Symmetric Reverse Bi-Derivations on Prime Rings

Abstract

Let R be a 2, 3-torsion free prime ring. Let D: (.,.): R × R → R and dbe a symmetric reverse bi-derivation and the trace of D respectively. If is commuting orcentralizing on R. Then D = 0.Let D1: (.,.): R × R → R, D2: (.,.): R × R → R aresymmetric reverse bi-derivations and B (.,.): R × R → R be a symmetric bi-additive mapping. If D1(d2 (x), x) = 0 and d1(d2(x)) = f(x), for all ∈, where, and are the traces of D1, D2 and B. In this case either D1 = 0or = D2 = 0.

Full text

Research J. Pharm. and Tech. 9(9): September 2016
1496
ISSN 0974-3618 (Print) www.rjptonline.org
0974-360X (Online)
RESEARCH ARTICLE
Symmetric Reverse Bi-Derivations on Prime Rings
Dr. C. Jaya Subba Reddy, M. Ramakrishna Naik
Department of Mathematics, S.V. University, Tirupati –517502, Andhra Pradesh, India.
*Corresponding Author E-mail: cjsreddysvu@gmail.com, ramsanthu950@gmail.com
ABSTRACT:
Let be a 2, 3-torsion free prime ring. Let :(. , . ): × 󱄳 and dbe a symmetric reverse bi-derivation and the
trace of D respectively.If is commuting orcentralizing on R.Then = 0.Let :(. , . ): × 󱄳 ,:(.,.): ×
󱄳aresymmetric reverse bi-derivations and (.,.): × 󱄳 be a symmetric bi-additive mapping. If
( ( ),)= 0 and ( ) = ( ), for all 󱆩, where ,and are the traces of ,and . In this
case either = 0or = 0.
KEY WORDS:Prime ring, Symmetric mapping, Trace, Symmetric bi-additive mapping, Symmetric bi-
derivation, Symmetric reverse bi-derivation.
INTRODUCTION:
The concept of a symmetric bi-derivation has been
introduced by Gy. Maksa in [6], [7]. A classical result in
the theory of centralizing mappings is a theorem first
proved by E. Posner[9]. J.Vukman [10], [11] has studied
some results concerning symmetric bi-derivations on
prime and semi prime rings. Jaya Subba Reddy. C et al
[1], [2], [3], [4], has studied some results concerning
reverse derivations on prime and semi prime rings. In
this paper we proved some results concerning to
symmetric reverse bi-derivations on prime rings.
Received on 08.05.2016 Modified on 27.05.2016
Accepted on 30.05.2016 © RJPT All right reserved
Research J. Pharm. and Tech 2016; 9(9):1496-1500.
DOI: 10.5958/0974-360X.2016.00291.2
Throughout this paper will be associative. We shall
denote by ( ) the center of a ring . Recall that a ring
is prime if = (0) implies that = 0or = 0.
We shall write[ , ] for 󰾴and use the
identities[,]=[,]+ [ , ],[,]=[,]+
[ , ].An additive map : 󱄳 is called derivation if
( ) =( ) + ( ), for all , 󱆩 .A mapping
(. , . ): × 󱄳 is said to be symmetric if (,)=
( , ), for all , 󱆩 .A mapping : 󱄳 defined
by ( ) = ( , ), where (.,.): × 󱄳 is a
symmetric mapping, is called a trace of B.
It is obvious that, in case (.,.): × 󱄳 is
symmetric mapping which is also bi-additive (i. e.
additive in both arguments) the trace of satisfies the
relation (+)=( ) +( ) + 2 ( , ), for
all , 󱆩 .We shall use also the fact that the trace of a
symmetric bi-additive mapping is an even function.A
symmetric bi-additive mapping (.,.): × 󱄳 is
called a symmetric bi-derivation if (,)=
(,)+ ( , ), for all , , 󱆩 .Obviously, in
this case also the relation (,)=(,)+
(,),for all , , 󱆩 . A symmetric bi-additive

