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''Krull domain''
 
''Krull domain''
  
A commutative [[Integral domain|integral domain]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055930/k0559301.png" /> with the following property: There exists a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055930/k0559302.png" /> of discrete valuations on the field of fractions (cf. [[Fractions, ring of|Fractions, ring of]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055930/k0559303.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055930/k0559304.png" /> such that: a) for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055930/k0559305.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055930/k0559306.png" />, except possibly a finite number of them, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055930/k0559307.png" />; and b) for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055930/k0559308.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055930/k0559309.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055930/k05593010.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055930/k05593011.png" />. Under these conditions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055930/k05593012.png" /> is said to be an essential valuation.
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A commutative [[Integral domain|integral domain]] $  A $
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with the following property: There exists a family $  ( v _ {i} ) _ {i \in I }  $
 +
of discrete valuations on the field of fractions (cf. [[Fractions, ring of|Fractions, ring of]]) $  K $
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of $  A $
 +
such that: a) for any $  x \in K \setminus  \{ 0 \} $
 +
and all $  i $,  
 +
except possibly a finite number of them, $  v _ {i} ( x) = 0 $;  
 +
and b) for any $  x \in K \setminus  \{ 0 \} $,  
 +
$  x \in A $
 +
if and only if $  v _ {i} ( x) \geq  0 $
 +
for all $  i \in I $.  
 +
Under these conditions, $  v _ {i} $
 +
is said to be an essential valuation.
  
Krull rings were first studied by W. Krull [[#References|[1]]], who called them rings of finite discrete principal order. They are the most natural class of rings in which there is a divisor theory (see also [[Divisorial ideal|Divisorial ideal]]; [[Divisor class group|Divisor class group]]). The ordered group of divisors of a Krull ring is canonically isomorphic to the ordered group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055930/k05593013.png" />. The essential valuations of a Krull ring may be identified with the set of prime ideals of height 1. A Krull ring is completely integrally closed. Any integrally-closed Noetherian integral domain, in particular a [[Dedekind ring|Dedekind ring]], is a Krull ring. The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055930/k05593014.png" /> of polynomials in infinitely many variables is an example of a Krull ring which is not Noetherian. In general, any [[Factorial ring|factorial ring]] is a Krull ring. A Krull ring is a factorial ring if and only if every prime ideal of height 1 is principal.
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Krull rings were first studied by W. Krull [[#References|[1]]], who called them rings of finite discrete principal order. They are the most natural class of rings in which there is a divisor theory (see also [[Divisorial ideal|Divisorial ideal]]; [[Divisor class group|Divisor class group]]). The ordered group of divisors of a Krull ring is canonically isomorphic to the ordered group $  \mathbf Z  ^ {(} I) $.  
 +
The essential valuations of a Krull ring may be identified with the set of prime ideals of height 1. A Krull ring is completely integrally closed. Any integrally-closed Noetherian integral domain, in particular a [[Dedekind ring|Dedekind ring]], is a Krull ring. The ring $  k [ X _ {1} \dots X _ {n} , . . . ] $
 +
of polynomials in infinitely many variables is an example of a Krull ring which is not Noetherian. In general, any [[Factorial ring|factorial ring]] is a Krull ring. A Krull ring is a factorial ring if and only if every prime ideal of height 1 is principal.
  
The class of Krull rings is closed under localization, passage to the ring of polynomials or formal power series, and also under integral closure in a finite extension of the field of fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055930/k05593015.png" />.
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The class of Krull rings is closed under localization, passage to the ring of polynomials or formal power series, and also under integral closure in a finite extension of the field of fractions $  K $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Krull,  "Allgemeine Bewertungstheorie"  ''J. Reine Angew. Math.'' , '''167'''  (1931)  pp. 160–196</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''2''' , Springer  (1975)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Krull,  "Allgemeine Bewertungstheorie"  ''J. Reine Angew. Math.'' , '''167'''  (1931)  pp. 160–196</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''2''' , Springer  (1975)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.M. Fossum,  "The divisor class group of a Krull domain" , Springer  (1973)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.M. Fossum,  "The divisor class group of a Krull domain" , Springer  (1973)</TD></TR></table>

Latest revision as of 22:15, 5 June 2020


Krull domain

A commutative integral domain $ A $ with the following property: There exists a family $ ( v _ {i} ) _ {i \in I } $ of discrete valuations on the field of fractions (cf. Fractions, ring of) $ K $ of $ A $ such that: a) for any $ x \in K \setminus \{ 0 \} $ and all $ i $, except possibly a finite number of them, $ v _ {i} ( x) = 0 $; and b) for any $ x \in K \setminus \{ 0 \} $, $ x \in A $ if and only if $ v _ {i} ( x) \geq 0 $ for all $ i \in I $. Under these conditions, $ v _ {i} $ is said to be an essential valuation.

Krull rings were first studied by W. Krull [1], who called them rings of finite discrete principal order. They are the most natural class of rings in which there is a divisor theory (see also Divisorial ideal; Divisor class group). The ordered group of divisors of a Krull ring is canonically isomorphic to the ordered group $ \mathbf Z ^ {(} I) $. The essential valuations of a Krull ring may be identified with the set of prime ideals of height 1. A Krull ring is completely integrally closed. Any integrally-closed Noetherian integral domain, in particular a Dedekind ring, is a Krull ring. The ring $ k [ X _ {1} \dots X _ {n} , . . . ] $ of polynomials in infinitely many variables is an example of a Krull ring which is not Noetherian. In general, any factorial ring is a Krull ring. A Krull ring is a factorial ring if and only if every prime ideal of height 1 is principal.

The class of Krull rings is closed under localization, passage to the ring of polynomials or formal power series, and also under integral closure in a finite extension of the field of fractions $ K $.

References

[1] W. Krull, "Allgemeine Bewertungstheorie" J. Reine Angew. Math. , 167 (1931) pp. 160–196
[2] O. Zariski, P. Samuel, "Commutative algebra" , 2 , Springer (1975)
[3] N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)

Comments

References

[a1] R.M. Fossum, "The divisor class group of a Krull domain" , Springer (1973)
How to Cite This Entry:
Krull ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Krull_ring&oldid=47530
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article