Krull ring
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 formula_1 be an integral domain and let formula_2 be the set of all prime ideals of formula_1 of height one, that is, the set of all prime ideals properly containing no nonzero prime ideal. Then formula_1  is a Krull ring if It is also possible to characterize Krull rings by mean of valuations only: An integral domain formula_10 is a Krull ring if there exists a family formula_11 of discrete valuations on the field of fractions formula_12 of formula_10 such that: The valuations formula_22 are called essential valuations of formula_10. The link between the two definitions is as follows: for every formula_24, one can associate a unique normalized valuation formula_25 of formula_12 whose valuation ring is formula_27. Then the set formula_28 satisfies the conditions of the equivalent definition. Conversely, if the set formula_29 is as above , and the formula_22 have been normalized, then formula_31 may be bigger than formula_32, but it "must" contain formula_32. In other words, formula_34 is the minimal set of normalized valuations satisfying the equivalent definition. There are other ways to introduce and define Krull rings. In fact, the theory of Krull ring is best exposed in synergy with the theory of "divisorial ideals". One of the best reference on the subject is "Lecture on Unique Factorization Domains" by P. Samuel. Properties. With the notations above, let formula_25 denote the normalized valuation corresponding to the valuation ring formula_27, formula_37 denote the set of  units of formula_10, and formula_12 its quotient field. The divisor class group of a Krull ring. Assume that formula_10 is a Krull domain and formula_12 is its quotient field. A prime divisor of formula_10 is a height 1 prime ideal of formula_10. The set of prime divisors of formula_10 will be denoted formula_134 in the sequel. A (Weil) divisor of formula_10 is a formal integral linear combination of prime divisors. They form an Abelian group, noted formula_136. A divisor of the form formula_137, for some non-zero formula_48 in formula_12, is called a principal divisor. The principal divisors of formula_10  form a subgroup of the group of divisors (it has been shown above that this group is isomorphic to formula_141, where formula_37 is the group of unities of formula_10). The quotient of the group of divisors by the subgroup of principal divisors is called the divisor class group of formula_10; it is usually denoted formula_145. Assume that formula_107 is a Krull domain containing formula_10. As usual, we say that a prime ideal formula_148 of formula_107 "lies above" a prime ideal formula_64 of formula_10 if formula_152; this is abbreviated in formula_153. Denote the ramification index of formula_154 over formula_25 by formula_156 , and by formula_157 the set of prime divisors of formula_107. Define the application formula_159 by (the above sum is finite since every formula_161 is contained in at most finitely many elements of formula_157). Let extend the application formula_163 by linearity to a linear application formula_164. One can now ask in what cases formula_163 induces a morphism formula_166. This leads to several results. For example, the following generalizes a theorem of Gauss: "The application formula_167 is bijective. In particular, if formula_10 is a unique factorization domain, then so is formula_169." The divisor class group of a Krull rings are also used  to setup 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"–"z") the divisor class group has order 2, generated by the divisor "y"="z", but the Picard subgroup is the trivial group.