The Transcendence Degree over a Ring
Commutative Algebra
2011-09-08 v1
Abstract
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.
Cite
@article{arxiv.1109.1391,
title = {The Transcendence Degree over a Ring},
author = {Gregor Kemper},
journal= {arXiv preprint arXiv:1109.1391},
year = {2011}
}