\"Uber 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 $M$, $\gamma_I(M) = \sum\{ \operatorname{Bi} f \,|\, f \in \operatorname{Hom}_R(I,M)\}$ is called the trace of $I$ in $M$. It is easy to see that $\operatorname{Ext}_R^1(R/I,M) = 0$ 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 $n \geq 1$. In the third part, we examine the dual construction $\kappa_I(M) = \bigcap \{ \operatorname{Ke} f \,|\, f\in \operatorname{Hom}_R(M,I^\circ) \}$ and reduce the main results about $\operatorname{Tor}_1^R(M, R/I) = 0$ and $\kappa_I(M) = M[I]$ to part 1 by considering the Matlis dual $M^\circ = \operatorname{Hom}_R(M, E)$ and the equalities $\gamma_I(M^\circ) = \operatorname{Ann}_{M^\circ}(\kappa_I(M))$, $\kappa_I(M^\circ) = \operatorname{Ann}_{M^\circ}(\gamma_I(M))$.