# Strongly nil-clean corner rings

Document Type: Research Paper

Author

Department of Mathematics‎, ‎University of Plovdiv‎, ‎Plovdiv 4000‎, ‎Bulgaria.

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 Ko\c{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 $n\times n$ matrix ring $\mathbb{M}_n(R)$ is nil-clean‎.

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### History

• Receive Date: 19 March 2016
• Revise Date: 02 June 2016
• Accept Date: 02 June 2016