Incommutative algebra, theKrull dimension of acommutative ringR, named afterWolfgang Krull, is thesupremum of the lengths of allchains ofprime ideals. The Krull dimension need not be finite even for aNoetherian ring. More generally the Krull dimension can be defined formodules over possiblynon-commutative rings as thedeviation of theposet of submodules.

The Krull dimension was introduced to provide an algebraic definition of thedimension of an algebraic variety: the dimension of theaffine variety defined by an idealI in apolynomial ringR is the Krull dimension ofR/I.

Afieldk has Krull dimension 0; more generally,k[x₁, ...,x_n] has Krull dimensionn. Aprincipal ideal domain that is not a field has Krull dimension 1. Alocal ring has Krull dimension 0 if and only if every element of itsmaximal ideal isnilpotent.

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${\displaystyle {\mathfrak{p}}_0⊊{\mathfrak{p}}_1⊊\dots ⊊{\mathfrak{p}}_n}$haslengthn. That is, the length is the number of strict inclusions, not the number of primes; these differ by 1. We define theKrull dimension of${\displaystyle R}$ to be the supremum of the lengths of all chains of prime ideals in${\displaystyle R}$.

Given a prime ideal${\displaystyle \mathfrak{p}}$ inR, we define theheight of${\displaystyle \mathfrak{p}}$, written${\displaystyle \mathrm{ht}(\mathfrak{p})}$, to be the supremum of the lengths of all chains of prime ideals contained in${\displaystyle \mathfrak{p}}$, meaning that${\displaystyle {\mathfrak{p}}_0⊊{\mathfrak{p}}_1⊊\dots ⊊{\mathfrak{p}}_n=\mathfrak{p}}$.^[1] In other words, the height of${\displaystyle \mathfrak{p}}$ is the Krull dimension of thelocalization ofR at${\displaystyle \mathfrak{p}}$. A prime ideal has height zero if and only if it is aminimal 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 aNoetherian ring, every prime ideal has finite height. Nonetheless, Nagata gave an example of a Noetherian ring of infinite Krull dimension.^[2] A ring is calledcatenary if any inclusion${\displaystyle \mathfrak{p}\subset \mathfrak{q}}$ of prime ideals can be extended to a maximal chain of prime ideals between${\displaystyle \mathfrak{p}}$ and${\displaystyle \mathfrak{q}}$, and any two maximal chains between${\displaystyle \mathfrak{p}}$and${\displaystyle \mathfrak{q}}$ have the same length. A ring is calleduniversally catenary if any finitely generated algebra over it is catenary. Nagata gave an example of a Noetherian ring which is not catenary.^[3]

In a Noetherian ring, a prime ideal has height at mostn if and only if it is aminimal prime ideal over an ideal generated byn elements (Krull's height theorem and its converse).^[4] It implies that thedescending 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.^[5]

More generally, the height of an idealI is the infimum of the heights of all prime ideals containingI. In the language ofalgebraic geometry, this is thecodimension of the subvariety of Spec(${\displaystyle R}$) corresponding toI.^[1]

It follows readily from the definition of thespectrum of a ring Spec(R), the space of prime ideals ofR equipped with theZariski topology, that the Krull dimension ofR is equal to the dimension of its spectrum as atopological space, meaning the supremum of the lengths of all chains ofirreducible closed subsets. This follows immediately from theGalois connection between ideals ofR and closed subsets of Spec(R) and the observation that, by the definition of Spec(R), each prime ideal${\displaystyle \mathfrak{p}}$ ofR corresponds to a generic point of the closed subset associated to${\displaystyle \mathfrak{p}}$ by the Galois connection.

