Semiderivations and Commutativity in Prime Rings

Abstract

AbstractA semiderivation of a ring R is an additive mapping f:R → R together with a function g:R → 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 ∊ R. Motivating examples are derivations and mappings of the form x → x — g(x), g a ring endomorphism. A semiderivation f of R is centralizing on an ideal U if [f(u), u] is central for all u ∊ U. For R prime of char. ≠2, U a nonzero ideal of R, and 0 ≠ f a semiderivation of R we prove: (1) if f is centralizing on U then either R is commutative or f is essentially one of the motivating examples, (2) if [f(U), f(U) ] is central then R is commutative.