Rings such that, for each unit $u$, $u^n-1$ belongs to the $Δ(R)$
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| Format: | Preprint |
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2024
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| _version_ | 1866909389441466368 |
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| author | Danchev, Peter Javan, Arash Hasanzadeh, Omid Doostalizadeh, Mina Moussavi, Ahmad |
| author_facet | Danchev, Peter Javan, Arash Hasanzadeh, Omid Doostalizadeh, Mina Moussavi, Ahmad |
| contents | We study in-depth those rings $R$ for which, there exists a fixed $n\geq 1$, such that $u^n-1$ lies in the subring $Δ(R)$ of $R$ for every unit $u\in R$. We succeeded to describe for any $n\geq 1$ all reduced $π$-regular $(2n-1)$-$Δ$U rings by showing that they satisfy the equation $x^{2n}=x$ as well as to prove that the property of being exchange and clean are tantamount in the class of $(2n-1)$-$Δ$U rings. These achievements considerably extend results established by Danchev (Rend. Sem. Mat. Univ. Pol. Torino, 2019) and Koşan et al. (Hacettepe J. Math. \& Stat., 2020). Some other closely related results of this branch are also established. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_09416 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rings such that, for each unit $u$, $u^n-1$ belongs to the $Δ(R)$ Danchev, Peter Javan, Arash Hasanzadeh, Omid Doostalizadeh, Mina Moussavi, Ahmad Rings and Algebras Representation Theory 16S34, 16U60 We study in-depth those rings $R$ for which, there exists a fixed $n\geq 1$, such that $u^n-1$ lies in the subring $Δ(R)$ of $R$ for every unit $u\in R$. We succeeded to describe for any $n\geq 1$ all reduced $π$-regular $(2n-1)$-$Δ$U rings by showing that they satisfy the equation $x^{2n}=x$ as well as to prove that the property of being exchange and clean are tantamount in the class of $(2n-1)$-$Δ$U rings. These achievements considerably extend results established by Danchev (Rend. Sem. Mat. Univ. Pol. Torino, 2019) and Koşan et al. (Hacettepe J. Math. \& Stat., 2020). Some other closely related results of this branch are also established. |
| title | Rings such that, for each unit $u$, $u^n-1$ belongs to the $Δ(R)$ |
| topic | Rings and Algebras Representation Theory 16S34, 16U60 |
| url | https://arxiv.org/abs/2411.09416 |