Exercise 21, § 2

(a) Let $(E_j)_{j \in J}$ be a finite family of A-modules and for all $j \in J$ let $I_j$ be an injective envelope of $E_j$; show that $\bigoplus_{j \in J} I_j$ is an injective envelope of $\bigoplus_{j \in J} E_j$.

(b) An A-module M is called indecomposable if it is distinct from $\{0\}$ and it is not the direct sum of two submodules distinct from $\{0\}$. Show that if I is an injective A-module, the following properties are equivalent: $(\alpha)$ $\{0\}$ is an irreducible submodule in I. $(\beta)$ I is indecomposable. $(\gamma)$ I is an injective envelope of each of its submodules $\neq 0$. (Use (a) and Exercise 18 (e).) $(\delta)$ I is isomorphic to the injective envelope $I(A/q)$, where q is an irreducible left ideal of A.

Moreover, when this is so, for all $x \neq 0$ in I, $\operatorname{Ann}(x)$ is an irreducible left ideal of A and I is isomorphic to $I(A/\operatorname{Ann}(x))$.

Deduce that for the injective envelope of an A-module E to be indecomposable, it is necessary and sufficient that $\{0\}$ be an irreducible submodule of E.

(c) Give an example of an indecomposable $\mathbf{Z}$-module E 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 and $I = \bigoplus_{\lambda \in L} I_\lambda$ a decomposition of I as a direct sum of a finite family of indecomposable injective submodules; for all $\lambda \in L$ let $J_\lambda = \bigoplus_{\mu \neq \lambda} I_\mu$ and let $N_\lambda = J_\lambda \cap E$. Show that $(N_\lambda)_{\lambda \in L}$ is a reduced irreducible decomposition of $\{0\}$ in E (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 above procedure uniquely up to isomorphism (use Exercise 16 (b)).