Krull dimension

Let $𝑅$ be a commutative ring and $𝔭 ◃ 𝑅$ be a prime ideal. The height $h t 𝔭$ of $𝔭$ is the supremum of the lengths of strictly increasing sequences of prime ideals culminating in $𝔭$

𝔭 0 ⊊ 𝔭 1 ⊊ \dots ⊊ 𝔭 h t 𝔭 = 𝔭

and the Krull dimension $d i m 𝑅$ of $𝑅$ is the supremum of the height of all prime ideals.

Properties

Krull dimension of an integral domain

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