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arXiv · 1109.1391

The Transcendence Degree over a Ring

Gregor Kemper
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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.

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Gregor Kemper. 2011-09-07. The Transcendence Degree over a Ring. https://arxiv.org/abs/1109.1391

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