For questions about or related to the Krull dimension, which counts the length of the longest chain of prime ideals of a ring under inclusion.
The Krull dimension of a commutative ring $R$ is defined to be the supremum of the lengths of chains of prime ideals in $R$. Given a chain of prime ideals
$$p_0 \subsetneq p_1 \subsetneq \dots \subsetneq p_n$$
we define the length of this chain to be $n$ (that is, $n$ is the number of strict inclusions). The Krull dimension is the supremum of the quantity $n$ over all such chains.
A field has Krull dimension $0$, and any principal ideal domain that is not a field has Krull dimension $1$. It is not necessary that a ring has finite Krull dimension, even if the ring is Noetherian.
If $M$ is an $R$-module, we define the Krull dimension of $M$ to be
$$\dim_R M = \dim(R/\operatorname{Ann}_R(M))$$
where $\operatorname {Ann}_R(M)$ is the annihilator in $R$ of $M$.
Reference: Krull dimension.
