From Wikipedia we have for a ring and a group
a new ring written as
that is defined on the functions
with finite support and addition and multiplication defined as follows.
and
Here, stands for a (formal) sum of elements of
with coefficients taken from the ring
. This makes sense as long as the functions in question have finite support, i.e.
.
The group ring is written . It is clear that the group ring is also a module over the ring
. For an element
let us define the following mapping (action) on the group ring
This is nothing else but a representation of the group on the linear mappings of the group ring as a module.
If is a field the group ring is called a group algebra over the field. In this case the group action from above is nothing else but a representation of the group on the linear mappings of the group algebra as a vector space.
As an example let us investigate the (multiplicative) group of the -th unit roots,
. Then we have for an arbitrary ring
an element of
:
with
For a product of 2 elements we get
(1)
taking and analogously for
. The coefficients of the product are the result of a convolution of the two
-element series
and
.