On $FGS$-Modules

Abstract

We consider $R$ a non-necessarily commutative ring with unity $1\neq 0$ and $M$ a module over $R$. By using the category $\sigma[M]$ we introduce the notion of $FGS$-module. The latter generalizes the notion of $FGS$-ring. In this paper we fix the ring $R$ and study $M$ for which every hopfian module of $\sigma[M]$ becomes finitely generated. These kinds of modules are said to be $FGS$-modules. Some properties of $FGS$-module, a characterization of semisimple $FGS$-module and of serial $FGS$-module over a duo ring have been obtained.