	PROPN
iajs-828	26	14	+	+	CCONJ
iajs-828	26	15	xd(y	xd(y	NOUN
iajs-828	26	16	)	)	PUNCT
iajs-828	26	17	,	,	PUNCT
iajs-828	26	18	for	for	ADP
iajs-828	26	19	all	all	DET
iajs-828	26	20	x	x	NOUN
iajs-828	26	21	,	,	PUNCT
iajs-828	26	22	y	y	PROPN
iajs-828	26	23	r	r	PROPN
iajs-828	26	24	and	and	CCONJ
iajs-828	26	25	we	we	PRON
iajs-828	26	26	say	say	VERB
iajs-828	26	27	that	that	SCONJ
iajs-828	26	28	d	d	NOUN
iajs-828	26	29	is	be	AUX
iajs-828	26	30	a	a	DET
iajs-828	26	31	jordan	jordan	PROPN
iajs-828	26	32	derivation	derivation	NOUN
iajs-828	26	33	if	if	SCONJ
iajs-828	26	34	d(x	d(x	PROPN
iajs-828	26	35	2)=d(x)x+xd(x	2)=d(x)x+xd(x	NUM
iajs-828	26	36	)	)	PUNCT
iajs-828	26	37	,	,	PUNCT
iajs-828	26	38	for	for	ADP
iajs-828	26	39	all	all	DET
iajs-828	26	40	xr	xr	NOUN
iajs-828	26	41	.	.	PUNCT
iajs-828	26	42	definition	definition	NOUN
iajs-828	26	43	1.9:[3	1.9:[3	NUM
iajs-828	26	44	]	]	PUNCT
iajs-828	26	45	let	let	VERB
iajs-828	26	46	r	r	PRON
iajs-828	26	47	be	be	AUX
iajs-828	26	48	a	a	DET
iajs-828	26	49	ring	ring	NOUN
iajs-828	26	50	.	.	PUNCT
iajs-828	27	1	an	an	DET
iajs-828	27	2	additive	additive	ADJ
iajs-828	27	3	mapping	mapping	NOUN
iajs-828	27	4	d	d	NOUN
iajs-828	27	5	:	:	PUNCT
iajs-828	27	6	rr	rr	ADV
iajs-828	27	7	is	be	AUX
iajs-828	27	8	called	call	VERB
iajs-828	27	9	a	a	DET
iajs-828	27	10	(	(	PUNCT
iajs-828	27	11	,)-derivation	,)-derivation	NOUN
iajs-828	27	12	,	,	PUNCT
iajs-828	27	13	where	where	SCONJ
iajs-828	27	14	,	,	NOUN
iajs-828	27	15	:	:	PUNCT
iajs-828	27	16	rr	rr	ADV
iajs-828	27	17	are	be	AUX
iajs-828	27	18	two	two	NUM
iajs-828	27	19	mappings	mapping	NOUN
iajs-828	27	20	of	of	ADP
iajs-828	27	21	r	r	NOUN
iajs-828	27	22	,	,	PUNCT
iajs-828	27	23	if	if	SCONJ
iajs-828	27	24	d(xy)=	d(xy)=	PROPN
iajs-828	27	25	d(x)σ(y	d(x)σ(y	NOUN
iajs-828	27	26	)	)	PUNCT
iajs-828	28	1	+	+	NUM
iajs-828	28	2	(x	(x	X
iajs-828	28	3	)	)	PUNCT
iajs-828	28	4	d(y	d(y	NOUN
iajs-828	28	5	)	)	PUNCT
iajs-828	28	6	,	,	PUNCT
iajs-828	28	7	for	for	ADP
iajs-828	28	8	all	all	DET
iajs-828	28	9	x	x	NOUN
iajs-828	28	10	,	,	PUNCT
iajs-828	28	11	y	y	PROPN
iajs-828	28	12			PROPN
iajs-828	28	13	r	r	NOUN
iajs-828	28	14	,	,	PUNCT
iajs-828	28	15	and	and	CCONJ
iajs-828	28	16	we	we	PRON
iajs-828	28	17	say	say	VERB
iajs-828	28	18	that	that	SCONJ
iajs-828	28	19	d	d	NOUN
iajs-828	28	20	is	be	AUX
iajs-828	28	21	a	a	DET
iajs-828	28	22	jordan	jordan	PROPN
iajs-828	28	23	(	(	PUNCT
iajs-828	28	24	σ,)-derivation	σ,)-derivation	PROPN
iajs-828	28	25	if	if	SCONJ
iajs-828	28	26	d(x	d(x	PROPN
iajs-828	28	27	2	2	NUM
iajs-828	28	28	)	)	PUNCT
iajs-828	29	1	=	=	NOUN
iajs-828	29	2	d(x	d(x	NOUN
iajs-828	29	3	)	)	PUNCT
iajs-828	29	4	σ(x)+	σ(x)+	NOUN
iajs-828	29	5	(x	(x	NOUN
iajs-828	29	6	)	)	PUNCT
iajs-828	29	7	d(x	d(x	NOUN
iajs-828	29	8	)	)	PUNCT
iajs-828	29	9	,	,	PUNCT
iajs-828	29	10	for	for	ADP
iajs-828	29	11	all	all	DET
iajs-828	29	12	x	x	PRON
iajs-828	29	13			PROPN
iajs-828	29	14	r.	r.	PROPN
iajs-828	29	15	definition	definition	NOUN
iajs-828	29	16	1.10:[4	1.10:[4	NUM
iajs-828	29	17	]	]	PUNCT
iajs-828	29	18	let	let	VERB
iajs-828	29	19	r	r	PRON
iajs-828	29	20	be	be	AUX
iajs-828	29	21	a	a	DET
iajs-828	29	22	ring	ring	NOUN
iajs-828	29	23	,	,	PUNCT
iajs-828	29	24	additive	additive	ADJ
iajs-828	29	25	mappings	mapping	NOUN
iajs-828	29	26	d	d	NOUN
iajs-828	29	27	,	,	PUNCT
iajs-828	29	28	g	g	NOUN
iajs-828	29	29	:	:	PUNCT
iajs-828	29	30	rr	rr	ADV
iajs-828	29	31	is	be	AUX
iajs-828	29	32	called	call	VERB
iajs-828	29	33	s	s	NOUN
iajs-828	29	34	-	-	PUNCT
iajs-828	29	35	derivation	derivation	ADJ
iajs-828	29	36	pair	pair	NOUN
iajs-828	29	37	(	(	PUNCT
iajs-828	29	38	d	d	NOUN
iajs-828	29	39	,	,	PUNCT
iajs-828	29	40	g	g	NOUN
iajs-828	29	41	)	)	PUNCT
iajs-828	29	42	if	if	SCONJ
iajs-828	29	43	satisfies	satisfy	VERB
iajs-828	29	44	the	the	DET
iajs-828	29	45	following	follow	VERB
iajs-828	29	46	equations	equation	NOUN
iajs-828	29	47	:	:	PUNCT
iajs-828	29	48	d(xy	d(xy	NUM
iajs-828	29	49	)	)	PUNCT
iajs-828	29	50	=	=	SYM
iajs-828	30	1	d(x)y	d(x)y	PROPN
iajs-828	30	2	+	+	NUM
iajs-828	30	3	xg(y	xg(y	NOUN
iajs-828	30	4	)	)	PUNCT
iajs-828	30	5	,	,	PUNCT
iajs-828	30	6	for	for	ADP
iajs-828	30	7	all	all	DET
iajs-828	30	8	x	x	NOUN
iajs-828	30	9	,	,	PUNCT
iajs-828	30	10	y	y	PROPN
iajs-828	30	11			PROPN
iajs-828	30	12	r.	r.	PROPN
iajs-828	30	13	g(xy)=	g(xy)=	PROPN
iajs-828	30	14	g(x)y	g(x)y	PROPN
iajs-828	30	15	+	+	CCONJ
iajs-828	30	16	xd(y	xd(y	NUM
iajs-828	30	17	)	)	PUNCT
iajs-828	30	18	,	,	PUNCT
iajs-828	30	19	for	for	ADP
iajs-828	30	20	all	all	DET
iajs-828	30	21	x	x	NOUN
iajs-828	30	22	,	,	PUNCT
iajs-828	30	23	y	y	PROPN
iajs-828	30	24			PROPN
iajs-828	30	25	r.	r.	PROPN
iajs-828	30	26	and	and	CCONJ
iajs-828	30	27	is	be	AUX
iajs-828	30	28	called	call	VERB
iajs-828	30	29	jordan	jordan	PROPN
iajs-828	30	30	s	s	PROPN
iajs-828	30	31	-	-	PUNCT
iajs-828	30	32	derivation	derivation	NOUN
iajs-828	30	33	pair	pair	NOUN
iajs-828	30	34	if	if	SCONJ
iajs-828	30	35	:	:	PUNCT
iajs-828	30	36	d(x	d(x	PROPN
iajs-828	30	37	2)=d(x)x	2)=d(x)x	NUM
iajs-828	30	38	+	+	CCONJ
iajs-828	30	39	xg(x	xg(x	NOUN
iajs-828	30	40	)	)	PUNCT
iajs-828	30	41	,	,	PUNCT
iajs-828	30	42	for	for	ADP
iajs-828	30	43	all	all	DET
iajs-828	30	44	x	x	PROPN
iajs-828	30	45			PROPN
iajs-828	30	46	r.	r.	PROPN
iajs-828	30	47	g(x2)=g(x)x	g(x2)=g(x)x	PROPN
iajs-828	30	48	+	+	CCONJ
iajs-828	30	49	xd(x	xd(x	NOUN
iajs-828	30	50	)	)	PUNCT
iajs-828	30	51	,	,	PUNCT
iajs-828	30	52	for	for	ADP
iajs-828	30	53	all	all	DET
iajs-828	30	54	x	x	PRON
iajs-828	30	55			PROPN
iajs-828	30	56	r.	r.	PROPN
iajs-828	30	57	example	example	PROPN
iajs-828	31	1	1.11:[4	1.11:[4	NUM
iajs-828	31	2	]	]	X
iajs-828	31	3	let	let	VERB
iajs-828	31	4	r	r	PRON
iajs-828	31	5	be	be	AUX
iajs-828	31	6	a	a	DET
iajs-828	31	7	non	non	X
iajs-828	31	8	commutative	commutative	ADJ
iajs-828	31	9	ring	ring	NOUN
iajs-828	31	10	and	and	CCONJ
iajs-828	31	11	let	let	VERB
iajs-828	31	12	a	a	DET
iajs-828	31	13	,	,	PUNCT
iajs-828	31	14	b	b	NOUN
iajs-828	31	15			PROPN
iajs-828	31	16	r	r	NOUN
iajs-828	31	17	,	,	PUNCT
iajs-828	31	18	such	such	ADJ
iajs-828	31	19	that	that	SCONJ
iajs-828	31	20	xa	xa	PROPN
iajs-828	31	21	=	=	PROPN
iajs-828	31	22	xb=0	xb=0	PROPN
iajs-828	31	23	,	,	PUNCT
iajs-828	31	24	for	for	SCONJ
iajs-828	31	25	all	all	DET
iajs-828	31	26	x	x	PRON
iajs-828	31	27			PROPN
iajs-828	31	28	r.	r.	PROPN
iajs-828	31	29	define	define	VERB
iajs-828	31	30	d	d	X
iajs-828	31	31	,	,	PUNCT
iajs-828	31	32	g	g	NOUN
iajs-828	31	33	:	:	PUNCT
iajs-828	31	34	rr	rr	ADV
iajs-828	31	35	,	,	PUNCT
iajs-828	31	36	as	as	SCONJ
iajs-828	31	37	follows	follow	VERB
iajs-828	31	38	:	:	PUNCT
iajs-828	31	39	d(x	d(x	NOUN
iajs-828	31	40	)	)	PUNCT
iajs-828	32	1	=	=	SYM
iajs-828	32	2	ax	ax	NOUN
iajs-828	32	3	,	,	PUNCT
iajs-828	32	4	g(x	g(x	NOUN
iajs-828	32	5	)	)	PUNCT
iajs-828	33	1	=	=	VERB
iajs-828	33	2	bx	bx	NOUN
iajs-828	33	3	then	then	ADV
iajs-828	33	4	(	(	PUNCT
iajs-828	33	5	d	d	X
iajs-828	33	6	,	,	PUNCT
iajs-828	33	7	g	g	NOUN
iajs-828	33	8	)	)	PUNCT
iajs-828	33	9	is	be	AUX
iajs-828	33	10	a	a	DET
iajs-828	33	11	s	s	NOUN
iajs-828	33	12	-	-	PUNCT
iajs-828	33	13	derivation	derivation	ADJ
iajs-828	33	14	pair	pair	NOUN
iajs-828	33	15	of	of	ADP
iajs-828	33	16	r.	r.	PROPN
iajs-828	33	17	remark	remark	PROPN
iajs-828	33	18	1.12:[4	1.12:[4	PROPN
iajs-828	33	19	]	]	X
iajs-828	33	20	every	every	DET
iajs-828	33	21	s	s	NOUN
iajs-828	33	22	-	-	PUNCT
iajs-828	33	23	derivation	derivation	ADJ
iajs-828	33	24	pair	pair	NOUN
iajs-828	33	25	is	be	AUX
iajs-828	33	26	a	a	DET
iajs-828	33	27	jordan	jordan	PROPN
iajs-828	33	28	s	s	PROPN
iajs-828	33	29	-	-	PUNCT
iajs-828	33	30	derivation	derivation	NOUN
iajs-828	33	31	pair	pair	NOUN
iajs-828	33	32	,	,	PUNCT
iajs-828	33	33	but	but	CCONJ
iajs-828	33	34	the	the	DET
iajs-828	33	35	converse	converse	NOUN
iajs-828	33	36	is	be	AUX
iajs-828	33	37	in	in	ADP
iajs-828	33	38	general	general	ADJ
iajs-828	33	39	not	not	PART
iajs-828	33	40	true	true	ADJ
iajs-828	33	41	.	.	PUNCT
iajs-828	34	1	the	the	DET
iajs-828	34	2	following	follow	VERB
iajs-828	34	3	example	example	NOUN
iajs-828	34	4	illustrates	illustrate	VERB
iajs-828	34	5	this	this	DET
iajs-828	34	6	remark	remark	NOUN
iajs-828	34	7	.	.	PUNCT
iajs-828	35	1	ibn	ibn	PROPN
iajs-828	35	2	alhaitham	alhaitham	PROPN
iajs-828	35	3	j.	j.	PROPN
iajs-828	35	4	for	for	ADP
iajs-828	35	5	pure	pure	ADJ
iajs-828	35	6	&	&	CCONJ
iajs-828	35	7	appl	appl	PROPN
iajs-828	35	8	.	.	PUNCT
iajs-828	36	1	sci	sci	PROPN
iajs-828	36	2	.	.	PUNCT
iajs-828	36	3	vol.24	vol.24	NOUN
iajs-828	36	4	(	(	PUNCT
iajs-828	36	5	3	3	NUM
iajs-828	36	6	)	)	PUNCT
iajs-828	36	7	2011	2011	NUM
iajs-828	36	8	example	example	NOUN
iajs-828	36	9	1.13:[4	1.13:[4	NUM
iajs-828	36	10	]	]	PUNCT
iajs-828	36	11	let	let	VERB
iajs-828	36	12	r	r	PRON
iajs-828	36	13	be	be	AUX
iajs-828	36	14	a	a	DET
iajs-828	36	15	2	2	NUM
iajs-828	36	16	-	-	PUNCT
iajs-828	36	17	torsion	torsion	NOUN
iajs-828	36	18	free	free	ADJ
iajs-828	36	19	non	non	ADJ
iajs-828	36	20	commutative	commutative	ADJ
iajs-828	36	21	ring	ring	NOUN
iajs-828	36	22	,	,	PUNCT
iajs-828	36	23	and	and	CCONJ
iajs-828	36	24	let	let	VERB
iajs-828	36	25	a	a	DET
iajs-828	36	26			NOUN
iajs-828	36	27	r	r	NOUN
iajs-828	36	28	,	,	PUNCT
iajs-828	36	29	such	such	ADJ
iajs-828	36	30	that	that	SCONJ
iajs-828	36	31	xax	xax	PROPN
iajs-828	36	32	=	=	PROPN
iajs-828	36	33	0	0	PROPN
iajs-828	36	34	,	,	PUNCT
iajs-828	36	35	for	for	ADP
iajs-828	36	36	all	all	DET
iajs-828	36	37	x	x	ADJ
iajs-828	36	38			NOUN
iajs-828	36	39	r	r	NOUN
iajs-828	36	40	,	,	PUNCT
iajs-828	36	41	but	but	CCONJ
iajs-828	36	42	xay	xay	PROPN
iajs-828	36	43	≠	≠	PROPN
iajs-828	36	44	0	0	NUM
iajs-828	36	45	,	,	PUNCT
iajs-828	36	46	for	for	ADP
iajs-828	36	47	some	some	PRON
iajs-828	36	48	(	(	PUNCT
iajs-828	36	49	x	x	PROPN
iajs-828	36	50	≠	≠	PROPN
iajs-828	36	51	y	y	PROPN
iajs-828	36	52	)	)	PUNCT
iajs-828	36	53			PROPN
iajs-828	36	54	r.	r.	VERB
iajs-828	36	55	an	an	DET
iajs-828	36	56	additive	additive	ADJ
iajs-828	36	57	pair	pair	NOUN
iajs-828	36	58	d	d	NOUN
iajs-828	36	59	,	,	PUNCT
iajs-828	36	60	g	g	NOUN
iajs-828	36	61	:	:	PUNCT
iajs-828	36	62	r	r	NOUN
iajs-828	36	63	r	r	NOUN
iajs-828	36	64	is	be	AUX
iajs-828	36	65	defined	define	VERB
iajs-828	36	66	as	as	ADP
iajs-828	36	67	d(x	d(x	PROPN
iajs-828	36	68	)	)	PUNCT
iajs-828	37	1	=	=	PUNCT
iajs-828	37	2	xa	xa	PROPN
iajs-828	38	1	+	+	NOUN
iajs-828	38	2	ax	ax	ADJ
iajs-828	38	3	,	,	PUNCT
iajs-828	38	4	g(x	g(x	NOUN
iajs-828	38	5	)	)	PUNCT
iajs-828	38	6	=	=	PUNCT
iajs-828	39	1	[	[	X
iajs-828	39	2	x	x	X
iajs-828	39	3	,	,	PUNCT
iajs-828	39	4	a	a	X
iajs-828	39	5	]	]	X
iajs-828	39	6	then	then	ADV
iajs-828	39	7	(	(	PUNCT
iajs-828	39	8	d	d	X
iajs-828	39	9	,	,	PUNCT
iajs-828	39	10	g	g	NOUN
iajs-828	39	11	)	)	PUNCT
iajs-828	39	12	is	be	AUX
iajs-828	39	13	jordan	jordan	PROPN
iajs-828	39	14	s	s	PROPN
iajs-828	39	15	-	-	PUNCT
iajs-828	39	16	derivation	derivation	NOUN
iajs-828	39	17	pair	pair	NOUN
iajs-828	39	18	,	,	PUNCT
iajs-828	39	19	but	but	CCONJ
iajs-828	39	20	not	not	PART
iajs-828	39	21	a	a	DET
iajs-828	39	22	s	s	NOUN
iajs-828	39	23	-	-	PUNCT
iajs-828	39	24	derivation	derivation	ADJ
iajs-828	39	25	pair	pair	NOUN
iajs-828	39	26	.	.	PUNCT
iajs-828	40	1	definition	definition	NOUN
iajs-828	40	2	1.14:[5	1.14:[5	NUM
iajs-828	40	3	]	]	X
iajs-828	40	4	a	a	DET
iajs-828	40	5	ring	ring	NOUN
iajs-828	40	6	r	r	NOUN
iajs-828	40	7	is	be	AUX
iajs-828	40	8	said	say	VERB
iajs-828	40	9	to	to	PART
iajs-828	40	10	be	be	AUX
iajs-828	40	11	a	a	DET
iajs-828	40	12	commutator	commutator	NOUN
iajs-828	40	13	right	right	NOUN
iajs-828	40	14	(	(	PUNCT
iajs-828	40	15	resp	resp	NOUN
iajs-828	40	16	.	.	PUNCT
