Editing Krull's principal ideal theorem

{{short description|Theorem in commutative algebra}}
In [[commutative algebra]], '''Krull's principal ideal theorem''', named after [[Wolfgang Krull]] (1899–1971),  gives a bound on the [[height (ring theory)|height]] of a [[principal ideal]] in a commutative [[Noetherian ring]].  The theorem is sometimes referred to by its German name, ''Krulls Hauptidealsatz'' (from ''{{Wikt-lang|en|Haupt-}}'' ("Principal") + ''{{Wikt-lang|en|ideal}}'' +  ''{{Wikt-lang|en|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 [[ideal (ring theory)|ideal]]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.<ref>{{harvnb|Eisenbud|1995|loc=Corollary 10.5.}}</ref>

The principal ideal theorem and the generalization, the height theorem, both follow from the [[fundamental theorem of dimension theory (algebra)|fundamental theorem of dimension theory]] in commutative algebra (see also below for the direct proofs). Bourbaki's ''[[Éléments de mathématique|Commutative Algebra]]'' gives a direct proof. Kaplansky's ''Commutative Rings'' includes a proof due to [[David Rees (mathematician)|David Rees]].

== Proofs ==
=== Proof of the principal ideal theorem ===
Let <math>A</math> be a Noetherian ring, ''x'' an element of it and <math>\mathfrak{p}</math> a minimal prime over ''x''. Replacing ''A'' by the [[Localization (commutative algebra)#Localization at primes|localization]] <math>A_\mathfrak{p}</math>, we can assume <math>A</math> is local with the maximal ideal <math>\mathfrak{p}</math>. Let <math>\mathfrak{q} \subsetneq \mathfrak{p}</math> be a strictly smaller prime ideal and let <math>\mathfrak{q}^{(n)} = \mathfrak{q}^n A_{\mathfrak{q}} \cap A</math>, which is a <math>\mathfrak{q}</math>-[[primary ideal]] called the ''n''-th [[symbolic power (mathematics)|symbolic power]] of <math>\mathfrak{q}</math>. It forms a descending chain of ideals <math>A \supset \mathfrak{q} \supset \mathfrak{q}^{(2)} \supset \mathfrak{q}^{(3)} \supset \cdots</math>. Thus, there is the descending chain of ideals <math>\mathfrak{q}^{(n)} + (x)/(x)</math> in the ring <math>\overline{A} = A/(x)</math>. Now, the [[radical of an ideal|radical]] <math>\sqrt{(x)}</math> is the intersection of all minimal prime ideals containing <math>x</math>; <math>\mathfrak{p}</math> is among them. But <math>\mathfrak{p}</math> is a unique maximal ideal and thus <math>\sqrt{(x)} = \mathfrak{p}</math>. Since <math>(x)</math> contains some power of its radical, it follows that <math>\overline{A}</math> is an Artinian ring and thus the chain <math>\mathfrak{q}^{(n)} + (x)/(x)</math> stabilizes and so there is some ''n'' such that <math>\mathfrak{q}^{(n)} + (x) = \mathfrak{q}^{(n+1)} + (x)</math>. It implies:
:<math>\mathfrak{q}^{(n)} = \mathfrak{q}^{(n+1)} + x \, \mathfrak{q}^{(n)}</math>,

