We introduce the concept of (strongly) $ss$-radical supplemented modules. We prove that if a submodule $N$ of $M$ is strongly $ss$-radical supplemented and $Rad(M/N)=M/N$, then $M$ is strongly $ss$-radical supplemented. For a left good ring $R$, we show that $Rad(R)\subseteq Soc(_{R}R)$ if and only if every left $R$-module is $ss$-radical supplemented. We characterize the rings over which all modules are strongly $ss$-radical supplemented. We also prove that over a left $WV$-ring every supplemented module is $ss$-supplemented.

In this work, we introduce $ps$-supplement submodules as a kind of supplements. Specifically, a submodule V of a module M is referred to as a $ps$-supplement if there exists another submodule $U\subseteq M$ such that $M=U+V$ and the intersection $U\cap V$ is projective semisimple. We demonstrate that, over an arbitrary ring, the collection $\mathcal{PS}$ of short exact sequences of the form $0⟶M\stackrel{\psi }{⟶}N\stackrel{\varphi }{⟶}K⟶0,$ where $\mathrm{Im}\left(\psi \right)$ is a $ps$-supplement in N, constitutes a proper class. In addition, we explore several homological aspects of this class, including the symmetry of the relations between the modules. Finally, we provide that a ring R is a left SI-ring with essential socle if and only if every left R-module is $ps$-supplemented.