Let M $M$ be an R-module,where R $R$-module,where $R$ is a hereditary(or hereditary (or cohomological dimension less or equal to 1).Take E(R) $E(R)$ to be injective hull of R, $R$, then we have the essential extension i:R^I--->E(R)^I $i:R^I\rightarrowtail E(R)^I$ (product I $I$ times) and we also have p:R^I--->>M $p:R^I\twoheadrightarrow M$ is epimorpshim. Then I take the push forward of these two morphisms i $i$ and p, $p$, denote the push out by (N,f,g),where f:M---->N $(N,f,g)$, where $f:M\to N$ and g:E(R)^I--->>N)
it $g:E(R)^I\twoheadrightarrow N$.
It is clear that N $N$ is an injective module(because module (because it is image of E(R)^I $E(R)^I$ and R $R$ is hereditary) and M--->N $f:M\to N$ is injective morphism. However, N $N$ is not necessarily injective hull of M $M$ because in general, essential extension does not commutes with colimit.
My question is: can we give some conditions to R $R$ or other extra conditions to make N is injective Hull hull of M.$M$?
In general, I know it is not true, but it seems that it gives a approximation of "functor":M--->E(M)functor" $M \to E(M)$