The following is taken from: is a continuation of the following question
Background:
Definition 1. A short exact sequence of the form
$(f,E,g)\equiv 0\to A\xrightarrow{f}E\xrightarrow{g}B\to 0$
is called an extension of $A$ by $B.$
Let $G,H$ be groups and let
$G^*=\{(g,e_G)\mid g\in G, e_H=\text{identity of } H\}$
$H^*=\{(e_G,h)\mid h\in H, e_G=\text{identity of } G\}$
Definition 2. If $G\cong {G^*}\trianglelefteq E$ and $E/G^*\cong H,$ we say that $E$ is an extension of $G$ by $H$.
Clearly $G\times H$ is always an extension of $G$ by $H.$ On the other hand not every extension of $G$ by $H$ need be isomorphic to $G\times H.$ For consider the following example: $\mathbb{Z}_4$ contains the n ormal group $G^*=\langle |\underline{2}| \rangle\cong \mathbb{Z}_2,$ and hence $\mathbb{Z}_4$ is an extension of $\mathbb{Z}_2$ by $\mathbb{Z}_2.$ However $Z_4$ is a cyclic group, but $\mathbb{Z}_2 + \mathbb{Z}_2$ is not; hence they are not isomorphic.
Definition 3. An $R-$module $N$ is said to be an extension of an $R$-module $M$ if there is as monomorphism $f:M\to N.$ Such an extension is an essential extension if every non-zero submodule of $N$ has a non-zero intersection with $\text{Im }f.$
Questions:
I know that I can use definition 2 above to undertand definition 1 for the case of module extension of $A$ by $B.$ I also know that definition 3 is not equivalent to definition 1 in both concepts and wording. But what is the relation between extension and essential extension. What I mean by relation is in the sense are there theorems that says under certain conditions, one of the definition implies the other.
I looked up the notion of "extension of A by B" or ones with variable letters changed. I only found an exposition of it in the book "Rings and Categories of Modules" by Fuller and Anderson, but with in the same text, I can't locate the concept of "essential extension." Other text that discusses "essential extension" don't discuss the notion of "extension of A by B." The concept of essenntial extension is introduced in the chapter on injective modules which is nine chapters ahead in the text I am using. Defintion 1 appears in the electronic edition of the same text in chapter 3 on exact sequence of modules.
Thank you in advance
Definition 3 gives you an exact sequence $0\to M\to N$ that you are free to complete to $0\to M\to N\to M/N\to 0$ for something satisfying Definition 1.
And conversely you can truncate the short exact sequence from Definition 1 to consider the leftmost module extended to the middle module in Definition 3. For that reason the two are pretty much the same. Depending on context, one or the other definition may be preferable.
Being an essential extension is a refinement on this which requires the image of $f$ be "dense" in some sense in $N$. If $f$ were onto then it would be an isomorphism, so that is too strong. So it makes sense to look at submodules that are "large" inside of $M$. In fact, some texts uses "large" as a synonym for essential (and small as a synonym for superfluous submodules.)