Resumen de Characterization and examples of commutative iso-Artinian rings

Asghar Daneshvar, Kamran Divaani-Aazar

• 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.