Definition 5

The Krull dimension, or simply dimension, of a (commutative) ring A is by definition the Krull dimension of the topological space Spec(A) (II, § 4, No. 3, Def. 4); it is denoted by \dim(A). If p is a prime ideal of A, the dimension of A at p is by definition the number \dim_p(\operatorname{Spec}(A)); it is denoted by \dim_p(A).

The mapping $p \mapsto V(p)$ is a decreasing bijection of the set of prime ideals of $A$ onto the set of irreducible closed subsets of $\mathrm{Spec}(A)$ (loc. cit., Cor. 2 to Prop. 14). The dimension of $A$ is therefore the least upper bound of the set of lengths of chains of prime ideals of $A$; it is equal to $-\infty, +\infty$ or to a positive integer.

Let $p \in \mathrm{Spec}(A)$; the sets $\mathrm{Spec}(A)_f$, where $f$ runs through $A$, form a basis for the topology of $\mathrm{Spec}(A)$, and $p$ belongs to the open set $\mathrm{Spec}(A)_f$ if and only if $f$ does not belong to $p$. Consequently, $\dim_p(A)$ is the greatest lower bound of the numbers $\dim(A_f)$, where $f$ runs through $A - p$ (II, § 5, No. 1, Prop. 1).