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Global Journal of Pure and Applied Mathematics * ISSN 0973-1768 Vol.12, No.3 (2016) P.382-385
@Research India Publications * http://www.ripublication.com/gjpam.htm
382
LIE IDEALS WITH LEFT GENERALIZED JORDAN
TRIPLE DERIVATIONS IN SEMI PRIME RINGS
Dr. C. Jaya subba Reddy
Department of Mathematics,
S.V.University, Tirupati -517502,
Andhra Pradesh, India.
S.Vasantha Kumar
Department of Mathematics,
S.V.University, Tirupati -517502,
Andhra Pradesh, India.
S.Mallikarjuna Rao
Department of Mathematics,
S.V.University, Tirupati -517502,
Andhra Pradesh, India.
Abstract: Let be a 2-torsion free semiprime ring,
a square-closed lie ideal of , and an
additive mapping then is a left generalized Jordan triple
derivation on associated with a Jordan triple derivation
iff is a left generalized Jordan derivation on
associated with a Jordan derivation and let be a 2-
torsion free semiprime ring, a square-closed lie
ideal of an additive map such that .
If is a left generalized Jordan triple derivation
associated with a Jordan triple derivation then is a
left generalized derivation on associated with a
derivation .
Keywords: Semiprime ring, Derivation, Left Generalized
derivation, Left generalized Jordan derivation, Jordan
triple derivation, Centralizer, Lie ideal and Commutator.
INTRODUCTION
Every derivation is a Jordan derivation. In general,
the converse is not true. A classical result of Herstein
[4] asserts that any Jordan derivation on a prime ring
of characteristic different from two is a derivation. A
brief proof of the Herstein theorem can be found in
Bresar and Vukman [2].Cusack [3] has generalized
Herstein theorem to 2-torsion free semiprime ring.
Bresar [1] has proved that any Jordan triple
derivation on a 2-torsion free semiprime ring is a
derivation. Zalar [9] has proved that any left(right)
Jordan centralizer on a semiprime ring is a
centralizer. The concept of generalized derivation has
been introduced by Bresar in [1] and also the concept
of generalized Jordan derivation and generalized
Jordan triple derivation have been introduced by Jing
and Lu in [7]. Vukman [8] has shown that any
generalized Jordan derivation on 2-torsion free
semiprime ring is a generalized derivation, and that
any generalized Jordan triple derivation on 2-torsion
free semiprime ring is a generalized derivation.
Hongan.et.all [5] proved every Jordan triple
(resp.generalized Jordan triple) derivation on Lie
ideal L is a derivation on L(resp.generalized
derivation on L).Hongan and Rehman in [6] also
extended their results to generalized Jordan triple
derivations on lie ideals in semiprime rings.
Motivated by above work we extended our results to
left generalized Jordan triple derivations on lie ideals
in semiprime rings.
Preliminaries:Throughout this Paper, let will
denote an associative ring with center . be a
non-zero Lie ideal of . An additive map is
a derivation (resp. Jordan derivation) , if
(resp.
,holds for all .An additive map is
right generalized derivation (resp.a right generalized
Jordan derivation), if
(resp. holds for all
. Left generalized derivation (resp. a left
generalized Jordan derivation), if
(resp. holds for all
. An additive map is right
generalized Jordan triple derivation, if
holds for all .
Left generalized Jordan triple derivation
holds for all .
An additive mapping is called a left(right)
centralizer in case
,holds for all pairs . An additive mapping
is called a two-sided centralizer if is
both a left and right centralizer. An additive mapping
is called a left(right) Jordan centralizer in
case ,holds for all
A
r
ing
is
called
s
emi p
r
ime i
f
implie
s
,
f
o
r
all
in
.
while the symbol
will denote the commutator
Lemma 1: Let be a 2-torsion free semiprime ring,
a square closed lie ideal of and ‘a’ a
fixed element of . If for all ,
then .
Proof: Given that . (1)
Substituting for in equation (1) and using (1),
we have
Implies
Implies . (2)
Global Journal of Pure and Applied Mathematics * ISSN 0973-1768 Vol.12, No.3 (2016) P.382-385
@Research India Publications * http://www.ripublication.com/gjpam.htm
383
So,we get . (3)
And substituting for in equation (2), we get
. (4)
Subtracting equation (3) from equation (4), we have
Implies
Implies By [ 5 ,Corollary 2.1]
We have for all
i.e.,
Lemma 2: Let be a ring, a square-closed
lie ideal of , and an additive mapping such
that , for all and
(1) If is 2-torsion free semiprime, then
, for all
(2) If is prime of then
,for all
Proof: (1) Suppose that , for all
. (5)
Substituting for in equation (5) and using
equation (5), we have
Implies
Implies for all (6)
Replacing by in equation (5) and using
equation (5), we have
,for all
(7)
Furthermore, substituting for in
equation (6), we have
Substituting equation (7) in the above equation, we
get
And so, we have for all
By [5, corollary 2.1(1)], we have
, for all by lemma 1
, for all . (8)
(2) Since for all by equation (8),
, for all
By [5, lemma 2.1], we have or
for all
Hence, we have
for all
By Breuer’s trick, we conclude that
or for
all
And so, for all
Lemma 3: Let be a ring a lie ideal of and
a derivation on such that
If then
Theorem 1: Let be a 2-torsion free semiprime ring,
a square-closed lie ideal of ,
and an additive mapping then the
following are equivalent:
(1) is a left generalized Jordan triple derivation on
associated with a Jordan triple derivation
(2) is a left generalized Jordan derivation on
associated with a Jordan derivation
Proof:
Since is a left generalized Jordan triple derivation
on .
