The centralizer of an $I$-matrix in $M_2(R/I)$, $R$ a UFD

TL;DR

This paper characterizes the centralizer of $I$-matrices in $2\times 2$ matrices over quotient rings of a UFD, revealing structural differences from PID cases and limitations for higher dimensions.

Contribution

It introduces the concept of $I$-matrices in $M_2(R/I)$ and provides a concrete description of their centralizers, highlighting distinctions between UFDs and PIDs.

Findings

Centralizer described as sum of two subrings.

All matrices are $I$-matrices if $R$ is a PID.

Description fails for some matrices over rings with zero divisors.

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 $R$, is introduced. We obtain 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 $M_2(R/I)$, 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 $\hat B$. It turns out that if $R$ is a PID, then every matrix in $M_2(R/I)$ is an $I$-matrix. However, this is not the case if $R$ is a UFD in general. Moreover, for every factor ring $R/I$ with zero divisors and every $n\ge 3$ there is a matrix for which the mentioned concrete description is not valid.