Unique representation domains, II

TL;DR

This paper introduces *-unique representation domains (URDs), generalizing unique factorization domains, and characterizes their properties, including their behavior under various ring extensions and their inclusion of Krull type and weakly Matlis domains.

Contribution

It defines *-URDs, provides conditions for their characterization, and explores their stability under ring extensions, unifying several classes of integral domains.

Findings

URDs include Krull type and weakly Matlis domains.

Conditions for *-ideal to be a unique *-product are established.

URD property extends to certain overrings and polynomial extensions.

Abstract

Given a star operation * of finite type, we call a domain R a *-unique representation domain (*-URD) if each *-invertible *-ideal of R can be uniquely expressed as a *-product of pairwise *-comaximal ideals with prime radical. When * is the t-operation we call the *-URD simply a URD. Any unique factorization domain is a URD. Generalizing and unifying results due to Zafrullah and Brewer-Heinzer, we give conditions for a *-ideal to be a unique *-product of pairwise *-comaximal ideals with prime radical and characterize *-URDs. We show that the class of URDs includes rings of Krull type, the generalized Krull domains introduced by El Baghdadi and weakly Matlis domains whose t-spectrum is treed. We also study when the property of being a URD extends to some classes of overrings, such as polynomial extensions, rings of fractions and rings obtained by the D+XD_S[X] construction.