A characterization of the quaternions using commutators

TL;DR

This paper characterizes the structure of certain non-commutative rings by showing they are division algebras related to quaternions under specific conditions involving commutators.

Contribution

It provides a novel characterization of rings with commutator properties, demonstrating they are division algebras akin to quaternions when localized at their center.

Findings

Rings with specified commutator properties have no zero divisors.

Such rings are quaternion division algebras when localized at their center (if char ≠ 2).

The assumptions do not require finite dimensionality.

Abstract

Let $R$ be an associative ring with ${\bf 1}$ which is not commutative. Assume that any non-zero commutator $v\in R$ satisfies: $v^2$ is in the center of $R$ and $v$ is not a zero-divisor. (Note that our assumptions do not include finite dimensionality.) We prove that $R$ has no zero divisors, and that if ${\rm char(R)}\ne 2,$ then the localization of $R$ at its center is a quaternion division algebra.