Group Rings

Let be a finite group. The group ring of is the free abelian group on the elements of equipped with multiplication given by the group structure on . Note that is a commutative ring if and only if  is commutative.

For example, the group ring of the cyclic group of order  is the free -module on , and the multiplication is induced by extended linearly. For example, in we have

You might think that is isomorphic to the ring of integers of , but you would be wrong, since the ring of integers is isomorphic to as abelian group, but is isomorphic to as abelian group. (Note that is a quadratic extension of  .)

William Stein 2012-09-24