A Study On a New Generalization of $\delta$-Supplemented Modules

Abstract

For any ring $S$ and an $S$-module $W$, a submodule $G$ of $W$ is termed \emph{co$_\delta$-coatomic} if the quotient module $W/G$ is $\delta$-coatomic. In this study, we introduce the term ($\oplus$-)\emph{co$_\delta$-coatomically $\delta$-supplemented module}, or shortly ($\oplus$-)\emph{co$_\delta$-$\delta$-supplemented module} to describe a module $W$ where each co$_\delta$-coatomic submodule has a $\delta$-supplement (that is a direct summand) in $W$. Furthermore, a module $W$ is identified as \emph{co$_\delta$-coatomically $\delta$-semiperfect}, or shortly \emph{co$_\delta$-$\delta$-semiperfect}, provided each $\delta$-coatomic quotient module of $W$ has a projective $\delta$-cover. It has been proved that over a $\delta$-semiperfect ring $S$, the module $_{S}S$ is $\oplus_{\delta}$-co-coatomically supplemented if and only if $_{S}S$ is co$_\delta$-$\delta$-semiperfect if and only if $_{S}S$ is $\oplus$-co$_\delta$-$\delta$-supplemented.