On multiplicative (generalized)-derivations in prime and semiprime rings

Let R be a ring. A map ${F : R rightarrow R}$ F : R → R is called a multiplicative (generalized)-derivation if F(xy) = F(x)y + xg(y) is fulfilled for all ${x, y in R}$ x , y ∈ R where ${g : R rightarrow R}$ g : R → R is any map (not necessarily derivation). The main objective of the present paper is to study the following situations: (i) ${F(xy) pm xy in Z}$ F ( xy ) ± xy ∈ Z , (ii) ${F(xy) pm yx in Z}$ F ( xy ) ± yx ∈ Z , (iii) ${F(x)F(y) pm xy in Z}$ F ( x ) F ( y ) ± xy ∈ Z and (iv) ${F(x)F(y) pm yx in Z}$ F ( x ) F ( y ) ± yx ∈ Z for all x, y in some appropriate subset of R. Moreover, some examples are also given.