On expressing matrices over Z2 as the sum of an idempotent and a nilpotent

We prove that for every n×n matrix A over the field Z2 there exists an idempotent matrix E such that (A−E)4=0. This result, strengthening the result of Breaz, Călugăreanu, Danchev and Micu, allows us to construct a nil-clean ring that is not strongly π-regular, which answers an open question of Diesl.