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

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The value group of a valuation ring

Definition

Let $V \subseteq K$ be a valuation ring (Valuation rings). Its value group is the quotient abelian group

K^{\times} / V^{\times},

written additively, where the group law is induced by multiplication in $K^{\times}$ .

For $x \in K^{\times}$ , write $\overline{x}$ for its class in $K^{\times} / V^{\times}$ . The intended order is

\overline{x} \leq \overline{y} ⟺ y / x \in V .

The next theorem checks that this order is well defined, total, and translation-invariant.

Depends on

Valuation rings

Used by

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, Proposition 8.6 (standard reference, not scraped)
Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., (26.11) (standard reference, not scraped)