On the Borel complexity of modules

We prove that among all countable, commutative rings R (with unit) the theory of R-modules is not Borel complete if and only if there are only countably many non-isomorphic countable R-modules. From the proof, we obtain a succinct proof that the class of torsion free abelian groups is Borel complete.
The results above follow from some general machinery that we expect to have applications in other algebraic settings. Here, we also show that for an arbitrary countable ring R, the class of left R-modules equipped with an endomorphism is Borel complete; as is the class of left R-modules equipped with predicates for four submodules. This is joint work with D. Ulrich.