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 dimR/0 : ^{J ∞} is the maximum number of analytically independent elements in an arbitrary ideal J of a local ring R and that dimB ≤ dimA if B ⊂ A are (not necessarily finitely generated) subalgebras of a finitely generated algebra over a Noetherian Jacobson ring.