I'm looking for an example of of two isomorphic abelian groups, which are not isomorphic $R$-modules for some ring $R$.
I suppose we can just the same abelian group $M$ twice, and use a different operation $R\times M \rightarrow M$ so the $R$ modules aren't isomorphic. I can't think of such a group $M$ and ring $R$ to make this possible, though. Any ideas? Thanks.