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

\mathfrak p_0 \subsetneq \mathfrak p_1 \subsetneq \cdots \subsetneq \mathfrak p_n

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

Equivalent characterizations

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

Remarks

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

\dim R = \sup_{\mathfrak p\in \operatorname{Spec}(R)} \operatorname{ht}(\mathfrak p).

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

Examples

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 $\dim R=0$ .

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 $\dim k[x]=1$ .

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.

Search Knowlpedia

Examples

Examples

Basic properties

Examples