Canonical: https://knowlpedia.com/algebra-rings/artin-wedderburn-theorem/

# Artin–Wedderburn theorem

Semisimple Artinian rings are exactly finite products of matrix rings over division rings.

**Artin–Wedderburn theorem**: A ring \(R\) is [semisimple Artinian](https://knowlpedia.com/algebra-rings/artinian-semisimple-ring/) if and only if there exist positive integers \(n_1,\dots,n_t\) and [division rings](https://knowlpedia.com/algebra-rings/division-ring/) \(D_1,\dots,D_t\) such that
\[
R \ \cong\ \prod_{i=1}^t M_{n_i}(D_i),
\]
where \(M_{n_i}(D_i)\) is the [matrix ring](https://knowlpedia.com/algebra-rings/matrix-ring/) of size \(n_i\) over \(D_i\). In particular, \(R\) is [simple](https://knowlpedia.com/algebra-rings/simple-ring/) Artinian if and only if \(R\cong M_n(D)\) for some division ring \(D\).

## Remarks

This is the structural classification underpinning [semisimple rings](https://knowlpedia.com/algebra-rings/semisimple-ring/) and explains why representation-theoretic decompositions are controlled by matrix blocks.