Krull dimension

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.