On Krull overrings of a Noetherian domain

Abstract

An integral domain D is called a Krull ring if D = nfa VO where { V, } is a set of rank one discrete valuation rings with the property that for each nonzero x in D, x Va = VO, for all but a finite number of the Va. If D is a Krull ring and { Pa } is the set of primes of height one (minimal primes) of D, then each localization DPa is a rank one discrete valuation ring and D = nfaDp is an irredundant representation of D. The set { Dp. } is called the set of essential valuation rings for D. We use the notation E(D) to denote the set of essential valuation rings of a Krull ring D. Important examples of Krull rings are noetherian integrally closed domains and unique factorization domains. The one dimensional Krull rings are precisely the Dedekind domains. Hence one dimensional Krull rings are noetherian. An example of a three dimensional nonnoetherian Krull ring is provided by Nagata's example [6, p. 207] of a three dimensional noetherian domain whose derived normal ring is not noetherian; for it is known that the derived normal ring of a noetherian domain is Krull [6, p. 118]. Our purpose here is to show that a rather extensive class of two dimensional Krull rings are noetherian.' If R is an integral domain with quotient field K we call a domain D such that RCDCK an overring of R. We show that if R is a two dimensional noetherian domain, then each Krull overring of R is noetherian. Since the derived normal ring of a two dimensional noetherian domain is again noetherian [6, p. 120], our problem is reduced to showing that a Krull overring of a two dimensional noetherian Krull ring is noetherian. Important arguments used in the proof were obtained from [8].