A note on $n$-centralizer finite rings

TL;DR

This paper characterizes finite rings with exactly n centralizers for n up to 7, providing a classification of such rings based on their centralizer structure.

Contribution

It offers a complete characterization of n-centralizer finite rings for n ≤ 7, extending understanding of their algebraic structure.

Findings

Classified all 2-centralizer finite rings.

Identified structures of 3- to 7-centralizer finite rings.

Provided explicit descriptions for each n-centralizer case.

Abstract

Let $R$ be a finite ring and let $\Cent(R)$ denote the set of all distinct centralizers of $R$. $R$ is called an $n$-centralizer ring if $|Cent(R)| = n$. In this paper, we characterize $n$-centralizer finite rings for $n \leq 7$.