LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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Localizing a Dedekind domain at a nonzero prime gives a DVR

Statement

Let $R$ be a Dedekind domain and let $p \subset R$ be a nonzero prime ideal. Then $R_{p}$ is a discrete valuation ring.

Facts & Assumptions

Given: A Dedekind domain $R$ and a nonzero prime ideal $p$ .

[F1]

A Dedekind domain is a Noetherian integrally closed domain of Krull dimension $1$ (Dedekind domains).

[L1]

In a Noetherian integrally closed domain, the localisation at a height-one prime is a discrete valuation ring (Height-one localizations of normal Noetherian domains are DVRs).

Proof

technique · direct

1.1

Because $R$ is a domain, $(0) ⊊ p$ is a strict prime chain. Since [F1] gives $\dim R = 1$ , no longer strict chain ending at $p$ exists. Therefore $p$ has height $1$ .

F1givenalgebra

2.1

The ring $R$ is Noetherian and integrally closed by [F1], and step 1.1 shows that $p$ is height one. Hence [L1] applies and gives that $R_{p}$ is a discrete valuation ring.

F1L1step 1.1∎

Depends on

Used by

Dependency tree · two levels

11 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

J. S. Milne, A Primer of Commutative Algebra, §20 (standard reference, not scraped)
Mircea Mustata, Introduction to Commutative Algebra, §8.5 (standard reference, not scraped)