Transcendence degree over an arbitrary commutative ring

Abstract

Let S = R[X, ,..., X,] be a polynomial ring in n indeterminates over a commutative ring R with identity. It is obvious that properties of R affect the properties of S. However, it would also seem that some behavior of S as an R-algebra is influenced by the polynomial indeterminates. Specifically the author has been in situations where one wants to claim that a particular element of S is an indeterminate over R or some ring between R and S. While in each special circumstance, the elements turned out to have the desired properties, it was the feeling of the author that some general results ought to hold. It is the purpose of this paper to study the notion of algebraic independence of elements of R[X, ,..., X,,]. A better title for the paper might be “Why there is no notion of transcendence degree over arbitrary commutative rings.” It is not difficult to find bad examples and the positive results may not be surprising. However, the author feels it is of some use to catalogue what can happen. Definitions are overdue, and are formulated in terms of ordered subsets of s= R[X, )...) X,].