Exercise 28, § 7

Let $\rho : K \to A$ be an injective homomorphism of a field $K$ into a ring $A$ and let $E$ be a left vector space over $K$. For the canonical mapping $(E^*)_{(A)} \to (E_{(A)})^*$ to be bijective, it is necessary and sufficient that $E$ be finite-dimensional or that $A$, considered as a right vector $K$-space, be finite-dimensional (use Exercise 25 (b)). When one of these two cases holds, there exists a canonical bijection $(E^{**})_{(A)} \to (E_{(A)})^{**}$.