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International Research Journal of Pure Algebra-5(10), 2015, 160-163
Available online through www.rjpa.info ISSN 2248–9037
International Research Journal of Pure Algebra-Vol.-5(10), Oct. – 2015 160
LIE IDEALS AND LEFT JORDAN GENERALIZED DERIVATIONS OF PRIME RINGS
C. JAYA SUBBA REDDY*1, S. MALLIKARJUNA RAO2
1,2Department of Mathematics, S. V. University, Tirupati-517502, Andhra Pradesh, India.
(Received On: 13-09-15; Revised & Accepted On: 08-10-15)
ABSTRACT
Let be a ring and a nonempty subset of . An additive mapping : is called Left generalized derivation
(Left Jordan derivation) on .If there exists a derivation : such that ()=()+()(respect to left
Jordan generalized derivation (2)=()+()) holds for all ,in . Suppose that is a 2- torsion free
prime ring and a non zero Lie ideal of such that 2in for all in. In this paper we proved that if is a left
Jordan generalized derivation on , then is a left generalized derivation on
Key words: Prime ring, Derivation, Generalized derivation, Left generalized derivation, Jordan generalizedderivation,
Left Jordan generalized derivation, Lie ideals.
INTRODUCTION
Throughout this paper will denote an associative ring with the Centre(). Recall that prime if = (0) implise
that = 0or= 0. As usual [,] will denote the commutator . An additive subgroup of is said to be Lie
ideal of if [,] for all , .We shall make use of Commutator identities: [,]=[,]+[,]
and [,]=[,]+[,]. An additive mapping: is called a derivation (resp. Jordan derivation) if
()=()+(), ((2)=()+()), holds for all , .An additive mapping : is called a
generalized derivation (resp.Jordan generalized derivation) if there exists a derivation : such that
()=()+() (resp.(2)=()+()) for all , . An additive mapping : is called a
left generalized derivation (resp.left Jordan generalized derivation) if there exists a derivation : Suchthat
()=()+() (resp.(2)=()+()) for all , . Obviously, every derivation is a Jordan
derivation. The converse is need not be true in general. Afamous result due to Herstein [11] states that every Jordan
derivation on 2-torsion free prime ring is a derivation. A brief proof of this result is presented in [8]. Further, Awtar [5]
generalized this result on Lie ideals. Hvala [12] states that every generalized derivation on a ring is a Jordan
generalized derivation. But the converse statement does not holds in general. In [4] states that ifis a 2-torsion free
prime ring and a non zero Lie ideal of such that every Jordan generalized derivation on is a generalized
derivation. The aim of present paper is a 2-torsion free prime ring and a non zero lie ideal of such that every left
Jordan generalized derivation on is a left generalized derivation .
We begin with the following result which is essentially proved in [6].
Lemma 1: If is a Lie ideal of a 2-torsion free prime ring and , such that =(0), then
= 0 = 0.
We define a mapping :2 such that (,)=() () ()
Now we see that (,+)=(,)+(,)and (+,)=(,)+(,) for all ,, .
By the definition
(,+)=(+) ()(+) (+)
=( +) () () () ()
=() () ()+() () ()
=(,)+(,)
(,+)=(,)+(,) for all ,, .
*Corresponding Author: Dr. C. Jaya Subba Reddy*1
C. Jaya Subba Reddy*1, S. Mallikarjuna Rao2 /
Lie Ideals and Left Jordan Generalized Derivations of Prime Rings / IRJPA- 5(10), Oct.-2015.
© 2015, RJPA. All Rights Reserved 161
Similarly we have prove that (+,)=(,)+(,) for all ,, .
More over if is zero then is left generalized derivation on .
Lemma 2: Let be a 2- torsion free ring and be a non zero Lie ideal of such that 2 , for all . If
: is an additive mapping satisfying (2)=()+() for all then
1. ( +)=()+()+()+(), for all , .
2. ()=() +()+(), for all , .
3. (+. )=() +()+()+() +()+()for all ,, .
Proof:
) (+)2=((+)(+)) = (+)(+)+(+)(+)
=()+()+()+()+()+()+()+().
(+)2=()+()+()+()+()+()+()+(). (1)
On the other hand, we have
(+)2=((+)(+) )
= (2+ + +2) = (2) + ( +) + (2)
=()+()+( +)+()+().
(+)2= ()+()+( +)+()+(). (2)
From (1) & (2), we have
()+()+()+()+()+()+()+()
=()+()+( +)+()+()
( +)=()+()+()+().
ii) Let =(( +)+( +)).
