Krull-Azikuzi theorem: Difference between revisions
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* <math>R</math> be a [[Noetherian ring|Noetherian]] [[integral domain]] of [[Krull dimension]] 1 | * <math>R</math> be a [[Noetherian ring|Noetherian]] [[integral domain]] of [[Krull dimension]] 1 | ||
* <math>K</math> be the [[field of fractions]] of < | * <math>K</math> be the [[field of fractions]] of <math>R</math> | ||
* <math>L</math> be a finite extension field of <math>K</math> | * <math>L</math> be a finite extension field of <math>K</math> | ||
* <math>S</math> be a subring of <math>L</math> that contains <math>R</math> | * <math>S</math> be a subring of <math>L</math> that contains <math>R</math> | ||
Latest revision as of 16:26, 12 May 2008
Statement
Let:
- be a Noetherian integral domain of Krull dimension 1
- be the field of fractions of
- be a finite extension field of
- be a subring of that contains
Then the following hold:
- is Noetherian
- The Krull dimension of is at most 1
- Given any nonzero ideal of , there are only finitely many ideals of containing that
In particular, the integral closure of in is Noetherian