We will give a method of producing a Polish module over an arbitrary subring of $\mathbb Q$ from an ideal of subsets of $\mathbb N$ and a sequence in $\mathbb N$. The method allows us to construct two Polish $\mathbb Q$-vector spaces, $U$ and $V$, such that
-- both $U$ and $V$ embed into $\mathbb R$, with $\mathbb R$ being considered a Polish $\mathbb Q$-vector space with its standard topology, but
-- $U$ does not embed into $V$ and $V$ does not embed into $U$,
where by an embedding we understand a continuous $\mathbb Q$-linear injection. This construction answers a question of Frisch and Shinko.
This is joint work with Slawek Solecki.