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DOI: https://doi.org/10.33048/smzh.2019.60.202

Abstract: Under study are the rings whose every element is a sum of a nilpotent and a $q$-potent that commute with one another. We describe the rings whose every element is a sum of $k$ idempotents (for some $k\in\mathbb{N}$) and a nilpotent that commute with one another.

Keywords: clean rings, strongly clean rings, strongly $q$-nil-clean rings.

Funding agency	Grant number
Russian Foundation for Basic Research	18-41-160024
The author was funded by the Russian Foundation for Basic Research (Grant 18-41-160024).

Received: 21.05.2018
Revised: 08.10.2018
Accepted: 17.10.2018

English version:
Siberian Mathematical Journal, 2019, Volume 60, Issue 2, Pages 197–208
DOI: https://doi.org/10.1134/S0037446619020022

Bibliographic databases:

Citation: A. N. Abyzov, “Strongly $q$-nil-clean rings”, Sibirsk. Mat. Zh., 60:2 (2019), 257–273; Siberian Math. J., 60:2 (2019), 197–208

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This publication is cited in the following articles:

1. Cimpean A., “M-Nil-Clean Companion Matrices”, Electron. J. Linear Algebra, 35 (2019), 626–632          
2. P. Danchev, “Representing matrices over fields as square-zero matrices and diagonal matrices”, Chebyshevskii sb., 21:3 (2020), 84–88    
3. A. N. Abyzov, D. T. Tapkin, “Rings over which matrices are sums of idempotent and $q$-potent matrices”, Siberian Math. J., 62:1 (2021), 1–13          
4. A. N. Abyzov, D. T. Tapkin, “When is every matrix over a ring the sum of two tripotents?”, Linear Alg. Appl., 630 (2021), 316–325          
5. Peter V. Danchev, “Certain Properties of Square Matrices over Fields with Applications to Rings”, rev.colomb.mat, 54:2 (2021), 109  
6. Ruhollah Barati, Ahmad Mousavi, Adel Abyzov, “Rings whose elements are sums of m-potents and nilpotents”, Communications in Algebra, 50:10 (2022), 4437  
7. A. N. Abyzov, D. T. Tapkin, “On rings with xn − x nilpotent”, J. Algebra Appl., 21:06 (2022)  
8. Alexander Diesl, “Sums of commuting potent and nilpotent elements in rings”, J. Algebra Appl., 22:05 (2023)  
9. Adel N. Abyzov, Peter V. Danchev, Daniel T. Tapkin, “Rings with xn + x or xn − x nilpotent”, J. Algebra Appl., 22:01 (2023)  

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles