Flat Ideals and Stability in Integral Domains

TL;DR

This paper introduces the concept of quasi-stable ideals in integral domains, exploring their properties and the implications for domains where all nonzero fractional ideals are quasi-stable, addressing longstanding questions on flatness.

Contribution

It defines quasi-stable ideals and studies their properties, providing new insights into flatness and stability in integral domains.

Findings

Characterization of quasi-stable ideals in integral domains

Analysis of domains where all nonzero fractional ideals are quasi-stable

Answers to open questions on flatness raised by Glaz and Vasconcelos

Abstract

We introduce the concept of \textit{quasi-stable} ideal in an integral domain $D$ (a nonzero fractional ideal $I$ of $D$ is quasi-stable if it is flat in its endomorphism ring $(I \colon I)$) and study properties of domains in which each nonzero fractional ideal is quasi-stable. We investigate some questions about flatness that were raised by S. Glaz and W.V. Vasconcelos in their 1977 paper \cite{GV}.