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Remark 110.34.3. The example above where $R = \prod _{n \in \mathbf{N}} \mathbf{F}_2$ and $I$ the ideal generated by the idempotents $e_ n = (1, 1, \ldots , 1, 0, \ldots )$ also gives us an example $M = I$ of a non-finitely generated projective module whose dual is finitely generated and even finite free. Namely, using the strategy in the proof of Lemma 110.34.1 the reader shows that $\mathop{\mathrm{Hom}}\nolimits _ R(I, R) \cong R$.

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