模的抽象初等类的格
The lattice of abstract elementary classes of modules
AI总结：
本文研究环 $R$ 上以 $R$-模类为底层的抽象初等类构成的格 $\mathscr{L}_{R}$，分析其格论性质、纯性上下的子格，特殊化到整数环时还得到了该格的若干具体性质。
AI中文摘要：
设 $R$ 为环。我们将以所有 $R$-模构成的类为底层类、且强子模型关系介于子模与直和项关系之间的抽象初等类组织为格 $\mathscr{L}_{R}$，按反向包含关系排序。我们确立了 $\mathscr{L}_{R}$ 的基本格论性质，并研究其两个自然子格：纯性以下与纯性以上的子格。在纯性以下，我们分离出由一阶 pp-公式定义的关系，这些关系满足合并性、温顺性与稳定性；在纯性以上，我们引入由无穷 pp-公式定义的关系，并证明了一个宽泛的稳定性结果。将其特殊化到阿贝尔群的情形，我们证明格 $\mathscr{L}_{\mathbb{Z}}$ 具有如下性质：存在非正句法的强子模型关系、包含一个不可数反链与一个真类大小的严格递增链，且在纯性以上的宽泛区域中合并性不成立。
英文摘要：
Let $R$ be a ring. We organize the abstract elementary classes whose underlying class is the class of all $R$-modules and whose strong submodel relation lies between the submodule and direct summand relations into a lattice $\mathscr{L}_{R}$, ordered by reverse inclusion. We establish the basic lattice-theoretic properties of $\mathscr{L}_{R}$ and investigate its two natural sublattices, below and above purity. Below purity, we isolate relations defined by first-order pp-formulas for which amalgamation, tameness, and stability hold. Above purity, we introduce relations defined by infinitary pp-formulas and prove a broad stability result. Specializing to abelian groups, we show that the lattice $\mathscr{L}_{\mathbb{Z}}$ has the following properties: it has a strong submodel relation that is not positive syntactic, it contains an uncountable antichain and a strictly increasing proper-class-sized chain, and it has a broad region above purity where amalgamation fails.
