Rings With uⁿ-1 Nilpotent For Each Unit u

2023/11/25 by Danchev, Peter, Javan, Arash, Moussavi, Ahmad · 1 citation
#16S34 #16U60 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2311.15018

Abstract

We continue the study in-depth of the so-called n-UU rings for any n≥ 1, that were defined by the first-named author in Toyama Math. J. (2017) as those rings R for which uⁿ-1 is always a nilpotent for every unit u∈ R. Specifically, for any n≥ 2, we prove that a ring is strongly n-nil-clean if, and only if, it is simultaneously strongly π-regular and an (n-1)-UU ring. This somewhat extends results due to Diesl in J. Algebra (2013), Abyzov in Sib. Math. J. (2019) and Cui-Danchev in J. Algebra Appl. (2020). Moreover, our results somewhat improves the ones obtained by Koşan et al. in Hacettepe J. Math. Stat. (2020).

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