Height of a prime ideal

Remarks

The height is an element of $\set{0, 1, 2, \dots} \cup \set{\infty}$, and $\operatorname{ht} P = \dim R_P$, the Krull dimension of the localization at $P$. Example: in $k[x, y]$ over a field $k$, $\operatorname{ht}(x, y) = 2$, because of the chain $(0) \subsetneq (x) \subsetneq (x, y)$, and the dimension of $k[x, y]$ is $2$. Non-example of a chain ending at $(x, y)$: $(x^2) \subsetneq (x) \subsetneq (x, y)$ does not give height $3$, because $(x^2)$ is not prime. The prime ideals of height $0$ are the minimal prime ideals of $R$. For any ideal $I \ne R$, the height of $I$ is the least height of a prime ideal that contains $I$. One has $\operatorname{ht} P + \dim R/P \le \dim R$.