Definition 15.23.1. Let $R$ be a domain. We say an $R$-module $M$ is *reflexive* if the natural map

\[ j : M \longrightarrow \mathop{\mathrm{Hom}}\nolimits _ R(\mathop{\mathrm{Hom}}\nolimits _ R(M, R), R) \]

which sends $m \in M$ to the map sending $\varphi \in \mathop{\mathrm{Hom}}\nolimits _ R(M, R)$ to $\varphi (m) \in R$ is an isomorphism.

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