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 multiplicity $n_i$ can be calculated as
$$
    n_i
  = \frac{\dim_k S_i}{\dim_k \operatorname{End}_R(S_i)} \cdotp
$$
This can be seen in (at least) two ways:
First:
If $M$ is any finite-dimensional $R$-module and $M \cong \bigoplus_i S_i^{\oplus n_i}$ is a decomposition into pairwise non-isomorphic simple $A$-modules $S_i$ then
\begin{align*}
  \operatorname{Hom}_R(M, S_i)
  \cong
  \operatorname{Hom}_R\left( \bigoplus_j S_j^{\oplus n_j}, S_i \right)
  &\cong
  \prod_j \operatorname{Hom}_R(S_j, S_i)^{n_j}
  \\
  &\cong
  \operatorname{Hom}_R(S_i, S_i)^{n_i}
  =
  \operatorname{End}_R(S_i)^{n_i}
\end{align*}
as $k$-vector spaces, where we used for the third isomorphim that $\operatorname{Hom}_R(S_j, S_i) = 0$ for all $j \neq i$ by Schur’s lemma.
It follows that
$$
    \dim_k \operatorname{Hom}_R(M,S_i)
  = n_i \dim_k \operatorname{End}_R(S_i)
$$
and therefore that
$$
    n_i
  = \frac{\dim_k \operatorname{Hom}_R(M,S_i)}{\dim_k \operatorname{End}_R(S_i)} \cdotp
$$
For $M = R$ we have that $\operatorname{Hom}_R(R, S_i) \cong S_i$ as $k$-vector spaces and therefore
$$
    n_i
  = \frac{\dim_k \operatorname{Hom}_R(R,S_i)}{\dim_k \operatorname{End}_R(S_i)}
  = \frac{\dim_k S_i}{\dim_k \operatorname{End}_R(S_i)} \cdotp
$$
Second:
By the theorem of Artin-Wedderburn we have that
$$
         R
  \cong  \operatorname{Mat}_{n_1}(D_1)
         \times \dotsb \times
         \operatorname{Mat}_{n_r}(D_r).
$$
for skew fields $D_1, \dotsc, D_r$ over $k$.
Then $D_1^{n_1}, \dotsc, D_r^{n_r}$ is a set of representatives for the isomorphism classes of simple $R$-modules, $D_i^{n_i}$ appears with multiplicity $n_i$ in $R$ and $\operatorname{End}_R(D_i^{n_i}) \cong D_i^\mathrm{op}$ for all $i = 1, \dotsc, r$.
We may assume that $S_i = D_i^{n_i}$ for all $i$.
It then follows that
$$
    \dim_k S_i
  = \dim_k D_i^{n_i}
  = n_i \dim_k D_i
  = n_i \dim_k D_i^{\mathrm{op}}
  = n_i \dim_k \operatorname{End}_R(S_i).
$$
