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Global Journal of Pure and Applied Mathematics * ISSN 0973-1768 Vol.12, No.3 (2016) P.116-118
@Research India Publications * http://www.ripublication.com/gjpam.htm
116
Lie Ideals With Left Generalized
Jordan Derivations In Primerings
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.
Dr.K.Madhusudhan Reddy
Department of Mathematics,
VIT University, Vellore,
TamilNadu, India.
Abstract: Let be a non-commutative prime ring with
characteristic not two, is a non central lie ideal of
such that for all .If be a left generalized
Jordan derivation on , then is a left generalized
derivation on .
Keywords: Prime ring, Derivation, Left Generalized
derivation, Left generalized Jordan derivation, Lie ideal
and Commutator. INTRODUCTION
The notation of generalized derivation of prime ring
was introduced by B.Hvala in [1]. A. Classical
result of I.N.Herstein states that every Jordan
derivation of prime ring with characteristic not two is
a derivation in [2]. A brief proof of this theorem can
be found in [3]. Latter on, this result was generalized
on lie ideals of such that for all in
[6] and generalized derivations of prime ring in [4].
In [5] O.Golbasi proved some results in generalized
Jordan derivation on prime rings with lie ideals. For
all these concerns we proved that if be a left
generalized Jordan derivation on , then is a left
generalized derivation on .
Preliminaries: Throughout this Paper, let be a
prime ring with its charectaristic not two. 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 .
A near ring is said to be prime if , for
all , implies or .while the
symbol will denote the commutator
Lemma1: For all the following
statements hold:
;
;
Proof:
,for all (1)
On the other hand, we have
)
=
.
, for all (2)
From (1) and (2), we have
.
.
for all
ii) Let .
On one hand, we have
Global Journal of Pure and Applied Mathematics * ISSN 0973-1768 Vol.12, No.3 (2016) P.116-118
@Research India Publications * http://www.ripublication.com/gjpam.htm
117
, for all (3)
On the other hand, we have
.
,
for all (4)
From (3) and (4), we have
Since is a 2-torsion free, we get
for all
Linearizing (ii) by replacing by
From L.H.S.
, for all (5)
From R.H.S
(6)
From (5) and (6), we get
for all
Remark 1:
We Introduce abbreviation
, for all
Observe also by lemma1, we have
And so that
From above we get ,
, for all (7)
Lemma 2: For all then
Proof: Replace by in Lemma1 ,and
using the fact that we get
, for all (8)
On the other hand,we get
, for all (9)
Comparing equations (8) and (9),we have
for all
Therefore, we get
, for all .
, for all .
Global Journal of Pure and Applied Mathematics * ISSN 0973-1768 Vol.12, No.3 (2016) P.116-118
@Research India Publications * http://www.ripublication.com/gjpam.htm
118
Theorem 1: Let be a non-commutative prime ring
with characteristic not two, is a non central lie
idealof such that , for all .If be a
left generalized Jordan derivation on , then is a
left generalized derivation on .
Proof : From Lemma 1 , we have
,
for all (10)
Replacing by and by in equation (10) ,
we get
, for all (11)
On the other hand, we have
,
for all (12)
Comparing equations (11) and (12) , we obtain
,
for all
, for all
, for all
Using the equation (7), that is , for all
we get
, for all
Since is prime ring, we have either or
.
Since is non-commutative, we have
Thus we have , for all .
This completes the proof.
References:
[1] B.Hvala: Generalized derivations in rings,
Comm.Algebra,26,No.4,1988,1147-1166.
[2] I.N.Herstein: Jordan Derivations of prime rings,
Proc. Amer.Math.Soc.,8,1957,1104-1110.
[3] M.Bresar, J.Vukman: Jordan Derivations on
Prime rings, Bull. Austral.Math.Soc.,37,1988,
321-322.
[4] M.Ashraf, N.Rehman: On Jordan Generalized
Derivations in Rings, Math. J.Okayama
Univ.,42, 2000,7-9.
[5] O.Golbasi, Neset Aydin: On Lie Ideals of
Prime rings with Generalized Jordan
Derivation,East Asian Math.J.21, No.1,
2005,21-26.
[6] R.Awtar: Lie ideal and Jordan Derivations of
Prime Rings,Proc.Amer.Math.Soc.,1984,9-14.
