Very clean matrices over local rings

TL;DR

This paper characterizes when 2x2 matrices over local rings are very clean, providing necessary and sufficient conditions, especially over commutative local rings, with applications to power series matrices.

Contribution

It offers a complete characterization of very clean 2x2 matrices over local rings, including commutative cases and applications to power series.

Findings

Complete characterization over commutative local rings

Necessary and sufficient conditions for 2x2 matrices over local rings

Applications to matrices over power series

Abstract

An element $a\in R$ is very clean provided that there exists an idempotent $e\in R$ such that $ae=ea$ and either $a-e$ or $a+e$ is invertible. A ring $R$ is very clean in case every element in $R$ is very clean. We explore the necessary and sufficient conditions under which a triangular $2\times 2$ matrix ring over local rings is very clean. The very clean $2\times 2$ matrices over commutative local rings are completely determined. Applications to matrices over power series are also obtained.