Let \(R\) be a commutative ring and \(G = \ts{g_1, \ldots, g_n}\) a finite group. The group ring \(RG\) is the free \(R\)-module with basis the elements of \(G\), \[ RG\coloneqq\ts{ \sum_{i=1}^n a_i g_i \st a_i\in R }, \] with the multiplication that extends the group law \(R\)-bilinearly: \((ag)(bh)\coloneqq(ab)(gh)\) for \(a, b\in R\) and \(g, h\in G\).
The group ring \(RG\) is a ring with identity \(1_R 1_G\). If \(R\neq 0\), then \(RG\) is commutative if and only if \(G\) is abelian.
See [@DF04].