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I'd like to find a ring $A$ and an $A$-module $E$ such that $E^*\neq\{0\}$ and the canonical mapping of $E$ into $E^{**}$ is not injective.

As a hint (it's an exercise in Bourbaki's Algebra) I have "consider a module containing an element whose annihilator contains an element which is not a divisor of zero".

I'm not sure how the hint is supposed to help: there are $\mathbf{Z}$-modules with the property from the hint ($\mathbf{Z}/n\mathbf{Z}$, $\mathbf{Q}/\mathbf{Z}$), but their dual is zero. Unfortunately, my repertoire of interesting modules is very limited.

Can somebody help me along? Another hint or an example without proof would be fine.

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up vote 4 down vote accepted

Recall that forming duals is additive (i.e. is compatible with taking direct sums). So try combining an example of a module with a non-zero dual and a module with zero dual via a direct sum.

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Ah, so the dual of the $\mathbf{Z}$-module $\mathbf{Q}\oplus\mathbf{Z}$ is isomorphic to $\mathbf{Z}$ and $(1,0)$ lies in the kernel of the canonical mapping. Thank you very much! – Stefan Walter Aug 19 '12 at 13:05

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