Some Transcendence Degree Questions

Abstract

This paper addresses some open questions from [1]. The general concern is to study the behavior of transcendence degree over an arbitrary commutative ring R with identity with particular interest in those R algebras which are contained in a polynomial ring over R. To be precise, let S = R[X _n ] where [X _n ] = {x ₁,…,x _n{, a set of independent indeterminates and the objects of interest are R-algebras B $\subseteq$ S with B finitely generated over R. The paper gives two generalizations of theorems from [1] which give conditions which guarantee that [S : B] + [B : R] = n. ([C : D] denotes the size of a maximal set contained in C which is algebraically independent over D.) S is a B-algebra about which little can be said except that S is finitely generated over R. If [S:B] = d, it does not follow that there exists a minimal generating set of S over B containing d algebraically independent elements. In fact, it is easy to construct examples where S can be minimally generated over B by algebraic elements, but [S:B] >0. This paper will show that such pathological behavior cannot happen for the R-algebras B.