Krull dimension

Remarks

For the zero ring, which has no prime ideals, the dimension is set to $-1$ or $-\infty$ by convention. Example: $\dim \Z = 1$; the chain $(0) \subsetneq (p)$ for a prime $p$ has length $1$, and no chain is longer, since every nonzero prime ideal of $\Z$ is maximal. Non-example of a chain: $(0) \subsetneq (6) \subsetneq (3)$ does not show $\dim \Z \ge 2$, because $(6)$ is not a prime ideal. For a field $k$, $\dim k[x_1, \dots, x_n] = n$, and the polynomial ring $k[x_1, x_2, \dots]$ in infinitely many variables has infinite dimension. A Noetherian ring can still have infinite dimension, by an example of Nagata.