On Divisor Topology of Commutative Rings
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917926792069120 |
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| author | Yiğit, Uğur Koç, Suat |
| author_facet | Yiğit, Uğur Koç, Suat |
| contents | Let $R\ $be an integral domain and $R^{\#}$ the set of all nonzero nonunits of $R.\ $For every elements $a,b\in R^{\#},$ we define $a\sim b$ if and only if $aR=bR,$ that is, $a$ and $b$ are associated elements. Suppose that $EC(R^{\#})$ is the set of all equivalence classes of $R^{\#}\ $according to $\sim$.$\ $Let $U_{a}=\{[b]\in EC(R^{\#}):b\ $divides $a\}$ for every $a\in R^{\#}.$ Then we prove that the family $\{U_{a}\}_{a\in R^{\#}}$ becomes a basis for a topology on $EC(R^{\#}).\ $This topology is called divisor topology of $R\ $and denoted by $D(R).\ $We investigate the connections between the algebraic properties of $R\ $and the topological properties of$\ D(R)$. In particular, we investigate the seperation axioms on $D(R)$, first and second countability axioms, connectivity and compactness on $D(R)$. We prove that for atomic domains $R,\ $the divisor topology $D(R)\ $is a Baire space. Also, we characterize valution domains $R$ in terms of nested property of $D(R).$ In the last section, we introduce a new topological proof of the infinitude of prime elements in a UFD and integers by using the topology $D(R)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10577 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Divisor Topology of Commutative Rings Yiğit, Uğur Koç, Suat Commutative Algebra General Topology 13A15 Let $R\ $be an integral domain and $R^{\#}$ the set of all nonzero nonunits of $R.\ $For every elements $a,b\in R^{\#},$ we define $a\sim b$ if and only if $aR=bR,$ that is, $a$ and $b$ are associated elements. Suppose that $EC(R^{\#})$ is the set of all equivalence classes of $R^{\#}\ $according to $\sim$.$\ $Let $U_{a}=\{[b]\in EC(R^{\#}):b\ $divides $a\}$ for every $a\in R^{\#}.$ Then we prove that the family $\{U_{a}\}_{a\in R^{\#}}$ becomes a basis for a topology on $EC(R^{\#}).\ $This topology is called divisor topology of $R\ $and denoted by $D(R).\ $We investigate the connections between the algebraic properties of $R\ $and the topological properties of$\ D(R)$. In particular, we investigate the seperation axioms on $D(R)$, first and second countability axioms, connectivity and compactness on $D(R)$. We prove that for atomic domains $R,\ $the divisor topology $D(R)\ $is a Baire space. Also, we characterize valution domains $R$ in terms of nested property of $D(R).$ In the last section, we introduce a new topological proof of the infinitude of prime elements in a UFD and integers by using the topology $D(R)$. |
| title | On Divisor Topology of Commutative Rings |
| topic | Commutative Algebra General Topology 13A15 |
| url | https://arxiv.org/abs/2409.10577 |