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 counterexamples or proofs for finitely generated or finitely presented $M$. See https://mathoverflow.net/questions/76000/are-there-non-reflexive-modules-isomorphic-to-their-bi-dual for a counter example for infinitely generated $M$.
For a positive answer in special cases see https://mathoverflow.net/questions/288878/characterisation-of-reflexive-modules .