# 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 converse is also true: if a prime ideal has height *n*, then it is a minimal prime ideal over an ideal generated by *n* elements.^{[1]}

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 Rings* includes a proof due to David Rees.

## Proofs[edit]

### Proof of the principal ideal theorem[edit]

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 (which says a finitely generated module *M* is zero if for some ideal *I* contained in the radical), we get ; i.e., and thus . Using Nakayama's lemma again, and is an Artinian ring; thus, the height of is zero.

### Proof of the height theorem[edit]

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 .

## References[edit]

- Eisenbud, David (1995).
*Commutative Algebra with a View Toward Algebraic Geometry*. Graduate Texts in Mathematics.**150**. Springer-Verlag. doi:10.1007/978-1-4612-5350-1. ISBN 0-387-94268-8. - Matsumura, Hideyuki (1970),
*Commutative Algebra*, New York: Benjamin, see in particular section (12.I), p. 77 - http://www.math.lsa.umich.edu/~hochster/615W10/supDim.pdf