What is a semisimple algebra? State the structure theorem for semisimple algebras.
Solution
A ring \(A\) is semisimple if the left \(A\)-module \({}_AA\) is semisimple, i.e. a direct sum of simple submodules. Equivalently, every left \(A\)-module is semisimple; equivalently, every short exact sequence of left \(A\)-modules splits.
The Artin–Wedderburn theorem states that a ring is semisimple iff it is isomorphic to a finite product \[ A\cong\prod_{i=1}^r M_{n_i}(D_i), \] where each \(D_i\) is a division ring and \(n_i\ge1\).
For a finite-dimensional algebra over a field \(k\), the division rings \(D_i\) are finite-dimensional over \(k\). In particular, if \(k\) is algebraically closed and \(A\) is finite-dimensional over \(k\), then each \(D_i=k\), so \[ A\cong\prod_{i=1}^r M_{n_i}(k). \]
The integers \(r,n_i\) and the division rings \(D_i\) are determined uniquely up to permutation and isomorphism.