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'graded algebra'
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| Title of object: |
graded algebra |
| Canonical Name: |
GradedAlgebra |
| Type: |
Definition |
| Created on: |
2002-06-07 12:00:00 |
| Modified on: |
2007-08-07 22:20:25 |
| Classification: |
msc:16W50 |
Revision comment (for changes between this and next version):
| Changes for correction #12960 ('definition is inconsistent'). |
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:
An algebra $A$ is \emph{graded} if it is a graded module and satisfies
$$A^p \cdot A^q \subseteq A^{p+q}$$
where $A^i$, $i \in \mathbb{N}$, are abelian subgroups of $A$.
More generally, one can replace $\mathbb{N}$ by a monoid or semigroup $G$.
In which case, $A$ is called a $G$-graded algebra.
A graded algebra then is the same thing as an $\mathbb{N}$-graded algebra.
Examples of graded algebras include the polynomial ring $k[X]$ being an $\mathbb{N}$-graded $k$-algebra, and the exterior algebra. |
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