When is the essential extension commutes with colimits(or push forward)

Question

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" $M \to E(M)$

Answer 1

First, I want to point out that in general there is no surjection from a direct product of copies of $R$ to an arbitrary module $M$. For example, if $R=\mathbb{Z}$ and $M=\mathbb{Z}^{(\omega)}$ (a countable direct sum of copies of $\mathbb{Z}$), then there is no surjection $\mathbb{Z}^I\to M$ according to the paper "Extension of a theorem on direct products of slender modules" by John D. O'Neill. [There are probably much simpler examples, but this will do.]

Thus, in general, there is no map $p$, and thus no pushout $N$.

Second, maps defined from $R^{I}$ are notoriously difficult to understand, and often depend on cardinality considerations for the set $I$. In particular, when $R=\mathbb{Z}$, things get really weird when $|I|$ is measurable.

That all said, assume $p$ does exist. Since, as you pointed out, $N$ is an injective module, we just need to examine when $N$ is an essential extension of $M$ (viewing $M$ as a submodule). One obvious situation when this holds is when $R$ itself is a (right) self-injective ring, for in that case $N=M$ is already injective. Of course, this case is somewhat trivial since self-injective hereditary rings are already semisimple.

More generally, write

$$N=M\oplus E(R)^I/\langle (p(x),-i(x))\ :\ x\in R^I \rangle.$$

For each $e\in E(R)^I\setminus\{0\}$, let $X_e:=\{r\in R\ :\ er\in R^I\}$. The set $E_e=\{er\ :\ r\in X_eR\}$ is a submodule of $R^I$. Given any element $\overline{(m,e)}\in N\setminus M$, we need $\overline{(m,e)}R\cap M\neq (0)$. Thus, we need some element $r\in X_e$ such that $mr\neq -p(er)$. In other words, we do not want $p|_{E_e}$ to extend to a map $eR\to M$. Thus, whatever condition you impose will need to restrict what types of maps $p$ are available.