# Krull ring

Krull domain

A commutative integral domain $A$ with the following property: There exists a family $( v _ {i} ) _ {i \in I }$ of discrete valuations on the field of fractions (cf. Fractions, ring of) $K$ of $A$ such that: a) for any $x \in K \setminus \{ 0 \}$ and all $i$, except possibly a finite number of them, $v _ {i} ( x) = 0$; and b) for any $x \in K \setminus \{ 0 \}$, $x \in A$ if and only if $v _ {i} ( x) \geq 0$ for all $i \in I$. Under these conditions, $v _ {i}$ is said to be an essential valuation.

Krull rings were first studied by W. Krull [1], who called them rings of finite discrete principal order. They are the most natural class of rings in which there is a divisor theory (see also Divisorial ideal; Divisor class group). The ordered group of divisors of a Krull ring is canonically isomorphic to the ordered group $\mathbf Z ^ {(} I)$. The essential valuations of a Krull ring may be identified with the set of prime ideals of height 1. A Krull ring is completely integrally closed. Any integrally-closed Noetherian integral domain, in particular a Dedekind ring, is a Krull ring. The ring $k [ X _ {1} \dots X _ {n} , . . . ]$ of polynomials in infinitely many variables is an example of a Krull ring which is not Noetherian. In general, any factorial ring is a Krull ring. A Krull ring is a factorial ring if and only if every prime ideal of height 1 is principal.

The class of Krull rings is closed under localization, passage to the ring of polynomials or formal power series, and also under integral closure in a finite extension of the field of fractions $K$.

#### References

 [1] W. Krull, "Allgemeine Bewertungstheorie" J. Reine Angew. Math. , 167 (1931) pp. 160–196 [2] O. Zariski, P. Samuel, "Commutative algebra" , 2 , Springer (1975) [3] N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)