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?
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