Krull dimension

Let $R$ be a commutative ring . The Krull dimension of $R$, denoted $\dim R$, is the supremum of integers $n\ge 0$ for which there exists a strictly increasing chain of prime ideals

in $R$. If no such finite supremum exists, one writes $\dim R = \infty$.

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 one has

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

Examples

1. Fields and Artinian rings have dimension $0$.
   If $k$ is a field , the only prime ideal is $(0)$, so $\dim k=0$. More generally, if $R$ is an Artinian ring , then every prime ideal is maximal and there are no nontrivial chains of primes, so $\dim R=0$.

2. 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, hence $\dim k[x]=1$.

3. Polynomial rings.
   For a field $k$, the polynomial ring $k[x_1,\dots,x_n]$ has Krull dimension $n$. For instance, in $k[x,y]$ one has the chain

   $$(0)\subsetneq (x)\subsetneq (x,y),$$

   showing $\dim k[x,y]\ge 2$, and in fact equality holds.