2011/09/07 by Gregor Kemper, Kemper, Gregor
Mathematics · #13A15 (Secondary) #13C15 (Primary) 13E05 #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #History and Theory of Mathematics #Mathematical and Theoretical Analysis #math.AC #msc:13A15 #msc:13C15 #msc:13E05
paper · pdf · doi:10.48550/arxiv.1109.1391
arxiv created 2011/09/07 · openalex publication_date 2011/09/07 · arxiv updated 2011/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.