A characterization of the quaternions using commutators

Abstract: Let

R

be an associative ring with

which is not commutative. Assume that any non-zero commutator

v i n 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

m c h a r (R) e 2,

then the localization of

R

at its center is a quaternion division algebra.