In
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
, Krull's principal ideal theorem, named after
Wolfgang Krull
Wolfgang Krull (26 August 1899 – 12 April 1971) was a German mathematician who made fundamental contributions to commutative algebra, introducing concepts that are now central to the subject.
Krull was born and went to school in Baden-Baden. H ...
(1899–1971), gives a bound on the
height
Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is).
For example, "The height of that building is 50 m" or "The height of an airplane in-flight is ab ...
of a
principal ideal
In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where ...
in a commutative
Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
. 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 In mathematics, especially in commutative algebra, certain prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principal ideal theorem use minimal primes.
Definitio ...
over ''I'' has height at most one.
This theorem can be generalized to
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
s 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.
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
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
'' gives a direct proof. Kaplansky's ''Commutative Rings'' includes a proof due to
David Rees David or Dai Rees may refer to:
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* Dave Rees (born 1969), American drummer for SNFU and Wheat Chiefs
* David Rees (cartoonist) (born 1972), American cartoonist and televis ...
.
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
The concept of symbolic power, also known as symbolic domination (''domination symbolique'' in French language) or symbolic violence, was first introduced by French sociologist Pierre Bourdieu to account for the tacit, almost unconscious modes of ...
of
. It forms a descending chain of ideals
. Thus, there is the descending chain of ideals
in the ring
. Now, the
radical
Radical may refer to:
Politics and ideology Politics
* Radical politics, the political intent of fundamental societal change
*Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe an ...
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 In mathematics, more specifically abstract algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem — governs the interaction between the Jacobson radical of a ring (typically a commutative ring) an ...
(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
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
*
* {{Citation , last1=Matsumura , first1=Hideyuki , title=Commutative Algebra , publisher=Benjamin , location=New York , year=1970, see in particular section (12.I), p. 77
*http://www.math.lsa.umich.edu/~hochster/615W10/supDim.pdf
Commutative algebra
Ideals (ring theory)
Theorems in ring theory