Lemma 47.7.2. Let $(R, \mathfrak m, \kappa )$ be a local ring. Let $E$ be the injective hull of $\kappa$. Let $M$ be a $\mathfrak m$-power torsion $R$-module with $n = \dim _\kappa (M[\mathfrak m]) < \infty$. Then $M$ is isomorphic to a submodule of $E^{\oplus n}$.

Proof. Observe that $E^{\oplus n}$ is the injective hull of $\kappa ^{\oplus n} = M[\mathfrak m]$. Thus there is an $R$-module map $M \to E^{\oplus n}$ which is injective on $M[\mathfrak m]$. Since $M$ is $\mathfrak m$-power torsion the inclusion $M[\mathfrak m] \subset M$ is an essential extension (for example by Lemma 47.2.4) we conclude that the kernel of $M \to E^{\oplus n}$ is zero. $\square$

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