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 dt, if dt2(U)=0 for a Lie ideal U of R then dt(U) = 0. Carini (Rend Circ Mat Palermo 34:122–126, 1985) extended this result for an arbitrary derivation d, proving that d2(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 F2(U) = (0) , then d3(U) = (0) and (d2(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) ⊆ CR(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) ⊆ CR(U). © 2016, The Managing Editors.