A Characterization of the Quaternions Using Commutators

Let R be an associative ring with 1, which is not commutative. Assume that any non-zero commutator v ∈ R satisfies: v2 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 char(R) 6= 2, then the localization of R at its center is a quaternion division algebra.