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On Long-Term Problems in Multiplicative Ideal Theory and Factorization Theory

Alfred Geroldinger, Hwankoo Kim, K. Alan Loper

TL;DR

The survey addresses long-standing open problems in multiplicative ideal theory and factorization theory by synthesizing star-operations, valuation theory, and homological methods across commutative domains and monoids. It surveys structural frameworks such as Mori/Krull domains, $w$-modules, and generalized rings of integer-valued polynomials, while highlighting transfer Krull phenomena and connections to additive combinatorics. The work identifies core goals, including characterizations of intersections of valuation domains, classifications between $\\mathbb{Z}[X]$ and $\\mathbb{Q}[X]$, and the development of homological analogues (Gorenstein, cotorsion) within multiplicative contexts, aiming to guide future research directions. Overall, the paper emphasizes how valuation theory, star-operations, and homological approaches jointly illuminate arithmetic finiteness, factorization behavior, and the transfer of problems to tractable combinatorial settings, with potential practical impact on understanding domains arising in number theory and algebraic geometry.

Abstract

In this survey article we discuss key open problems which could serve as a guidance for further research directions of multiplicative ideal theory and factorization theory.

On Long-Term Problems in Multiplicative Ideal Theory and Factorization Theory

TL;DR

The survey addresses long-standing open problems in multiplicative ideal theory and factorization theory by synthesizing star-operations, valuation theory, and homological methods across commutative domains and monoids. It surveys structural frameworks such as Mori/Krull domains, -modules, and generalized rings of integer-valued polynomials, while highlighting transfer Krull phenomena and connections to additive combinatorics. The work identifies core goals, including characterizations of intersections of valuation domains, classifications between and , and the development of homological analogues (Gorenstein, cotorsion) within multiplicative contexts, aiming to guide future research directions. Overall, the paper emphasizes how valuation theory, star-operations, and homological approaches jointly illuminate arithmetic finiteness, factorization behavior, and the transfer of problems to tractable combinatorial settings, with potential practical impact on understanding domains arising in number theory and algebraic geometry.

Abstract

In this survey article we discuss key open problems which could serve as a guidance for further research directions of multiplicative ideal theory and factorization theory.

Paper Structure

This paper contains 17 sections, 14 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Intersecting Valuation Domains
  4. Classification of Integral Domains between $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$
  5. Generalizing Kronecker Function Domains
  6. Coherent Domains
  7. Theory of $w$-modules
  8. Homological versus Elementary Methods
  9. Gorenstein Multiplicative Ideal Theory
  10. Cotorsion Theory
  11. Factorization Theory in Mori Monoids
  12. Some Long-Term Arithmetic Problems
  13. Half-Factoriality
  14. Finiteness of Arithmetic Invariants
  15. The Transfer Krull Property
  16. ...and 2 more sections

Key Result

Theorem 3.1

Every integrally closed domain $D$ (not a field) can be expressed as the intersection of all the valuation domains between $D$ and its quotient field.

Theorems & Definitions (27)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 4.1: L-T09
  • Theorem 4.2: Pe25a
  • Theorem 6.1
  • Definition 7.1
  • Definition 9.1
  • Example 9.2
  • Definition 9.3
  • Definition 10.1
  • ...and 17 more