There has been a number of publications on how the structure of \(R\) is related to the behavior of derivations on \(R\). For example, I. N. Herstein [Contemp. Math. 13, 162–171 (1982; Zbl 0503.16002)] proved that if \(R\) is a ring with center \(Z(R)\), \(n\) is a positive integer, and \(d\) is a derivation of \(R\) such that \(d(x)^n\in Z(R)\) for all \(x\in R\), then \(R\subseteq M_2(K)\), the ring of \(2\times 2\) matrices over a field \(K\).
In this paper, the author aims to investigate the relation between the structure of prime rings and generalized skew derivations with an annihilating condition and obtains the following result.
Let \(R\) be a prime ring of characteristic different from 2, let \(Q_r\) be its right Martindale quotient ring, and let \(C\) be its extended centroid. Suppose that \(F\) is a generalized skew derivation of \(R\), \(L\) a non-central Lie ideal of \(R\), \(0\not =a\in R\) and \(m\geq 0\), \(s\geq 1\) fixed integers. If \(a\big (u^mF(u)u^n)^s=0\) for all \(u\in L\), then either \(R\subseteq M_2(C)\), the ring of \(2\times 2\) matrices over \(R\), or \(m=0\) and there exists \(b\in Q_r\) such that \(F(x)=bx\) for any \(x\in R\) with \(ab=0\).