Krull's principal ideal theorem
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 nonimplications Get help on looking up commutative unital ring property implications/nonimplications

This fact is an application of the following pivotal fact/result/idea: Nakayama's lemma
View other applications of Nakayama's lemma OR Read a survey article on applying Nakayama's lemma
Contents
Statement
Symbolic statement
Let be a Noetherian and . Let be a minimal element among prime ideals containing . Then, the codimension of is at most 1.
Propertytheoretic 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
Proof
Starting assumptions
In the above setup, we show that if is a prime ideal in contained inside , then the codimension of , which is the same as the dimension of is zero. This will show that the codimension of is at most 1. The crucial thing we shall use is that .
First note that we can replace by , so we may assume that is a maximal ideal in . We now begin the proof.
Argument setup
Since is minimal over , we see that in the ring , the ideal is the unique maximal ideal of a local ring, and is also a minimal prime ideal. Thus, is a local Artinian ring with unique maximal ideal .
Hence, in , consider the descending chain , where denotes the symbolic power of . This descending chain stabilizes, so we get:
In particular, we can find and such that:
This yields , so since , we get
Nakayama's lemma
The above reasoning shows that:
Now consider the module . The above equation yields that:
But since , we see that is in the Jacobson radical of , so Nakayama's lemma yields that . Thus .
We now apply Nakayama's lemma in the localization at , to conclude that:
This yields that isa zerodimensional ring.