A short characterization of the Octonions

TL;DR

This paper characterizes octonion division algebras through properties of alternative rings, proving conditions under which such rings are division algebras and exploring related algebraic structures.

Contribution

It provides new characterizations of octonion division algebras based on properties of alternative rings and extends results to rings with 3-torsion.

Findings

Proper alternative rings with no 3-torsion and non-zero commutators are free of zero-divisors.

Under certain conditions, the central quotient of such rings is an octonion division algebra.

The paper discusses characterizations of octonion division algebras and cases involving 3-torsion in the additive group.

Abstract

In this paper we prove that if $R$ is a proper alternative ring whose additive group has no $3$-torsion and whose non-zero commutators are not zero-divisors, then $R$ has no zero-divisors. It follows from a theorem of Bruck and Kleinfeld that if, in addition, the characteristic of $R$ is not $2,$ then the central quotient of $R$ is an octonion division algebra over some field. We include other characterizations of octonion division algebras and we also deal with the case where $(R,+)$ has $3$-torsion.