# Krull dimension

The **Krull dimension** of a commutative ring is the supremum of the lengths of chains of prime ideals in the ring, where a chain

a strictly increasing chain of prime ideals \[ \mathfrak{p}_0 \subsetneq \mathfrak{p}_1 \subsetneq \cdots \subsetneq \mathfrak{p}_n, \] in which each \(\mathfrak{p}_i\) differs from \(\mathfrak{p}_{i+1}\)

of strict inclusions has length \( n \). Equivalently, it is the Krull dimension of the topological space \( \mathrm{Spec}(R) \), the supremum over nonnegative integers \( n \) for which such a chain exists.<sup>[1](https://stacks.math.columbia.edu/tag/00KD)</sup> The concept, introduced by [Wolfgang Krull](https://www.edgepedia.org/wolfgang-krull) in 1928 for commutative Noetherian rings, translates geometric dimension into purely algebraic terms and is the standard notion of dimension in commutative algebra and algebraic geometry.<sup>[2](https://mathshistory.st-andrews.ac.uk//Biographies/Krull/)</sup>

| Key fact | Detail |
|---|---|
| Definition | Supremum of lengths of strictly increasing chains of prime ideals; equivalently, the Krull dimension of the topological space \( \mathrm{Spec}(R) \)<sup>[1](https://stacks.math.columbia.edu/tag/00KD)</sup> |
| Origin | Defined by Wolfgang Krull in 1928 for commutative Noetherian rings, in *Primidealketten in allgemeinen Ringbereichen*<sup>[2](https://mathshistory.st-andrews.ac.uk//Biographies/Krull/)</sup><sup> • </sup><sup>[3](https://digi.hadw-bw.de/view/sbhadwmnkl_1928_7/0003)</sup> |
| Height | The height of a prime \( \mathfrak{p} \) is \( \dim(R_{\mathfrak{p}}) \); the Krull dimension of \( R \) is the supremum of the heights of its maximal primes<sup>[1](https://stacks.math.columbia.edu/tag/00KD)</sup> |
| Polynomial rings | \( \dim k[x_1, \ldots, x_d] = d \), realized by the chain \( (0) \subsetneq (x_1) \subsetneq \cdots \subsetneq (x_1, \ldots, x_d) \)<sup>[4](https://androma.org/theorems/2951)</sup> |
| Finite generation | For a finitely generated domain over a field \( k \), the dimension equals the transcendence degree of its fraction field over \( k \)<sup>[5](https://www.jmilne.org/math/CourseNotes/AG10.pdf)</sup> |
| Height theorem | In a Noetherian ring, an ideal generated by \( r \) elements has height at most \( r \); a nonzero principal ideal has height one<sup>[6](https://encyclopediaofmath.org/wiki/Height_of_an_ideal)</sup> |
| Dimension zero | A Noetherian ring has dimension 0 exactly when it is Artinian<sup>[1](https://stacks.math.columbia.edu/tag/00KD)</sup> |

## Definition and basic properties

Prime ideals of \( R \) form a chain when they can be arranged as \( \mathfrak{p}_0 \subset \mathfrak{p}_1 \subset \cdots \subset \mathfrak{p}_n \), where each \( \mathfrak{p}_i \) is prime and distinct from \( \mathfrak{p}_{i+1} \); the chain's length is \( n \), counting the strict inclusions.<sup>[1](https://stacks.math.columbia.edu/tag/00KD)</sup> The Krull dimension of \( R \) is the supremum of the integers \( n \geq 0 \) arising this way. The definition is topological in origin: prime ideals of \( R \) correspond, by an inclusion-reversing bijection, to irreducible closed subsets of \( \mathrm{Spec}(R) \), so chains of primes are chains of irreducible [closed sets](https://www.edgepedia.org/closed-set).<sup>[1](https://stacks.math.columbia.edu/tag/00KD)</sup>

The **height** of a prime ideal \( \mathfrak{p} \) is the dimension of the [local ring](https://www.edgepedia.org/local-ring) \( R_{\mathfrak{p}} \), that is, the supremum of lengths of chains of primes contained in \( \mathfrak{p} \).<sup>[1](https://stacks.math.columbia.edu/tag/00KD)</sup> Because a maximal ideal can always be appended to the end of any chain, the Krull dimension of \( R \) equals the supremum of the heights of its maximal primes.<sup>[1](https://stacks.math.columbia.edu/tag/00KD)</sup> The co-height of \( \mathfrak{p} \), the length of chains above it, satisfies \( \mathrm{ht}(\mathfrak{p}) + \mathrm{coht}(\mathfrak{p}) \leq \dim R \), with equality for example when \( R \) is a local [Cohen–Macaulay ring](https://www.edgepedia.org/cohen-macaulay-ring).<sup>[6](https://encyclopediaofmath.org/wiki/Height_of_an_ideal)</sup>

