On a generalization of regular rings with central nilpotents

Document Type : Research Paper

Authors

Department of Mathematics, College of Sciences, Shiraz University, Shiraz, Iran

Abstract

‎A ring $R$ is called $\pi$-regular if‎, ‎for every $x\in R$‎, ‎there exists $y\in R$ such that $x^n=x^nyx^n$ for some positive integer $n$‎. ‎Here‎, ‎we shall give some characterizations of $\pi$-regular rings in which nilpotent elements lie in the center‎. ‎It is shown that these rings can be formulated in a way motivated by recent works of P‎. ‎Danchev‎, ‎leading to new insights into $\pi$-regular rings and providing partial answers to a question posed by him‎. ‎In the end‎, ‎we aim to classify this class of rings‎, ‎up to an isomorphism‎.

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References

[1] D. D. Anderson, S. Chun and J. R. Juett, Module-theoretic generalization of commutative von Neumann regular rings, Comm. Algebra, 47 (2019), 4713–4728.
[2] A. Alahmadi, S. K. Jain and A. Leroy, Regular elements determined by generalized inverses, J. Algebra Appl., 18 (2019), Article ID 1950128.
[3] P. Ara and J. Claramunt, Uniqueness of the von Neumann continuous factor, Canad. J. Math., 70 (2018), 961–982.
[4] P. Danchev, A generalization of π-regular rings, Turk. J. Math., 43 (2019), 702–711.
[5] P. Danchev, A symmetric generalization of π-regular rings, Ricerche di Matematica, (2021), 1–12.
[6] P. Danchev and J. Ster, Generalizing π-regular rings, Taiwan. J. Math., 19 (2015), 1577–1592.
[7] I. N. Herstein, A generalization of a theorem of Jacobson III, Amer. J. Math., 75 (1953), 105–111.
[8] D. Khurana and P. P. Nielsen, Perspectivity and von Neumann regularity, Comm. Algebra, 49 (2021), 5483–5499.
[9] N. H. McCoy, Generalized regular rings, Bull. Amer. Math. Soc., 45 (1939), 175–178.
[10] A. R. Nasr-Isfahani and A. Moussavi, On a quotient of polynomial rings, Comm. Algebra, 38 (2010), 567–575.
[11] W. K. Nicholson, Strongly clean rings and Fitting’s lemma, Comm. Algebra., 27 (1999), 3583–3592.
[12] T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Math., vol. 131, New York: Springer-Verlag, 1991.
[13] A. Tuganbaev, Rings Close to Regular, Mathematics and Its Applications, vol. 545, Springer Science & Business Media, 2002.
[14] Y. Utumi, On ξ-rings, Proc. Jap. Acad. Ser. A, 33 (1957), 63–66.
Journal of Algebra and Related Topics
Volume 14, Issue 1
August 2026
Pages 37-48

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