Let $R$ be a commutative non-Noetherian ring and $m$ a maximal ideal. My question is whether $E(R)_m = E(R_m)$ holds, i.e. whether the localization of the injective hull $E(R)$ of $R$ is the injective hull $E(R_m)$ of $R_m$. This is true in the Noetherian case. Dade has some examples of injective modules that are not anymore injective after their localization by a prime ideal in the non-Noetherian case. But what about the injective hull of the ring itself localized by a maximal ideal? Is $E(R)_m$ at least an injective $R_m$-module?

Let $R$ be a commutative non-Noetherian ring and $m$ a maximal ideal. My question is whether the localization $E(R)_m$ of the injective hull $E(R)$ of $R$ is an injective $R_m$-module. This is true in the Noetherian case. Dade has some examples of injective modules that are not anymore injective after their localization by a prime ideal in the non-Noetherian case. But what about the injective hull of the ring itself localized by a maximal ideal?