On Boolean Lattices of Module Classes

We introduce new big lattices of classes of R-modules induced by suitable preradicals, the big lattice of σ-hereditary classes, the big lattice of σ-cohereditary classes, and the big lattice of σ-open classes. We prove that their Skeletons are boolean lattices which generalize and extend the lattice of natural classes, the lattice of conatural classes and the Skeleton of the lattice of hereditary torsion theories, respectively. We also introduce the lattice of σ-hereditary torsion theories, for a exact preradical σ. In case that σ be the identity preradical, we obtain the usual lattice of hereditary torsion classes.

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“…We refer to [1][2][3][4]9], for basic results about hereditary, cohereditary, natural, conatural and open classes.…”

Section: Preradicals and Lattices Of Module Classesmentioning

confidence: 99%

Big lattices of module classes induced by preradicals

Cerda-León

Rincón-Mejía

2019

São Paulo J. Math. Sci.

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“…In [3] it is shown that every conatural class is a hereditary torsion class if and only if every simple module is parainjective and that every natural class is cohereditary if and only if every simple module is paraprojective.…”

Section: Introductionmentioning

confidence: 99%

On retractability and its relationships with lattices associated to a ring

Domínguez-López

Garrido-Jiménez

Mejía

et al. 2021

Hacettepe Journal of Mathematics and Statistics

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The purpose of this work is threefold. First, we explore some relationships between retractability and some lattices of classes of modules. Secondly, we weaken the hypothesis of a result of Ohtake characterizing rings over which all radicals are left exact. In the last section of this work, we introduce a binary relation between modules that produce a Galois connection between the lattice of natural classes and the lattice of conatural classes, and we obtain some results about it.

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On lattices associated to rings with respect to a preradical

Cerda-León

Rincón-Mejía

2022

International Electronic Journal of Algebra

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We introduce some new lattices of classes of modules with respect to appropriate preradicals. We introduce some concepts associated with these lattices, such as the σ-semiartinian rings, the σ-retractable modules, the σ-V -rings, the σ-max rings. We continue to study σ-torsion theories, σ-open classes, σ-stable classes. We prove some theorems that extend some known results. Our results fall into well known situations when the preradical σ is chosen as the identity preradical.

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On Boolean Lattices of Module Classes

Cited by 3 publications

References 3 publications

Big lattices of module classes induced by preradicals

Big lattices of module classes induced by preradicals

On retractability and its relationships with lattices associated to a ring

On lattices associated to rings with respect to a preradical

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