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A left module $M$ over an arbitrary ring is called an $\mathrm{ℛ𝒟}$-module (or an $\mathrm{ℛ𝒮}$-module) if every submodule $N$ of $M$ with $\mathrm{Rad}\left(M\right)\subseteq N$ is a direct summand of (a supplement in, respectively) $M$. In this paper, we investigate the various properties of $\mathrm{ℛ𝒟}$-modules and $\mathrm{ℛ𝒮}$-modules. We prove that $M$ is an $\mathrm{ℛ𝒟}$-module if and only if $M=\mathrm{Rad}\left(M\right)\oplus X$, where $X$ is semisimple. We show that a finitely generated $\mathrm{ℛ𝒮}$-module is semisimple. This gives us the characterization of semisimple rings in terms of $\mathrm{ℛ𝒮}$-modules. We completely determine the structure of these...