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.

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