# 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.:

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

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.

Title group ring GroupRing 2013-03-22 12:13:27 2013-03-22 12:13:27 djao (24) djao (24) 8 djao (24) Definition msc 20C05 msc 20C07 msc 16S34 group algebra