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.KeywordsPrime IdealPolynomial RingHomomorphic ImageMinimal GenerateZero DivisorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.