Semiderivations of prime rings

Abstract

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