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

Discrete valuation rings Definition
The value group of a valuation ring Definition
A nondiscrete rank-one valuation from incommensurate values Example
A valuation ring is local Lemma
Characterizations of valuation rings 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)