Multiplicative Relationships of Subrings and their Applications to Factorization
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866913922664103936 |
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| author | Moles, Grant |
| author_facet | Moles, Grant |
| contents | When studying the properties of a ring $R$, it is often useful to compare $R$ to other rings whose properties are already known. In this paper, we define three ways in which a subring $R$ might be compared to a larger ring $T$: being associated, being ideal-preserving, or being locally associated. We then explore how these properties of a subring might be leveraged to give information about $R$, including applications to the field of factorization. Of particular interest is the result that an order in a number field is associated if and only if it is both ideal-preserving and locally associated. We conclude with a discussion of how these properties are realized in the case of orders in a number field and how such orders might be found. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_24031 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiplicative Relationships of Subrings and their Applications to Factorization Moles, Grant Commutative Algebra When studying the properties of a ring $R$, it is often useful to compare $R$ to other rings whose properties are already known. In this paper, we define three ways in which a subring $R$ might be compared to a larger ring $T$: being associated, being ideal-preserving, or being locally associated. We then explore how these properties of a subring might be leveraged to give information about $R$, including applications to the field of factorization. Of particular interest is the result that an order in a number field is associated if and only if it is both ideal-preserving and locally associated. We conclude with a discussion of how these properties are realized in the case of orders in a number field and how such orders might be found. |
| title | Multiplicative Relationships of Subrings and their Applications to Factorization |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2506.24031 |