Several operations leave the dimension controlled. Localization and quotient cannot increase Krull dimension.<sup>[7](https://www.researchgate.net/publication/228359411_CONSTRUCTIVE_KRULL_DIMENSION_I_INTEGRAL_EXTENSIONS)</sup> Quotienting by the nilradical changes nothing, since every prime contains the nilradical, so \( \dim(R) = \dim(R/\mathfrak{N}) \).<sup>[5](https://www.jmilne.org/math/CourseNotes/AG10.pdf)</sup> Integral extensions preserve dimension: if \( A \subset B \) is an integral extension of rings, then \( \dim(A) = \dim(B) \).<sup>[4](https://androma.org/theorems/2951)</sup> For a [Noetherian ring](https://www.edgepedia.org/noetherian-ring) \( A \), \( \dim(A) = \sup(\dim(A_{\mathfrak{m}})) \) over maximal ideals \( \mathfrak{m} \).<sup>[5](https://www.jmilne.org/math/CourseNotes/AG10.pdf)</sup>

## Examples

A field has dimension 0: its only prime ideal is \( (0) \), so no chain of positive length exists. The polynomial ring \( k[x_1, \ldots, x_d] \) over a field has dimension exactly \( d \). The chain

\[ (0) \subsetneq (x_1) \subsetneq (x_1, x_2) \subsetneq \cdots \subsetneq (x_1, \ldots, x_d) \]

has length \( d \), and no longer chain exists.<sup>[4](https://androma.org/theorems/2951)</sup> The integers \( \mathbb{Z} \), like \( k[x] \), have dimension 1, with chain \( (0) \subsetneq (p) \) for a prime number \( p \).<sup>[4](https://androma.org/theorems/2951)</sup>

Dimension zero behaves rigidly. A Noetherian ring of dimension 0 is Artinian, and conversely every [Artinian ring](https://www.edgepedia.org/artinian-ring) is Noetherian of dimension zero; equivalently, such a ring is a finite product of Artinian local rings, and its spectrum is a finite discrete space.<sup>[1](https://stacks.math.columbia.edu/tag/00KD)</sup>

## Height theorems

Krull's principal ideal theorem, proved in the 1928 paper that introduced the dimension concept, states that in a Noetherian ring the height of a nonzero principal ideal is one.<sup>[2](https://mathshistory.st-andrews.ac.uk//Biographies/Krull/)</sup><sup> • </sup><sup>[6](https://encyclopediaofmath.org/wiki/Height_of_an_ideal)</sup> The general form bounds height by the number of generators: an ideal generated by \( r \) elements in a Noetherian ring has height at most \( r \), and conversely a prime ideal of height \( r \) is minimal over some ideal generated by \( r \) elements.<sup>[6](https://encyclopediaofmath.org/wiki/Height_of_an_ideal)</sup> In particular, every ideal in a Noetherian ring has finite height, though the co-height need not be finite.<sup>[6](https://encyclopediaofmath.org/wiki/Height_of_an_ideal)</sup> A concrete consequence from the Stacks project: if \( \mathfrak{p} \) is minimal over a principal ideal \( (x) \) in a Noetherian ring, its height is 0 or 1, and if \( \mathfrak{q} \) is minimal over \( (\mathfrak{p}, x) \), no prime lies strictly between them.<sup>[1](https://stacks.math.columbia.edu/tag/00KD)</sup>

## Connection with algebraic geometry

For an affine variety over a field, the Krull dimension of its coordinate ring equals the geometric dimension of the variety.<sup>[8](https://websites.umich.edu/~mmustata/ag0523.pdf)</sup> The bridge is the Noether normalization theorem: a finitely generated \( k \)-algebra \( A \) whose reduction is a domain contains a polynomial ring \( k[x_1, \ldots, x_r] \) over which \( A \) is a [finitely generated module](https://www.edgepedia.org/finitely-generated-module), and \( r \) equals the transcendence degree of the fraction field of \( A \) over \( k \).<sup>[5](https://www.jmilne.org/math/CourseNotes/AG10.pdf)</sup> Every maximal chain of distinct prime ideals in \( A \) then has length \( \mathrm{trdeg}_k(A) \), every maximal ideal has that height, and \( \dim(A) = \mathrm{trdeg}_k(A) \).<sup>[5](https://www.jmilne.org/math/CourseNotes/AG10.pdf)</sup> Geometrically, normalization presents the variety as a finite covering of affine \( r \)-space.<sup>[9](https://www.mat.uniroma3.it/users/fontana/documents/clarck-commutative%20algebra.pdf)</sup> For a smooth complex variety, the Krull dimension of the coordinate ring equals the topological dimension as a real manifold divided by 2, since each algebraic dimension corresponds to two real dimensions.<sup>[10](https://syntagmatic.github.io/moonshine/noether/14-noether-normalization.html)</sup>

