It is well known that, given a commutative ring $R$ and an $R$-module $M$, the dual module $M^\vee = \operatorname{Hom}_R(M, R)$ does not always satisfy $M^\vee \cong M \ (1)$, and not even $M^{\vee \vee} \cong M$ (the latter is satisfied, for example, when $M$ is finitely generated projective). My question is, are there any known conditions for $(1)$ to be satisfied? I am primarily investigating finitely generated $M$ over a commutative finite principal ideal ring (not necessarily local or a domain), but would welcome a more general result, if it exists. In the case that conditions for an isomorphism are nonexistent or too deep, is there a known way to at least compare the lengths of $M^\vee$ and $M$ as $R$-modules? In the case that mainly interests me (finitely generated over a finite PIR), I am thinking that they should have equal length, but could not find a proper proof.

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