The Transcendence Degree over a Ring

TL;DR

This paper generalizes the concept of transcendence degree from algebras over fields to those over rings, establishing its equality with Krull dimension for finitely generated algebras over Noetherian Jacobson rings.

Contribution

It introduces a new definition of transcendence degree over rings and proves its equivalence to Krull dimension in a broad algebraic setting.

Findings

Transcendence degree equals Krull dimension over Noetherian Jacobson rings.

New algebraic dependence definition based on algebraic equations with specific leading coefficients.

Extension of classical field-based results to ring-based algebraic structures.

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.