Alphabeta MathDefinitionDefinition: 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.
Dedekind domains
Definition
A Dedekind domain is a commutative ring such that:
- is a Noetherian ring (Left and right Noetherian rings);
- is an integrally closed domain (Integral closure in an extension ring and integrally closed domains);
- has Krull dimension (Krull dimension of a nonzero ring).
Thus this page uses the one-dimensional normal-domain convention, so fields are excluded.
Depends on
Used by
- The integral closure of a Dedekind domain in a finite separable extension is Dedekind Corollary
- Prime-ideal valuations on fractional ideals Definition
- The divisor group of a Dedekind domain Definition
- Localizing a Dedekind domain at a nonzero prime gives a DVR Lemma
- Prime-ideal valuations of a fractional ideal have finite support and add under products Lemma
- A localization of a Dedekind domain is Dedekind or a field Theorem
- Equivalent local characterizations of Dedekind domains Theorem
- Every nonzero fractional ideal of a Dedekind domain is invertible Theorem
Dependency tree · two levels
9 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. S. Milne, A Primer of Commutative Algebra, §20 (standard reference, not scraped)
- Mircea Mustata, Introduction to Commutative Algebra, §8.5 (standard reference, not scraped)