Artin–Wedderburn theorem

Remarks

Group the minimal left ideals of $R$ by isomorphism type; there are finitely many types $S_1, \dots, S_k$, and $R \cong S_1^{n_1} \oplus \cdots \oplus S_k^{n_k}$ as left modules. Then $R^{\mathrm{op}} \cong \End_R(R)$ is the product of the rings $\End_R(S_i^{n_i}) \cong M_{n_i}(\End_R(S_i))$. By Schur's lemma each $\End_R(S_i)$ is a division ring, and passing to opposite rings gives the result. For example, the group algebra $\C[S_3]$ of the symmetric group is isomorphic to $\C \times \C \times M_2(\C)$. If $R$ is a finite-dimensional semisimple algebra over an algebraically closed field $K$, every $D_i$ equals $K$.

Going deeper

For a student

The theorem reduces the study of semisimple rings, and so of the representations of finite groups in characteristic zero, to division rings and matrix sizes. The factors $M_{n_i}(D_i)$ correspond to the isomorphism classes of simple modules, and $n_i$ is the multiplicity of the $i$-th simple module in $R$. For $R = \C[G]$ with $G$ finite this gives $\abs{G} = n_1^2 + \cdots + n_k^2$, where $k$ is the number of irreducible representations. For a simple Artinian ring there is only one factor, which is Wedderburn's theorem $R \cong M_n(D)$.