Intersection_and_Sum_of_Submodules

Let $\left({G, +, \circ}\right)_R$ be an $R$-module. Let $H$ and $K$ be submodules of $G$. Then $H + K$ and $H \cap K$ are also submodules of $G$. The intersection of any set of submodules of $G$ is a submodule. Thus if $S \subseteq G$, the intersection of all submodules of $G$ containing $S$ is the smallest submodule of $G$ containing $S$.