将环表示为阿贝尔群的自同态环
Representing a ring as the endomorphism ring of an abelian group
查看机构详情
- School of Mathematics, Institute for Research in Fundamental Sciences (IPM)（基础科学研究院数学学院）
- Einstein Institute of Mathematics, The Hebrew University of Jerusalem（耶路撒冷希伯来大学爱因斯坦数学研究所）
- Department of Mathematics, Rutgers University（罗格斯大学数学系）
机构由 AI 辅助整理，请以论文原文为准。
浏览论文内容
中文总结 AI 辅助
本文在ZFC中通过结合κ-框架与Shelah超黑箱，构造了自同态环同构于给定环的μ^+-自由模，部分解决了Göbel和Trlifaj关于环表示的问题。
中文摘要 AI 辅助
我们研究了几乎自由阿贝尔群的Baer实现问题，重点关注在强自由性条件下环可被表示为自同态环的程度。在早期依赖于额外集合论原理（如菱形原理或强黑箱）的工作基础上，我们开发了新方法，显著削弱了这些假设。主要结果在ZFC（假设一个温和的基数算术配置）中获得：对于强极限奇异基数μ，满足μ^+ < 2^μ < 2^{μ^+}，并且对于一大类无挠自由环，我们构造了μ^+-自由模，其自同态环同构于给定环。这为Göbel和Trlifaj的问题提供了一个实质性的部分解。关键创新在于将κ-框架构造与Shelah的超黑箱相结合，实现了精细的对角化，从而在保持高度自由性的同时消除了非平凡自同态。
英文摘要
We investigate Baer's realization problem for almost free abelian groups, focusing on the extent to which rings can be represented as endomorphism rings under strong freeness conditions. Building on earlier work that relied on additional set-theoretic principles such as the diamond or strong black boxes, we develop new methods that significantly weaken these assumptions. The main result is obtained in ZFC (assuming a mild cardinal arithmetic configuration): for a strong limit singular cardinal $μ$ with $μ^+ < 2^μ< 2^{μ^+}$, and for a wide class of cotorsion-free rings, we construct $μ^+$-free modules whose endomorphism rings are isomorphic to the given ring. This provides a substantial partial solution to a problem of Göbel and Trlifaj. The key innovation is the integration of $κ$-frame constructions with Shelah's Super Black Box, enabling a delicate diagonalization that eliminates nontrivial endomorphisms while preserving high degrees of freeness.
