A Note On Square-free Commuting Probabilities of Finite Rings

TL;DR

This paper proves that the probability of two elements commuting in a finite ring cannot be expressed as a fraction with a square-free denominator, settling a previously conjectured mathematical property.

Contribution

It establishes a key restriction on the form of commuting probabilities in finite rings, confirming a conjecture by Buckley and MacHale.

Findings

Commuting probability cannot have a square-free denominator.

Resolved a conjecture in finite ring theory.

Provides new insights into algebraic structure probabilities.

Abstract

It is shown that the commuting probability of a finite ring cannot be a fraction with square-free denominator, resolving a conjecture of Buckley and MacHale.