# Krull dimension

If $R$ is a ring, the *Krull dimension ^{}* (or simply dimension

^{}) of $R$, $dimR$ is the supremum

^{}of all integers $n$ such that there is an increasing sequence of prime ideals

^{}${\U0001d52d}_{0}\u228a\mathrm{\cdots}\u228a{\U0001d52d}_{n}$ of length $n$ in $R$.

If $X$ is a topological space^{}, the Krull dimension (or simply dimension) of $X$, $dimX$ is the supremum of all integers $n$ such that there is a decreasing sequence of irreducible^{} closed subsets ${F}_{0}\u228b\mathrm{\cdots}\u228b{F}_{n}$ of $X$.

Title | Krull dimension |
---|---|

Canonical name | KrullDimension |

Date of creation | 2013-03-22 12:03:27 |

Last modified on | 2013-03-22 12:03:27 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 8 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 54-00 |

Synonym | dimension (Krull) |

Related topic | HeightOfAPrimeIdeal |

Related topic | Dimension3 |