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

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.