Intersection of Submodules is Submodule/General Result

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 $S$ be a set of submodules of $M$.

Then the intersection $\ds \bigcap S$ is a submodule of $M$.