Spahn new page Krull dimension

Krull dimension, dimension of an affine scheme

Definition

Let $R$ be a ring.

The Krull dimension of $R$ is defined to be the supremum of the number of strict inclusions of prime ideals in $R$.

Examples

• The Krull dimension of a field is $0$.

• An integral domain is a field iff its Krull dimension is zero.

• The Krull dimension of a PID which is not a field is $1$.

• The Krull dimension of $k[X_1,\dots,X_n]$ for a field $k$ is $n$.

• The Krull dimension of $R[X]$ for a noetherian ring of Krull dimension $d$ is $d+1$. This is not always true if $R$ is not noetherian.