Krull dimension

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Template:Curing-dimension notion

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

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.