由一个理想$I \subset R$构造$R$ -模II
Über die von einem Ideal $I \subset R$ erzeugten $R$-Moduln II
摘要：
设$(R, \mathfrak m)$是交换Noether局部环，$I$是$R$的理想.令$\mathcal{P}$是所有由$I$生成的$R$ -模块$M$的类（即存在满态$I^{(Λ)} \twoheadrightarrow M$），令$\mathcal{S}$是所有由$I^{\circ}$共生成的$R$ -模块$N$的类（即存在具有$I^{\circ} = \operatorname{Hom}_R(I,E)$的单态$N \hookrightarrow (I^{\circ})^Λ$）.我们给出了$\mathcal{P}$和$\mathcal{S}$中所有内射模和平坦模的完整刻画。我们证明了$(\mathcal{S},\mathcal{P})$在Mehdi-Prest（2015）的意义下形成对偶对，并且$\mathcal{P}$在纯子模下总是闭的。我们确定了所有理想$I$，$\mathcal{P}$在子模下是封闭的，$\mathcal{S}$在因子模下是封闭的，$\mathcal{P}$（分别是）在子模下是封闭的。$\mathcal{S}$）在组扩展下关闭。在最后一节中，我们检查所有$R$ -模块$M$的子模块$γ(M) = \sum\{U \subset M \,|\, U \in \mathcal{P}\}$和$κ(M) = \bigcap \{V \subset M \,|\, M/V \in \mathcal{S}\}$，并在特殊情况下指定它们的显式结构。
英文摘要：
Let $(R, \mathfrak m)$ be a commutative noetherian local ring and $I$ an ideal of $R$. Let $\mathcal{P}$ be the class of all $I$-generated $R$-modules $M$ (i.e. there is an epimorphism $I^{(Λ)} \twoheadrightarrow M$) and let $\mathcal{S}$ be the class of all $I^{\circ}$-cogenerated $R$-modules $N$ (i.e. there is a monomorphism $N \hookrightarrow (I^{\circ})^Λ$ with $I^{\circ} = \operatorname{Hom}_R(I,E)$). We give a complete description of all injective and flat modules in $\mathcal{P}$ and $\mathcal{S}$. We show that $(\mathcal{S},\mathcal{P})$ forms a dual pair in the sense of Mehdi--Prest(2015) and that $\mathcal{P}$ is always closed under pure submodules. We determine all ideals $I$ for which $\mathcal{P}$ is closed under submodules, $\mathcal{S}$ is closed under factor modules and $\mathcal{P}$ (resp. $\mathcal{S}$) is closed under group extensions. In the last section, we examine the submodules $γ(M) = \sum\{U \subset M \,|\, U \in \mathcal{P}\}$ and $κ(M) = \bigcap \{V \subset M \,|\, M/V \in \mathcal{S}\}$ for all $R$-modules $M$, and we specify their explicit structure in special cases.
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