Proposition 19

Let E be an A-module, the direct sum of a family $(M_\lambda)_{\lambda \in L}$ of submodules. If, for each $\lambda \in L$, $S_\lambda$ is a free subset (resp. generating set, basis) of $M_\lambda$, then $S = \bigcup_{\lambda \in L} S_\lambda$ is a free subset (resp. generating set, basis) of E.

The proposition follows from the definitions and the relation $A_s^{(S)} = \bigoplus_{\lambda \in L} A_s^{(S_\lambda)}$ (associativity of direct sums, cf. no. 6).