graded algebra GradedAlgebra 2002-06-07 12:00:00 2007-09-15 18:12:51 Definition % 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 An algebra $A$ over a graded ring $B$ is \emph{graded} if it is itself a graded ring and a graded module over $B$ such that $$A^p \cdot A^q \subseteq A^{p+q}$$ where $A^i$, $i \in \mathbb{N}$, are submodules 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.