On one hand, we have
=()( +)+( +)+( +)+( +)()
=() +() +()+()+()+()+()+()+()+() +()
+()
=() +() +()+2() + ()+()) + (()+() + ()+())
+() + (). (3)
On the other hand, we have
=(2+ 2 +2).
=(2)+2() + 2() + ()2+(2)
= (( )+())+2() + 2() + ()2+(()+())
=( )+() +2()+ 2()+()2+()+().(4)
From (3) & (4), we have
() +() +()+2() + ()+()) + () +()+()2+()+()
+()=( ) +()+2() + 2() + ()2+()+()
2()= 2() + 2()+ 2 ()
Since is 2-torsion free, we get
()=() +()+().
() Linearizing (ii) by replacing by +
(+)(+)=(+)(+)+(+)()(+)+(+)(+)
From L.H.S
(+)(+)=( +)(+)
=( + + +)
=() + () + () + ()
=() +()+()+( +)+() +()+()
C. Jaya Subba Reddy*1, S. Mallikarjuna Rao2 /
Lie Ideals and Left Jordan Generalized Derivations of Prime Rings / IRJPA- 5(10), Oct.-2015.
© 2015, RJPA. All Rights Reserved 162
(+)(+)=() +()+()+( +)+() +()+(). (5)
From R.H.S
(+)(+)+(+)()(+)+(+)(+)
= (() + ())( +)+ (() + ())(+)+( +)(() + ())
=() +() +() +() +()+()+()+()+ ()
+()+()+(). (6)
From (5) & (6), we get
() +()+()+( +)+() +()+()
=() +() +() +() +()+()+()+()+ ()
+()+()+()
( +) = () +()+()+ () +()+().
Lemma 3: Let be a 2-torsion free ring and be a nonzero Lie ideal of such that 2 for all .If :
is an additive mapping satisfying (2)=()+(), for all , then (,)[,]= 0, for all
,, .
Proof: Let =( +)
=() +() +()+() +()
= () +() +() +()+() +() +()
+(). (7)
On the other hand
=(()+())
=() +()+()+() +()+()
= () +() +() +()+()+() +()
+() +()+(). (8)
From (7) & (8), we have
() +() +() +()+() +() +()()+()
= () +() +() +()+()+() +()
+() +()+()
()+()=()+()+()+()
() () ()+() () ()= 0
(,)+(,)= 0
We know that
(,)=(,)
( )(,)= 0
[,](,)= 0 for all ,,
MAIN RESULT
The main result of the present paper as follows
Theorem 1: Let be a 2-torsion free prime ring and a non zero Lie ideal of such that 2 for all . If is
an additive mapping of into itself satisfying (2)=()+() for all , then ()=()+()
for all .
Proof: If is commutative lie ideal of i.e [,]= 0 for all, , then use the same argument as used in the proof
of lemma 1. 3 of [11], .Now by lemma 2(iii) we have
( +)=() + () + ()+() +()+(). (9)
C. Jaya Subba Reddy*1, S. Mallikarjuna Rao2 /
Lie Ideals and Left Jordan Generalized Derivations of Prime Rings / IRJPA- 5(10), Oct.-2015.
© 2015, RJPA. All Rights Reserved 163
Since 2 ,for all we find that + for all , . This yields that 2 for all , . As the
ideal is commutative, in view of lemma 2 (i) we have
2( +)=(2)+2()
2( +)= 2(()+()+() +()
This shows that for all , .
( +)=() +()+()+() +(). (10)
Using (9) & (10) and using the fact of = we obtain
() + () + ()+() +()+()
=() +()+()+() +()
() () ()= 0
(,)= 0 for all ,, . (11)
Now, replacing by [,] in (11) and using (11), we get(,)= 0 for all ,, and and hence
(,)= 0 for all , . since 0 and is prime the above expression yields that (,)= 0 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
[,] (,)= 0 for all , . Thus in view of lemma 1, we find that for each pair , either [,]= 0 or
(,)= 0 for . Let 1={ |[,]= 0} 2= { |(,)= 0 }. Hence 1 and 2 are additive
subgroups of whose union is By Brauer’s trick, we have either =1 or =2. Again by using the same
method we find that either = { |=1} or = { |=2}. Since is non-commutative, we find that
(,)= 0, for all , i.e. is left generalized derivation on .
Corollary: let be a 2-torsion free prime ring and : be a left Jordan generalized derivation. Then is a left
generalized derivation on .
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Source of Support: Nil, Conflict of interest: None Declared
[Copy right © 2015, RJPA. All Rights Reserved. This is an Open Access article distributed under the terms of
the International Research Journal of Pure Algebra (IRJPA), which permits unrestricted use, distribution,
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