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.