Borel complexity of modules

2022/09/14 by Laskowski, Michael C., Ulrich, Danielle S.
#03E15 13C05 #Commutative Algebra (math.AC) #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2209.06898

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.

Related

• The classification problem for extensions of torsion-free abelian groups, I
• Torsion-Free Abelian Groups are Borel Complete
• Complexity test modules
• Kaplansky Test Problems for R-Modules in ZFC
• A proof of the Borel completeness of torsion free abelian groups