Rings Whose Invertible Elements Are Weakly Nil-Clean
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911769934430208 |
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| author | Danchev, Peter Hasanzadeh, Omid Javan, Arash Moussavi, Ahmad |
| author_facet | Danchev, Peter Hasanzadeh, Omid Javan, Arash Moussavi, Ahmad |
| contents | We study those rings in which all invertible elements are weakly nil-clean calling them {\it UWNC rings}. This somewhat extends results due to Karimi-Mansoub et al. in Contemp. Math. (2018), where rings in which all invertible elements are nil-clean were considered abbreviating them as {\it UNC rings}. Specifically, our main achievements are that the triangular matrix ring ${\rm T}_n(R)$ over a ring $R$ is UWNC precisely when $R$ is UNC. Besides, the notions UWNC and UNC do coincide when $2 \in J(R)$. We also describe UWNC $2$-primal rings $R$ by proving that $R$ is a ring with $J(R) = {\rm Nil}(R)$ such that $U(R)=\pm 1+{\rm Nil}(R)$. In particular, the polynomial ring $R[x]$ over some arbitrary variable $x$ is UWNC exactly when $R$ is UWNC. Some other relevant assertions are proved in the present direction as well. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_11461 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rings Whose Invertible Elements Are Weakly Nil-Clean Danchev, Peter Hasanzadeh, Omid Javan, Arash Moussavi, Ahmad Rings and Algebras Representation Theory We study those rings in which all invertible elements are weakly nil-clean calling them {\it UWNC rings}. This somewhat extends results due to Karimi-Mansoub et al. in Contemp. Math. (2018), where rings in which all invertible elements are nil-clean were considered abbreviating them as {\it UNC rings}. Specifically, our main achievements are that the triangular matrix ring ${\rm T}_n(R)$ over a ring $R$ is UWNC precisely when $R$ is UNC. Besides, the notions UWNC and UNC do coincide when $2 \in J(R)$. We also describe UWNC $2$-primal rings $R$ by proving that $R$ is a ring with $J(R) = {\rm Nil}(R)$ such that $U(R)=\pm 1+{\rm Nil}(R)$. In particular, the polynomial ring $R[x]$ over some arbitrary variable $x$ is UWNC exactly when $R$ is UWNC. Some other relevant assertions are proved in the present direction as well. |
| title | Rings Whose Invertible Elements Are Weakly Nil-Clean |
| topic | Rings and Algebras Representation Theory |
| url | https://arxiv.org/abs/2401.11461 |