# \"Uber die von einem Ideal $I \subset R$ erzeugten $R$-Moduln II

## TL;DR

This paper characterizes injective and flat modules generated or cogenerated by an ideal in a noetherian local ring, establishing duality, closure properties, and explicit submodule structures in this context.

## Contribution

It provides a complete description of injective and flat modules within classes generated or cogenerated by an ideal, and explores their duality and closure properties.

## Key findings

- Characterization of injective and flat modules in classes ormed by ideal I
- Establishment of duality between classes ormed by ideal I
- Identification of conditions for closure under submodules, factor modules, and extensions

## Abstract

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 $I^{(\Lambda)} \twoheadrightarrow M$) and let $\mathcal{S}$ be the class of all $I^{\circ}$-cogenerated $R$-modules $N$ (i.e. there is a monomorphism $N \hookrightarrow (I^{\circ})^{\Lambda}$ 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 $\gamma(M) = \sum\{U \subset M \,|\, U \in \mathcal{P}\}$ and $\kappa(M) = \bigcap \{V \subset M \,|\, M/V \in \mathcal{S}\}$ for all $R$-modules $M$, and we specify their explicit structure in special cases.