Group Rings
For any finite group 
the group ring 
of is the group of formal linear combinations of elements of , with coefficients in 
and multiplication defined on the basis elements and then extended linearly:
Even though it is called a group ring this important object actually has the structure of a -algebra. That is it is both a ring, and a finite dimensional vector space over . In reading what follows it is important to keep clear in your mind when we are treating 
as a ring, and when we are treating it as a vector space over .
A representation of of a a -algebra 
on a finite dimensional vector space 
is an algebra homomorphism:
Here 
is the space of endomorphisms of . If is of dimension 
then once a basis has been chosen 
may be thought of as the algebra of by matrices.
Every representation 
of a group on a vector space lifts naturally to a representation 
of the algebra on the same vector space.
Since these two things are really “the same” in a certain category theoretical sense which I don’t really want to go into just now (I’m starting to sound like a physicist, aren’t I?), we usually drop the hat and swap back and forth freely between the representation of the group, and the representation of the associated group ring.
Now, there are two natural representations of on the group ring thought of as a vector space. Firstly, right multiplication:
given by:
and secondly left multiplication by the inverse:
given by:
The linear map:
is an intertwiner between these two representations. In other words, upto conjugation they are really the same representation
On the dual space 
of linear functions on we have:
given by:
One can check that:
Since the left and right actions clearly commute, we actually have a representation:
given by:
Similarly we have:
given by:
The linear map:
is an intertwiner between these two representations of 
.
The inverse map is given by:
We have all the ingredients now to prove the Artin-Wedderburn theorem for semisimple rings.