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.