Two Simple Examples
Let G = Z3, the cyclic group of three elements with generator a and identity element . An element r of C may be written as
where z0, z1 and z2 are in C, the complex numbers. Writing a different element s as
their sum is
and their product is
Notice that the identity element of G induces a canonical embedding of the coefficient ring (in this case C) into C; however strictly speaking the multiplicative identity element of C is where the first 1 comes from C and the second from G. The additive identity element is of course zero.
When G is a non-commutative group, one must be careful to preserve the order of the group elements (and not accidentally commute them) when multiplying the terms.
A different example is that of the Laurent polynomials over a ring R: these are nothing more or less than the group ring of the infinite cyclic group Z over R.
Read more about this topic: Group Ring
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