# Dedekind domains with finitely many primes are PIDs

A commutative ring in which there are only finitely many maximal ideals^{} is known as a semi-local ring. For such rings, the property of being a Dedekind domain^{} and of being a principal ideal domain^{} coincide.

###### Theorem.

A Dedekind domain in which there are only finitely many prime ideals^{} is a principal ideal domain.

This result is sometimes proven using the chinese remainder theorem^{} or, alternatively, it follows directly from the fact that invertible ideals in semi-local rings are principal.

Suppose that $R$ is a Dedekind domain such as the ring of algebraic integers in a number field. Although there are infinitely many prime ideals in such a ring, we can use the result that localizations of Dedekind domains are Dedekind and apply the above theorem to localizations^{} of $R$.

In particular, if $\U0001d52d$ is a nonzero prime ideal, then ${R}_{\U0001d52d}\equiv {(R\setminus \U0001d52d)}^{-1}R$ is a Dedekind domain with a unique nonzero prime ideal, so the theorem shows that it is a principal ideal domain.

Title | Dedekind domains with finitely many primes are PIDs |
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Canonical name | DedekindDomainsWithFinitelyManyPrimesArePIDs |

Date of creation | 2013-03-22 18:35:18 |

Last modified on | 2013-03-22 18:35:18 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 6 |

Author | gel (22282) |

Entry type | Theorem |

Classification | msc 13F05 |

Classification | msc 11R04 |

Related topic | DivisorTheory |