Editing Krull dimension (section)

==Schemes==
{{see also|Dimension of a scheme}}
It follows readily from the definition of the [[spectrum of a ring]] Spec(''R''), the space of prime ideals of ''R'' equipped with the [[Zariski topology]], that the Krull dimension of ''R'' is equal to the dimension of its spectrum as a [[topological space]], meaning the supremum of the lengths of all chains of [[Irreducible space|irreducible]] closed subsets.  This follows immediately from the [[Galois connection]] between ideals of ''R'' and closed subsets of Spec(''R'') and the observation that, by the definition of Spec(''R''), each prime ideal <math>\mathfrak{p}</math> of ''R'' corresponds to a generic point of the closed subset associated to <math>\mathfrak{p}</math> by the Galois connection.