GENERALIZED SKEW DERIVATIONS WITH ANNIHILATING ENGEL CONDITIONS

Abstract

Let $R$ be a noncommutative prime ring and $a\in R$. Suppose that $f$ is a right generalized $\beta$-derivation of $R$ such that $a[f(x),x]_k=0$ for all $x\in R$, where $k$ is a fixed positive integer. Then $a=0$ or there exists $s\in C$ such that $f(x)=sx$ for all $x\in R$ except when $R=M_2(GF(2))$.