Theorem 3

Let I be an injective A-module.

a) If I is the injective envelope of a noetherian A-module M, I is direct sum of a finite family of indecomposable (injective) submodules.

b) If A is left noetherian, I is direct sum of a family of indecomposable (injective) submodules.

c) If I is direct sum of indecomposable (injective) submodules, there exists one and only one family of cardinals $(a_E)_{E \in \mathscr{I}}$, such that I is isomorphic to

Let us first note that c) follows from prop. 14 (X, p. 21) and from VIII, § 1, No. 7, theorem 2. Let us prove a).

Let N be a submodule of M whose injective envelopes are direct sum of a finite family of indecomposable submodules, and maximal for this property (there exists one since M is noetherian). According to remark 2 (X, p. 21), there exists a submodule I₁ of I which is an injective envelope of N. If I₁ = I, the proof is complete; otherwise, let $J$ be a direct factor complement of $I_1$ in $I$. Then $J$ is the injective envelope of the noetherian module $J\cap M$ (loc. cit.), hence possesses a nonzero indecomposable injective submodule $J'$ (lemma 4). Then $I_1+J'$ is injective, direct sum of a finite family of indecomposable submodules, and the injective envelope of the submodule $(I_1+J')\cap M$ of $M$ which strictly majorizes $N$, whence a contradiction.

Suppose $A$ is left noetherian and let us prove b). Let $X$ be the set sum of the sets $\operatorname{Hom}_A(E,I)$ for $E\in\mathscr{J}$. To each subset $Y$ of $X$ associate an $A$-module $E_Y$ and an $A$-homomorphism $f_Y:E_Y\to I$ in the following way: $Y$ is the sum of a family $(Y(E))_{E\in\mathscr{J}}$ where $Y(E)\subset\operatorname{Hom}_A(E,I)$, one sets

and the component of $f$ on the direct factor of $E_Y$ corresponding to the element $y$ of $Y(E)\subset\operatorname{Hom}_A(E,I)$ is $y:E\to I$. Let $Y$ be a subset of $X$, such that $f_Y$ is injective and that $Y$ is maximal for this property (such a subset exists according to E, III, p. 20); it suffices to prove that $f_Y$ is bijective. Otherwise, let $J$ be a direct factor complement of the injective submodule $\operatorname{Im}(f_Y)$ of $I$; since $J$ is nonzero, it possesses a nonzero noetherian submodule (because A is supposed noetherian), hence also a nonzero injective submodule $J'$ which is the injective envelope of a noetherian module. According to a), $J'$ is direct sum of a finite nonempty family of indecomposable submodules. There therefore exists a finite nonempty subset $Y'$ of $X$ such that $f_{Y'}$ maps bijectively $E_{Y'}$ onto $J'$. Since $\operatorname{Im}(f_Y)\cap J'=0$, one has $Y\cap Y'=\varnothing$ and $f_{Y\cup Y'}$ is injective; this contradicts the maximal character of $Y$ and completes the proof.