Let $\mathscr{S}$ be the set of subsets $I'$ of I such that the sum of the family consisting of the modules N and $S_i$ for $i$ in $I'$ is direct. The set $\mathscr{S}$ is of finite character: a subset J of I belongs to $\mathscr{S}$ if and only if the same holds for every finite subset of J. Hence, the set $\mathscr{S}$ has a maximal element J (Set Theory, III, §4, No. 5, p. 171). Set $N'= N +\sum_{j\in J}S_j$. Let $i$ be in I - J. Since J is maximal in $\mathscr{S}$, the set $J\cup \{i\}$ does not belong to $\mathscr{S}$, so that $S_i\cap N'\not= 0$. Since $S_i$ is a simple module, we have $S_i\cap N'= S_i$. We therefore have $S_i\subset N'$ for every $i\in I$, whence $N'= M$. This completes the proof.

It suffices to apply Theorem 1 to the case N = 0.

The condition is necessary by Theorem 1.

Conversely, suppose that every submodule of M admits a supplement. Let $M'$ be the sum of the simple submodules of M, and let $M''$ be a supplement of $M'$ in M. Suppose that we have $M'\not= M$ and therefore $M''\not= 0$. Let N be a nonzero monogenous submodule of $M''$. By Proposition 3 of VIII, p. 49, there exists a maximal submodule P of N. Let Q be a submodule supplementary to P in M. Then $N\cap Q$ is a submodule of N, supplementary to P in N, hence isomorphic to $N/P$ (II, §1, No. 9, p. 210, Proposition 13). Consequently, $N\cap Q$ is a simple submodule of $M''$, contrary to the definition of $M'$.

We therefore have $M'= M$, and the module M is semisimple by Corollary 1.

Choose J as in Theorem 1. The module $N'=\bigoplus_{j\in J}S_j$ is supplementary to N in M; it is therefore isomorphic to $M/N$. Moreover, the submodules N and $\bigoplus_{i\in I-J}S_i$ of M are both supplementary to $N'$ and therefore isomorphic to $M/N'$.

If M is simple, then E is a field by the corollary of Proposition 2 of VIII, p. 47.

If E is a field, then the module M is indecomposable (VIII, p. 31, Proposition 4, a)). Since it is moreover semisimple, it is simple.

(i) There exists a set I such that M is isomorphic to $S^{(I)}$.

(ii) The module M is the direct sum of a family of submodules isomorphic to S.

(iii) The module M is the sum of a family of submodules isomorphic to S.

(iv) There exists a left vector space V over the field D such that the A-module M is isomorphic to $S\otimes_DV$.

The equivalence of (i) and (ii) is immediate, and that of (ii) and (iii) follows from Theorem 1 of VIII, p. 56, applied to the case N = 0. Every left vector space over D is isomorphic to a vector space of the form $D_s^{(I)}$, where I is a set (II, §7, No. 1, p. 292, Theorem 1). Since the tensor product commutes with direct sums, (i) is equivalent to (iv).

The first assertion follows from the definitions, the second from Corollary 3 of VIII, p. 56.

b) Let V be a left vector space over the field D. The D-linear mapping $\beta_V: V\rightarrow$ Hom$_A(S,S\otimes_DV)$ defined by $\beta_V(v)(s) =s\otimes v$ (VIII, p. 59) is bijective.

Denote the left D-vector space Hom$_A(S,M)$ by $\mathscr{H}(M)$. The A-module M is, by assumption, the direct sum of a family of submodules isomorphic to S. The A-module S is monogenous; to prove that the mapping $\alpha_M$ is bijective, it therefore suffices to consider the case M = S (VIII, p. 60, Remark 3). Now, $\mathscr{H}(S)$ is simply the D-vector space $D_s$, and $\alpha_S$ is simply the isomorphism $\iota : S\otimes_DD_s\rightarrow S$ defined by $\iota (s\otimes d) =sd$.

Likewise, to prove b), it suffices to consider the case $V = D_s$. Since the mapping $\alpha_S$ is bijective, the mapping $\beta_{D_s}=\beta_{\mathscr{H}(S)}$ is too (VIII, p. 60, Remark 2).

a) The mapping $W\mapsto \alpha (S\otimes_DW)$ is an isomorphism of ordered sets from $\mathscr{D}_D(V)$ to $\mathscr{D}_A(M)$.

b) The inverse isomorphism sends a submodule N of M to the linear subspace of V consisting of the elements $v$ such that $\alpha (s\otimes v)$ belongs to N for every $s\in S$.

For $W\in \mathscr{D}_D$(V), set $\varphi (W) =\alpha (S\otimes_DW)$. For $N\in \mathscr{D}_A$(M), denote by $\psi (N)$ the set of elements $v\in V$ such that $\alpha (s\otimes v)\in N$ for every $s\in S$. This defines two mappings $\varphi :\mathscr{D}_D(V)\rightarrow \mathscr{D}_A(M)$ and $\psi :\mathscr{D}_A(M)\rightarrow \mathscr{D}_D(V)$. They are clearly increasing.

