TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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one dimensional regular local rings are dvrs

Statement

A nonzero Noetherian local ring of dimension one is regular if and only if it is a discrete valuation ring. Fields are excluded from the term DVR.

Facts & Assumptions

Given: The objects and hypotheses in the statement. We work with the Axiom of Choice; cited dependent-choice and resolution-existence hypotheses are retained.

[F1]

regular local domain induction: Every regular local ring is an integral domain.

[F2]

embedding dimension is minimal maximal ideal generator number: For a nonzero Noetherian local ring $(R, m, k)$ , $edim R$ is the least number of generators of $m$ .

[F3]

Equivalent characterizations of a DVR: Let $R$ be a nonfield domain. The following are equivalent. 1. $R$ is a discrete valuation ring. 2. $R$ is a Noetherian valuation ring. 3. $R$ is a one-dimensional Noetherian local integrally closed domain. 4. $R$ is a local principal ideal domain with nonzero maximal ideal.

[F4]

The Krull intersection is the $(1 - a)$ -torsion submodule, and it vanishes in the Jacobson-radical case: The first clause below is choice-free; the second uses the published Jacobson-radical unit criterion and therefore inherits its Axiom-of-Choice boundary. Let $R$ be a Noetherian commutative ring, let $I \subset R$ be an ideal, and let $M$ be a finite $R$ -module. Put $K : = ⋂_{n \geq 0} I^{n} M .$ Then: 1. $K$ is exactly the set of elements $m \in M$ for which $(1 - a) m = 0$ for some $a \in I$ ; 2. if $I \subseteq J (R)$ , then $K = 0$ .

Proof

1.1

If $R$ is regular of dimension one, it is a domain and $m = (t)$ for a nonzero nonunit $t$ . Krull intersection gives for any $a \neq 0$ a largest $n$ with $a \in (t^{n})$ ; writing $a = t^{n} u$ , maximality makes $u$ a unit.

F1F2F4

2.1

In a nonzero ideal choose an element with least such exponent $n$ . Every other nonzero element has exponent at least $n$ , so the ideal is $(t^{n})$ . The zero ideal is principal as well. Thus $R$ is a local PID with nonzero maximal ideal, and the stated DVR equivalence applies.

F3step 1.1algebra

3.1

Conversely, a DVR is a nonfield local PID of dimension one. Its maximal ideal $(t)$ is nonzero, and $t \notin (t^{2})$ by cancellation in a domain. Therefore its embedding dimension is one and it is regular.

F3F2algebra∎

Depends on

Used by

regular local ring satisfies r one Corollary
serre normality criterion two directions Corollary
dvrs as regular local rings Example
normal domain implies r one Lemma
polynomial local regularity fibre step Lemma
serre normality criterion Theorem

Dependency tree · two levels

26 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

Example 12.10, p.116 (standard reference, not scraped)