Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Promine ...
, the Krull dimension of a commutative ring
''R'', named after
Wolfgang Krull (26 August 1899 – 12 April 1971) was a German mathematician who made fundamental contributions to commutative algebra, introducing concepts that are now central to the subject.
Krull was born and went to school in Baden-Baden. ...
, is the
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
of the lengths of all chains of prime ideals
. The Krull dimension need not be finite even for a Noetherian ring
. More generally the Krull dimension can be defined for module
s over possibly non-commutative rings as the deviation
of the poset of submodules.
The Krull dimension was introduced to provide an algebraic definition of the
dimension of an algebraic variety
In mathematics and specifically in algebraic geometry, the dimension of an algebraic variety may be defined in various equivalent ways.
Some of these definitions are of geometric nature, while some other are purely algebraic and rely on commutat ...
: the dimension of the affine variety
defined by an ideal ''I'' in a polynomial ring
''R'' is the Krull dimension of ''R''/''I''.
''k'' has Krull dimension 0; more generally, ''k'' 1, ..., ''x''''n''">'x''1, ..., ''x''''n''
has Krull dimension ''n''. A
principal ideal domain
In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, ...
that is not a field has Krull dimension 1. A local ring
has Krull dimension 0 if and only if every element of its maximal ideal
There are several other ways that have been used to define the dimension of a ring. Most of them coincide with the Krull dimension for Noetherian rings, but can differ for non-Noetherian rings.
We say that a chain of prime ideals of the form
has length n. That is, the length is the number of strict inclusions, not the number of primes; these differ by 1. We define the Krull dimension of
to be the supremum of the lengths of all chains of prime ideals in
Given a prime
in ''R'', we define the of
, to be the supremum of the lengths of all chains of prime ideals contained in
, meaning that
. In other words, the height of
is the Krull dimension of the localization
of ''R'' at
. A prime ideal has height zero if and only if it is a minimal prime ideal
. The Krull dimension of a ring is the supremum of the heights of all maximal ideals, or those of all prime ideals. The height is also sometimes called the codimension, rank, or altitude of a prime ideal.
In a Noetherian ring
, every prime ideal has finite height. Nonetheless, Nagata gave an example of a Noetherian ring of infinite Krull dimension. A ring is called catenary
if any inclusion
of prime ideals can be extended to a maximal chain of prime ideals between
, and any two maximal chains between
have the same length. A ring is called universally catenary
if any finitely generated algebra over it is catenary. Nagata gave an example of a Noetherian ring which is not catenary.
In a Noetherian ring, a prime ideal has height at most ''n'' if and only if it is a minimal prime ideal
over an ideal generated by ''n'' elements ( Krull's height theorem
and its converse). It implies that the descending chain condition
holds for prime ideals in such a way the lengths of the chains descending from a prime ideal are bounded by the number of generators of the prime.
More generally, 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(
) corresponding to I.
It follows readily from the definition of the
spectrum of a ring
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with th ...
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 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
of ''R'' corresponds to a generic point of the closed subset associated to
by the Galois connection.
* The dimension of a polynomial ring
over a field ''k'' 1, ..., ''x''''n''">'x''1, ..., ''x''''n''
is the number of variables ''n''. In the language of algebraic geometry
, this says that the affine space of dimension ''n'' over a field has dimension ''n'', as expected. In general, if ''R'' is a
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
ring of dimension ''n'', then the dimension of ''R'' 'x''
is ''n'' + 1. If the Noetherian hypothesis is dropped, then ''R'' 'x''
can have dimension anywhere between ''n'' + 1 and 2''n'' + 1.
* For example, the ideal