group ring

group ring (plural group rings)

1. (algebra) Given ring R with identity not equal to zero, and group ${\displaystyle G=\{g_{1},g_{2},...,g_{n}\}}$ , the group ring RG has elements of the form ${\displaystyle a_{1}g_{1}+a_{2}g_{2}+...+a_{n}g_{n}}$ (where ${\displaystyle a_{i}\in R}$ ) such that the sum of ${\displaystyle a_{1}g_{1}+a_{2}g_{2}+...+a_{n}g_{n}}$ and ${\displaystyle b_{1}g_{1}+b_{2}g_{2}+...+b_{n}g_{n}}$ is ${\displaystyle (a_{1}+b_{1})g_{1}+(a_{2}+b_{2})g_{2}+...+(a_{n}+b_{n})g_{n}}$ and the product is ${\displaystyle \sum _{k=1}^{n}\left(\sum _{g_{i}g_{j}=g_{k}}a_{i}b_{j}\right)g_{k}}$ .