iajs-828	41	1	left	left	ADJ
iajs-828	41	2	)	)	PUNCT
iajs-828	41	3	nonzero	nonzero	PROPN
iajs-828	41	4	divisior	divisior	PROPN
iajs-828	41	5	,	,	PUNCT
iajs-828	41	6	if	if	SCONJ
iajs-828	41	7	there	there	PRON
iajs-828	41	8	exists	exist	VERB
iajs-828	41	9	elements	element	NOUN
iajs-828	41	10	a	a	PRON
iajs-828	41	11	and	and	CCONJ
iajs-828	41	12	b	b	NOUN
iajs-828	41	13	of	of	ADP
iajs-828	41	14	r	r	NOUN
iajs-828	41	15	,	,	PUNCT
iajs-828	41	16	such	such	ADJ
iajs-828	41	17	that	that	SCONJ
iajs-828	41	18	c[a	c[a	NOUN
iajs-828	41	19	,	,	PUNCT
iajs-828	41	20	b	b	X
iajs-828	41	21	]	]	X
iajs-828	41	22	=	=	SYM
iajs-828	41	23	0	0	NUM
iajs-828	41	24	(	(	PUNCT
iajs-828	41	25	resp	resp	NOUN
iajs-828	41	26	.	.	PUNCT
iajs-828	42	1	[	[	X
iajs-828	42	2	a	a	DET
iajs-828	42	3	,	,	PUNCT
iajs-828	42	4	b]c	b]c	NOUN
iajs-828	42	5	=	=	SYM
iajs-828	42	6	0	0	NUM
iajs-828	42	7	)	)	PUNCT
iajs-828	42	8	implies	imply	VERB
iajs-828	42	9	c=0	c=0	VERB
iajs-828	42	10	,	,	PUNCT
iajs-828	42	11	for	for	ADP
iajs-828	42	12	every	every	DET
iajs-828	42	13	c	c	PROPN
iajs-828	43	1			PROPN
iajs-828	43	2	r.	r.	PROPN
iajs-828	43	3	§	§	PROPN
iajs-828	43	4	2	2	NUM
iajs-828	43	5	(	(	PUNCT
iajs-828	43	6	,)-s	,)-s	NOUN
iajs-828	43	7	-	-	NOUN
iajs-828	43	8	derivation	derivation	NOUN
iajs-828	43	9	pairs	pair	NOUN
iajs-828	43	10	in	in	ADP
iajs-828	43	11	this	this	DET
iajs-828	43	12	section	section	NOUN
iajs-828	43	13	,	,	PUNCT
iajs-828	43	14	we	we	PRON
iajs-828	43	15	will	will	AUX
iajs-828	43	16	introduce	introduce	VERB
iajs-828	43	17	the	the	DET
iajs-828	43	18	definition	definition	NOUN
iajs-828	43	19	of	of	ADP
iajs-828	43	20	(	(	PUNCT
iajs-828	43	21	,)-strongly	,)-strongly	ADV
iajs-828	43	22	derivation	derivation	NOUN
iajs-828	43	23	pair	pair	NOUN
iajs-828	43	24	,	,	PUNCT
iajs-828	43	25	and	and	CCONJ
iajs-828	43	26	we	we	PRON
iajs-828	43	27	denoted	denote	VERB
iajs-828	43	28	by	by	ADP
iajs-828	43	29	(	(	PUNCT
iajs-828	43	30	,)-s	,)-s	ADJ
iajs-828	43	31	-	-	ADJ
iajs-828	43	32	derivation	derivation	NOUN
iajs-828	43	33	pair	pair	NOUN
iajs-828	43	34	,	,	PUNCT
iajs-828	43	35	and	and	CCONJ
iajs-828	43	36	jordan	jordan	PROPN
iajs-828	43	37	(	(	PUNCT
iajs-828	43	38	,)-strongly	,)-strongly	ADV
iajs-828	43	39	derivation	derivation	NOUN
iajs-828	43	40	pair	pair	NOUN
iajs-828	43	41	and	and	CCONJ
iajs-828	43	42	we	we	PRON
iajs-828	43	43	denoted	denote	VERB
iajs-828	43	44	by	by	ADP
iajs-828	43	45	jordan	jordan	PROPN
iajs-828	43	46	(	(	PUNCT
iajs-828	43	47	,)-s	,)-s	ADJ
iajs-828	43	48	-	-	ADJ
iajs-828	43	49	derivation	derivation	ADJ
iajs-828	43	50	pair	pair	NOUN
iajs-828	43	51	,	,	PUNCT
iajs-828	43	52	also	also	ADV
iajs-828	43	53	we	we	PRON
iajs-828	43	54	will	will	AUX
iajs-828	43	55	give	give	VERB
iajs-828	43	56	the	the	DET
iajs-828	43	57	relation	relation	NOUN
iajs-828	43	58	between	between	ADP
iajs-828	43	59	them	they	PRON
iajs-828	43	60	.	.	PUNCT
iajs-828	44	1	where	where	SCONJ
iajs-828	44	2	,	,	NOUN
iajs-828	44	3	:	:	PUNCT
iajs-828	44	4	r	r	NOUN
iajs-828	44	5			NOUN
iajs-828	44	6	r	r	NOUN
iajs-828	44	7	are	be	AUX
iajs-828	44	8	two	two	NUM
iajs-828	44	9	mappings	mapping	NOUN
iajs-828	44	10	on	on	ADP
iajs-828	44	11	r.	r.	PROPN
iajs-828	44	12	now	now	ADV
iajs-828	44	13	,	,	PUNCT
iajs-828	44	14	in	in	ADP
iajs-828	44	15	this	this	DET
iajs-828	44	16	section	section	NOUN
iajs-828	44	17	we	we	PRON
iajs-828	44	18	introduce	introduce	VERB
iajs-828	44	19	the	the	DET
iajs-828	44	20	principle	principle	ADJ
iajs-828	44	21	definition	definition	NOUN
iajs-828	44	22	.	.	PUNCT
iajs-828	45	1	definition	definition	NOUN
iajs-828	45	2	2.1	2.1	NUM
iajs-828	45	3	let	let	VERB
iajs-828	45	4	r	r	PRON
iajs-828	45	5	be	be	AUX
iajs-828	45	6	a	a	DET
iajs-828	45	7	ring	ring	NOUN
iajs-828	45	8	,	,	PUNCT
iajs-828	45	9	additive	additive	ADJ
iajs-828	45	10	mappings	mapping	NOUN
iajs-828	46	1	d	d	NOUN
iajs-828	46	2	,	,	PUNCT
iajs-828	46	3	g	g	NOUN
iajs-828	46	4	:	:	PUNCT
iajs-828	46	5	rr	rr	ADV
iajs-828	46	6	is	be	AUX
iajs-828	46	7	called	call	VERB
iajs-828	46	8	(	(	PUNCT
iajs-828	46	9	,)-s	,)-s	ADJ
iajs-828	46	10	-	-	ADJ
iajs-828	46	11	derivation	derivation	ADJ
iajs-828	46	12	pair	pair	NOUN
iajs-828	46	13	(	(	PUNCT
iajs-828	46	14	d	d	NOUN
iajs-828	46	15	,	,	PUNCT
iajs-828	46	16	g	g	NOUN
iajs-828	46	17	)	)	PUNCT
iajs-828	46	18	where	where	SCONJ
iajs-828	46	19	,	,	NOUN
iajs-828	46	20	:	:	PUNCT
iajs-828	46	21	rr	rr	ADV
iajs-828	46	22	are	be	AUX
iajs-828	46	23	two	two	NUM
iajs-828	46	24	mappings	mapping	NOUN
iajs-828	46	25	of	of	ADP
iajs-828	46	26	r	r	NOUN
iajs-828	46	27	,	,	PUNCT
iajs-828	46	28	if	if	SCONJ
iajs-828	46	29	satisfy	satisfy	VERB
iajs-828	46	30	the	the	DET
iajs-828	46	31	following	follow	VERB
iajs-828	46	32	equations	equation	NOUN
iajs-828	46	33	:	:	PUNCT
iajs-828	46	34	d(xy)=d(x)σ(y	d(xy)=d(x)σ(y	NUM
iajs-828	46	35	)	)	PUNCT
iajs-828	47	1	+	+	NUM
iajs-828	47	2	(x)g(y	(x)g(y	NOUN
iajs-828	47	3	)	)	PUNCT
iajs-828	47	4	,	,	PUNCT
iajs-828	47	5	for	for	ADP
iajs-828	47	6	all	all	DET
iajs-828	47	7	x	x	NOUN
iajs-828	47	8	,	,	PUNCT
iajs-828	47	9	y	y	PROPN
iajs-828	47	10			PROPN
iajs-828	47	11	r.	r.	PROPN
iajs-828	47	12	g(xy)=g(x)σ(y	g(xy)=g(x)σ(y	PROPN
iajs-828	47	13	)	)	PUNCT
iajs-828	48	1	+	+	SYM
iajs-828	48	2	(x)d(y	(x)d(y	NUM
iajs-828	48	3	)	)	PUNCT
iajs-828	48	4	,	,	PUNCT
iajs-828	48	5	for	for	ADP
iajs-828	48	6	all	all	DET
iajs-828	48	7	x	x	NOUN
iajs-828	48	8	,	,	PUNCT
iajs-828	48	9	y	y	PROPN
iajs-828	48	10			PROPN
iajs-828	48	11	r.	r.	PROPN
iajs-828	48	12	and	and	CCONJ
iajs-828	48	13	is	be	AUX
iajs-828	48	14	called	call	VERB
iajs-828	48	15	jordan	jordan	PROPN
iajs-828	48	16	(	(	PUNCT
iajs-828	48	17	,)-s	,)-s	ADJ
iajs-828	48	18	-	-	ADJ
iajs-828	48	19	derivation	derivation	NOUN
iajs-828	48	20	pair	pair	NOUN
iajs-828	48	21	if	if	SCONJ
iajs-828	48	22	:	:	PUNCT
iajs-828	48	23	d(x	d(x	NOUN
iajs-828	48	24	2)=d(x)σ(x	2)=d(x)σ(x	NUM
iajs-828	48	25	)	)	PUNCT
iajs-828	48	26	+	+	CCONJ
iajs-828	48	27	(x)g(x	(x)g(x	PROPN
iajs-828	48	28	)	)	PUNCT
iajs-828	48	29	,	,	PUNCT
iajs-828	48	30	for	for	ADP
iajs-828	48	31	all	all	DET
iajs-828	48	32	x	x	PROPN
iajs-828	48	33			PROPN
iajs-828	48	34	r.	r.	PROPN
iajs-828	48	35	g(x2)=g(x)σ(x	g(x2)=g(x)σ(x	PROPN
iajs-828	48	36	)	)	PUNCT
iajs-828	48	37	+	+	CCONJ
iajs-828	48	38	(x)d(x	(x)d(x	NUM
iajs-828	48	39	)	)	PUNCT
iajs-828	48	40	,	,	PUNCT
iajs-828	48	41	for	for	ADP
iajs-828	48	42	all	all	DET
iajs-828	48	43	x	x	PRON
iajs-828	48	44			PROPN
iajs-828	48	45	r.	r.	NOUN
iajs-828	48	46	the	the	DET
iajs-828	48	47	following	follow	VERB
iajs-828	48	48	example	example	NOUN
iajs-828	48	49	explains	explain	VERB
iajs-828	48	50	the	the	DET
iajs-828	48	51	principle	principle	ADJ
iajs-828	48	52	definition	definition	NOUN
iajs-828	48	53	:	:	PUNCT
iajs-828	48	54	example	example	NOUN
iajs-828	48	55	2.2	2.2	NUM
iajs-828	48	56	let	let	VERB
iajs-828	48	57	r	r	PRON
iajs-828	48	58	be	be	AUX
iajs-828	48	59	a	a	DET
iajs-828	48	60	non	non	X
iajs-828	48	61	commutative	commutative	ADJ
iajs-828	48	62	ring	ring	NOUN
iajs-828	48	63	and	and	CCONJ
iajs-828	48	64	let	let	VERB
iajs-828	48	65	a	a	DET
iajs-828	48	66	,	,	PUNCT
iajs-828	48	67	b	b	NOUN
iajs-828	48	68			PROPN
iajs-828	48	69	r	r	NOUN
iajs-828	48	70	,	,	PUNCT
iajs-828	48	71	such	such	ADJ
iajs-828	48	72	that	that	DET
iajs-828	48	73	(x	(x	NOUN
iajs-828	48	74	)	)	PUNCT
iajs-828	48	75	a=	a=	PROPN
iajs-828	48	76	(x)b	(x)b	X
iajs-828	48	77	=	=	SYM
iajs-828	48	78	0	0	PROPN
iajs-828	48	79	,	,	PUNCT
iajs-828	48	80	for	for	SCONJ
iajs-828	48	81	all	all	DET
iajs-828	48	82	x	x	PRON
iajs-828	48	83			PROPN
iajs-828	48	84	r.	r.	PROPN
iajs-828	48	85	define	define	VERB
iajs-828	48	86	d	d	X
iajs-828	48	87	,	,	PUNCT
iajs-828	48	88	g	g	NOUN
iajs-828	48	89	:	:	PUNCT
iajs-828	49	1	r	r	NOUN
iajs-828	49	2			NOUN
iajs-828	49	3	r	r	NOUN
iajs-828	49	4	as	as	SCONJ
iajs-828	49	5	follows	follow	VERB
iajs-828	49	6	:	:	PUNCT
iajs-828	49	7	d(x	d(x	NUM
iajs-828	49	8	)	)	PUNCT
iajs-828	49	9	=	=	PUNCT
iajs-828	49	10	a	a	DET
iajs-828	49	11	(x	(x	PROPN
iajs-828	49	12	)	)	PUNCT
iajs-828	49	13	,	,	PUNCT
iajs-828	49	14	g(x	g(x	NOUN
iajs-828	49	15	)	)	PUNCT
iajs-828	50	1	=	=	SYM
iajs-828	50	2	b	b	NOUN
iajs-828	50	3	(x	(x	PROPN
iajs-828	50	4	)	)	PUNCT
iajs-828	50	5	,	,	PUNCT
iajs-828	50	6	for	for	ADP
iajs-828	50	7	all	all	DET
iajs-828	50	8	x	x	ADJ
iajs-828	50	9			NOUN
iajs-828	50	10	r	r	NOUN
iajs-828	50	11	where	where	SCONJ
iajs-828	50	12			NOUN
iajs-828	50	13	,	,	PUNCT
iajs-828	50	14			NOUN
iajs-828	50	15	:	:	PUNCT
iajs-828	50	16	r	r	NOUN
iajs-828	50	17			NOUN
iajs-828	50	18	r	r	NOUN
iajs-828	50	19	are	be	AUX
iajs-828	50	20	two	two	NUM
iajs-828	50	21	endomorphism	endomorphism	NOUN
iajs-828	50	22	mappings	mapping	NOUN
iajs-828	50	23	.	.	PUNCT
iajs-828	51	1	then	then	ADV
iajs-828	51	2	(	(	PUNCT
iajs-828	51	3	d	d	X
iajs-828	51	4	,	,	PUNCT
iajs-828	51	5	g	g	NOUN
iajs-828	51	6	)	)	PUNCT
iajs-828	51	7	is	be	AUX
iajs-828	51	8	a	a	DET
iajs-828	51	9	(	(	PUNCT
iajs-828	51	10	,)-s	,)-s	ADJ
iajs-828	51	11	-	-	ADJ
iajs-828	51	12	derivation	derivation	ADJ
iajs-828	51	13	pair	pair	NOUN
iajs-828	51	14	of	of	ADP
iajs-828	51	15	r.	r.	PROPN
iajs-828	51	16	let	let	VERB
iajs-828	51	17	x	x	PRON
iajs-828	51	18	,	,	PUNCT
iajs-828	51	19	y	y	PROPN
iajs-828	51	20			PROPN
iajs-828	51	21	r	r	NOUN
iajs-828	51	22	,	,	PUNCT
iajs-828	51	23	so	so	ADV
iajs-828	51	24	:	:	PUNCT
iajs-828	51	25	d(xy	d(xy	NUM
iajs-828	51	26	)	)	PUNCT
iajs-828	51	27	=	=	SYM
iajs-828	51	28	aσ(xy	aσ(xy	PROPN
iajs-828	51	29	)	)	PUNCT
iajs-828	51	30	=	=	SYM
iajs-828	51	31	aσ(x)σ(y	aσ(x)σ(y	NOUN
iajs-828	51	32	)	)	PUNCT
iajs-828	51	33	=	=	SYM
iajs-828	51	34	aσ(x)σ(y	aσ(x)σ(y	ADJ
iajs-828	51	35	)	)	PUNCT
iajs-828	52	1	+	+	CCONJ
iajs-828	52	2	(x	(x	NOUN
iajs-828	52	3	)	)	PUNCT
iajs-828	52	4	bσ(y	bσ(y	NUM
iajs-828	52	5	)	)	PUNCT
iajs-828	53	1	=	=	NOUN
iajs-828	53	2	d(x)σ(y	d(x)σ(y	NOUN
iajs-828	53	3	)	)	PUNCT
iajs-828	54	1	+	+	NUM
iajs-828	54	2	(x)g(y	(x)g(y	PRON
iajs-828	54	3	)	)	PUNCT
iajs-828	54	4	ibn	ibn	PROPN
iajs-828	54	5	alhaitham	alhaitham	NOUN
iajs-828	54	6	j.	j.	PROPN
iajs-828	54	7	for	for	ADP
iajs-828	54	8	pure	pure	ADJ
iajs-828	54	9	&	&	CCONJ
iajs-828	54	10	appl	appl	PROPN
iajs-828	54	11	.	.	PUNCT
iajs-828	55	1	sci	sci	PROPN
iajs-828	55	2	.	.	PUNCT
iajs-828	55	3	vol.24	vol.24	NOUN
iajs-828	55	4	(	(	PUNCT
iajs-828	55	5	3	3	NUM
iajs-828	55	6	)	)	PUNCT
iajs-828	55	7	2011	2011	NUM
iajs-828	55	8	also	also	ADV
iajs-828	55	9	:	:	PUNCT
iajs-828	55	10	g(xy	g(xy	X
iajs-828	55	11	)	)	PUNCT
iajs-828	55	12	=	=	SYM
iajs-828	55	13	bσ(xy	bσ(xy	NOUN
iajs-828	55	14	)	)	PUNCT
iajs-828	55	15	=	=	SYM
iajs-828	55	16	bσ(x)σ(y	bσ(x)σ(y	NOUN
iajs-828	55	17	)	)	PUNCT
iajs-828	55	18	=	=	SYM
iajs-828	55	19	bσ(x)σ(y	bσ(x)σ(y	NOUN
iajs-828	55	20	)	)	PUNCT
iajs-828	55	21	+	+	NUM
iajs-828	55	22	(x	(x	NOUN
iajs-828	55	23	)	)	PUNCT
iajs-828	55	24	aσ(y	aσ(y	NOUN
iajs-828	55	25	)	)	PUNCT
iajs-828	55	26	=	=	SYM
iajs-828	55	27	g(x)σ(y	g(x)σ(y	NOUN
iajs-828	55	28	)	)	PUNCT
iajs-828	55	29	+	+	SYM
iajs-828	55	30	(x)d(y	(x)d(y	NUM
iajs-828	55	31	)	)	PUNCT
iajs-828	55	32	hence	hence	ADV
iajs-828	55	33	(	(	PUNCT
iajs-828	55	34	d	d	X
iajs-828	55	35	,	,	PUNCT
iajs-828	55	36	g	g	NOUN
iajs-828	55	37	)	)	PUNCT
iajs-828	55	38	is	be	AUX
iajs-828	55	39	a	a	DET
iajs-828	55	40	(	(	PUNCT
iajs-828	55	41	,)-s	,)-s	ADJ
iajs-828	55	42	-	-	ADJ
iajs-828	55	43	derivation	derivation	ADJ
iajs-828	55	44	pair	pair	NOUN
iajs-828	55	45	.	.	PUNCT
iajs-828	56	1	remark	remark	VERB
iajs-828	56	2	2.3	2.3	NUM
iajs-828	56	3	every	every	DET
iajs-828	56	4	(	(	PUNCT
iajs-828	56	5	,)-s	,)-s	ADJ
iajs-828	56	6	-	-	ADJ
iajs-828	56	7	derivation	derivation	NOUN
iajs-828	56	8	pair	pair	NOUN
