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

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Invertible fractional ideals are exactly the rank-one projective modules

Statement

Assume the Axiom of Choice. Let $R$ be a domain with fraction field $K$ . A nonzero fractional ideal of $R$ is invertible if and only if, as an $R$ -module, it is finite projective and becomes $K$ after tensoring with $K$ . Equivalently, the finite projective $R$ -modules of constant rank one are precisely the invertible fractional ideals inside $K$ .

Facts & Assumptions

Given: A domain $R$ with fraction field $K$ .

[F1]

Invertibility of a fractional ideal means $I (R : I) = R$ (Invertible fractional ideals).

[L1]

A fractional ideal is invertible exactly when it is finite projective and locally principal at every maximal ideal (Equivalent characterizations of invertible fractional ideals).

[L2]

Every finite projective module with $P \otimes_{R} K ≅ K$ is isomorphic to a fractional ideal (A finite rank-one projective module embeds as a fractional ideal).

Proof

technique · direct

1.1

Let $I \subseteq K$ be an invertible fractional ideal. Then [L1] makes $I$ finite projective and locally free of rank one at every maximal ideal. After tensoring with the field $K$ , the module $I \otimes_{R} K$ is a one-dimensional $K$ -vector space, hence isomorphic to $K$ . Thus invertible fractional ideals are finite rank-one projectives.

F1L1givenalgebra

1.2

Conversely, let $P$ be a finite projective module with $P \otimes_{R} K ≅ K$ . By [L2], it is isomorphic to a fractional ideal $I \subseteq K$ . For every maximal ideal $m$ , the localisation $P_{m}$ is a free rank-one module over the local ring $R_{m}$ , so $I_{m}$ is principal. Therefore [L1] makes $I$ invertible.

L1L2givenalgebra

2.1

Steps 1.1 and 1.2 prove the equivalence between invertible fractional ideals and finite projective modules of constant rank one.

step 1.1step 1.2∎

Depends on

Used by

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Sources

J. P. May, Notes on Dedekind Rings (standard reference, not scraped)
The Stacks Project, Section 10.78: Finite projective modules (standard reference, not scraped)