Codimension of an ideal: Difference between revisions
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==Definition== | ==Definition== | ||
Let <math>R</math> be a [[commutative unital ring]] and <math>I</math> an [[ideal]] in <math>R</math>. The '''codimension''' of <math>I</math> is defined as follows: | Let <math>R</math> be a [[commutative unital ring]] and <math>I</math> an [[ideal]] in <math>R</math>. The '''codimension''' or '''height''' of <math>I</math> is defined as follows: | ||
* If <math>I</math> is a [[prime ideal]], it is defined as the [[Krull dimension]] of the [[localization at a prime ideal|localization]] <math>R_I</math> | * If <math>I</math> is a [[prime ideal]], it is defined as the [[Krull dimension]] of the [[localization at a prime ideal|localization]] <math>R_I</math> | ||
* Otherwise, it is defined to be the minimum of the Krull dimensions of [[prime ideal]]s containing <math>I</math> | * Otherwise, it is defined to be the minimum of the Krull dimensions of [[prime ideal]]s containing <math>I</math> | ||
Latest revision as of 16:19, 12 May 2008
Definition
Let be a commutative unital ring and an ideal in . The codimension or height of is defined as follows:
- If is a prime ideal, it is defined as the Krull dimension of the localization
- Otherwise, it is defined to be the minimum of the Krull dimensions of prime ideals containing
