On commutativity of prime rings with skew derivations

Abstract

Let $\mathcal{R}$ be a prime ring of Char$(\mathcal{R})\ne 2$ and $m\ne 1$ be a positive integer. If $S$ is a nonzero skew derivation with an associated automorphism $\mathcal{T}$ of $\mathcal{R}$ such that $([S([a,b]),[a,b]]{)}^m=[S([a,b]),[a,b]]$ for all $a,b\in \mathcal{R}$, then $\mathcal{R}$ is commutative.