Definition 35.4.9. Let $R$ be a ring. Define the contravariant functor $C$ $: \text{Mod}_ R \to \text{Mod}_ R$ by setting

$C(M) = \mathop{\mathrm{Hom}}\nolimits _{\textit{Ab}}(M, \mathbf{Q}/\mathbf{Z}),$

with the $R$-action on $C(M)$ given by $rf(s) = f(rs)$.

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