On $\ast $-Semi Homogeneous Domains

TL;DR

This paper introduces the concept of $igast$-Semi Homogeneous Domains, characterizing their structure through $igast$-homog ideals and prime ideal families, unifying various well-known domain classes.

Contribution

It defines $igast$-SHDs, explores their properties, and shows how to adapt the theory to special cases, connecting several classical domain types.

Findings

$igast$-SHDs include h-local, Krull, UFD, and independent rings of Krull type.

Every $igast$-SHD has a family of prime ideals with a locally finite intersection.

The theory of $igast$-homog ideals can be modified for specific subclasses.

Abstract

Let $\ast $ be a finite character star operation defined on an integral domain $D.$ Call a nonzero $\ast $-ideal $I$ of finite type a $\ast $ -homogeneous ($\ast $-homog) ideal, if $I\subsetneq D$ and $(J+K)^{\ast }\neq D$ for every pair $D\supsetneq J,K\supseteq I$ of proper $\ast $ -ideals of finite type$.$ Call an integral domain $D$ a $\ast $-Semi Homogeneous Domain ($\ast $-SHD) if every proper principal ideal $xD$ of $D$ is expressible as a $\ast $-product of finitely many $\ast $-homog ideals. We show that a $\ast $-SHD contains a family $\mathcal{F}$ of prime ideals such that (a) $D=\cap_{P\in \mathcal{F}}D_{P},$ a locally finite intersection and (b) no two members of $\mathcal{F}$ contain a common non zero prime ideal. The $\ast $-SHDs include h-local domains, independent rings of Krull type, Krull domains, UFDs etc. We show also that we can modify the definition of the $\ast $-homog ideals to get a theory of each special case of a $\ast $-SH domain.