Francisco Raggi

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.

PRIME AND IRREDUCIBLE PRERADICALS

In this paper we study prime preradicals, irreducible preradicals, ∧-prime preradicals, prime submodules and diuniform modules. We study some relations between these concepts, using the lattice structure of preradicals developed in previous papers. In particular, we give a characterization of prime preradicals using an operator named the relative annihilator. We also characterize prime submodules by means of prime preradicals. We give some characterizations of rings that have certain conditions on prime radicals and on irreducible preradicals, such as left local left V-rings, as well as 1-spr rings, which we introduce.

THE LATTICE STRUCTURE OF PRERADICALS II: PARTITIONS

We continue the study of the preradicals of a ring in the lattice point of view. We introduce several interesting preradicals associated to a given preradical and some partitions of the whole lattice in terms of preradicals. As an application, we also give some classification theorems.

BASIC PRERADICALS AND MAIN INJECTIVE MODULES

We consider those injective modules that determine every left exact preradical and we call them main injective modules. We construct a main injective module for every ring and we prove some of its properties. In particular we give a characterization, in terms of main injective modules, of rings with a dimension defined by a filtration in the lattice of left exact preradicals. We define also the concept of basic preradical and prove some of its properties. In particular we prove that the class of all basic preradicals is a set, giving a bijective correspondence with the set of all left exact preradicals.

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.

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.

MAIN MODULES AND SOME CHARACTERIZATIONS OF RINGS WITH GLOBAL CONDITIONS ON PRERADICALS

Main injective modules, which determine every left exact preradical, were introduced in a former work. In this paper, we consider those modules which determine every preradical and we call them main modules. We prove that a main module exists if and only if the lattice of preradicals R-pr is a set, and in this case we give a general construction. Some properties of main modules are proven. We also prove some characterizations of rings for which (a) every preradical is left exact, (b) every preradical is idempotent, (c) every preradical is a radical, (d) every preradical is a t-radical, (e) every preradical which is not the identity functor is prime. These characterizations relate to semisimple artinian rings, rings that are a direct product of a finite number of simple rings, left V-rings, simple rings, among others. In order to illustrate the theory introduced in this paper, several examples are provided.

Main Injective Rings

We refer to those injective modules that determine every left exact preradical and that we called main injective modules in a preceding article, and we consider left main injective rings, which as left modules are main injective modules. We prove some properties of these rings, and we characterize QF-rings as those rings which are left and right main injective.