Generalized derivations with annihilating and centralizing Engel conditions on Lie ideals

Abstract

Let \(R\) be a non-commutative prime ring, with center \(Z(R)\), extended centroid \(C\) and let \(F\) be a non-zero generalized derivation of \(R\). Denote by \(L\) a non-central Lie ideal of \(R\). If there exists \(0\ne a\in R\) such that \(a[F(x),x]_k\in Z(R)\) for all \(x\in L\), where \(k\) is a fixed integer, then one of the followings holds: (1) either there exists \(\lambda \in C\) such that \(F(x)=\lambda x\) for all \(x\in R\), (2) or \(R\) satisfies \(s_4\), the standard identity in \(4\) variables, and \(char(R)=2\); (3) or \(R\) satisfies \(s_4\) and there exist \(q\in U, \gamma \in C\) such that \(F(x)=qx+xq+\gamma x\).