Theorem in commutative algebra
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 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.
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 containing
;
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 of
, it follows that
cannot contain all the
; relabeling the subscripts, say,
. Since every prime ideal containing
is between
and
,
and thus we can write for each
,
![{\displaystyle x_{i}^{r_{i}}=y_{i}+a_{i}x_{1}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/159d3d5c73819531fcc25750589285b2da51af1e)
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
.