Notes on Automorphisms of Prime Rings

Abstract

Let $R$ be a prime ring, $L$ a noncentral Lie ideal and $σ$ a nontrivial automorphism of $R$ such that $u^sσ(u)u^t = 0$ for all $u ∈ L$, where $s$, $t$ are fixed non-negative integers. If either char$R > s + t$ or char$R = 0$, then $R$ satisfies $s_4$, the standard identity in four variables. We also examine the identity $(σ([x, y])−[x, y])^n = 0$ for all $x, y ∈ I$, where $I$ is a nonzero ideal of $R$ and $n$ is a fixed positive integer. If either char$R > n$ or char$R = 0$, then $R$ is commutative.