Generalized derivations on Lie ideals in semiprime rings

Abstract

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 \(d(U)\subseteq Z(R)\). 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 \(d(R), F(R)\subseteq C_{R}(U)\). 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 \(u, v \in U\), then \(d(U), g(U)\subseteq C_{R}(U)\).