Remark

(2) Let E, F be two A-modules such that E is the direct sum of a finite family $(M_i)_{1 \leq i \leq m}$ of submodules and F the direct sum of a finite family $(N_j)_{1 \leq j \leq n}$ of submodules. Then it is known (no. 6, Corollary 1 to Proposition 6) that $\mathrm{Hom}_A(E, F)$ is canonically identified with the product $\prod_{i,j} \mathrm{Hom}_A(M_i, N_j)$; to be precise, to a family $(u_{ji})$, where $u_{ji} \in \mathrm{Hom}_A(M_i, N_j)$ there corresponds the linear mapping $u : E \to F$ defined as follows. It suffices to define the restriction of $u$ to each of the $M_i$ and for each $x_i \in M_i$,

Now let G be a third A-module, the direct sum of a finite family $(P_k)_{1 \leq k \leq p}$ of submodules; let $v$ be a linear mapping of F into G and let $(v_{kj}) \in \prod_{j,k} \mathrm{Hom}_A(N_j, P_k)$ be the family corresponding to it canonically. For all $x_i \in M_i$, Thus it is seen that if we write the family $(w_{ki})$ corresponds canonically to the composite linear mapping $w = v \circ u$ from E to G (cf. § 10, no. 5).