  For a Noetherian local ring $(R,{\mathfrak m})$ with $\mathfrak p\in
\Spec(R)$ we denote $E_R(R/\mathfrak p)$ by the $R$-injective hull of
$R/\mathfrak p$. We will show that it has an $\hat{R}^\mathfrak p$-module
structure and there is an isomorphism $E_R(R/\mathfrak p)\cong
E_{\hat{R}^\mathfrak p}(\hat{R}^\mathfrak p/\mathfrak p\hat{R}^\mathfrak p)$
where $\hat{R}^\mathfrak p$ stands for the $\mathfrak p$-adic completion of
$R$. Moreover for a complete Cohen-Macaulay ring $R$ the module
$D(E_R(R/\mathfrak p))$ is isomorphic to $\hat{R}_\mathfrak{p}$ provided that
$\dim(R/\mathfrak p)=1$ and $D(\cdot)$ denotes the Matlis dual functor
$\Hom_R(\cdot, E_R(R/\mathfrak m))$. Here $\hat{R}_\mathfrak{p}$ denotes the
completion of ${R_\mathfrak p}$ with respect to the maximal ideal $\mathfrak
pR_\mathfrak p$. These results extend those of Matlis (see \cite{m}) shown in
the case of the maximal ideal ${\mathfrak m}$.

Commutative Algebra