height of a prime ideal

Let R be a commutative ring and 𝔭 a prime idealMathworldPlanetmathPlanetmathPlanetmath of R. The height of 𝔭 is the supremumPlanetmathPlanetmath of all integers n such that there exists a chain


of distinct prime ideals. The height of 𝔭 is denoted by h⁑(𝔭).

h⁑(𝔭) is also known as the rank of 𝔭 and the codimension of 𝔭.

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

sup⁑{h⁑(𝔭)βˆ£π”­β’Β prime in ⁒R}.
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