LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passaudited 2026-08-31

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The ideal class group quotient is well defined

Statement

Let $R$ be a Dedekind domain. The nonzero principal fractional ideals form a subgroup of the fractional-ideal group, and multiplication descends to a well-defined product on $Cl (R)$ .

Facts & Assumptions

Given: A Dedekind domain $R$ and nonzero fractional ideals $I, J$ .

[F1]

The ideal class group is defined as the quotient by nonzero principal fractional ideals (The ideal class group).

[F2]

Invertibility is expressed by fractional-ideal multiplication inside one common fraction field (Invertible fractional ideals).

Proof

technique · direct

1.1

If $(a)$ and $(b)$ are principal fractional ideals with $a, b \in K^{\times}$ , then $(a) (b) = (a b)$ and $(a)^{- 1} = (a^{- 1})$ . Therefore the principal fractional ideals form a subgroup.

F1F2givenalgebra

2.1

If $I^{'} = (a) I$ and $J^{'} = (b) J$ , then $I^{'} J^{'} = (a b) (I J)$ . So changing representatives by principal multiples changes the product by another principal factor. Hence the class of $I J$ depends only on the classes of $I$ and $J$ .

F1F2step 1.1algebra∎

Depends on

Used by

Cited to discharge well-definedness by The ideal class group.

Dependency tree · two levels

4 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

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