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

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

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  $\mathbf {Q}$.)