Krull dimension
From Commalg
Template:Curing-dimension notion
Contents
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 be a commutative unital ring. The Krull dimension of
, denoted
is the supremum over all
for which there exist strictly descending chains of prime ideals:
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
variables over a field, has dimension
, while the polynomial ring in
variables over a principal ideal domain (or Dedekind domain) which is not a field, has dimension
- For a Noetherian local ring, the Krull dimension equals the degree of its Hilbert-Samuel polynomial.