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.

Valuations on a field

Definition

Let $K$ be a field and let $Γ$ be a totally ordered abelian group (Totally ordered abelian groups). Adjoin a symbol $\infty$ with $γ < \infty, γ + \infty = \infty + γ = \infty, \infty + \infty = \infty .$ for every $γ \in Γ$ .

A valuation on $K$ with value group in $Γ$ is a map

v : K \to Γ \cup {\infty}

such that for all $x, y \in K$ ,

v (x) = \infty ⟺ x = 0,

v (x y) = v (x) + v (y),

and

v (x + y) \geq \min {v (x), v (y)} .

When $x \neq 0$ , the value $v (x)$ lies in $Γ$ . The displayed laws imply $v (1) = 0$ and $v (x^{- 1}) = - v (x)$ for $x \in K^{\times}$ .

Depends on

Totally ordered abelian groups

Used by

Discrete valuations Definition
A nondiscrete rank-one valuation from incommensurate values Example
A rank-two valuation ring that is not a DVR Example
A valuation ring is recovered from its value group Theorem

Dependency tree · one level

1 result within one dependency step 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 8.1 and Remark 8.2 (standard reference, not scraped)
Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., (26.11) (standard reference, not scraped)