Idempotents and Units of Matrix Rings over Polynomial Rings

Year 2017, Volume: 22 Issue: 22, 147 - 169, 11.07.2017

Pramod Kanwar , Meenu Khatkar R. K. Sharma

https://doi.org/10.24330/ieja.325941

Cited By: 3

Abstract

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 $M_2(\mathbb{Z}_3)$.
 

Keywords

Idempotent, unit

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