The Krull-Akizuki Theorem concerns a one-dimensional reduced Noetherian ring A with total ring of fractions K: if L is a finite extension of K and B is a reduced ring with A contained in B contained in L, then B is itself a Noetherian ring of dimension at most one, and every nonzero ideal of B has finite index over A. One consequence is that the integral closure of a Dedekind domain in a finite extension of its field of fractions is again a Dedekind domain, a fact that does not extend directly to higher dimension, where the analogous statement instead becomes the Mori-Nagata theorem.
Facts
StatementLet A be a one-dimensional reduced noetherian ring with total ring of fractions K, and L a finite extension of K. If A is contained in B, B is contained in L and B is reduced, then B is a noetherian ring of dimension at most one, and for every nonzero ideal I of B, B/I is finite over A. 2 Classification
Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Krull-Akizuki theorem (Wikipedia)
Proved By
Source Krull-Akizuki theorem (Wikipedia)
Sources
1. Wikipedia: Krull-Akizuki theorem
WikipediaLead section, statement-form referenceQuote, Lead section, statement-form reference
If A ⊂ B ⊂ L and B is reduced, then B is a noetherian ring of dimension at most one.
2. Krull-Akizuki theorem (Wikipedia)
Intro, sentences 1-2
then B is a noetherian ring of dimension at most one.
- In Branch: Commutative Algebra, Lead sentence
Proved By: Wolfgang Krull, Lead paragraph
In commutative algebra, the Krull-Akizuki theorem states the following: Let A be a one-dimensional reduced noetherian ring, K its total ring of fractions.
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