group ring

For any group $G$, the group ring $\mathbb{Z}[G]$ is defined to be the ring whose additive group is the abelian group of formal integer linear combinations of elements of $G$, and whose multiplication operation is defined by multiplication in $G$, extended $\mathbb{Z}$–linearly to $\mathbb{Z}[G]$.

More generally, for any ring $R$, the group ring of $G$ over $R$ is the ring $R[G]$ whose additive group is the abelian group of formal $R$–linear combinations of elements of $G$, i.e.:

$$R[G]:=\left\{\sum _{i=1}^n{r}_i{g}_i\right|r_i\in R,g_i\in G\},$$

and whose multiplication operation is defined by $R$–linearly extending the group multiplication operation of $G$. In the case where $K$ is a field, the group ring $K[G]$ is usually called a group algebra.