On the centralizer of an $I$-matrix in $M_2(R/I)$, $I$ a principal ideal and $R$ a UFD

TL;DR

This paper characterizes the centralizer of an $I$-matrix in $M_2(R/I)$ for a principal ideal $I$, providing conditions for subring inclusion and a formula for the size of the centralizer when $R/I$ is finite.

Contribution

It extends previous work by explicitly solving conditions for subring relations and calculating the centralizer size in the principal ideal case.

Findings

Conditions for $ ext{S}_1 ext{ and } ext{S}_2$ inclusion relations.

Explicit formula for the number of elements in the centralizer when $R/<k>$ is finite.

Generalization of centralizer description in $M_2(R/I)$ for principal ideals.

Abstract

The concept of an $I$-matrix in the full $2\times 2$ matrix ring $M_2(R/I)$, where $R$ is an arbitrary UFD and $I$ is a nonzero ideal in $R$, was introduced in \cite{mar}. Moreover a concrete description of the centralizer of an $I$-matrix $\hat B$ in $M_2(R/I)$ as the sum of two subrings $\mathcal S_1$ and $\mathcal S_2$ of $M_2(R/I)$ was also given, where $\mathcal S_1$ is the image (under the natural epimorphism from $M_2(R)$ to $M_2(R/I)$) of the centralizer in $M_2(R)$ of a pre-image of $\hat B$, and where the entries in $\mathcal S_2$ are intersections of certain annihilators of elements arising from the entries of $\hat B$. In the present paper, we obtain results for the case when $I$ is a principal ideal $<k>$, $k\in R$ a nonzero nonunit. Mainly we solve two problems. Firstly we find necessary and sufficient conditions for when $\mathcal S_1\subseteq\mathcal S_2$, for when $\mathcal S_2\subseteq \mathcal S_1$ and for when $\mathcal S_1=\mathcal S_2$. Secondly we provide a formula for the number of elements in the centralizer of $\hat B$ for the case when $R/<k>$ is finite.