A Note on n-Commuting Generalized Skew Derivations on Prime Rings

Let \(\mathcal {R}\) be a prime ring, \(\mathcal {Q}_r\) the right Martindale quotient ring of \(\mathcal {R}\), \(\mathcal {C}\) the extended centroid of \(\mathcal {R}\), \(\mathcal {I}\) a noncentral ideal of \(\mathcal {R}\), F a nonzero generalized skew derivation of \(\mathcal {R}\), and \(m,n,s \ge 1\) be fixed integers, such that \([F(u^m)u^n,u^s]=0\), for all \(u \in \mathcal {I}\). If either \(char(R)=0\) or \(char(R)=p\ne 2\) and \(p\not \mid s\), then there exists \(a \in \mathcal {Q}_r\) such that \(F(x)=xa\), for all \(x\in \mathcal {R}\).