Corollary 1

Let M be a semisimple A-module and N a submodule of M. The following properties are equivalent:

(i) There exists a unique submodule supplementary to N in M.

(ii) We have Hom$_A(M/N,N) = 0$.

(iii) There exists a subset Λ of $\mathscr{S}$ such that $N =\bigoplus_{\lambda\in\Lambda}M_{\lambda}$.

Choose a submodule $N'$ supplementary to N in M (VIII, p. 56, Corollary 2). If we identify M with $N'\times N$, then the submodules of M supplementary to N are the graphs of the A-linear mappings from $N'$ to N. Since $N'$ is isomorphic to $M/N$, we have proved the equivalence of properties (i) and (ii). By Proposition 5, b), the group Hom$_A(N',N)$ is isomorphic to the group $\prod_{\lambda\in\mathscr{S}}$ Hom$_A(N'_{\lambda},N_{\lambda})$. It is zero if and only if for every $\lambda \in \mathscr{S}$, we have $N_{\lambda}= 0$ or $N'_{\lambda}= 0$ (VIII, p. 61, Remark), that is, $N_{\lambda}= 0$ or $N_{\lambda}= M_{\lambda}$. This proves the equivalence of properties (ii) and (iii).