Some properties of weakly r-supplemented modules

Abstract 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=M₁+M₂+. . . +M₍ where each M₈ (i=1, 2,. . . , n) is weakly r-supplemented, 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.