Über die von einem Ideal $I\subset R$ erzeugten $R$-Moduln III

Abstract

Let $(R,\mathfrak{m})$ be a commutative noetherian local ring and $I$ an ideal of $R$. For every $R$-module $M$, ${\gamma }_I(M)=\sum \{\mathrm{Bi}f|f\in {\mathrm{Hom}}_R(I,M)\}$ is called the trace of $I$ in $M$. It is easy to see that ${\mathrm{Ext}}_R^1(R/I,M)=0$ always implies $IM={\gamma }_I(M)$. If the second condition holds for all ideals $I$ of $R$, we say that $M$ is excellent. In part 1, we show a number of conditions for these modules, which are well-known for injective modules. In the second part, we examine the special case $M=R$. In particular, we show that for every prime ideal $\mathfrak{p}$ the equality $\mathfrak{p}={\gamma }_{\mathfrak{p}}(R)$ holds iff $R_{\mathfrak{p}}$ is not a discrete valuation ring. From the results by Matlis (1973) about 1-dimensional local CM-rings and with the help of the first neighborhood ring $\Lambda$, it follows immediately that ${\gamma }_{{\mathfrak{m}}^n}(R)={\Lambda }^{-1}$ for almost all $n\ge 1$. In the third part, we examine the dual construction ${\kappa }_I(M)=\bigcap \{\mathrm{Ke}f|f\in {\mathrm{Hom}}_R(M,I^{\circ })\}$ and reduce the main results about ${\mathrm{Tor}}_1^R(M,R/I)=0$ and ${\kappa }_I(M)=M[I]$ to part 1 by considering the Matlis dual $M^{\circ }={\mathrm{Hom}}_R(M,E)$ and the equalities ${\gamma }_I(M^{\circ })={\mathrm{Ann}}_{M^{\circ }}({\kappa }_I(M))$, ${\kappa }_I(M^{\circ })={\mathrm{Ann}}_{M^{\circ }}({\gamma }_I(M))$.

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