The reciprocal complements of classes of integral domains
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916464620994560 |
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| author | Guerrieri, Lorenzo |
| author_facet | Guerrieri, Lorenzo |
| contents | Given an integral domain $D$ with quotient field $\mathcal{Q}(D)$, the reciprocal complement of $D$ is the subring $R(D)$ of $\mathcal{Q}(D)$ whose elements are all the sums $\frac{1}{d_1}+\ldots+\frac{1}{d_n} $ for $d_1, \ldots, d_n$ nonzero elements of $D$. In this article we study problems related with prime ideals, localizations and Krull dimension of rings of the form $R(D)$ and we describe the reciprocal complements of classes of domains, including semigroup algebras and $ D+ \mathfrak{m} $ constructions. We also characterize when $R(D)$ is a DVR. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_00616 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The reciprocal complements of classes of integral domains Guerrieri, Lorenzo Commutative Algebra 13G05, 13A15, 13B30, 13F99, 20M25 Given an integral domain $D$ with quotient field $\mathcal{Q}(D)$, the reciprocal complement of $D$ is the subring $R(D)$ of $\mathcal{Q}(D)$ whose elements are all the sums $\frac{1}{d_1}+\ldots+\frac{1}{d_n} $ for $d_1, \ldots, d_n$ nonzero elements of $D$. In this article we study problems related with prime ideals, localizations and Krull dimension of rings of the form $R(D)$ and we describe the reciprocal complements of classes of domains, including semigroup algebras and $ D+ \mathfrak{m} $ constructions. We also characterize when $R(D)$ is a DVR. |
| title | The reciprocal complements of classes of integral domains |
| topic | Commutative Algebra 13G05, 13A15, 13B30, 13F99, 20M25 |
| url | https://arxiv.org/abs/2411.00616 |