Nielsen-Soelberg 群环的整环问题：交换情形，附认证球检验
The domain question for the Nielsen-Soelberg group rings: the commutative case, with certified ball checks
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中文总结 AI 辅助
本文证明对任意交换整环 R，Nielsen-Soelberg 三个群环 R[G_i] 均为整环，利用虚拟幂零性与 Kropholler-Linnell-Moody 定理，并补充 DRAT 认证的球内无零因子检验，将开放问题限定于非交换系数情形。
中文摘要 AI 辅助
Nielsen 和 Soelberg 构造了三个无挠群 G_1、G_2、G_3，它们带有不含唯一乘积的 8 元集合，并提问：对于任意整环 R，群环 R[G_i] 是否为整环。我们证明，对于每个交换整环 R，这三个群的答案都是肯定的：每个 G_i 都是虚拟幂零的，因此 Kropholler、Linnell 和 Moody 的定理适用于任意域、任意特征，而交换系数可归结为分式域。除 G_2 的相关有限指数子群的幂零结构外，所有要素均见于文献，该结构由我们姊妹论文（arXiv:2607.19687）的认证计算模型提供。因此，该问题仅在非交换系数整环情形下保持开放，而唯一乘积机制——唯一已知的与环无关的机制——正是这些群被构造为所缺乏的。作为补充，我们报告了机器可检查、DRAT 认证的验证结果：F_2[G_i] 在定义生成集的显式球内，两个支撑集均无零因子，该结果仅通过命题推理获得，独立于 K-理论机制；我们精确陈述了这些证书的贡献与局限。
英文摘要
Nielsen and Soelberg exhibited three torsion-free groups G_1, G_2, G_3 carrying 8-element sets without unique products, and asked whether any of the group rings R[G_i], R a domain, is a domain. We record that for every commutative domain R the answer is affirmative for all three groups: each G_i is virtually nilpotent, so the theorem of Kropholler, Linnell and Moody applies over every field, in every characteristic, and commutative coefficients reduce to the fraction field. Every ingredient is in the literature except the nilpotent structure of the relevant finite-index subgroup of G_2, which is supplied by the certified computational model of our companion paper (arXiv:2607.19687). The question therefore remains open exactly for noncommutative coefficient domains, where the unique-product mechanism -- the only known ring-independent one -- is precisely what these groups are constructed to lack. As a complement we report machine-checkable, DRAT-certified verifications that F_2[G_i] has no zero divisors with both supports in explicit balls of the defining generating sets, obtained by propositional reasoning alone, independent of the K-theoretic machinery; we state precisely what these certificates do and do not add.
