The classification of dp-minimal integral domains

Christian D’elbée; Yatir Halevi; Will Johnson

doi:10.2140/mt.2025.4.131

The classification of dp-minimal integral domains

Christian D’elbée
, Yatir Halevi
, Will Johnson

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

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 ⊇ R, a proper ideal I in O, and a finite subring R₀ ⊆ O/I such that R is the preimage of R₀ in O.

Original language	English
Pages (from-to)	131-162
Number of pages	32
Journal	Model Theory
Volume	4
Issue number	2
DOIs	https://doi.org/10.2140/mt.2025.4.131
State	Published - 1 Jan 2025
Externally published	Yes

Keywords

dp-minimality
integral domains
valued fields

ASJC Scopus subject areas

Algebra and Number Theory
Logic

Access to Document

10.2140/mt.2025.4.131

Cite this

@article{569214f2cbe54005a8c914c203994586,

title = "The classification of dp-minimal integral domains",

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 ⊇ R, a proper ideal I in O, and a finite subring R0 ⊆ O/I such that R is the preimage of R0 in O.",

keywords = "dp-minimality, integral domains, valued fields",

author = "Christian D{\textquoteright}elb{\'e}e and Yatir Halevi and Will Johnson",

note = "Publisher Copyright: {\textcopyright} 2025 MSP (Mathematical Sciences Publishers).",

year = "2025",

month = jan,

day = "1",

doi = "10.2140/mt.2025.4.131",

language = "English",

volume = "4",

pages = "131--162",

journal = "Model Theory",

issn = "2832-9058",

publisher = "Mathematical Sciences Publishers",

number = "2",

}

TY - JOUR

T1 - The classification of dp-minimal integral domains

AU - D’elbée, Christian

AU - Halevi, Yatir

AU - Johnson, Will

PY - 2025/1/1

Y1 - 2025/1/1

N2 - 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 ⊇ R, a proper ideal I in O, and a finite subring R0 ⊆ O/I such that R is the preimage of R0 in O.

AB - 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 ⊇ R, a proper ideal I in O, and a finite subring R0 ⊆ O/I such that R is the preimage of R0 in O.

KW - dp-minimality

KW - integral domains

KW - valued fields

UR - https://www.scopus.com/pages/publications/105018064017

U2 - 10.2140/mt.2025.4.131

DO - 10.2140/mt.2025.4.131

M3 - Article

AN - SCOPUS:105018064017

SN - 2832-9058

VL - 4

SP - 131

EP - 162

JO - Model Theory

JF - Model Theory

IS - 2

ER -

The classification of dp-minimal integral domains

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this