Some identities related to multiplicative (generalized)-derivations in prime and semiprime rings

Let R be a semiprime ring with center Z(R) and lambda a nonzero left ideal of R. A mapping F : R -> R (not necessarily additive) is said to be a multiplicative (generalized)-derivation on R, if there exists a map d (not necessarily an additive map or derivation) on R such that F(xy) = F(x)y + xd(y) holds for all x, y is an element of R. Suppose that F and G are two multiplicative (generalized)-derivations of R associated with the maps d and g respectively on R. Throughout this paper we study the following situations: (1) F([x, y]) + G(yx) + d(x)F(y) + xy is an element of Z(R), (2) F(xoy) + G(yx) + d(x)F(y) + xy is an element of Z(R), (3) F(xy) + G(yx) + d(x)F(y) +/- [x,y] is an element of Z(R), (4) F([x,y]) + G(xy) + d(x)F(y) + yx is an element of Z(R), (5) F(xoy) + G(xy) + d(x)F(y) + yx is an element of Z(R), (6) F([x, y]) + G(yx) + d(y)F(x) - xy is an element of Z(R), (7) F(x)F(y) - G(yx) - xy + yx is an element of Z(R); for all x, y is an element of lambda.