A completion of reduced commutative rings
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | |
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| _version_ | 1866916006850461696 |
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| author | Carai, Luca Kurtzhals, Miriam Moraschini, Tommaso |
| author_facet | Carai, Luca Kurtzhals, Miriam Moraschini, Tommaso |
| contents | A commutative ring is reduced when it can be embedded into a direct product of fields. While the category of reduced commutative rings plays a fundamental role in affine geometry, it exhibits several structural deficiencies: it admits nonregular monomorphisms and epimorphisms, lacks amalgamation, and is not equationally axiomatizable. In this paper, we simultaneously repair these defects via a canonical completion in which all monomorphisms become regular. This completion is obtained by adjoining weak inverses and weak prime roots, turning the class of reduced commutative rings into a discriminator variety. As a consequence, we obtain an explicit description of dominions in every class of reduced commutative rings containing all fields. This description is strikingly simple compared to that of dominions in the category of all commutative rings, as reflected in the Isbell-Mazet-Silver Zigzag Theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_12661 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A completion of reduced commutative rings Carai, Luca Kurtzhals, Miriam Moraschini, Tommaso Rings and Algebras Logic 03C05, 18A20, 12F99, 13A99 A commutative ring is reduced when it can be embedded into a direct product of fields. While the category of reduced commutative rings plays a fundamental role in affine geometry, it exhibits several structural deficiencies: it admits nonregular monomorphisms and epimorphisms, lacks amalgamation, and is not equationally axiomatizable. In this paper, we simultaneously repair these defects via a canonical completion in which all monomorphisms become regular. This completion is obtained by adjoining weak inverses and weak prime roots, turning the class of reduced commutative rings into a discriminator variety. As a consequence, we obtain an explicit description of dominions in every class of reduced commutative rings containing all fields. This description is strikingly simple compared to that of dominions in the category of all commutative rings, as reflected in the Isbell-Mazet-Silver Zigzag Theorem. |
| title | A completion of reduced commutative rings |
| topic | Rings and Algebras Logic 03C05, 18A20, 12F99, 13A99 |
| url | https://arxiv.org/abs/2605.12661 |