Editing Krull dimension (section)

==Explanation==
We say that a chain of prime ideals of the form
<math>\mathfrak{p}_0\subsetneq \mathfrak{p}_1\subsetneq \ldots \subsetneq \mathfrak{p}_n</math>
has '''length ''n'''''. That is, the length is the number of strict inclusions, not the number of primes; these differ by 1. We define the '''Krull dimension''' of <math>R</math> to be the supremum of the lengths of all chains of prime ideals in <math>R</math>.

Given a prime ideal <math>\mathfrak{p}</math> in ''R'', we define the '''{{visible anchor|height}}''' of <math>\mathfrak{p}</math>, written <math>\operatorname{ht}(\mathfrak{p})</math>, to be the supremum of the lengths of all chains of prime ideals contained in <math>\mathfrak{p}</math>, meaning that <math>\mathfrak{p}_0\subsetneq \mathfrak{p}_1\subsetneq \ldots \subsetneq \mathfrak{p}_n = \mathfrak{p}</math>.<ref name="matsumura30">Matsumura, Hideyuki: "Commutative Ring Theory", page 30–31, 1989</ref>  In other words, the height of <math>\mathfrak{p}</math> is the Krull dimension of the [[localization of a ring|localization]] of ''R'' at <math>\mathfrak{p}</math>. A prime ideal has height zero if and only if it is a [[minimal prime ideal]]. The Krull dimension of a ring is the supremum of the heights of all maximal ideals, or those of all prime ideals. The height is also sometimes called the codimension, rank, or altitude of a prime ideal.

In a [[Noetherian ring]], every prime ideal has finite height. Nonetheless, [[Masayoshi Nagata|Nagata]] gave an example of a Noetherian ring of infinite Krull dimension.<ref>Eisenbud, D. ''Commutative Algebra'' (1995). Springer, Berlin. Exercise 9.6.</ref> A ring is called '''[[catenary ring|catenary]]''' if any inclusion <math>\mathfrak{p}\subset \mathfrak{q}</math> of prime ideals can be extended to a maximal chain of prime ideals between <math>\mathfrak{p}</math> and <math>\mathfrak{q}</math>, and any two maximal chains between <math>\mathfrak{p}</math>
and <math>\mathfrak{q}</math> have the same length. A ring is called [[universally catenary]] if any finitely generated algebra over it is  catenary. Nagata gave an example of a Noetherian ring which is not catenary.<ref>Matsumura, H. ''Commutative Algebra'' (1970). Benjamin, New York. Example 14.E.</ref>

In a Noetherian ring, a prime ideal has height at most ''n'' if and only if it is a [[minimal prime ideal]] over an ideal generated by ''n'' elements ([[Krull's principal ideal theorem|Krull's height theorem]] and its converse).<ref>{{harvnb|Serre|2000|loc=Ch. III, § B.2, Theorem 1, Corollary 4.}}</ref> It implies that the [[descending chain condition]] holds for prime ideals in such a way the lengths of the chains descending from a prime ideal are bounded by the number of generators of the prime.<ref>{{harvnb|Eisenbud|1995|loc=Corollary 10.3.}}</ref>

More generally, the height of an ideal '''I'''  is the infimum of the heights of all prime ideals containing '''I'''.  In the language of [[algebraic geometry]], this is the [[codimension]] of the subvariety of Spec(<math>R</math>) corresponding to '''I'''.<ref name="matsumura30" />