Strongly Rad-clean Matrices over Commutative Local Rings

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.