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| {{mergefrom|Height (ring theory)|date=July 2013}}
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| In [[commutative algebra]], the '''Krull dimension''' of a [[commutative ring]] ''R'', named after [[Wolfgang Krull]], is the supremum 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 a poset|deviation]] of the poset of submodules.
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| The Krull dimension has been introduced to provide an algebraic definition of the [[dimension of an algebraic variety]]: the dimension of the [[affine variety]] defined by an ideal ''I'' in a [[polynomial ring]] ''R'' is the Krull dimension of ''R''/''I''.
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| A field ''k'' has Krull dimension 0; more generally, ''k''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>] has Krull dimension ''n''. A [[principal ideal domain]] 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.
| | Po pierwsze, użycie owej technologii wywołuje, iż właściwie plag po zwieńczeniu fazy projektowania możemy rozpocząć do zrobienia gotowego naocznie tworu. Dlaczego rwące prototypowanie staje się aż tak pokupne? W zeszłym czasie jeszcze głośniej istnieje w sektorze o dziewiczych technikach Rapid Prototyping.<br><br>ZAŚ jeszcze jedna decydująca wartość, technologia owa wytwarza nam szansa gwałtownego i gibkiego zweryfikowania błędów konstrukcyjno- wizualnych, co samemu honoruje zaoszczędzić czas a pieniądze. Po pozostałe, technologia kasuje nieodzowność skorzystania urządzeń CNC, które w efekcie są cenne natomiast czasochłonne.<br><br>Po trzecie, Rapid Prototyping pozwoli nam [http://szybkieprototypowanie.blox.pl/html szybkie prototypy] przesunięcie petentowi skończonego wizualnie a tworu. Ponieważ jego skorzystanie to niepowszechna ekonomizacja czasu natomiast pieniądzach. |
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| == Explanation ==
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| We say that a chain of prime ideals of the form
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| <math>\mathfrak{p}_0\subsetneq \mathfrak{p}_1\subsetneq \ldots \subsetneq \mathfrak{p}_n</math>
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| 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 <math>R</math> to be the supremum of the lengths of all chains of prime ideals in <math>R</math>.
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| Given a prime <math>\mathfrak{p}</math> in ''R'', we define the '''[[height (ring theory)|height]]''' of <math>\mathfrak{p}</math>, written <math>\operatorname{ht}(\mathfrak{p})</math>, to be the supremum of the lengths of all chains of prime ideals contained in <math>\mathfrak{p}</math>, meaning that <math>\mathfrak{p}_0\subsetneq \mathfrak{p}_1\subsetneq \ldots \subsetneq \mathfrak{p}_n\subseteq \mathfrak{p}</math>. In other words, the height of <math>\mathfrak{p}</math> is the Krull dimension of the [[localization of a ring|localization]] of ''R'' at <math>\mathfrak{p}</math>. 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.
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| In a Noetherian ring, every prime ideal has finite height. Nonetheless,
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| Nagata gave an example of a Noetherian ring of infinite Krull dimension.<ref>Eisenbud, D. ''Commutative Algebra'' (1995). Springer, Berlin. Exercise 9.6.</ref> A ring is called '''[[catenary ring|catenary]]''' if any inclusion <math>\mathfrak{p}\subset \mathfrak{q}</math> of prime ideals can be extended to a maximal chain of prime ideals between <math>\mathfrak{p}</math> and <math>\mathfrak{q}</math>, and any two maximal chains between <math>\mathfrak{p}</math>
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| and <math>\mathfrak{q}</math> have the same length. Nagata gave an example of a Noetherian ring which is not catenary.<ref>Matsumura, H. ''Commutative Algebra'' (1970). Benjamin, New York. Example 14.E.</ref>
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| ==Krull dimension and schemes==
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| It follows readily from the definition of the [[spectrum of a ring]] 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 <math>\mathfrak{p}</math> of ''R'' corresponds to a generic point of the closed subset associated to <math>\mathfrak{p}</math> by the Galois connection.
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| ==Examples==
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| * The dimension of a [[polynomial ring]] over a field ''k''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>] 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 ring|Noetherian]] 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.
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| * The ring of integers '''Z''' has dimension 1. More generally, any [[principal ideal domain]] that is not a field has dimension 1.
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| * An [[integral domain]] is a field if and only if its Krull dimension is zero. [[Dedekind domain]]s that are not fields (for example, [[discrete valuation ring]]s) have dimension one. A [[Noetherian ring|Noetherian]] ring is [[Artinian ring|Artinian]] if and only if its Krull dimension is 0.
