Consider the $R$-modules $A$ and $B$ and suppose we would like to classify extensions of $A$ by $B$, i.e we would like to classify all $R$-modules which fit in to the middle slot of the following SES, $$ 0 \to B \to ? \to A \to 0 $$ up to isomorphism of extension. We know such modules are classified by the group $Ext_R^1(A,B)$.
Suppose I know the generators of $Ext_R^1(A,B)$ as an $R$-module. How do I use these generators to recover extensions?
I am looking for a very easy example.
This should probably be a comment, but I don't have the rep.
Aluffi's Algebra Chapter 0, exercise VIII.6.22 walks through this in depth, giving quite a few hints. I'm sure there are other references, but that's where I remember seeing it done.