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mapping (. , . ): × 󱄳 is called a symmetric
reverse bi-derivation if (,)=(,)+
(,),for all , , 󱆩 .Obviously, in this case also
the relation (,)=(,)+(,),for
all , , 󱆩 . A mapping : 󱄳 is said to be
commuting on if [ ( ),]= 0, for all 󱆩. A
mapping : 󱄳 is said to be centralizing on if
[ ( ),]󱆩 ( ), for all 󱆩. A ring is said to be n-
torsion free if whenever = 0, with 󱆩, then = 0,
where is nonzero integer.
Lemma 1: [8, Lemma 1] Let : 󱄳 be a derivation,
where is a prime ring. Suppose that either (i) ( ) =
0 , for all 󱆩or (ii) ( ) = 0, for all 󱆩holds.
In both the cases we have = 0 or = 0.
Lemma 2: [5, Lemma 3.10] Let R be a 2-torsion free
prime ring and let a, b 󱆩be fixed elements. If +
= 0is fulfilled for all 󱆩, then either a=0 or b=0.
Theorem 1: Let be a 2-torsion free non commutative prime ring. Let (. , . ): × 󱄳 and be a symmetric
reverse bi-derivation and the trace of respectively.Suppose that is commuting on . Then in this case = 0.
Proof: We have d is commuting on R.
[ ( ),]= 0,for all 󱆩. (1)
By linearizing equation (1) in the above equation, we get
[ ( +), + ]= 0
[ ( ) +( ) + 2 (,), + ]= 0
[ ( ),]+[ ( ),]+[ ( ),]+[ ( ),]+ 2[ ( ,),]+ 2[ ( ,),]= 0
By using equation (1), we get
[ ( ),]+[ ( ),]+ 2[ ( ,),]+ 2[ ( ,),]= 0, for all , 󱆩 . (2)
We replace by 󱆳in (2), we get
[ (󱆳),]+[ ( ), 󱆳 ]+ 2[ (󱆳 , ), 󱆳 ]+ 2[ (󱆳 , ),]= 0
[ ( ),]󱆳[ ( ),]+ 2[ ( ,),]󱆳 2[ ( ,),]= 0, for all , 󱆩 . (3)
By adding (2) and (3), we get
[ ( ),]+ 2[ ( ,),]= 0 , for all , 󱆩 . (4)
We replace by in (4), we get
[ ( ),]+ 2[ ( ,),]= 0
[ ( ),]+[ ( ),]+ 2[ ( ) +(,),]= 0
[ ( ),]+[ ( ),]+ 2[ ( ), ] + 2[ (,),]= 0
[ ( ),]+[ ( ),]+ 2 [ ( ), ] + 2[ , ] ( ) + 2[ (,),]+ 2 ( , )[,]= 0
[ ( ),]+ 2[,] ( ) + 2[ ( ,),]= 0, for all , 󱆩 . (5)
We replace d(x) by ( ) in (4), we get
[ ( ) ,]+ 2[ ( ,),]= 0
[ ( ),]+( )[ , ] + 2[ ( ,),]= 0, for all , 󱆩 . (6)
We replace x by –x in (6), we get
[ (󱆳),](󱆳 ) + (󱆳)[󱆳 , ] + 2[ (󱆳 , ), (󱆳 )]= 0
󱆳[ ( ),]󱆳( )[ , ] + 2[ ( ,),]= 0, for all , 󱆩 . (7)
By adding (6) and (7), we get
4[ ( ,),]= 0
Since R is 2-torsion free,we get
[ ( ,),]= 0, for all , 󱆩 . (8)
By using (8) in (4), we get
[ ( ), ] = 0, for all , 󱆩 . (9)
By using (8), (9) in(5), we get
[ , ] ( ) = 0, for all , 󱆩 . (10)
We replace by in (10), we get
[ , ] ( ) = 0
[ , ] ( ) + [ , ] ( ) = 0
By using (10) in the above equation, we get
[ , ] ( ) = 0, since R is prime and noncommutative, which implies that ( ) = 0, for all , 󱆩 , which means
that ( , ) = 0, for all , 󱆩 .

