Krull dimension

Equivalently, $\dim R$ is the Krull dimension of the topological space $\operatorname{Spec}(R)$ with its Zariski topology.

The Krull dimension can also be expressed in terms of heights: for each prime $\mathfrak p$, its height $\operatorname{ht}(\mathfrak p)$ is the supremum of lengths of prime chains ending at $\mathfrak p$, and

Moreover, $\operatorname{ht}(\mathfrak p)$ is the dimension of the localization $R_{\mathfrak p}$.

1. Fields and nonzero Artinian rings have dimension $0$. If $k$ is a field, its only prime ideal is $(0)$, so $\dim k=0$. More generally, every prime ideal in a nonzero commutative Artinian ring is maximal, so no strict chain of prime ideals has positive length.

1. Dimension $1$: $\mathbb Z$ and $k[x]$. In $\mathbb Z$ there are chains $(0)\subsetneq(p)$, but no longer chains, so $\dim\mathbb Z=1$. Similarly, for a field $k$, the ring $k[x]$ has chains $(0)\subsetneq(f)$, with $f$ irreducible, but no longer ones.

1. Polynomial rings. For a field $k$, the polynomial ring $k[x_1,\dots,x_n]$ has Krull dimension $n$. For instance, in $k[x,y]$,
   $$(0)\subsetneq (x)\subsetneq (x,y),$$
   has length $2$, and no longer prime chain exists.