Definition 8

In an A-module E, two submodules $M_1, M_2$ are said to be supplementary if E is the direct sum of $M_1$ and $M_2$.

Proposition 11 of no. 8 shows that, for $M_1$ and $M_2$ to be supplementary, it is necessary and sufficient that $M_1 + M_2 = E$ and $M_1 \cap M_2 = \{0\}$ (cf. I, § 4, no. 9, Proposition 15).