Alternative rings whose associators are not zero-divisors

TL;DR

This paper proves that alternative rings with associators not being zero-divisors have no zero divisors, and under certain conditions, their central quotient forms an octonion division algebra.

Contribution

It establishes a new link between associator properties and zero divisor absence in alternative rings, extending previous results to a broader class.

Findings

Alternative rings with non-zero-divisor associators have no zero divisors.

Under characteristic not 2, their central quotient is an octonion division algebra.

Provides conditions under which alternative rings are division algebras.

Abstract

The purpose of this short note is to prove that if $R$ is an alternative ring whose associators are not zero-divisors, then $R$ has no zero divisors. By a result of Bruck and Kleinfeld, if, in addition, the characteristic of $R$ is not $2,$ then the central quotient of $R$ is an octonion division algebra over some field.