On generalization of $⊕$-cofinitely supplemented modules

Abstract

We study the properties of ⊕-cofinitely radical supplemented modules, or, briefly, $cgs^{⊕}$-modules. It is shown that a module with summand sum property (SSP) is $cgs^{⊕}$ if and only if $M/w \text{Loc}^{⊕} M$ ($w \text{Loc}^{⊕} M$ is the sum of all $w$-local direct summands of a module $M$) does not contain any maximal submodule, that every cofinite direct summand of a UC-extending $cgs^{⊕}$-module is $cgs^{⊕}$, and that, for any ring $R$, every free $R$-module is $cgs^{⊕}$ if and only if $R$ is semiperfect.