  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 characterize the maximum number of analytically independent
elements in an arbitrary ideal of a local ring and that dim B is not greater
than dim A if B is a subalgebra of A and A is a (not necessarily finitely
generated) subalgebra of a finitely generated algebra over a Noetherian
Jacobson ring.