Let N be a submodule of M. It is isotypical of type S (VIII, p. 61, Proposition 2). Set $W =\psi (N)$. By Proposition 3, b) of VIII, p. 62, the A-linear mappings $h: S\rightarrow M$ are simply the mappings $s\mapsto \alpha (s\otimes v)$, where $v$ runs through V. Those with image contained in N are the mappings $s\mapsto \alpha (s\otimes w)$ where $w$ runs through W; their images generate N because N is isotypical of type S. We therefore have $\alpha (S\otimes_DW) = N$, that is, $\varphi (\psi (N)) = N$. This proves that $\varphi \circ \psi$ is the identity mapping on $\mathscr{D}_A(M)$. In particular, $\varphi$ is surjective and $\psi$ is injective.

To finish the proof, it suffices to prove that the mapping $\varphi$ is injective. Let W and $W'$ be linear subspaces of V such that $\varphi (W) =\varphi (W')$. The vector spaces $S\otimes_DW$ and $S\otimes_DW'$ coincide when viewed as linear subspaces of $S\otimes_DV$. Choose a nonzero linear form $f$ on the D-vector space S, and let $g: S\otimes_DV\rightarrow V$ be the group homomorphism defined by $g(s\otimes v) =f(s)v$. We have $W =g(S\otimes_DW) =g(S\otimes_DW') = W'$, so $\varphi$ is injective.

a) The mapping $N\mapsto$ Hom$_A(S,N)$ is an isomorphism of ordered sets from $\mathscr{D}_A(M)$ to $\mathscr{D}_D$(Hom$_A(S,M)$).

b) The inverse bijection sends a linear subspace W of Hom$_A(S,M)$ to the submodule $\sum_{h\in W}h(S)$ of M.

This is a reformulation of Theorem 2 when we take $(V, \alpha )$ to be the canonical description of M.

Indeed, by Theorem 2, every A-submodule N of $S\otimes_DV$ is equal to $S\otimes_DW$, where W is the linear subspace of V consisting of the elements $v$ such that $s\otimes v$ belongs to N for every $s\in S$.

$S\otimes_DV^{\alpha}$ // M

$S\otimes_D{V'}^{\alpha'}$ // $M'$.

The mapping $f\mapsto \widetilde{f}$ from Hom$_D(V,V')$ to Hom$_A(M,M')$ is a group isomorphism.

It suffices to prove that the $\mathbf{Z}$-linear mapping $u\mapsto 1_S\otimes u$ from Hom$_D(V,V')$ to Hom$_A(S\otimes_DV,S\otimes_DV')$ is bijective. In the notation of No.3 applied to the $(A$, D)-bimodule S, this corresponds to showing that the mapping

$\mathscr{T}:$ Hom$_D(V,V')\longrightarrow$ Hom$_A(\mathscr{T}(V),\mathscr{T}(V'))$

is bijective. But by Remark 1 of VIII, p. 60, since the adjunction isomorphism (VIII, p. 59) is bijective, this corresponds to showing that the mapping that sends $u$ to $\beta_{V'}\circ u$ is bijective, which follows from the fact that the D-linear mapping $\beta_{V'}$ is bijective (VIII, p. 62, Proposition 3, b)).

We keep the notation of Theorem 3. Let $M''$ be an isotypical A-module of type S, and let $(V'', \alpha '')$ be a description of $M''$. For every $f\in$ Hom$_D(V,V')$ and every $g\in$ Hom$_D(V',V'')$, we have $g]\circ f=\widetilde{g}\circ \widetilde{f}$. In particular, for $M = M'$, $V = V'$, and $\alpha =\alpha '$, the mapping $f\mapsto \widetilde{f}$ from End$_D(V)$ to End$_A(M)$ is a ring isomorphism.

Because of the commutativity of diagram (I) of VIII, p. 59, this corollary follows from Theorem 3, applied to the canonical descriptions of M and $M'$.

a) The module M is the direct sum of the family $(M_{\lambda})_{\lambda\in\mathscr{S}}$ of its isotypical components.

b) Suppose that M is the direct sum of a family $(N_i)_{i\in I}$ of simple submodules. For every $\lambda \in \mathscr{S}$, let $I(\lambda )$ be the set of indices $i\in I$ such that $N_i$ is of class $\lambda$. We have $M_{\lambda}=\bigoplus_{i\in I(\lambda)}N_i$.

c) If M is finitely generated, then the set of $\lambda \in \mathscr{S}$ such that $M_{\lambda}\not= 0$ is finite.

d) For every submodule N of M and every $\lambda \in \mathscr{S}$, we have $N_{\lambda}= N\cap M_{\lambda}$ and $(M/N)_{\lambda}= (M_{\lambda}+ N)/N$.

