Height (ring theory)

In commutative algebra, the height of a prime ideal $\mathfrak{p}$ in a ring $R$ is the number of strict inclusions in the longest chain of prime ideals contained in $\mathfrak{p}$ .^[1] Then the height of an ideal I is the infimum of the heights of all prime ideals containing I. In the language of algebraic geometry, this is the codimension of the subvariety of Spec( $R$ ) corresponding to I.^[2]

It is not true that every maximal chain of prime ideals with common endpoints has the same length; the first counterexample was found by Masayoshi Nagata. The existence of such an ideal is usually considered pathological and is ruled out by an assumption that the ring is catenary.

Many conditions on rings impose conditions on the heights of certain ideals or on all ideals of certain heights. Some notable conditions are:

A ring is catenary if and only if for every two prime ideals $\mathfrak{p}$ ⊆ $\mathfrak{q}$ , every saturated chain of strict inclusions $\mathfrak{p}= \mathfrak{p}_{1} \subsetneq \mathfrak{p}_{2} \cdots \subsetneq \mathfrak{p}_{h}=\mathfrak{q}$ has the same length $h$ .
A ring is universally catenary if and only if any finitely generated algebra over it is catenary.
A local ring is Cohen–Macaulay if and only if for any ideal I the height and depth of I with respect to I are equal.
A Noetherian integral domain is a unique factorization domain if and only if every height 1 prime ideal is principal.^[3]

In a Noetherian ring, Krull's height theorem says that the height of an ideal generated by n elements is no greater than n.

^ Matsumura, Hideyuki: "Commutative Ring Theory", page 30–31, 1989
^ Matsumura, Hideyuki: "Commutative Ring Theory", page 30–31, 1989
^ Hartshorne,Robin:"Algebraic Geometry", page 7,1977