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 Wolfgang Krull (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.
- 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.
- 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).
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.
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- Hazewinkel, Michiel, ed. (2001), "Krull ring", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Krull, Wolfgang (1931), "Allgemeine Bewertungstheorie", J. Reine Angew. Math., 167: 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
- Samuel, Pierre (1964), Murthy, M. Pavman, ed., Lectures on unique factorization domains, Tata Institute of Fundamental Research Lectures on Mathematics, 30, Bombay: Tata Institute of Fundamental Research, MR 0214579