Lemma 47.5.8. Let R be a Noetherian ring. Let E be an indecomposable injective R-module. Then there exists a prime ideal \mathfrak p of R such that E is the injective hull of \kappa (\mathfrak p).
Proof. Let \mathfrak p be the prime ideal found in Lemma 47.5.6. Say \mathfrak p = (f_1, \ldots , f_ r). Pick a nonzero element x \in \bigcap \mathop{\mathrm{Ker}}(f_ i : E \to E), see Lemma 47.5.6. Then (R_\mathfrak p)x is a module isomorphic to \kappa (\mathfrak p) inside E. We conclude by Lemma 47.5.6. \square
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