Elementary proofs of ring commutativity theorems

Abstract

Jacobson's commutativity theorem says that a ring is commutative if, for each $x$, $x^n = x$ for some $n > 1$. Herstein's generalization says that the condition can be weakened to $x^n-x$ being central. In both theorems, $n$ may depend on $x$. In this paper, in certain cases where $n$ is a fixed constant, we find equational proofs of each theorem. For the odd exponent cases $n = 2k+1$ of Jacobson's theorem, our main tool is a lemma stating that for each $x$, $x^k$ is central. For Herstein's theorem, we consider the cases $n=4$ and $n=8$, obtaining proofs with the assistance of the automated theorem prover Prover9.