Group ring

Summary

The group ring $R[G]$ of a group $G$ over a ring $R$ is the ring of finite formal sums $\sum_g a_g g$ with coefficients in $R$. Such sums are multiplied using the group law. It turns questions about $G$ and its actions into questions about modules over one ring. When $R$ is a field, it is the group algebra, the basic object of representation theory. It is the special case of the monoid ring in which the monoid is a group.