# associates

Two elements in a ring with unity are *associates* or associated elements of each other
if one can be obtained from the other by multiplying by some unit,
that is, $a$ and $b$ are associates if there is a unit $u$ such that
$a=bu$. Equivalently, one can say that two associates are divisible by each other.

The binary relation^{} “is an associate of” is an equivalence relation^{}
on any ring with unity. For example, the equivalence class^{} of the
unity of the ring consists of all units of the ring.

Examples. In the ring $\mathbb{Z}$ of the rational integers, only opposite numbers $\pm n$ are associates. Among the polynomials, the associates of a polynomial are gotten by multiplying the polynomial by an element belonging to the coefficient ring in question (and being no zero divisor^{}).

In an integral domain^{}, two elements are associates if and only if they
generate the same principal ideal^{}.

Title | associates |
---|---|

Canonical name | Associates |

Date of creation | 2013-03-22 11:56:31 |

Last modified on | 2013-03-22 11:56:31 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 10 |

Author | drini (3) |

Entry type | Definition |

Classification | msc 13-00 |

Classification | msc 16-00 |

Related topic | Ring |

Related topic | Unit |

Defines | associate |

Defines | associated element |