GroupRing 2002-01-23 14:24:20 2002-11-06 12:08:53 Definition group algebra % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here For any group $G$, the {\em 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 {\em 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.: $$ R[G] := \left\{\left. \sum_{i=1}^n r_i g_i\ \right|\ r_i \in R,\ g_i \in G\right\}, $$ 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 {\em group algebra}.