Strongly nil-clean corner rings

Abstract

We show that if $R$ is a ring with an arbitrary idempotent $e$ such that $eRe$ and $(1-e)R(1-e)$ are both strongly nil-clean rings‎, ‎then $R/J(R)$ is nil-clean‎. ‎In particular‎, ‎under certain additional circumstances‎, ‎$R$ is also nil-clean‎. ‎These results somewhat improves on achievements due to Diesl in J‎. ‎Algebra (2013) and to Koc{s}an-Wang-Zhou in J‎. ‎Pure Appl‎. ‎Algebra (2016)‎. ‎In addition‎, ‎we also give a new transparent proof of the main result of Breaz-Calugareanu-Danchev-Micu in Linear Algebra Appl‎. ‎(2013) which says that if $R$ is a commutative nil-clean ring‎, ‎then the full $ntimes n$ matrix ring $mathbb{M}_n(R)$ is nil-clean‎.