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. 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 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. It is also possible to characterize Krull rings by mean of valuations only: An integral domain is a Krull ring if there exists a family of discrete valuations on the field of fractions of such that: for any and all , except possibly a finite number of them, ; for any , belongs to if and only if for all . The valuations are called essential valuations of . The link between the two definitions is as follows: for every , one can associate a unique normalized valuation of whose valuation ring is . Then the set satisfies the conditions of the equivalent definition. Conversely, if the set is as above, and the have been normalized, then may be bigger than , but it must contain . In other words, is the minimal set of normalized valuations satisfying the equivalent definition. There are other ways to introduce and define Krull rings. The theory of Krull rings can be exposed in synergy with the theory of divisorial ideals. One of the best references on the subject is Lecture on Unique Factorization Domains by P. Samuel. With the notations above, let denote the normalized valuation corresponding to the valuation ring , denote the set of units of , and its quotient field. An element belongs to if, and only if, for every . Indeed, in this case, for every , hence ; by the intersection property, .