# 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.:

$$R[G]:=\left\{\sum _{i=1}^{n}{r}_{i}{g}_{i}\right|{r}_{i}\in R,{g}_{i}\in G\},$$ |

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 |
---|---|

Canonical name | GroupRing |

Date of creation | 2013-03-22 12:13:27 |

Last modified on | 2013-03-22 12:13:27 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 8 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 20C05 |

Classification | msc 20C07 |

Classification | msc 16S34 |

Defines | group algebra |