Transcendence degree over an arbitrary commutative ring
This paper considers some candidates for a definition of transcendence degree in commutative rings, along with their consequences and limitations. Let S be a polynomial algebra in n variables over R. The transcendence degree of an ordered subset \(\{y_ 1,...,y_ m\}\) of S over R is denoted [R[y]:R] and defined to be \(\sum\) \(a_ k\quad where\) \(a_ k=0\) if \(y_ k\) is R-algebraic over \(R[y_ 1,...,y_{k-1}]\) and \(a_ k=1\) otherwise. If B is an R-subalgebra of S, the complete transcendence degrees \([B:R]_ C\) and \([S:B]_ C\) are defined as the maximum transcendence degrees of ordered subsets of B and S respectively. The generating transcendence degrees \([B:R]_ G\) and \([S:B]_ G\) are defined as the maximum transcendence degrees of ordered subsets belonging to minimal generating sets. Examples show that generating transcendence degree can be strictly smaller than complete transcendence degree. Also, for both types of degrees, the degree of B over R plus the degree of S over B is bounded by n. If \([B/P:R/P]=k\) for all associated primes P of 0 in R, and B(Nil R)S\(=(Nil R)B\), then if S can be generated over B by n-k elements, \([S:B]_ G=[S:B]_ C=n-k\). If R is reduced with a finite number of minimal primes and B is finitely generated over R, then \([B:R]_ G=[B:R]_ C=\min \{[B/P:R/P]\}.\) Combining, if the degree [B/P:R/P] is a constant k for all minimal primes P then \([B:R]_ G=[B:R]_ C\) and \([S:B]_ G=[S:B]_ C=n-k\).
transcendence degree in commutative rings
scientific article; zbMATH DE number 3948435