Commutativity conditions on derivations and Lie ideals $\sigma$-prime rings

TL;DR

This paper investigates conditions under which derivations on $\sigma$-prime rings are trivial or central, focusing on commutativity and Lie ideal properties, extending understanding of ring derivation behaviors.

Contribution

It establishes new conditions linking derivations and Lie ideals in $\sigma$-prime rings, showing when derivations must be zero or central.

Findings

Derivations centralize on $U$ imply $d=0$ or $U$ is central.

If $d$ annihilates commutators in $U$, then $d=0$ or $U$ is central.

Commuting derivations with specific properties lead to trivial or central derivations.

Abstract

Let $R$ be a 2-torsion free $\sigma$-prime ring, $U$ a nonzero square closed $\sigma$-Lie ideal of $R$ and let $d$ be a derivation of $R$. In this paper it is shown that: 1) If $d$ is centralizing on $U$, then $d = 0$ or $U \subseteq Z(R)$. 2) If either $d([x, y]) = 0$ for all $x, y \in U$, or $[d(x), d(y)] = 0$ for all $x, y \in U$ and $d$ commutes with $\sigma$ on $U$, then $d = 0$ or $U \subseteq Z(R)$.