NONADDITIVE STRONG COMMUTATIVITY PRESERVING DERIVATIONS AND ENDOMORPHISMS

Let S be a nonempty subset of a ring R. A map f : R -> R is called strong commutativity preserving on S if [f(x), f (y)] = [x, y] for all x, y is an element of S, where the symbol [x, y] denotes xy - yx. Bell and Daif proved that if a derivation D of a semiprime ring R is strong commutativity preserving on a nonzero right ideal rho of R, then rho subset of Z, the center of R. Also they proved that if an endomorphism T of a semiprime ring R is strong commutativity preserving on a nonzero two-sided ideal I of R and not identity on the ideal I boolean OR T-1 (I), then R contains a nonzero central ideal. This short note shows that the conclusions of Bell and Daif are also true without the additivity of the derivation D and the endomorphism T.