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Let be an integral domain with quotient field . Call an overring of a subring of containing as a subring. A family of overrings of is called a defining family of , if . Call an overring a sublocalization of , if has a defining family consisting of rings of fractions of . Sublocalizations and their intersections exhibit interesting examples of semistar or star operations [D. D. Anderson, Star operations induced by overrings, Comm. Algebra 16 (1988) 2535-2553]. We show as a consequence of our work that domains that are locally finite intersections of Prüfer -multiplication (respectively, Mori) sublocalizations turn out to be Prüfer -multiplication domains (PvMDs) (respectively, Mori); in particular, for the Mori domain case, we reobtain a special case of Théorème 1 of [J. Querré, Intersections d'anneaux intègers, J. Algebra 43 (1976) 55-60] and Proposition 3.2 of [N. Dessagnes, Intersections d'anneaux de Mori - exemples, Port. Math. 44 (1987) 379-392]. We also show that, more than the finite character of the defining family, it is the finite character of the star operation induced by the defining family that causes the interesting results. As a particular case of this theory, we provide a purely algebraic approach for characterizing P MDs as a subclass of the class of essential domains (see also Theorem 2.4 of [C. A. Finocchiaro and F. Tartarone, On a topological characterization of Prüfer -multiplication domains among essential domains, preprint (2014), arXiv:1410.4037]). [ABSTRACT FROM AUTHOR]

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]

Abstract: We provide a complete solution to the problem of extending arbitrary semistar operations of an integral domain D to semistar operations of the polynomial ring . As an application, we show that one can reobtain the main results of the papers in Chang and Fontana (2007, 2011) [1,2] concerning the problem in the special cases of stable semistar operations of finite type or semistar operations defined by families of overrings. Finally, we investigate the behavior of the polynomial extensions of the most important and classical operations such as , , , and operations. [Copyright &y& Elsevier]

Call a semistar operation ⊛ on the polynomial domain D[X] an extension (respectively, a strict extension) of a semistar operation ☆ defined on an integral domain D, with quotient field K, if E☆ = (E[X])⊛ ∩ K (respectively, E☆[X] = (E[X])⊛) for all nonzero D-submodules E of K. In this article, we study the general properties of the above defined extensions and link our work with earlier efforts, centered on the stable semistar operation case, at defining semistar operations on D[X] that are 'canonical' extensions (or, 'canonical' strict extensions) of semistar operations on D. [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]

In 1994, Matsuda and Okabe introduced the notion of semistar operation, extending the "classical" concept of star operation. In this paper, we introduce and study the notions of semistar linkedness and semistar flatness which are natural generalizations, to the semistar setting, of their corresponding "classical" concepts. As an application, among other results, we obtain a semistar version of Davis' and Richman's overring-theoretical theorems of characterization of Prüfer domains for Prüfer semistar multiplication domains. [ABSTRACT FROM AUTHOR]

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]