Exercise 19, § 2

(a) Let E, F be two a-modules, M an injective submodule of E ⊕ F. Let I be an injective envelope of M ∩ E in M, J a direct complement of I in M. Now prove that the restriction to I (resp. J) of the canonical projection of E ⊕ F onto E (resp. F) is injective (compose with the projection of I into E an extension E → I of the injection M ∩ E → I).

(b) Let E be an a-module, M a submodule of E, maximal in the set of injective submodules of E; M admits in E a direct complement N which contains no injective submodule ≠ {0}. Now prove that for every injective submodule P of E, the image of P by the projection of E onto M (for the decomposition of E into direct sum M ⊕ N) is an injective envelope of P ∩ M in M (use (a)). If M' is a second submodule of E, maximal in the set of injective submodules of E, now prove that there exists an automorphism of E transforming M into M' and leaving the elements of N invariant.