当R不满足pp-可定义子群的DCC时，R-模的Borel完备性
Borel completeness of $R$-modules when $R$ fails the DCC on pp-definable subgroups
浏览论文内容
中文总结 AI 辅助
该研究证明了非左完全可数环的R-模理论是Borel完备的，刻画了可数单环的Borel完备性，引入了相关理想与包络构造，强化了TFAB的Borel完备性结论。
中文摘要 AI 辅助
我们证明：对于任意可数环R（不一定交换），若其对应的左R-模{}_R R存在严格递减的pp-可定义子群序列，则无限维直和的理论Th(R^(ω))是Borel完备的。由此我们得出：若R是可数且非左完全的，则R-模的理论是Borel完备的，并且我们给出了哪些可数单环具有Borel完备理论的完整刻画。一个特殊情况是，完全理论Th(ℤ^(ω))是Borel完备的，这强化了关于无挠阿贝尔群理论TFAB的Borel完备性的现有证明。该证明还相对于所选的pp-链引入了一个合适的双边理想L^R，以及有限生成（f.g.）包络的概念，对于可数环和满足T=T^ℵ₀的理论中的可数参数集，这些包络存在且在同构意义下唯一。这些构造在模的模型论中可能具有独立意义。
英文摘要
We prove that for any countable ring $R$ (not necessarily commutative), if the associated left $R$-module ${}_R R$ has a strictly descending sequence of pp-definable subgroups, then the theory $Th(R^{(ω)})$ of the infinite dimensional direct sum is Borel complete. From this, we conclude that if $R$ is countable and not left perfect, then the theory of $R$-modules is Borel complete, and we give a full characterization of which countable simple rings have Borel complete theories. One special case is that the complete theory $Th({\mathbb Z}^{(ω)})$ is Borel complete, which strengthens the existing proofs of the Borel completeness of TFAB, the theory of torsion free abelian groups. The proof also introduces, relative to the chosen pp-chain, a proper two-sided ideal $L^R$, and a notion of f.g. hulls which, for countable rings and countable parameter sets in theories satisfying $T=T^{\aleph_0}$ exist and are unique up to isomorphism. These constructions may be of independent interest in the model theory of modules.
