## Krull dimension, dimension of an affine scheme +-- {: .num_defn} ###### 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 $R$. =-- +-- {: .num_remark} ###### 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. =--