词条 | Krull ring |
释义 |
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 domains, which are exactly the Krull domains of dimension at most 1. In this article, a ring is commutative and has unity. Formal definitionLet 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
PropertiesA 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
The divisor class group of a Krull ringA (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. References1. ^{{Cite web|url = http://eom.springer.de/k/k055930.htm|title = Krull ring - Encyclopedia of Mathematics|website = eom.springer.de|access-date = 2016-04-14}} 2. ^Bourbaki, 7.1, no 10, Proposition 16. 3. ^{{Cite book|url = https://books.google.com/books?id=APPtnn84FMIC|title = Integral Closure of Ideals, Rings, and Modules|last = Huneke|first = Craig|last2 = Swanson|first2 = Irena|author2-link= Irena Swanson |date = 2006-10-12|publisher = Cambridge University Press|isbn = 9780521688604|language = en}}
2 : Ring theory|Commutative algebra |
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