quality:corruption
Dimension of a commutative ring

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 p 0 ⊊ p 1 ⊊ ⋯ ⊊ p n {\displaystyle {\mathfrak {p}}_{0}\subsetneq {\mathfrak {p}}_{1}\subsetneq \cdots \subsetneq {\mathfrak {p}}_{n}}. It turns out that the polynomial ring k {\displaystyle k} over a field k has dimension n. The fundamental theorem of dimension theory states that the following numbers coincide for a noetherian local ring (R, m) {\displaystyle (R,{\mathfrak {m}})}:

A commutative ring R is said to be catenary if for every pair of prime ideals p ⊂ p ′ {\displaystyle {\mathfrak {p}}\subset {\mathfrak {p}}'}, there exists a finite chain of prime ideals p = p 0 ⊊ ⋯ ⊊ p n = p ′ {\displaystyle {\mathfrak {p}}={\mathfrak {p}}_{0}\subsetneq \cdots \subsetneq {\mathfrak {p}}_{n}={\mathfrak {p}}'} 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 p {\displaystyle {\mathfrak {p}}} and p ′ {\displaystyle {\mathfrak {p}}'} 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 p {\displaystyle {\mathfrak {p}}},

where ht p {\displaystyle \operatorname {ht} {\mathfrak {p}}} is the height of p {\displaystyle {\mathfrak {p}}}.

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 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 notion.