On the iso-Artinianness of one-dimensional Noetherian rings

Abstract

Let $R$ be a one-dimensional commutative Noetherian ring with a unique minimal prime $\mathfrak{p}$ such that $D=R/\mathfrak{p}$ is a principal ideal domain. We give a complete characterization of the iso-Artinian property in this class, that is, the following conditions are equivalent: $R$ is iso-Artinian; $\mathfrak{p}{R}_{\mathfrak{p}}=0$; $le{n}_R(\mathfrak{p})<\infty$; $\mathfrak{p}/{\mathfrak{p}}^2$ is torsion over $D$. As a consequence, we completely answer the Question~3.9 of Daneshvar and Divaani-Aazar: under their hypotheses $MinR⊊AssR$, the ring is iso-Artinian precisely when its nilradical has finite length, and it is non-iso-Artinian precisely when the conormal module has positive rank.