IDEMPOTENTS IN CERTAIN MATRIX RINGS OVER POLYNOMIAL RINGS

Abstract

We determine the forms of the nontrivial idempotents in the ring of $2\times 2$ matrices over the polynomial rings $\mathbb{Z}_{pq}[x]$ and $\mathbb{Z}_{p^2}[x]$, where $p$ and $q$ are any primes. Any such idempotent in the stated rings will be of a form in our list. Our work generalizes the results of Kanwar, Khatkar and Sharma (2017) who identified the forms of idempotents in $M_2(\mathbb{Z}_{2p}[x])$ and $M_2(\mathbb{Z}_{3p}[x])$.