Lie Ideals With Multiplicative (Generalized)-Derivations of Semiprime Rings

Let R be a 2-torsion free semiprime ring, U a square-closed Lie ideal of R, f a multiplicative (generalized)-derivation with the additive map d of R: In the presentpaper, we shall prove that R contains a nonzero central ideal if any one of the followingholds: i) F(x)F(y)±[x; y] ∈ Z, ii) F(x)F(y)±(x◦y) ∈ Z, iii) F([x; y]) = ±(xy ± yx), iv) F(x ◦ y) = ±(xy ± yx), v) F(xy) ± F(x)F(y) = 0, vi) F(xy) ± F(y)F(x) = 0 for all x; y ∈ U.