Lie Ideals of Semiprime Rings with Generalized Derivations

Let R be a 2- torsion free semiprime ring, U a noncentral square-closed Lie ideal of R. A map F:R→R is called a generalized derivations if there exists a derivation d:R→R such that F(xy)=F(x)y+xd(y), for all x,y∈R. In the present paper, we shall prove that h is commuting map on U if any one of the following holds: i) F(u)u=±uG(u), ii) [F(u),v]=±[u,G(v)], iii) F(u)°v=± u°G(v), iv) [F(u),v]=±u°G(v), v)F([u,v])=[F(u),v]+[d(v),u] for all u,v∈U, where G:R→R is a generalized derivation associated with the derivation h:R→R