Ring theory in o-minimal structures
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909524252688384 |
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| author | Conversano, Annalisa |
| author_facet | Conversano, Annalisa |
| contents | We develop a general ring theory in the o-minimal setting culminating in a description of all the definable rings in an arbitrary o-minimal structure. We show that every definably connected ring with non-trivial multiplication defines an infinite field and it is essentially semialgebraic. A surprisingly strong correspondence between definably connected rings and finite-dimensional associative $\mathbb{R}$-algebras is established. Every ideal of a definable unital ring is definable, from which it follows that every definable unital ring is Artinian and Noetherian. If a definable ring $R$ is not unital, we give necessary and sufficient conditions for $R$ to embed in a definable unital ring as an ideal. Moreover, when this is the case, we provide the smallest such definable unital ring $R^{\wedge}$, its definable unitazation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_02215 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ring theory in o-minimal structures Conversano, Annalisa Logic Group Theory Rings and Algebras Representation Theory We develop a general ring theory in the o-minimal setting culminating in a description of all the definable rings in an arbitrary o-minimal structure. We show that every definably connected ring with non-trivial multiplication defines an infinite field and it is essentially semialgebraic. A surprisingly strong correspondence between definably connected rings and finite-dimensional associative $\mathbb{R}$-algebras is established. Every ideal of a definable unital ring is definable, from which it follows that every definable unital ring is Artinian and Noetherian. If a definable ring $R$ is not unital, we give necessary and sufficient conditions for $R$ to embed in a definable unital ring as an ideal. Moreover, when this is the case, we provide the smallest such definable unital ring $R^{\wedge}$, its definable unitazation. |
| title | Ring theory in o-minimal structures |
| topic | Logic Group Theory Rings and Algebras Representation Theory |
| url | https://arxiv.org/abs/2503.02215 |