Research ArticlesNo Access

Rings in which elements are sums of tripotents and nilpotents

Marjan Sheibani Abdolyousefi

Farzanegan Semnan University, Semnan, Iran

E-mail Address: m.sheibani1@gmail.com

Search for more papers by this author

and

Huanyin Chen

Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, P. R. China

E-mail Address: huanyinchen@aliyun.com

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0219498818500421Cited by:3 (Source: Crossref)

Abstract

A ring $R$ is strongly 2-nil-clean if every element in $R$ is the sum of a tripotent and a nilpotent that commute. We prove that a ring $R$ is strongly 2-nil-clean if and only if $R$ is a strongly feebly clean 2-UU ring if and only if $R$ is an exchange 2-UU ring. Furthermore, we characterize strongly 2-nil-clean ring via involutions. We show that a ring $R$ is strongly 2-nil-clean if and only if every element in $R$ is the sum of an idempotent, an involution and a nilpotent that commute.

Communicated by L. Bokut

Keywords:

AMSC: 16U99, 16E50, 13B99

References

1. N. Arora and S. Kundu, Commutative feebly clean rings, J. Algebra Appl. 16 (2017) 1750128, 14 pp., https://doi.org/10.1142/S0219498817501286 Link, Web of Science, Google Scholar
2. H. Chen, Rings Related Stable Range Conditions, Series in Algebra, Vol. 11 (World Scientific, Hackensack, NJ, 2011). Link, Google Scholar
3. H. Chen and M. Sheibani, Strongly 2-nil-clean rings, J. Algebra Appl. 16 (2017) 1750178, 12 pp., https://doi.org/10.1142/S021949881750178X Web of Science, Google Scholar
4. P. V. Danchev and T. Y. Lam, Rings with unipotent units, Publ. Math. Debrecen 88 (2016) 449–466. Web of Science, Google Scholar
5. A. J. Diesl, Nil clean rings, J. Algebra 383 (2013) 197–211. Web of Science, Google Scholar
6. Y. Hirano, H. Tominaga and A. Yaqub, 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 (1988) 33–40. Web of Science, Google Scholar
7. Y. Hirano and H. Tominaga, Rings in which every element is a sum of two idempotents, Bull. Aust. Math. Soc. 37 (1988) 161–164. Web of Science, Google Scholar
8. M. T. Kosan, Z. Wang and Y. Zhou, Nil-clean and strongly nil-clean rings, J. Pure Appl. Algebra (2015), http://dx.doi.org/10.1016/j.jpaa.2015.07.009. Web of Science, Google Scholar
9. J. Levitzki, On the structure of algebraic algebras and related rings, Trans. Amer. Math. Soc. 74 (1953) 384–409. Web of Science, Google Scholar
10. W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977) 269–278. Web of Science, Google Scholar
11. A. Stancu, A note on commutative weakly nil clean rings, J. Algebra Appl. 15 (2016). https://doi.org/10.1142/S0219498816200012 Link, Web of Science, Google Scholar
12. Z. L. Ying, T. Kosan and Y. Zhou, Rings in which every element is a sum of two tripotents, Canad. Math. Bull., http://dx.doi.org/10.4153/CMB-2016-009-0. Google Scholar
13. Y. Zhou, Rings in which elements are sum of nilpotents, idempotents and nilpotents, J. Algebra Appl., 16 (2017). https://doi.org/10.1142/S0219498818500093 Google Scholar