Representations of A Group Ring
A module M over R is then the same as a linear representation of G over the field R. There is no particular reason to limit R to be a field here. However, the classical results were obtained first when R is the complex number field and G is a finite group, so this case deserves close attention. It was shown that R is a semisimple ring, under those conditions, with profound implications for the representations of finite groups. More generally, whenever the characteristic of the field R does not divide the order of the finite group G, then R is semisimple (Maschke's theorem).
When G is a finite abelian group, the group ring is commutative, and its structure is easy to express in terms of roots of unity. When R is a field of characteristic p, and the prime number p divides the order of the finite group G, then the group ring is not semisimple: it has a non-zero Jacobson radical, and this gives the corresponding subject of modular representation theory its own, deeper character.
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