Let R be a ring such that, for each element a of R, there exists a positive integer n(a) > 1, depending on a, such that an(a)=a. Jacobson proved that such a ring R is necessarily commutative and the purpose of this paper is to give a proof of Jacobson’s Theorem that does not involve the use of the axiom of choice.

We take this opportunity to point out that all rings are assumed to be associative and that nothing beyond elementary ring theory is assumed in the proof to follow. Such ring theory can be found, for example, in [1].