On \tilde{Spec}(M) Topology of Module M over Commutative Rings
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| Format: | Preprint |
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2025
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| _version_ | 1866912723250446336 |
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| author | Erdemir, Dilara Koç, Suat Tekir, Ünsal Buğday, Mesut |
| author_facet | Erdemir, Dilara Koç, Suat Tekir, Ünsal Buğday, Mesut |
| contents | Let R be a commutative ring with unity and M be an R-module. In this study, we construct the \tilde{Spec}(M) topology using the prime spectrum of module M and multiplicatively closed subsets of R with the closed sets \tilde{V}(S)={P \in Spec(M) : (P : M) \cap S_i \neq \emptyset for all i \in I} with the open sets \tilde{D}(S_i):={P \in Spec(M) : (P : M) \cap S_i = \emptyset} where S = {S_i}_{i \in I} is a family of multiplicatively closed subsets of R. We investigate connections between the algebraic properties of R-module M and the topological properties of \tilde{Spec}(M). We examine specifically the separation axioms, connectivity, nested and Lindelöf property together with quasi-compactness as well as the isolated, closure, interior and limit points of tilde{Spec}(M). Moreover, in the last section, we provide an example of a Lindelöf space which is not quasi-compact by means of \tilde{Spec}(M). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_17465 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On \tilde{Spec}(M) Topology of Module M over Commutative Rings Erdemir, Dilara Koç, Suat Tekir, Ünsal Buğday, Mesut General Topology 13C99, 13J99, 54B99, 54D05 Let R be a commutative ring with unity and M be an R-module. In this study, we construct the \tilde{Spec}(M) topology using the prime spectrum of module M and multiplicatively closed subsets of R with the closed sets \tilde{V}(S)={P \in Spec(M) : (P : M) \cap S_i \neq \emptyset for all i \in I} with the open sets \tilde{D}(S_i):={P \in Spec(M) : (P : M) \cap S_i = \emptyset} where S = {S_i}_{i \in I} is a family of multiplicatively closed subsets of R. We investigate connections between the algebraic properties of R-module M and the topological properties of \tilde{Spec}(M). We examine specifically the separation axioms, connectivity, nested and Lindelöf property together with quasi-compactness as well as the isolated, closure, interior and limit points of tilde{Spec}(M). Moreover, in the last section, we provide an example of a Lindelöf space which is not quasi-compact by means of \tilde{Spec}(M). |
| title | On \tilde{Spec}(M) Topology of Module M over Commutative Rings |
| topic | General Topology 13C99, 13J99, 54B99, 54D05 |
| url | https://arxiv.org/abs/2511.17465 |