Semistar Dedekind domains

Abstract

Let D be an integral domain and ★ a semistar operation on D. As a generalization of the notion of Noetherian domains to the semistar setting, we say that D is a ★-Noetherian domain if it has the ascending chain condition on the set of its quasi-★-ideals. On the other hand, as an extension the notion of Prüfer domain (and of Prüfer v-multiplication domain), we say that D is a Prüfer ★-multiplication domain (P★MD, for short) if D M is a valuation domain, for each quasi-★ f -maximal ideal M of D. Finally, recalling that a Dedekind domain is a Noetherian Prüfer domain, we define a ★-Dedekind domain to be an integral domain which is ★-Noetherian and a P★MD. For the identity semistar operation d, this definition coincides with that of the usual Dedekind domains and when the semistar operation is the v-operation, this notion gives rise to Krull domains. Moreover, Mori domains not strongly Mori are ★-Dedekind for a suitable spectral semistar operation. Examples show that ★-Dedekind domains are not necessarily integrally closed nor one-dimensional, although they mimic various aspects, varying according to the choice of ★, of the “classical” Dedekind domains. In any case, a ★-Dedekind domain is an integral domain D having a Krull overring T (canonically associated to D and ★) such that the semistar operation ★ is essentially “univocally associated” to the v-operation on T. In the present paper, after a preliminary study of ★-Noetherian domains, we investigate the ★-Dedekind domains. We extend to the ★-Dedekind domains the main classical results and several characterizations proven for Dedekind domains. In particular, we obtain a characterization of a ★-Dedekind domain by a property of decomposition of any semistar ideal into a “semistar product” of prime ideals. Moreover, we show that an integral domain D is a ★-Dedekind domain if and only if the Nagata semistar domain Na( D,★) is a Dedekind domain. Several applications of the general results are given for special cases of the semistar operation ★.