Semistar Dedekind domains

Let D be an integral domain and star, filled a semistar operation on D. As a generalization of the notion of Noetherian domains to the semistar setting, we say that D is a star, filled-Noetherian domain if it has the ascending chain condition on the set of its quasi-star, filled-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 star, filled-multiplication domain (Pstar, filledMD, for short) if DM is a valuation domain, for each quasi-star, filledf-maximal ideal M of D. Finally, recalling that a Dedekind domain is a Noetherian Prüfer domain, we define a star, filled-Dedekind domain to be an integral domain which is star, filled-Noetherian and a Pstar, filledMD. 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 star, filled-Dedekind for a suitable spectral semistar operation. Examples show that star, filled-Dedekind domains are not necessarily integrally closed nor one-dimensional, although they mimic various aspects, varying according to the choice of star, filled, of the “classical” Dedekind domains. In any case, a star, filled-Dedekind domain is an integral domain D having a Krull overring T (canonically associated to D and star, filled) such that the semistar operation star, filled is essentially “univocally associated” to the v-operation on T. In the present paper, after a preliminary study of star, filled-Noetherian domains, we investigate the star, filled-Dedekind domains. We extend to the star, filled-Dedekind domains the main classical results and several characterizations proven for Dedekind domains. In particular, we obtain a characterization of a star, filled-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 star, filled-Dedekind domain if and only if the Nagata semistar domain Na(D,star, filled) is a Dedekind domain. Several applications of the general results are given for special cases of the semistar operation star, filled.