  For a finitely generated algebra over a field, the transcendence degree is
known to be equal to the Krull dimension. The aim of this paper is to
generalize this result to algebras over rings. A new definition of the
transcendence degree of an algebra A over a ring R is given by calling elements
of A algebraically dependent if they satisfy an algebraic equation over R whose
trailing coefficient, with respect to some monomial ordering, is 1. The main
result is that for a finitely generated algebra over a Noetherian Jacobson
ring, the transcendence degree is equal to the Krull dimension.