iajs-828	56	9	is	be	AUX
iajs-828	56	10	a	a	DET
iajs-828	56	11	jordan	jordan	PROPN
iajs-828	56	12	(	(	PUNCT
iajs-828	56	13	,)-s	,)-s	ADJ
iajs-828	56	14	-	-	ADJ
iajs-828	56	15	derivation	derivation	NOUN
iajs-828	56	16	pair	pair	NOUN
iajs-828	56	17	,	,	PUNCT
iajs-828	56	18	but	but	CCONJ
iajs-828	56	19	the	the	DET
iajs-828	56	20	converse	converse	NOUN
iajs-828	56	21	is	be	AUX
iajs-828	56	22	in	in	ADP
iajs-828	56	23	general	general	ADJ
iajs-828	56	24	not	not	PART
iajs-828	56	25	true	true	ADJ
iajs-828	56	26	.	.	PUNCT
iajs-828	57	1	the	the	DET
iajs-828	57	2	following	follow	VERB
iajs-828	57	3	example	example	NOUN
iajs-828	57	4	illustrates	illustrate	VERB
iajs-828	57	5	this	this	PRON
iajs-828	57	6	:	:	PUNCT
iajs-828	57	7	example	example	NOUN
iajs-828	57	8	2.4	2.4	NUM
iajs-828	57	9	let	let	VERB
iajs-828	57	10	r	r	PRON
iajs-828	57	11	be	be	AUX
iajs-828	57	12	a	a	DET
iajs-828	57	13	2	2	NUM
iajs-828	57	14	-	-	PUNCT
iajs-828	57	15	torsion	torsion	NOUN
iajs-828	57	16	free	free	ADJ
iajs-828	57	17	non	non	ADJ
iajs-828	57	18	commutative	commutative	ADJ
iajs-828	57	19	ring	ring	NOUN
iajs-828	57	20	,	,	PUNCT
iajs-828	57	21	and	and	CCONJ
iajs-828	57	22	let	let	VERB
iajs-828	57	23	a	a	DET
iajs-828	57	24			NOUN
iajs-828	57	25	r	r	NOUN
iajs-828	57	26	,	,	PUNCT
iajs-828	57	27	such	such	ADJ
iajs-828	57	28	that	that	DET
iajs-828	57	29	(x	(x	NOUN
iajs-828	57	30	)	)	PUNCT
iajs-828	57	31	a	a	DET
iajs-828	57	32	(x	(x	PROPN
iajs-828	57	33	)	)	PUNCT
iajs-828	57	34	=	=	SYM
iajs-828	57	35	0	0	NUM
iajs-828	57	36	,	,	PUNCT
iajs-828	57	37	for	for	ADP
iajs-828	57	38	all	all	DET
iajs-828	57	39	x	x	ADJ
iajs-828	57	40			NOUN
iajs-828	57	41	r	r	NOUN
iajs-828	57	42	,	,	PUNCT
iajs-828	57	43	but(x	but(x	PROPN
iajs-828	57	44	)	)	PUNCT
iajs-828	57	45	a	a	DET
iajs-828	57	46	(y	(y	NOUN
iajs-828	57	47	)	)	PUNCT
iajs-828	57	48	≠	≠	PROPN
iajs-828	57	49	0	0	NUM
iajs-828	57	50	,	,	PUNCT
iajs-828	57	51	for	for	ADP
iajs-828	57	52	some	some	DET
iajs-828	57	53	(	(	PUNCT
iajs-828	57	54	x	x	PROPN
iajs-828	57	55	≠	≠	PROPN
iajs-828	57	56	y	y	PROPN
iajs-828	57	57	)	)	PUNCT
iajs-828	57	58			PROPN
iajs-828	57	59	r.	r.	PROPN
iajs-828	57	60	define	define	VERB
iajs-828	57	61	an	an	DET
iajs-828	57	62	additive	additive	ADJ
iajs-828	57	63	pair	pair	NOUN
iajs-828	57	64	d	d	NOUN
iajs-828	57	65	,	,	PUNCT
iajs-828	57	66	g	g	NOUN
iajs-828	57	67	:	:	PUNCT
iajs-828	58	1	r	r	NOUN
iajs-828	58	2			NOUN
iajs-828	58	3	r	r	NOUN
iajs-828	58	4	,	,	PUNCT
iajs-828	58	5	as	as	SCONJ
iajs-828	58	6	follows	follow	VERB
iajs-828	58	7	:	:	PUNCT
iajs-828	58	8	d(x	d(x	NOUN
iajs-828	58	9	)	)	PUNCT
iajs-828	58	10	=	=	SYM
iajs-828	58	11	(x	(x	X
iajs-828	58	12	)	)	PUNCT
iajs-828	59	1	a	a	DET
iajs-828	59	2	+	+	NOUN
iajs-828	59	3	a	a	DET
iajs-828	59	4	(x	(x	PROPN
iajs-828	59	5	)	)	PUNCT
iajs-828	59	6	,	,	PUNCT
iajs-828	59	7	g(x	g(x	NOUN
iajs-828	59	8	)	)	PUNCT
iajs-828	59	9	=	=	SYM
iajs-828	59	10	(x)aa(x	(x)aa(x	NOUN
iajs-828	59	11	)	)	PUNCT
iajs-828	59	12	,	,	PUNCT
iajs-828	59	13	for	for	ADP
iajs-828	59	14	all	all	DET
iajs-828	59	15	x	x	PRON
iajs-828	59	16			PROPN
iajs-828	59	17	r.	r.	NOUN
iajs-828	59	18	where	where	SCONJ
iajs-828	59	19	,	,	NOUN
iajs-828	59	20	:	:	PUNCT
iajs-828	59	21	r	r	NOUN
iajs-828	59	22			NOUN
iajs-828	59	23	r	r	NOUN
iajs-828	59	24	are	be	AUX
iajs-828	59	25	two	two	NUM
iajs-828	59	26	endomorphism	endomorphism	NOUN
iajs-828	59	27	mappings	mapping	NOUN
iajs-828	59	28	.	.	PUNCT
iajs-828	60	1	then	then	ADV
iajs-828	60	2	(	(	PUNCT
iajs-828	60	3	d	d	X
iajs-828	60	4	,	,	PUNCT
iajs-828	60	5	g	g	NOUN
iajs-828	60	6	)	)	PUNCT
iajs-828	60	7	is	be	AUX
iajs-828	60	8	a	a	DET
iajs-828	60	9	jordan	jordan	PROPN
iajs-828	60	10	(	(	PUNCT
iajs-828	60	11	,)-s	,)-s	ADJ
iajs-828	60	12	-	-	ADJ
iajs-828	60	13	derivation	derivation	NOUN
iajs-828	60	14	pair	pair	NOUN
iajs-828	60	15	,	,	PUNCT
iajs-828	60	16	but	but	CCONJ
iajs-828	60	17	not	not	PART
iajs-828	60	18	a	a	DET
iajs-828	60	19	(	(	PUNCT
iajs-828	60	20	,)-s	,)-s	ADJ
iajs-828	60	21	-	-	ADJ
iajs-828	60	22	derivation	derivation	ADJ
iajs-828	60	23	pair	pair	NOUN
iajs-828	60	24	.	.	PUNCT
iajs-828	61	1	let	let	VERB
iajs-828	61	2	x	x	PRON
iajs-828	61	3	,	,	PUNCT
iajs-828	61	4	y	y	PROPN
iajs-828	61	5			PROPN
iajs-828	61	6	r	r	NOUN
iajs-828	61	7	,	,	PUNCT
iajs-828	61	8	so	so	ADV
iajs-828	61	9	:	:	PUNCT
iajs-828	61	10	d(x	d(x	PROPN
iajs-828	61	11	2)=(x2	2)=(x2	NOUN
iajs-828	61	12	)	)	PUNCT
iajs-828	61	13	a	a	DET
iajs-828	61	14	+	+	NOUN
iajs-828	61	15	aσ(x2	aσ(x2	NOUN
iajs-828	61	16	)	)	PUNCT
iajs-828	61	17	d(x)σ(x	d(x)σ(x	PROPN
iajs-828	61	18	)	)	PUNCT
iajs-828	61	19	+	+	CCONJ
iajs-828	61	20	(x)g(x	(x)g(x	X
iajs-828	61	21	)	)	PUNCT
iajs-828	62	1	=	=	SYM
iajs-828	62	2	(	(	PUNCT
iajs-828	62	3	(x)a	(x)a	X
iajs-828	62	4	+	+	NOUN
iajs-828	62	5	aσ(x	aσ(x	NUM
iajs-828	62	6	)	)	PUNCT
iajs-828	62	7	)	)	PUNCT
iajs-828	63	1	σ(x	σ(x	X
iajs-828	63	2	)	)	PUNCT
iajs-828	63	3	+	+	CCONJ
iajs-828	63	4	(x)((x)a	(x)((x)a	NOUN
iajs-828	63	5	–	–	PUNCT
iajs-828	63	6	aσ(x	aσ(x	NUM
iajs-828	63	7	)	)	PUNCT
iajs-828	63	8	)	)	PUNCT
iajs-828	64	1	=	=	X
iajs-828	64	2	(x)aσ(x	(x)aσ(x	X
iajs-828	64	3	)	)	PUNCT
iajs-828	65	1	+	+	CCONJ
iajs-828	65	2	aσ(x)σ(x	aσ(x)σ(x	X
iajs-828	65	3	)	)	PUNCT
iajs-828	66	1	+	+	CCONJ
iajs-828	66	2	(x)(x)a	(x)(x)a	PROPN
iajs-828	66	3	(x)aσ(x	(x)aσ(x	PROPN
iajs-828	66	4	)	)	PUNCT
iajs-828	67	1	=	=	PUNCT
iajs-828	67	2	(x2)a	(x2)a	PROPN
iajs-828	67	3	+	+	SYM
iajs-828	67	4	aσ(x2	aσ(x2	NOUN
iajs-828	67	5	)	)	PUNCT
iajs-828	67	6	hence	hence	ADV
iajs-828	67	7	d(x2	d(x2	NOUN
iajs-828	67	8	)	)	PUNCT
iajs-828	67	9	=	=	SYM
iajs-828	67	10	d(x)σ(x	d(x)σ(x	X
iajs-828	67	11	)	)	PUNCT
iajs-828	67	12	+	+	NUM
iajs-828	67	13	(x	(x	X
iajs-828	67	14	)	)	PUNCT
iajs-828	67	15	g(x	g(x	NOUN
iajs-828	67	16	)	)	PUNCT
iajs-828	67	17	also	also	ADV
iajs-828	67	18	:	:	PUNCT
iajs-828	67	19	g(x	g(x	NOUN
iajs-828	67	20	2	2	X
iajs-828	67	21	)	)	PUNCT
iajs-828	67	22	=	=	SYM
iajs-828	67	23	(x2)a	(x2)a	PROPN
iajs-828	67	24	–	–	PUNCT
iajs-828	67	25	aσ(x2	aσ(x2	NOUN
iajs-828	67	26	)	)	PUNCT
iajs-828	67	27	=	=	SYM
iajs-828	67	28	g(x)σ(x	g(x)σ(x	NOUN
iajs-828	67	29	)	)	PUNCT
iajs-828	67	30	+	+	CCONJ
iajs-828	67	31	(x)d(x	(x)d(x	X
iajs-828	67	32	)	)	PUNCT
iajs-828	67	33	thus	thus	ADV
iajs-828	67	34	,	,	PUNCT
iajs-828	67	35	(	(	PUNCT
iajs-828	67	36	d	d	X
iajs-828	67	37	,	,	PUNCT
iajs-828	67	38	g	g	NOUN
iajs-828	67	39	)	)	PUNCT
iajs-828	67	40	is	be	AUX
iajs-828	67	41	jordan	jordan	PROPN
iajs-828	67	42	(	(	PUNCT
iajs-828	67	43	,)-s	,)-s	ADJ
iajs-828	67	44	-	-	ADJ
iajs-828	67	45	derivation	derivation	ADJ
iajs-828	67	46	pair	pair	NOUN
iajs-828	67	47	.	.	PUNCT
iajs-828	68	1	now	now	ADV
iajs-828	68	2	,	,	PUNCT
iajs-828	68	3	we	we	PRON
iajs-828	68	4	show	show	VERB
iajs-828	68	5	that	that	SCONJ
iajs-828	68	6	(	(	PUNCT
iajs-828	68	7	d	d	X
iajs-828	68	8	,	,	PUNCT
iajs-828	68	9	g	g	NOUN
iajs-828	68	10	)	)	PUNCT
iajs-828	68	11	is	be	AUX
iajs-828	68	12	not	not	PART
iajs-828	68	13	(	(	PUNCT
iajs-828	68	14	,)-s	,)-s	ADJ
iajs-828	68	15	-	-	ADJ
iajs-828	68	16	derivation	derivation	ADJ
iajs-828	68	17	pair	pair	NOUN
iajs-828	68	18	.	.	PUNCT
iajs-828	69	1	d(xy	d(xy	NUM
iajs-828	69	2	)	)	PUNCT
iajs-828	69	3	=	=	SYM
iajs-828	70	1	(xy)a	(xy)a	PROPN
iajs-828	70	2	+	+	CCONJ
iajs-828	70	3	aσ(xy	aσ(xy	ADJ
iajs-828	70	4	)	)	PUNCT
iajs-828	70	5	d(x)σ(y	d(x)σ(y	NOUN
iajs-828	70	6	)	)	PUNCT
iajs-828	71	1	+	+	NUM
iajs-828	71	2	(x)g(y	(x)g(y	NOUN
iajs-828	71	3	)	)	PUNCT
iajs-828	71	4	=	=	PUNCT
iajs-828	72	1	x)a	x)a	PROPN
iajs-828	72	2	+	+	CCONJ
iajs-828	72	3	aσ(x	aσ(x	NUM
iajs-828	72	4	)	)	PUNCT
iajs-828	72	5	)	)	PUNCT
iajs-828	72	6	σ(y	σ(y	NOUN
iajs-828	72	7	)	)	PUNCT
iajs-828	73	1	+	+	CCONJ
iajs-828	73	2	(x)((y)a	(x)((y)a	NOUN
iajs-828	73	3	–	–	PUNCT
iajs-828	73	4	a	a	DET
iajs-828	73	5	σ(y	σ(y	NOUN
iajs-828	73	6	)	)	PUNCT
iajs-828	73	7	)	)	PUNCT
iajs-828	74	1	=	=	NOUN
iajs-828	74	2	(x)aσ(y	(x)aσ(y	NUM
iajs-828	74	3	)	)	PUNCT
iajs-828	74	4	+	+	CCONJ
iajs-828	74	5	aσ(x)σ(y	aσ(x)σ(y	ADJ
iajs-828	74	6	)	)	PUNCT
iajs-828	75	1	+	+	NUM
iajs-828	75	2	(x)(y)a	(x)(y)a	NOUN
iajs-828	75	3	(x)aσ(y	(x)aσ(y	NUM
iajs-828	75	4	)	)	PUNCT
iajs-828	76	1	=	=	SYM
iajs-828	76	2	(xy)a	(xy)a	PROPN
iajs-828	76	3	+	+	CCONJ
iajs-828	76	4	aσ(xy	aσ(xy	ADJ
iajs-828	76	5	)	)	PUNCT
iajs-828	76	6	hence	hence	ADV
iajs-828	76	7	d(xy)=d(x	d(xy)=d(x	VERB
iajs-828	76	8	)	)	PUNCT
iajs-828	76	9	σ(y	σ(y	NOUN
iajs-828	76	10	)	)	PUNCT
iajs-828	77	1	+	+	CCONJ
iajs-828	77	2	(x)g(y	(x)g(y	NOUN
iajs-828	77	3	)	)	PUNCT
iajs-828	78	1	but	but	CCONJ
iajs-828	78	2	:	:	PUNCT
iajs-828	78	3	g(xy)=	g(xy)=	NOUN
iajs-828	78	4	g(x)σ(y	g(x)σ(y	NOUN
iajs-828	78	5	)	)	PUNCT
iajs-828	79	1	+	+	SYM
iajs-828	79	2	(x)d(y	(x)d(y	NOUN
iajs-828	79	3	)	)	PUNCT
iajs-828	79	4	=(	=(	NOUN
iajs-828	79	5	(x)a	(x)a	PROPN
iajs-828	79	6	–	–	PUNCT
iajs-828	79	7	aσ(x))σ(y)+	aσ(x))σ(y)+	VERB
iajs-828	79	8	(x)((y)a	(x)((y)a	NOUN
iajs-828	79	9	+	+	CCONJ
iajs-828	79	10	aσ(y	aσ(y	NUM
iajs-828	79	11	)	)	PUNCT
iajs-828	79	12	)	)	PUNCT
iajs-828	80	1	=	=	SYM
iajs-828	80	2	(x)aσ(y	(x)aσ(y	NUM
iajs-828	80	3	)	)	PUNCT
iajs-828	80	4	–	–	PUNCT
iajs-828	80	5	aσ(x)σ(y	aσ(x)σ(y	ADJ
iajs-828	80	6	)	)	PUNCT
iajs-828	81	1	+	+	NUM
iajs-828	81	2	(x)(y)a	(x)(y)a	NOUN
iajs-828	81	3	+	+	CCONJ
iajs-828	81	4	(x)aσ(y	(x)aσ(y	NUM
iajs-828	81	5	)	)	PUNCT
iajs-828	81	6	=	=	SYM
iajs-828	81	7	(xy)a	(xy)a	PROPN
iajs-828	81	8	–	–	PUNCT
iajs-828	81	9	aσ(xy	aσ(xy	NOUN
iajs-828	81	10	)	)	PUNCT
iajs-828	81	11	+2x)aσ(y	+2x)aσ(y	NOUN
iajs-828	81	12	)	)	PUNCT
iajs-828	81	13	on	on	ADP
iajs-828	81	14	the	the	DET
iajs-828	81	15	other	other	ADJ
iajs-828	81	16	hand	hand	NOUN
iajs-828	81	17	:	:	PUNCT
iajs-828	81	18	g(xy)=(xy)a	g(xy)=(xy)a	PROPN
iajs-828	81	19	aσ(xy	aσ(xy	PROPN
iajs-828	81	20	)	)	PUNCT
iajs-828	81	21	since	since	SCONJ
iajs-828	81	22	(x)aσ(y)≠0	(x)aσ(y)≠0	NOUN
iajs-828	81	23	,	,	PUNCT
iajs-828	81	24	for	for	ADP
iajs-828	81	25	some	some	DET
iajs-828	81	26	x≠y	x≠y	ADJ
iajs-828	81	27	r	r	NOUN
iajs-828	81	28	,	,	PUNCT
iajs-828	81	29	the	the	DET
iajs-828	81	30	two	two	NUM
iajs-828	81	31	expressions	expression	NOUN
iajs-828	81	32	are	be	AUX
iajs-828	81	33	not	not	PART
iajs-828	81	34	equal	equal	ADJ
iajs-828	81	35	,	,	PUNCT
iajs-828	81	36	hence	hence	ADV
iajs-828	81	37	we	we	PRON
iajs-828	81	38	get	get	VERB
iajs-828	81	39	(	(	PUNCT
iajs-828	81	40	d	d	NOUN
iajs-828	81	41	,	,	PUNCT
iajs-828	81	42	g	g	NOUN
iajs-828	81	43	)	)	PUNCT
iajs-828	81	44	is	be	AUX
iajs-828	81	45	not	not	PART
iajs-828	81	46	(	(	PUNCT
iajs-828	81	47	,)-s	,)-s	ADJ
iajs-828	81	48	-	-	ADJ
iajs-828	81	49	derivation	derivation	ADJ
iajs-828	81	50	pair	pair	NOUN
iajs-828	81	51	.	.	PUNCT
iajs-828	82	1	ibn	ibn	PROPN
iajs-828	82	2	alhaitham	alhaitham	PROPN
iajs-828	82	3	j.	j.	PROPN
iajs-828	82	4	for	for	ADP
iajs-828	82	5	pure	pure	ADJ
iajs-828	82	6	&	&	CCONJ
iajs-828	82	7	appl	appl	PROPN
iajs-828	82	8	.	.	PUNCT
iajs-828	83	1	sci	sci	PROPN
iajs-828	83	2	.	.	PUNCT
iajs-828	83	3	vol.24	vol.24	NOUN
iajs-828	83	4	(	(	PUNCT
iajs-828	83	5	3	3	NUM
iajs-828	83	6	)	)	PUNCT
iajs-828	83	7	2011	2011	NUM
iajs-828	83	8	proposition	proposition	NOUN
iajs-828	83	9	2.5	2.5	NUM
iajs-828	83	10	let	let	VERB
iajs-828	83	11	r	r	PRON
iajs-828	83	12	be	be	AUX
iajs-828	83	13	a	a	DET
iajs-828	83	14	semiprime	semiprime	NOUN
iajs-828	83	15	ring	ring	NOUN
iajs-828	83	16	.	.	PUNCT
iajs-828	84	1	suppose	suppose	VERB
iajs-828	84	2	that	that	SCONJ
iajs-828	84	3	,	,	PROPN
iajs-828	84	4	are	be	AUX
iajs-828	84	5	automorphisms	automorphism	NOUN
iajs-828	84	6	of	of	ADP
iajs-828	84	7	r.	r.	PROPN
iajs-828	84	8	if	if	SCONJ
iajs-828	84	9	r	r	NOUN
iajs-828	84	10	admits	admit	VERB
