= Artin–Wedderburn theorem
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Every finite-dimensional semisimple algebra is a finite product of matrix algebras over division algebras. In particular, every <central simple algebra> over $k$ is isomorphic to $M_r(D)$ for a finite-dimensional central division algebra $D$ over $k$. The integer $r$ and the isomorphism class of $D$ are unique.
Indeed, decompose the right regular module of a finite-dimensional semisimple algebra $A$ as
$$
A_A\cong\bigoplus_{i=1}^sS_i^{n_i},
$$
where the $S_i$ are pairwise nonisomorphic <simple modules>. The <Schur lemma> makes $D_i=\operatorname{End}_A(S_i)$ a <division ring> and gives $\operatorname{Hom}_A(S_i,S_j)=0$ for $i\ne j$. Left multiplication identifies
$$
A\cong\operatorname{End}_A(A_A)
\cong\prod_{i=1}^sM_{n_i}(D_i).
$$
Conversely, the column module over $M_n(D)$ is simple and the regular module is a direct sum of $n$ copies of it, so finite products of such matrix rings are semisimple. The simple-module decomposition also proves uniqueness up to permuting the factors.
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