Rings With $u^n-1$ Nilpotent For Each Unit $u$
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866914664524283904 |
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| author | Danchev, Peter Javan, Arash Moussavi, Ahmad |
| author_facet | Danchev, Peter Javan, Arash Moussavi, Ahmad |
| contents | We continue the study in-depth of the so-called $n$-UU rings for any $n\geq 1$, that were defined by the first-named author in Toyama Math. J. (2017) as those rings $R$ for which $u^n-1$ is always a nilpotent for every unit $u\in R$. Specifically, for any $n\geq 2$, 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 Ko$ş$an et al. in Hacettepe J. Math. Stat. (2020). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_15018 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rings With $u^n-1$ Nilpotent For Each Unit $u$ Danchev, Peter Javan, Arash Moussavi, Ahmad Rings and Algebras 16S34, 16U60 We continue the study in-depth of the so-called $n$-UU rings for any $n\geq 1$, that were defined by the first-named author in Toyama Math. J. (2017) as those rings $R$ for which $u^n-1$ is always a nilpotent for every unit $u\in R$. Specifically, for any $n\geq 2$, 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 Ko$ş$an et al. in Hacettepe J. Math. Stat. (2020). |
| title | Rings With $u^n-1$ Nilpotent For Each Unit $u$ |
| topic | Rings and Algebras 16S34, 16U60 |
| url | https://arxiv.org/abs/2311.15018 |