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We investigate, from a topological point of view, the classes of spectral semistar operations and of eab semistar operations, following methods recently introduced in [11,13] . We show that, in both cases, the subspaces of finite type operations are spectral spaces in the sense of Hochster and, moreover, that there is a distinguished class of overrings strictly connected to each of the two types of collections of semistar operations. We also prove that the space of stable semistar operations is homeomorphic to the space of Gabriel–Popescu localizing systems, endowed with a Zariski-like topology, extending to the topological level a result established in [14] . As a side effect, we obtain that the space of localizing systems of finite type is also a spectral space. Finally, we show that the Zariski topology on the set of semistar operations is the same as the b -topology defined recently by B. Olberding [37,38] . [ABSTRACT FROM AUTHOR]

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 DM 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 ★. [Copyright &y& Elsevier]

Abstract: Let be a ring homomorphism and let be an ideal of . In this paper, we study the amalgamation of with along with respect to (denoted by ), a construction that provides a general frame for studying the amalgamated duplication of a ring along an ideal, introduced and studied by D’Anna and Fontana in 2007, and other classical constructions (such as the , the and the constructions). In particular, we completely describe the prime spectrum of the amalgamated duplication and we give bounds for its Krull dimension. [Copyright &y& Elsevier]