MULTIPLICATIVE GENERALIZED DERIVATIONS ON IDEALS IN SEMIPRIME RINGS

Abstract

Let R be a ring and I is a nonzero ideal of R. A mapping F : R -> R is called a multiplicative generalized derivation if there exists a mapping g : R -> R such that F(xy) = F(x)y + xg(y), for all x, y is an element of R. In the present paper, we shall prove that R contains a nonzero central ideal if any one of the following holds: i) F([x, y]) = 0, vii) F([x, y]) = +/-[F(x), y], ii) F(x circle y) = 0, viii) F(x circle y) = +/-(F(x)circle y), iii) F([x, y]) = +/-[x, y], ix) F(xy) +/- xy is an element of Z, iv) F(x circle y) =+/-(x circle y), x) F(xy)+/- yx is an element of Z, v) F([x, y]) = +/-(x circle y), xi) F(xy) +/- [x, y]is an element of Z, vi) F(x circle y) = +/-[x, y], xii) F(xy) +/- (x circle y)is an element of Z, for all x, y is an element of I. (C) 2016 Mathematical Institute Slovak Academy of Sciences