# an Artinian integral domain is a field

Let $R$ be an integral domain^{}, and assume that $R$ is Artinian.

Let $a\in R$ with $a\ne 0$. Then $R\supseteq aR\supseteq {a}^{2}R\supseteq \mathrm{\cdots}$.

As $R$ is Artinian, there is some $n\in \mathbb{N}$ such that ${a}^{n}R={a}^{n+1}R$. There exists $r\in R$ such that ${a}^{n}={a}^{n+1}r$, that is, ${a}^{n}1={a}^{n}(ar)$. But ${a}^{n}\ne 0$ (as $R$ is an integral domain), so we have $1=ar$. Thus $a$ is a unit.

Therefore, every Artinian integral domain is a field.

Title | an Artinian integral domain is a field |
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Canonical name | AnArtinianIntegralDomainIsAField |

Date of creation | 2013-03-22 12:49:37 |

Last modified on | 2013-03-22 12:49:37 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 13 |

Author | yark (2760) |

Entry type | Theorem |

Classification | msc 16P20 |

Classification | msc 13G05 |

Related topic | AFiniteIntegralDomainIsAField |