# 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.

## Proofs

### 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 .

## References

- 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