The union of a chain of submodules is a submodule

Let R be a ring with 1 and let M be a left R-module. Let \{M_t\}_{t \in T} be a nonempty chain of left R-submodules of M; that is, if i \leq j, then M_i \subseteq M_j. Prove that \bigcup_T M_t is a submodule of M.


We use the submodule criterion.

Certainly 0 \in M_i for some i. Thus 0 \in \bigcup M_t, and hence \bigcup M_t \neq 0.

Now suppose a,b \in \bigcup M_t and r \in R. Now a \in M_i and b \in M_j for some i and j. Suppose without loss of generality that i \leq j; then a,b \in M_j. Since M_j is a submodule, a+rb \in M_j. Thus a+rb \in \bigcup M_t.

Hence \bigcup M_t is a submodule of M.

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