Characterization and examples of commutative iso-Artinian rings
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Daneshvar, Asghar [1] ; Divaani-Aazar, Kamran [1]
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[1] University of Tehran
University of Tehran
Irán
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- Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 69, Nº 2, 2025, págs. 403-414
- Idioma: inglés
- DOI: 10.5565/publmat6922507
- Enlaces
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Resumen
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.
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