The nonempty intersection of submodules is a submodule

Let $R$ be a ring with 1 and let $M$ be a left $R$-module. Let $\{M_t\}_T$ be a family of left $R$-submodules of $M$ with $T \neq \emptyset$. Prove that $\bigcap_T M_t$ is a submodule of $M$.

We use the submodule criterion.

Note that for all $t$, $0 \in M_t$. Because $T \neq \emptyset$, we have $0 \in \bigcap_T M_t$. Now suppose $a,b \in \bigcap M_t$ and $r \in R$. Since each $M_t$ is a submodule, we have $a+rb \in M_t$ for all $t$, and thus $a+rb \in \bigcap M_t$.

So $\bigcap M_t \subseteq M$ is a submodule.