iajs-828	84	11	a	a	DET
iajs-828	84	12	(	(	PUNCT
iajs-828	84	13	,)-sderivation	,)-sderivation	NOUN
iajs-828	84	14	pair	pair	NOUN
iajs-828	84	15	(	(	PUNCT
iajs-828	84	16	d	d	NOUN
iajs-828	84	17	,	,	PUNCT
iajs-828	84	18	g	g	NOUN
iajs-828	84	19	)	)	PUNCT
iajs-828	84	20	,	,	PUNCT
iajs-828	84	21	such	such	ADJ
iajs-828	84	22	that	that	PRON
iajs-828	84	23	d(x	d(x	NOUN
iajs-828	84	24	)	)	PUNCT
iajs-828	84	25	g(y)=0	g(y)=0	NOUN
iajs-828	84	26	(	(	PUNCT
iajs-828	84	27	resp	resp	NOUN
iajs-828	84	28	.	.	PUNCT
iajs-828	85	1	g(x	g(x	NOUN
iajs-828	85	2	)	)	PUNCT
iajs-828	85	3	d(y	d(y	NOUN
iajs-828	85	4	)	)	PUNCT
iajs-828	86	1	=	=	NOUN
iajs-828	86	2	0	0	NUM
iajs-828	86	3	)	)	PUNCT
iajs-828	86	4	,	,	PUNCT
iajs-828	86	5	for	for	ADP
iajs-828	86	6	all	all	DET
iajs-828	86	7	x	x	NOUN
iajs-828	86	8	,	,	PUNCT
iajs-828	86	9	y	y	PROPN
iajs-828	86	10			PROPN
iajs-828	86	11	r	r	NOUN
iajs-828	86	12	,	,	PUNCT
iajs-828	86	13	then	then	ADV
iajs-828	86	14	d=0	d=0	PROPN
iajs-828	86	15	(	(	PUNCT
iajs-828	86	16	resp	resp	NOUN
iajs-828	86	17	.	.	PUNCT
iajs-828	87	1	g=0	g=0	PUNCT
iajs-828	87	2	)	)	PUNCT
iajs-828	87	3	.	.	PUNCT
iajs-828	88	1	proof	proof	NOUN
iajs-828	88	2	we	we	PRON
iajs-828	88	3	have	have	VERB
iajs-828	88	4	d(x)g(y)=0	d(x)g(y)=0	VERB
iajs-828	88	5	,	,	PUNCT
iajs-828	88	6	for	for	ADP
iajs-828	88	7	all	all	DET
iajs-828	88	8	x	x	NOUN
iajs-828	88	9	,	,	PUNCT
iajs-828	88	10	y	y	PROPN
iajs-828	88	11			PROPN
iajs-828	88	12	r_____(1	r_____(1	PROPN
iajs-828	88	13	)	)	PUNCT
iajs-828	88	14	replacing	replace	VERB
iajs-828	88	15	yx	yx	NOUN
iajs-828	88	16	for	for	ADP
iajs-828	88	17	y	y	PROPN
iajs-828	88	18	in	in	ADP
iajs-828	88	19	(	(	PUNCT
iajs-828	88	20	1	1	NUM
iajs-828	88	21	)	)	PUNCT
iajs-828	88	22	and	and	CCONJ
iajs-828	88	23	using	use	VERB
iajs-828	88	24	(	(	PUNCT
iajs-828	88	25	1	1	NUM
iajs-828	88	26	)	)	PUNCT
iajs-828	88	27	,	,	PUNCT
iajs-828	88	28	we	we	PRON
iajs-828	88	29	have	have	VERB
iajs-828	88	30	:	:	PUNCT
iajs-828	88	31	d(x)g(yx)=0	d(x)g(yx)=0	ADJ
iajs-828	88	32	,	,	PUNCT
iajs-828	88	33	for	for	ADP
iajs-828	88	34	all	all	DET
iajs-828	88	35	x	x	NOUN
iajs-828	88	36	,	,	PUNCT
iajs-828	88	37	y	y	PROPN
iajs-828	88	38			PROPN
iajs-828	88	39	r.	r.	PROPN
iajs-828	88	40	d(x)(g(y)σ(x	d(x)(g(y)σ(x	PROPN
iajs-828	88	41	)	)	PUNCT
iajs-828	89	1	+	+	CCONJ
iajs-828	89	2	(y)d(x))=0	(y)d(x))=0	ADJ
iajs-828	89	3	,	,	PUNCT
iajs-828	89	4	for	for	ADP
iajs-828	89	5	all	all	DET
iajs-828	89	6	x	x	NOUN
iajs-828	89	7	,	,	PUNCT
iajs-828	89	8	y	y	PROPN
iajs-828	89	9			PROPN
iajs-828	89	10	r.	r.	PROPN
iajs-828	89	11	d(x)g(y)σ(x	d(x)g(y)σ(x	PROPN
iajs-828	89	12	)	)	PUNCT
iajs-828	89	13	+	+	CCONJ
iajs-828	89	14	d(x)(y)d(x)=0	d(x)(y)d(x)=0	X
iajs-828	89	15	,	,	PUNCT
iajs-828	89	16	for	for	ADP
iajs-828	89	17	all	all	DET
iajs-828	89	18	x	x	NOUN
iajs-828	89	19	,	,	PUNCT
iajs-828	89	20	y	y	PROPN
iajs-828	89	21	r	r	PROPN
iajs-828	89	22	.	.	PUNCT
iajs-828	90	1	d(x)(y)d(x)=0	d(x)(y)d(x)=0	NUM
iajs-828	90	2	,	,	PUNCT
iajs-828	90	3	for	for	ADP
iajs-828	90	4	all	all	DET
iajs-828	90	5	x	x	NOUN
iajs-828	90	6	,	,	PUNCT
iajs-828	90	7	y	y	PROPN
iajs-828	90	8			PROPN
iajs-828	90	9	r._____(2	r._____(2	PROPN
iajs-828	90	10	)	)	PUNCT
iajs-828	90	11	by	by	ADP
iajs-828	90	12	semiprimeness	semiprimeness	NOUN
iajs-828	90	13	of	of	ADP
iajs-828	90	14	r	r	NOUN
iajs-828	90	15	,	,	PUNCT
iajs-828	90	16	(	(	PUNCT
iajs-828	90	17	2	2	X
iajs-828	90	18	)	)	PUNCT
iajs-828	90	19	gives	give	VERB
iajs-828	90	20	:	:	PUNCT
iajs-828	90	21	d(x)=0	d(x)=0	X
iajs-828	90	22	,	,	PUNCT
iajs-828	90	23	for	for	ADP
iajs-828	90	24	all	all	DET
iajs-828	90	25	xr	xr	NOUN
iajs-828	90	26	.	.	PUNCT
iajs-828	91	1	if	if	SCONJ
iajs-828	91	2	we	we	PRON
iajs-828	91	3	have	have	VERB
iajs-828	91	4	g(x)d(y)=0	g(x)d(y)=0	NOUN
iajs-828	91	5	,	,	PUNCT
iajs-828	91	6	for	for	ADP
iajs-828	91	7	all	all	DET
iajs-828	91	8	x	x	NOUN
iajs-828	91	9	,	,	PUNCT
iajs-828	91	10	y	y	PROPN
iajs-828	91	11			PROPN
iajs-828	91	12	r_____(3	r_____(3	VERB
iajs-828	91	13	)	)	PUNCT
iajs-828	91	14	replacing	replace	VERB
iajs-828	91	15	yx	yx	NOUN
iajs-828	91	16	for	for	ADP
iajs-828	91	17	y	y	PROPN
iajs-828	91	18	in	in	ADP
iajs-828	91	19	(	(	PUNCT
iajs-828	91	20	3	3	NUM
iajs-828	91	21	)	)	PUNCT
iajs-828	91	22	and	and	CCONJ
iajs-828	91	23	using	use	VERB
iajs-828	91	24	(	(	PUNCT
iajs-828	91	25	3	3	NUM
iajs-828	91	26	)	)	PUNCT
iajs-828	91	27	,	,	PUNCT
iajs-828	91	28	we	we	PRON
iajs-828	91	29	have	have	AUX
iajs-828	91	30	:	:	PUNCT
iajs-828	91	31	g(x)d(yx)=0	g(x)d(yx)=0	VERB
iajs-828	91	32	,	,	PUNCT
iajs-828	91	33	for	for	ADP
iajs-828	91	34	all	all	DET
iajs-828	91	35	x	x	NOUN
iajs-828	91	36	,	,	PUNCT
iajs-828	91	37	y	y	PROPN
iajs-828	91	38	r.	r.	PROPN
iajs-828	91	39	g(x)(d(y)σ(x	g(x)(d(y)σ(x	PROPN
iajs-828	91	40	)	)	PUNCT
iajs-828	92	1	+	+	CCONJ
iajs-828	92	2	(y)g(x))=0	(y)g(x))=0	ADJ
iajs-828	92	3	,	,	PUNCT
iajs-828	92	4	for	for	ADP
iajs-828	92	5	all	all	DET
iajs-828	92	6	x	x	NOUN
iajs-828	92	7	,	,	PUNCT
iajs-828	92	8	y	y	PROPN
iajs-828	92	9			PROPN
iajs-828	92	10	r.	r.	PROPN
iajs-828	92	11	g(x)d(y)σ(x	g(x)d(y)σ(x	PROPN
iajs-828	92	12	)	)	PUNCT
iajs-828	93	1	+	+	CCONJ
iajs-828	93	2	g(x	g(x	NOUN
iajs-828	93	3	)	)	PUNCT
iajs-828	93	4	(y)g(x)=0	(y)g(x)=0	NUM
iajs-828	93	5	,	,	PUNCT
iajs-828	93	6	for	for	ADP
iajs-828	93	7	all	all	DET
iajs-828	93	8	x	x	NOUN
iajs-828	93	9	,	,	PUNCT
iajs-828	93	10	y	y	PROPN
iajs-828	93	11			PROPN
iajs-828	93	12	r.	r.	PROPN
iajs-828	93	13	g(x)(y)g(x)=0	g(x)(y)g(x)=0	PROPN
iajs-828	93	14	,	,	PUNCT
iajs-828	93	15	for	for	AUX
iajs-828	93	16	all	all	DET
iajs-828	93	17	x	x	NOUN
iajs-828	93	18	,	,	PUNCT
iajs-828	93	19	y	y	PROPN
iajs-828	93	20			PROPN
iajs-828	93	21	r_____(4	r_____(4	ADV
iajs-828	93	22	)	)	PUNCT
iajs-828	93	23	by	by	ADP
iajs-828	93	24	semiprimeness	semiprimeness	NOUN
iajs-828	93	25	of	of	ADP
iajs-828	93	26	r	r	NOUN
iajs-828	93	27	,	,	PUNCT
iajs-828	93	28	(	(	PUNCT
iajs-828	93	29	4	4	X
iajs-828	93	30	)	)	PUNCT
iajs-828	93	31	gives	give	VERB
iajs-828	93	32	:	:	PUNCT
iajs-828	93	33	g(x)=0	g(x)=0	NOUN
iajs-828	93	34	,	,	PUNCT
iajs-828	93	35	for	for	ADP
iajs-828	93	36	all	all	PRON
iajs-828	93	37	x	x	PRON
iajs-828	93	38			PROPN
iajs-828	93	39	r.	r.	PROPN
iajs-828	93	40	proposition	proposition	PROPN
iajs-828	93	41	2.6	2.6	NUM
iajs-828	93	42	let	let	VERB
iajs-828	93	43	r	r	PRON
iajs-828	93	44	be	be	AUX
iajs-828	93	45	a	a	DET
iajs-828	93	46	semiprime	semiprime	NOUN
iajs-828	93	47	ring	ring	NOUN
iajs-828	93	48	.	.	PUNCT
iajs-828	94	1	suppose	suppose	VERB
iajs-828	94	2	that	that	SCONJ
iajs-828	94	3	,	,	PROPN
iajs-828	94	4	are	be	AUX
iajs-828	94	5	automorphisms	automorphism	NOUN
iajs-828	94	6	of	of	ADP
iajs-828	94	7	r.	r.	PROPN
iajs-828	94	8	if	if	SCONJ
iajs-828	94	9	r	r	NOUN
iajs-828	94	10	admits	admit	VERB
iajs-828	94	11	a	a	DET
iajs-828	94	12	(	(	PUNCT
iajs-828	94	13	,)-sderivation	,)-sderivation	NOUN
iajs-828	94	14	pair	pair	NOUN
iajs-828	94	15	(	(	PUNCT
iajs-828	94	16	d	d	NOUN
iajs-828	94	17	,	,	PUNCT
iajs-828	94	18	g	g	NOUN
iajs-828	94	19	)	)	PUNCT
iajs-828	94	20	,	,	PUNCT
iajs-828	94	21	such	such	ADJ
iajs-828	94	22	that	that	SCONJ
iajs-828	94	23	d(x)=	d(x)=	PROPN
iajs-828	94	24	±	±	NOUN
iajs-828	94	25	(x	(x	PROPN
iajs-828	94	26	)	)	PUNCT
iajs-828	94	27	(	(	PUNCT
iajs-828	94	28	resp	resp	NOUN
iajs-828	94	29	.	.	PUNCT
iajs-828	95	1	g(x)=±	g(x)=±	VERB
iajs-828	95	2	(x	(x	NOUN
iajs-828	95	3	)	)	PUNCT
iajs-828	95	4	)	)	PUNCT
iajs-828	95	5	,	,	PUNCT
iajs-828	95	6	for	for	ADP
iajs-828	95	7	all	all	DET
iajs-828	95	8	x	x	ADJ
iajs-828	95	9			NOUN
iajs-828	95	10	r	r	NOUN
iajs-828	95	11	,	,	PUNCT
iajs-828	95	12	then	then	ADV
iajs-828	95	13	g=0	g=0	NOUN
iajs-828	95	14	(	(	PUNCT
iajs-828	95	15	resp	resp	NOUN
iajs-828	95	16	.	.	PUNCT
iajs-828	96	1	d=0	d=0	X
iajs-828	96	2	)	)	PUNCT
iajs-828	96	3	.	.	PUNCT
iajs-828	97	1	proof	proof	NOUN
iajs-828	97	2	we	we	PRON
iajs-828	97	3	have	have	VERB
iajs-828	97	4	d(x)=σ(x	d(x)=σ(x	PROPN
iajs-828	97	5	)	)	PUNCT
iajs-828	97	6	,	,	PUNCT
iajs-828	97	7	for	for	ADP
iajs-828	97	8	all	all	DET
iajs-828	97	9	x	x	ADJ
iajs-828	97	10			PROPN
iajs-828	97	11	r_____(1	r_____(1	PROPN
iajs-828	97	12	)	)	PUNCT
iajs-828	97	13	replacing	replace	VERB
iajs-828	97	14	x	x	PUNCT
iajs-828	97	15	by	by	ADP
iajs-828	97	16	xy	xy	PROPN
iajs-828	97	17	in	in	ADP
iajs-828	97	18	(	(	PUNCT
iajs-828	97	19	1	1	NUM
iajs-828	97	20	)	)	PUNCT
iajs-828	97	21	and	and	CCONJ
iajs-828	97	22	using	use	VERB
iajs-828	97	23	(	(	PUNCT
iajs-828	97	24	1	1	NUM
iajs-828	97	25	)	)	PUNCT
iajs-828	97	26	,	,	PUNCT
iajs-828	97	27	we	we	PRON
iajs-828	97	28	get	get	VERB
iajs-828	97	29	:	:	PUNCT
iajs-828	97	30	d(xy	d(xy	NUM
iajs-828	97	31	)	)	PUNCT
iajs-828	97	32	=	=	SYM
iajs-828	97	33	(xy	(xy	NOUN
iajs-828	97	34	)	)	PUNCT
iajs-828	97	35	,	,	PUNCT
iajs-828	97	36	for	for	ADP
iajs-828	97	37	all	all	DET
iajs-828	97	38	x	x	NOUN
iajs-828	97	39	,	,	PUNCT
iajs-828	97	40	y	y	PROPN
iajs-828	97	41			PROPN
iajs-828	97	42	r.	r.	PROPN
iajs-828	97	43	d(x)σ(y	d(x)σ(y	NOUN
iajs-828	97	44	)	)	PUNCT
iajs-828	97	45	+	+	CCONJ
iajs-828	97	46	(x)g(y)=σ(xy	(x)g(y)=σ(xy	NOUN
iajs-828	97	47	)	)	PUNCT
iajs-828	97	48	,	,	PUNCT
iajs-828	97	49	for	for	ADP
iajs-828	97	50	all	all	DET
iajs-828	97	51	x	x	NOUN
iajs-828	97	52	,	,	PUNCT
iajs-828	97	53	y	y	PROPN
iajs-828	97	54			PROPN
iajs-828	97	55	r.	r.	PROPN
iajs-828	97	56	σ(x)σ(y)+(x)g(y)=	σ(x)σ(y)+(x)g(y)=	PROPN
iajs-828	97	57	σ(x)σ(y	σ(x)σ(y	NUM
iajs-828	97	58	)	)	PUNCT
iajs-828	97	59	,	,	PUNCT
iajs-828	97	60	for	for	ADP
iajs-828	97	61	all	all	DET
iajs-828	97	62	x	x	NOUN
iajs-828	97	63	,	,	PUNCT
iajs-828	97	64	y	y	PROPN
iajs-828	97	65			PROPN
iajs-828	97	66	r.	r.	PROPN
iajs-828	97	67	(x)g(y)=0	(x)g(y)=0	PROPN
iajs-828	97	68	,	,	PUNCT
iajs-828	97	69	for	for	ADP
iajs-828	97	70	all	all	DET
iajs-828	97	71	x	x	NOUN
iajs-828	97	72	,	,	PUNCT
iajs-828	97	73	y	y	PROPN
iajs-828	97	74			NOUN
iajs-828	97	75	r	r	NOUN
iajs-828	97	76	_	_	PUNCT
iajs-828	98	1	_	_	PUNCT
iajs-828	99	1	_	_	PUNCT
iajs-828	100	1	_	_	PUNCT
iajs-828	100	2	(	(	PUNCT
iajs-828	100	3	2	2	X
iajs-828	100	4	)	)	PUNCT
iajs-828	100	5	left	leave	VERB
iajs-828	100	6	multiplication	multiplication	NOUN
iajs-828	100	7	of	of	ADP
iajs-828	100	8	(	(	PUNCT
iajs-828	100	9	2	2	NUM
iajs-828	100	10	)	)	PUNCT
iajs-828	100	11	by	by	ADP
iajs-828	100	12	g(y	g(y	NOUN
iajs-828	100	13	)	)	PUNCT
iajs-828	100	14	,	,	PUNCT
iajs-828	100	15	leads	lead	VERB
iajs-828	100	16	to	to	ADP
iajs-828	100	17	:	:	PUNCT
iajs-828	100	18	g(y)(x)g(y)=0	g(y)(x)g(y)=0	NUM
iajs-828	100	19	,	,	PUNCT
iajs-828	100	20	for	for	ADP
iajs-828	100	21	all	all	DET
iajs-828	100	22	x	x	NOUN
iajs-828	100	23	,	,	PUNCT
iajs-828	100	24	y	y	PROPN
iajs-828	100	25			NOUN
iajs-828	100	26	r	r	NOUN
iajs-828	100	27	_	_	PUNCT
iajs-828	101	1	_	_	PUNCT
iajs-828	102	1	_	_	PUNCT
iajs-828	103	1	_	_	PUNCT
iajs-828	104	1	_	_	PUNCT
iajs-828	104	2	(	(	PUNCT
iajs-828	104	3	3	3	NUM
iajs-828	104	4	)	)	PUNCT
iajs-828	104	5	by	by	ADP
iajs-828	104	6	semiprimeness	semiprimeness	NOUN
iajs-828	104	7	of	of	ADP
iajs-828	104	8	r	r	NOUN
iajs-828	104	9	,	,	PUNCT
iajs-828	104	10	(	(	PUNCT
iajs-828	104	11	3	3	X
iajs-828	104	12	)	)	PUNCT
iajs-828	104	13	gives	give	VERB
iajs-828	104	14	:	:	PUNCT
iajs-828	104	15	g(y)=0	g(y)=0	NOUN
iajs-828	104	16	,	,	PUNCT
iajs-828	104	17	for	for	ADP
iajs-828	104	18	all	all	DET
iajs-828	104	19	y	y	PROPN
iajs-828	104	20			PROPN
iajs-828	104	21	r.	r.	PROPN
iajs-828	104	22	ibn	ibn	PROPN
iajs-828	104	23	alhaitham	alhaitham	PROPN
iajs-828	104	24	j.	j.	PROPN
iajs-828	104	25	for	for	ADP
iajs-828	104	26	pure	pure	ADJ
iajs-828	104	27	&	&	CCONJ
iajs-828	104	28	appl	appl	PROPN
iajs-828	104	29	.	.	PUNCT
iajs-828	105	1	sci	sci	PROPN
