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 in 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 definition

Let A be an integral domain and let P be the set of all prime ideals of A of height one, that is, the set of all prime ideals properly containing no nonzero prime ideal. Then A is a Krull ring if It is also possible to characterize Krull rings by mean of valuations only: An integral domain A is a Krull ring if there exists a family of discrete valuations on the field of fractions K of A such that: The valuations v_i are called essential valuations of A. The link between the two definitions is as follows: for every, one can associate a unique normalized valuation of K whose valuation ring is. Then the set satisfies the conditions of the equivalent definition. Conversely, if the set is as above, and the v_i have been normalized, then \mathcal V' may be bigger than \mathcal V, but it must contain \mathcal V. In other words, \mathcal V is the minimal set of normalized valuations satisfying the equivalent definition.

Properties

With the notations above, let denote the normalized valuation corresponding to the valuation ring, U denote the set of units of A, and K its quotient field.

Examples

The divisor class group of a Krull ring

Assume that A is a Krull domain and K is its quotient field. A prime divisor of A is a height 1 prime ideal of A. The set of prime divisors of A will be denoted P(A) in the sequel. A (Weil) divisor of A is a formal integral linear combination of prime divisors. They form an Abelian group, noted D(A). A divisor of the form, for some non-zero x in K, is called a principal divisor. The principal divisors of A form a subgroup of the group of divisors (it has been shown above that this group is isomorphic to A^\times /U, where U is the group of unities of A). The quotient of the group of divisors by the subgroup of principal divisors is called the divisor class group of A; it is usually denoted C(A). Assume that B is a Krull domain containing A. As usual, we say that a prime ideal \mathfrak P of B lies above a prime ideal \mathfrak p of A if ; this is abbreviated in. Denote the ramification index of over by, and by P(B) the set of prime divisors of B. Define the application by (the above sum is finite since every is contained in at most finitely many elements of P(B)). Let extend the application j by linearity to a linear application. One can now ask in what cases j induces a morphism. This leads to several results. For example, the following generalizes a theorem of Gauss: ''The application is bijective. In particular, if A is a unique factorization domain, then so is A[X].'' The divisor class group of a Krull rings are also used to set up powerful descent methods, and in particular the Galoisian descent.

Cartier divisor

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.