On generalized commuting probability of finite rings

TL;DR

This paper investigates the probability that the commutator of two randomly selected elements in a finite ring equals a specific element, extending the understanding of algebraic structures and their probabilistic properties.

Contribution

It introduces a generalized framework for commuting probabilities in finite rings, focusing on the likelihood of commutators equaling a given element, which is a novel extension of existing concepts.

Findings

Derived formulas for the probability of commutators equaling a fixed element.

Analyzed how the structure of the ring influences these probabilities.

Provided examples illustrating the application of the theoretical results.

Abstract

Let $R$ be a finite ring and $r \in R$. The aim of this paper is to study the probability that the commutator of a randomly chosen pair of elements of $R$ equals $r$.