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The main result from this note provides a constructive characterization of the valuative dimension, which bears a strong analogy to Lombardi's [7] constructive characterization of the Krull dimension. While Lombardi's characterization uses the lexicographic monomial order, ours uses the graded (reverse) lexicographic order or, in fact, any graded rational monomial order. Apart from this, the paper contains some related results and some examples which readers may find illuminating. [ABSTRACT FROM AUTHOR]

Abstract: 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 is the maximum number of analytically independent elements in an arbitrary ideal J of a local ring R and that if are (not necessarily finitely generated) subalgebras of a finitely generated algebra over a Noetherian Jacobson ring. [Copyright &y& Elsevier]