Locally principal ideals and finite character

TL;DR

This paper explores the relationship between locally principal ideals and finite character in domains, investigating conditions under which locally principal ideals are invertible and when finite character can be characterized by this property.

Contribution

It provides new insights into the conditions linking locally principal ideals and finite character, including recent results and open problems.

Findings

Locally principal nonzero ideals are invertible in domains with finite character.

The paper surveys recent progress on the converse problem.

Conditions under which the converse holds are identified and discussed.

Abstract

It is well-known that if R is a domain with finite character, each locally principal nonzero ideal of R is invertible. We address the problem of understanding when the converse is true and survey some recent results.