Remark

(1) Let $M, N$ be two supplementary submodules in an $A$-module $E$, $p, q$ the projectors of $E$ onto $M$ and $N$ respectively, corresponding to the decomposition of $E$ as a direct sum of $M$ and $N$. It is known (no. 6, Corollary 1 to Proposition 6) that, for every $A$-module $F$, the mapping $(u, v) \mapsto u \circ p + v \circ q$ is an isomorphism of

onto $\operatorname{Hom}_A(E, F)$. The image of $\operatorname{Hom}_A(M, F)$ under this isomorphism is the set of linear mappings $w : E \to F$ such that $w(x) = 0$ for all $x \in N$.

(2) If $M, N$ are two submodules of $E$ such that $M \cap N$ is a direct factor of $M$ and $N$, then $M \cap N$ is also a direct factor of $M + N$: if $P$ (resp. $Q$) is a supplement of $M \cap N$ in $M$ (resp. $N$), $M + N$ is the direct sum of $M \cap N, P$ and $Q$, as is immediately verified.