Exercise 19, § 2

(a) Let E, F be two A-modules and M an injective submodule of $E \oplus F$. Let I be an injective envelope of $M \cap E$ in M and J a supplementary submodule of I in M. Show that the restriction to I (resp. J) of the canonical projection of $E \oplus F$ onto E (resp. F) is injective (compose with the projection of I into E an extension $E \to I$ of the injection $M \cap E \to I$).

(b) Let E be an A-module and M a submodule of E, maximal in the set of injective submodules of E; M admits in E a supplementary submodule N which contains no injective submodule $\neq \{0\}$. Show that for every injective submodule P of E, the image of P under the projection of E onto M (under the decomposition of E as a direct sum $M \oplus N$) is an injective envelope of $P \cap M$ in M (use (a)). If $M'$ is another submodule of E which is maximal in the set of injective submodules of E, show that there exists an automorphism of E transforming M into $M'$ and leaving invariant the elements of N.