---
tag: R1F271180N
kind: statement
role: theorem
name: Artin–Wedderburn theorem
msc: [16K99, 16D60, 16P20]
uses: [NY25HXA478, 0FHVPRAE60, 0EA5BTA0CX]
about: [NY25HXA478]
sources:
  - url: https://en.wikipedia.org/wiki/Wedderburn%E2%80%93Artin_theorem
same_as:
  wikidata: Q776578
  wikipedia: Wedderburn–Artin theorem
---

## Statement

Let $R$ be a nonzero [[semisimple ring|NY25HXA478]].
Then there are integers $k \ge 1$ and $n_1, \dots, n_k \ge 1$ and [[division rings|0EA5BTA0CX]] $D_1, \dots, D_k$ such that
$$R \cong M_{n_1}(D_1) \times \cdots \times M_{n_k}(D_k),$$
where $M_n(D)$ is the [[matrix ring|0FHVPRAE60]] of $n \times n$ matrices over $D$.
The number $k$ is unique, and the pairs $(n_i, D_i)$ are unique up to order and up to isomorphism of the $D_i$.
Conversely, every such finite product is a semisimple ring.

## Remarks

Group the minimal left ideals of $R$ by isomorphism type; there are finitely many types $S_1, \dots, S_k$, and $R \cong S_1^{n_1} \oplus \cdots \oplus S_k^{n_k}$ as left modules.
Then $R^{\mathrm{op}} \cong \End_R(R)$ is the product of the rings $\End_R(S_i^{n_i}) \cong M_{n_i}(\End_R(S_i))$.
By Schur's lemma each $\End_R(S_i)$ is a division ring, and passing to opposite rings gives the result.
For example, the group algebra $\C[S_3]$ of the symmetric group is isomorphic to $\C \times \C \times M_2(\C)$.
If $R$ is a finite-dimensional semisimple algebra over an algebraically closed field $K$, every $D_i$ equals $K$.

## Explanations

### L1 human

Semisimple rings, which look abstract, are all built from matrices.
Each one is a product of a few matrix rings, and the entries of each matrix ring come from a division ring.
So knowing the division rings gives all semisimple rings.

### L2 human

The theorem reduces the study of semisimple rings, and so of the representations of finite groups in characteristic zero, to division rings and matrix sizes.
The factors $M_{n_i}(D_i)$ correspond to the isomorphism classes of simple modules, and $n_i$ is the multiplicity of the $i$-th simple module in $R$.
For $R = \C[G]$ with $G$ finite this gives $\abs{G} = n_1^2 + \cdots + n_k^2$, where $k$ is the number of irreducible representations.
For a simple Artinian ring there is only one factor, which is Wedderburn's theorem $R \cong M_n(D)$.