group ring

English

group ring (plural group rings)

(algebra) 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 $\sum _{k=1}^{n}\left(\sum _{g_{i}g_{j}=g_{k}}a_{i}b_{j}\right)g_{k}$ .

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