Let $R$ be a Noetherian ring and suppose that we can write $1 = e_1 + e_2 + \dots + e_n$ where the $e_i$ are pairwise orthogonal idempotents. Let $S = e_1 S e_1$, and consider the right $S$-modules $e_iRe_1$. Then we have $e_jRe_i \subset \text{Hom}_S(e_iRe_1,e_jRe_1)$ where $e_iRe_j$ gives rise to morphisms by left multiplication. Is it necessarily the case that this inclusion is actually an inequality, and if not, are there any general conditions under which it is an equality?
Edit: It seems that $\text{Hom}_S(e_i R e_1, e_j R e_1) \cong \text{Hom}_R(e_iRe_1 \otimes_S e_1R, e_jR)$ and that $\text{Hom}_R(e_iR,e_jR) = e_jRe_i$ so it suffices to show that $e_iRe_1 \otimes_S e_1R \cong e_i R$. This seems plausible, perhaps if there's some domain-like condition on $R$ or $S$.
Second edit: I suppose it might be useful if I included the specific case of interest to me. Let $R$ be the path algebra of a finite quiver $Q$ with vertices labelled by $\{1, \dots , n\}$, let the $e_i$ be the vertex idempotents, and set $S = e_1Re_1$. In this setting, is the claim true?