Intersection of Set of Submodules containing Subset is Smallest Submodule

Theorem
Let $R$ be a ring.

Let $\struct {G, +_G}$ be an abelian group.

Let $M = \struct {G, +, \circ}_R$ be an $R$-module.

Let $S \subset M$ be a subset of $M$.

Let $T$ be the intersection of all submodules of $M$ which contain $S$ as a subset.

Then the intersection $\bigcap T$ is the smallest submodule of $M$ containing $S$.