Group Rings

Let $ G$ be a finite group. The group ring $ \mathbf{Z}[G]$ of $ G$ is the free abelian group on the elements of $ G$ equipped with multiplication given by the group structure on $ G$. Note that $ \mathbf{Z}[G]$ is a commutative ring if and only if $ G$ is commutative.

For example, the group ring of the cyclic group $ C_n=\langle a\rangle$ of order $ n$ is the free $ \mathbf {Z}$-module on $ 1,a,\ldots, a^{n-1}$, and the multiplication is induced by $ a^i a^j = a^{i+j} = a^{i + j \pmod{n}}$ extended linearly. For example, in $ \mathbf{Z}[C_3]$ we have

$\displaystyle (1 + 2 a)(1 - a^2) = 1 - a^2 + 2a - 2 a^3 = 1 + 2a - a^2 - 2 = -1 + 2a - a^2.
$

You might think that $ \mathbf{Z}[C_3]$ is isomorphic to the ring $ \mathbf{Z}[\zeta_3]$ of integers of $ \mathbf{Q}(\zeta_3)$, but you would be wrong, since the ring of integers is isomorphic to $ \mathbf{Z}^2$ as abelian group, but $ \mathbf{Z}[C_3]$ is isomorphic to $ \mathbf{Z}^3$ as abelian group. (Note that $ \mathbf{Q}(\zeta_3)$ is a quadratic extension of  $ \mathbf {Q}$.)



William Stein 2012-09-24