Semiderivations of prime rings

Abstract:

A semiderivation of a ring $R$ is an additive mapping $f:R \to R$ together with a function $g:R \to R$ such that $f(xy) = f(x)g(y) + xf(y) = f(x)y + g(x)f(y)$ and $f(g(x)) = g(f(x))$ for all $x,y \in R$. We prove that the only semiderivations of prime rings are derivations and mappings of the form $f(x) = \lambda (x - g(x))$, where $g$ is an endomorphism and $\lambda$ is an element in the extended centroid.