Research ArticleNo Access

On rings with $x^{n} - x$ nilpotent

A. N. Abyzov

https://orcid.org/0000-0002-9809-2091

N.I. Lobachevsky Institute of Mathematics and Mechanics, Kazan Federal University, Kazan 420008, Russian Federation

E-mail Address: Adel.Abyzov@kpfu.ru

Corresponding author.

Search for more papers by this author

and

D. T. Tapkin

N.I. Lobachevsky Institute of Mathematics and Mechanics, Kazan Federal University, Kazan 420008, Russian Federation

E-mail Address: danil.tapkin@yandex.ru

Search for more papers by this author

https://doi.org/10.1142/S0219498822501110Cited by:5 (Source: Crossref)

Abstract

For $n \geq 2$ and for a ring $R,$ the notation $P_{n} (R)$ means that $r^{n} - r$ is nilpotent for all $r \in R$ . In this paper, rings $R$ for which $P_{n} (R)$ holds are completely characterized for any integers $n \geq 2$ . This answers a question which was raised in [T. Kosan, Y. Zhou, T. Yildirim, Rings with $x^{n} - x$ nilpotent, J. Algebra Appl.19(4) (2020) 2050065].

Communicated by A. Facchini

Keywords:

AMSC: 15A24, 15B33, 16U99

References

1. W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977) 269–278. Crossref, Web of Science, Google Scholar
2. W. K. Nicholson, Strongly clean rings and Fitting’s lemma, Commun. Algebra 27(8) (1999) 3583–3592. Crossref, Web of Science, Google Scholar
3. A. J. Diesl, Nil clean rings, J. Algebra 383 (2013) 197–211. Crossref, Web of Science, Google Scholar
4. H. Chen and M. Sheibani, Strongly 2-nil-clean rings, J. Algebra Appl. 16(9) (2017) 1750178. Link, Web of Science, Google Scholar
5. H. Chen and M. Sheibani, Rings in which elements are sums of tripotents and nilpotents, J. Algebra Appl. 17(3) (2018) 1850042. Link, Web of Science, Google Scholar
6. Y. Zhou, Rings in which elements are sum of nilpotents, idempotents and nilpotents, J. Algebra Appl. 217(1) (2018) 1850009. Link, Web of Science, Google Scholar
7. S. Breaz, G. Calugareanu, P. Danchev and T. Micu, Nil-clean matrix rings, Linear Algebra Appl. 439(10) (2013) 3115–3119. Crossref, Web of Science, Google Scholar
8. A. N. Abyzov and I. I. Mukhametgaliev, On some matrix analogs of the little fermat theorem, Math. Notes 101(2) (2017) 187–192. Crossref, Web of Science, Google Scholar
9. A. N. Abyzov, Strongly $q$ -nil-clean rings, Siberian Math. J. 60(2) (2019) 197–208. Crossref, Web of Science, Google Scholar
10. T. Kosan, Y. Zhou and T. Yildirim, Rings with $x^{n} - x$ nilpotent. J. Algebra Appl. 19(4) (2020) 2050065. Link, Web of Science, Google Scholar
11. 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
12. A. A. Tuganbaev, Rings Close to Regular (Springer Netherlands (Kluwer) Dordrecht-Boston-London, 2002), 362 pp. Google Scholar
13. T. Y. Lam, A First Course in Noncommutative Rings, 2nd edn. Graduate Texts in Mathematics, 131. (Springer-Verlag, New York, 2001). Crossref, Google Scholar
14. Z. Ying, T. Kosan and Y. Zhou, Rings in which every element is a sum of two tripotents, Can. Math. Bull. 59(3) (2016) 661–672. Crossref, Web of Science, Google Scholar
15. J. Matczuk, Conjugate (nil) clean rings and Köthes problem, J. Algebra Appli. 16(2) (2017) 1750073. https://doi.org/10.1142/S0219498817500736 Link, Web of Science, Google Scholar
16. J. Krempa, Logical connections between some open problems concerning nil rings, Fund. Math. 76(2) (1972) 121–130. Crossref, Google Scholar