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Krull dimension, dimension of an affine scheme

Definition

Let RR be a ring.

The Krull dimension of RR is defined to be the supremum of the number of strict inclusions of prime ideals in RR.

Examples
  • The Krull dimension of a field is 00.

  • An integral domain is a field iff its Krull dimension is zero.

  • The Krull dimension of a PID which is not a field is 11.

  • The Krull dimension of k[X 1,,X n]k[X_1,\dots,X_n] for a field kk is nn.

  • The Krull dimension of R[X]R[X] for a noetherian ring of Krull dimension dd is d+1d+1. This is not always true if RR is not noetherian.

Created on August 20, 2012 at 16:20:59. See the history of this page for a list of all contributions to it.