Matrix nil-clean factorizations over abelian rings

TL;DR

This paper characterizes when matrix rings over abelian rings are nil-clean, linking this property to the Boolean nature of the quotient over the Jacobson radical and nilpotency of matrices over the radical.

Contribution

It extends previous results by providing a complete characterization of nil-clean matrix rings over abelian rings.

Findings

Matrix rings over abelian rings are nil-clean iff the quotient over the Jacobson radical is Boolean and matrices over the radical are nilpotent.

The result generalizes earlier work by Breaz et al. and Koşan et al.

Provides a clear criterion connecting ring properties to matrix nil-cleanliness.

Abstract

A ring $R$ is nil-clean if every element in $R$ is the sum of an idempotent and a nilpotent. A ring $R$ is abelian if every idempotent is central. We prove that if $R$ is abelian then $M_n(R)$ is nil-clean if and only if $R/J(R)$ is Boolean and $M_n(J(R))$ is nil. This extend the main results of Breaz et al. ~\cite{BGDT} and that of Ko\c{s}an et al.~\cite{KLZ}.