• LEFT MULTIPLICATIVE GENERALIZED DERIVATIONS IN SEMIPRIME RINGS

Abstract

Let R be a ring with center Z(R). An additive mapping F: R→R is said to be a left multiplicative generalized derivation if there exists a derivation d: R→R such thatF(xy)=xF(y)+d(x)y for all x, y ∈ R (the map d is called the derivation associated with F). In the present note we prove that if a semiprime ring R admits a generalized derivation F, d is the nonzero associated derivation of F, satisfying certain polynomial constraints on a nonzero ideal I, then R contains a nonzero central ideal.