graded algebra

An algebra $A$ over a graded ring $B$ is 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.

Title	graded algebra
Canonical name	GradedAlgebra
Date of creation	2013-03-22 12:45:47
Last modified on	2013-03-22 12:45:47
Owner	mhale (572)
Last modified by	mhale (572)
Numerical id	8
Author	mhale (572)
Entry type	Definition
Classification	msc 16W50
Related topic	GradedModule
Related topic	SuperAlgebra
Related topic	LieSuperalgebra
Related topic	LieSuperalgebra3