发表机构
School of Mathematics and Statistics, Tianshui Normal University; School of Mathematical Sciences, Sichuan Normal University（天水师范学院数学与统计学院; 四川师范大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明 Baer 模均投射的环不一定是 $\tau_q$-半单环，通过非约化 Noether 环和约化凝聚 Bézout 环两类反例，推翻了原有猜想。
AI 中文摘要
对于交换环 $R$，若对挠 $R$-模 $T$ 有 $\Ext_R^1(B,T)=0$，则称 $R$-模 $B$ 为 Baer 模。已知当 $R$ 是 $\tau_q$-半单环（即其全商环是半单的环）时，每个 Baer 模都是投射的。曾有猜想认为其逆命题刻画了 $\tau_q$-半单环。我们证明该逆命题以两种本质上不同的方式失效。首先，在非约化 Noether 情形中，这种失效已是系统性的。若 $D$ 是非域的 Dedekind 整环且 $R=D[\varepsilon]/(\varepsilon^2)$，则每个 Baer $R$-模都是投射的，尽管 $R$ 不是约化的，因此不是 $\tau_q$-半单的。其次，即使在约化环中，事实上在约化凝聚 Bézout 环中，该逆命题也失效。主要工具是一个可数环判据。对于可数 Noether 整环 $D$，令 $\EC(D)$ 为 $D$ 上最终常值序列构成的环。若 $D$ 不是域，则每个 Baer $\EC(D)$-模都是投射的，而 $\EC(D)$ 是约化的且具有可数无穷多个极小素理想，因此不是 $\tau_q$-半单的。
英文摘要
For a commutative ring $R$, we call an $R$-module $B$ Baer if $\Ext_R^1(B,T)=0$ for torsion $R$-module $T$. It is known that every Baer module is projective when $R$ is $τ_q$-semisimple, i.e., rings whose total rings of quotients are semisimple. And it was conjectured that the converse characterizes $τ_q$-semisimple rings. We show that the converse fails in two substantially different ways. First, the failure is already systematic in the non-reduced Noetherian case. If $D$ is a Dedekind domain which is not a field and $R=D[\varepsilon]/(\varepsilon^2)$, then every Baer $R$-module is projective, although $R$ is not reduced, and thus is not $τ_q$-semisimple. Second, the converse fails even among reduced rings, and in fact among reduced coherent Bézout rings. The main tool is a countable-ring criterion. For a countable Noetherian domain $D$, let $\EC(D)$ be the ring of eventually constant sequences over $D$. If $D$ is not a field, then every Baer $\EC(D)$-module is projective, whereas $\EC(D)$ is reduced and has countably infinitely many minimal prime ideals, and thus is not $τ_q$-semisimple.
