%0 Journal Article
%T The Transcendence Degree over a Ring
%A Gregor Kemper
%J Mathematics
%D 2011
%I arXiv
%X 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.
%U http://arxiv.org/abs/1109.1391v1