Artin–Wedderburn theorem

The Artin–Wedderburn theorem says that a ring \(R\) is semisimple Artinian if and only if \[ R\cong \prod_{i=1}^r M_{n_i}(D_i), \] where each \(D_i\) is a division ring and each \(n_i\ge1\). The factors are unique up to permutation and isomorphism of the division rings.

For the factor \(M_{n_i}(D_i)\), the unique simple left module up to isomorphism is the column module \(D_i^{n_i}\). Thus the number \(r\) is the number of isomorphism classes of simple left \(R\)-modules, and as a left module \[ R\cong \bigoplus_{i=1}^r S_i^{\oplus n_i}. \] Moreover \[ \operatorname{End}_R(S_i)\cong D_i^{\mathrm{op}}. \]

If \(A\) is a finite-dimensional semisimple algebra over an algebraically closed field \(k\), then every finite-dimensional division \(k\)-algebra is \(k\): for \(a\) in such a division algebra, the subalgebra \(k[a]\) is finite-dimensional, so \(a\) is algebraic over \(k\); its minimal polynomial over the algebraically closed field \(k\) is linear, hence \(a\in k\). Therefore \[ A\cong \prod_{i=1}^r M_{n_i}(k). \] For example, Maschke’s theorem gives, for a finite group \(G\) over \(\mathbb C\), \[ \mathbb C[G]\cong \prod_i M_{d_i}(\mathbb C), \qquad |G|=\sum_i d_i^2. \]