A guide to closure operations in commutative algebra

TL;DR

This survey reviews various closure operations on ideals in commutative algebra, highlighting their structural properties and interrelations across different types of closures.

Contribution

It provides a comprehensive overview of multiple closure operations, connecting tools and concepts across different areas within commutative algebra.

Findings

Summarizes key properties of radical, tight, and integral closures.

Explores relationships between different closure operations.

Highlights the use of tools from one area to analyze others.

Abstract

This article is a survey of closure operations on ideals in commutative rings, with an emphasis on structural properties and on using tools from one part of the field to analyze structures in another part. The survey is broad enough to encompass the radical, tight closure, integral closure, basically full closure, saturation with respect to a fixed ideal, and the v-operation, among others.