Über 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 ${\mathfrak{m}}^{n-1}\cdot I\not\subset {\mathrm{Ann}}_R(M)\cdot I$ ($\mathrm{length}(M)=n\ge 1$). 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 $M\cong C/\mathrm{So}(C)$ ($C$ 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.

9 pages, in German