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Theorem 2:Let R be a 2,3-torsion freenoncommutative prime ring. Let D(.,.):R×R󱄳and d be a symmetric
reverse bi-derivation and the trace of D respectively. Suppose that d is centralizing on R. Then in this caseD=0.
Proof: We have[ ( ), ] 󱆩 ( ), for all 󱆩 . (11)
By linearizing (11), we get
[ ( +), + ]󱆩 ( )
[ ( ) + ( ) + 2 ( , ), + ] 󱆩 ( )
[ ( ),]+[ ( ),]+[ ( ),]+[ ( ),]+ 2[ ( ,),]+ 2[ ( , ), ] 󱆩 ( )
By using (11) in the above equation, we get
[ ( ),]+[ ( ),]+ 2[ ( ,),]+ 2[ ( ,),]󱆩( ),for all , 󱆩 . (12)
We replace x by –x in (12), we get
[ (󱆳),]+[ ( ), (󱆳 )]+ 2[ (󱆳 , ), (󱆳 )]+ 2[ (󱆳 , ), ] 󱆩 ( )
[ ( ),]󱆳[ ( ),( )] + 2[ ( ,),( )] 󱆳 2[ ( ,),]󱆩( ), for all , 󱆩 . (13)
By adding (12) and (13), we get
2[ ( ),]+ 4[ (,), ] 󱆩 ( )
[ ( ),]+ 2[ ( ,),]󱆩( ),for all x,y󱆩. (14)
We replace y by in (14), we get
[ ( ),]+ 2[ ( ,),]󱆩( )
[ ( ),]+[ ( ),]+ 2[ ( , ) + ( , ), ] 󱆩 ( )
[ ( ),]+[ ( ),]+ 2[ ( ) ,]+ 2[ ( ), ] 󱆩 ( )
[ ( ),]+[ ( ),]+ 2 ( )[ ,]+ 2[ ( ),]+ 2[,] ( ) + 2 [ ( ), ] 󱆩 ( )
3[ ( ),]+ 3[ ( ),]󱆩( ),for all 󱆩. (15)
By using (11), we have [( ), ] 󱆩 ( )
Which implies [ ( ),], = 0
[ ( ),]=[ ( ),], for all 󱆩(16)
By using (16) in (15), we get
6[ ( ),]󱆩( )
[ ( ),]󱆩( )
[[ ( ),], ] = 0
[ ( ),][ ,]+[ ( ),], = 0
[ ( ),][ ,]= 0,for all , 󱆩 . (17)
We replace y by yr in (17),we get
[ ( ),][ ,]= 0
[ ( ),][ ,]+[ ( ),] [ ,]= 0
By using (17) in the above equation, we get
[ ( ),] [ ,]= 0, for all , , 󱆩 (18)
We replace by ( ) in (18), we get
[ ( ),] [ , ( )]= 0
Since R is primering which implies that [ ( ),]= 0, by using theorem 1 we can complete the proof.
Theorem 3: Let R be a 2-torsion free prime ring, suppose there exist symmetric reverse bi-derivations :(.,.): ×
󱄳, and :(.,.): × 󱄳 such that ( (x),x)=0 holds for all x󱆩,where denotes the trace of .In
this case either =0 or =0.
Proof:We have ( (x),x)=0, for all 󱆩. (19)
We replace x by x+in (19), we get
( ( + ), + ) = 0
(( ) +( ) + 2 ( , ), + ) = 0
( ( ),)+( ( ),)+( ( ),)+ ( (), )+ 2 ( ( , ), )+2 ( ( ,),)= 0
By using (19) in above equation, we get
( ( ),)+( ( ),)+ 2 ( ( , ), )+2 ( ( ,),)= 0,for all , 󱆩 . (20)
We replace x by -x in (20), we get
( (󱆳),)+( ( ), 󱆳 )+ 2 ( (󱆳 , ), 󱆳 )+2 ( (󱆳 , ),)= 0
( ( ),)󱆳( ( ),)+ 2 ( ( ,),)󱆳 2 ( ( ,),)= 0, for all , 󱆩 . (21)

