Class Of R-modules Research Articles

Weak-Injective Modules

ABSTRACT Pure-injective and RD-injective R-modules over domains R have been investigated by many authors. We introduce another class of R-modules, called weak-injective modules, which turn out to be useful in addressing several unanswered questions between the two classes of modules. We also find that this class is an envelope class over any domain, giving a partial answer to the existence of envelope classes in the hierarchy of injective and divisible modules. Communicated by I. Swanson.

Modules with the Direct Summand Sum Property

The present work gives some characterizations of R-modules with the direct summand sum property (in short DSSP), that is of those R-modules for which the sum of any two direct summands, so the submodule generated by their union, is a direct summand, too. General results and results concerning certain classes of R-modules (injective or projective) with this property, over several rings, are presented.

On some classes of R-modules and congruences in R-tors

On some classes of R-modules and congruences in R-tors

Modules which are extending relative to module classes

Given a ring Rand a class χ of right R-modules, a right R-module Mcan be an extending module relative to χ in two different ways. Various general properties of such extending modules are given and, in case of specific classes of R-modules, we characterize them.

Quasi-Tilting Modules and Counter Equivalences

Given two ringsRandS, we study the category equivalences T⇄Y, where T is a torsion class ofR-modules and Y is a torsion-free class ofS-modules. These equivalences correspond to quasi-tilting triples (R,V,S), whereRVSis a bimodule which has, “locally,” a tilting behavior. Comparing this setting with tilting bimodules and, more generally, with the torsion theory counter equivalences introduced by Colby and Fuller, we prove a local version of the Tilting Theorem for quasi-tilting triples. A whole section is devoted to examples in case of algebras over a field.

THE LARGEST NON-TRIVIAL TORSION CLASS OF MODULES

ABSTRACT In this paper we investigate the following two classes of left R-modules: N(P) ={A|A has no non-zero direct summand P e P} and H(p) = {A} if B ⋚ A with B e N(P), then B = 0}, where P is a class of projective R-modules. We demonstrate that N(p) is, in general, not a torsion class but that H(P) is always a torsionfree class. We also investigate those classes P and rings R for which N(P) is the largest non-trivial torsion class of R-modules.

A generalization of finite dimensionality for modules

Let R be a ring with identity. Let C be a class of R-modules which is closed under submodules and isomorphic images. Define a submodule C of an R-module M to be a C-submodule of M if C ϵ C. An R-module M is said to be C-finite dimensional if it does not contain an infinite direct sum of non-zero C-submodules of M. Theorem: Let M be a C-finite dimensional R-module. Then there is a uniform bound (the C-dimension of M) on the number of non-zero C-submodules in a direct sum of submodules of M. When C = M R , we recover the definition of dimension in the sense of Goldie. When C is the class of torsion-free modules relative to a kernel functor σ, we derive the formula: dim M = σ-dim M + dim (σ( M)) where for an R-module N, dim N is the dimension of N in the sense of Goldie and σ-dim N is the dimension of N relative to the class of σ-torsion- free modules. A special case gives a new interpretation of rank of a module as defined by Goldie.

TTF-classes over perfect rings

For a ring R with unit, let RM denote the category of unitary left R-modules. Following S.E. Dickson [3], a(non-empty) class of R-modules is a torsion class in rM if is closed under factors, extensions, and direct sums. If, in addition, is closed under submodules, then is said to be hereditary.