Krull's principal ideal theorem
In commutative algebra, Krull's principal ideal theorem, named after Wolfgang Krull (1899–1971), gives a bound on the height of a principal ideal in a commutative Noetherian ring. The theorem is sometimes referred to by its German name, Krulls Hauptidealsatz (Satz meaning "proposition" or "theorem").
Precisely, if R is a Noetherian ring and I is a principal, proper ideal of R, then each minimal prime ideal over I has height at most one.
This theorem can be generalized to ideals that are not principal, and the result is often called Krull's height theorem. This says that if R is a Noetherian ring and I is a proper ideal generated by n elements of R, then each minimal prime over I has height at most n.
The principal ideal theorem and the generalization, the height theorem, both follow from the fundamental theorem of dimension theory in commutative algebra (see also below for the direct proofs). Bourbaki's Commutative Algebra gives a direct proof. Kaplansky's Commutative ring includes a proof due to David Rees.
Proof of the principal ideal theorem
Let be a Noetherian ring, x an element of it and a minimal prime over x. Replacing A by the localization , we can assume is local with the maximal ideal . Let be a strictly smaller prime ideal and let , which is a -primary ideal called the n-th symbolic power of . It forms a descending chain of ideals . Thus, there is the descending chain of ideals in the ring . Now, the radical is the intersection of all minimal prime ideals containg ; is among them. But is a unique maximal ideal and thus . Since contains some power of its radical, it follows that is an Artinian ring and thus the chain stabilizes and so there is some n such that . It implies:
from the fact is -primary (if is in , then with and . Since is minimal over , and so implies is in .) Now, quotienting out both sides by yields . Then, by Nakayama's lemma, letting , we get that both sides are zero and , thus . Using Nakayama's lemma again, and is an Artinian ring; thus, the height of is zero.
Proof of the height theorem
Krull’s height theorem can be proved as a consequence of the principal ideal theorem by induction on the number of elements. Let be elements in , a minimal prime over and a prime ideal such that there is no prime strictly between them. Replacing by the localization we can assume is a local ring; note we then have . By minimality, cannot contain all the ; relabeling the subscripts, say, . Since every prime ideal containing is between and , and thus we can write for each ,
with and . Now we consider the ring and the corresponding chain in it. If is a minimal prime over , then contains and thus ; that is to say, is a minimal prime over and so, by Krull’s principal ideal theorem, is a minimal prime (over zero); is a minimal prime over . By inductive hypothesis, and thus .
- Matsumura, Hideyuki (1970), Commutative Algebra, New York: Benjamin, see in particular section (12.I), p. 77