Corollary 2

Let M be an A-module. The following two conditions are equivalent:

(i) Every submodule of M admits a unique supplementary submodule.

(ii) The module M is the direct sum of a family $(S_i)_{i\in I}$ of simple, pairwise nonisomorphic modules.

Suppose that M satisfies these conditions. Then, for every submodule N of M, there exists a unique subset J of I such that we have $N =\bigoplus_{j\in J}S_j$ and every simple submodule of M is equal to one of the $S_i$.

Conditions (i) and (ii) both imply that M is semisimple.

Suppose that condition (i) is satisfied. Let $\lambda \in \mathscr{S}$. By the equivalence of (i) and (iii) in Corollary 1, every submodule of $M_{\lambda}$ is zero or equal to $M_{\lambda}$. Consequently, $M_{\lambda}$ is zero or simple, and (ii) follows from the fact that M is the direct sum of the family $(M_{\lambda})_{\lambda\in\mathscr{S}}$.

Conversely, if condition (ii) is satisfied, then $M_{\lambda}$ is zero or simple for every $\lambda \in \mathscr{S}$. If N is a submodule of M, then $N\cap M_{\lambda}$ is equal to 0 or $M_{\lambda}$ for every $\lambda \in \mathscr{S}$. Since we have $N =\bigoplus_{\lambda\in\mathscr{S}}(N\cap M_{\lambda})$ (VIII, p. 66, Corollary), the submodule N has property (iii) of Corollary 1 and therefore admits a unique supplementary submodule in M. This proves that (ii) implies (i), as well as the last assertions of the corollary.

Let M be an A-module and S a simple A-module. Denote the opposite ring of the field End$_A(S)$ by D, and view S as an $(A$, D)-bimodule. Then Hom$_A(S,M)$ is a left vector space over D, and Hom$_A(M,S)$ is a right vector space over D. The dual of the left D-vector space Hom$_A(S,M)$ is a right vector space over D (II, §2, No. 3, p. 232, Definition 2). For every $u\in$ Hom$_A(M,S)$, the mapping $h(u) :v\mapsto u\circ v$ from Hom$_A(S,M)$ to Hom$_A(S,S) = D$ is a linear form on the left D-vector space Hom$_A(S,M)$.