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