Some Properties of Weakly R-Supplemented Modules

In this study, we introduce and examine the concept of weakly r-supplemented modules and establish several fundamental properties concerning them. It is shown that if the radical of a weakly supplemented module M serves as a supplement submodule in M, then M is necessarily weakly r-supplemented. We also demonstrate that every factor module, every homomorphic image, and every r-small cover of a weakly r-supplemented module inherit the weakly r-supplemented property. Moreover, if a module M can be written as the sum M = M-1 + M-2 + ... + M-n where each Mi (i = 1, 2, ..., n) is weakly rsupplemented, then M itself enjoys the same property. Finally, it is established that whenever M is weakly r-supplemented, any finitely M-generated R-module is also weakly r-supplemented.