Lemma 4

Let M be a nonzero noetherian A-module, I an injective envelope of M. Then I possesses an indecomposable injective submodule.

One may obviously suppose that M is a submodule of I. Let N be a submodule of M, such that I is not an injective envelope of N and maximal for this property. According to remark 2 (X, p. 21), there exists a submodule I₁ of I which is an injective envelope of N; then I₁ is a direct factor in I, let J be a direct factor complement. One has J ≠ 0, let us prove that J is indecomposable. If J′ is a nonzero direct factor submodule of J, one has J′ ∩ M ≠ 0 and

The submodule N′ = (J′ ∩ M) + N of M is direct sum of J′ ∩ M and N, hence strictly contains N. Moreover N′ is contained in the submodule J′ + I₁ of J, which is direct sum of J′ and I₁, hence injective. According to the maximal character of N, this implies J′ + I₁ = I, hence J′ = J, and J is indecomposable.

Let $\mathscr{I}$ denote the set (X, p. 21, cor. 1) of classes of indecomposable injective A-modules.

Recall (X, p. 17, cor. 1) that, if A is left noetherian, every A-module direct sum of injective A-modules is injective.