a b s t r a c tLet f : A → B be a ring homomorphism and let J be an ideal of B. In this paper, we study the amalgamation of A with B along J with respect to f (denoted by A f J), 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 A + XB[X ], the A + XB[[X ]] and the D + M constructions). In particular, we completely describe the prime spectrum of the amalgamated duplication and we give bounds for its Krull dimension.

We consider properties and applications of a compact, Hausdorff topology called the "ultrafilter topology" defined on an arbitrary spectral space and we observe that this topology coincides with the constructible topology. If K is a field and A a subring of K, we show that the space Zar(K|A) of all valuation domains, having K as the quotient field and containing A, (endowed with the Zariski topology) is a spectral space by giving in this general setting the explicit construction of a ring whose Zariski spectrum is homeomorphic to Zar(K|A). We extend results regarding spectral topologies on the spaces of all valuation domains and apply the theory developed to study representations of integrally closed domains as intersections of valuation overrings. As a very particular case, we prove that two collections of valuation domains of K with the same ultrafilter closure represent, as an intersection, the same integrally closed domain. © 2013 American Mathematical Society

Let f W A ! B be a ring homomorphism and J an ideal of B. In this paper, we initiate a systematic study of a new ring construction called the "amalgamation of A with B along J with respect to f ". This construction finds its roots in a paper by J. L. Dorroh appeared in 1932 and 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 A C XBOEX and A C XBOEOEX constructions, the CPI-extensions of Boisen and Sheldon, the D C M constructions and the Nagata's idealization.

Abstract. We consider properties and applications of a new topology, called the Zariski topology, on the space SStar(A) of all the semistar operations on an integral domain A. We prove that the set of all overrings of A, endowed with the classical Zariski topology, is homeomorphic to a subspace of SStar(A). The topology on SStar(A) provides a general theory, through which we see several algebraic properties of semistar operation as very particular cases of our construction. Moreover, we show that the subspace SStar f (A) of all the semistar operations of finite type on A is a spectral space.

Let X be the prime spectrum of a ring. In [Fo-Lo] the authors define a topology on X by using ultrafilters and they show that this topology is precisely the constructible topology. In this paper we generalize the construction given in [Fo-Lo] and, starting from a set X and a collection of subsets F of X, we define by using ultrafilters a topology on X in which F is a collection of clopen sets. We use this construction for giving a new characterization of spectral spaces and several examples of spectral spaces.