Does reflexivity of module $M^{**}$ implies that $M$ is reflexive as well?

Question

We have module $M$ over commutative ring $R$ and we know that its double dual $M^{**}$ is reflexive.

Is $M$ reflexive?

PS. Reflexivity of some module $P$ means that the natural map $P\to P^{**}$ is an isomorphism. (Yep assumption is such that $M^{**}\cong M^{****}$)

It may look out of context, but I came across such question when working with module of sections of some vector bundle.

Answer 1

Not necessarily. For instance, if $R=\mathbb{Z}$ and $M=\mathbb{Z}/2$, then $M^{**}=0$ is reflexive but $M$ is not.