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In commutative algebra, a '''Krull ring''' or '''Krull domain''' is a [[commutative ring]] with a well behaved theory of prime factorization. They were introduced by {{harvs|txt|authorlink=Wolfgang Krull|first=Wolfgang |last=Krull|year=1931}}. They are a higher-dimensional generalization of [[Dedekind domain]]s, which are exactly the Krull domains of dimension at most 1. | |||
In this article, a ring is commutative and has unity. | |||
==Formal definition== | |||
Let <math> A </math> be an [[integral domain]] and let <math> P </math> be the set of all [[prime ideal]]s of <math> A </math> of [[height (ring theory)|height]] one, that is, the set of all prime ideals properly containing no nonzero prime ideal. Then <math> A </math> is a '''Krull ring''' if and only if | |||
# <math> A_{\mathfrak{p}} </math> is a [[discrete valuation ring]] for all <math> \mathfrak{p} \in P </math>, | |||
#<math> A </math> is the intersection of these discrete valuation rings (considered as subrings of the quotient field of <math> A </math>). | |||
#Any nonzero element of <math> A </math> is contained in only a finite number of height 1 prime ideals. | |||
== Properties == | |||
A Krull domain is a [[unique factorization domain]] if and only if every prime ideal of height one is principal.<ref>http://eom.springer.de/k/k055930.htm</ref> | |||
Let ''A'' be a [[Zariski ring]] (e.g., a local noetherian ring). If the completion <math>\widehat{A}</math> is a Krull domain, then ''A'' is a Krull domain.<ref>Bourbaki, 7.1, no 10, Proposition 16.</ref> | |||
==Examples== | |||
# Every [[integrally closed domain|integrally closed]] [[noetherian ring|noetherian]] [[integral domain|domain]] is a Krull ring. In particular, [[Dedekind domain]]s are Krull rings. Conversely Krull rings are integrally closed, so a Noetherian domain is Krull if and only if it is integrally closed. | |||
# If <math> A </math> is a Krull ring then so is the [[polynomial ring]] <math> A[x] </math> and the [[Power series ring|formal power series ring]] <math> A[[x]] </math>. | |||
# The polynomial ring <math>R[x_1, x_2, x_3, \ldots]</math> in infinitely many variables over a [[unique factorization domain]] <math> R </math> is a Krull ring which is not noetherian. In general, any unique factorization domain is a Krull ring. | |||
# Let <math> A </math> be a [[Noetherian ring|Noetherian]] [[integral domain|domain]] with [[quotient field]] <math> K </math>, and <math> L </math> be a [[field extension|finite algebraic extension]] of <math> K </math>. Then the [[integral closure]] of <math> A </math> in <math> L </math> is a Krull ring ([[Mori–Nagata theorem]]).<ref>http://books.google.com/books?id=APPtnn84FMIC&lpg=PA83&ots=2L9MiWbIYZ&dq=krull%20akizuki&pg=PA85#v=onepage&q=krull%20akizuki&f=false</ref> | |||
==The divisor class group of a Krull ring== | |||
A (Weil) divisor of a Krull ring ''A'' is a formal integral linear combination of the height 1 prime ideals, and these form a group ''D''(''A''). A divisor of the form div(''x'') for some non-zero ''x'' in ''A'' is called a principal divisor, and the principal divisors form a subgroup of the group of divisors. The quotient of the group of divisors by the subgroup of principal divisors is called the '''divisor class group''' of ''A''. | |||
A [[Cartier divisor]] of a Krull ring is a locally principal (Weil) divisor. The Cartier divisors form a subgroup of the group of divisors containing the principal divisors. The quotient of the Cartier divisors by the principal divisors is a subgroup of the divisor class group, isomorphic to the [[Picard group]] of invertible sheaves on Spec(''A''). | |||
Example: in the ring ''k''[''x'',''y'',''z'']/(''xy''–''z''<sup>2</sup>] the divisor class group has order 2, generated by the divisor ''y''=''z'', but the Picard subgroup is the trivial group. | |||
==References== | |||
{{reflist}} | |||
*{{cite book |author=N. Bourbaki |title=Commutative algebra}} | |||
* {{springer|title=Krull ring|id=p/k055930}} | |||
