Narrow Results

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.

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.

In this paper we describe the lattice of preradicals over any local uniserial ring (equivalently, any local artinian principal ideal ring). This lattice is isomorphic to a lattice of binary sequences of length n, where n is the length of the composition series for the ring, as a left and as a right module. We prove some of its properties: it is finite having cardinality 2ⁿ, it is distributive, self-dual and it is graded having rank . We also describe the correspondent posets of irreducible, join-irreducible, prime and coprime elements.

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.

In this paper, some mappings to and from R-tors are introduced, and sufficient (or necessary) conditions for their being lattice isomorphisms are established.

We continue the study of Galois connections between the lattices of preradicals of two rings $A$ and $B$ induced by an adjoint pair of functors between the categories $A$-Mod and $B$-Mod. In this paper, we focus on the functor triple induced by any ring homomorphism $f:A\to B$, and particularly when it is a ring epimorphism. We give additional results when the epimorphism is flat and when it is projective.

We introduce continuity filters as a generalization of Gabriel filters, still with the possibility of defining modules of quotients, which we present and give some properties about them. We describe continuity filters on principal ideal domains, and finite products of rings. We also study continuity filters with a least element.