A characterization of the quaternions using commutators

Abstract

Let $R$ be an associative ring with $1$ 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 $\mathrm{char}(\mathrm{R})\ne 2,$ then the localization of $R$ at its center is a quaternion division algebra.

4 pages