I am asking because I was thinking about this problem:
Can we determine the simple modules of a semisimple ring from its product decomposition given by the Wedderburn-Artin theorem?
Let me state the theorem first
Thm (Wedderburn-Artin) Let $R$ be a left semisimple ring. Then $R\cong M_{n_1}(D_1)\times\cdots\times M_{n_r}(D_r)$ for division rings $D_i$ and the pairs $(n_i,D_i)$ are uniquely determined. Moreover, there are exactly $r$ isomorphism classes of simple $R$-modules.
Now suppose the pairs $(n_i,D_i)$ are distinct. Then their simple modules are respectively $V_i=D_i^{n_i}$ and are non isomorphic to each other. Note that they are also simple over $R$ (by projecting $R$ to $M_{n_i}(D_i)$ first and then act on the modules). Since they are nonisomorphic, by the Wedderburn-Artin theorem they are exactly all the simple modules of $R$ up to isomorphism.
However, suppose $(n_i,D_i)=(n_j,D_j)$ for some $i\neq j$. Then there are less than $r$ simple modules obtained as above. How can we determine the simple modules from the decomposition now? Or can such case not happen at all?