DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 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.

Products, colons, and inverse candidates for fractional ideals

Definition

Let $I$ and $J$ be fractional ideals of a domain $R$ with fraction field $K$ . Their product is

I J : = {\sum_{r = 1}^{n} x_{r} y_{r} : n \geq 1, x_{r} \in I, y_{r} \in J} \subseteq K,

and their colon is

(I : J) : = {x \in K : x J \subseteq I} .

The usual inverse candidate is

I^{- 1} : = (R : I) .

The next lemma checks that these constructions stay inside the world of fractional ideals.

Depends on

Fractional ideals

Used by

Invertible fractional ideals Definition
Computing an inverse fractional ideal explicitly Example
The basic operations on fractional ideals are well defined Lemma

Dependency tree · two levels

2 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)
Mircea Mustata, Introduction to Commutative Algebra, §8.5 (standard reference, not scraped)