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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 toBirkhoff's Completeness Theorem, if $n$ is fixed, there must be an equationalproof of this theorem. But equational proofs have only appeared for some valuesof $n$ so far. This paper is about finding such a proof in general. We are ableto make a reduction to the case that $n$ is a prime power $p^k$ and the ringhas characteristic $p$. We then prove the special cases $k=1$ and $k=2$. Thegeneral case is reduced to a series of constructive Wedderburn Theorems, whichwe can prove in many special cases. Several examples of equational proofs arediscussed in detail.

Equational proofs of Jacobson's Theorem