Non trivial idempotents of the matrix rings over polynomial ring Z_{pqr}[x]

TL;DR

This paper classifies all non-trivial idempotents in the 2x2 matrix ring over the polynomial ring with coefficients in Z_{pqr}, extending previous work to a more complex algebraic structure.

Contribution

It provides a complete classification of idempotents in the specified matrix ring over Z_{pqr}[x], a novel extension of earlier classifications.

Findings

All idempotents are classified into specific classes.

The classification extends previous results to polynomial rings over composite moduli.

The work enhances understanding of algebraic structures over polynomial rings.

Abstract

In this paper, we study the non trivial idempotents of the $2 \times 2$ matrix ring over the polynomial ring $\mathbb{Z}_{pqr}[x]$ for distinct primes $p, q $ and $r$ greater than $3$. We have classified all the idempotents of this matrix ring into several classes such that any idempotent must belong to one of these classes. This work is extension of the work done in $[1]$.