Matrix rings over a principal ideal domain in which elements are nil-clean

Abstract

An element of a ring $R$ is called nil-clean if it is the sum of an idempotent and a nilpotent element. A ring is called nil-clean if each of its elements is nil-clean. S. Breaz et al. in [1] proved their main result that the matrix ring $\mathbb{M}_{ n}(F)$ over a field $F$ is nil-clean if and only if $F\cong \mathbb{F}_2$, where $\mathbb{F}_2$ is the field of two elements. M. T. Ko\c{s}an et al. generalized this result to a division ring. In this paper, we show that the $n\times n$ matrix ring over a principal ideal domain $R$ is a nil-clean ring if and only if $R$ is isomorphic to $\mathbb{F}_2$. Also, we show that the same result is true for the $2\times 2$ matrix ring over an integral domain $R$. As a consequence, we show that for a commutative ring $R$, if $\mathbb{M}_{ 2}(R)$ is a nil-clean ring, then dim$R=0$ and char${R}/{J(R)}=2$.