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'group ring'
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| Title of object: |
group ring |
| Canonical Name: |
GroupRing |
| Type: |
Definition |
| Created on: |
2002-01-23 14:24:20-05 |
| Modified on: |
2002-04-19 01:17:11-04 |
| Classification: |
msc:20C05, msc:20C07, msc:16S34 |
Revision comment (for changes between this and next version):
| Changes for correction #1129 ('R need not be commutative'). |
Preamble:
% 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 |
Content:
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 commutative ring $R$, the {\em group algebra} 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 $G$. |
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