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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S. Breaz, G. Calugareanu, P. Danchev and T. Micu, Nil-clean matrix rings, Linear Algebra Appl. 439 (2013) 3115--3119.

W. Chen, A question on strongly clean rings, Comm. Algebra 34 (2006), no. 7, 1--4.

P.V. Danchev and T.Y. Lam, Rings with unipotent units, Publ. Math. Debrecen 88 (2016), no. 3-4, 449--466.

A.J. Diesl, Nil clean rings, J. Algebra 383 (2013) 197--211.

J. Han and W.K. Nicholson, Extensions of clean rings, Comm. Algebra 29 (2001), no. 6, 2589--2595.

T. Kosan, Z. Wang and Y. Zhou, Nil-clean and strongly nil-clean rings, J. Pure Appl. Algebra 220 (2016), no. 2, 633--646.

T.Y. Lam, A First Course in Noncommutative Rings, Grad. Texts in Math. 131, Springer-Verlag, 2nd edition, New York, 2001.

N.H. McCoy, Prime ideals in general rings, Amer. J. Math. 71 (1948) 823--833.

W.K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977) 269--278.

J. Ster, Corner rings of a clean ring need not be clean, Comm. Algebra 40 (2012), no. 5, 1595--1604.

J. Ster, Rings in which nilpotents form a subring, Carpath. J. Math. 32 (2016) 251--258