# unit

Let $R$ be a ring with multiplicative identity^{} $1$. We say that $u\in R$ is an unit (or unital) if $u$ divides $1$ (denoted $u\mid 1$). That is, there exists an $r\in R$ such that $1=ur=ru$.

Notice that $r$ will be the multiplicative inverse (in the ring) of $u$, so we can characterize the units as those elements of the ring having multiplicative inverses.

In the special case that $R$ is the ring of integers^{} of an algebraic number field^{} $K$, the units of $R$ are sometimes called the algebraic units of $K$ (and also the units of $K$). They are determined by Dirichlet’s unit theorem.

Title | unit |

Canonical name | Unit |

Date of creation | 2013-03-22 11:56:28 |

Last modified on | 2013-03-22 11:56:28 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 15 |

Author | drini (3) |

Entry type | Definition |

Classification | msc 16B99 |

Synonym | unital |

Related topic | Associates^{} |

Related topic | Prime |

Related topic | Ring |

Related topic | UnitsOfQuadraticFields |

Defines | algebraic unit |