In this paper, we introduce strongly generalized (weakly) δ-supplemented modules. We call a module strongly generalized (weakly) δ-supplemented (briefly δ-SGS (δ-SWGS)) if every submodule containing the δ-radical has a (weak) δ-supplement. The first part of this paper investigates various properties of δ-SGS modules. We prove that δ-SGS modules are closed under factor modules and finite sums. Using these modules, we show that a ring R is δ-semiperfect if and only if every left R-module is a δ-SGS module. The second part of this paper establishes some properties of δ-SWGS modules.
