Remark 4

Let M and N be two A-modules, with components $(M_i)$ and $(N_i)$. Let $u \in \mathrm{Hom}_A(M, N)$ be an A-linear map from M to N; then for all i and for all $m \in M_i$ we have $u(m) \in N_i$, so u induces an $A_i$-linear map $u_i \in \mathrm{Hom}_{A_i}(M_i, N_i)$. It is easy to check that the map $u \mapsto (u_i)$ is an isomorphism of Z-modules (resp. of A-modules when A is commutative)