Two generalizations of Krull domains

TL;DR

This paper introduces two new generalizations of Krull domains, called $ ext{ extasteriskcentered}$-almost IRKTs and $ ext{ extasteriskcentered}$-AGKDs, characterized via $ ext{ extasteriskcentered}$-homogeneous ideals, expanding the theory of non-integrally closed domains.

Contribution

The paper defines and characterizes $ ext{ extasteriskcentered}$-almost IRKTs and $ ext{ extasteriskcentered}$-AGKDs, broadening the class of Krull domain generalizations without requiring integral closure.

Findings

Characterization of $ ext{ extasteriskcentered}$-almost IRKTs via $ ext{ extasteriskcentered}$-super-SH domains.

Characterization of $ ext{ extasteriskcentered}$-AGKDs as type 1 $ ext{ extasteriskcentered}$-almost super-SH domains.

Equivalence of $ ext{ extasteriskcentered}$-afg SH domains with $ ext{ extasteriskcentered}$-IRKT and AGCD-domain.

Abstract

In this paper we introduce two new generalizations of Krull domains: $\ast$-almost independent rings of Krull type ($\ast$-almost IRKTs) and $\ast$-almost generalized Krull domains ($\ast$-AGKDs), neither of which need be integrally closed. We characterize them using certain types of $\ast$-homogeneous ideals. To do this we introduce $\ast$-almost super-homogeneous ideals and $\ast$-almost super-SH domains. We prove that a domain $D$ is a $\ast$-almost IRKT if and only if $D$ is a $\ast$-almost super-SH domain and that a domain is a $\ast$-AGKD if and only if $D$ is a type 1 $\ast$-almost super-SH domain. Further, we study $\ast$-almost factorial general-SH domains ($\ast$-afg SH domains) and we prove that a domain $D$ is a $\ast$-afg-SH domain if and only if $D$ is a $\ast$-IRKT and an AGCD-domain.