On Divisor Topology of Modules over Domains
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866910980029546496 |
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| author | Tekir, Ünsal Yiğit, Uğur Buğday, Mesut Koç, Suat |
| author_facet | Tekir, Ünsal Yiğit, Uğur Buğday, Mesut Koç, Suat |
| contents | Let $M\ $be a module over a domain $R$ and $M^{\#}=\{0\neq m\in M:Rm\neq M\}$ be the set of all nonzero nongenerators of $M.\ $Consider following equivalence relation $\sim$ on $M^{\#}$ as follows: for every $m,n\in M^{\#},\ m\sim n$ if and only if $Rm=Rn.\ $Let $EC(M^{\#})$ be the set of all equivalence classes of $M^{\#}$ with respect to $\sim$. In this paper, we construct a topology on $EC(M^{\#})$ which is called divisor topology of $M\ $and denoted by $D(M).$ Actually, $D(M)$ is extension of the divisor topology $D(R)$ over domains in the sense of Yiğit and Koc to modules. We investigate separation axioms $T_{i}$ for every $0\leq i\leq5,$ first and second countability, connectivity, compactness, nested property, and Noetherian property on $D(M)$. Also, we characterize some important classes of modules such as uniserial modules, simple modules, vector spaces, and finitely cogenerated modules in terms of $D(M)$. Furthermore, we prove that $D(M)$ is a Baire space for factorial modules. Finally, we introduce and study pseudo simple modules which is a new generalization of simple modules, and use them to determine when $D(M)$ is a discrete space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_01179 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Divisor Topology of Modules over Domains Tekir, Ünsal Yiğit, Uğur Buğday, Mesut Koç, Suat Commutative Algebra General Topology Let $M\ $be a module over a domain $R$ and $M^{\#}=\{0\neq m\in M:Rm\neq M\}$ be the set of all nonzero nongenerators of $M.\ $Consider following equivalence relation $\sim$ on $M^{\#}$ as follows: for every $m,n\in M^{\#},\ m\sim n$ if and only if $Rm=Rn.\ $Let $EC(M^{\#})$ be the set of all equivalence classes of $M^{\#}$ with respect to $\sim$. In this paper, we construct a topology on $EC(M^{\#})$ which is called divisor topology of $M\ $and denoted by $D(M).$ Actually, $D(M)$ is extension of the divisor topology $D(R)$ over domains in the sense of Yiğit and Koc to modules. We investigate separation axioms $T_{i}$ for every $0\leq i\leq5,$ first and second countability, connectivity, compactness, nested property, and Noetherian property on $D(M)$. Also, we characterize some important classes of modules such as uniserial modules, simple modules, vector spaces, and finitely cogenerated modules in terms of $D(M)$. Furthermore, we prove that $D(M)$ is a Baire space for factorial modules. Finally, we introduce and study pseudo simple modules which is a new generalization of simple modules, and use them to determine when $D(M)$ is a discrete space. |
| title | On Divisor Topology of Modules over Domains |
| topic | Commutative Algebra General Topology |
| url | https://arxiv.org/abs/2506.01179 |