Ratliff-Rush Closure of Ideals in Integral Domains

TL;DR

This paper explores the properties of Ratliff-Rush closure of ideals within integral domains, providing characterizations of integrally closed domains where all ideals are Ratliff-Rush and detailing the closure in valuation domains.

Contribution

It offers a complete characterization of integrally closed domains with all ideals Ratliff-Rush and describes the closure process in valuation domains.

Findings

Characterization of integrally closed domains where every ideal is Ratliff-Rush

Description of Ratliff-Rush closure in valuation domains

Complete description of the closure process in specific domains

Abstract

This paper studies the Ratliff-Rush closure of ideals in integral domains. By definition, the Ratliff-Rush closure of an ideal $I$ of a domain $R$ is the ideal given by $\tilde{I}:=\bigcup(I^{n+1}:_{R}I^{n})$ and an ideal $I$ is said to be a Ratliff-Rush ideal if $\tilde{I}=I$. We completely characterize integrally closed domains in which every ideal is a Ratliff-Rush ideal and we give a complete description of the Ratliff-Rush closure of an ideal in a valuation domain.