Given a semisimple ring $R$, we have a $R$-module isomorphism $R \cong n_1V_1 \oplus n_2V_2 \oplus\cdots \oplus n_lV_l$ where $V_1, V_2, \cdots, V_l$ are nonisomorphic simple $R$-modules.
By Wedderburn-Artin Theorem we have a ring isomorphism: $$R \cong End_R(R)^{op} \cong End_R(n_1V_1 \oplus n_2V_2 \oplus\cdots \oplus n_lV_l)^{op}$$ $$\cong End_R(n_1V_1)^{op}\times End_R(n_2V_2)^{op} \times\cdots \times End_R(n_lV_l)^{op}$$ $$\cong M_{n_1}(End_R(V_1))^{op} \times M_{n_2}(End_R(V_2))^{op} \times \cdots \times M_{n_l}(End_R(V_l))^{op}$$ $$\cong M_{n_1}(End_R(V_1)^{op}) \times M_{n_2}(End_R(V_2)^{op}) \times \cdots \times M_{n_l}(End_R(V_l)^{op})$$
From the ring isomorphism we can obtain another $R$-module isomorphism $R \cong n_1V'_1 \oplus n_2V'_2 \oplus\cdots \oplus n_lV'_l$, where $V'_i$ is $R$-isomorphic to the space of matrices in $M_{n_i}(End_R(V_i)^{op})$ with every entry zero except those in the first column. We can see the $V'_i$s are again nonisomorphic simple $R$-modules.
Question: Is $V_i$ isomorphic to $V'_i$ as $R$-module?
I know $V_i$ must be isomorphic to one of the $V'_j$s since $V_i$ is a smiple $R$-module, but it is hard to show $V_i$ is exactly isomorphic to $V'_i$. I have tried tracking the $M_{n_1}(End_R(V_1)^{op}) \times M_{n_2}(End_R(V_2)^{op}) \times \cdots \times M_{n_l}(End_R(V_l)^{op})$-module structure of $n_1V_1 \oplus n_2V_2 \oplus\cdots \oplus n_lV_l$, but I cannot get anything useful from it.