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

Properties of chains of prime ideals in an amalgamated algebra along an ideal

Let $f:A \to B$ 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 ${A\Join^fJ}$), 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.Comment: J. Pure Appl. Algebra (to appear

Star-Invertibility and tt-finite character in Integral Domains

Let $A$ be an integral domain. We study new conditions on families of integral ideals of $A$ in order to get that $A$ is of $t$-finite character (i.e., each nonzero element of $A$ is contained in finitely many $t$-maximal ideals). We also investigate problems connected with the local invertibility of ideals.Comment: 16 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

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

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