Semiderivations and Commutativity in Semiprime Rings

Let R be a semiprime ring. An additive mapping f : R → R is called a semiderivation if there exists 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 2 R. In the present paper we investigate commutativity of R satisfying any one of the properties (i) [f(x); f(y)] = 0, (ii) [f(x); f(y)] = [x; y], (iii) [f(x); d(y)] = [x; y], d is a derivation on R, or (iv) f([x; y]) = [x; y], for all x; y in some appropriate subset of R. Also we extend two results of Bell and Martindale from prime rings to semiprime rings.