Some notes on topological rings and their groups of units
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866912386067202048 |
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| author | Tarizadeh, Abolfazl |
| author_facet | Tarizadeh, Abolfazl |
| contents | If $R$ is a topological ring then $R^{\ast}$, the group of units of $R$, with the subspace topology is not necessarily a topological group. This leads us to the following natural definition: By an \emph{absolute topological ring} we mean a topological ring such that its group of units with the subspace topology is a topological group. We prove that every commutative ring with the $I$-adic topology is an absolute topological ring. Next, we prove that if $I$ is an ideal of a ring $R$ then for the $I$-adic topology over $R$ we have $π_{0}(R)=R/(\bigcap\limits_{n\geqslant1}I^{n})=t(R)$ where $π_{0}(R)$ is the space of connected components of $R$ and $t(R)$ is the space of irreducible closed subsets of $R$. We observed that the main result of Koh \cite{kwangil} as well as its corrected form \cite[Chap II, \S12, Theorem 12.1]{Ursul} are not true, and then we corrected this result in the right way. In the Wikipedia pages, it is claimed that ``the identity component of a topological group is always a characteristic subgroup'', we also provide a counterexample to this claim. Finally, we fix a gap in the proof of the fact that every epimorphism of the category of Hausdorff topological spaces has a dense image. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_02634 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Some notes on topological rings and their groups of units Tarizadeh, Abolfazl Commutative Algebra Algebraic Geometry Group Theory 13Jxx, 13J20, 16W80, 14A05 If $R$ is a topological ring then $R^{\ast}$, the group of units of $R$, with the subspace topology is not necessarily a topological group. This leads us to the following natural definition: By an \emph{absolute topological ring} we mean a topological ring such that its group of units with the subspace topology is a topological group. We prove that every commutative ring with the $I$-adic topology is an absolute topological ring. Next, we prove that if $I$ is an ideal of a ring $R$ then for the $I$-adic topology over $R$ we have $π_{0}(R)=R/(\bigcap\limits_{n\geqslant1}I^{n})=t(R)$ where $π_{0}(R)$ is the space of connected components of $R$ and $t(R)$ is the space of irreducible closed subsets of $R$. We observed that the main result of Koh \cite{kwangil} as well as its corrected form \cite[Chap II, \S12, Theorem 12.1]{Ursul} are not true, and then we corrected this result in the right way. In the Wikipedia pages, it is claimed that ``the identity component of a topological group is always a characteristic subgroup'', we also provide a counterexample to this claim. Finally, we fix a gap in the proof of the fact that every epimorphism of the category of Hausdorff topological spaces has a dense image. |
| title | Some notes on topological rings and their groups of units |
| topic | Commutative Algebra Algebraic Geometry Group Theory 13Jxx, 13J20, 16W80, 14A05 |
| url | https://arxiv.org/abs/2303.02634 |