Equational proofs of Jacobson's Theorem

Abstract

A classical theorem by Jacobson says that a ring in which every element $x$ satisfies the equation $x^n=x$ for some $n>1$ is commutative. According to Birkhoff's Completeness Theorem, if $n$ is fixed, there must be an equational proof of this theorem. But equational proofs have only appeared for some values of $n$ so far. This paper is about finding such a proof in general. We are able to make a reduction to the case that $n$ is a prime power $p^k$ and the ring has characteristic $p$. We then prove the special cases $k=1$ and $k=2$. The general case is reduced to a series of constructive Wedderburn Theorems, which we can prove in many special cases. Several examples of equational proofs are discussed in detail.