A module $M$ over a general ring R is called reflexive in case the canonical map $e_M: M \rightarrow M^{**}$ is an isomorphism, when $(-)^{*}:=Hom_A(-,A)$. Is $M$ reflexive iff $M \cong M^{**}$ for some isomorphism?
I am interested in examples or proofs for general $M$ but also the cases when $M$ is finitely generated or finitely presented.
For a positive answer in special cases see https://mathoverflow.net/questions/288878/characterisation-of-reflexive-modules .