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| * An [[integral extension]] of a ring has the same dimension as the ring does.
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| * 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''.<ref>http://mathoverflow.net/questions/79959/krull-dimension-transcendence-degree</ref> The equality holds if ''R'' is finitely generated as algebra (for instance by the [[noether normalization lemma]]).
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| * Let ''R'' be a noetherian ring, ''I'' an ideal and <math>\operatorname{gr}_I(R) = \oplus_0^\infty I^k/I^{k+1}</math> be the [[associated graded ring]] (geometers call it the ring of the [[normal cone]] of ''I''.) Then <math>\operatorname{dim} \operatorname{gr}_I(R)</math> is the supremum of the heights of maximal ideals of ''R'' containing ''I''.<ref>{{harvnb|Eisenbud|2004|loc=Exercise 13.8}}</ref>
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| * A Noetherian local ring is called a [[Cohen–Macaulay ring]] if its dimension is equal to its [[Depth (ring theory)|depth]]. A [[regular local ring]] is an example of such a ring.
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| ==Krull dimension of a module==
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| 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]]. That is, we define it by the formula:
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| :<math>\operatorname{dim}_R M := \operatorname{dim}( R/\operatorname{Ann}_R(M))</math>
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| where Ann<sub>''R''</sub>(''M''), the [[annihilator (ring theory)|annihilator]], is the kernel of the natural map R → End<sub>''R''</sub>(M) of ''R'' into the ring of ''R''-linear endomorphisms of ''M''.
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| In the language of [[Scheme (mathematics)|schemes]], finitely generated modules are interpreted as [[coherent sheaves]], or generalized finite rank [[vector bundles]].
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| ==Krull dimension for non-commutative rings==
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| The Krull dimension of a module over a possibly non-commutative ring is defined as the [[deviation of a poset|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.<ref>McConnell, J.C. and Robson, J.C. ''Noncommutative Noetherian Rings'' (2001). Amer. Math. Soc., Providence. Corollary 6.4.8.</ref> The two definitions can be different for commutative rings which are not Noetherian.
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| == See also ==
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| *[[Dimension theory (algebra)]]
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| *[[Regular local ring]]
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| *[[Hilbert function]]
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| *[[Krull's principal ideal theorem]]
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| ==Notes==
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| {{reflist}}
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| ==Bibliography==
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| * [[Irving Kaplansky]], ''Commutative rings (revised ed.)'', [[University of Chicago Press]], 1974, ISBN 0-226-42454-5. Page 32.
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| * {{cite book | author1=L.A. Bokhut' | author2=I.V. L'vov | author3=V.K. Kharchenko | chapter=I. Noncommuative rings | editor1-first=A.I. | editor1-last=Kostrikin | editor1-link=A.I. Kostrikin | editor2-first=I.R. | editor2-last=Shafarevich | editor2-link=Igor Shafarevich | title=Algebra II | series=Encyclopaedia of Mathematical Sciences | volume=18 | publisher=[[Springer-Verlag]] | year=1991 | isbn=3-540-18177-6 }} Sect.4.7.
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| * {{Citation | last1=Eisenbud | first1=David | author1-link=David Eisenbud | title=Commutative algebra with a view toward algebraic geometry | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-94268-1 | mr=1322960 | year=1995 | volume=150}}
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| [[Category:Commutative algebra]]
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| [[Category:Dimension]]
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Po pierwsze, użycie owej technologii wywołuje, iż właściwie plag po zwieńczeniu fazy projektowania możemy rozpocząć do zrobienia gotowego naocznie tworu. Dlaczego rwące prototypowanie staje się aż tak pokupne? W zeszłym czasie jeszcze głośniej istnieje w sektorze o dziewiczych technikach Rapid Prototyping.
ZAŚ jeszcze jedna decydująca wartość, technologia owa wytwarza nam szansa gwałtownego i gibkiego zweryfikowania błędów konstrukcyjno- wizualnych, co samemu honoruje zaoszczędzić czas a pieniądze. Po pozostałe, technologia kasuje nieodzowność skorzystania urządzeń CNC, które w efekcie są cenne natomiast czasochłonne.
Po trzecie, Rapid Prototyping pozwoli nam szybkie prototypy przesunięcie petentowi skończonego wizualnie a tworu. Ponieważ jego skorzystanie to niepowszechna ekonomizacja czasu natomiast pieniądzach.