Therefore, we have
,for all . (9)
In equation (9), we take as a Jordan triple derivation
on .
Since is a 2-torsion free semiprime ring, so in view
of [5, theorem 2.1] is a derivation on .
Now we write .
Then we have
, for all .
In other words, is a Jordan triple right centralizer on
.
Since is a 2-torsion free semiprime ring, one can
conclude that is a Jordan right centralizer by [5,
theorem 3.1].
Hence is of the form , where is a
derivation and is a Jordan right centralizer on .
Hence is a left generalized Jordan derivation on .
, Suppose that .
(10)
Replacing by in (10) and using (10),
we have
.
(11)
Global Journal of Pure and Applied Mathematics * ISSN 0973-1768 Vol.12, No.3 (2016) P.382-385
@Research India Publications * http://www.ripublication.com/gjpam.htm
384
Replacing by in (11) and using (11), we
have
.
(12)
On the other hand, substituting for in equation
(11) and adding to both sides, we have
(13)
From equation (12) and equation (13), we get
Since be 2-torsion free semi prime ring then
(or)
Since is a derivation, is a Jordan triple derivation
and is a generalized left Jordan triple derivation on
associated with a Jordan triple derivation.
Theorem 2: Let be a 2-torsion free semiprime
ring, a square-closed lie ideal of
an additive map such that . If is
a left generalized Jordan triple derivation associated
with a Jordan triple derivation then is a left
generalized derivation on associated with a
derivation .
Proof: Suppose that there exists a Jordan triple
derivation on such that
,
(14)
Since is a 2-torsion free ring, is a derivation by
[5, theorem 2.1]
Substituting for in equation (14) and
using (14), we have
(15)
Now we set and we shall
compute it in two different ways using equation (14),
we have
(16)
Using (15),
(17)
From equation (16) and equation (17), we get
Global Journal of Pure and Applied Mathematics * ISSN 0973-1768 Vol.12, No.3 (2016) P.382-385
@Research India Publications * http://www.ripublication.com/gjpam.htm
385
. (18)
Now putting .(19)
Substituting equation (19) in equation (18), we have
, for all .
(20)
By the way, is a generalized left derivation on
associated with a derivation by theorem 1, and so
, for all . (21)
Substituting for in equation (21), we have
,
for all (22)
Now using equation (19), we have
, for all . (23)
Substituting equation (23) in equation (20), we have
, for all .
By [5, lemma 2.4], we get , for all
and
We have for all by
[5 corollary 2.1(3)] and
So, we get by lemma 1
Now, we put and ,
we will compute that in two different ways.
Using equation (22), we have
. (24)
Because of , using equation (15), we have
. (25)
From equation (24) and equation (25), we have
Using equation (19), we have
.
And so, we have .
Since is semi prime, .
That is, , for all
By Theorems 1 and 2 we obtain the following result
which explains the relationships of left generalized
Jordan triple derivations, left generalized derivations
and left generalized Jordan derivations.
Corollary 1: Let be a 2 – torsion free semiprime
ring , a square-closed lie ideal of and let
be an additive map such that
,then the followings are equivalent:
(1) is a left generalized Jordan triple
derivation on associated with a Jordan
triple derivation .
(2) is a left generalized derivation on
associated with a derivation .
(3) is a left generalized Jordan derivation on
associated with a derivation .
References:
[1] Bresar.M.: On the distance of the
compositions of two derivations to the
generalized derivations, lasgow, Math.J.,
33(1991),89-93.
[2] Bresar.M, Vukman J.: Jordan derivations on
prime rings, Bull. Austral.Math.Soc.,37
(1988),321-322.
[3] Cusack.J.: Jordan derivations on rings,
Proc.Amer.Math.Soc. 53 (1975),321-324.
[4] Herstein I.N.: Jordan derivations of prime
rings, Proc.Amer.Math.Soc.,8 (1957),1104-
1119.
[5] Hongan.M, Rehman.N. and Al-Omary,
R.M.:Lie ideals and Jordan triple derivations
in rings, Rend.Sem.Mat.Iniv.Padova,
125(2011),147-156.
[6] Hongan.M and Rehman.N.: A note on the
Generalized Jordan triple derivations on lie
ideals in semi prime rings, Sarajevo Journal
of Mathematics,vol.9(21)(2013),29-36.
[7] Jing.W and Lu.S.:Generalized Jordan
derivations on prime rings and standard
operator algebras, Taiwanese J.Math.,
7(2003),605-613.
[8] Vukman.J.: A note on generalized
derivations of semiprime rings,
Taiwan.J.Math.,11(2007),367-370.
[9] Zalar.B.: On centralizers of semiprime rings,
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