A note on the Matlis dual of a certain injective hull

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