José Ríos Montes

THE LATTICE STRUCTURE OF PRERADICALS

Abstract In this paper we study the lattice of all preradicals on a ring R. We describe this lattice, we prove that it is an atomic and coatomic lattice and we describe the atoms and coatoms. We also give characterizations of simple Artinian, semisimple Artinian, and V-rings in terms of preradicals.

ON THE LATTICES OF NATURAL AND CONATURAL CLASSES IN R-MOD

In studying rings and modules, some kinds of lattices of module classes have been used to obtain information about the internal structure of the ring and about the Module Categories associated. The lattice R-tors of hereditary torsion theories in R-mod has been extensively studied. The properties of this lattice have been used to characterize many important classes of rings and module categories; see Refs. (1) and (2) for details about this lattice. Other important lattice of classes of modules is the lattice of Natural classes introduced by Dauns (3). This lattice has been used, among other things, for classifying into types the objects of certain subcategories of R-mod. The structure properties of the lattice of natural classes and the interplay between the properties of this lattice and the structure of R-mod have been studied in (4,5). In very recent papers (6–8) two new kinds of complete lattices have been introduced and studied, namely the prenatural classes lattice in R-mod (6,7) and the open classes lattice. The lattice of open classes is closely connected with the theory of Serre subcategories of R-mod (8).

THE LATTICE STRUCTURE OF HEREDITARY PRETORSION CLASSES

In this paper we continue the investigation of the lattice structure of hereditary pretorsion classes [(Comm. Algebra 1994, 22, 3613–3627; ibid. 1995, 23, 4173–4188)]. We show the existence of pseudocomplements and study right supplements for every hereditary pretorsion class. Moreover we investigate relations between these concepts and characterize a class of modules by means of these relations.

Prime Submodules and Local Gabriel Correspondence in σ[M]

We consider the concept of prime submodule defined by Raggi et al. [7]. We find equivalent conditions for a module M progenerator in σ[M], with τ M -Gabriel dimension, to have a one-to-one correspondence between the set of isomorphism classes of indecomposable τ-torsion free injective modules in σ[M] and the set of τ-pure submodules prime in M, where τ is a hereditary torsion theory in σ[M]. Also we give a relation between the concept of prime M-ideal given by Beachy and the concept of prime submodule in M. We obtain that if M is progenerator in σ[M], then these concepts are equivalent.

ON SOME LATTICES OF MODULE CLASSES

In this work we continue the discussion of the conatural classes introduced in [1]. We prove that the collection R-conat of all conatural classes of left modules over a ring R is a set, and it is a boolean lattice. Afterwards we study relationships between some lattices of module classes: R-conat and R-tors, R-her and R-quot. As a consequence of the developed theory we obtain characterizations of left MAX rings and artinian principal ideal rings.

Krull Dimension and Classical Krull Dimension of Modules

Using the concept of prime submodule defined by Raggi et al. in [16], for M ∈ R-Mod we define the concept of classical Krull dimension relative to a hereditary torsion theory τ ∈M-tors. We prove that if M is progenerator in σ[M], τ ∈M-tors such that M has τ-Krull dimension then cl.K τdim (M) ≤ k τ(M). Also we show that if M is noetherian, τ-fully bounded, progenerator of σ[M], and M ∈ 𝔽τ, then cl·K τdim (M) = k τ(M).

On the Atomic Dimension in Module Categories

ABSTRACT This article is concerned with the study of atomic dimension defined on the category of left modules over a ring R. A module theoretic as well as torsion theoretic characterization of rings with atomic dimension is provided. The atomic and Gabriel dimensions are compared and necessary and sufficient conditions for these two dimensions to coincide are established.

On Semiprime Goldie Modules

For an R-module M, projective in σ[M] and satisfying ascending chain condition on left annihilators, the authors introduce the concept of Goldie module. The authors also use the concept of semiprime module defined by Raggi et al. in [15] to give necessary and sufficient conditions for an R-module M, to be a semiprime Goldie module. This theorem is a generalization of Goldies theorem for semiprime left Goldie rings. Moreover, the authors prove that M is a semiprime (prime) Goldie module if and only if the ring S = EndR(M) is a semiprime (prime) right Goldie ring. Also, we study the case when M is a duo module.

Torsion theoretic dimensions and relative Gabriel correspondence

Let τ be an hereditary torsion theory. For a ring with τ-Gabriel dimension, we find necessary and sufficient conditions for the existence of a bijective correspondence between the τ-torsionfree injective modules and the τ-closed prime ideals. As an application, new characterizations of fully bounded noetherian rings are obtained.

Spectral torsion theories in module categories

Let M be an R-module and α[M] the category of all R-modules that are subgenerated by M. For any hereditary torsion theory T in α[M], the class of quotient modules defines a Grothendieck category denoted by ∊T[M]. T is called spectral if ∊T[M] is a spectral category (every short exact sequence in∊T[M] splits). The paper is devoted to the investigation of such torsion theories. Various characterizations are obtained and properties of the endomorphism rings of the corresponding quotient modules are studied. The final sectlon is concerned with the role of simple and semisimple modules with respect to spectral torsion theories. Our presentation generalizes and extends related results for spectral torsion theories in the category of all R-modules R-Mod.