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Let $R$ be a noetherian local ring with maximal ideal $\mathcal m$ and denote by $E$ the injective hull of the residue field $k$.

Then, as an $R-$module, what is the support of $E$?

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The support of $E$ is just $\lbrace \mathfrak{m} \rbrace$.

For, the only associated prime of $E=E(R/\mathfrak{m})$ is $\mathfrak{m}$ (Lemma 3.2.7 in Herzog, Bruns "Cohen-Macaulay rings") and since $\text{Ass}_R(E)$ and $\text{Supp}_R(E)$ share the same minimal elements, the assertion follows.

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