Group ring over splitting field is direct sum of matrix rings for each irreducible representation
Suppose is a finite group and is a splitting field for . Suppose the irreducible representations of are and the degrees of irreducible representations are respectively. Then, the group ring is a semisimple Artinian ring and is expressible as a direct sum of matrix rings over as follows:
where is the ring of matrices over . Note that each is a simple ring and therefore corresponds to a minimal two-sided ideal in the decomposition.
Over non-splitting fields
Even if is not a splitting field, it is true that if the characteristic of does not divide the order of , is a semisimple Artinian ring. By the Artin-Wedderburn theorem, it splits as a direct sum of matrix rings over division rings (in an essentially unique fashion), where each division ring has in its center. If the division rings that we need to use are not fields, then we have the Schur index phenomenon.