On commutativity of prime rings with skew derivations

TL;DR

This paper proves that under certain conditions involving skew derivations and automorphisms, a prime ring must be commutative, extending understanding of ring structure constraints.

Contribution

It establishes a new criterion linking skew derivations and automorphisms to the commutativity of prime rings.

Findings

Prime rings with specific skew derivation conditions are commutative.

The paper generalizes previous results on ring commutativity.

Conditions involve automorphisms and powers of commutators.

Abstract

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