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