Transcendence degree over an arbitrary commutative ring
type: Item
instance of: scholarly article
title: Transcendence degree over an arbitrary commutative ring
zbMATH Open document ID: 0591.13003
DOI: 10.1016/0021-8693(86)90100-6
author: Eloise Hamann
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: 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 \([S:B]_ G=[S:B]_ C=n-k\).
Mathematics Subject Classification ID: 13B02
Mathematics Subject Classification ID: 13E10
zbMATH DE Number: 3948435
zbMATH Keywords: transcendence degree in commutative rings
reviewed by: Andy R. Magid
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90100-6
OpenAlex ID: W1971313501
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description: scientific article; zbMATH DE number 3948435