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I know that the definition of reflexive module is that the $R$-module $M$ should be isomomorphic to its double dual $M^{**}$ via the canonical map $M\rightarrow M^{**}$.

I'd like to know an example of an $R$-module $M$ such that it's isomorphic to $M^{**}$ but the canonical map $M\rightarrow M^{**}$ is not an isomorphism. Do you have such an example?

(I know that for Banach spaces such example exists, but I don't know it.)

(I put also the tag banach-spaces, maybe it's helpful to know the example for Banach spaces.)

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    For Banach spaces, one example is James's Construction: See http://mathoverflow.net/questions/43986/are-there-non-reflexive-vector-spaces-isomorphic-to-their-bi-dual2011-09-20

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