论文标题
一些稳定的非元素模块类别
Some stable non-elementary classes of modules
论文作者
论文摘要
Fisher [FIS75]和Baur [Bau75]在七十年代独立地表明，如果$ t $是一种完整的一阶理论，扩展了模块的理论，那么$ t $的$ t $类别具有纯嵌入的模型是稳定的。在[maz4，2.12]中，询问对于任何抽象的基本类$（k，\ leq_p）$是否相同，使得$ k $是一类模块，而$ \ leq_p $是纯的subpodule关系。在本文中，我们给出了这是真的的一些实例： $ \ textbf {theorem。} $假设$ r $是与统一的关联戒指。令$（k，\ leq_p）$为AEC，以至于$ k \ subseteq r \ text {-mod} $，$ k $在有限直接总和下关闭，然后： - 如果$ k $在纯净的信封下关闭，则$（k，\ leq_p）$为$λ$ - 稳定性，每$λ\ geq ls（k）$，这样$λ^{| r | + \ \ aleph_0} =λ$。 - 如果$ k $在纯的子模块和纯表现图像下关闭，则$（k，\ leq_p）$是$λ$ - 稳定的每个$λ$，因此$λ^{| r | + \ \ aleph_0} =λ$。 - 假设$ r $是von Neumann常规。如果$ k $在子模型下关闭并且具有任意大型型号，则$（k，\ leq_p）$为$λ$ - 稳定每个$λ$，以便$λ^{| r | + \ \ aleph_0} =λ$。 作为这些结果的应用，我们通过超惊人性提供了Noetherian环，纯净的隔离环，Dedekind域和田地的新特征。此外，我们展示了如何使用这些结果来显示稳定性层次结构的良好与良好性层次结构之间的联系。 另一个应用是在几类模块中相对于纯嵌入的通用模型的存在。其中，平坦模块类别和注射扭转模块类别。
Fisher [Fis75] and Baur [Bau75] showed independently in the seventies that if $T$ is a complete first-order theory extending the theory of modules, then the class of models of $T$ with pure embeddings is stable. In [Maz4, 2.12], it is asked if the same is true for any abstract elementary class $(K, \leq_p)$ such that $K$ is a class of modules and $\leq_p$ is the pure submodule relation. In this paper we give some instances where this is true: $\textbf{Theorem.}$ Assume $R$ is an associative ring with unity. Let $(K, \leq_p)$ be an AEC such that $K \subseteq R\text{-Mod}$ and $K$ is closed under finite direct sums, then: - If $K$ is closed under pure-injective envelopes, then $(K, \leq_p)$ is $λ$-stable for every $λ\geq LS(K)$ such that $λ^{|R| + \aleph_0}= λ$. - If $K$ is closed under pure submodules and pure epimorphic images, then $(K, \leq_p)$ is $λ$-stable for every $λ$ such that $λ^{|R| + \aleph_0}= λ$. - Assume $R$ is Von Neumann regular. If $K$ is closed under submodules and has arbitrarily large models, then $(K, \leq_p)$ is $λ$-stable for every $λ$ such that $λ^{|R| + \aleph_0}= λ$. As an application of these results we give new characterizations of noetherian rings, pure-semisimple rings, dedekind domains, and fields via superstability. Moreover, we show how these results can be used to show a link between being good in the stability hierarchy and being good in the axiomatizability hierarchy. Another application is the existence of universal models with respect to pure embeddings in several classes of modules. Among them, the class of flat modules and the class of injective torsion modules.
