A module $M$ over a general ring A is called reflexive in case the canonical evaluation 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 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 Characterisation of reflexive modules .

  • $\begingroup$ I would expect that this is something which should have been thought about in the literature already so maybe there is an article about such things? $\endgroup$ – Mare Dec 20 '17 at 13:10

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