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

Abstract

For any ring S and an S-moduleW, a submodule G ofW is termed coδ-coatomic if the quotient module W/G is δ-coatomic. It is demonstrated that a ring S is left δ-coatomic if and only if each simple left S-module is singular, and this is equivalent to stating that each coatomic left S-module is δ-coatomic. We introduce the term (⊕-)coδ-coatomically δ-supplemented module, or shortly (⊕-)coδ-δ-supplemented module to describe a module W where each coδ-coatomic submodule possesses a δ-supplement (that is a direct summand) in W. Furthermore, a module W is identified as coδ-coatomically δ-semiperfect, or shortly coδ-δ-semiperfect, provided each δ-coatomic quotient module of W possesses a projective δ-cover. This paper explores various properties of these modules. Specifically, it is proven that over a δ-semiperfect ring S, the S-module SS is ⊕δ-co-coatomically supplemented if and only if SS is coδ-δ- semiperfect, if and only if SS is ⊕-coδ-δ-supplemented.