Editing Group ring (section)

==Examples==
1. Let {{nowrap|1=''G'' = ''C''<sub>3</sub>}}, the [[cyclic group]] of order 3, with generator <math>a</math> and identity element 1<sub>''G''</sub>. An element ''r'' of '''C'''[''G''] can be written as

:<math>r = z_0 1_G + z_1 a + z_2 a^2\,</math>

where ''z''<sub>0</sub>, ''z''<sub>1</sub> and ''z''<sub>2</sub> are in '''C''', the [[complex numbers]]. This is the same thing as a [[polynomial ring]] in variable <math>a</math> such that <math>a^3=a^0=1</math> i.e. '''C'''[''G''] is isomorphic to the ring '''C'''[<math>a</math>]/<math>(a^3-1)</math>.

Writing a different element ''s'' as <math>s=w_0 1_G +w_1 a +w_2 a^2</math>, their sum is

:<math>r + s = (z_0+w_0) 1_G + (z_1+w_1) a + (z_2+w_2) a^2\,</math>

and their product is

:<math>rs = (z_0w_0 + z_1w_2 + z_2w_1) 1_G  +(z_0w_1 + z_1w_0 + z_2w_2)a +(z_0w_2 + z_2w_0 + z_1w_1)a^2.</math>

Notice that the identity element 1<sub>''G''</sub> of ''G'' induces a canonical embedding of the coefficient ring (in this case '''C''') into '''C'''[''G'']; however strictly speaking the multiplicative identity element of '''C'''[''G''] is 1⋅1<sub>''G''</sub> where the first ''1'' comes from '''C''' and the second from ''G''. The additive identity element is 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.

2. The ring of [[Laurent polynomial]]s over a ring ''R'' is the group ring of the [[infinite cyclic group]] '''Z''' over ''R''.

3. Let ''Q'' be the [[quaternion group]] with elements <math>\{e, \bar{e}, i, \bar{i}, j, \bar{j}, k, \bar{k}\}</math>. Consider the group ring '''R'''''Q'', where '''R''' is the set of real numbers. An arbitrary element of this group ring is of the form

:<math>x_1 \cdot e + x_2 \cdot \bar{e} + x_3 \cdot i + x_4 \cdot \bar{i} + x_5 \cdot j + x_6 \cdot \bar{j} + x_7 \cdot k + x_8 \cdot \bar{k}</math>
where <math>x_i </math> is a real number.

Multiplication, as in any other group ring, is defined based on the group operation. For example,

:<math>\begin{align} \big(3 \cdot e + \sqrt{2} \cdot i \big)\left(\frac{1}{2} \cdot \bar{j}\right) &= (3 \cdot e)\left(\frac{1}{2} \cdot \bar{j}\right) + (\sqrt{2} \cdot i)\left(\frac{1}{2} \cdot \bar{j}\right)\\
&= \frac{3}{2} \cdot \big((e)(\bar{j})\big) + \frac{\sqrt{2}}{2} \cdot \big((i)(\bar{j})\big)\\
&= \frac{3}{2} \cdot \bar{j} + \frac{\sqrt{2}}{2} \cdot \bar{k}
\end{align}.</math>

Note that '''R'''''Q'' is not the same as the skew field of [[quaternions]] over '''R'''. This is because the skew field of quaternions satisfies additional relations in the ring, such as <math>-1 \cdot i = -i</math>, whereas in the group ring '''R'''''Q'', <math>-1\cdot i</math> is not equal to <math>1\cdot \bar{i}</math>. To be more specific, the group ring '''R'''''Q'' has dimension 8 as a real [[vector space]], while the skew field of quaternions has dimension 4 as a [[real vector space]].

4. Another example of a non-abelian group ring is <math>\mathbb{Z}[\mathbb{S}_3]</math> where <math>\mathbb{S}_3</math> is the symmetric group on 3 letters. This is not an integral domain since we have <math>[1 - (12)]*[1+(12)] = 1 -(12)+(12) -(12)(12) = 1 - 1 = 0</math> where the element <math>(12)\in \mathbb{S}_3</math> is the [[Cyclic_permutation#Transpositions|transposition]] that swaps 1 and 2. Therefore the group ring need not be an integral domain even when the underlying ring is an integral domain.