ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2019. Vol. 29

A Note on Commutative Nil-Clean Corners in Unital Rings

P. V. Danchev

We shall prove that if R is a ring with a family of orthogonal idempotents {ei}ni=1 having sum 1 such that each corner subring eiRei is commutative nil-clean, then R is too nil-clean, by showing that this assertion is actually equivalent to the statement established by Breaz-Călugăreanu-Danchev-Micu in Lin. Algebra & Appl. (2013) that if R is a commutative nil-clean ring, then the full matrix ring Mn(R) is also nil-clean for any size n. Likewise, the present proof somewhat supplies our recent result in Bull. Iran. Math. Soc. (2018) concerning strongly nil-clean corner rings as well as it gives a new strategy for further developments of the investigated theme.

About the Authors

Peter V. Danchev, Ph.D. (Math.), Prof., Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev st., bl. 8, Sofia, 1113, Bulgaria, e-mail: danchev@math.bas.bg; pvdanchev@yahoo.com

For citation

Danchev P.V. A Note on Commutative Nil-Clean Corners in Unital Rings. The Bulletin of Irkutsk State University. Series Mathematics, 2019, vol. 29, pp. 3-9. https://doi.org/10.26516/1997-7670.2019.29.3

nil-clean rings, nilpotents, idempotents, corners
16U99; 16E50; 13B99
  1. Breaz S., Călugăreanu G., Danchev P., Micu T. Nil-clean matrix rings, Lin. Algebra & Appl., 2013, vol. 439, no. 10, pp. 3115–3119. https://doi.org/10.1016/j.laa.2013.08.027
  2. Danchev P.V. Strongly nil-clean corner rings, Bull. Iran. Math. Soc., 2017, vol. 43, no. 5, pp. 1333–1339.
  3. Danchev P.V. Semi-boolean corner rings, Internat. Math. Forum, 2017, vol. 12, no. 16, pp. 795–802. https://doi.org/10.12988/imf.2017.7655
  4. Danchev P.V. On corner subrings of unital rings, Internat. J. Contemp. Math. Sci., 2018, vol. 13, no. 2, pp. 59–62. https://doi.org/10.12988/ijcms.2018.812
  5. Danchev P.V. Corners of invo-clean unital rings, Pure Math. Sci., 2018, vol. 7, no. 1, pp. 27–31. https://doi.org/10.12988/pms.2018.877
  6. Danchev P.V. Feebly nil-clean unital rings, Proc. Jangjeon Math. Soc., 2018, vol. 21, no. 1, pp. 155–165.
  7. Danchev P.V., Lam T.Y. Rings with unipotent units, Publ. Math. Debrecen, 2016, vol. 88, no. 3–4, pp. 449–466. https://doi.org/10.5486/PMD.2016.7405
  8. Danchev P.V., McGovern W.Wm. Commutative weakly nil clean unital rings, J. Algebra, 2015, vol. 425, no. 5, pp. 410–422. https://doi.org/10.1016/j.jalgebra.2014.12.003
  9. Diesl A.J. Nil clean rings, J. Algebra, 2013, vol. 383, no. 11, pp. 197–211. https://doi.org/10.1016/j.jalgebra.2013.02.020
  10. Kosan M.T., Lee T.K., Zhou Y. When is every matrix over a division ring a sum of an idempotent and a nilpotent?, Lin. Algebra & Appl., 2014, vol. 450, no. 11, pp. 7–12. https://doi.org/10.1016/j.laa.2014.02.047
  11. Lam T.Y. A First Course in Noncommutative Rings, Second Edition, Graduate Texts in Math., 2001, vol. 131, Springer-Verlag, Berlin-Heidelberg-New York.

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