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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2310.04116 |
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| _version_ | 1866917944130273280 |
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| author | Fujita, Masato Kageyama, Masaru |
| author_facet | Fujita, Masato Kageyama, Masaru |
| contents | We study quasi-quadratic modules in a pseudo-valuation domain $A$ whose strict units admit a square root. Let $\mathfrak X_R^N$ denote the set of quasi-quadratic modules in an $R$-module $N$, where $R$ is a commutative ring. It is known that there exists a unique overring $B$ of $A$ such that $B$ is a valuation ring with the valuation group $(G,\leq)$ and the maximal ideal of $B$ coincides with that of $A$. Let $F$ be the residue field of $B$. In the above setting, we found a one-to-one correspondence between $\mathfrak X_A^A$ and a subset of $\prod_{g \in G,g \geq e} \mathfrak X_{F_0}^F$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_04116 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quasi-quadratic modules in pseudo-valuation domain Fujita, Masato Kageyama, Masaru Commutative Algebra Primary 13J30, Secondary 12J10 We study quasi-quadratic modules in a pseudo-valuation domain $A$ whose strict units admit a square root. Let $\mathfrak X_R^N$ denote the set of quasi-quadratic modules in an $R$-module $N$, where $R$ is a commutative ring. It is known that there exists a unique overring $B$ of $A$ such that $B$ is a valuation ring with the valuation group $(G,\leq)$ and the maximal ideal of $B$ coincides with that of $A$. Let $F$ be the residue field of $B$. In the above setting, we found a one-to-one correspondence between $\mathfrak X_A^A$ and a subset of $\prod_{g \in G,g \geq e} \mathfrak X_{F_0}^F$. |
| title | Quasi-quadratic modules in pseudo-valuation domain |
| topic | Commutative Algebra Primary 13J30, Secondary 12J10 |
| url | https://arxiv.org/abs/2310.04116 |