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The classification of dp-minimal integral domains

Christian d'Elbée, Yatir Halevi, Will Johnson

TL;DR

This work provides a complete classification of dp-minimal integral domains, showing they are either dp-minimal fields, dp-minimal valuation rings, or arise from a dp-minimal valuation overring via a finite quotient, with the overring and quotient data definable in the domain. The authors develop a two-pronged proof: first verify the constructive direction where the preimage construction yields dp-minimal domains; then, for nontrivial cases, reduce to a dp-minimal field K=Frac(R) and analyze K under ACVF-like, pCF-like, or RCVF-like trichotomy, eliminating rogue phenomena through m^{00}-dichotomy and external definability arguments. The paper introduces and leverages tools such as external definability of valuation rings, 00-connectedness of prime ideals, and the logic topology on R^{00}-quotients, to tightly constrain the structure and ensure the reduction to the preimage construction. Additional results include the dp-minimal extension to more general rings, a trichotomy for dp-minimal fields, and a decomposition principle for dp-minimal rings into a henselian local component and a finite component, clarifying the landscape of dp-minimal algebraic structures with potential implications for model-theoretic algebra and valued-field theory.

Abstract

We classify dp-minimal integral domains, building off the existing classification of dp-minimal fields and dp-minimal valuation rings. We show that if R is a dp-minimal integral domain, then R is a field or a valuation ring or arises from the following construction: there is a dp-minimal valuation overring O extending R, a proper ideal I in O, and a finite subring S in O/I such that R is the preimage of S in O.

The classification of dp-minimal integral domains

TL;DR

This work provides a complete classification of dp-minimal integral domains, showing they are either dp-minimal fields, dp-minimal valuation rings, or arise from a dp-minimal valuation overring via a finite quotient, with the overring and quotient data definable in the domain. The authors develop a two-pronged proof: first verify the constructive direction where the preimage construction yields dp-minimal domains; then, for nontrivial cases, reduce to a dp-minimal field K=Frac(R) and analyze K under ACVF-like, pCF-like, or RCVF-like trichotomy, eliminating rogue phenomena through m^{00}-dichotomy and external definability arguments. The paper introduces and leverages tools such as external definability of valuation rings, 00-connectedness of prime ideals, and the logic topology on R^{00}-quotients, to tightly constrain the structure and ensure the reduction to the preimage construction. Additional results include the dp-minimal extension to more general rings, a trichotomy for dp-minimal fields, and a decomposition principle for dp-minimal rings into a henselian local component and a finite component, clarifying the landscape of dp-minimal algebraic structures with potential implications for model-theoretic algebra and valued-field theory.

Abstract

We classify dp-minimal integral domains, building off the existing classification of dp-minimal fields and dp-minimal valuation rings. We show that if R is a dp-minimal integral domain, then R is a field or a valuation ring or arises from the following construction: there is a dp-minimal valuation overring O extending R, a proper ideal I in O, and a finite subring S in O/I such that R is the preimage of S in O.
Paper Structure (20 sections, 35 theorems, 45 equations)

This paper contains 20 sections, 35 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Sketch of the proof
  3. Preliminaries
  4. Notation
  5. Dp-minimal valuation rings and fields
  6. Constructing dp-minimal integral domains
  7. Tools
  8. Basic facts about NIP and dp-minimal domains
  9. 00-connectedness of prime ideals
  10. External definability of $\vee$-definable valuation rings and $R^{00}$
  11. The $\sqrt{\mathfrak{m}^{00}}$ dichotomy
  12. The topological ring $R/R^{00}$
  13. Subrings from additive subgroups
  14. A trichotomy for dp-minimal fields
  15. Proof of the main theorem
  16. ...and 5 more sections

Key Result

Theorem 1

Let $R$ be an integral domain. Then $R$ is dp-minimal if and only if one of the following holds: Moreover, in (3) the ring $\mathcal{O}$, ideal $I$, and subring $R_0$ can be chosen to be definable in $R$.

Theorems & Definitions (83)

  • Theorem
  • Example 1.1
  • Example 1.2
  • Proposition : Proposition \ref{['P:vee valuation ring is ext def']}
  • Proposition : Proposition \ref{['P:ext-val-3']}
  • proof
  • Proposition 3.2
  • proof
  • Remark 3.3
  • proof
  • ...and 73 more