Extending a theorem of Herstein

TL;DR

This paper extends Herstein's results on simple rings to just infinite associative algebras, showing that certain Lie algebra structures derived from them are also just infinite under specific characteristic conditions.

Contribution

It generalizes Herstein's theorems from simple rings to the broader class of just infinite algebras, particularly in the context of Lie algebra structures.

Findings

Lie algebra $[A,A]/(Zigcap[A,A])$ is just infinite for characteristic not 2,3,5

Extends Herstein's results to just infinite algebras

Supports the notion that just infinite algebras generalize simple rings

Abstract

Just infinite algebras have been considered from various perspectives; a common thread in these treatments is that the notion of just infinite is an extension of the notion of simple. We reinforce this generalization by considering some well-known results of Herstein regarding simple rings and their Lie and Jordan structures and extend these results to their just infinite analogues. In particular, we prove that if A is a just infinite associative algebra, of characteristic not 2,3, or 5, then the Lie algebra $[A,A]/(Z\cap[A,A])$ is also just infinite (where Z denotes the center of A).