Lie ideals and derivations of exceptional prime rings

TL;DR

This paper extends Herstein's theorem to simple rings, characterizes additive subgroups and derivations in exceptional prime rings, and explores identities involving Lie ideals to deepen understanding of ring structure and derivations.

Contribution

It generalizes Herstein's theorem to all simple rings, including exceptional prime rings, and characterizes derivations satisfying specific identities in these rings.

Findings

Extended Herstein's theorem to arbitrary simple rings.

Characterized additive subgroups in exceptional prime rings.

Described derivations satisfying identities on Lie ideals.

Abstract

A prime ring $R$ with extended centroid $C$ is said to be exceptional if both $\text{\rm char}\,R=2$ and $\dim_CRC=4$. Herstein characterized additive subgroups $A$ of a nonexceptional simple ring $R$ satisfying $\big[A, [R, R]\big]\subseteq A$. In 1972 Lanski and Montgomery extended Herstein's theorem to nonexceptional prime rings. In the paper we first extend Herstein's theorem to arbitrary simple rings. For the prime case, let $R$ be an exceptional prime ring with center $Z(R)$. It is proved that if $A$ is a noncentral additive subgroup of $R$ satisfying $\big[A, L\big]\subseteq A$ for some nonabelian Lie ideal $L$ of $R$, then $\beta Z(R)\subseteq A$ for some nonzero $\beta\in Z(R)$, and either $AC=Ca+C$ for some $a\in A\setminus Z(R)$ with $a^2\in Z(R)$ or $[RC, RC]\subseteq AC$. Secondly, we study certain generalized linear identities satisfied by Lie ideals and then completely characterize derivations $\delta, d$ of $R$ satisfying $\delta d(L)\subseteq Z(R)$ for $L$ a Lie ideal of $R$.