Height (ring theory)

The dimension or, more precisely Krull dimension (after Wolfgang Krull ) of a commutative ring with unit element is the ideological dimension of its associated in algebraic geometry variety, or more generally, the associated schema.

This article deals with commutative algebra. In particular, all rings considered are commutative and have an identity element. Ring homomorphisms form elements from one to one elements. For more details see Commutative Algebra.

Definition

The height of a prime ideal is the maximum length of an ascending chain of prime ideals

The height is then. If there is no maximum length, the prime ideal has infinite height.

The dimension of a ring is the supremum of the heights of its prime ideals.

Properties

  • In a Noetherian ring, every prime ideal has a finite height. But there are Noetherian rings of infinite dimension.
  • In Noetherian local rings is the dimension equal to the smallest possible cardinality of a definition of ideal, in particular finite.
  • The height of a prime ideal is equal to the codimension of the corresponding closed subset of the spectrum of the ring.
  • Krull's principal ideal theorem says that the level of prime ideals of a noetherian ring, which are minimal over a principal ideal (ie, contain it and in respect of this property are minimal), may be at most 1. More generally, the level of prime ideals of Noetherian rings which are minimal over an ideal that can be generated by r elements, at most r.

Examples

  • . Maximum ascending chains of prime ideals have the form
  • An integral domain is one-dimensional if and only if every nonzero prime ideal is maximal. Each Dedekind ring is a one-dimensional integral domain.
  • Sink and all other artinian rings are zero-dimensional.
  • The formula
  • The Going up theorem states that

Topological version

The dimension discussed here can be generalized to the Krull dimension of topological spaces by the Primidealketten is replaced by chains closed, irreducible subsets. Then, the dimension of the ring is no more than the dimension of its spectrum Krull.

Krulldimension Spectrum of a ring Affine space
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