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:
“Young children make only the simple assumption: “This is life—you go along....” He stands ready to go along with whatever adults seem to want. He stands poised, trying to figure out what they want. The young child is almost at the mercy of adults—it is so important to him to please.”
—James L. Hymes, Jr. (20th century)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)