Characterization and examples of commutative iso-Artinian rings
Noetherian rings have played a fundamental role in commutative algebra, algebraic number theory, and algebraic geometry. Along with their dual, Artinian rings, they have many generalizations, including the notions of iso-Noetherian and iso-Artinian rings. In this paper, we prove that the Krull dimen...
| Autores: | , |
|---|---|
| Tipo de recurso: | artículo |
| Fecha de publicación: | 2025 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:318142 |
| Acceso en línea: | https://ddd.uab.cat/record/318142 https://dx.doi.org/urn:doi:10.5565/PUBLMAT6922507 |
| Access Level: | acceso abierto |
| Palabra clave: | Dedekind ring Iso-artinian ring Marot ring Perfect ring Principal ideal domain Prüfer ring Subperfect ring |
| Sumario: | Noetherian rings have played a fundamental role in commutative algebra, algebraic number theory, and algebraic geometry. Along with their dual, Artinian rings, they have many generalizations, including the notions of iso-Noetherian and iso-Artinian rings. In this paper, we prove that the Krull dimension of every iso-Artinian ring is at most one. We then use this result to provide a characterization of iso-Artinian rings. Specifically, we prove that a ring R is iso-Artinian if and only if R is uniquely isomorphic to the direct product of a finite number of rings of the following types: (i) Artinian local rings; (ii) non-Noetherian iso-Artinian local rings with a nilpotent maximal ideal; (iii) non-field principal ideal domains; (iv) Noetherian iso-Artinian rings A with Min A being a singleton and Min A ( Ass A; (v) non-Noetherian iso-Artinian rings A with Min A being a singleton and Min A ( Ass A; (vi) non-Noetherian iso-Artinian rings A with a unique element in Min A that is not maximal, and Min A = Ass A. Several examples of these types of rings are also provided. |
|---|