Kaplansky decompositions of Polish modules

Abstract

Let $R$ be a countable ring. Given an $R$-module $A$, we call a decomposition $A = \bigoplus_{i \in I} N_i$ a Kaplansky decomposition if each $N_i$ is countable. We characterize the uncountable Polish $R$-modules that admit a Kaplansky decomposition: they are exactly the modules of the form $B \oplus M^ω$, where $B$ and $M$ are countable and $M$ is $Σ$-algebraically compact. The countable summand $B$ may moreover be taken to be an elementary submodule satisfying a closure condition, which makes $M^ω$ unique up to isomorphism, and hence an invariant of $A$. We use this to characterize the countable rings admitting a free uncountable Polish $R$-module, generalizing results of Shelah and Solecki. This class of rings has a purely ring-theoretic description: it consists exactly of the countable left perfect and right coherent rings, i.e. the rings identified by Chase's theorem on products of projective modules. We observe that these are also exactly the countable $F$-rings, i.e. those countable rings $R$ for which $R^ω$ is free. Finally, we give a ring-theoretic characterization of the countable rings $R$ for which there exists an uncountable projective Polish $R$-module.