= Dual module
{title2=$M^*$}
{wiki}
For a left <module> $M$ over a <ring> $R$, its <dual module> is $M^*=\operatorname{Hom}_R(M,R)$. It is naturally a <right module>, with
$$
(f\cdot r)(m)=f(m)r.
$$
Right multiplication preserves left linearity: $f(am)r=af(m)r$. If $R$ is commutative, this can equally be regarded as a left $R$-module, with the usual pointwise scalar action.
For a finite <free module> with a specified basis, its coordinate functionals form a dual basis. Over $\mathbb Z$, the dual of $\mathbb Z/n\mathbb Z$ is zero for $n>1$, since an additive map from a finite group to the torsion-free group $\mathbb Z$ is zero. Duality therefore need not recover every module.
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