Artin–Wedderburn theorem

Artin–Wedderburn theorem: A ring $R$ is semisimple Artinian if and only if there exist positive integers $n_1,\dots,n_t$ and division rings $D_1,\dots,D_t$ such that

where $M_{n_i}(D_i)$ is the matrix ring of size $n_i$ over $D_i$. In particular, $R$ is simple Artinian if and only if $R\cong M_n(D)$ for some division ring $D$.

This is the structural classification underpinning semisimple rings and explains why representation-theoretic decompositions are controlled by matrix blocks.

Proof sketch: Semisimplicity implies that the regular module $R_R$ decomposes into a finite direct sum of simple modules; Artinian hypotheses ensure finiteness and allow lifting idempotents. One shows $R$ is isomorphic to a product of endomorphism rings of simple modules, and each such endomorphism ring is a matrix ring over a division ring by Schur-type arguments. Conversely, finite products of matrix rings over division rings are semisimple and Artinian by explicit module decompositions.