Some Transcendence Degree Questions

  • Eloise Hamann
Conference paper
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 15)


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.


Prime Ideal Polynomial Ring Homomorphic Image Minimal Generate Zero Divisor 
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    E. Hamann, Transcendence degree over an arbitrary commutative ring, J. Algebra 101 (1986), 110–119.MathSciNetMATHCrossRefGoogle Scholar
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    M. Nag at a, “Local Rings,” Tracts in Pure and Applied Mathematics, no. 13, Interscience, New York, 1962.Google Scholar
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    O. Zariski and P. Samuel, “Commutative Algebra,” vol. I, Van Nostrand, New York, 1958.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1989

Authors and Affiliations

  • Eloise Hamann
    • 1
  1. 1.Department of MathematicsSan Jose State UniversitySan JoseUSA

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