20 research outputs found
Spectral spaces and ultrafilters
Let be the prime spectrum of a ring. In [arXiv:0707.1525] the authors
define a topology on 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 and a
collection of subsets of , we define by using ultrafilters a
topology on in which 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 be a ring homomorphism and let be an
ideal of . In this paper we study Pr\"ufer like conditions in the
amalgamation of with along , with respect to , a ring
construction introduced in 2009 by D'Anna, Finocchiaro and Fontana.Comment: 17 pages. To appear in Houston Journal of Mathematic
A topological version of Hilbert's Nullstellensatz
We prove that the space of radical ideals of a ring , 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, 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 is a prime ideal of the semigroup
. One of the purposes of this paper is to study, from a topological
point of view, the space \scal(R) of prime semigroups of . 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 is an integral domain -- the topology on \scal(R) with the
Zariski topology on the set of overrings of . Furthermore, we investigate
the relationship between \scal(R) and the space
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
and we characterize when \scal( R) is
canonically homeomorphic to , both in general and
when \spec(R) is a Noetherian space. In particular, we obtain that, when
is a B\'ezout domain, \scal( R) is canonically homeomorphic both to
and to the space \overr(R) of the overrings of
(endowed with the Zariski topology). Finally, we compare the space
with the space \scal(R(T)) of semigroup primes
of the Nagata ring , 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
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 of families
of commutative rings. We show that every
maximal ideal is induced by an ultrafilter on the Boolean algebra . If every 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 . If every
is a Pr\"ufer domain, we depict all prime ideals of . 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 be a field and let be a subring of . We consider properties
and applications of a compact, Hausdorff topology called the "ultrafilter
topology" defined on the space Zar of all valuation domains having
as quotient field and containing . We show that the ultrafilter topology
coincides with the constructible topology on the abstract Riemann-Zariski
surface Zar. 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