Double dual module
= Double dual module
{title2=$M^{**}$}
The <double dual module> is obtained by dualizing the <dual module> again. For a left <module> $M$ over a possibly noncommutative <ring> $R$, the correct sided formula is
$$
M^{**}=\operatorname{Hom}_{R^{\mathrm{op}}}\bigl(\operatorname{Hom}_R(M,R),R\bigr).
$$
It is again a left module. The <evaluation homomorphism> sends $m$ to the functional $f\mapsto f(m)$; right linearity of this functional follows from $(f\cdot r)(m)=f(m)r$.
Over a <commutative ring>, both Hom operations can be written with subscript $R$. Evaluation is an isomorphism for <finite projective modules>, while a <reflexive module> is defined by requiring it to be an isomorphism. It need not even be injective in general: the double dual of $\mathbb Z/n\mathbb Z$ is zero when $n>1$.