Year: 2011
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Kemper, Gregor
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1642581802