Strongly Rad-clean Matrices over Commutative Local Rings

TL;DR

This paper characterizes when 2x2 matrices over commutative local rings are strongly rad-clean, providing a complete criterion based on ring and matrix properties.

Contribution

It offers a complete characterization of strongly rad-clean matrices over commutative local rings, focusing on 2x2 matrices and their algebraic conditions.

Findings

Complete characterization of strongly rad-clean 2x2 matrices over local rings

Conditions involving idempotents and units in the ring

Criteria based on the Jacobson radical and ring elements

Abstract

An element $a\in R$ is provided that there exists an idempotent $e\in R$ such that $a-e\in U(R), ae=ea$ and $eae\in J(eRe)$. In this article, we investigate strongly rad-clean matrices over a commutative local ring. We completely determine when a $2\times 2$ matrix over a commutative local ring is strongly rad-clean.