Since M is semisimple, it is the sum of the family $(M_{\lambda})_{\lambda\in\mathscr{S}}$; let us prove that this sum is direct. Let $\lambda \in \mathscr{S}$. Denote the sum of the family $(M_\mu)_{\mu\in\mathscr{S}-\{\lambda\}}$ by $M'_{\lambda}$. The module $M'_{\lambda}$ is the direct sum of a family of simple modules not isomorphic to $\lambda$ (VIII, p. 56, Theorem 1). By Corollary 3 of Theorem 1 of VIII, p. 56, $M'_{\lambda}$ does not contain any simple submodules of class $\lambda$. We consequently have $M_{\lambda}\cap M'_{\lambda}= 0$. Assertion a) is therefore proved. By construction, we have $M_{\lambda}\supset \bigoplus_{i\in I(\lambda)}N_i$, so assertion b) follows from Remark 1 of II, §1, No. 8, p. 208.

Assertion c) follows from a) and Proposition 23 of II, §1, No. 12, p. 221.

Let N be a submodule of M and $\lambda \in \mathscr{S}$. The isotypical component $N_{\lambda}$ of type $\lambda$ of N is contained in $M_{\lambda}$ and $M_{\lambda}\cap N\subset N_{\lambda}$. The intersection $N\cap M_{\lambda}$ is therefore the isotypical component of N of type $\lambda$.

For every $\lambda \in \mathscr{S}$, the module $M_{\lambda}+N/N$ is isomorphic to $M_{\lambda}/(M_{\lambda}\cap N)$. It is therefore isotypical of type $\lambda$ and contained in $(M/N)_{\lambda}$. The last assertion then follows from a) and II, §1, No. 8, p. 208, Remark 1.

Since N and $M/N$ are semisimple (VIII, p. 56, Corollary 3), the corollary follows from Proposition 4, d).

The support of a semisimple A-module M is the set of classes $\lambda$ of simple A-modules such that the isotypical component of M of type $\lambda$ is nonzero. The support of a finitely generated semisimple A-module is finite.

a) Let $f: M\rightarrow N$ be a homomorphism. For every $\lambda \in \mathscr{S},f$ induces a homomorphism $f_{\lambda}$ from $M_{\lambda}$ to $N_{\lambda}$; if M is semisimple and $f$ surjective, then each of the homomorphisms $f_{\lambda}$ is surjective.

b) Suppose that M is semisimple. The mapping $f\mapsto (f_{\lambda})_{\lambda\in\mathscr{S}}$ is a group isomorphism from Hom$_A(M,N)$ to $\prod_{\lambda\in\mathscr{S}}$ Hom$_A(M_{\lambda},N_{\lambda})$. When M is equal to N, the mapping is a ring isomorphism from End$_A(M)$ to $\prod_{\lambda\in\mathscr{S}}$ End$_A(M_{\lambda})$.

For every $\lambda \in \mathscr{S}$, the submodule $f(M_{\lambda})$ of N is isomorphic to a quotient of an isotypical module of type $\lambda$; it is therefore isotypical of type $\lambda$ and, consequently, contained in $N_{\lambda}$.

Suppose that M is semisimple and $f$ surjective. Then $f$ induces an isomorphism from $M/$ Ker($f$) to N that sends $(M_{\lambda}$+Ker($f$))$/$ Ker($f$) to $f(M_{\lambda})$. By Proposition 4 of VIII, p. 65, we have $N_{\lambda}=f(M_{\lambda})$, which completes the proof of a).

The mapping considered in b) is clearly a group homomorphism, and it is a ring homomorphism when M is equal to N. Let $(f_{\lambda})_{\lambda\in\mathscr{S}}$ be an element of $\prod_{\lambda\in\mathscr{S}}$ Hom(M$_{\lambda},N_{\lambda}$). The unique element of its inverse image under the mapping in b) is the homomorphism $f: M\rightarrow N$ defined by

for every $(x_{\lambda})_{\lambda\in\mathscr{S}}\in \bigoplus_{\lambda}M_{\lambda}$.

(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).

(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$_A(S,M)$.

Let $u\in$ Hom$_A(M,S),v\in$ Hom$_A(S,M)$, and $d\in D$. We have

This proves that the mapping $h$ is D-linear. It is simply the mapping given by $u\mapsto$ Hom(1$_S, u$) from Hom$_A(M,S)$ to Hom$_D$(Hom$_A(S,M)$, Hom$_A(S,S)$). To prove that it is bijective, by Proposition 5, b) of VIII, p. 66, it suffices to treat the case when M is isotypical of type S; we can then apply the corollary of VIII, p. 65.