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 $.

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