Characterization and examples of commutative isoartinian rings

Abstract

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 isonoetherian and isoartinian rings. In this paper, we prove that the Krull dimension of every isoartinian ring is at most one. We then use this result to provide a characterization of isoartinian rings. Specifically, we prove that a ring $R$ is isoartinian 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 isoartinian local rings with a nilpotent maximal ideal; (iii) non-field principal ideal domains; (iv) Noetherian isoartinian rings $A$ with $\Min A$ being a singleton and $\Min A \subsetneq \Ass A$; (v) non-Noetherian isoartinian rings $A$ with $\Min A$ being a singleton and $\Min A \subsetneq \Ass A$; (vi) non-Noetherian isoartinian 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.