• The dimension of apolynomial ring over a fieldk[x₁, ...,x_n] is the number of variablesn. In the language ofalgebraic geometry, this says that the affine space of dimensionn over a field has dimensionn, as expected. In general, ifR is aNoetherian ring of dimensionn, then the dimension ofR[x] isn + 1. If the Noetherian hypothesis is dropped, thenR[x] can have dimension anywhere betweenn + 1 and 2n + 1.
• For example, the ideal${\displaystyle \mathfrak{p}=(y^2-x,y)\subset \mathbb{C}[x,y]}$ has height 2 since we can form the maximal ascending chain of prime ideals${\displaystyle (0)={\mathfrak{p}}_0⊊(y^2-x)={\mathfrak{p}}_1⊊(y^2-x,y)={\mathfrak{p}}_2=\mathfrak{p}}$.
• Given an irreducible polynomial${\displaystyle f\in \mathbb{C}[x,y,z]}$, the ideal${\displaystyle I=(f^3)}$ is not prime (since${\displaystyle f\cdot f^2\in I}$, but neither of the factors are), but we can easily compute the height since the smallest prime ideal containing${\displaystyle I}$ is just${\displaystyle (f)}$.
• The ring of integersZ has dimension 1. More generally, anyprincipal ideal domain that is not a field has dimension 1.
• Anintegral domain is a field if and only if its Krull dimension is zero.Dedekind domains that are not fields (for example,discrete valuation rings) have dimension one.
• The Krull dimension of thezero ring is typically defined to be either${\displaystyle -\mathrm{\infty }}$ or${\displaystyle -1}$. The zero ring is the only ring with a negative dimension.
• A ring isArtinian if and only if it isNoetherian and its Krull dimension is ≤0.
• Anintegral extension of a ring has the same dimension as the ring does.
• LetR be an algebra over a fieldk that is an integral domain. Then the Krull dimension ofR is less than or equal to the transcendence degree of the field of fractions ofR overk.^[6] The equality holds ifR is finitely generated as an algebra (for instance by theNoether normalization lemma).
• LetR be a Noetherian ring,I an ideal and${\displaystyle {\mathrm{gr}}_I(R)=\underset{k=0}{\overset{\mathrm{\infty }}{⨁}}{I}^k/I^{k+1}}$ be theassociated graded ring (geometers call it the ring of thenormal cone ofI). Then${\displaystyle \dim {\mathrm{gr}}_I(R)}$ is the supremum of the heights of maximal ideals ofR containingI.^[7]
• A commutative Noetherian ring of Krull dimension zero is a direct product of a finite number (possibly one) oflocal rings of Krull dimension zero.
• A Noetherian local ring is called aCohen–Macaulay ring if its dimension is equal to itsdepth. Aregular local ring is an example of such a ring.
• ANoetherianintegral domain is aunique factorization domain if and only if every height 1 prime ideal is principal.^[8]
• For a commutative Noetherian ring the three following conditions are equivalent: being areduced ring of Krull dimension zero, being a field or adirect product of fields, beingvon Neumann regular.

IfR is a commutative ring, andM is anR-module, we define the Krull dimension ofM to be the Krull dimension of the quotient ofR makingM afaithful module. That is, we define it by the formula:

${\displaystyle {\dim }_{R}M:=\dim (R/{\mathrm{Ann}}_R(M))}$

where Ann_R(M), theannihilator, is the kernel of the natural map R → End_R(M) ofR into the ring ofR-linear endomorphisms ofM.

In the language ofschemes, finitely generated modules are interpreted ascoherent sheaves, or generalized finite rankvector bundles.

The Krull dimension of a module over a possibly non-commutative ring is defined as thedeviation of the poset of submodules ordered by inclusion. For commutative Noetherian rings, this is the same as the definition using chains of prime ideals.^[9] The two definitions can be different for commutative rings which are not Noetherian.

• Analytic spread
• Dimension theory (algebra)
• Gelfand–Kirillov dimension
• Hilbert function
• Homological conjectures in commutative algebra
• Krull's principal ideal theorem

• Irving Kaplansky,Commutative rings (revised ed.),University of Chicago Press, 1974,ISBN 0-226-42454-5. Page 32.
• L.A. Bokhut'; I.V. L'vov; V.K. Kharchenko (1991). "I. Noncommuative rings". InKostrikin, A.I.;Shafarevich, I.R. (eds.).Algebra II. Encyclopaedia of Mathematical Sciences. Vol. 18.Springer-Verlag.ISBN 3-540-18177-6. Sect.4.7.
• Eisenbud, David (1995),Commutative algebra with a view toward algebraic geometry,Graduate Texts in Mathematics, vol. 150, Berlin, New York:Springer-Verlag,ISBN 978-0-387-94268-1,MR 1322960
• Hartshorne, Robin (1977),Algebraic Geometry,Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag,ISBN 978-0-387-90244-9,MR 0463157
• Matsumura, Hideyuki (1989),Commutative Ring Theory, Cambridge Studies in Advanced Mathematics (2nd ed.),Cambridge University Press,ISBN 978-0-521-36764-6
• Serre, Jean-Pierre (2000).Local Algebra. Springer Monographs in Mathematics (in German).doi:10.1007/978-3-662-04203-8.ISBN 978-3-662-04203-8.OCLC 864077388.

• Commutative algebra
• Dimension

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