# 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 |