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.

Discrete valuation rings

Definition

Let $K$ be a field and let $v$ be a discrete valuation on $K$ (Discrete valuations). Its valuation ring is

V_{v} : = {x \in K : v (x) \geq 0} .

This is a valuation ring in the sense of Valuation rings.

A discrete valuation ring (DVR) is a subring $V \subseteq K$ of the form $V = V_{v}$ for some discrete valuation $v$ on $K$ . Because $v$ is surjective, there is an element of value $1$ , so $V$ has a nonunit and is therefore not a field.

Depends on

Used by

Uniformising parameters Definition
The p-adic valuation ring Example
Every nonzero fraction is a unit times a power of a uniformiser Theorem

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

M. Mustata, Commutative Algebra, Definition 8.8 (standard reference, not scraped)
Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., (23.1) (standard reference, not scraped)