Suppose $X, Y, Z$ are modules over some ring $R$ and for which we have the following short exact sequence,
$0\to Z\stackrel{i}{\to} Y\stackrel{p}{\to} X\to 0$.
Then there is a class $c\in Ext^1(X,Z)$ representing this SES. In particular, I know $Ext^1(X,Z)$ is an abelian group under the Baer sum (denoted by $+$ for ease). Moreover, the class $-c$ is represented by the exact sequence,
$0\to Z\stackrel{i}{\to} Y\stackrel{-p}{\to} X\to 0$.
Clearly, then, $Ext^1(X,\,Z)$ is telling us something about the object in the middle $\textit{and}\:$ the maps of the exact sequence. I was wondering if anyone could shed some more light on just what it is telling us.
Further, suppose I know $Ext^1(X,\,Z)\cong\mathbb{Z}/2$, then if $Y_1, Y_2\not\cong X\oplus Z$, then I know $Y_1\cong Y_2$ (by the Five Lemma). However, surely I also know that if $Ext^1(X,\,Z)\cong\mathbb{Z}/3$, and $Y_1,\,Y_2$ are as above then this also implies $Y_1\cong Y_2$. For suppose $Y_1$ (say) exists in the SES,
$0\to Z\stackrel{i}{\to} Y_1\stackrel{p}{\to} X\to 0$
represented by the class $c\in Ext^1(X,\,Z)$. Then the inverse class $-c$ represents the SES
$0\to Z\stackrel{i}{\to} Y_1\stackrel{-p}{\to} X\to 0$.
Now, if $Y_1,\,Y_2$ belong in the same class, then they are isomorphic (again by the Five Lemma). If they are not, then $Y_2$ is represented by the class $-c$ and is therefore also isomorphic to $Y_1$. Is this correct?
If this is true, then can we not say something about how many modules can exist (up to isomorphism) in the middle? So if $Ext^1(X,\,Z)\cong\mathbb{Z}/2$ or $\mathbb{Z}/3$ then we have (up to isomprhism) two modules - the split module and the non-split module. If $Ext^1(X,\,Z)\cong\mathbb{Z}/5$ then we have (up to isomorphism) three modules etc.