Commutative Algebra pp 265-277 | Cite as

# 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*.

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### References

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*Transcendence degree over an arbitrary commutative ring*, J. Algebra**101**(1986), 110–119.MathSciNetMATHCrossRefGoogle Scholar - 2.M. Nag at a, “Local Rings,” Tracts in Pure and Applied Mathematics, no. 13, Interscience, New York, 1962.Google Scholar
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