iajs-828	105	2	.	.	PUNCT
iajs-828	105	3	vol.24	vol.24	NOUN
iajs-828	105	4	(	(	PUNCT
iajs-828	105	5	3	3	NUM
iajs-828	105	6	)	)	PUNCT
iajs-828	105	7	2011	2011	NUM
iajs-828	105	8	similarly	similarly	ADV
iajs-828	105	9	,	,	PUNCT
iajs-828	105	10	we	we	PRON
iajs-828	105	11	can	can	AUX
iajs-828	105	12	show	show	VERB
iajs-828	105	13	if	if	SCONJ
iajs-828	105	14	d(x)=-(x	d(x)=-(x	NOUN
iajs-828	105	15	)	)	PUNCT
iajs-828	105	16	,	,	PUNCT
iajs-828	105	17	for	for	ADP
iajs-828	105	18	all	all	DET
iajs-828	105	19	x	x	ADJ
iajs-828	105	20			NOUN
iajs-828	105	21	r	r	NOUN
iajs-828	105	22	,	,	PUNCT
iajs-828	105	23	then	then	ADV
iajs-828	105	24	g=0	g=0	NOUN
iajs-828	105	25	in	in	ADP
iajs-828	105	26	the	the	DET
iajs-828	105	27	same	same	ADJ
iajs-828	105	28	way	way	NOUN
iajs-828	105	29	,	,	PUNCT
iajs-828	105	30	if	if	SCONJ
iajs-828	105	31	g(x)=±	g(x)=±	NOUN
iajs-828	105	32	(x	(x	NOUN
iajs-828	105	33	)	)	PUNCT
iajs-828	105	34	,	,	PUNCT
iajs-828	105	35	for	for	ADP
iajs-828	105	36	all	all	DET
iajs-828	105	37	xr	xr	PROPN
iajs-828	105	38	,	,	PUNCT
iajs-828	105	39	then	then	ADV
iajs-828	105	40	d=0	d=0	PROPN
iajs-828	105	41	.	.	PUNCT
iajs-828	105	42	proposition	proposition	NOUN
iajs-828	105	43	2.7	2.7	NUM
iajs-828	105	44	let	let	VERB
iajs-828	105	45	r	r	NOUN
iajs-828	105	46	be	be	AUX
iajs-828	105	47	any	any	DET
iajs-828	105	48	ring	ring	NOUN
iajs-828	105	49	and	and	CCONJ
iajs-828	105	50	,	,	NOUN
iajs-828	105	51	are	be	AUX
iajs-828	105	52	two	two	NUM
iajs-828	105	53	mappings	mapping	NOUN
iajs-828	105	54	on	on	ADP
iajs-828	105	55	r.	r.	PROPN
iajs-828	105	56	then	then	ADV
iajs-828	105	57	1if	1if	PROPN
iajs-828	106	1	(	(	PUNCT
iajs-828	106	2	d	d	X
iajs-828	106	3	,	,	PUNCT
iajs-828	106	4	g	g	NOUN
iajs-828	106	5	)	)	PUNCT
iajs-828	106	6	is	be	AUX
iajs-828	106	7	a	a	DET
iajs-828	106	8	(	(	PUNCT
iajs-828	106	9	,)-s	,)-s	ADJ
iajs-828	106	10	-	-	ADJ
iajs-828	106	11	derivation	derivation	NOUN
iajs-828	106	12	pair	pair	NOUN
iajs-828	106	13	on	on	ADP
iajs-828	106	14	r	r	NOUN
iajs-828	106	15	,	,	PUNCT
iajs-828	106	16	then	then	ADV
iajs-828	106	17	d+g	d+g	PROPN
iajs-828	106	18	is	be	AUX
iajs-828	106	19	a	a	DET
iajs-828	106	20	(	(	PUNCT
iajs-828	106	21	,)-derivation	,)-derivation	NOUN
iajs-828	106	22	.	.	PUNCT
iajs-828	107	1	2if	2if	NOUN
iajs-828	107	2	(	(	PUNCT
iajs-828	107	3	d	d	NOUN
iajs-828	107	4	,	,	PUNCT
iajs-828	107	5	g	g	NOUN
iajs-828	107	6	)	)	PUNCT
iajs-828	107	7	is	be	AUX
iajs-828	107	8	a	a	DET
iajs-828	107	9	jordan	jordan	PROPN
iajs-828	107	10	(	(	PUNCT
iajs-828	107	11	,)-s	,)-s	ADJ
iajs-828	107	12	-	-	ADJ
iajs-828	107	13	derivation	derivation	NOUN
iajs-828	107	14	pair	pair	NOUN
iajs-828	107	15	on	on	ADP
iajs-828	107	16	r	r	NOUN
iajs-828	107	17	,	,	PUNCT
iajs-828	107	18	then	then	ADV
iajs-828	107	19	d+g	d+g	PROPN
iajs-828	107	20	is	be	AUX
iajs-828	107	21	a	a	DET
iajs-828	107	22	jordan	jordan	PROPN
iajs-828	107	23	(	(	PUNCT
iajs-828	107	24	,)-derivation	,)-derivation	NOUN
iajs-828	107	25	.	.	PUNCT
iajs-828	108	1	proof	proof	NOUN
iajs-828	108	2	1we	1we	ADV
iajs-828	108	3	have	have	AUX
iajs-828	108	4	(	(	PUNCT
iajs-828	108	5	d	d	NOUN
iajs-828	108	6	,	,	PUNCT
iajs-828	108	7	g	g	NOUN
iajs-828	108	8	)	)	PUNCT
iajs-828	108	9	is	be	AUX
iajs-828	108	10	a	a	DET
iajs-828	108	11	(	(	PUNCT
iajs-828	108	12	,)-s	,)-s	ADJ
iajs-828	108	13	-	-	ADJ
iajs-828	108	14	derivation	derivation	ADJ
iajs-828	108	15	pair	pair	NOUN
iajs-828	108	16	,	,	PUNCT
iajs-828	108	17	so	so	ADV
iajs-828	108	18	d(xy)=	d(xy)=	PROPN
iajs-828	108	19	d(x	d(x	NOUN
iajs-828	108	20	)	)	PUNCT
iajs-828	108	21	(y)x)g(y	(y)x)g(y	PROPN
iajs-828	108	22	)	)	PUNCT
iajs-828	108	23	,	,	PUNCT
iajs-828	108	24	for	for	ADP
iajs-828	108	25	all	all	DET
iajs-828	108	26	x	x	NOUN
iajs-828	108	27	,	,	PUNCT
iajs-828	108	28	y	y	PROPN
iajs-828	108	29			NOUN
iajs-828	108	30	r	r	NOUN
iajs-828	108	31	_	_	PUNCT
iajs-828	109	1	_	_	PUNCT
iajs-828	110	1	_	_	PUNCT
iajs-828	111	1	_	_	PUNCT
iajs-828	112	1	_	_	PUNCT
iajs-828	112	2	(	(	PUNCT
iajs-828	112	3	1	1	X
iajs-828	112	4	)	)	PUNCT
iajs-828	112	5	g(xy)=	g(xy)=	NOUN
iajs-828	112	6	g(x	g(x	NOUN
iajs-828	112	7	)	)	PUNCT
iajs-828	112	8	(y)x)d(y	(y)x)d(y	PROPN
iajs-828	112	9	)	)	PUNCT
iajs-828	112	10	,	,	PUNCT
iajs-828	112	11	for	for	ADP
iajs-828	112	12	all	all	DET
iajs-828	112	13	x	x	NOUN
iajs-828	112	14	,	,	PUNCT
iajs-828	112	15	y	y	PROPN
iajs-828	112	16			NOUN
iajs-828	112	17	r	r	NOUN
iajs-828	113	1	_	_	PUNCT
iajs-828	114	1	_	_	PUNCT
iajs-828	115	1	_	_	PUNCT
iajs-828	116	1	_	_	PUNCT
iajs-828	117	1	_	_	PUNCT
iajs-828	117	2	(	(	PUNCT
iajs-828	117	3	2	2	NUM
iajs-828	117	4	)	)	PUNCT
iajs-828	117	5	by	by	ADP
iajs-828	117	6	adding	add	VERB
iajs-828	117	7	(	(	PUNCT
iajs-828	117	8	1	1	NUM
iajs-828	117	9	)	)	PUNCT
iajs-828	117	10	and	and	CCONJ
iajs-828	117	11	(	(	PUNCT
iajs-828	117	12	2	2	NUM
iajs-828	117	13	)	)	PUNCT
iajs-828	117	14	,	,	PUNCT
iajs-828	117	15	we	we	PRON
iajs-828	117	16	get	get	VERB
iajs-828	117	17	(	(	PUNCT
iajs-828	117	18	d+g)(xy)=(d+g)(x)(y)+(x)(d+g)(y	d+g)(xy)=(d+g)(x)(y)+(x)(d+g)(y	ADJ
iajs-828	117	19	)	)	PUNCT
iajs-828	117	20	hence	hence	ADV
iajs-828	117	21	d+g	d+g	PROPN
iajs-828	117	22	is	be	AUX
iajs-828	117	23	a	a	DET
iajs-828	117	24	(	(	PUNCT
iajs-828	117	25	,)-derivation	,)-derivation	NOUN
iajs-828	117	26	2we	2we	NOUN
iajs-828	117	27	have	have	AUX
iajs-828	117	28	(	(	PUNCT
iajs-828	117	29	d	d	NOUN
iajs-828	117	30	,	,	PUNCT
iajs-828	117	31	g	g	NOUN
iajs-828	117	32	)	)	PUNCT
iajs-828	117	33	is	be	AUX
iajs-828	117	34	a	a	DET
iajs-828	117	35	jordan	jordan	PROPN
iajs-828	117	36	(	(	PUNCT
iajs-828	117	37	,)-s	,)-s	ADJ
iajs-828	117	38	-	-	ADJ
iajs-828	117	39	derivation	derivation	ADJ
iajs-828	117	40	pair	pair	NOUN
iajs-828	117	41	,	,	PUNCT
iajs-828	117	42	so	so	SCONJ
iajs-828	118	1	d(x	d(x	PROPN
iajs-828	118	2	2)=d(x)(x	2)=d(x)(x	NUM
iajs-828	118	3	)	)	PUNCT
iajs-828	118	4	+	+	CCONJ
iajs-828	118	5	(x)g(x	(x)g(x	PROPN
iajs-828	118	6	)	)	PUNCT
iajs-828	118	7	,	,	PUNCT
iajs-828	118	8	for	for	ADP
iajs-828	118	9	all	all	DET
iajs-828	118	10	x	x	NOUN
iajs-828	118	11			NOUN
iajs-828	118	12	r	r	NOUN
iajs-828	118	13	_	_	PUNCT
iajs-828	119	1	_	_	PUNCT
iajs-828	120	1	_	_	PUNCT
iajs-828	121	1	_	_	PUNCT
iajs-828	122	1	_	_	PUNCT
iajs-828	122	2	(	(	PUNCT
iajs-828	122	3	3	3	NUM
iajs-828	122	4	)	)	PUNCT
iajs-828	122	5	g(x2)=g(x)(x	g(x2)=g(x)(x	NOUN
iajs-828	122	6	)	)	PUNCT
iajs-828	122	7	+	+	CCONJ
iajs-828	122	8	(x)d(x	(x)d(x	NUM
iajs-828	122	9	)	)	PUNCT
iajs-828	122	10	,	,	PUNCT
iajs-828	122	11	for	for	ADP
iajs-828	122	12	all	all	DET
iajs-828	122	13	x	x	NOUN
iajs-828	122	14			NOUN
iajs-828	122	15	r	r	NOUN
iajs-828	123	1	_	_	PUNCT
iajs-828	124	1	_	_	PUNCT
iajs-828	125	1	_	_	PUNCT
iajs-828	126	1	_	_	PUNCT
iajs-828	127	1	_	_	PUNCT
iajs-828	127	2	(	(	PUNCT
iajs-828	127	3	4	4	NUM
iajs-828	127	4	)	)	PUNCT
iajs-828	127	5	by	by	ADP
iajs-828	127	6	adding	add	VERB
iajs-828	127	7	(	(	PUNCT
iajs-828	127	8	3	3	NUM
iajs-828	127	9	)	)	PUNCT
iajs-828	127	10	and	and	CCONJ
iajs-828	127	11	(	(	PUNCT
iajs-828	127	12	2	2	NUM
iajs-828	127	13	)	)	PUNCT
iajs-828	127	14	,	,	PUNCT
iajs-828	127	15	we	we	PRON
iajs-828	127	16	get	get	VERB
iajs-828	127	17	(	(	PUNCT
iajs-828	127	18	d+g)(x	d+g)(x	NOUN
iajs-828	127	19	2	2	NUM
iajs-828	127	20	)	)	PUNCT
iajs-828	127	21	=(	=(	NOUN
iajs-828	127	22	d+g)(x)(x	d+g)(x)(x	NOUN
iajs-828	127	23	)	)	PUNCT
iajs-828	127	24	+	+	NUM
iajs-828	127	25	(x)(d+g)(x	(x)(d+g)(x	NOUN
iajs-828	127	26	)	)	PUNCT
iajs-828	127	27	,	,	PUNCT
iajs-828	127	28	for	for	ADP
iajs-828	127	29	all	all	DET
iajs-828	127	30	x	x	PRON
iajs-828	127	31			PROPN
iajs-828	127	32	r.	r.	PROPN
iajs-828	127	33	hence	hence	ADV
iajs-828	127	34	d+g	d+g	PROPN
iajs-828	127	35	is	be	AUX
iajs-828	127	36	a	a	DET
iajs-828	127	37	jordan	jordan	PROPN
iajs-828	127	38	(	(	PUNCT
iajs-828	127	39	,)-derivation	,)-derivation	NOUN
iajs-828	127	40	.	.	PUNCT
iajs-828	128	1	§	§	NOUN
iajs-828	128	2	3	3	NUM
iajs-828	128	3	relation	relation	NOUN
iajs-828	128	4	between	between	ADP
iajs-828	128	5	(	(	PUNCT
iajs-828	128	6			PROPN
iajs-828	128	7	,	,	PUNCT
iajs-828	128	8	)-s	)-s	NOUN
iajs-828	128	9	-	-	PUNCT
iajs-828	128	10	derivation	derivation	NOUN
iajs-828	128	11	pairs	pair	NOUN
iajs-828	128	12	and	and	CCONJ
iajs-828	128	13	(	(	PUNCT
iajs-828	128	14			PROPN
iajs-828	128	15	,	,	PUNCT
iajs-828	128	16	)-derivations	)-derivation	NOUN
iajs-828	128	17	in	in	ADP
iajs-828	128	18	this	this	DET
iajs-828	128	19	section	section	NOUN
iajs-828	128	20	,	,	PUNCT
iajs-828	128	21	we	we	PRON
iajs-828	128	22	study	study	VERB
iajs-828	128	23	prime	prime	ADJ
iajs-828	128	24	rings	ring	NOUN
iajs-828	128	25	,	,	PUNCT
iajs-828	128	26	semiprime	semiprime	NOUN
iajs-828	128	27	rings	ring	NOUN
iajs-828	128	28	,	,	PUNCT
iajs-828	128	29	and	and	CCONJ
iajs-828	128	30	rings	ring	NOUN
iajs-828	128	31	that	that	PRON
iajs-828	128	32	have	have	VERB
iajs-828	128	33	a	a	DET
iajs-828	128	34	commutator	commutator	NOUN
iajs-828	128	35	left	leave	VERB
iajs-828	128	36	nonzero	nonzero	PROPN
iajs-828	128	37	divisor	divisor	NOUN
iajs-828	128	38	with	with	ADP
iajs-828	128	39	(	(	PUNCT
iajs-828	128	40	,)-s	,)-s	ADJ
iajs-828	128	41	-	-	ADJ
iajs-828	128	42	derivation	derivation	ADJ
iajs-828	128	43	pair	pair	NOUN
iajs-828	128	44	,	,	PUNCT
iajs-828	128	45	to	to	PART
iajs-828	128	46	obtain	obtain	VERB
iajs-828	128	47	a	a	DET
iajs-828	128	48	(	(	PUNCT
iajs-828	128	49	,)-derivation	,)-derivation	NOUN
iajs-828	128	50	.	.	PUNCT
iajs-828	129	1	theorem	theorem	VERB
iajs-828	129	2	3.1	3.1	NUM
iajs-828	129	3	let	let	VERB
iajs-828	129	4	r	r	PRON
iajs-828	129	5	be	be	AUX
iajs-828	129	6	a	a	DET
iajs-828	129	7	2	2	NUM
iajs-828	129	8	-	-	PUNCT
iajs-828	129	9	torsion	torsion	NOUN
iajs-828	129	10	free	free	ADJ
iajs-828	129	11	semiprime	semiprime	NOUN
iajs-828	129	12	ring	ring	NOUN
iajs-828	129	13	,	,	PUNCT
iajs-828	129	14	and	and	CCONJ
iajs-828	129	15	(	(	PUNCT
iajs-828	129	16	d	d	NOUN
iajs-828	129	17	,	,	PUNCT
iajs-828	129	18	g	g	NOUN
iajs-828	129	19	)	)	PUNCT
iajs-828	129	20	be	be	VERB
iajs-828	129	21	a	a	DET
iajs-828	129	22	(	(	PUNCT
iajs-828	129	23	,)-s	,)-s	ADJ
iajs-828	129	24	-	-	ADJ
iajs-828	129	25	derivation	derivation	NOUN
iajs-828	129	26	pair	pair	NOUN
iajs-828	129	27	on	on	ADP
iajs-828	129	28	r	r	NOUN
iajs-828	129	29	,	,	PUNCT
iajs-828	129	30	then	then	ADV
iajs-828	130	1	d	d	PROPN
iajs-828	130	2	and	and	CCONJ
iajs-828	130	3	g	g	PROPN
iajs-828	130	4	are	be	AUX
iajs-828	130	5	(	(	PUNCT
iajs-828	130	6	,)-derivations	,)-derivation	NOUN
iajs-828	130	7	.	.	PUNCT
iajs-828	131	1	where	where	SCONJ
iajs-828	131	2	,	,	NOUN
iajs-828	131	3	are	be	AUX
iajs-828	131	4	automorphisms	automorphism	NOUN
iajs-828	131	5	of	of	ADP
iajs-828	131	6	r.	r.	PROPN
iajs-828	131	7	proof	proof	NOUN
iajs-828	131	8	suppose	suppose	VERB
iajs-828	131	9	that	that	SCONJ
iajs-828	131	10	(	(	PUNCT
iajs-828	131	11	d	d	X
iajs-828	131	12	,	,	PUNCT
iajs-828	131	13	g	g	NOUN
iajs-828	131	14	)	)	PUNCT
iajs-828	131	15	is	be	AUX
iajs-828	131	16	(	(	PUNCT
iajs-828	131	17	,)-s	,)-s	ADJ
iajs-828	131	18	-	-	ADJ
iajs-828	131	19	derivation	derivation	ADJ
iajs-828	131	20	pair	pair	NOUN
iajs-828	131	21	.	.	PUNCT
iajs-828	132	1	then	then	ADV
iajs-828	132	2	:	:	PUNCT
iajs-828	132	3	d(xyx)=d(x(yx	d(xyx)=d(x(yx	X
iajs-828	132	4	)	)	PUNCT
iajs-828	132	5	)	)	PUNCT
iajs-828	133	1	=	=	SYM
iajs-828	133	2	d(x)yx	d(x)yx	X
iajs-828	133	3	)	)	PUNCT
iajs-828	134	1	+	+	CCONJ
iajs-828	134	2	(x)g(yx	(x)g(yx	X
iajs-828	134	3	)	)	PUNCT
iajs-828	134	4	,	,	PUNCT
iajs-828	134	5	for	for	ADP
iajs-828	134	6	all	all	DET
iajs-828	134	7	x	x	NOUN
iajs-828	134	8	,	,	PUNCT
iajs-828	134	9	y	y	PROPN
iajs-828	134	10			NOUN
iajs-828	134	11	r	r	NOUN