from the fact <math>\mathfrak{q}^{(n)}</math> is <math>\mathfrak{q}</math>-primary (if <math>y</math> is in <math>\mathfrak{q}^{(n)}</math>, then <math>y = z + ax</math> with <math>z \in \mathfrak{q}^{(n+1)}</math> and <math>a \in A</math>. Since <math>\mathfrak{p}</math> is minimal over <math>x</math>, <math>x \not\in \mathfrak{q}</math> and so <math>ax \in \mathfrak{q}^{(n)}</math> implies <math>a</math> is in <math>\mathfrak{q}^{(n)}</math>.) Now, quotienting out both sides by   <math>\mathfrak{q}^{(n+1)}</math>  yields <math>\mathfrak{q}^{(n)}/\mathfrak{q}^{(n+1)} = (x)\mathfrak{q}^{(n)}/\mathfrak{q}^{(n+1)}</math>. Then, by [[Nakayama's lemma]] (which says a finitely generated module ''M'' is zero if <math>M = IM</math> for some ideal ''I'' contained in the radical), we get <math>M = \mathfrak{q}^{(n)}/\mathfrak{q}^{(n+1)} = 0</math>; i.e., <math>\mathfrak{q}^{(n)} = \mathfrak{q}^{(n+1)}</math> and thus <math>\mathfrak{q}^{n} A_{\mathfrak{q}} = \mathfrak{q}^{n+1} A_{\mathfrak{q}}</math>. Using Nakayama's lemma again, <math>\mathfrak{q}^{n} A_{\mathfrak{q}} = 0</math> and <math>A_{\mathfrak{q}}</math> is an Artinian ring; thus, the height of <math>\mathfrak{q}</math> is zero. <math>\square</math>

=== 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 <math>x_1, \dots, x_n</math> be elements in <math>A</math>, <math>\mathfrak{p}</math> a minimal prime over <math>(x_1, \dots, x_n)</math> and <math>\mathfrak{q} \subsetneq \mathfrak{p}</math> a prime ideal such that there is no prime strictly between them. Replacing <math>A</math> by the localization <math>A_{\mathfrak{p}}</math> we can assume <math>(A, \mathfrak{p})</math> is a local ring; note we then have <math>\mathfrak{p} = \sqrt{(x_1, \dots, x_n)}</math>. By minimality of <math>\mathfrak{p}</math>, it follows that <math>\mathfrak{q}</math> cannot contain all the <math>x_i</math>; relabeling the subscripts, say, <math>x_1 \not\in \mathfrak{q}</math>. Since every prime ideal containing <math>\mathfrak{q} + (x_1)</math> is between <math>\mathfrak{q}</math> and <math>\mathfrak{p}</math>, <math>\sqrt{\mathfrak{q} + (x_1)} = \mathfrak{p}</math> and thus we can write for each <math>i \ge 2</math>,
:<math>x_i^{r_i} = y_i + a_i x_1</math>
with <math>y_i \in \mathfrak{q}</math> and <math>a_i \in A</math>. Now we consider the ring <math>\overline{A} = A/(y_2, \dots, y_n)</math> and the corresponding chain <math>\overline{\mathfrak{q}} \subset \overline{\mathfrak{p}}</math> in it. If <math>\overline{\mathfrak{r}}</math> is a minimal prime over <math>\overline{x_1}</math>, then <math>\mathfrak{r}</math> contains <math>x_1, x_2^{r_2}, \dots, x_n^{r_n}</math> and thus <math>\mathfrak{r} = \mathfrak{p}</math>; that is to say, <math>\overline{\mathfrak{p}}</math> is a minimal prime over <math>\overline{x_1}</math> and so, by Krull's principal ideal theorem, <math>\overline{\mathfrak{q}}</math> is a minimal prime (over zero); <math>\mathfrak{q}</math> is a minimal prime over <math>(y_2, \dots, y_n)</math>. By inductive hypothesis, <math>\operatorname{ht}(\mathfrak{q}) \le n-1</math> and thus <math>\operatorname{ht}(\mathfrak{p}) \le n</math>. <math>\square</math>

==References==
{{reflist}}
* {{cite book|author-link=David Eisenbud|last1=Eisenbud|first1=David|title=Commutative Algebra with a View Toward Algebraic Geometry|series=Graduate Texts in Mathematics|volume=150|publisher=Springer-Verlag|year=1995|isbn=0-387-94268-8|doi=10.1007/978-1-4612-5350-1}}
* {{Citation | last1=Matsumura | first1=Hideyuki | title=Commutative Algebra | publisher=Benjamin | location=New York | year=1970}}, see in particular section (12.I), p.&nbsp;77
*[http://www.math.lsa.umich.edu/~hochster/615W10/supDim.pdf ]

[[Category:Commutative algebra]]
[[Category:Ideals (ring theory)]]
[[Category:Theorems in ring theory]]