Modules and abelian groups with a bounded domain of injectivity

In this work, impecunious modules are introduced as modules whose injectivity domains are contained in the class of all pure-split modules. This notion gives a generalization of both poor modules and pure-injectively poor modules. Properties involving impecunious modules as well as examples that show the relations between impecunious modules, poor modules and pure-injectively poor modules are given. Rings over which every module is impecunious are right pure-semisimple. A commutative ring over which there is a projective semisimple impecunious module is proved to be semisimple artinian. Moreover, the characterization of impecunious abelian groups is given. It states that an abelian group M is impecunious if and only if for every prime integer p, M has a direct summand isomorphic to Z(p)(n) for some positive integer n. Consequently, an example of an impecunious abelian group which is neither poor nor pure-injectively poor is given so that the generalization defined is proper.