GENERALIZED DERIVATIONS WITH NILPOTENT VALUES ON MULTILINEAR POLYNOMIALS

Abstract

Let $R$ be a prime ring without nonzero nil one-sided ideals. Suppose that $g$ is a generalized derivation of $R$ and that $f(X_{1},\cdots,X_{k})$ is a multilinear polynomial not central-valued on $R$ such that $g(f(x_{1},\cdots,x_{k}))$ is nilpotent for all $x_{1},\cdots,x_{k}$ in some nonzero ideal of $R$. Then $g=0$.