On expressing matrices over \(\mathbb{Z}_2\) as the sum of an idempotent and a nilpotent (Q1638006)

An element \(a\) in a ring \(R\) is nil-clean if it is the sum of an idempotent and a nilpotent. In [\textit{S. Breaz} et al., Linear Algebra Appl. 439, No. 10, 3115--3119 (2013; Zbl 1355.16023)] it was proved that every \(n\times n\) matrix over \(\mathbb{Z}_2\) is nil-clean. The main result of the present paper extends this result, proving that every \(n\times n\) matrix over \(\mathbb{Z}_2\) is the sum of an idempotent matrix and a nilpotent matrix with index \(\leq 4\). Furthermore, a ring in which every element is nil-clean that is not strongly \(\pi\)-regular is constructed. This answers an open question of \textit{A. J. Diesl} [J. Algebra 383, 197--211 (2013; Zbl 1296.16016)].