Let $A$ be a Noetherian ring, $\mathfrak{p},\mathfrak{q}\subset A$ distinct prime ideals of the same height, $N$ an $A_\mathfrak{p}$-module of finite length. Then is it true that $\operatorname{Hom}_A(N,E(A/\mathfrak{q}))=0,$ where $E(A/\mathfrak{q})$ is the injective hull of $A/\mathfrak{q}$?
The main trouble here is that $N$ may not be finitely generated over $A$. If it were, I would use formulas like \begin{eqnarray} \operatorname{Ass}\operatorname{Hom}_A(N,E(A/\mathfrak{q}))&=&\operatorname{Supp}_AN\cap\operatorname{Ass}E(A/\mathfrak{q})\ &=&V(\mathfrak{p})\cap\lbrace\mathfrak{q}\rbrace\ &=&\emptyset. \end{eqnarray}