GENERALIZED DERIVATIONS WITH ANNIHILATOR CONDITIONS IN PRIME RINGS

Abstract

Let $R$ be a noncommutative primering with its Utumi ring of quotients $U$, $C=Z(U)$ the extendedcentroid of $R$, $F$ a generalized derivation of $R$ and $I$ anonzero ideal of $R$. Suppose that there exists $0\neq a\in R$ such that $a(F([x,y])^n-[x,y])=0$ for all $x,y \in I$, where $n\geq 1$ is a fixedinteger. Then either $n=1$ and $F(x)=bx$ for all $x\in R$ with$a(b-1)=0$ or $n\geq 2$ and one of the following holds:

1. char $(R)\neq 2$, $R\subseteq M_2(C)$, $F(x)=bx$ for all$x\in R$ with $a(b-1)=0$ (In this case $n$ is an odd integer);

2. char $(R)= 2$, $R\subseteq M_2(C)$ and $F(x)=bx+[c,x]$ forall $x\in R$ with $a(b^n-1)=0$.