Quasi Divisor Topology of Modules over Domains
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908563296747520 |
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| author | Buğday, Mesut Erdemir, Dilara Tekir, Ünsal Koç, Suat |
| author_facet | Buğday, Mesut Erdemir, Dilara Tekir, Ünsal Koç, Suat |
| contents | Let $E$ be a module over a domain $A$, and $W(E)^{\#}=W(E)-ann(E)$ where $W(E)=\{a\in A:aE\neq E\}$. We define an equivalence relation $\sim$ on $W(E)^{\#}$ as follows: $a\sim b$ if and only if $aE=bE$ for any $a,b\in W(E)^{\#}$ and denote $EC(W(E)^{\#})$ to be the set of all equivalence classes $[a]$ of $W(E)^{\#}$. We first show that the family $\{U_a\}_{a\in W(E)^\#}$ generates a topology which we called the quasi divisor topology of $A$-module $E$ denoted by $qD_A(E)$ where $U_{a}=\{[b]\in EC(W(E)^{\#}):\ aE\subseteq bE\}$ for every $a\in W(E)^{\#}$. This paper examines the connections between topological properties of the quasi divisor topology $qD_{A}(E)$ and algebraic properties of $A$-module $E$. These include each separation axioms, compactness, connectedness and first and second countability. Also, we characterize some important class of rings/modules such as divisible modules and uniserial modules by means of $qD_{A}(E)$. Furthermore, we introduce quasi second modules and study its algebraic properties to decide when $qD_A(E)$ is a $T_1$-space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_23743 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasi Divisor Topology of Modules over Domains Buğday, Mesut Erdemir, Dilara Tekir, Ünsal Koç, Suat Commutative Algebra General Topology Let $E$ be a module over a domain $A$, and $W(E)^{\#}=W(E)-ann(E)$ where $W(E)=\{a\in A:aE\neq E\}$. We define an equivalence relation $\sim$ on $W(E)^{\#}$ as follows: $a\sim b$ if and only if $aE=bE$ for any $a,b\in W(E)^{\#}$ and denote $EC(W(E)^{\#})$ to be the set of all equivalence classes $[a]$ of $W(E)^{\#}$. We first show that the family $\{U_a\}_{a\in W(E)^\#}$ generates a topology which we called the quasi divisor topology of $A$-module $E$ denoted by $qD_A(E)$ where $U_{a}=\{[b]\in EC(W(E)^{\#}):\ aE\subseteq bE\}$ for every $a\in W(E)^{\#}$. This paper examines the connections between topological properties of the quasi divisor topology $qD_{A}(E)$ and algebraic properties of $A$-module $E$. These include each separation axioms, compactness, connectedness and first and second countability. Also, we characterize some important class of rings/modules such as divisible modules and uniserial modules by means of $qD_{A}(E)$. Furthermore, we introduce quasi second modules and study its algebraic properties to decide when $qD_A(E)$ is a $T_1$-space. |
| title | Quasi Divisor Topology of Modules over Domains |
| topic | Commutative Algebra General Topology |
| url | https://arxiv.org/abs/2509.23743 |