# A uniform characterization of the octonions and the quaternions using commutators

Erwin Kleinfeld, Yoav Segev

Research output: Working paper/PreprintPreprint

## 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.
Original language English Published - 8 Dec 2021

## Keywords

• math.RA
• math.GR
• Primary: 12E15, 17D05, Secondary: 11R52, 17A35

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