Modules and abelian groups with a bounded domain of injectivity

Abstract

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 [Formula: see text] is impecunious if and only if for every prime integer [Formula: see text], [Formula: see text] has a direct summand isomorphic to [Formula: see text] for some positive integer [Formula: see text]. 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.

A Hyperstructural Approach to Semisimplicity

“…Moreover, in recent years, types of injectivity and projectivity have been studied with the help of (semi)simple modules. Refer to the books [10,11,[23][24][25] and the papers [26][27][28][29][30][31][32][33][34] for detailed information.…”

Rings with modules having a restricted injectivity domain

“…This definition gives a natural opposite to injectivity of modules, since only injective modules have the class of all modules as their injectivity domain. Recently, many studies have been conducted concerning poor modules along with their generalizations and restrictions (see [2][3][4]7,9]).…”

“…In [7], modules whose injectivity domain is contained in the class of all pure-split modules are introduced as impecunious modules, where a module M is called puresplit if all its pure submodules are direct summands of M. Starting point of defining such modules was the fact that semisimple modules are pure-split and therefore a generalization of poor modules is obtained.…”