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

Authors : Emine Önal Kır

Pages : 114-127

Doi:10.33484/sinopfbd.1411952

View : 36 | Download : 41

Publication Date : 2024-06-29

Article Type : Research

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.
Keywords : eş$_\\delta$-eşatom alt modül, eş$_\\delta$-eşatom $\\delta$-tümlenmiş modül, $\\oplus$-eş$_\\delta$-eşatom $\\delta$-tümlenmiş modül, eş$_\\delta$-eşatom $\\delta$-yarı mükemmel modül