*{{Citation | last1=Krull | first1=Wolfgang | author1-link=Wolfgang Krull | title=Allgemeine Bewertungstheorie | url=http://gdz.sub.uni-goettingen.de/en/dms/load/img/?IDDOC=260807 | year=1931 | journal= J. Reine Angew. Math. | volume=167 | pages=160–196}} | |||
* Hideyuki Matsumura, ''Commutative Algebra''. Second Edition. Mathematics Lecture Note Series, 56. Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. xv+313 pp. ISBN 0-8053-7026-9 | |||
* Hideyuki Matsumura, ''Commutative Ring Theory''. Translated from the Japanese by M. Reid. Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press, Cambridge, 1986. xiv+320 pp. ISBN 0-521-25916-9 | |||
*{{Citation | last1=Samuel | first1=Pierre | author1-link=Pierre Samuel | editor1-last=Murthy | editor1-first=M. Pavman | title=Lectures on unique factorization domains | url=http://www.math.tifr.res.in/~publ/ln/ | publisher=Tata Institute of Fundamental Research | location=Bombay | series=Tata Institute of Fundamental Research Lectures on Mathematics | id={{MR|0214579}} | year=1964 | volume=30}} | |||
[[Category:Ring theory]] | |||
[[Category:Commutative algebra]] | |||
Latest revision as of 00:16, 14 January 2014
In commutative algebra, a Krull ring or Krull domain is a commutative ring with a well behaved theory of prime factorization. They were introduced by Template:Harvs. They are a higher-dimensional generalization of Dedekind domains, which are exactly the Krull domains of dimension at most 1.
In this article, a ring is commutative and has unity.
Formal definition
Let be an integral domain and let be the set of all prime ideals of of height one, that is, the set of all prime ideals properly containing no nonzero prime ideal. Then is a Krull ring if and only if
- is a discrete valuation ring for all ,
- is the intersection of these discrete valuation rings (considered as subrings of the quotient field of ).
- Any nonzero element of is contained in only a finite number of height 1 prime ideals.
Properties
A Krull domain is a unique factorization domain if and only if every prime ideal of height one is principal.[1]
Let A be a Zariski ring (e.g., a local noetherian ring). If the completion is a Krull domain, then A is a Krull domain.[2]
Examples
- Every integrally closed noetherian domain is a Krull ring. In particular, Dedekind domains are Krull rings. Conversely Krull rings are integrally closed, so a Noetherian domain is Krull if and only if it is integrally closed.
- If is a Krull ring then so is the polynomial ring and the formal power series ring .
- The polynomial ring in infinitely many variables over a unique factorization domain is a Krull ring which is not noetherian. In general, any unique factorization domain is a Krull ring.
- Let be a Noetherian domain with quotient field , and be a finite algebraic extension of . Then the integral closure of in is a Krull ring (Mori–Nagata theorem).[3]
The divisor class group of a Krull ring
A (Weil) divisor of a Krull ring A is a formal integral linear combination of the height 1 prime ideals, and these form a group D(A). A divisor of the form div(x) for some non-zero x in A is called a principal divisor, and the principal divisors form a subgroup of the group of divisors. The quotient of the group of divisors by the subgroup of principal divisors is called the divisor class group of A.
A Cartier divisor of a Krull ring is a locally principal (Weil) divisor. The Cartier divisors form a subgroup of the group of divisors containing the principal divisors. The quotient of the Cartier divisors by the principal divisors is a subgroup of the divisor class group, isomorphic to the Picard group of invertible sheaves on Spec(A).
Example: in the ring k[x,y,z]/(xy–z2] the divisor class group has order 2, generated by the divisor y=z, but the Picard subgroup is the trivial group.
References
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To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Hideyuki Matsumura, Commutative Algebra. Second Edition. Mathematics Lecture Note Series, 56. Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. xv+313 pp. ISBN 0-8053-7026-9
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