Let $R$ be integral domain and $p \neq 0$ a prime ideal. It's well know 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 inj hulls known going beyong this 'baby' example $\mathbb{Z}/p$? What if we impose additional restrictive assumtions on $R$ like beeing local, regular or factorial etc? This is an exact copy of same mse question I asked a week ago.

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.