Let $(R, m, k)$ be a Noetherian local ring. Let $E\_R(k)$ be the injective hull of $k$ as an $R$-module. It is well known that $E\_R(k)$ is Artinian and is an $(R, \hat{R})$-bimodule (see, Brodmann-Sharp: local cohomology, 1998), where $\hat R$ is the completion of $R$. My question is


**Question**: Is $E\_R(k)$ injective as an $\hat R$-module. In this is the case $E\_R(k) = E\_{\hat{R}}(k)$.



Is $E_R(k) = E_{\hat{R}}(k)$?
You can see Theorem 18.6 in Commutative ring theory by H.Matsumura.