The height of a prime ideal matches codimension: the height of \( \mathfrak{p} \) equals the codimension of the variety it defines, while the co-height equals that variety's dimension.<sup>[6](https://encyclopediaofmath.org/wiki/Height_of_an_ideal)</sup> A dimension 1 Noetherian regular local ring can be read as a germ of a curve.<sup>[11](https://math.stanford.edu/~vakil/0506-216/216class12.pdf)</sup>

## Beyond the Noetherian setting

Krull originally defined the dimension for Noetherian rings, where chains terminate and the supremum is a maximum.<sup>[2](https://mathshistory.st-andrews.ac.uk//Biographies/Krull/)</sup> For general rings the supremum may be infinite, and the notion extends verbatim. The topological version applies to any space: any nonempty [Hausdorff space](https://www.edgepedia.org/hausdorff-space) has Krull dimension 0, since its irreducible closed subsets are single points.<sup>[12](https://www.math.mcgill.ca/darmon/courses/18-19/algebra2/kunz.pdf)</sup> The definition also reaches large function rings. The ring of holomorphic functions on a connected complex manifold has Krull dimension at least the continuum whenever it is positive.<sup>[13](https://arxiv.org/abs/1512.04647)</sup> For the ring \( \mathcal{S}(M) \) of semialgebraic functions on a semialgebraic set \( M \), \( \dim \mathcal{S}(M) = \dim M \), and the height of the maximal ideal of a point \( p \) equals the local dimension of \( M \) at \( p \).<sup>[14](https://arxiv.org/abs/1306.4109)</sup> For finitely generated graded commutative algebras, the Krull dimension coincides with the rate of growth of the algebra's graded pieces.<sup>[15](https://people.tamu.edu/~sjw/pub/gsm204.pdf)</sup>

## References

1. [The Stacks Project authors, Section 10.60: Dimension (tag 00KD), The Stacks Project](https://stacks.math.columbia.edu/tag/00KD)
2. [Wolfgang Krull (1899–1971), MacTutor History of Mathematics, University of St Andrews](https://mathshistory.st-andrews.ac.uk//Biographies/Krull/)
3. [W. Krull, Primidealketten in allgemeinen Ringbereichen, Sitzungsberichte der Heidelberger Akademie der Wissenschaften (1928)](https://digi.hadw-bw.de/view/sbhadwmnkl_1928_7/0003)
4. [Dimension of Finitely Generated Domains, Androma](https://androma.org/theorems/2951)
5. [J. S. Milne, Algebraic Geometry, course notes](https://www.jmilne.org/math/CourseNotes/AG10.pdf)
6. [Height of an ideal, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Height_of_an_ideal)
7. [Constructive Krull Dimension I: Integral Extensions, ResearchGate](https://www.researchgate.net/publication/228359411_CONSTRUCTIVE_KRULL_DIMENSION_I_INTEGRAL_EXTENSIONS)
8. [M. Mustață, Algebraic Geometry lecture notes, University of Michigan](https://websites.umich.edu/~mmustata/ag0523.pdf)
9. [Noether Normalization Theorem, Commutative Algebra notes, Università Roma Tre](https://www.mat.uniroma3.it/users/fontana/documents/clarck-commutative%20algebra.pdf)
10. [Noether Normalization, Invariance: The Mathematics of Emmy Noether](https://syntagmatic.github.io/moonshine/noether/14-noether-normalization.html)
11. [R. Vakil, Foundations of Algebraic Geometry, class notes, Stanford](https://math.stanford.edu/~vakil/0506-216/216class12.pdf)
12. [E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry, lecture notes](https://www.math.mcgill.ca/darmon/courses/18-19/algebra2/kunz.pdf)
13. [Krull dimensions of rings of holomorphic functions, arXiv:1512.04647](https://arxiv.org/abs/1512.04647)
14. [On the Krull dimension of rings of semialgebraic functions, arXiv:1306.4109](https://arxiv.org/abs/1306.4109)
15. [Support Varieties for Finite-Dimensional Algebras, AMS Graduate Studies in Mathematics 204](https://people.tamu.edu/~sjw/pub/gsm204.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Ring theory › Commutative algebra › Dimension theory*

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