Krull dimension

Definition

Symbol-free definition

The Krull dimension of a commutative unital ring is the supremum of lengths of descending chains of distinct prime ideals.

Definition with symbols

Let $R$ be a commutative unital ring. The Krull dimension of $R$, denoted $dim(R)$ is the supremum over all $n$ for which there exist strictly descending chains of prime ideals:

$P_0 \supset P_1 \supset \ldots \supset P_n$

Related ring properties

• Zero-dimensional ring is a ring whose Krull dimension is zero. Particular examples of such rings are Artinian rings and completely local rings.
• Any integral domain which is not a field must have dimension at least one. A one-dimensional domain is an integral domain which has Krull dimension at most one. Any principal ideal domain, and more generally, any Dedekind domain, is one-dimensional.
• A finite-dimensional ring is a ring with finite Krull dimension; a finite-dimensional domain is an integral domain with finite Krull dimension.

Facts

• The polynomial ring over any Noetherian ring of finite dimension, has dimension one more than the original ring. Thus, we see that the polynomial ring in $n$ variables over a field, has dimension $n$, while the polynomial ring in $n$ variables over a principal ideal domain (or Dedekind domain) which is not a field, has dimension $n + 1$
• For a Noetherian local ring, the Krull dimension equals the degree of its Hilbert-Samuel polynomial.