Carmelo Antonio Finocchiaro

Some topological considerations on semistar operations

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.

Spectral Spaces and Ultrafilters

In memory of my father. Let X be the prime spectrum of a ring. In Fontana and Loper [5] the authors define a topology on X by using ultrafilters and show that this topology is precisely the constructible topology. In this paper we generalize the construction given in Fontana and Loper [5] 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.

Ultrafilter and Constructible Topologies on Spaces of Valuation Domains

Let K be a field, and let A be a subring of K. We consider properties and applications of a compact, Hausdorff topology called the “ultrafilter topology” defined on the space Zar(K | A) of all valuation domains having K as quotient field and containing A. We show that the ultrafilter topology coincides with the constructible topology on the abstract Riemann-Zariski surface Zar(K | A). We extend results regarding distinguished spectral topologies on spaces of valuation domains.

Some Closure Operations in Zariski-Riemann Spaces of Valuation Domains: A Survey

In this survey we present several results concerning various topologies that were introduced in recent years on spaces of valuation domains.

Invertibility of ideals in Prüfer extensions

ABSTRACT Using the general approach to invertibility for ideals in ring extensions given by Knebush and Zhang in [9], we investigate about connections between faithfully flatness and invertibility for ideals in rings with zero divisors.

Corrigendum to “New algebraic properties of an amalgamated algebra along an ideal”

ABSTRACT At some point, after publication, we realized that Proposition 4.1(2) and Theorem 4.4 in [2] hold under the assumption (not explicitly declared) that B = f(A)+J. Furthermore, we provide here the exact value for the embedding dimension of A⋈fJ, also when B≠f(A)+J, under the hypothesis that J is finitely generated as an ideal of the ring f(A)+J.