CHARACTERIZATIONS OF SOME CLASSES OF RINGS VIA LOCALLY SUPPLEMENTED MODULES

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.