# supercommutative

Let $R$ be a ${\mathbb{Z}}_{2}$-graded ring^{} (or more generally, an associative algebra). We say that $R$ is supercommutative if for any homogeneous elements $a$ and $b\in R$:

$$ab={(-1)}^{\mathrm{deg}a\mathrm{deg}b}ba.$$ |

In other words, even homogeneous elements are in the center of the ring, and odd homogeneous elements anti-commute.

Common examples of supercommutative rings are the exterior algebra^{} of a module over a commutative ring (in particular, a vector space^{}) and the cohomology ring of a topological space (both with the standard grading by degree reduced mod 2).

Title | supercommutative |
---|---|

Canonical name | Supercommutative |

Date of creation | 2013-03-22 13:25:18 |

Last modified on | 2013-03-22 13:25:18 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 7 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 16W50 |

Synonym | graded-commutative |

Synonym | supercommutative algebra |

Synonym | even element |

Synonym | odd element |

Related topic | SuperAlgebra |