Editing Krull dimension (section)

==Examples==

* The dimension of a [[polynomial ring]] over a field ''k''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>] is the number of variables ''n''. In the language of [[algebraic geometry]], this says that the affine space of dimension ''n'' over a field has dimension ''n'', as expected. In general, if ''R'' is a [[Noetherian ring|Noetherian]] ring of dimension ''n'', then the dimension of ''R''[''x''] is  ''n'' + 1. If the Noetherian hypothesis is dropped, then ''R''[''x''] can have dimension anywhere between ''n'' + 1 and 2''n'' + 1.
* For example, the ideal <math>\mathfrak{p} = (y^2 - x, y) \subset \mathbb{C}[x,y]</math> has height 2 since we can form the maximal ascending chain of prime ideals<math>(0)=\mathfrak{p}_0 \subsetneq (y^2 - x)= \mathfrak{p}_1 \subsetneq (y^2 - x, y) = \mathfrak{p}_2 = \mathfrak{p}</math>.
* Given an irreducible polynomial <math>f \in \mathbb{C}[x,y,z]</math>, the ideal <math>I = (f^3)</math> is not prime (since <math>f\cdot f^2 \in I</math>, but neither of the factors are), but we can easily compute the height since the smallest prime ideal containing <math>I</math> is just <math>(f)</math>.
* The ring of integers '''Z''' has dimension 1. More generally, any [[principal ideal domain]] that is not a field has dimension 1.
* An [[integral domain]] is a field if and only if its Krull dimension is zero. [[Dedekind domain]]s that are not fields (for example, [[discrete valuation ring]]s) have dimension one.
* The Krull dimension of the [[zero ring]] is typically defined to be either <math>-\infty</math> or <math>-1</math>. The zero ring is the only ring with a negative dimension.
* A ring is [[Artinian ring|Artinian]] if and only if it is [[Noetherian ring|Noetherian]] and its Krull dimension is ≤0.
* An [[integral extension]] of a ring has the same dimension as the ring does.
* Let ''R'' be an algebra over a field ''k'' that is an integral domain. Then the Krull dimension of ''R'' is less than or equal to the transcendence degree of the field of fractions of ''R'' over ''k''.<ref>[https://mathoverflow.net/q/79959 Krull dimension less or equal than transcendence degree?]</ref> The equality holds if ''R'' is finitely generated as an algebra (for instance by the [[Noether normalization lemma]]).
* Let ''R'' be a Noetherian ring, ''I'' an ideal and <math>\operatorname{gr}_I(R) = \bigoplus_{k=0}^\infty I^k/I^{k+1}</math> be the [[associated graded ring]] (geometers call it the ring of the [[normal cone]] of ''I''). Then <math>\operatorname{dim} \operatorname{gr}_I(R)</math> is the supremum of the heights of maximal ideals of ''R'' containing ''I''.<ref>{{harvnb|Eisenbud|1995|loc=Exercise 13.8}}</ref>
* A commutative Noetherian ring of Krull dimension zero is a direct product of a finite number (possibly one) of [[local ring]]s of Krull dimension zero.
* A Noetherian local ring is called a [[Cohen–Macaulay ring]] if its dimension is equal to its [[Depth (ring theory)|depth]]. A [[regular local ring]] is an example of such a ring.
* A Noetherian [[integral domain]] is a [[unique factorization domain]] if and only if every height 1 prime ideal is principal.<ref>Hartshorne, Robin: "Algebraic Geometry", page 7, 1977</ref>
* For a commutative Noetherian ring the three following conditions are equivalent: being a [[reduced ring]] of Krull dimension zero, being a field or a [[direct product]] of fields, being [[von Neumann regular ring|von Neumann regular]].