Wedderburn's theorem on simple Artinian rings

Remarks

Let $L$ be a minimal left ideal of $R$, which exists because $R$ is Artinian. The sum of the right translates $La$ is a nonzero two-sided ideal, so it equals $R$. Hence $R$ is a finite direct sum of copies of $L$, say $n$ of them. Then $R^{\mathrm{op}} \cong \End_R(R) \cong M_n(\End_R(L))$, and $\End_R(L)$ is a division ring by Schur's lemma, so $R \cong M_n(D)$ with $D = \End_R(L)^{\mathrm{op}}$. For example, a finite-dimensional simple algebra over $\R$ is $M_n(\R)$, $M_n(\C)$ or $M_n(\mathbb{H})$. The Artinian hypothesis is needed: the Weyl algebra over $\C$ is simple and is not a matrix ring over a division ring.

Going deeper

For a student

The theorem is the case of one factor in the Artin–Wedderburn theorem. It is the starting point of the theory of central simple algebras: a central simple algebra over a field $K$ is $M_n(D)$ for a central division algebra $D$ over $K$. This is why the Brauer group, which collects these $D$, classifies central simple algebras up to matrix size. Uniqueness of $D$ follows because $D^{\mathrm{op}}$ is the endomorphism ring of the unique simple left module of $R$.