Generalized Dedekind domains

Abstract

A Dedekind domain is a Noetherian Priifer domain. By weakening some of the finiteness properties of Dedekind domains, one often obtains classes of Priifer domains having good multiplicative properties which are interesting in their own right. For example, almost Dedekind domains were defined by R. Gilmer by relaxing the condition that each proper ideal has finitely many minimal primes [21] and the study of these domains led to the introduction of the T^t-property and of some related conditions [18, 24, 25, 35, 36, 39]. The class of generalized Dedekind domains is complementary to that of almost Dedekind domains, in the sense that a domain which is at the same time almost Dedekind and generalized Dedekind is in fact a Dedekind domain. Generalized Dedekind domains were defined by N. Popescu by means of localizing systems of ideals [39]; but in this context they can be defined more naturally as Priifer domains with the properties that no nonzero prime ideal is idempotent and each proper ideal has finitely many minimal primes [39, 40]. Thus, roughly speaking, generalized Dedekind domains are "Dedekind domains of dimension greater than one". These domains behave nicely with respect to several important ringtheoretic and ideal-theoretic conditions introduced in the last few decades, since the appearance of the first version of Gilmer's Multiplicative Ideal Theory [23]. Their study gives the opportunity to bring into evidence the connections among all these properties and to provide a unified point of view of various apparently unrelated results. Generalized Dedekind domains are the subject of [11, Chapter V]. In the present survey paper I report the latest developments and give direct proofs of the main results. However, I do not consider module-theoretic properties; for this aspect of the theory the reader is referred to [14]. Recently the class of generalized Dedekind domains has been enlarged, within Priifer u-multiplication domains, to the class of the so called generalized Krull domains [5]. Multiplicative properties of generalized Krull domains are studied in [5, 6, 7, 8, 18, 19].