Krull's principal ideal theorem: Difference between revisions
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{{ | {{curing property implication}} | ||
==Statement== | ==Statement== | ||
===Symbolic statement=== | |||
Let <math>R</math> be a [[Noetherian ring|Noetherian]] and <math>x \in R</math>. Let <math>P</math> be a minimal prime ideal among those containing <math>x</math>. Then, the codimension of <math>P</math> is at most 1. | Let <math>R</math> be a [[Noetherian ring|Noetherian]] and <math>x \in R</math>. Let <math>P</math> be a minimal prime ideal among those containing <math>x</math>. Then, the codimension of <math>P</math> is at most 1. | ||
===Property-theoretic statement=== | |||
The [[property of commutative unital rings]] of being a [[Noetherian ring]] is stronger than the property of being a [[ring satisfying PIT]]. | |||
==Generalizations== | ==Generalizations== | ||
* [[Krull's height theorem]]: This is often also called the ''final version'' of the principal ideal theorem. | * [[Krull's height theorem]]: This is often also called the ''final version'' of the principal ideal theorem. | ||
* [[Determinantal ideal theorem]]: This generalizes the principal ideal theorem to the ideal generated by the determinants of minors of a matrix | * [[Determinantal ideal theorem]]: This generalizes the principal ideal theorem to the ideal generated by the determinants of minors of a matrix | ||
Revision as of 23:43, 7 January 2008
This article gives the statement and possibly, proof, of an implication relation between two commutative unital ring properties. That is, it states that every commutative unital ring satisfying the first commutative unital ring property must also satisfy the second commutative unital ring property
View all commutative unital ring property implications | View all commutative unital ring property non-implications |Get help on looking up commutative unital ring property implications/non-implications
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Statement
Symbolic statement
Let be a Noetherian and . Let be a minimal prime ideal among those containing . Then, the codimension of is at most 1.
Property-theoretic statement
The property of commutative unital rings of being a Noetherian ring is stronger than the property of being a ring satisfying PIT.
Generalizations
- Krull's height theorem: This is often also called the final version of the principal ideal theorem.
- Determinantal ideal theorem: This generalizes the principal ideal theorem to the ideal generated by the determinants of minors of a matrix