# Krull’s principal ideal theorem

Let $R$ be a Noetherian ring^{}, and $P$ be a prime ideal^{} minimal over a principal ideal^{} $(x)$.
Then the height (http://planetmath.org/HeightOfAPrimeIdeal) of $P$, that is, the dimension (http://planetmath.org/KrullDimension) of ${R}_{P}$, is less than 1.
More generally, if $P$ is a minimal prime of an ideal generated by $n$ elements, the height of $P$ is less than $n$.

Title | Krull’s principal ideal theorem |
---|---|

Canonical name | KrullsPrincipalIdealTheorem |

Date of creation | 2013-03-22 13:12:08 |

Last modified on | 2013-03-22 13:12:08 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 5 |

Author | drini (3) |

Entry type | Theorem |

Classification | msc 13C15 |