iajs-828	134	12	_	_	PUNCT
iajs-828	135	1	_	_	PUNCT
iajs-828	136	1	_	_	PUNCT
iajs-828	137	1	_	_	PUNCT
iajs-828	138	1	_	_	PUNCT
iajs-828	138	2	(	(	PUNCT
iajs-828	138	3	1	1	NUM
iajs-828	138	4	)	)	PUNCT
iajs-828	138	5	that	that	PRON
iajs-828	138	6	is	be	AUX
iajs-828	138	7	:	:	PUNCT
iajs-828	138	8	d(xyx)=d(x)(yx	d(xyx)=d(x)(yx	X
iajs-828	138	9	)	)	PUNCT
iajs-828	139	1	+	+	CCONJ
iajs-828	139	2	(x)g(y)(x	(x)g(y)(x	NOUN
iajs-828	139	3	)	)	PUNCT
iajs-828	139	4	+	+	NUM
iajs-828	139	5	(x)(y)d(x	(x)(y)d(x	NOUN
iajs-828	139	6	)	)	PUNCT
iajs-828	139	7	,	,	PUNCT
iajs-828	139	8	for	for	ADP
iajs-828	139	9	all	all	DET
iajs-828	139	10	x	x	NOUN
iajs-828	139	11	,	,	PUNCT
iajs-828	139	12	y	y	PROPN
iajs-828	139	13			PROPN
iajs-828	139	14	r____(2	r____(2	NOUN
iajs-828	139	15	)	)	PUNCT
iajs-828	139	16	also	also	ADV
iajs-828	139	17	:	:	PUNCT
iajs-828	139	18	ibn	ibn	PROPN
iajs-828	139	19	alhaitham	alhaitham	NOUN
iajs-828	139	20	j.	j.	PROPN
iajs-828	139	21	for	for	ADP
iajs-828	139	22	pure	pure	ADJ
iajs-828	139	23	&	&	CCONJ
iajs-828	139	24	appl	appl	PROPN
iajs-828	139	25	.	.	PUNCT
iajs-828	140	1	sci	sci	PROPN
iajs-828	140	2	.	.	PUNCT
iajs-828	140	3	vol.24	vol.24	NOUN
iajs-828	140	4	(	(	PUNCT
iajs-828	140	5	3	3	NUM
iajs-828	140	6	)	)	PUNCT
iajs-828	140	7	2011	2011	NUM
iajs-828	140	8	d(xyx)=d((xy)x)=d(xy)(x	d(xyx)=d((xy)x)=d(xy)(x	NOUN
iajs-828	140	9	)	)	PUNCT
iajs-828	140	10	+	+	X
iajs-828	140	11	(xy)g(x	(xy)g(x	ADJ
iajs-828	140	12	)	)	PUNCT
iajs-828	140	13	,	,	PUNCT
iajs-828	140	14	for	for	ADP
iajs-828	140	15	all	all	DET
iajs-828	140	16	x	x	NOUN
iajs-828	140	17	,	,	PUNCT
iajs-828	140	18	y	y	PROPN
iajs-828	140	19			PROPN
iajs-828	140	20	r_____(3	r_____(3	VERB
iajs-828	140	21	)	)	PUNCT
iajs-828	140	22	that	that	PRON
iajs-828	140	23	is	be	AUX
iajs-828	140	24	:	:	PUNCT
iajs-828	140	25	d(xyx)=d(x)(y)(x	d(xyx)=d(x)(y)(x	NOUN
iajs-828	140	26	)	)	PUNCT
iajs-828	140	27	+	+	CCONJ
iajs-828	140	28	(x)g(y)(x	(x)g(y)(x	NOUN
iajs-828	140	29	)	)	PUNCT
iajs-828	141	1	+	+	CCONJ
iajs-828	141	2	(xy)g(x	(xy)g(x	ADJ
iajs-828	141	3	)	)	PUNCT
iajs-828	141	4	,	,	PUNCT
iajs-828	141	5	for	for	ADP
iajs-828	141	6	all	all	DET
iajs-828	141	7	x	x	NOUN
iajs-828	141	8	,	,	PUNCT
iajs-828	141	9	y	y	PROPN
iajs-828	141	10			PROPN
iajs-828	141	11	r___(4	r___(4	PROPN
iajs-828	141	12	)	)	PUNCT
iajs-828	141	13	from	from	ADP
iajs-828	141	14	(	(	PUNCT
iajs-828	141	15	2	2	NUM
iajs-828	141	16	)	)	PUNCT
iajs-828	141	17	and	and	CCONJ
iajs-828	141	18	(	(	PUNCT
iajs-828	141	19	4	4	NUM
iajs-828	141	20	)	)	PUNCT
iajs-828	141	21	,	,	PUNCT
iajs-828	141	22	we	we	PRON
iajs-828	141	23	get	get	VERB
iajs-828	141	24	:	:	PUNCT
iajs-828	141	25	(xy)(d(x)-g(x))=0	(xy)(d(x)-g(x))=0	ADV
iajs-828	141	26	,	,	PUNCT
iajs-828	141	27	for	for	ADP
iajs-828	141	28	all	all	DET
iajs-828	141	29	x	x	NOUN
iajs-828	141	30	,	,	PUNCT
iajs-828	141	31	y	y	PROPN
iajs-828	141	32			NOUN
iajs-828	141	33	r_____(5	r_____(5	NOUN
iajs-828	141	34	)	)	PUNCT
iajs-828	141	35	replace	replace	NOUN
iajs-828	141	36	(y	(y	NOUN
iajs-828	141	37	)	)	PUNCT
iajs-828	141	38	by	by	ADP
iajs-828	141	39	(	(	PUNCT
iajs-828	141	40	d(x)-g(x	d(x)-g(x	NOUN
iajs-828	141	41	)	)	PUNCT
iajs-828	141	42	)	)	PUNCT
iajs-828	141	43	(y	(y	NOUN
iajs-828	141	44	)	)	PUNCT
iajs-828	141	45	(x	(x	PROPN
iajs-828	141	46	)	)	PUNCT
iajs-828	141	47	in	in	ADP
iajs-828	141	48	(	(	PUNCT
iajs-828	141	49	5	5	NUM
iajs-828	141	50	)	)	PUNCT
iajs-828	141	51	,	,	PUNCT
iajs-828	141	52	we	we	PRON
iajs-828	141	53	get	get	VERB
iajs-828	141	54	:	:	PUNCT
iajs-828	141	55	(x)(d(x)-g(x))(y)(x)(d(x)-g(x))=0	(x)(d(x)-g(x))(y)(x)(d(x)-g(x))=0	ADJ
iajs-828	141	56	,	,	PUNCT
iajs-828	141	57	for	for	ADP
iajs-828	141	58	all	all	DET
iajs-828	141	59	x	x	NOUN
iajs-828	141	60	,	,	PUNCT
iajs-828	141	61	y	y	PROPN
iajs-828	141	62			PROPN
iajs-828	141	63	r_____(6	r_____(6	NOUN
iajs-828	141	64	)	)	PUNCT
iajs-828	141	65	since	since	SCONJ
iajs-828	141	66	r	r	NOUN
iajs-828	141	67	is	be	AUX
iajs-828	141	68	semiprime	semiprime	NOUN
iajs-828	141	69	,	,	PUNCT
iajs-828	141	70	we	we	PRON
iajs-828	141	71	get	get	VERB
iajs-828	141	72	:	:	PUNCT
iajs-828	141	73	(x)d(x)=(x)g(x	(x)d(x)=(x)g(x	NUM
iajs-828	141	74	)	)	PUNCT
iajs-828	141	75	,	,	PUNCT
iajs-828	141	76	for	for	ADP
iajs-828	141	77	all	all	DET
iajs-828	141	78	x	x	ADJ
iajs-828	141	79			NOUN
iajs-828	141	80	r_____(7	r_____(7	PROPN
iajs-828	141	81	)	)	PUNCT
iajs-828	142	1	it	it	PRON
iajs-828	142	2	follows	follow	VERB
iajs-828	142	3	that	that	SCONJ
iajs-828	142	4	:	:	PUNCT
iajs-828	142	5	d(x	d(x	PROPN
iajs-828	142	6	2	2	NUM
iajs-828	142	7	)	)	PUNCT
iajs-828	142	8	=	=	NOUN
iajs-828	142	9	d(x)(x)+(x)d(x	d(x)(x)+(x)d(x	NOUN
iajs-828	142	10	)	)	PUNCT
iajs-828	142	11	,	,	PUNCT
iajs-828	142	12	for	for	ADP
iajs-828	142	13	all	all	DET
iajs-828	142	14	x	x	ADJ
iajs-828	142	15			NOUN
iajs-828	142	16	r_____(8	r_____(8	PROPN
iajs-828	142	17	)	)	PUNCT
iajs-828	142	18	and	and	CCONJ
iajs-828	142	19	:	:	PUNCT
iajs-828	142	20	g(x2)=g(x)(x)+(x)g(x	g(x2)=g(x)(x)+(x)g(x	X
iajs-828	142	21	)	)	PUNCT
iajs-828	142	22	,	,	PUNCT
iajs-828	142	23	for	for	ADP
iajs-828	142	24	all	all	DET
iajs-828	142	25	x	x	ADJ
iajs-828	142	26			NOUN
iajs-828	142	27	r_____(9	r_____(9	NOUN
iajs-828	142	28	)	)	PUNCT
iajs-828	142	29	thus	thus	ADV
iajs-828	142	30	,	,	PUNCT
iajs-828	142	31	by	by	ADP
iajs-828	142	32	using	use	VERB
iajs-828	142	33	[	[	X
iajs-828	142	34	3	3	NUM
iajs-828	142	35	,	,	PUNCT
iajs-828	142	36	theorem	theorem	VERB
iajs-828	142	37	2.3.7	2.3.7	NOUN
iajs-828	142	38	]	]	PUNCT
iajs-828	142	39	,	,	PUNCT
iajs-828	142	40	we	we	PRON
iajs-828	142	41	obtain	obtain	VERB
iajs-828	142	42	that	that	PRON
iajs-828	142	43	d	d	PROPN
iajs-828	142	44	and	and	CCONJ
iajs-828	142	45	g	g	PROPN
iajs-828	142	46	are	be	AUX
iajs-828	142	47	(	(	PUNCT
iajs-828	142	48	,)-derivations	,)-derivation	NOUN
iajs-828	142	49	on	on	ADP
iajs-828	142	50	r.	r.	PROPN
iajs-828	142	51	theorem	theorem	VERB
iajs-828	142	52	3.2	3.2	NUM
iajs-828	142	53	let	let	VERB
iajs-828	142	54	r	r	NOUN
iajs-828	142	55	be	be	AUX
iajs-828	142	56	a	a	DET
iajs-828	142	57	prime	prime	NOUN
iajs-828	142	58	,	,	PUNCT
iajs-828	142	59	and	and	CCONJ
iajs-828	142	60	(	(	PUNCT
iajs-828	142	61	d	d	NOUN
iajs-828	142	62	,	,	PUNCT
iajs-828	142	63	g	g	NOUN
iajs-828	142	64	)	)	PUNCT
iajs-828	142	65	be	be	VERB
iajs-828	142	66	a	a	DET
iajs-828	142	67	(	(	PUNCT
iajs-828	142	68	,)-s	,)-s	ADJ
iajs-828	142	69	-	-	ADJ
iajs-828	142	70	derivation	derivation	NOUN
iajs-828	142	71	pair	pair	NOUN
iajs-828	142	72	on	on	ADP
iajs-828	142	73	r	r	NOUN
iajs-828	142	74	,	,	PUNCT
iajs-828	142	75	then	then	ADV
iajs-828	142	76	d	d	PROPN
iajs-828	142	77	and	and	CCONJ
iajs-828	142	78	g	g	PROPN
iajs-828	142	79	are	be	AUX
iajs-828	142	80	(	(	PUNCT
iajs-828	142	81	,)derivations	,)derivation	NOUN
iajs-828	142	82	.	.	PUNCT
iajs-828	143	1	where	where	SCONJ
iajs-828	143	2	,	,	NOUN
iajs-828	143	3	are	be	AUX
iajs-828	143	4	automorphisms	automorphism	NOUN
iajs-828	143	5	of	of	ADP
iajs-828	143	6	r.	r.	PROPN
iajs-828	143	7	proof	proof	NOUN
iajs-828	143	8	since	since	SCONJ
iajs-828	143	9	(	(	PUNCT
iajs-828	143	10	d	d	NOUN
iajs-828	143	11	,	,	PUNCT
iajs-828	143	12	g	g	NOUN
iajs-828	143	13	)	)	PUNCT
iajs-828	143	14	is	be	AUX
iajs-828	143	15	(	(	PUNCT
iajs-828	143	16	,)-s	,)-s	ADJ
iajs-828	143	17	-	-	ADJ
iajs-828	143	18	derivation	derivation	NOUN
iajs-828	143	19	pair	pair	NOUN
iajs-828	143	20	,	,	PUNCT
iajs-828	143	21	we	we	PRON
iajs-828	143	22	have	have	AUX
iajs-828	143	23	(	(	PUNCT
iajs-828	143	24	see	see	VERB
iajs-828	143	25	how	how	SCONJ
iajs-828	143	26	relation	relation	NOUN
iajs-828	143	27	(	(	PUNCT
iajs-828	143	28	5	5	NUM
iajs-828	143	29	)	)	PUNCT
iajs-828	143	30	was	be	AUX
iajs-828	143	31	obtained	obtain	VERB
iajs-828	143	32	from	from	ADP
iajs-828	143	33	relation	relation	NOUN
iajs-828	143	34	(	(	PUNCT
iajs-828	143	35	1	1	NUM
iajs-828	143	36	)	)	PUNCT
iajs-828	143	37	in	in	ADP
iajs-828	143	38	the	the	DET
iajs-828	143	39	proof	proof	NOUN
iajs-828	143	40	of	of	ADP
iajs-828	143	41	theorem	theorem	ADJ
iajs-828	143	42	3.1	3.1	NUM
iajs-828	143	43	)	)	PUNCT
iajs-828	143	44	(xy)(d(x)-g(x)=0	(xy)(d(x)-g(x)=0	VERB
iajs-828	143	45	,	,	PUNCT
iajs-828	143	46	for	for	ADP
iajs-828	143	47	all	all	DET
iajs-828	143	48	x	x	NOUN
iajs-828	143	49	,	,	PUNCT
iajs-828	143	50	y	y	PROPN
iajs-828	143	51			PROPN
iajs-828	143	52	r_____(1	r_____(1	PROPN
iajs-828	143	53	)	)	PUNCT
iajs-828	143	54	and	and	CCONJ
iajs-828	143	55	,	,	PUNCT
iajs-828	143	56	by	by	ADP
iajs-828	143	57	primeness	primeness	NOUN
iajs-828	143	58	of	of	ADP
iajs-828	143	59	r	r	NOUN
iajs-828	143	60	,	,	PUNCT
iajs-828	143	61	we	we	PRON
iajs-828	143	62	get	get	VERB
iajs-828	143	63	:	:	PUNCT
iajs-828	143	64	d(x)=g(x	d(x)=g(x	NUM
iajs-828	143	65	)	)	PUNCT
iajs-828	143	66	,	,	PUNCT
iajs-828	143	67	for	for	ADP
iajs-828	143	68	all	all	DET
iajs-828	143	69	x	x	ADJ
iajs-828	143	70			PROPN
iajs-828	143	71	r_____(2	r_____(2	PROPN
iajs-828	143	72	)	)	PUNCT
iajs-828	143	73	and	and	CCONJ
iajs-828	143	74	hence	hence	ADV
iajs-828	143	75	d	d	PROPN
iajs-828	143	76	and	and	CCONJ
iajs-828	143	77	g	g	PROPN
iajs-828	143	78	are	be	AUX
iajs-828	143	79	(	(	PUNCT
iajs-828	143	80	,)-derivations	,)-derivation	NOUN
iajs-828	143	81	on	on	ADP
iajs-828	143	82	r.	r.	PROPN
iajs-828	143	83	theorem	theorem	VERB
iajs-828	143	84	3.3	3.3	NUM
iajs-828	143	85	let	let	VERB
iajs-828	143	86	r	r	PRON
iajs-828	143	87	be	be	AUX
iajs-828	143	88	a	a	DET
iajs-828	143	89	ring	ring	NOUN
iajs-828	143	90	which	which	PRON
iajs-828	143	91	has	have	VERB
iajs-828	143	92	a	a	DET
iajs-828	143	93	commutator	commutator	NOUN
iajs-828	143	94	left	leave	VERB
iajs-828	143	95	nonzero	nonzero	PROPN
iajs-828	143	96	divisor	divisor	NOUN
iajs-828	143	97	and	and	CCONJ
iajs-828	143	98	(	(	PUNCT
iajs-828	143	99	d	d	NOUN
iajs-828	143	100	,	,	PUNCT
iajs-828	143	101	g	g	NOUN
iajs-828	143	102	)	)	PUNCT
iajs-828	143	103	be	be	VERB
iajs-828	143	104	a	a	DET
iajs-828	143	105	(	(	PUNCT
iajs-828	143	106	,)-s	,)-s	ADJ
iajs-828	143	107	-	-	ADJ
iajs-828	143	108	derivation	derivation	NOUN
iajs-828	143	109	pair	pair	NOUN
iajs-828	143	110	on	on	ADP
iajs-828	143	111	r	r	NOUN
iajs-828	143	112	,	,	PUNCT
iajs-828	143	113	then	then	ADV
iajs-828	143	114	d	d	PROPN
iajs-828	143	115	and	and	CCONJ
iajs-828	143	116	g	g	PROPN
iajs-828	143	117	are	be	AUX
iajs-828	143	118	(	(	PUNCT
iajs-828	143	119	,)-derivations.where	,)-derivations.where	PROPN
iajs-828	143	120	,	,	PROPN
iajs-828	143	121	are	be	AUX
iajs-828	143	122	automorphisms	automorphism	NOUN
iajs-828	143	123	of	of	ADP
iajs-828	143	124	r.	r.	PROPN
iajs-828	143	125	proof	proof	NOUN
iajs-828	143	126	1	1	NUM
iajs-828	143	127	.	.	PUNCT
iajs-828	144	1	that	that	PRON
iajs-828	144	2	is	be	AUX
iajs-828	144	3	we	we	PRON
iajs-828	144	4	have	have	VERB
iajs-828	144	5	:	:	PUNCT
iajs-828	144	6	2	2	X
iajs-828	144	7	.	.	X
iajs-828	144	8	d(yx	d(yx	NOUN
iajs-828	144	9	2)=d(y)(x2	2)=d(y)(x2	NOUN
iajs-828	144	10	)	)	PUNCT
iajs-828	144	11	+	+	CCONJ
iajs-828	144	12	(y)g(x2	(y)g(x2	NUM
iajs-828	144	13	)	)	PUNCT
iajs-828	144	14	,	,	PUNCT
iajs-828	144	15	for	for	ADP
iajs-828	144	16	all	all	DET
iajs-828	144	17	x	x	NOUN
iajs-828	144	18	,	,	PUNCT
iajs-828	144	19	y	y	PROPN
iajs-828	144	20			PROPN
iajs-828	144	21	r____(1	r____(1	PROPN
iajs-828	144	22	)	)	PUNCT
iajs-828	144	23	3	3	NUM
iajs-828	144	24	.	.	X
iajs-828	145	1	that	that	PRON
iajs-828	145	2	is	be	AUX
iajs-828	145	3	:	:	PUNCT
iajs-828	145	4	4	4	X
iajs-828	145	5	.	.	X
iajs-828	145	6	d(yx	d(yx	NOUN
iajs-828	145	7	2	2	NUM
iajs-828	145	8	)	)	PUNCT
iajs-828	146	1	=	=	NOUN
iajs-828	146	2	d(y)(x	d(y)(x	NOUN
iajs-828	146	3	2	2	NUM
iajs-828	146	4	)	)	PUNCT
