ANNIHILATOR CONDITIONS OF MULTIPLICATIVE REVERSE DERIVATIONS ON PRIME RINGS

Abstract

 Let $R$ be a ring, a mapping $F:R\rightarrow R$ together with a mapping $d:R\rightarrow R$

is called a multiplicative (generalized)-reverse derivation if

$F(xy)=F(y)x+yd(x)$ for all $x,y\in R$. The aim of this note is to

investigate the commutativity of prime rings admitting

multiplicative (generalized)-reverse derivations. Precisely, it is

proved that for some nonzero element $a$ in $R$ the conditions:

$a(F(xy)\pm xy)=0$, $a(F(x)F(y)\pm xy)=0$, $a(F(xy)\pm

F(y)F(x))=0$, $a(F(x)F(y)\pm yx)=0$, $a(F(xy)\pm yx)=0$ are

sufficient for the commutativity of $R$. Moreover, we describe the

possible forms of generalized reverse derivations of prime rings.

Keywords

• Prime ring,multiplicative (generalized)-reverse derivation,generalized reverse derivation,annihilator conditions

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