Borel complexity of modules

TL;DR

This paper investigates the complexity of classifying countable modules over countable rings, showing a dichotomy where the classification problem is either very simple or as complex as possible (Borel complete).

Contribution

It establishes a dichotomy for the Borel complexity of countable modules over countable rings and provides new proofs of Borel completeness for related classes.

Findings

Class of countable R-modules is either countably many types or Borel complete.

Proves Borel completeness of torsion-free abelian groups (TFAB).

Shows classes of modules with endomorphisms or submodules are Borel complete.

Abstract

We prove that for a countable, commutative ring $R$, the class of countable $R$-modules either has only countably many isomorphism types, or else it is Borel complete. The machinery gives a succinct proof of the Borel completeness of TFAB, the class of torsion-free abelian groups. We also prove that for any countable ring $R$, both the class of left $R$-modules endowed with an endomorphism and the class of left $R$-modules with four named submodules are Borel complete.