Let $R$ be integral domain and $p \neq 0$ a prime ideal. It's well known that in category of $R/p$ modules the injective hull of $R/p$ is $K=\operatorname{Frac}(R/p)$. Is there a successful theory known treating injective hull of $R/p$ considered as $R$-module? If $R= \mathbb{Z}$ then it is known that the inj hull of $\mathbb{Z}/p$ as $\mathbb{Z}$-modukle is $\mathbb{Z}(p^\infty)=(\mathbb{Z}[1/p])/\mathbb{Z}$. Is there any fruitful theory known classifying the structure of injective hulls known going beyong this 'baby' example $\mathbb{Z}/p$? What if we impose additional restrictive assumtions on $R$ like being local, regular or factorial etc? This is an exact copy of same mse question I asked a week ago.
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3$\begingroup$ If $p$ is maximal, $R/p$ is Artinian and its injective hull as $R$-module is the same as its injective hull as $R_p$-module or equivalently $\widehat{R_p}$, and this is the Matlis dual of $\widehat{R_p}$ as $\widehat{R_p}$-module. $\endgroup$– YCorCommented Mar 11, 2021 at 17:30
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$\begingroup$ do you know where a proof of this result can be found? $\endgroup$– user267839Commented Mar 11, 2021 at 17:41
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$\begingroup$ I'm not aware of a reference. I think this follows from basic properties of Matlis duality. $\endgroup$– YCorCommented Mar 11, 2021 at 18:08
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$\begingroup$ For noetherian rings there is the paper of Matlis from 1958. $\endgroup$– Friedrich KnopCommented Mar 11, 2021 at 18:51
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$\begingroup$ Chapters 2, 3 of Hochster's Local Cohomology are one good reference for this stuff. $\endgroup$– PrimeRibeyeDealCommented Jul 19, 2022 at 21:18
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