1,720,978 research outputs found
Pretorsion theories, stable category and preordered sets
No full textWe show that in the category of preordered sets there is a natural notion of pretorsion theory, in which the partially ordered sets are the torsion-free objects and the sets endowed with an equivalence relation are the torsion objects. Correspondingly, it is possible to construct a stable category factoring out the objects that are both torsion and torsion-free
Some remarks on Prüfer rings with zero-divisors
Full text linkLet A be the fiber product R×TB, where B→T is a surjective ring homomorphism with regular kernel and R⊆T is a ring extension where T is an overring of R. In this paper we provide a characterization of when A has distinguished Prüfer-like properties and new constructions of Prüfer rings with zero-divisors. Furthermore we give examples of homomorphic images of Prüfer rings that are Prüfer without assuming that the kernel of the surjection is regular. Finally we provide some remarks on the ideal theory of pre-Prüfer rings
Some applications of the ultrafilter topology on spaces of valuation domains, Part I - Actes des rencontres du CIRM, 2 no. 2 - Societé Mathématique de France
No full textOn quotients of Rees algebras
No full textWe study ring theory properties of some quotients of Rees algebras, A(f) (a), that extend results of [2] and [3]. In particular, we use pullback constructions to describe the prime spectrum of A(f) (a). Some questions that we discuss in this paper remain open in general. (c) 2022 Elsevier B.V. All rights reserved
Topological properties of semigroup primes of a commutative ring
No full textA semigroup prime of a commutative ring R is a prime ideal of the semigroup (R, ·). One of the purposes of this paper is to study, from a topological point of view, the space S(R) of prime semigroups of R. We show that, under a natural topology introduced by B. Olberding in 2010, S(R) is a spectral space (after Hochster), spectral extension of Spec(R) , and that the assignment R↦ S(R) induces a contravariant functor. We then relate—in the case R is an integral domain—the topology on S(R) with the Zariski topology on the set of overrings of R. Furthermore, we investigate the relationship between S(R) and the space X(R) consisting of all nonempty inverse-closed subspaces of Spec(R) , which has been introduced and studied in Finocchiaro et al. (submitted). In this context, we show that S(R) is a spectral retract of X(R) and we characterize when S(R) is canonically homeomorphic to X(R) , both in general and when Spec(R) is a Noetherian space. In particular, we obtain that, when R is a Bézout domain, S(R) is canonically homeomorphic both to X(R) and to the space Overr(R) of the overrings of R (endowed with the Zariski topology). Finally, we compare the space X(R) with the space S(R(T)) of semigroup primes of the Nagata ring R(T), providing a canonical spectral embedding X(R) ↪ S(R(T)) which makes X(R) a spectral retract of S(R(T))
Spectral spaces of semistar operations
No full textWe investigate, from a topological point of view, the classes of spectral semistar operations and of eab semistar operations, following methods recently introduced in [11,13]. We show that, in both cases, the subspaces of finite type operations are spectral spaces in the sense of Hochster and, moreover, that there is a distinguished class of overrings strictly connected to each of the two types of collections of semistar operations. We also prove that the space of stable semistar operations is homeomorphic to the space of Gabriel-Popescu localizing systems, endowed with a Zariski-like topology, extending to the topological level a result established in [14]. As a side effect, we obtain that the space of localizing systems of finite type is also a spectral space. Finally, we show that the Zariski topology on the set of semistar operations is the same as the b-topology defined recently by B. Olberding [37,38]
Distinguished classes of ideal spaces and their topological properties
No full textWe consider the set of all the ideals of a ring, endowed with the coarse lower topology. The aim of this paper is to study topological properties of distinguished subspaces of this space and detect the spectrality of some of them
New distinguished classes of spectral spaces: A survey
No full textIn the present survey paper, we present several new classes of Hochster’s spectral spaces “occurring in nature,” actually in multiplicative ideal theory, and not linked to or realized in an explicit way by prime spectra of rings. The general setting is the space of the semistar operations (of finite type), endowed with a Zariski-like topology, which turns out to be a natural topological extension of the space of the overrings of an integral domain, endowed with a topology introduced by Zariski. One of the key tool is a recent characterization of spectral spaces, based on the ultrafilter topology, given in Finocchiaro, Commun Algebra, 42:1496-1508, 2014, [15]. Several applications are also discussed
A topological version of Hilbert's Nullstellensatz
No full textWe 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 non-empty Zariski closed subspaces of Spec(R), endowed with a Zariski-like topology
Abstractly constructed prime spectra
No full textThe main purpose of this paper is a wide generalization of one of the results abstract algebraic geometry begins with, namely of the fact that the prime spectrum Spec (R) of a unital commutative ring R is always a spectral (= coherent) topological space. In this generalization, which includes several other known ones, the role of ideals of R is played by elements of an abstract complete lattice L equipped with a binary multiplication with xy⩽ x∧ y for all x, y∈ L. In fact when no further conditions on L are required, the resulting space can be and is only shown to be sober, and we discuss further conditions sufficient to make it spectral. This discussion involves establishing various comparison theorems on so-called prime, radical, solvable, and locally solvable elements of L; we also make short additional remarks on semiprime elements. We consider categorical and universal-algebraic applications involving general theory of commutators, and an application to ideals in what we call the commutative world. The cases of groups and of non-commutative rings are briefly considered separately
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