Finitely Star Regular Domains

TL;DR

This paper introduces finitely star regular domains, a new class of integral domains characterized by the behavior of finite type star operations on overrings, and explores their properties and distinctions from star regular domains.

Contribution

It defines finitely star regular domains, distinguishes them from star regular domains, and extends classical results to this new class, including constructions from pullback diagrams.

Findings

Finitely star regular domains are distinct from star regular domains.

Classical results on star regularity are extended to finitely star regularity.

Constructed examples of finitely star regular domains that are not star regular.

Abstract

Let $R$ be an integral domain, $Star(R)$ the set of all star operations on $R$ and $StarFC(R)$ the set of all star operations of finite type on $R$. Then $R$ is said to be star regular if $|Star(T)|\leq |Star(R)|$ for every overring $T$ of $R$. In this paper we introduce the notion of finitely star regular domain as an integral domain $R$ such that $|StarFC(T)|\leq |StarFC(R)|$ for each overring $T$ of $R$. First, we show that the notions of star regular and finitely star regular domains are completely different and do not imply each other. Next, we extend/generalize well-known results on star regularity in Noetherian and Pr\"ufer contexts to finitely star regularity. Also we handle the finite star regular domains issued from classical pullback constructions to construct finitely star regular domains that are not star regular and enriches the literature with a such class of domains.