Rings over which every matrix is the sum of two idempotents and a nilpotent

TL;DR

This paper explores rings where every matrix can be expressed as the sum of two idempotents and a nilpotent, focusing on properties of strongly 2-nil-clean rings and their matrix rings.

Contribution

It establishes conditions under which matrix rings over certain rings are 2-nil-clean, extending the class of rings with this property.

Findings

Strongly 2-nil-clean rings have matrices that are 2-nil-clean.

Matrix rings over strongly 2-nil-clean rings of bounded index are 2-nil-clean.

Many classes of rings are identified where every matrix is the sum of two idempotents and a nilpotent.

Abstract

A ring $R$ is (strongly) 2-nil-clean if every element in $R$ is the sum of two idempotents and a nilpotent (that commute). Fundamental properties of such rings are discussed. Let $R$ be a 2-primal ring. If $R$ is strongly 2-nil-clean, we show that $M_n(R)$ is 2-nil-clean for all $n\in {\Bbb N}$. We also prove that the matrix ring is 2-nil-clean for a strongly 2-nil-clean ring of bounded index. These provide many classes of rings over which every matrix is the sum of two idempotents and a nilpotent.