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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 introduce a new general construction, denoted by R ⋈ E, called the amalgamated duplication of a ring R along an R-module E, that we assume to be an ideal in some overring of R. (Note that, when E2 = 0, R ⋈ E coincides with the Nagata's idealization R ⋉ E.). After discussing the main properties of the amalgamated duplication R ⋈ E in relation with pullback-type constructions, we restrict our investigation to the study of R ⋈ E when E is an ideal of R. Special attention is devoted to the ideal-theoretic properties of R ⋈ E and to the topological structure of its prime spectrum. [ABSTRACT FROM AUTHOR]

Call a domain R an sQQR-domain if each simple overring of R, i.e., each ring of the form R[u] with u in the quotient field of R, is an intersection of localizations of R. We characterize Prüfer domains as integrally closed sQQR-domains. In the presence of certain finiteness conditions, we show that the sQQR-property is very strong; for instance, a Mori sQQR-domain must be a Dedekind domain. We also show how to construct sQQR-domains which have (non-simple) overrings which are not intersections of localizations. [ABSTRACT FROM AUTHOR]