Group ring of finite group over field of characteristic not dividing its order is semisimple Artinian
Suppose is a finite group and is a field whose characteristic does not divide the order of . Then, the group ring (i.e., the group ring of over is a semisimple Artinian ring. In particular, it is a finite-dimensional semisimple algebra over .
- Artin-Wedderburn theorem that tells us further that the group ring must therefore be a direct sum of matrix rings over division rings.
- Group ring over splitting field is direct sum of matrix rings for each irreducible representation: In the case of a splitting field, the group ring is a direct sum of matrix rings over the field itself, with the sizes of the matrices corresponding to the degrees of irreducible representations.