Abstract
A ring R is unit nil-clean if, for any \(a\in R\), there exists a unit \(u\in R\), such that ua is the sum of an idempotent and a nilpotent. In this paper, we completely describe the structure of unit nil-clean rings. We thereby provide a large class of rings over which every square matrix is equivalent to the sum of idempotent and nilpotent matrices. Furthermore, the uniqueness is determined by the abelian property.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abdolyousefi, M.S., Chen, H.: Matrices over Zhou nil-clean rings. Comm. Algebra 46, 1527–1533 (2018)
Bossaller, D.P.: On a Generalization of Clean Rings, M.A. Thesis, Saint Louis University, Saint Louis (2013)
Breaz, S., Galugareanu, G., Danchev, P., Micu, T.: Nil-clean matrix rings. Linear Algebra Appl. 439, 3115–3119 (2013)
Călugăreanu, G.: UN-rings. J. Algebra Appl. 15, 1650182 (2016). https://doi.org/10.1142/S0219498816501826. (9 pages)
Chen, H.: Rings Related Stable Range Conditions, Series in Algebra 11. World Scientific, Hackensack (2011)
Chen, H., Sheibani, M.: Strongly 2-nil-clean rings. J. Algebra Appl. 16, 1750178 (2017). https://doi.org/10.1142/S021949881750178X. (12 pages)
Danchev, P.V., McGovern, W.W.: Commutative weakly nil clean unital rings. J. Algebra 425, 410–422 (2015)
Diesl, A.J.: Nil clean rings. J. Algebra 383, 197–211 (2013)
Goodearl, K.R.: Von Neumann Regular Rings. Pitman, Krieger, Malabar (1991)
Hirano, Y., Tominaga, H., Yaqub, A.: On rings in which every element is uniquely expressible as a sum of a nilpotent element and a certain potent element. Math. J. Okayama Univ. 30, 33–40 (1988)
Kosan, M.T., Wang, Z., Zhou, Y.: Nil-clean and strongly nil-clean rings. J. Pure Appl. Algebra. (2018). https://doi.org/10.1016/j.jpaa.2015.07.009
McGovern, W.W., Raja, S., Sharp, A.: Communicative nil clean group rings. J. Algebra Appl. (2014). https://doi.org/10.1142/S0219498815500942
Nicholson, W.K.: Lifting idempotents and exchange rings. Trans. Am. Math. Soc. 229, 269–278 (1977)
Nicholson, W.K.: Strongly clean rings and fitting’s lemma. Comm. Algebra 27(8), 3583–3592 (1999)
Ying, Z.L., Kosan, T., Zhou, Y.: Rings in which every element is a sum of two tripotents. Canad. Math. Bull. (2016). https://doi.org/10.4153/CMB-2016-009-0
Acknowledgements
The authors are grateful to the referee for his/her careful reading and valuable remarks that helped them to improve the presentation of this work tremendously. H. Chen was supported by the Natural Science Foundation of Zhejiang Province, China (No. LY17A010018).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Abdolyousefi, M.S., Ashrafi, N. & Chen, H. On Unit Nil-Clean Rings. Mediterr. J. Math. 16, 100 (2019). https://doi.org/10.1007/s00009-019-1369-z
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-019-1369-z