Polish 模的 Kaplansky 分解
Kaplansky decompositions of Polish modules
- The Ohio State University at Lima（俄亥俄州立大学利马分校）
- University of Torino（都灵大学）
- The Hebrew University of Jerusalem（耶路撒冷希伯来大学）
- Rutgers University（罗格斯大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文刻画了 admits Kaplansky 分解的不可数 Polish 模，并利用该刻画推广了 Shelah 和 Solecki 的结果，给出了 admits 自由不可数 Polish 模的可数环的纯环论描述。
中文摘要 AI 辅助
设 $R$ 为可数环。给定一个 $R$-模 $A$，若每个 $N_i$ 都是可数的，则称分解 $A = \bigoplus_{i \in I} N_i$ 为 Kaplansky 分解。我们刻画了 admits 一个 Kaplansky 分解的不可数 Polish $R$-模：它们恰好是形如 $B \oplus M^{\omega}$ 的模，其中 $B$ 和 $M$ 是可数的，且 $M$ 是 $\Sigma$-代数紧的。可数直和项 $B$ 还可取为满足闭包条件的初等子模，这使得 $M^{\omega}$ 在同构意义下唯一，从而成为 $A$ 的一个不变量。我们利用这一点来刻画 admits 一个自由不可数 Polish $R$-模的可数环，推广了 Shelah 和 Solecki 的结果。这类环具有纯环论描述：它恰好由可数左完全且右凝聚的环组成，即 Chase 关于投射模乘积的定理所识别的环。我们观察到这些也正是可数 $F$-环，即那些使得 $R^{\omega}$ 为自由模的可数环 $R$。最后，我们给出了存在不可数投射 Polish $R$-模的可数环 $R$ 的环论刻画。
英文摘要
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.
