Generalized derivations on Lie ideals in semiprime rings

Herstein (J Algebra 14:561–571, 1970 ) proved that given a semiprime 2-torsion free ring R and an inner derivation d_t , if d_t^2(U)=0 for a Lie ideal U of R then d_t(U)=0 . Carini (Rend Circ Mat Palermo 34:122–126, 1985 ) extended this result for an arbitrary derivation d , proving that d^2(U)=0 implies d(U)⊆ Z(R) . The aim of this paper is to extend the results mentioned above for right (resp. left) generalized derivations. Precisely, we prove that if R admits a right generalized derivation F associated with a derivation d such that F^2(U) = (0) , then d^3(U)= (0) and (d^2(U))^2= (0) . Furthermore, if F is also a left generalized derivation on U , then d(U)=F(U)=(0) , and d(R), F(R)⊆ C_R(U) . On the other hand, if ( F , d ), ( G , g ) are, respectively, right and left generalized derivations that satisfy F(u)v=uG(v) for all u, v ∈ U , then d(U), g(U)⊆ C_R(U) .