# opposite ring

If $R$ is a ring, then we may construct the opposite ring ${R}^{op}$ which has the same underlying abelian group^{} structure^{}, but with multiplication in the opposite order: the product of ${r}_{1}$ and ${r}_{2}$ in ${R}^{op}$ is ${r}_{2}{r}_{1}$.

If $M$ is a left $R$-module, then it can be made into a right ${R}^{op}$-module, where a module element $m$, when multiplied on the right by an element $r$ of ${R}^{op}$, yields the $rm$ that we have with our left $R$-module action on $M$. Similarly, right $R$-modules can be made into left ${R}^{op}$-modules.

If $R$ is a commutative ring, then it is equal to its own opposite ring.

Similar constructions occur in the opposite group and opposite category.

Title | opposite ring |
---|---|

Canonical name | OppositeRing |

Date of creation | 2013-03-22 11:51:14 |

Last modified on | 2013-03-22 11:51:14 |

Owner | antizeus (11) |

Last modified by | antizeus (11) |

Numerical id | 7 |

Author | antizeus (11) |

Entry type | Definition |

Classification | msc 16B99 |

Classification | msc 17A01 |

Related topic | DualCategory |

Related topic | NonCommutativeRingsOfOrderFour |