Submission date: 14 September 2022

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.

Mathematics Subject Classification: 03E15 13C05

Keywords and phrases:

Full text arXiv 2209.06898: pdf, ps.