Polish modules over subrings of $\mathbb{Q}$

Abstract

We 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 $\mathbb{R}$ 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. In fact, our method produces a large number of incomparable with respect to embeddings Polish $\mathbb{Q}$-vector spaces.