Borel completeness of $R$-modules when $R$ fails the DCC on pp-definable subgroups

Abstract

We prove that for any countable ring $R$ (not necessarily commutative), if the associated left $R$-module ${}_R R$ has a strictly descending sequence of pp-definable subgroups, then the theory $Th(R^{(ω)})$ of the infinite dimensional direct sum is Borel complete. From this, we conclude that if $R$ is countable and not left perfect, then the theory of $R$-modules is Borel complete, and we give a full characterization of which countable simple rings have Borel complete theories. One special case is that the complete theory $Th({\mathbb Z}^{(ω)})$ is Borel complete, which strengthens the existing proofs of the Borel completeness of TFAB, the theory of torsion free abelian groups. The proof also introduces, relative to the chosen pp-chain, a proper two-sided ideal $L^R$, and a notion of f.g. hulls which, for countable rings and countable parameter sets in theories satisfying $T=T^{\aleph_0}$ exist and are unique up to isomorphism. These constructions may be of independent interest in the model theory of modules.