JOURNAL OF ALGEBRA 14, 333-340 (1970) Non Finiteness in Finite Dimensional Krull Domains PAUL EAKIN AND WILLIAM HEINZER Department of Mathematics, Louisiana State University, Baton Rouge, Louti 70803 Communicated by D. Rees Received February 24, 1969 1. INTRODUCTION Finitenessis an intrinsic aspect of the study of Krull domains. They are defined by a family of Noetherian valuation rings with a finitenesscondition and they satisfy the ascendingchain condition (a.c.c.) on integral divisorial ideals[8, p. 51.In fact, if adis a divisorial ideal in a Krull domain D, then Oc has the form D : [D : (x, y)] for somex, y in fl [ 1, p. 831.If K is a field and {X,},“r a collection of indeterminatesover K then K[{X,}EJ is a Krull ring. Thus, Krull rings need be neither finite-dimensional nor Noetherian. This example and a number of simple modifications, e.g., K[{X,X,}], while being infinite-dimensional and Krull, retain the property that each minimal prime has finitely generated primary ideals. Examples of finite-dimensional non- Noetherian Krull rings are not so easy to construct. The best known example is due to Nagata [6, p. 2071.This is his example of a three-dimensionallocal domain whose derived normal ring is not Noetherian. Since the derived normal ring of a Noetherian domain is Krull [6, p. 1181, this provides an example. Were this the only way of obtaining finite-dimensional Krull rings it would follow that every two-dimensional Krull ring is Noetherian; for the derived normal ring of a two-dimensional Noetherian domain is again Noetherianl [6, p. 1201. Ana lysis of Nagata’sexample showsthat it also has the property that each minimal prime has finitely generated primary ideals. Our purposehere is to discuss two methods for constructing finite-dimen- sional, non-Noetherian Krull rings. In doing so, we show the existenceof (1) a three-dimensional Krull ring with a minimal prime P such that no P primary ideal is finitely generated and (2) a two-dimensional, quasilocal, non-Noetherian Krull domain.* 1 In fact, any Krull ring which liea between a two-dimensional Noetherian domain and its quotient field is Noetherian [4]. s Since the writing of this paper we have lcamed of a paper by Nagarajan [Groups acting on Noetherian rings, Niew. Arch. VOOT Wircudr 16 (1968) 25-291 where another 333