Let $k$ be a field , let $R$ be a finite dimensional semisimple algebra over $k$ ; is it true that $\exists t \in \mathbb N$ and $n_1,...,n_t \in \mathbb N$ such that $R$ is isomorphic with $\oplus_{i=1}^t M(n_i,k)$ as a $k$- algebra ? If not true in general , can we characterize such semisimple algebras over a given field $k$ which can be written as a direct sum of finitely many matrix algebras over $k$ ?
Note that this requires the semisimple algebra to have an apparently more "nice" structure than is implied by the Artin-Wedderburn Theorem .