A Study On a New Generalization of δ-Supplemented Modules

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