On the structure of semiderivations in prime rings

Abstract:

Let $R$ be a prime ring. By a semiderivation associated with a function $g:R \to R$, we mean an additive mapping $f:R \to R$ such that, for all $x,y \in R,f(xy) = f(x)g(y) + xf(y) = f(x)y + g(x)f(y)$ and $f(g(x)) = g(f(x))$. It is known that $g$ must necessarily be a ring endomorphism. Here it is shown that $f$ must be an ordinary derivation or of the form $f(x) = \lambda (x - g(x))$ for all $x \in R$, where $\lambda$ is an element of the extended centroid of $R$.