S. Shelah

Kaplansky decompositions of Polish modules

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 $B \oplus M^{\omega}$, where...