Remark

(1) Let $(M_\iota)_{\iota \in I}, (N_\iota)_{\iota \in I}$ be two families of submodules of a module $E$, with the same indexing set; suppose that $E$ is both the direct sum of the family $(M_\iota)$ and of the family $(N_\iota)$ and that $N_\iota \subset M_\iota$ for all $\iota \in I$. Then $N_\iota = M_\iota$ for all $\iota \in I$, as follows immediately from no. 6, Corollary 1 to Proposition 7 applied to the canonical injections $f_\iota : N_\iota \to M_\iota$.