  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.
