Proposition 3

a) Let M be an isotypical A-module of type S. The A-linear mapping $\alpha_M: S\otimes_D$Hom$_A(S,M)\rightarrow M$ characterized by $\alpha_M(s\otimes f) =f(s)$ (VIII, p. 59) is bijective.

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).