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.