Commuting probabilities of $n$-centralizer finite rings

TL;DR

This paper investigates the commuting probabilities of finite rings with a specific number of centralizers, providing explicit calculations for certain classes of $n$-centralizer rings.

Contribution

It computes the commuting probability for finite rings with a given number of centralizers, extending understanding of their algebraic structure.

Findings

Calculated $ ext{Pr}(R)$ for specific $n$-centralizer finite rings

Established relationships between the number of centralizers and commuting probabilities

Provided explicit formulas for certain classes of $n$-centralizer rings

Abstract

Let $R$ be a finite ring. The commuting probability of $R$, denoted by $\Pr(R)$, is the probability that any two randomly chosen elements of $R$ commute. $R$ is called an $n$-centralizer ring if it has $n$ distinct centralizers. In this paper, we compute $\Pr(R)$ for some $n$-centralizer finite rings.