Matrix nil-clean factorizations over abelian rings

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}.