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Is there infinite generated reflexive module?

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Yes, over the ring of integers. The free abelian group on countably many generators clearly has, as its dual, the direct product of countably many copies of $\mathbb Z$. The dual of the latter is, by a theorem of Specker (1950) [maybe already in a paper of Baer, 1937] again the free abelian group we started with, and the canonical embedding is an isomorphism.

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Thank you very much! So over an Artin ring all reflexive modules are finitely generated?? – wangzp Apr 5 '13 at 11:51

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