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

The concept of an $I$-matrix in the full $2\times 2$ matrix ring $M_2(R\mathrm{/}I)$, 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 $\widehat{B}$ in $M_2(R\mathrm{/}I)$ as the sum of two subrings ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$ of $M_2(R\mathrm{/}I)$ was also given, where ${\mathcal{S}}_1$ is the image (under the natural epimorphism from $M_2(R)$ to $M_2(R\mathrm{/}I)$) of the centralizer in $M_2(R)$ of a pre-image of $\widehat{B}$, and where the entries in ${\mathcal{S}}_2$ are intersections of certain annihilators of elements arising from the entries of $\widehat{B}$. 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 $\mathcal S_2\subseteq \mathcal S_1$andforwhen$\mathcal S_1=\mathcal S_2$. Secondly we provide a formula for the number of elements in the centralizer of $\widehat{B}$ for the case when $R\mathrm{/}<k>$ is finite.