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arXiv · 2112.04250

A uniform characterization of the octonions and the quaternions using commutators

Erwin Kleinfeld·Yoav Segev
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Abstract

Let $R$ be a ring with ${\bf 1}$ which is not commutative. Assume that a non-zero commutator in $R$ is not a zero divisor. Assume further that either $R$ is alternative, but not associative, or $R$ is associative and any commutator $v\in R$ satisfies: $v^2$ is in the center of $R.$ We prove that $R$ has no zero divisors. Furthermore, if $\text{char}(R)\ne 2,$ then the localization of $R$ at its center is an octonion division algebra, if $R$ is alternative and a quaternion division algebra, if $R$ is associative. Our proof in both cases is essentially the same and it is elementary and rather self contained.

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Erwin Kleinfeld, Yoav Segev. 2021-12-08. A uniform characterization of the octonions and the quaternions using commutators. https://arxiv.org/abs/2112.04250

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