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
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This fact is an application of the following pivotal fact/result/idea: Nakayama's lemma
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- 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
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.
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
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 zero-dimensional ring.