Remark

With the exception of no. 7, Proposition 9, all the results of nos. 2 to 10 are valid for arbitrary commutative groups with operators, submodules (resp. quotient modules) being replaced in the statements by stable subgroups (resp. quotient groups by stable subgroups); we also make the convention of calling homomorphisms of groups with operators "linear mappings". The corollaries to no. 7, Proposition 9 are also valid for commutative groups with operators: this is obvious for Corollaries 4 and 5, as also for Corollary 2, since $\alpha \left( \sum_{i \in I} x_i \right) = \sum_{i \in I} \alpha x_i$ for every operator $\alpha$, and Corollary 3 follows from it. As for Corollary 1, it suffices to note that if N is a stable subgroup of F containing $u(S)$, $u^{-1}(N)$ is a stable subgroup of E containing S, hence $u^{-1}(N)$ contains M and therefore $u(M) \subset N$.