Characterization of $2\times 2$ nil-clean matrices over integral domains

Abstract

Let $R$ be any ring with identity. An element $a \in R$ is called nil-clean, if $a=e+n$ where $e$ is an idempotent element and $n$ is a nil-potent element. In this paper we give necessary and sufficient conditions for a $2\times 2$ matrix over an integral domain $R$ to be nil-clean.

Keywords

• Nil-clean matrix,Idempotent matrix,Nil-potent matrix,Diophantine equation

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