In commutative algebra

Commutative algebra

Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...

, the height of a prime ideal

Prime ideal

In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...

  in a ring

Ring (mathematics)

In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

  is the number of strict inclusions in the longest chain of prime ideal

Prime ideal

In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...

s contained in . 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

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, this is the codimension

Codimension

In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and also to submanifolds in manifolds, and suitable subsets of algebraic varieties.The dual concept is relative dimension.-Definition:...

 of the subvariety of Spec() corresponding to I.

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

Masayoshi Nagata

Masayoshi Nagata was a Japanese mathematician, known for his work in the field of commutative algebra....

. 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 ⊆ , every saturated chain of strict inclusions has the same length .
• A ring is universally catenary if and only if any finitely generated algebra over it is catenary.
• A local ring
  Local ring
  In abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...
  
   is Cohen–Macaulay if and only if for any ideal I the height and depth
  Depth (ring theory)
  In commutative and homological algebra, depth is an important invariant of rings and modules. Although depth can be defined more generally, the most common case considered is the case of modules over a commutative Noetherian local ring. In this case, the depth of a module is related with its...
  
   of I with respect to I are equal.
• A Noetherian
  Noetherian
  In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects; in particular,* Noetherian group, a group that satisfies the ascending chain condition on subgroups...
  
   integral domain is a unique factorization domain
  Unique factorization domain
  In mathematics, a unique factorization domain is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements , analogous to the fundamental theorem of arithmetic for the integers...
  
   if and only if every height 1 prime ideal is principal.

In a Noetherian ring

Noetherian ring

In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...

, Krull's height theorem

Krull's principal ideal theorem

In commutative algebra, Krull's principal ideal theorem, named after Wolfgang Krull , gives a bound on the height of a principal ideal in a Noetherian ring...

says that the height of an ideal generated by n elements is no greater than n.