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.: $$ R[G] := \left\{\left. \sum_{i=1}^n r_i g_i\ \right|\ r_i \in R,\ g_i \in G\right\}, $$ 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.