  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 $\gamma_I(M^\circ) =
\operatorname{Ann}_{M^\circ}(\kappa_I(M))$, $\kappa_I(M^\circ) =
\operatorname{Ann}_{M^\circ}(\gamma_I(M))$.
