Corollary 2

Tag 0DQD, permanent. Cite it as [Bourbaki, Tag 0DQD].

Algebra, chapter X, § 1. Compléments d’algèbre linéaire

Let M be an A-module, I an injective envelope of M. In order that I be indecomposable, it is necessary and sufficient that 0 not be the intersection of two nonzero submodules of M.

The condition is necessary according to prop. 14 ((i) $\Rightarrow$ (ii)). Conversely, if I is direct sum of the nonzero submodules $I_1$ and $I_2$ , one has:

I_1 \cap M \neq 0,\quad I_2 \cap M \neq 0\quad \text{et}\quad (I_1 \cap M) \cap (I_2 \cap M) = 0 .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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