This should be considered as very general question regarding the extension group $Ext^i _A (R,S)$, in particular where $i=1$, for $R$ and $S$, a pair of given objects in an abelian category $A$. For the sake of concreteness, let's assume modules over an associative finite dimensional algebra A, over an algebraically closed field $K$ (if this setting possibly helps to address the question).
I wish to know under which additional assumptions on the setting (possibly the weakest set of them), we could derive information about $Ext^1 _A (R,S)$ from the self-extensions $Ext^1 _A (R,R)$ and $Ext^1 _A (S,S)$ and the missing part.
It is clear to me that in the current setting, there is not much one could say just based on the latter extension groups. But, my question is exactly about the missing part of this puzzle.
Knowing about $Hom _A (R,S)$ seems to strong to me. I know that having $Ext^i _A$ as derived functors from $Hom _A $ might make it impossible to say something very useful, but many questions seem stupid at the beginning (if not at the end either)!