CHARACTERIZATIONS OF SOME CLASSES OF RINGS VIA LOCALLY SUPPLEMENTED MODULES

Year 2020, , 178 - 193, 07.01.2020

Farid Kourki Rachid Tribak

https://doi.org/10.24330/ieja.663060

Abstract

We introduce the notion of locally supplemented modules (i.e., modules for which every finitely generated submodule is supplemented). We show that a module $M$ is locally supplemented if and only if $M$ is a sum of local submodules. We characterize several classes of rings in terms of locally supplemented modules. Among others, we prove that a ring $R$ is a Camillo ring if and only if every finitely embedded $R$-module is locally supplemented. It is also shown that a ring $R$ is a Gelfand ring if and only if every $R$-module having a finite Goldie dimension is locally supplemented.

Keywords

Camillo ring, Gelfand ring, locally supplemented module, semiperfect ring, supplemented module

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