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

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Local DVRs at the nonzero primes force global normality

Statement

Assume the Axiom of Choice. Let $R$ be a Noetherian domain. If $R_{p}$ is a discrete valuation ring for every nonzero prime ideal $p$ of $R$ , then $R$ is integrally closed.

Facts & Assumptions

Given: A Noetherian domain $R$ such that $R_{p}$ is a discrete valuation ring for every nonzero prime ideal $p$ .

[F1]

The fraction field $Frac (R)$ is formed by inverting the nonzero elements of the domain $R$ (The field of fractions $Frac (D) = (D ∖ {0})^{- 1} D$ of an integral domain).

[L1]

A nonfield domain is a DVR exactly when it is a one-dimensional Noetherian local integrally closed domain (Equivalent characterizations of a DVR).

[L2]

A domain is integrally closed if and only if each prime localisation is integrally closed (A domain is integrally closed if and only if its prime localisations are, equivalently if and only if its maximal localisations are).

Proof

technique · direct

1.1

At the zero prime, localisation inverts every nonzero element of $R$ , so $R_{(0)} = Frac (R)$ by [F1]. A field is integrally closed, so $R_{(0)}$ is integrally closed.

F1givenalgebra

1.2

If $p \neq (0)$ is prime, the hypothesis makes $R_{p}$ a discrete valuation ring. By [L1], every discrete valuation ring is integrally closed. Hence each nonzero-prime localisation of $R$ is integrally closed.

L1given

2.1

Steps 1.1 and 1.2 show that every prime localisation of $R$ is integrally closed. Therefore [L2] gives that $R$ itself is integrally closed.

L2step 1.1step 1.2∎

Depends on

Used by

Equivalent local characterizations of Dedekind domains Theorem

Dependency tree · two levels

25 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)