Assume we have a complete regular local ring $R$ and an $R$-algebra $S$.

Is there a class of such algebras $S$ with the following property:

Given two $S$-modules $M,N$, then the maps induced by the forgetful functor $S-Mod \rightarrow R-Mod$ give injections $Ext^i_S(M,N)\rightarrow Ext_R^i(M,N)$?

If this question is too broad, here are the special cases i'm especially interested in:

We start with $R=\mathbb{C}[[x,y]]$, on the algebra side one may choose $S=M_2(R)$ or the subalgebra of the matrices given by: \begin{pmatrix}R &R \\ xR &R \end{pmatrix} and one may choose $N=S$ and $M=S/T$ to be an finite length quotient of $S$, i.e there is a sequence $0\rightarrow T\rightarrow S \rightarrow M\rightarrow 0$. Further more the case $i=2$ would be enough.

So what i'm really interested in is: Is the map $ Ext_S^2(S/T,S)\rightarrow Ext_R^2(S/T,S)$ injective in these cases?

Inducing injective maps on the $Ext$-groups is, i think, more than being a faithful functor. Maybe this property has already been studied and has its own name? I don't see any sequences which relate these two groups, so that one could see this by doing some kind of diagram chasing.

Any hints or ideas are welcome!

surjectiveon Exts is called anhomological epimorphism(these were studied by Auslander, Platzek and Todorov a while ago). I've never seen the injective variant. $\endgroup$