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
Famous quotes containing the words simple and/or examples:
“If we do take statements to be the primary bearers of truth, there seems to be a very simple answer to the question, what is it for them to be true: for a statement to be true is for things to be as they are stated to be.”
—J.L. (John Langshaw)
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold peoples attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)