On some extensions of strongly unit nil-clean rings
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929234088296448 |
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| author | Barati, Ruhollah |
| author_facet | Barati, Ruhollah |
| contents | An element $x \in R$ is considered (strongly) nil-clean if it can be expressed as the sum of an idempotent $e \in R$ and a nilpotent $b \in R$ (where $eb = be$). If for any $x \in R$, there exists a unit $u \in R$ such that $ux$ is (strongly) nil-clean, then $R$ is called a (strongly) unit nil-clean ring. It is worth noting that any unit-regular ring is strongly unit nil-clean. In this note, we provide a characterization of the unit regularity of a group ring, along with an additional condition. We also fully characterize the unit-regularity of the group ring $\mathbb{Z}_nG$ for every $n > 1$. Additionally, we discuss strongly unit nil-cleanness in the context of Morita contexts, matrix rings, and group rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_02261 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On some extensions of strongly unit nil-clean rings Barati, Ruhollah Rings and Algebras An element $x \in R$ is considered (strongly) nil-clean if it can be expressed as the sum of an idempotent $e \in R$ and a nilpotent $b \in R$ (where $eb = be$). If for any $x \in R$, there exists a unit $u \in R$ such that $ux$ is (strongly) nil-clean, then $R$ is called a (strongly) unit nil-clean ring. It is worth noting that any unit-regular ring is strongly unit nil-clean. In this note, we provide a characterization of the unit regularity of a group ring, along with an additional condition. We also fully characterize the unit-regularity of the group ring $\mathbb{Z}_nG$ for every $n > 1$. Additionally, we discuss strongly unit nil-cleanness in the context of Morita contexts, matrix rings, and group rings. |
| title | On some extensions of strongly unit nil-clean rings |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2402.02261 |