Elementary proofs of ring commutativity theorems

TL;DR

The paper addresses the problem of proving Jacobson's and Herstein's commutativity theorems in the fixed-exponent setting, i.e., for rings where a single $n eq 1$ satisfies $x^n = x$ for all $x$. It introduces an elementary equational approach, including a new lemma for odd $n=2k+1$ that $x^k$ is central, and provides explicit equational proofs for several small exponents ($n=2,3,4,5,7$) via reduced-ring and commutator techniques. For Herstein's theorem, it treats $n=2,4,8$, developing an ad(x) (adjoint) framework and leveraging automated proving (Prover9) to obtain proofs that are then humanized; the section culminates in a complete commutativity result for these fixed exponents. The results illuminate how fixed-exponent rings behave under equational reasoning and demonstrate the effective use of automated tools in constructing transparent proofs, with reduced and reversible rings playing a key auxiliary role. The work suggests avenues for extending equational methods to broader fixed-exponent cases and clarifies structural aspects of $n$-potent rings.

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.

Theorems & Definitions (49)

• lemma 1
• proof
• lemma 2
• proof
• lemma 3
• proof
• lemma 4
• proof
• lemma 5
• proof