Injective hull and some Hom Let $R$ be a commutative ring with unit. Suppose $P\in Spec(R)$ and let $E=E(R/P)$ be the injective hull of $R/P$. What can we say about $Hom_R(R/P, E)$. We know that $R/m\cong Hom_R(R/m, E)$, where $m$ is a maximal ideal of $R$.
 A: For general commutative rings, the structure of injective modules can be quite complicated and I cannot answer to your question in general. Anyway, one can say many thing about injective modules over commutative Noetherian rings. Some amazing results in this area are now classical (I'm reading them from a very nice paper of Eben Matlis from 1958, I strongly recomend you that paper).
After decomposing injective modules in direct sums of indecomposable injectives, Matlis passes to describe the internal structure of the indecomposable injectives, that is, of the modules of the form $E(R/P)$ for some prime ideal $P$. In particular, he shows that the submodule $A_1=\{x\in E(R/P): Px=0\}$ is isomorphic to $K(P)$ (the field of fractions of $R/P$). So you have:
$$\hom_R(R/P,E(R/P))\cong\hom_R(R/P,A_1)=\hom_R(R/P,K(P))\cong K(P).\ \ \ \ (*)$$
The case you mention about maximal ideals comes from the fact that $K(m)\cong R/m$ in case $m$ is maximal. 
If you drop the Noetherian assumption, then there may be indecomposable injectives which are not of the form $E(R/P)$ for some prime $P$. I do not know if you can still prove something like (*)... maybe you can try to read Matlis' paper and try to see if something fails in there without Noetherianity (what you need is part (4) of Theorem 3.4 in that paper). 
For non-commutative Noetherian rings there is probably something similar you can say, at least for FBN rings (such restriction seams reasonale as it allows you to say that any indecomposable injective is -a summand of a module- of the form $E(R/P)$, in other words, there is a Gabriel correspondence)... if you are interested to these metters I suggest you to look in the book of Jategaonkar, it is quite complete even if, in my opinion, not so easily accessible. 
