TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-31

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A Noetherian valuation ring is a field or a DVR

Statement

Let $V$ be a valuation ring. Then $V$ is Noetherian if and only if $V$ is a field or a discrete valuation ring.

Facts & Assumptions

Given: A valuation ring $V$ .

[F1]

A ring is Noetherian by the definition fixed earlier (Left and right Noetherian rings).

[L1]

In a valuation ring the ideals are linearly ordered and every finitely generated ideal is principal (Characterizations of valuation rings).

[L2]

For a nonfield domain, being a Noetherian valuation ring is equivalent to being a DVR (Equivalent characterizations of a DVR).

Proof

technique · direct

1.1

Suppose $V$ is Noetherian. A valuation ring is a domain, so if $V$ is not a field then [L2] applies and shows that $V$ is a DVR. Thus a Noetherian valuation ring is a field or a DVR.

F1L2given

2.1

Conversely, every field is Noetherian because its only ideals are $(0)$ and the whole ring. If $V$ is a DVR, then [L2] applied in the forward direction shows that it is a Noetherian valuation ring. Hence $V$ is Noetherian exactly in the two stated cases.

L1L2algebra∎

Depends on

Used by

A rank-two valuation ring that is not a DVR Example

Dependency tree · two levels

18 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, Proposition 8.13 (standard reference, not scraped)
Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., Exercise (26.15)(2) (standard reference, not scraped)