Strong Commutativity Preserving Skew Derivations in Semiprime Rings

Abstract

Let R be a semiprime ring of characteristic different from 2, C its extended centroid, Z(R) its center, F and G non-zero skew derivations of R with associated automorphism \(\alpha \) and m, n positive integers such that $$\begin{aligned}{}[F(x),G(y)]_m=[x,y]^n ~\mathrm{for \, all}~x,y \in R. \end{aligned}$$ Then R is commutative.