iajs-828	146	5	+	+	NUM
iajs-828	146	6	(y)g(x)(x)+(y)(x)d(x	(y)g(x)(x)+(y)(x)d(x	NOUN
iajs-828	146	7	)	)	PUNCT
iajs-828	146	8	,	,	PUNCT
iajs-828	146	9	for	for	ADP
iajs-828	146	10	all	all	DET
iajs-828	146	11	x	x	NOUN
iajs-828	146	12	,	,	PUNCT
iajs-828	146	13	y	y	PROPN
iajs-828	146	14			PROPN
iajs-828	146	15	r__(2	r__(2	PROPN
iajs-828	146	16	)	)	PUNCT
iajs-828	146	17	5	5	NUM
iajs-828	146	18	.	.	PUNCT
iajs-828	147	1	on	on	ADP
iajs-828	147	2	the	the	DET
iajs-828	147	3	other	other	ADJ
iajs-828	147	4	hand	hand	NOUN
iajs-828	147	5	:	:	PUNCT
iajs-828	147	6	6	6	X
iajs-828	147	7	.	.	X
iajs-828	147	8	d(yx	d(yx	SYM
iajs-828	147	9	2)=d(yx)(x)+	2)=d(yx)(x)+	NUM
iajs-828	147	10	(yx)g(x	(yx)g(x	NOUN
iajs-828	147	11	)	)	PUNCT
iajs-828	147	12	,	,	PUNCT
iajs-828	147	13	for	for	ADP
iajs-828	147	14	all	all	DET
iajs-828	147	15	x	x	NOUN
iajs-828	147	16	,	,	PUNCT
iajs-828	147	17	y	y	PROPN
iajs-828	147	18			PROPN
iajs-828	147	19	r_____(3	r_____(3	VERB
iajs-828	147	20	)	)	PUNCT
iajs-828	147	21	7	7	NUM
iajs-828	147	22	.	.	NOUN
iajs-828	147	23	8	8	NUM
iajs-828	147	24	.	.	PUNCT
iajs-828	148	1	d(yx	d(yx	NOUN
iajs-828	148	2	2	2	NUM
iajs-828	148	3	)	)	PUNCT
iajs-828	148	4	=	=	NOUN
iajs-828	148	5	d(y)(x	d(y)(x	NOUN
iajs-828	148	6	2	2	NUM
iajs-828	148	7	)	)	PUNCT
iajs-828	148	8	+	+	PUNCT
iajs-828	148	9	(y)g(x)(x	(y)g(x)(x	NOUN
iajs-828	148	10	)	)	PUNCT
iajs-828	148	11	+	+	CCONJ
iajs-828	148	12	(y)(x)g(x	(y)(x)g(x	NOUN
iajs-828	148	13	)	)	PUNCT
iajs-828	148	14	,	,	PUNCT
iajs-828	148	15	for	for	ADP
iajs-828	148	16	all	all	DET
iajs-828	148	17	x	x	NOUN
iajs-828	148	18	,	,	PUNCT
iajs-828	148	19	y	y	PROPN
iajs-828	148	20			PROPN
iajs-828	148	21	r_____(4	r_____(4	ADV
iajs-828	148	22	)	)	PUNCT
iajs-828	148	23	9	9	NUM
iajs-828	148	24	.	.	X
iajs-828	148	25	from	from	ADP
iajs-828	148	26	(	(	PUNCT
iajs-828	148	27	2	2	NUM
iajs-828	148	28	)	)	PUNCT
iajs-828	148	29	and	and	CCONJ
iajs-828	148	30	(	(	PUNCT
iajs-828	148	31	4	4	NUM
iajs-828	148	32	)	)	PUNCT
iajs-828	148	33	,	,	PUNCT
iajs-828	148	34	we	we	PRON
iajs-828	148	35	obtain	obtain	VERB
iajs-828	148	36	:	:	PUNCT
iajs-828	148	37	ibn	ibn	NOUN
iajs-828	148	38	alhaitham	alhaitham	NOUN
iajs-828	148	39	j.	j.	PROPN
iajs-828	148	40	for	for	ADP
iajs-828	148	41	pure	pure	ADJ
iajs-828	148	42	&	&	CCONJ
iajs-828	148	43	appl	appl	PROPN
iajs-828	148	44	.	.	PUNCT
iajs-828	149	1	sci	sci	PROPN
iajs-828	149	2	.	.	PUNCT
iajs-828	149	3	vol.24	vol.24	NOUN
iajs-828	149	4	(	(	PUNCT
iajs-828	149	5	3	3	NUM
iajs-828	149	6	)	)	PUNCT
iajs-828	149	7	2011	2011	NUM
iajs-828	149	8			NUM
iajs-828	149	9	(y)((x)d(x)-(x)g(x))=0	(y)((x)d(x)-(x)g(x))=0	VERB
iajs-828	149	10	,	,	PUNCT
iajs-828	149	11	for	for	ADP
iajs-828	149	12	all	all	DET
iajs-828	149	13	x	x	NOUN
iajs-828	149	14	,	,	PUNCT
iajs-828	149	15	y	y	PROPN
iajs-828	149	16			NOUN
iajs-828	149	17	r_____(5	r_____(5	NOUN
iajs-828	149	18	)	)	PUNCT
iajs-828	149	19	replacing	replace	VERB
iajs-828	149	20	y	y	PRON
iajs-828	149	21	by	by	ADP
iajs-828	149	22	yr	yr	PROPN
iajs-828	149	23	in	in	ADP
iajs-828	149	24	(	(	PUNCT
iajs-828	149	25	5	5	NUM
iajs-828	149	26	)	)	PUNCT
iajs-828	149	27	,	,	PUNCT
iajs-828	149	28	to	to	PART
iajs-828	149	29	get	get	VERB
iajs-828	149	30	:	:	PUNCT
iajs-828	149	31	(yr)((x)d(x)-(x)g(x))=0	(yr)((x)d(x)-(x)g(x))=0	ADJ
iajs-828	149	32	,	,	PUNCT
iajs-828	149	33	for	for	SCONJ
iajs-828	149	34	all	all	DET
iajs-828	149	35	x	x	NOUN
iajs-828	149	36	,	,	PUNCT
iajs-828	149	37	y	y	PROPN
iajs-828	149	38	,	,	PUNCT
iajs-828	149	39	r	r	NOUN
iajs-828	149	40			PROPN
iajs-828	149	41	r_____(6	r_____(6	NOUN
iajs-828	149	42	)	)	PUNCT
iajs-828	149	43	again	again	ADV
iajs-828	149	44	,	,	PUNCT
iajs-828	149	45	left	leave	VERB
iajs-828	149	46	multiplying	multiplying	NOUN
iajs-828	149	47	of	of	ADP
iajs-828	149	48	(	(	PUNCT
iajs-828	149	49	5	5	NUM
iajs-828	149	50	)	)	PUNCT
iajs-828	149	51	by	by	ADP
iajs-828	149	52	(r	(r	PROPN
iajs-828	149	53	)	)	PUNCT
iajs-828	149	54	,	,	PUNCT
iajs-828	149	55	to	to	PART
iajs-828	149	56	get	get	VERB
iajs-828	149	57	:	:	PUNCT
iajs-828	149	58	(r)(y)((x)d(x)-(x)g(x))=0	(r)(y)((x)d(x)-(x)g(x))=0	NOUN
iajs-828	149	59	,	,	PUNCT
iajs-828	149	60	for	for	ADP
iajs-828	149	61	all	all	DET
iajs-828	149	62	x	x	NOUN
iajs-828	149	63	,	,	PUNCT
iajs-828	149	64	y	y	PROPN
iajs-828	149	65	,	,	PUNCT
iajs-828	149	66	r	r	NOUN
iajs-828	149	67			NOUN
iajs-828	149	68	r_____(7	r_____(7	PROPN
iajs-828	149	69	)	)	PUNCT
iajs-828	149	70	subtracting	subtract	VERB
iajs-828	149	71	(	(	PUNCT
iajs-828	149	72	7	7	NUM
iajs-828	149	73	)	)	PUNCT
iajs-828	149	74	from	from	ADP
iajs-828	149	75	(	(	PUNCT
iajs-828	149	76	6	6	NUM
iajs-828	149	77	)	)	PUNCT
iajs-828	149	78	,	,	PUNCT
iajs-828	149	79	we	we	PRON
iajs-828	149	80	get	get	VERB
iajs-828	149	81	:	:	PUNCT
iajs-828	150	1	[	[	X
iajs-828	150	2	(y),(r)]((x)d(x)-(x)g(x))=0	(y),(r)]((x)d(x)-(x)g(x))=0	X
iajs-828	150	3	,	,	PUNCT
iajs-828	150	4	for	for	ADP
iajs-828	150	5	all	all	DET
iajs-828	150	6	x	x	NOUN
iajs-828	150	7	,	,	PUNCT
iajs-828	150	8	y	y	PROPN
iajs-828	150	9	,	,	PUNCT
iajs-828	150	10	r	r	NOUN
iajs-828	150	11			NOUN
iajs-828	150	12	r_____(8	r_____(8	PROPN
iajs-828	150	13	)	)	PUNCT
iajs-828	150	14	since	since	SCONJ
iajs-828	150	15	r	r	NOUN
iajs-828	150	16	has	have	VERB
iajs-828	150	17	a	a	DET
iajs-828	150	18	commutator	commutator	NOUN
iajs-828	150	19	left	leave	VERB
iajs-828	150	20	nonzero	nonzero	PROPN
iajs-828	150	21	divisor	divisor	NOUN
iajs-828	150	22	,	,	PUNCT
iajs-828	150	23	we	we	PRON
iajs-828	150	24	get	get	VERB
iajs-828	150	25	:	:	PUNCT
iajs-828	150	26	(x)d(x)=(x)g(x	(x)d(x)=(x)g(x	NUM
iajs-828	150	27	)	)	PUNCT
iajs-828	150	28	,	,	PUNCT
iajs-828	150	29	for	for	ADP
iajs-828	150	30	all	all	DET
iajs-828	150	31	x	x	ADJ
iajs-828	150	32			NOUN
iajs-828	150	33	r_____(9	r_____(9	NOUN
iajs-828	150	34	)	)	PUNCT
iajs-828	150	35	linearizing	linearize	VERB
iajs-828	150	36	(	(	PUNCT
iajs-828	150	37	9	9	NUM
iajs-828	150	38	)	)	PUNCT
iajs-828	150	39	,	,	PUNCT
iajs-828	150	40	we	we	PRON
iajs-828	150	41	get	get	VERB
iajs-828	150	42	:	:	PUNCT
iajs-828	150	43	(x)d(y	(x)d(y	ADJ
iajs-828	150	44	)	)	PUNCT
iajs-828	150	45	+	+	NUM
iajs-828	150	46	(y)d(x)=	(y)d(x)=	NOUN
iajs-828	150	47	(x)g(y	(x)g(y	PRON
iajs-828	150	48	)	)	PUNCT
iajs-828	150	49	+	+	CCONJ
iajs-828	151	1	(y)g(x	(y)g(x	ADJ
iajs-828	151	2	)	)	PUNCT
iajs-828	151	3	,	,	PUNCT
iajs-828	151	4	for	for	SCONJ
iajs-828	151	5	all	all	DET
iajs-828	151	6	x	x	NOUN
iajs-828	151	7	,	,	PUNCT
iajs-828	151	8	y	y	PROPN
iajs-828	151	9			PROPN
iajs-828	151	10	r_____(10	r_____(10	PROPN
iajs-828	151	11	)	)	PUNCT
iajs-828	151	12	that	that	PRON
iajs-828	151	13	is	be	AUX
iajs-828	151	14	:	:	PUNCT
iajs-828	151	15	(x)(d	(x)(d	PROPN
iajs-828	151	16	-	-	PUNCT
iajs-828	151	17	g)(y	g)(y	PROPN
iajs-828	151	18	)	)	PUNCT
iajs-828	152	1	+	+	CCONJ
iajs-828	152	2	(y)(d	(y)(d	PROPN
iajs-828	152	3	-	-	PUNCT
iajs-828	152	4	g)(x)=0	g)(x)=0	PROPN
iajs-828	152	5	,	,	PUNCT
iajs-828	152	6	for	for	ADP
iajs-828	152	7	all	all	DET
iajs-828	152	8	x	x	NOUN
iajs-828	152	9	,	,	PUNCT
iajs-828	152	10	y	y	PROPN
iajs-828	152	11			PROPN
iajs-828	152	12	r_____(11	r_____(11	NOUN
iajs-828	152	13	)	)	PUNCT
iajs-828	152	14	replacing	replace	VERB
iajs-828	152	15	y	y	PRON
iajs-828	152	16	by	by	ADP
iajs-828	152	17	ry	ry	INTJ
iajs-828	152	18	in	in	ADP
iajs-828	152	19	(	(	PUNCT
iajs-828	152	20	11	11	NUM
iajs-828	152	21	)	)	PUNCT
iajs-828	152	22	,	,	PUNCT
iajs-828	152	23	to	to	PART
iajs-828	152	24	get	get	VERB
iajs-828	152	25	:	:	PUNCT
iajs-828	152	26	(x)(d	(x)(d	PROPN
iajs-828	152	27	-	-	PUNCT
iajs-828	152	28	g)(ry	g)(ry	NOUN
iajs-828	152	29	)	)	PUNCT
iajs-828	152	30	+	+	CCONJ
iajs-828	152	31	(ry)(d	(ry)(d	ADJ
iajs-828	152	32	-	-	ADJ
iajs-828	152	33	g)(x)=0	g)(x)=0	ADJ
iajs-828	152	34	,	,	PUNCT
iajs-828	152	35	for	for	ADP
iajs-828	152	36	all	all	DET
iajs-828	152	37	x	x	NOUN
iajs-828	152	38	,	,	PUNCT
iajs-828	152	39	y	y	PROPN
iajs-828	152	40	,	,	PUNCT
iajs-828	152	41	r	r	NOUN
iajs-828	152	42			NOUN
iajs-828	152	43	r_____(12	r_____(12	NOUN
iajs-828	152	44	)	)	PUNCT
iajs-828	152	45	again	again	ADV
iajs-828	152	46	,	,	PUNCT
iajs-828	152	47	left	leave	VERB
iajs-828	152	48	multiplying	multiplying	NOUN
iajs-828	152	49	of	of	ADP
iajs-828	152	50	(	(	PUNCT
iajs-828	152	51	11	11	NUM
iajs-828	152	52	)	)	PUNCT
iajs-828	152	53	by	by	ADP
iajs-828	152	54	(r	(r	PROPN
iajs-828	152	55	)	)	PUNCT
iajs-828	152	56	,	,	PUNCT
iajs-828	152	57	to	to	PART
iajs-828	152	58	get	get	VERB
iajs-828	152	59	:	:	PUNCT
iajs-828	152	60	(r)(x)(d	(r)(x)(d	NOUN
iajs-828	152	61	-	-	PUNCT
iajs-828	152	62	g)(y	g)(y	PROPN
iajs-828	152	63	)	)	PUNCT
iajs-828	153	1	+	+	CCONJ
iajs-828	153	2	(r)(y)(d	(r)(y)(d	NOUN
iajs-828	153	3	-	-	PUNCT
iajs-828	153	4	g)(x)=0	g)(x)=0	NOUN
iajs-828	153	5	,	,	PUNCT
iajs-828	153	6	for	for	ADP
iajs-828	153	7	all	all	DET
iajs-828	153	8	x	x	NOUN
iajs-828	153	9	,	,	PUNCT
iajs-828	153	10	y	y	PROPN
iajs-828	153	11	,	,	PUNCT
iajs-828	153	12	r	r	NOUN
iajs-828	153	13	r	r	NOUN
iajs-828	154	1	_	_	PUNCT
iajs-828	155	1	_	_	PUNCT
iajs-828	156	1	_	_	PUNCT
iajs-828	157	1	_	_	PUNCT
iajs-828	158	1	_	_	PUNCT
iajs-828	158	2	(	(	PUNCT
iajs-828	158	3	13	13	NUM
iajs-828	158	4	)	)	PUNCT
iajs-828	158	5	subtracting	subtract	VERB
iajs-828	158	6	(	(	PUNCT
iajs-828	158	7	12	12	NUM
iajs-828	158	8	)	)	PUNCT
iajs-828	158	9	from	from	ADP
iajs-828	158	10	(	(	PUNCT
iajs-828	158	11	13	13	NUM
iajs-828	158	12	)	)	PUNCT
iajs-828	158	13	,	,	PUNCT
iajs-828	158	14	we	we	PRON
iajs-828	158	15	get	get	VERB
iajs-828	158	16	:	:	PUNCT
iajs-828	158	17	(rx	(rx	ADJ
iajs-828	158	18	)	)	PUNCT
iajs-828	158	19	(	(	PUNCT
iajs-828	158	20	d	d	X
iajs-828	158	21	-	-	ADJ
iajs-828	158	22	g)(y	g)(y	ADJ
iajs-828	158	23	)	)	PUNCT
iajs-828	159	1	(x)(d	(x)(d	PROPN
iajs-828	159	2	-	-	PUNCT
iajs-828	159	3	g)(ry)=0	g)(ry)=0	NOUN
iajs-828	159	4	,	,	PUNCT
iajs-828	159	5	for	for	ADP
iajs-828	159	6	all	all	DET
iajs-828	159	7	x	x	NOUN
iajs-828	159	8	,	,	PUNCT
iajs-828	159	9	y	y	PROPN
iajs-828	159	10	,	,	PUNCT
iajs-828	159	11	r	r	NOUN
iajs-828	159	12			NOUN
iajs-828	159	13	r	r	NOUN
iajs-828	159	14	_	_	PUNCT
iajs-828	160	1	_	_	PUNCT
iajs-828	161	1	_	_	PUNCT
iajs-828	162	1	_	_	PUNCT
iajs-828	163	1	_	_	PUNCT
iajs-828	163	2	(	(	PUNCT
iajs-828	163	3	14	14	NUM
iajs-828	163	4	)	)	PUNCT
iajs-828	163	5	replacing	replace	VERB
iajs-828	163	6	x	x	PUNCT
iajs-828	163	7	by	by	ADP
iajs-828	163	8	sx	sx	PROPN
iajs-828	163	9	in	in	ADP
iajs-828	163	10	(	(	PUNCT
iajs-828	163	11	14	14	NUM
iajs-828	163	12	)	)	PUNCT
iajs-828	163	13	,	,	PUNCT
iajs-828	163	14	to	to	PART
iajs-828	163	15	get	get	VERB
iajs-828	163	16	:	:	PUNCT
iajs-828	163	17	(rsx)(d	(rsx)(d	NOUN
iajs-828	163	18	-	-	PUNCT
iajs-828	163	19	g)(y)-(sx)(d	g)(y)-(sx)(d	NOUN
iajs-828	163	20	-	-	PUNCT
iajs-828	163	21	g)(ry)=0	g)(ry)=0	NOUN
iajs-828	163	22	,	,	PUNCT
iajs-828	163	23	for	for	ADP
iajs-828	163	24	all	all	DET
iajs-828	163	25	x	x	NOUN
iajs-828	163	26	,	,	PUNCT
iajs-828	163	27	y	y	PROPN
iajs-828	163	28	,	,	PUNCT
iajs-828	163	29	r	r	NOUN
iajs-828	163	30	,	,	PUNCT
iajs-828	163	31	s	s	PART
iajs-828	163	32			NOUN
iajs-828	163	33	r	r	NOUN
iajs-828	163	34	_	_	PUNCT
iajs-828	164	1	_	_	PUNCT
iajs-828	165	1	_	_	PUNCT
iajs-828	166	1	_	_	PUNCT
iajs-828	167	1	_	_	PUNCT
iajs-828	167	2	(	(	PUNCT
iajs-828	167	3	15	15	NUM
iajs-828	167	4	)	)	PUNCT
iajs-828	167	5	also	also	ADV
iajs-828	167	6	,	,	PUNCT
iajs-828	167	7	left	leave	VERB
iajs-828	167	8	multiplying	multiplying	NOUN
iajs-828	167	9	of	of	ADP
iajs-828	167	10	(	(	PUNCT
iajs-828	167	11	14	14	NUM
iajs-828	167	12	)	)	PUNCT
iajs-828	167	13	by	by	ADP
iajs-828	167	14	(s	(	VERB
iajs-828	167	15	)	)	PUNCT
iajs-828	167	16	,	,	PUNCT
