9 research outputs found
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
w-Divisoriality in Polynomial Rings
We extend the Bass-Matlis characterization of local Noetherian divisorial
domains to the non-Noetherian case. This result is then used to study the
following question: If a domain D is w-divisorial, that is, if each w-ideal of
D is divisorial, then is D[X] automatically w-divisorial? We show that the
answer is yes if D is either integrally closed or Mori.Comment: 9 pages Comm. Algebr
Progress in Commutative Algebra 2
This is the second of two volumes of a state-of-the-art survey article collection which originates from three commutative algebra sessions at the 2009 Fall Southeastern American Mathematical Society Meeting at Florida Atlantic University. The articles reach into diverse areas of commutative algebra and build a bridge between Noetherian and non-Noetherian commutative algebra. These volumes present current trends in two of the most active areas of commutative algebra: non-noetherian rings (factorization, ideal theory, integrality), and noetherian rings (the local theory, graded situation, and interactions with combinatorics and geometry). This volume contains surveys on aspects of closure operations, finiteness conditions and factorization. Closure operations on ideals and modules are a bridge between noetherian and nonnoetherian commutative algebra. It contains a nice guide to closure operations by Epstein, but also contains an article on test ideals by Schwede and Tucker and more
Anneaux \`a diviseurs et anneaux de Krull (une approche constructive)
Nous pr\'esentons dans cet article une approche constructive, dans le style
de Bishop, de la th\'eorie des diviseurs et des anneaux de Krull. Nous
accordons une place centrale aux "anneaux \`a diviseurs," appel\'es PvMD dans
la litt\'erature anglaise. Les r\'esultats classiques sont obtenus comme
r\'esultats d'algorithmes explicites sans faire appel aux hypoth\`eses de
factorisation compl\`ete.
We give give an elementary and constructive version of the theory of
"Pr\"ufer v-Multiplication Domains" (which we call "anneaux \`a diviseurs" in
the paper) and Krull Domains. The main results of these theories are revisited
from a constructive point of view, following the Bishop style, and without
assuming properties of complete factorizations.Comment: in French, Quelques typos dans la bibliographie ont \'et\'e
corrig\'es par rapport \`a la version publi\'ee et la version 1 sur ArXi