Idempotents and Units of Matrix Rings over Polynomial Rings

The aim of this paper is to study idempotents and units in certain matrix rings over polynomial rings. More precisely, the conditions under which an element in $M_2(\mathbb{Z}_p[x])$ for any prime $p$, an element in $M_2(\mathbb{Z}_{2p}[x])$ for any odd prime $p$, and an element in $M_2(\mathbb{Z}_{3p}[x])$ for any prime $p$ greater than 3 is an idempotent are obtained and these conditions are used to give the form of idempotents in these matrix rings. The form of elements in $M_2(\mathbb{Z}_2[x])$ and elements in $M_2(\mathbb{Z}_3[x])$ that are units is also given. It is observed that unit group of these rings behave differently from the unit groups of $M_2(\mathbb{Z}_2)$ and $M_2(\mathbb{Z}_3)$.