# anti-isomorphism

Let $R$ and $S$ be rings and $f:R\u27f6S$ be a function such that $f({r}_{1}{r}_{2})=f({r}_{2})f({r}_{1})$ for all ${r}_{1},{r}_{2}\in R$.

If $f$ is a homomorphism^{} of the additive groups^{} of $R$ and $S$,
then $f$ is called an anti-homomorphsim.

If $f$ is a bijection and anti-homomorphism, then $f$ is called an anti-isomorphism.

If $f$ is an anti-homomorphism and $R=S$ then $f$ is called an anti-endomorphism.

If $f$ is an anti-isomorphism and $R=S$ then $f$ is called an anti-automorphism.

As an example, when $m\ne n$, the mapping that sends a matrix to its transpose^{}
(or to its conjugate transpose^{} if the matrix is complex) is an anti-isomorphism
of ${M}_{m,n}\to {M}_{n,m}$.

$R$ and $S$ are *anti-isomorphic* if there is an anti-isomorphism $R\to S$.

All of the things defined in this entry are also defined for groups.

Title | anti-isomorphism |
---|---|

Canonical name | Antiisomorphism |

Date of creation | 2013-03-22 16:01:08 |

Last modified on | 2013-03-22 16:01:08 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 15 |

Author | Mathprof (13753) |

Entry type | Definition |

Classification | msc 13B10 |

Classification | msc 16B99 |

Defines | anti-endomorphism |

Defines | anti-homomorphism |

Defines | anti-isomorphic |

Defines | anti-automorphism |