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 (from Haupt- ("Principal") + ideal + Satz ("theorem")).
Precisely, if R is a Noetherian ring and I is a principal, proper ideal of R, then each minimal prime ideal containing 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.
be a Noetherian ring, x an element of it and
a minimal prime over x.
Replacing A by the localization
is local with the maximal ideal
be a strictly smaller prime ideal and let
-primary ideal called the n-th symbolic power of
It forms a descending chain of ideals
Thus, there is the descending chain of ideals
is the intersection of all minimal prime ideals containing
is a unique maximal ideal and thus
contains some power of its radical, it follows that
is an Artinian ring and thus the chain
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
is an Artinian ring; thus, the height of
Krull’s height theorem can be proved as a consequence of the principal ideal theorem by induction on the number of elements.
a minimal prime over
is a local ring; note we then have
Since every prime ideal containing
is a minimal prime over
and so, by Krull’s principal ideal theorem,
By inductive hypothesis,