Multiplicities in the Regular Representation of a Semisimple Algebra Let $R$ be a finite dimensional associative algebra over a field $k$ and suppose that $R$ is semisimple, i.e., that we can express $R$ as a direct sum of left $R$-modules $$R\cong \oplus_i S_i^{\oplus n_i},$$ where the $S_i$ are non-isomorphic simple modules.
Fact: If $R=kG$ is the group algebra of a finite group $G$ over an algebraically closed field $k$ then the multiplicity $n_i$ is equal to the dimension of the simple module $S_i$: $$n_i=\dim_k S_i.$$
Question: At what level of generality does this fact remain true?

The general description of $R$ is given by the Artin-Wedderburn Theorem: $R$ is a product of matrix algebras over finite-dimensional division algebras over $k$.
If $k$ is algebraically closed, then the only finite-dimensional division algebra over $k$ is itself, and your theorem follows from the representation theory of the matrix algebras $M_{n \times n}(k)$.
