  Let $(R,\mathfrak{m})$ denote a local ring with $E = E_R(R/\mathfrak{m})$ the
injective hull of the residue field. Let $\mathfrak{p} \in \Spec R$ denote a
prime ideal with $\dim R/\mathfrak{p} = 1$, and let $E_R(R/\mathfrak{p})$ be
the injective hull of $R/\mathfrak{p}$. As the main result we prove that the
Matlis dual $\Hom_R(E_R(R/\mathfrak{p}), E)$ is isomorphic to
$\hat{R_{\mathfrak{p}}}$, the completion of $R_{\mathfrak{p}}$, if and only if
$R/\mathfrak{p}$ is complete. In the case of $R$ a one dimensional domain there
is a complete description of $Q \otimes_R \hat{R}$ in terms of the completion
$\hat{R}$.
