Structure theorem for simple rings

The simple case of the Artin–Wedderburn theorem says:

A ring \(R\) is simple Artinian if and only if \[ R\cong M_n(D) \] for some integer \(n\ge1\) and some division ring \(D\).

Here “simple” means that \(R\) has no two-sided ideals other than \(0\) and \(R\), and “Artinian” means that descending chains of left ideals terminate.

The converse is direct. The matrix ring \(M_n(D)\) is left and right Artinian because it is finite-dimensional over the division ring \(D\), and every nonzero two-sided ideal contains a nonzero matrix. Multiplying by matrix units then produces every matrix unit, so the ideal is all of \(M_n(D)\).

For the forward direction, the Artinian hypothesis gives a minimal nonzero left ideal \(L\). Schur’s lemma implies that \[ D=\operatorname{End}_R(L)^{\mathrm{op}} \] is a division ring. The regular left module \(R\) is a finite direct sum of minimal left ideals, and simplicity makes all of them isomorphic to \(L\): \[ {}_R R\cong L^{\oplus n}. \] Taking endomorphism rings gives \[ R^{\mathrm{op}}\cong\operatorname{End}_R(R) \cong M_n(\operatorname{End}_R(L)), \] and hence \[ R\cong M_n(D). \]