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Let $M$ be an $R$-module,where $R$ is a hereditary (or cohomological dimension less or equal to 1).Take $E(R)$ to be injective hull of $R$, then we have the essential extension $i:R^I\rightarrowtail E(R)^I$ (product $I$ times) and we also have $p:R^I\twoheadrightarrow M$ is epimorpshim. Then I take the push forward of these two morphisms $i$ and $p$, denote the push out by $(N,f,g)$, where $f:M\to N$ and $g:E(R)^I\twoheadrightarrow N$.

It is clear that $N$ is an injective module (because it is image of $E(R)^I$ and $R$ is hereditary) and $f:M\to N$ is injective morphism. However, $N$ is not necessarily injective hull of $M$ because in general, essential extension does not commutes with colimit.

My question is: can we give some conditions to $R$ or other extra conditions to make N is injective hull of $M$?

In general, I know it is not true, but it seems that it gives a approximation of "functor" $M \to E(M)$

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