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

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