Remark

(1) Let F be an A-module and C the centre of the ring A; as central homotheties commute with all homotheties, F has a bimodule structure whose rings of left operators are A and C. If E is another A-module, $\operatorname{Hom}_A(E, F)$ has therefore a canonical C-*module* structure (where, for $f \in \operatorname{Hom}_A(E, F)$ and $\gamma \in C$, $\gamma f$ is the homomorphism $x \mapsto \gamma f(x)$); if E', F' are two A-modules, $u : E' \to E, v : F \to F'$ two A-homomorphisms, the mapping $\operatorname{Hom}(u, v)$ is C-*linear*.

(2) Let E be a left A-module; as A has a canonical (A, A)-bimodule structure, so has the direct sum $A^{(T)}$ for any indexing set T; by the above, $\mathrm{Hom}_A(A_s^{(T)}, E)$ has a canonical left A-module structure arising from the right A-module structure on $A_s^{(T)}$: for $f \in \mathrm{Hom}_A(A_s^{(T)}, E)$ and $\alpha \in A$, $\alpha f$ is the linear mapping $x \mapsto f(x\alpha)$. Corollary 2 to Proposition 17 of no. 11 defines a canonical mapping $j_{E, T}$ from the product module $E^T$ to $\mathrm{Hom}_A(A_s^{(T)}, E)$, the image under $j_{E, T}$ of a family $(x_t)_{t \in T}$ being the linear mapping $f : A_s^{(T)} \to E$ such that $f(e_t) = x_t$ for all $t \in T$ (where $(e_t)$ is the canonical basis of $A_s^{(T)}$); it is known (loc. cit.) that $j_{E, T}$ is bijective and it follows from the definition given above of the A-module structure on $\mathrm{Hom}_A(A_s^{(T)}, E)$ that $j_{E, T}$ is A-linear. Finally, if $u : E \to F$ is an A-module homomorphism, the diagram

is commutative.

Note that when T consists of a single element, $j_E : E \to \mathrm{Hom}_A(A_s, E)$ is just the mapping $x \mapsto \theta_x$ defined in no. 2, Example 1.