I have recently been introduced to the Krull dimension of rings, and the definition I am using is consistent with that on wikipedia: the Krull dimension of a ring $R$ is "the supremum of the lengths of all chains of prime ideals in $R$".
However the only definitions I can find do not specify whether the prime ideals that can be used in said chains are proper or not.
Wikipedia claims that $\mathbb{Z}$ has Krull dimension 1. Would ${0}$ $\subseteq $ $<2>$ $\subseteq$ $\mathbb{Z}$ not be a counterexample giving a chain of length $2$, assuming we count from $0$? Why is clarification on proper ideals not given? Is it an obvious assumption that the ring itself is not included?