Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
add comment

1 Answer 1

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.

share|improve this answer
1  
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
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.