Group ring

English

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