Spectral spaces and ultrafilters

Let $X$ be the prime spectrum of a ring. In [arXiv:0707.1525] 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 [arXiv:0707.1525] and, starting from a set $X$ and a collection of subsets $\mathcal{F}$ of $X$, we define by using ultrafilters a topology on $X$ in which $\mathcal F$ is a collection of clopen sets. We use this construction for giving a new characterization of spectral spaces and several new examples of spectral spaces.Comment: 16 pages. To appear in Communications in Algebr

Pr\"ufer-like conditions on an amalgamated algebra along an ideal

Let $f:A\longrightarrow B$ be a ring homomorphism and let $\mathfrak b$ be an ideal of $B$. In this paper we study Pr\"ufer like conditions in the amalgamation of $A$ with $B$ along $\mathfrak b$, with respect to $f$, a ring construction introduced in 2009 by D'Anna, Finocchiaro and Fontana.Comment: 17 pages. To appear in Houston Journal of Mathematic

Topological properties of semigroup primes of a commutative ring

A semigroup prime of a commutative ring $R$ is a prime ideal of the semigroup $(R,\cdot)$. One of the purposes of this paper is to study, from a topological point of view, the space \scal(R) of prime semigroups of $R$. We show that, under a natural topology introduced by B. Olberding in 2010, \scal(R) is a spectral space (after Hochster), spectral extension of \Spec(R), and that the assignment R\mapsto\scal(R) induces a contravariant functor. We then relate -- in the case $R$ is an integral domain -- the topology on \scal(R) with the Zariski topology on the set of overrings of $R$. Furthermore, we investigate the relationship between \scal(R) and the space $\boldsymbol{\mathcal{X}}(R)$ consisting of all nonempty inverse-closed subspaces of \spec(R), which has been introduced and studied in C.A. Finocchiaro, M. Fontana and D. Spirito, "The space of inverse-closed subsets of a spectral space is spectral" (submitted). In this context, we show that \scal( R) is a spectral retract of $\boldsymbol{\mathcal{X}}(R)$ and we characterize when \scal( R) is canonically homeomorphic to $\boldsymbol{\mathcal{X}}(R)$, both in general and when \spec(R) is a Noetherian space. In particular, we obtain that, when $R$ is a B\'ezout domain, \scal( R) is canonically homeomorphic both to $\boldsymbol{\mathcal{X}}(R)$ and to the space \overr(R) of the overrings of $R$ (endowed with the Zariski topology). Finally, we compare the space $\boldsymbol{\mathcal{X}}(R)$ with the space \scal(R(T)) of semigroup primes of the Nagata ring $R(T)$, providing a canonical spectral embedding \xcal(R)\hookrightarrow\scal(R(T)) which makes \xcal(R) a spectral retract of \scal(R(T)).Comment: 21 page

A topological version of Hilbert's Nullstellensatz

We prove that the space of radical ideals of a ring $R$, endowed with the hull-kernel topology, is a spectral space, and that it is canonically homeomorphic to the space of the nonempty Zariski closed subspaces of Spec$(R)$, endowed with a Zariski-like topology.Comment: J. Algebra (to appear

Prime Ideals in Infinite Products of Commutative Rings

In this work we present descriptions of prime ideals and in particular of maximal ideals in products $R = \prod D_\lambda$ of families $(D_\lambda)_{\lambda \in \Lambda}$ of commutative rings. We show that every maximal ideal is induced by an ultrafilter on the Boolean algebra $\prod \mathcal{P}(\max(D_\lambda))$. If every $D_\lambda$ is in a certain class of rings including finite character domains and one-dimensional domains, then this leads to a characterization of the maximal ideals of $R$. If every $D_\lambda$ is a Pr\"ufer domain, we depict all prime ideals of $R$. Moreover, we give an example of a (optionally non-local or local) Pr\"ufer domain such that every non-zero prime ideal is of infinite height

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.Comment: Comm. Algebra (accepted for publication

Topology, intersections and flat modules

It is well known that, in general, multiplication by an ideal I does not commute with the intersection of a family of ideals, but that this fact holds if I is flat and the family is finite. We generalize this result by showing that finite families of ideals can be replaced by compact subspaces of a natural topological space, and that ideals can be replaced by submodules of an epimorphic extension of a base ring. As a particular case, we give a new proof of a conjecture by Glaz and Vasconcelos

The upper Vietoris topology on the space of inverse-closed subsets of a spectral space and applications

Given an arbitrary spectral space X, we consider the set X(X) of all nonempty subsets of X that are closed with respect to the inverse topology. We introduce a Zariski-like topology on X(X) and, after observing that it coincides the upper Vietoris topology, we prove that X(X) is itself a spectral space, that this construction is functorial, and that X(X) provides an extension of X in a more “complete” spectral space. Among the applications, we show that, starting from an integral domain D, X(Spec(D)) is homeomorphic to the (spectral) space of all the stable semistar operations of finite type on D