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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 introducedin \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 centralizerin $M_2(R)$ 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 entriesof $\hat B$. In the present paper, we obtain results for the case when $I$ is aprincipal ideal $<k>$, $k\in R$ a nonzero nonunit. Mainly we solve twoproblems. Firstly we find necessary and sufficient conditions for when$\mathcal S_1\subseteq\mathcal S_2$, for when $\mathcal S_2\subseteq \mathcalS_1$ and for when $\mathcal S_1=\mathcal S_2$. Secondly we provide a formulafor the number of elements in the centralizer of $\hat B$ for the case when$R/<k>$ is finite.

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