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 Characterisation of reflexive modules .