On a Functional Identity Involving Power Values of Generalized Skew Derivations on Lie Ideals

Let R be a prime ring, \(Q_r\) its right Martindale quotient ring and C its extended centroid. Suppose that F is a generalized skew derivation of R, L a non-central Lie ideal of R, \(m,n,s\ge 1\) positive fixed integers. If \( F(u)^n\biggl (F(u)^m-u^m\biggr )^s=0 \) for all \(u\in L\) , then there exists \(\lambda \in C\) such that \(F(x)=\lambda x\) , for any \(x\in R\) , with \(\lambda ^m=1\) , unless when \(char(R)=2\) and \(R\subseteq M_2(K)\) , the ring of \(2\times 2\) matrices over a field K. We will also provide a generalization of the previous result for semiprime rings.