In
commutative algebra
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. Prominent ...
, the Krull dimension of a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
''R'', named after
Wolfgang Krull
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. H ...
, is the
supremum
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 lo ...
of the lengths of all chains of
prime ideals. The Krull dimension need not be finite even for a
Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noeth ...
. More generally the Krull dimension can be defined for
modules 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
In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ideal. ...
defined by an ideal ''I'' in a
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variabl ...
''R'' is the Krull dimension of ''R''/''I''.
A
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''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 princi ...
that is not a field has Krull dimension 1. A
local ring In abstract algebra, more specifically 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 n ...
has Krull dimension 0 if and only if every element of its
maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals ...
is nilpotent.
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.
Explanation
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
, written
, 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 In mathematics, especially in commutative algebra, certain prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principal ideal theorem use minimal primes.
Definition ...
. 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
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noeth ...
, every prime ideal has finite height. Nonetheless, Nagata gave an example of a Noetherian ring of infinite Krull dimension. A ring is called
catenary
In physics and geometry, a catenary (, ) is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends in a uniform gravitational field.
The catenary curve has a U-like shape, superficia ...
if any inclusion
of prime ideals can be extended to a maximal chain of prime ideals between
and
, and any two maximal chains between
and
have the same length. A ring is called
universally catenary In mathematics, a commutative ring ''R'' is catenary if for any pair of prime ideals
:''p'', ''q'',
any two strictly increasing chains
:''p''=''p''0 ⊂''p''1 ... ⊂''p'n''= ''q'' of prime ideals
are contained in maximal strictly incre ...
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 In mathematics, especially in commutative algebra, certain prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principal ideal theorem use minimal primes.
Definition ...
over an ideal generated by ''n'' elements (
Krull's height theorem and its converse). It implies that the
descending chain condition In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings.Jacobson (2009), p. 142 and 147 These c ...
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
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
, this is the
codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.
For affine and projective algebraic varieties, the codimension equals ...
of the subvariety of Spec(
) corresponding to I.
Schemes
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 the ...
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 In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the fund ...
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.
Examples
* The dimension of a
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variabl ...
over a field ''k''
1, ..., ''x''''n''">'x''1, ..., ''x''''n''is the number of variables ''n''. In the language of
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
, 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 leng ...
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