NONADDITIVE STRONG COMMUTATIVITY PRESERVING DERIVATIONS AND ENDOMORPHISMS

Abstract

Let S be a nonempty subset of a ring R. A map <TEX>$f:R{\rightarrow}R$</TEX> is called strong commutativity preserving on S if [f(x), f(y)] = [x, y] for all <TEX>$x,y{\in}S$</TEX>, 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 <TEX>${\rho}$</TEX> of R, then <TEX>${\rho}{\subseteq}Z$</TEX>, 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 <TEX>$I{\cup}T^{-1}(I)$</TEX>, 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.