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

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Every nonzero fractional ideal of a Dedekind domain is invertible

Statement

Assume the Axiom of Choice. Every nonzero fractional ideal of a Dedekind domain is invertible.

Facts & Assumptions

Given: A Dedekind domain $R$ and a nonzero fractional ideal $I$ of $R$ .

[F1]

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

[L1]

A nonzero-prime localisation of a Dedekind domain is a DVR (Localizing a Dedekind domain at a nonzero prime gives a DVR).

[L2]

Every nonzero ideal of a DVR is principal (Ideals in a DVR are powers of the maximal ideal).

[L3]

A nonzero finitely generated fractional ideal is invertible exactly when all maximal localisations are principal (Equivalent characterizations of invertible fractional ideals).

Proof

technique · direct

1.1

Choose $0 \neq d \in R$ with $J : = d I \subseteq R$ . Then $J$ is a nonzero integral ideal of $R$ , so [F1] makes it finitely generated; hence the fractional ideal $I = d^{- 1} J$ is finitely generated as well. Let $m$ be a maximal ideal. If $J ⊈ m$ , then $J_{m} = R_{m}$ , so $I_{m} = d^{- 1} R_{m}$ is principal. If $J \subseteq m$ , then $m$ is a nonzero prime, [L1] makes $R_{m}$ a DVR, and [L2] makes the integral ideal $J_{m}$ principal. Hence $I_{m} = d^{- 1} J_{m}$ is principal in either case. Thus $I$ is finitely generated and every maximal localisation of $I$ is principal.

F1L1L2givenchoosealgebra

2.1

Applying [L3] to step 1.1 shows that $I$ is invertible.

L3step 1.1∎

Depends on

Used by

Prime-ideal valuations on fractional ideals Definition

Dependency tree · two levels

21 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. P. May, Notes on Dedekind Rings (standard reference, not scraped)
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)