Exercise 21, § 2

(a) Let (E_j)_{j ∈ J} be a finite family of A-modules, and for every j ∈ J, let I_j be an injective envelope of E_j; now prove that $\bigoplus_{j \in J} I_j$ is an injective envelope of $\bigoplus_{j \in J} E_j$. (b) One says that an A-module M is indecomposable if it is distinct from {0} and if it is not a direct sum of two submodules distinct from {0}. Now prove that if I is an injective A-module, the following properties are equivalent: α) {0} is an irreducible submodule in I. β) I is indecomposable. γ) I is an injective envelope of each of its submodules ≠ {0}. (Use a) and Exercise 18 e).) δ) I is isomorphic to the injective envelope I(A/q), where q is an irreducible left ideal in A. Moreover, when this is so, for every x ≠ 0 in I, Ann(x) is an irreducible left ideal in A, and I is isomorphic to I(A/Ann(x)).

Deduce that in order that the injective envelope of an A-module E be indecomposable, it is necessary and sufficient that {0} be an irreducible submodule in E. (c) Give an example of $\mathbf{Z}$-module E indecomposable such that its injective envelope $I(E)$ is decomposable (cf. VII, § 3, Exercise 5). (d) Let E be an A-module, I an injective envelope of E, $I = \bigoplus_{\lambda \in L} I_\lambda$ a decomposition of I as a direct sum of a finite family of indecomposable injective submodules; for every $\lambda \in L$, let $J_\lambda = \bigoplus_{\mu \neq \lambda} I_\mu$, and let $N_\lambda = J_\lambda \cap E$. Now prove that $(N_\lambda)_{\lambda \in L}$ is a reduced irreducible decomposition of {0} in E (II, p. 186, Exercise 16 b)) and that $I_\lambda$ is an injective envelope of $E/N_\lambda$. Conversely, every reduced irreducible decomposition of {0} in E can be obtained by the preceding procedure in one and only one way up to an isomorphism (use Exercise 16 b) of II, p. 186).