iajs-828	167	17	to	to	PART
iajs-828	167	18	get	get	VERB
iajs-828	167	19	:	:	PUNCT
iajs-828	167	20	(srx)(d	(srx)(d	NUM
iajs-828	167	21	-	-	PUNCT
iajs-828	167	22	g)(y)-(sx)(d	g)(y)-(sx)(d	NOUN
iajs-828	167	23	-	-	PUNCT
iajs-828	167	24	g)(ry)=0	g)(ry)=0	NOUN
iajs-828	167	25	,	,	PUNCT
iajs-828	167	26	for	for	SCONJ
iajs-828	167	27	all	all	DET
iajs-828	167	28	x	x	NOUN
iajs-828	167	29	,	,	PUNCT
iajs-828	167	30	y	y	PROPN
iajs-828	167	31	,	,	PUNCT
iajs-828	167	32	r	r	NOUN
iajs-828	167	33	,	,	PUNCT
iajs-828	167	34	s	s	PART
iajs-828	167	35			PROPN
iajs-828	167	36	r_____(16	r_____(16	NOUN
iajs-828	167	37	)	)	PUNCT
iajs-828	167	38	subtracting	subtract	VERB
iajs-828	167	39	(	(	PUNCT
iajs-828	167	40	16	16	NUM
iajs-828	167	41	)	)	PUNCT
iajs-828	167	42	from	from	ADP
iajs-828	167	43	(	(	PUNCT
iajs-828	167	44	15	15	NUM
iajs-828	167	45	)	)	PUNCT
iajs-828	167	46	,	,	PUNCT
iajs-828	167	47	we	we	PRON
iajs-828	167	48	get	get	VERB
iajs-828	167	49	:	:	PUNCT
iajs-828	168	1	[	[	X
iajs-828	168	2	(r),(s	(r),(s	NOUN
iajs-828	168	3	)	)	PUNCT
iajs-828	168	4	]	]	PUNCT
iajs-828	168	5	(x)(d	(x)(d	PROPN
iajs-828	168	6	-	-	PUNCT
iajs-828	168	7	g)(y)=0	g)(y)=0	NOUN
iajs-828	168	8	,	,	PUNCT
iajs-828	168	9	for	for	ADP
iajs-828	168	10	all	all	DET
iajs-828	168	11	x	x	NOUN
iajs-828	168	12	,	,	PUNCT
iajs-828	168	13	y	y	PROPN
iajs-828	168	14	,	,	PUNCT
iajs-828	168	15	r	r	NOUN
iajs-828	168	16	,	,	PUNCT
iajs-828	168	17	s	s	PART
iajs-828	168	18	r	r	NOUN
iajs-828	169	1	_	_	PUNCT
iajs-828	170	1	_	_	PUNCT
iajs-828	171	1	_	_	PUNCT
iajs-828	172	1	_	_	PUNCT
iajs-828	173	1	_	_	PUNCT
iajs-828	173	2	(	(	PUNCT
iajs-828	173	3	17	17	NUM
iajs-828	173	4	)	)	PUNCT
iajs-828	173	5	since	since	SCONJ
iajs-828	173	6	r	r	NOUN
iajs-828	173	7	has	have	VERB
iajs-828	173	8	a	a	DET
iajs-828	173	9	commutator	commutator	NOUN
iajs-828	173	10	left	leave	VERB
iajs-828	173	11	nonzero	nonzero	PROPN
iajs-828	173	12	divisor	divisor	NOUN
iajs-828	173	13	,	,	PUNCT
iajs-828	173	14	we	we	PRON
iajs-828	173	15	get	get	VERB
iajs-828	173	16	:	:	PUNCT
iajs-828	173	17	(x)(d	(x)(d	PROPN
iajs-828	173	18	-	-	PUNCT
iajs-828	173	19	g)(y)=0	g)(y)=0	NOUN
iajs-828	173	20	,	,	PUNCT
iajs-828	173	21	for	for	ADP
iajs-828	173	22	all	all	DET
iajs-828	173	23	x	x	NOUN
iajs-828	173	24	,	,	PUNCT
iajs-828	173	25	y	y	PROPN
iajs-828	173	26	r_____(18	r_____(18	PROPN
iajs-828	173	27	)	)	PUNCT
iajs-828	173	28	that	that	PRON
iajs-828	173	29	is	be	AUX
iajs-828	173	30	:	:	PUNCT
iajs-828	173	31	(x)d(y)=(x)g(y	(x)d(y)=(x)g(y	PROPN
iajs-828	173	32	)	)	PUNCT
iajs-828	173	33	,	,	PUNCT
iajs-828	173	34	for	for	ADP
iajs-828	173	35	all	all	DET
iajs-828	173	36	x	x	NOUN
iajs-828	173	37	,	,	PUNCT
iajs-828	173	38	y	y	PROPN
iajs-828	173	39			PROPN
iajs-828	173	40	r_____(19	r_____(19	PROPN
iajs-828	173	41	)	)	PUNCT
iajs-828	174	1	hence	hence	ADV
iajs-828	174	2	d	d	PROPN
iajs-828	174	3	and	and	CCONJ
iajs-828	174	4	g	g	PROPN
iajs-828	174	5	are	be	AUX
iajs-828	174	6	(	(	PUNCT
iajs-828	174	7	σ,)-derivations	σ,)-derivation	NOUN
iajs-828	174	8	.	.	PROPN
iajs-828	174	9	references	reference	NOUN
iajs-828	174	10	1	1	NUM
iajs-828	174	11	.	.	PUNCT
iajs-828	175	1	herstien	herstien	PROPN
iajs-828	175	2	,	,	PUNCT
iajs-828	175	3	i.n	i.n	PROPN
iajs-828	175	4	.	.	PROPN
iajs-828	175	5	,	,	PUNCT
iajs-828	175	6	(	(	PUNCT
iajs-828	175	7	1969	1969	NUM
iajs-828	175	8	)	)	PUNCT
iajs-828	175	9	,	,	PUNCT
iajs-828	175	10	topics	topic	NOUN
iajs-828	175	11	in	in	ADP
iajs-828	175	12	ring	ring	NOUN
iajs-828	175	13	theory	theory	NOUN
iajs-828	175	14	,	,	PUNCT
iajs-828	175	15	the	the	DET
iajs-828	175	16	university	university	NOUN
iajs-828	175	17	of	of	ADP
iajs-828	175	18	chicago	chicago	PROPN
iajs-828	175	19	press	press	PROPN
iajs-828	175	20	,	,	PUNCT
iajs-828	175	21	chicago	chicago	PROPN
iajs-828	175	22	.	.	PUNCT
iajs-828	176	1	2	2	X
iajs-828	176	2	.	.	X
iajs-828	176	3	ashraf	ashraf	PROPN
iajs-828	176	4	,	,	PUNCT
iajs-828	176	5	m.	m.	NOUN
iajs-828	176	6	,	,	PUNCT
iajs-828	176	7	ali	ali	PROPN
iajs-828	176	8	,	,	PUNCT
iajs-828	176	9	s.	s.	PROPN
iajs-828	176	10	and	and	CCONJ
iajs-828	176	11	haetinger	haetinger	PROPN
iajs-828	176	12	,	,	PUNCT
iajs-828	176	13	c.	c.	NOUN
iajs-828	176	14	,	,	PUNCT
iajs-828	176	15	(	(	PUNCT
iajs-828	176	16	2006	2006	NUM
iajs-828	176	17	)	)	PUNCT
iajs-828	176	18	,	,	PUNCT
iajs-828	176	19	"	"	PUNCT
iajs-828	176	20	on	on	ADP
iajs-828	176	21	derivations	derivation	NOUN
iajs-828	176	22	in	in	ADP
iajs-828	176	23	rings	ring	NOUN
iajs-828	176	24	and	and	CCONJ
iajs-828	176	25	their	their	PRON
iajs-828	176	26	applications	application	NOUN
iajs-828	176	27	"	"	PUNCT
iajs-828	176	28	,	,	PUNCT
iajs-828	176	29	the	the	DET
iajs-828	176	30	aligarh	aligarh	PROPN
iajs-828	176	31	bull	bull	PROPN
iajs-828	176	32	of	of	ADP
iajs-828	176	33	math	math	NOUN
iajs-828	176	34	.	.	PUNCT
iajs-828	176	35	,	,	PUNCT
iajs-828	176	36	25(2	25(2	NUM
iajs-828	176	37	)	)	PUNCT
iajs-828	176	38	,	,	PUNCT
iajs-828	176	39	79	79	NUM
iajs-828	176	40	-	-	SYM
iajs-828	176	41	107	107	NUM
iajs-828	176	42	.	.	PUNCT
iajs-828	177	1	3	3	X
iajs-828	177	2	.	.	X
iajs-828	177	3	hamdi	hamdi	PROPN
iajs-828	177	4	,	,	PUNCT
iajs-828	177	5	a.d	a.d	PROPN
iajs-828	177	6	.	.	PROPN
iajs-828	177	7	,	,	PUNCT
iajs-828	177	8	(	(	PUNCT
iajs-828	177	9	2007	2007	NUM
iajs-828	177	10	)	)	PUNCT
iajs-828	177	11	,	,	PUNCT
iajs-828	177	12	"	"	PUNCT
iajs-828	177	13	(	(	PUNCT
iajs-828	177	14	σ,)-derivations	σ,)-derivation	NOUN
iajs-828	177	15	on	on	ADP
iajs-828	177	16	prime	prime	ADJ
iajs-828	177	17	rings	ring	NOUN
iajs-828	177	18	"	"	PUNCT
iajs-828	177	19	,	,	PUNCT
iajs-828	177	20	msc	msc	PROPN
iajs-828	177	21	.	.	PROPN
iajs-828	178	1	thesis	thesis	PROPN
iajs-828	178	2	,	,	PUNCT
iajs-828	178	3	baghdad	baghdad	PROPN
iajs-828	178	4	university	university	PROPN
iajs-828	178	5	.	.	PUNCT
iajs-828	179	1	4	4	X
iajs-828	179	2	.	.	X
iajs-828	179	3	yass	yass	PROPN
iajs-828	179	4	,	,	PUNCT
iajs-828	179	5	s.	s.	PROPN
iajs-828	179	6	,	,	PUNCT
iajs-828	179	7	(	(	PUNCT
iajs-828	179	8	2010	2010	NUM
iajs-828	179	9	)	)	PUNCT
iajs-828	179	10	,	,	PUNCT
iajs-828	179	11	"	"	PUNCT
iajs-828	179	12	strongly	strongly	ADV
iajs-828	179	13	derivation	derivation	NOUN
iajs-828	179	14	pairs	pair	NOUN
iajs-828	179	15	on	on	ADP
iajs-828	179	16	prime	prime	ADJ
iajs-828	179	17	and	and	CCONJ
iajs-828	179	18	semiprime	semiprime	NOUN
iajs-828	179	19	rings	ring	NOUN
iajs-828	179	20	"	"	PUNCT
iajs-828	179	21	,	,	PUNCT
iajs-828	179	22	msc	msc	PROPN
iajs-828	179	23	.	.	PROPN
iajs-828	180	1	thesis	thesis	PROPN
iajs-828	180	2	,	,	PUNCT
iajs-828	180	3	baghdad	baghdad	PROPN
iajs-828	180	4	university	university	NOUN
iajs-828	180	5	.	.	PUNCT
iajs-828	181	1	5	5	X
iajs-828	181	2	.	.	X
iajs-828	181	3	cortes	corte	NOUN
iajs-828	181	4	,	,	PUNCT
iajs-828	181	5	w.	w.	NOUN
iajs-828	181	6	and	and	CCONJ
iajs-828	181	7	haetinger	haetinger	PROPN
iajs-828	181	8	,	,	PUNCT
iajs-828	181	9	c.	c.	NOUN
iajs-828	181	10	,	,	PUNCT
iajs-828	181	11	(	(	PUNCT
iajs-828	181	12	2005),"on	2005),"on	PROPN
iajs-828	181	13	jordan	jordan	PROPN
iajs-828	181	14	generalized	generalize	VERB
iajs-828	181	15	higher	high	ADJ
iajs-828	181	16	derivations	derivation	NOUN
iajs-828	181	17	in	in	ADP
iajs-828	181	18	rings	ring	NOUN
iajs-828	181	19	"	"	PUNCT
iajs-828	181	20	,	,	PUNCT
iajs-828	181	21	turkish	turkish	ADJ
iajs-828	181	22	j.	j.	PROPN
iajs-828	181	23	of	of	ADP
iajs-828	181	24	m	m	PROPN
iajs-828	181	25	ath	ath	PROPN
iajs-828	181	26	.	.	PROPN
iajs-828	181	27	,	,	PUNCT
iajs-828	181	28	29(1),1	29(1),1	PROPN
iajs-828	181	29	-	-	PUNCT
iajs-828	181	30	10	10	NUM
iajs-828	181	31	.	.	PUNCT
iajs-828	181	32	2011	2011	NUM
iajs-828	181	33	)	)	PUNCT
iajs-828	181	34	3	3	NUM
iajs-828	181	35	(	(	PUNCT
iajs-828	181	36	24المجلد	24المجلد	NUM
iajs-828	181	37	مجلة	مجلة	VERB
iajs-828	181	38	ابن	ابن	PROPN
iajs-828	181	39	الهیثم	الهیثم	PROPN
iajs-828	181	40	للعلوم	للعلوم	PROPN
iajs-828	181	41	الصرفة	الصرفة	PROPN
iajs-828	181	42	والتطبیقیة	والتطبیقیة	PROPN
iajs-828	181	43	على	على	PROPN
iajs-828	181	44	الحلقات	الحلقات	PROPN
iajs-828	181	45	)	)	PUNCT
iajs-828	181	46	,(-األشتقاقات	,(-األشتقاقات	NOUN
iajs-828	181	47	المزدوجة	المزدوجة	VERB
iajs-828	181	48	القویة	القویة	ADJ
iajs-828	181	49	سعید	سعید	ADJ
iajs-828	181	50	اكرام	اكرام	PROPN
iajs-828	181	51	احمد	احمد	VERB
iajs-828	181	52	الجامعة	الجامعة	PROPN
iajs-828	181	53	التكنولوجیة	التكنولوجیة	NOUN
iajs-828	181	54	2010تشرین	2010تشرین	NUM
iajs-828	181	55	الثاني	الثاني	PROPN
iajs-828	181	56	30	30	NUM
iajs-828	181	57	:	:	PUNCT
iajs-828	181	58	استلم	استلم	PROPN
iajs-828	181	59	البحث	البحث	VERB
iajs-828	181	60	في	في	ADP
iajs-828	181	61	2011شباط	2011شباط	NUM
iajs-828	181	62	27	27	NUM
iajs-828	181	63	:	:	PUNCT
iajs-828	181	64	قبل	قبل	NOUN
iajs-828	181	65	البحث	البحث	VERB
iajs-828	181	66	في	في	ADP
iajs-828	182	1	الخالصة	الخالصة	PROPN
iajs-828	182	2	واشتقاق	واشتقاق	NOUN
iajs-828	182	3	جوردان	جوردان	VERB
iajs-828	182	4	المزدوج	المزدوج	PROPN
iajs-828	182	5	القوي	القوي	PROPN
iajs-828	182	6	)	)	PUNCT
iajs-828	182	7	,(-في	,(-في	ADJ
iajs-828	182	8	هذا	هذا	NOUN
iajs-828	182	9	البحث	البحث	PROPN
iajs-828	182	10	قدمنا	قدمنا	PROPN
iajs-828	182	11	تعریف	تعریف	PROPN
iajs-828	182	12	األشتقاق	األشتقاق	NOUN
iajs-828	182	13	المزدوج	المزدوج	PROPN
iajs-828	182	14	القوي	القوي	PROPN
iajs-828	182	15	.	.	PUNCT
iajs-828	183	1	حلقة	حلقة	PROPN
iajs-828	183	2	تجمیعیة	تجمیعیة	VERB
iajs-828	183	3	rلتكن	rلتكن	NOUN
iajs-828	183	4	)	)	PUNCT
iajs-828	183	5	,	,	NOUN
iajs-828	183	6	(	(	PUNCT
iajs-828	183	7	ة	ة	NOUN
iajs-828	183	8	في	في	ADP
iajs-828	183	9	الحلقrكذلك	الحلقrكذلك	PROPN
iajs-828	183	10	،	،	PROPN
iajs-828	183	11	ندرس	ندرس	PROPN
iajs-828	183	12	الحلقات	الحلقات	PROPN
iajs-828	183	13	األولیة	األولیة	PROPN
iajs-828	183	14	،	،	PROPN
iajs-828	183	15	الحلقات	الحلقات	PROPN
iajs-828	183	16	شبه	شبه	PROPN
iajs-828	183	17	األولیة	األولیة	PROPN
iajs-828	183	18	،	،	PROPN
iajs-828	183	19	والحلقات	والحلقات	PROPN
iajs-828	183	20	التي	التي	PROPN
iajs-828	183	21	لها	لها	VERB
iajs-828	183	22	مبدل	مبدل	X
iajs-828	183	23	.	.	PUNCT
iajs-828	184	1	،	،	PROPN
iajs-828	184	2	ودراسة	ودراسة	PROPN
iajs-828	184	3	العالقة	العالقة	PROPN
iajs-828	184	4	بینهم	بینهم	ADJ
iajs-828	184	5	هما	هما	INTJ
iajs-828	184	6	,	,	PROPN
iajs-828	184	7	:	:	PUNCT
iajs-828	184	8	r	r	NOUN
iajs-828	184	9	rأي	rأي	VERB
iajs-828	184	10	ان	ان	ADV
iajs-828	184	11	)	)	PUNCT
iajs-828	184	12	.	.	PUNCT
iajs-828	185	1	,(-للحصول	,(-للحصول	PRON
iajs-828	185	2	على	على	PROPN
iajs-828	185	3	األشتقاق	األشتقاق	NOUN
iajs-828	185	4	)	)	PUNCT
iajs-828	185	5	,(-قاسم	,(-قاسم	NOUN
iajs-828	185	6	غیر	غیر	ADJ
iajs-828	185	7	صفري	صفري	ADJ
iajs-828	185	8	أیسر	أیسر	NOUN
iajs-828	185	9	مع	مع	INTJ
iajs-828	185	10	األشتقاق	األشتقاق	PROPN
iajs-828	185	11	المزدوج	المزدوج	PROPN
iajs-828	185	12	القوي	القوي	PROPN
iajs-828	185	13	.rدالتین	.rدالتین	PUNCT
iajs-828	185	14	على	على	PROPN
iajs-828	185	15	الحلقة	الحلقة	PROPN
iajs-828	185	16	:	:	PUNCT
iajs-828	185	17	الكلمات	الكلمات	VERB
iajs-828	185	18	المفتاحیة	المفتاحیة	ADV
iajs-828	185	19	.),(-	.),(-	PROPN
iajs-828	185	20	،	،	PROPN
iajs-828	185	21	اشتقاق	اشتقاق	PROPN
iajs-828	185	22	جوردان	جوردان	PROPN
iajs-828	185	23	المزدوج	المزدوج	PROPN
iajs-828	185	24	القوي	القوي	PROPN
iajs-828	185	25	)	)	PUNCT
iajs-828	185	26	,(-	,(-	PROPN
iajs-828	185	27	،	،	NOUN
iajs-828	185	28	األشتقاق	األشتقاق	PROPN
iajs-828	185	29	المزدوج	المزدوج	PROPN
iajs-828	185	30	القوي	القوي	PROPN
iajs-828	185	31	)	)	PUNCT
iajs-828	185	32	,(-حلقة	,(-حلقة	PROPN
iajs-828	185	33	اولیة	اولیة	PROPN
iajs-828	185	34	،	،	PROPN
iajs-828	185	35	حلقة	حلقة	PROPN
iajs-828	185	36	شبھ	شبھ	PROPN
iajs-828	185	37	اولیة	اولیة	PROPN
iajs-828	185	38	،	،	PROPN
iajs-828	185	39	مشتقة	مشتقة	ADJ