A dichotomy for Polish modules

TL;DR

This paper investigates the structure of Polish modules over rings with a proper norm, establishing a basis for non-countably generated modules and simplifying to a singleton basis when the ring is a division ring.

Contribution

It introduces a natural basis for Polish modules over certain rings, revealing a dichotomy based on the ring's properties and extending understanding of module classification.

Findings

Existence of a natural basis for non-countably generated Polish modules

Simplification to a singleton basis when the ring is a division ring

Conditions on the ring that enable this basis construction

Abstract

Let $R$ be a ring equipped with a proper norm. We show that under suitable conditions on $R$, there is a natural basis under continuous linear injection for the set of Polish $R$-modules which are not countably generated. When $R$ is a division ring, this basis can be taken to be a singleton.