DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Valuation rings

Definition

Let $K$ be a field. A subring $V \subseteq K$ is a valuation ring of $K$ if for every $x \in K^{\times}$ at least one of $x$ and $x^{- 1}$ belongs to $V$ .

Thus a valuation ring decides each nonzero element of the ambient field by membership: either the element itself is in the ring, or its inverse is.

Used by

Puiseux discs normalise a reduced plane curve germ Corollary
DVR uniqueness need not detect nonseparatedness Counterexample
Discrete valuation rings Definition
The value group of a valuation ring Definition
Valuative uniqueness diagram Definition
A nondiscrete rank-one valuation from incommensurate values Example
Valuative extension of projective coordinates Example
A local domain has a dominating valuation overring Lemma
A valuation ring is local Lemma
Closed gluing of two projective three-spaces is proper Lemma
Rational maps from a smooth curve to a proper scheme are morphisms Lemma
Separatedness implies valuative uniqueness Lemma
Valuation lifts detect universal closedness Lemma
The valuative criterion quantifies over all valuation rings Remark
Characterizations of valuation rings Theorem
Finite-dimensional projective space is proper over every base Theorem
Valuation rings are integrally closed Theorem

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

M. Mustata, Commutative Algebra, Definition 8.4 (standard reference, not scraped)
Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., Definition (26.1) (standard reference, not scraped)