Given a regular local ring $R$ and an $R$-algebras $S$, which is torsion free and finitely generated (even free if needed) as an $R$-module.

Assume we have a nontrivial surjective map $f: M \rightarrow T$, where $M$ is a projective $S$-module, finitely generated and torsion free, and $T$ is a torsion module over $S$. If $N$ denotes $ker(f)$, we get an exact sequence: $0\rightarrow N\rightarrow M\rightarrow T\rightarrow 0$.

Given another torsion module $Q$, when is the induced map $f^{\*}: Hom_S(M,Q)\rightarrow Hom_S(N,Q)$ non trivial, when is it trivial?

My first idea was to use the long exact $Ext$-sequence: Since $M$ is projective we have $Ext^1_S(M,Q)=0$, thus if $f^{\*}=0$, the sequence gives an isomorphism $Hom_S(N,Q)\cong Ext^1_S(T,Q)$.

So what can be said about the groups $Ext^1_S(T,Q)$? Are they always/sometimes/never trivial? Can we compute them if we assume that one of these moudles is a simple $S$-module? Are there other approaches to this question?