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

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.