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

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 is an element of R. The objective of the present paper is to study the following identities: (i) F(x) F(y) +/- [x, y] is an element of Z(R), (ii) F(x) F(y) +/- x omicron y is an element of Z(R), (iii) F([x, y]) +/- [x, y] is an element of Z(R), (iv) F(x omicron y) +/- (x omicron y) is an element of Z(R), (v) F([x, y]) +/- [F(x), y] is an element of Z(R), (vi) F(x omicron y) +/- (F(x) omicron y) is an element of Z(R), (vii) [F(x), y] +/- [G(y), x] is an element of Z(R), (viii) F([x, y]) +/- [F(x), F(y)] = 0, (ix) F(x omicron y) +/- (F(x) omicron F(y)) = 0, (x) F(xy) +/- [x, y] is an element of Z(R) and (xi) F(xy) +/- x omicron y is an element of 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