Krull dimension and monomial orders

We introduce the notion of independent sequences with respect to a monomial order by using the least terms of polynomials vanishing at the sequence. Our main result shows that the Krull dimension of a Noetherian ring is equal to the supremum of the length of independent sequences. The proof has led to other notions of independent sequences, which have interesting applications. For example, we can show that dim R/0 : J(infinity) is the maximum number of analytically independent elements hi an arbitrary ideal J of a local ring R and that dim B <= dim A if B subset of A are (not necessarily finitely generated) subalgebras of a finitely generated algebra over a Noetherian Jacobson ring. (C) 2013 Elsevier Inc. All rights reserved.