Multiplicative (generalized)-derivations and left ideals in semiprime rings

Abstract

Let R be a semiprime ring with center Z(R). A mapping F : R → R (not necessarily additive) is said to be a multiplicative (generalized)- derivation if there exists a map f : R → R (not necessarily a derivation nor an additive map) such that F (xy) = F (x)y + xf (y) holds for all x,y ∈ R. The objective of the present paper is to study the following identities: (i) F(x)F(y) ± [x,y] ∈ Z(R), (ii) F(x)F(y) ± x ◦ y ∈ Z(R), (iii) F([x,y]) ± [x,y] ∈ Z(R), (iv) F(x ◦ y) ± (x ◦ y) ∈ Z(R), (v) F([x,y]) ± [F(x),y] ∈ Z(R), (vi) F(x ◦ y) ± (F(x) ◦ y) ∈ Z(R), (vii) [F(x),y] ± [G(y),x] ∈ Z(R), (viii) F([x,y]) ± [F(x),F(y)] = 0, (ix) F(x◦y)±(F(x)◦F(y)) = 0, (x) F(xy)±[x,y] ∈ Z(R) and (xi) F(xy)±x◦y ∈ Z(R) for all x,y in some appropriate subset of R, where G : R → R is a multiplicative (generalized)-derivation associated with the map g : R → R.