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By adding (20) and (21), we get
( ( ),)+ 2 ( ( ,),)= 0, for all , 󱆩 . (22)
We replace by in (22), we get
( ( ),)+ 2 ( ( ,),)= 0
( ( ),)+( ( ),)+ 2 ( ( ,)+( ) ,)= 0
( ( ),)+ 2 ( ( ,),)+ 2 ( ( ) , ) = 0
( ( ),)+ 2 (,) ( ) + 2 ( ( ,),)
+2 (( ), )+ 2 (,) ( ) = 0
( ( ),)+ 2 (,) ( ) + 2 ( ( ,),)+ 2 (,) ( ) = 0,for all , 󱆩 . (23)
We multiply (22) by x on right hand side, we get
( ( ),)+ 2 ( ( ,),)= 0, for all , 󱆩 . (24)
We subtract (24) from (23), we get
( ( ),)󱆳( ( ),)+ 2 (,) ( ) + 2 (,) ( ) = 0
[,( ( ),)] + 2 (,) ( ) + 2 (,) ( ) = 0, for all , 󱆩 . (25)
Replace y by yx in (25), we get
[,( ( ),)] + 2 (,) ( ) + 2 (,) ( ) = 0
[,( ( ),)+( ( ),) ] + 2 (,) ( ) + 2 (,) ( ) + 2 (,) ( )
+ 2 (,) ( ) = 0
[,( ( ),)] + [,] ( ( ),)+ [ , ( ( ),)] +( ( ),)[ , ] + 2 (,) ( )
+ 2 (,) ( ) + 2 (,) ( ) + 2 (,) ( ) = 0
By using (19), we get
[ , ( ( ),)]+2(,) ( ) + 2 ( ) ( ) + 2 (,) ( ) + 2 ( ) ( ) = 0
{[ ,( ( ),)] + 2 (,) ( ) + 2 (,) ( )} + 2 ( ) ( ) + 2 ( ) ( ) = 0
By using (25) and R is 2-torsion and 3-torsion free, we get
( ) ( ) +( ) ( ) = 0, for all , 󱆩 . (26)
By using lemma 2, we get either ( ) = 0 or ( ) = 0, for all , 󱆩 .
Hence we complete the proof.
Theorem 4: Let R be a 2, 3-torsion free prime ring. Let (.,.):R× 󱄳 and (. , . ):R× 󱄳 besymmetric
reverse bi-derivations.Suppose there exists a symmetric bi-additive mapping B (. , . ):R× 󱄳 , such that
( ) =( ), holds for all 󱆩, where and are the traces of and ,respectively, and f is the trace
of B. Then either = 0 or =0.
Proof: We have ( ) =( ), for all 󱆩. (27)
Linearization of the relation (27), we get
(+)=(+)
( ( ) + ( )+ 2 ( , )) = () + ( )+ 2 ( , )
() + ( )+ 2 (,)+ 2 ( () + ( ), 2 (,)) = () + ( )+ 2 ( , )
()) + ( ( )+ 2 (), ( )+ 4 (,)+ 4 ( ),(,)+ 4 ( ),(,)
=() + ( )+ 2 ( , )
By using (27) in the above equation, we get
4 ( ( , )) + 2 (( ( ), ( )) + 4 ( ( ), ( , )) + 4 ( ( ), ( , )) = 2 ( , )
2(,)+ (( ( ), ( )) + 2 ( ( ), ( , )+ 2 ( ( ),(,)) = (,), for all , 󱆩 .
(28)
We replace x by –x in (28), we get
2(󱆳 , )+ (( (󱆳 ), ( )) + 2 ( (󱆳), (󱆳 , )+ 2 ( ( ),(󱆳 , )) = (󱆳 , )
2(,)+ (( ( ), ( )) 󱆳 2 ( ( ), ( , )󱆳 2 ( ( ),(,)) = 󱆳 (,)for all , 󱆩 .
(29)
Subtract (29) from (28), we get
4 ( ( ),(,))+4 ( ),(,)=2(,)
2 ( ( ),(,))+2 ( ),(,)=(,),for all , 󱆩 . (30)
We replace x by 2x in (30), we get
2 ( (2),(2 , ))+2 ( ),(2 , )=(2 , )

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16( ( ),(,))+4 ( ),(,)=2(2 , ),for all , 󱆩 . (31)
We subtract two times of (30) from (31), we get
12 ( ( ),(,))=0, for all , 󱆩 . (32)
Since R is 2, 3-torsion free ring, which leads to
(( ),(,))=0,for all , 󱆩 . (33)
We replace by and by in (33), we get
(( ),(,))=0, for all , 󱆩 . (34)
By using (33),and (34) in (30), we get
B(x, y)=0,for all , 󱆩 . (35)
By using (35) in (27), we get
( ) = 0,for all , 󱆩 . (36)
We replace y by yx in (33), we get
(( ),(,))=0
(( ),(,)+(,))=0
( ( ),())+( ),(,)=0
( ( ),) ( ) + ( ( ),())+( ),(,)+(,)(( ), ) = 0
By using (33) and (36) in the above equation, we get
( ( ),) ( ) +(,)(( ),) =0,for all , 󱆩 . (37)
We replace by in (37), we get
( ( ),) ( ) +(,)(( ),) =0
( ( ),) ( ) +( ( ),) ( )
+(,)(( ),)+(,)(( ),) =0
( ( ),) ( ) +(,)(( ),)
+(( ( ),) ( ) +(,)(( ),))=0
By using (37) in the above equation, we get
( ( ),) ( ) +(,)(( ),) =0
( ( ),) ( ) +( ) (( ),) =0, for all , 󱆩 .
By using lemma 2, we get either ( ( ),)= 0 which implies that by theorem 3, is central mapping on R or
( ) =0, which implies that is central mapping on R.
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