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International Journal of Research In Science & Engineering e-ISSN: 2394-8299
Special Issue –NCRAPAM March 2017 p-ISSN: 2394-8280
IJRISE| www.ijrise.org|editor@ijrise.org [151-155]
LIE IDEALS AND JORDAN GENERALIZED REVERSE
DERIVATIONS OF PRIME RINGS
A.Sivakameshwara Kumar1, C.Jaya Subba Reddy2, K.Madhusudhan Reddy3
kamesh1069@yahoo.com, cjsreddysvu@gmail.com, kmsrsku@gmail.com
1Research Scholar, Rayalaseema University, Kurnool, Andhrapradesh, India.
2Department of Mathematics, S.V.University, Tirupati -517502, Andhrapradesh, India.
3Department of Mathematics,VIT University, Vellore, TamilNadu, India.
Abstract: Let be a 2-torsion free prime ring and a nonzero lie ideal of such that
for all . In this paper it is shown that if is a Jordan generalized reverse
derivation on , then is a generalized reverse derivation on .
Key words: Lie ideals, Prime rings, Derivations, Jordan derivations, Generalized
derivations, Generalized Jordan derivations, Reverse derivations, Generalized reverse
derivations, Jordan generalized reverse derivations, Torsion free rings.
Preliminaries: Throughout this paper will denote an associative ring with centre .
Recall that is prime if implies that or . As usual will denote the
commutator . An additive subgroup of is said to be lie ideal of if
for all , . An additive mapping is said to be a derivation (resp. Jordan
derivation) if (resp. for all . An
additive mapping is said to be a generalized derivation (resp. Jordan generalized
derivation) associated with the derivation such that
(resp. for all . An additive mapping is called a
reverse derivation (resp. Jordan reverse derivation) if
, holds for all . An additive mapping is called a
generalized reverse derivation (resp.Jordan generalized reverse derivation) if there exists a
derivation such that (resp. ) for
all . Clearly, every generalized reverse derivation on a ring is a Jordan generalized
reverse derivation. But the converse statement does not hold in general. The aim of the
present paper is to establish another set of conditions under which every Jordan generalized
reverse derivation on a ring is a generalized reverse derivation. We shall make use of
commutator identities; and .
Introduction: Bresar and Vukman [8] have introduced the notion of a reverse derivations.
The reverse derivations on semi prime rings have been studied by Samman and Alyamani
[14]. Hvala [12] studied the properties of generalized derivations of prime rings. Posner [13]
which states that the existence of a nonzero centralizing derivation on a prime rings implies
that the ring has to be commutative. A famous result due to Herstein [11] states that every
Jordan derivation on a 2-torsion free prime ring is a derivation. A brief proof of this result is
presented in Bresar and Vukman [7]. Further, Awtar [4] generalized this result on lie ideals.
Mohammad Ashraf, Nadeem-UR-Rehman and Shakir Ali [3] studied on lie ideals and Jordan
generalized derivation of prime rings. In this paper we extended some results on lie ideals and
Jordan generalized reverse derivation of prime rings.
We begin with the following result which is essentially proved in [5].
Lemma 1: If is a lie ideal of a 2-torsion free prime ring and such that
, then or .
International Journal of Research In Science & Engineering e-ISSN: 2394-8299
Special Issue –NCRAPAM March 2017 p-ISSN: 2394-8280
IJRISE| www.ijrise.org|editor@ijrise.org [151-155]
We define a mapping such that .
Now we see that and , for
all .
By the definition
, for all .
, for all .
Moreover if is zero, then is generalized reverse derivation on .
Lemma 2: Let be a 2-torsion free ring and be a non zero lie ideal of such that
, for all . If is an additive mapping satisfying , for
all then
(i) , for all .
(ii) , for all .
(iii) ,
for
all .
Proof:
(i) Since , we find that , for all .
, for all .
(1)
(ii) Since , replacing by in equation (1), we get
(2)
On the other hand, we have
International Journal of Research In Science & Engineering e-ISSN: 2394-8299
Special Issue –NCRAPAM March 2017 p-ISSN: 2394-8280
IJRISE| www.ijrise.org|editor@ijrise.org [151-155]
(3)
Combining the above equations (2) and (3), we get
Since is a 2-torsion free ring, we get
, for all .
(iii) Linearizing (ii) by replacing by
(4)
On the other hand, we have
(5)
Combining the above equations (4) and (5), we get
.
Lemma 3: Let be a 2-torsion free ring and be a nonzero lie ideal of such that
, for all . If is an additive mapping satisfying
, for all then , for all .
Proof: Let
(6)
On the other hand, we have
(7)
Combining the above equations (6) and (7) and using the fact that , we obtain
International Journal of Research In Science & Engineering e-ISSN: 2394-8299
Special Issue –NCRAPAM March 2017 p-ISSN: 2394-8280
IJRISE| www.ijrise.org|editor@ijrise.org [151-155]
We know that
for all .
Theorem 1:Let be a 2-torsion free prime ring and a nonzero lie ideal of such that
, for all . If is an additive mapping satisfying
, for all , then , for all .
Proof:Let the mapping defined by
.
As we can see, is bi additive. If is a commutative lie ideal of , i.e., , for all
, then use the same arguments as used in the proof lemma 1.3 of [11], . Now,
by lemma 2 (iii), we have
(8)
Since for all , we find that for all . This yields that
for all . As the ideal is commutative, in view of lemma 2 (i) we have
This shows that for all
Since is a 2-torsion free ring, we get
(9)
Combining the above equations (8) and (9) and using the fact that , we obtain
, for all .
(10)
Now, replacing by in (10) and using (10) , we get , for all
and and hence , for all . Since and is prime the
above expression yields that , for all . Hence we get the required
result.
Hence, onward we shall assume that is a non-commutative lie ideal of i.e., .
By lemma 3, we have , for all i.e., ,
for all . Thus in view of lemma 1, we find that for each pair either
, for each . Let and
. Hence, and are additive subgroups of whose union is
.
International Journal of Research In Science & Engineering e-ISSN: 2394-8299
Special Issue –NCRAPAM March 2017 p-ISSN: 2394-8280
IJRISE| www.ijrise.org|editor@ijrise.org [151-155]
By Brauer’s trick, we have either or . Again by using the same method we
find that either or . Since is non-
commutative, we find that , for all i.e. is a generalized reverse
derivation on .
Corollary 1: Let be a 2-torsion free prime ring and be a Jordan generalized
reverse derivation. Then is a generalized reverse derivation on .
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[3] Ashraf.M , Rehman.N and Shakir Ali: On lie ideals and Jordan generalized derivations of
prime rings, Indian J. pure appl. Math. Vol.34,2 (2003), 291-294.
[4] Awtar.R: Lie ideals and Jordan derivations of prime rings, Proc. Amer. math. Soc.
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[5] Bergen.J, Herstein.I.N and Kerr.J.W: Lie ideals and derivations of prime rings, J. Algebra
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[10] Herstein.I.N: Jordan derivations of prime rings, Proc. Amer. math. Soc.Vol 8 (1957),
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[11] Herstein .I.N: Topics in ring theory, Univ. of Chicago Press, Chicago, 1969.
[12] Hvala.B: Generalized derivations in rings, Comm. Algebra, Vol.26, (1998), 1147-1166.
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