Editing Ring theory (section)

===Dimension of a commutative ring===
{{main|Dimension theory (algebra)}}
In this section, ''R'' denotes a commutative ring.  The [[Krull dimension]] of ''R'' is the supremum of the lengths ''n'' of all the chains of prime ideals <math>\mathfrak{p}_0 \subsetneq \mathfrak{p}_1 \subsetneq \cdots \subsetneq \mathfrak{p}_n</math>. It turns out that the polynomial ring <math>k[t_1, \cdots, t_n]</math> over a field ''k'' has dimension ''n''. The fundamental theorem of dimension theory states that the following numbers coincide for a noetherian local ring <math>(R, \mathfrak{m})</math>:<ref>{{harvnb|Matsumura|1989|loc=Theorem 13.4}}</ref>
*The Krull dimension of ''R''.
*The minimum number of the generators of the <math>\mathfrak{m}</math>-primary ideals.
*The dimension of the graded ring <math>\textstyle \operatorname{gr}_{\mathfrak{m}}(R) = \bigoplus_{k \ge 0} \mathfrak{m}^k/{\mathfrak{m}^{k+1}}</math> (equivalently, 1 plus the degree of its [[Hilbert polynomial]]).

A commutative ring ''R'' is said to be [[Catenary ring|catenary]] if for every pair of prime ideals <math>\mathfrak{p} \subset \mathfrak{p}'</math>, there exists a finite chain of prime ideals <math>\mathfrak{p} = \mathfrak{p}_0 \subsetneq \cdots \subsetneq \mathfrak{p}_n = \mathfrak{p}'</math> that is maximal in the sense that it is impossible to insert an additional prime ideal between two ideals in the chain, and all such maximal chains between <math>\mathfrak{p}</math> and  <math>\mathfrak{p}'</math> have the same length.  Practically all noetherian rings that appear in applications are catenary.  Ratliff proved that a noetherian local integral domain ''R'' is catenary if and only if for every prime ideal <math>\mathfrak{p}</math>,
:<math>\operatorname{dim}R = \operatorname{ht}\mathfrak{p} + \operatorname{dim}R/\mathfrak{p}</math>
where <math>\operatorname{ht}\mathfrak{p}</math> is the [[Height (ring theory)|height]] of <math>\mathfrak{p}</math>.<ref>{{harvnb|Matsumura|1989|loc=Theorem 31.4}}</ref>

If ''R'' is an integral domain that is a finitely generated ''k''-algebra, then its dimension is the [[transcendence degree]] of its field of fractions over ''k''. If ''S'' is an [[integral extension]] of a commutative ring ''R'', then ''S'' and ''R'' have the same dimension.

Closely related concepts are those of [[depth (ring theory)|depth]] and [[global dimension]]. In general, if ''R'' is a noetherian local ring, then the depth of ''R'' is less than or equal to the dimension of ''R''. When the equality holds, ''R'' is called a [[Cohen–Macaulay ring]]. A [[regular local ring]] is an example of a Cohen–Macaulay ring. It is a theorem of Serre that ''R'' is a regular local ring if and only if it has finite global dimension and in that case the global dimension is the Krull dimension of ''R''. The significance of this is that a global dimension is a [[homological algebra|homological]] notion.