# height of a prime ideal

Let $R$ be a commutative ring and $\mathrm{\pi \x9d\x94\xad}$ a prime ideal^{} of $R$. The height of $\mathrm{\pi \x9d\x94\xad}$ is the supremum^{} of all integers $n$ such that there exists a chain

$${\mathrm{\pi \x9d\x94\xad}}_{0}\beta \x8a\x82\mathrm{\beta \x8b\u2015}\beta \x8a\x82{\mathrm{\pi \x9d\x94\xad}}_{n}=\mathrm{\pi \x9d\x94\xad}$$ |

of distinct prime ideals. The height of $\mathrm{\pi \x9d\x94\xad}$ is denoted by $\mathrm{h}\beta \x81\u2018(\mathrm{\pi \x9d\x94\xad})$.

$\mathrm{h}\beta \x81\u2018(\mathrm{\pi \x9d\x94\xad})$ is also known as the rank of $\mathrm{\pi \x9d\x94\xad}$ and the codimension of $\mathrm{\pi \x9d\x94\xad}$.

The Krull dimension of $R$ is the supremum of the heights of all the prime ideals of $R$:

$$sup\beta \x81\u2018\{\mathrm{h}\beta \x81\u2018(\mathrm{\pi \x9d\x94\xad})\beta \x88\pounds \mathrm{\pi \x9d\x94\xad}\beta \x81\u2019\text{\Beta prime in\Beta}\beta \x81\u2019R\}.$$ |

Title | height of a prime ideal |
---|---|

Canonical name | HeightOfAPrimeIdeal |

Date of creation | 2013-03-22 12:49:25 |

Last modified on | 2013-03-22 12:49:25 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 10 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 14A99 |

Synonym | height |

Related topic | KrullDimension |

Related topic | Cevian |

Defines | rank of an ideal |

Defines | codimension of an ideal |