group ring

English

Noun

group ring (plural group rings)

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

Synonyms

• group algebra

Hypernyms

• algebra over a ring
• free module
• ring

Translations

Translations

• Portuguese: anel de grupo m