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

Abstract

Let R be any ring with identity. An element a ∈ 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 sucient conditions for a 2 × 2 matrix over an integral domain R to be nil-clean.