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

TL;DR

This paper characterizes when modules over a noetherian local ring are generated by an ideal, providing conditions for uniserial, cyclic modules, and exploring duality and generalizations of existing ideal concepts.

Contribution

It offers new criteria for ideal-generated modules, extends duality results, and generalizes the concept of basically full ideals in the context of noetherian local rings.

Findings

Uniserial modules are generated by an ideal under specific conditions.

Cyclic modules generated by the maximal ideal are characterized as either valuation rings or certain uniserial modules.

Duality conditions relate submodules and ideal generation, extending known ideal concepts.

Abstract

Let $(R, \mathfrak m)$ be a commutative noetherian local ring. We investigate under which conditions an $R$-module $M$ is generated by an ideal $I$, i.e. there exists an epimorphism $I^{(\Lambda)} \twoheadrightarrow M$. If $M$ is uniserial, i.e. $\mathcal{L}(M)$ is totally ordered and finite, this is equivalent to $\mathfrak{m}^{n-1} \cdot I \not\subset \operatorname{Ann}_R(M) \cdot I$ ($\operatorname{length}(M) = n \geq 1$). If $M$ is cyclic and $I = \mathfrak{m}$, this is equivalent to: Either it is $M \cong R/\mathfrak{p}$ ($R/\mathfrak{p}$ a discrete valuation ring) or $M \cong C/\operatorname{So}(C)$ ($C$ a uniserial $R$-module). If $A$ is free and $B$ is a submodule of $A$, then the Matlis dual $(A/B)^{\circ} = operatorname{Hom}_R(A/B, E)$ is $I$-generated if and only if $B = (IB) :_A I$. In the case $I = \mathfrak{m}$, this condition leads to the "basically full ideals" considered by Heinzer, Ratliff~Jr. and Rush. By studying the dual condition $M = I(M :_X I)$ in the last section, we can generalize some results of that work.