arXiv subjects

Martha Lizbeth Shaid Sandoval-Miranda

Publications and source records attributed to Martha Lizbeth Shaid Sandoval-Miranda.

4 recordsLinked to original sources

arXiv2025

Galois Connections and Preradicals in Abelian Categories

In this paper we define operations of preradicals of any abelian category. We define idempotent preradicals and radicals. We prove that every adjoint pair between abelian categories induces a Galois connection between the corresponding ordered collections of preradicals. If the abelian categories are bicomplete, we construct alpha and omega preradicals and study their respective preservation under the Galois connection. If a bicomplete abelian category is in addition locally small, then the corresponding collection of preradicals is a complete lattice. The Galois connection induced by an adjoint pair between locally small bicomplete abelian categories preserves, respectively, idempotent preradicals and radicals.

math.CT↗

arXiv2024

A point-free version of torsionfree classes and the Goldie torsion theory

Torsion theories are a pinnacle in the theory of abelian categories. They are a generalization of torsion abelian groups and in this generalization one of the most studied is that whose torsionfree class consists of nonsingular modules. To introduce the concept of singular interval we use the symmetric idea of torsion theories, that is the torsion class determines the torsionfree class and vice-versa, thus to introduce nonsingular intervals over an upper-continuous modular complete lattice, (a.k.a idiom, a.k.a modular preframe) we define the concept of \emph{division free} set. We introduce the division free set of nonsingular intervals which defines a division set of singular intervals in a canonical way. Several properties of division free sets and some consequences of nonsingular intervals are explored allowing us to develop a small part of a point-free nonsingular theory.

math.RA↗

arXiv2020

On strongly harmonic and Gelfand modules

We introduce the notions of Strongly harmonic and Gelfand module, as a generalization of the well-known ring theoretic case. We prove some properties of these modules and we give a characterization via their lattice of submodules and their space of maximal submodules. It is also observed that, under some assumptions, the space of maximal submodules of a strongly harmonic module constitutes a compact Hausdorff space whose frame of open sets is isomorphic to the frame $Ψ(M)$ defined in [arXiv:1612.07407]. Finally, we mention some open questions that arose during this investigation.

math.RA↗

arXiv2020

Gabriel localization in functor categories

P. Gabriel showed that for an unital ring $R$, there exists a bijective correspondence between the set of Gabriel filters of $R$ and the set of Giraud subcategories of $\mathrm{Mod}(R)$ (see \cite[Lemme 1]{Gabriel1} on page 412). In this paper we prove analogous of Gabriel's result: for a small preadditive category $\mathcal{C}$, there exists a bijective correspondence between the Gabriel filters of $\mathcal{C}$ and Giraud subcategories of $\mathrm{Mod}(\mathcal{C})$.

math.CT↗