Weakly Poor Modules

Abstract

In this paper, weakly poor modules are introduced as modules whose injectivity domains are contained in the class of all copure-split modules. This notion gives a generalization of both poor modules and copure-injectively poor modules. Properties involving weakly poor modules as well as examples that show the relations between weakly poor modules, poor modules, impecunious modules and copure-injectively poor modules are given. Rings over which every module is weakly poor are right CDS. A ring over which there is a cyclic projective weakly poor module is proved to be weakly poor. Moreover, the characterizations of weakly poor abelian groups is given. It states that an abelian group $A$ is weakly poor if and only if $A$ is impecunious if and only if for every prime integer $p$, $A$ has a direct summand isomorphic to $\mathbb{Z}_{p^{n}}$ for some positive integer $n$. Consequently, an example of a weakly poor abelian group which is neither poor nor copure-injectively poor is given so that the generalization defined is proper.

Keywords

• Copure-injective modules
• copure-split modules
• (weakly) poor modules
• CDS rings

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