Assume we have a noncommuative ring $R$ with exactly 2 non-isomorphic simple left modules $S_1$ and $S_2$ (up to isomorphism) and an $R$-bimodule $M$, which switches the simples, i.e. $M\otimes_R S_1=S_2$ and $M\otimes_R S_2=S_1$.
Then we have $Hom_R(S_i,M\otimes_R S_i)=0$ by Schur's lemma ($*$).
Now assume $F$ is an arbitrary left $R$-module of finite length. What can be said about $Hom_R(F,M\otimes_R F)$ with the help of ($*$)?
If $F$ is a quotient of $R$, i.e. we have a surjection $R\rightarrow F \rightarrow 0$, then we get an injection $0\rightarrow Hom_R(F,M\otimes_R F)\rightarrow Hom_R(R,M\otimes_R F)$. Is there a method, using ($*$) to decide whether this is even an isomorphism?
So for example, if $F$ is one of the simples or a direct sum of one of the simples, then $Hom_R(F,M\otimes_R F)=0$ so the map is not an isomorphism. Otherwise both simple modules occur at least once in a Jordan-Hölder composition series. What can be said in this case?
Or is
If this question is too broad? Does this strongly depend on , the ring $R$? If necessary i can specify R$ i'm interetested in is the following subring of $R$, M_2(A)$:
\begin{pmatrix}A &A \\ xA &A \end{pmatrix}
where $M$ etcA$ is a complete regular local ring of dimension 2, and $x\in A$ s.t. $A=\mathbb{C}[[x,y]]$. The $R$-bimodule $M$ is given as the following $R$-submodule in $M_2(Quot(A))$:
\begin{pmatrix}A &x^{-1}A \\ A &A \end{pmatrix}
