The opposite of projectivity by proper classes

Abstract

In the last decade, two new approaches have been introduced for the analysis of the projectivity of modules. In this paper, the projectivity of modules has been studied with a new approach by using projectively generated proper classes. As an opposite to projectivity, a module [Formula: see text] is said to be [Formula: see text]-indigent if the projectively generated proper class by [Formula: see text] is consisting of exactly the split short exact sequences. We consider rings over which every (finitely generated, finitely presented, cyclic, simple) right [Formula: see text]-modules are either projective or [Formula: see text]-indigent.