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OR:

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. Promine ...
, the Krull dimension of a commutative ring ''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. ...
, 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 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 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 defined by an ideal ''I'' in a polynomial ring ''R'' is the Krull dimension of ''R''/''I''. A field ''k'' has Krull dimension 0; more generally, ''k'' '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 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 $\mathfrak_0\subsetneq \mathfrak_1\subsetneq \ldots \subsetneq \mathfrak_n$ 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 $R$ to be the supremum of the lengths of all chains of prime ideals in $R$. Given a prime $\mathfrak$ in ''R'', we define the of $\mathfrak$, written $\operatorname\left(\mathfrak\right)$, to be the supremum of the lengths of all chains of prime ideals contained in $\mathfrak$, meaning that $\mathfrak_0\subsetneq \mathfrak_1\subsetneq \ldots \subsetneq \mathfrak_n = \mathfrak$. In other words, the height of $\mathfrak$ is the Krull dimension of the localization of ''R'' at $\mathfrak$. 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 $\mathfrak\subset \mathfrak$ of prime ideals can be extended to a maximal chain of prime ideals between $\mathfrak$ and $\mathfrak$, and any two maximal chains between $\mathfrak$ and $\mathfrak$ 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($R$) 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 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 $\mathfrak$ of ''R'' corresponds to a generic point of the closed subset associated to $\mathfrak$ by the Galois connection.

# Examples

* The dimension of a polynomial ring over a field ''k'' '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
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 dimension 1. * An
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural se ...
is a field if and only if its Krull dimension is zero. Dedekind domains that are not fields (for example,
discrete valuation ring In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions: # ''R'' ...
s) have dimension one. * The Krull dimension of the
zero ring In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rng of square zero, i.e., a rng in which fo ...
is typically defined to be either $-\infty$ or $-1$. The zero ring is the only ring with a negative dimension. * A ring is Artinian if and only if it is
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 ...
and its Krull dimension is ≤0. * An
integral extension In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that :b^n + a_ b^ + \cdots + a_1 b + a_0 = 0. That is to say, ''b'' is ...
of a ring has the same dimension as the ring does. * Let ''R'' be an algebra over a field ''k'' that is an integral domain. Then the Krull dimension of ''R'' is less than or equal to the transcendence degree of the field of fractions of ''R'' over ''k''. The equality holds if ''R'' is finitely generated as an algebra (for instance by the Noether normalization lemma). * Let ''R'' be a Noetherian ring, ''I'' an ideal and $\operatorname_I\left(R\right) = \oplus_0^\infty I^k/I^$ be the
associated graded ring In mathematics, the associated graded ring of a ring ''R'' with respect to a proper ideal ''I'' is the graded ring: :\operatorname_I R = \oplus_^\infty I^n/I^. Similarly, if ''M'' is a left ''R''-module, then the associated graded module is the gr ...
(geometers call it the ring of the normal cone of ''I''.) Then $\operatorname \operatorname_I\left(R\right)$ is the supremum of the heights of maximal ideals of ''R'' containing ''I''. * A commutative Noetherian ring of Krull dimension zero is a direct product of a finite number (possibly one) of local rings of Krull dimension zero. * A Noetherian local ring is called a Cohen–Macaulay ring if its dimension is equal to its depth. A regular local ring is an example of such a ring. * 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 ...
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural se ...
is a
unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is a ...
if and only if every height 1 prime ideal is principal. * For a commutative Noetherian ring the three following conditions are equivalent: being a reduced ring of Krull dimension zero, being a field or a
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
of fields, being von Neumann regular.

# Of a module

If ''R'' is a commutative ring, and ''M'' is an ''R''-module, we define the Krull dimension of ''M'' to be the Krull dimension of the quotient of ''R'' making ''M'' a
faithful module In mathematics, the annihilator of a subset of a module over a ring is the ideal formed by the elements of the ring that give always zero when multiplied by an element of . Over an integral domain, a module that has a nonzero annihilator is ...
. That is, we define it by the formula: :$\dim_R M := \dim\left( R/\operatorname_R\left(M\right)\right)$ where Ann''R''(''M''), the annihilator, is the kernel of the natural map R → End''R''(M) of ''R'' into the ring of ''R''-linear endomorphisms of ''M''. In the language of schemes, finitely generated modules are interpreted as
coherent sheaves In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refe ...
, or generalized finite rank
vector bundles In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every ...
.

# For non-commutative rings

The Krull dimension of a module over a possibly non-commutative ring is defined as the deviation of the poset of submodules ordered by inclusion. For commutative Noetherian rings, this is the same as the definition using chains of prime ideals.McConnell, J.C. and Robson, J.C. ''Noncommutative Noetherian Rings'' (2001). Amer. Math. Soc., Providence. Corollary 6.4.8. The two definitions can be different for commutative rings which are not Noetherian.