Proposition 5

For every ring A, the mapping which associates with every $\xi \in A$ the linear form $\eta \mapsto \eta \xi$ on $A_s$ is an isomorphism of $A_d$ onto the dual of $A_s$.

It is the particular case of the canonical isomorphism $E \to \mathrm{Hom}_A(A_s, E)$ of § 1, no. 14, Remark 2, corresponding to $E = A_s$; the commutativity of diagram (50) of § 1, no. 14, shows that we have here an isomorphism of right A-modules.

If $A_d$ is identified with the dual of $A_s$ by means of the isomorphism of Proposition 5, the canonical bilinear form on $A_s \times A_d$ is then expressed by

Similarly, the dual of $A_d$ is canonically identified with $A_s$, the canonical bilinear form on $A_d \times A_s$ being expressed by