Rings of polynomials
Authors:
A. Evyatar and A. Zaks
Journal:
Proc. Amer. Math. Soc. 25 (1970), 559-562
MSC:
Primary 13.93
DOI:
https://doi.org/10.1090/S0002-9939-1970-0258820-3
MathSciNet review:
0258820
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Abstract: For an algebra $R$ over a field $k$, with residue field $K$ to be a ring of polynomials in one variable over $k$ it is necessary that $\operatorname {tr} \cdot \deg \;K/k = 1$. We prove that under the hypothesis $\operatorname {tr} \cdot \deg \;K/k = 1$ is a ring of Krull-dimension at most one. This is used to derive sufficient conditions for $R$ to be a ring of polynomials in one variable over $k$.
- A. Evyatar and A. Zaks, Purely transcendental subfields of $k(x_{1},\cdots ,x_{n})$, Proc. Amer. Math. Soc. 22 (1969), 582–586. MR 242811, DOI https://doi.org/10.1090/S0002-9939-1969-0242811-4 P. Samuel, Lectures in commutative algebra, Brandeis University, 1964/65.
- Oscar Zariski and Pierre Samuel, Commutative algebra, Volume I, The University Series in Higher Mathematics, D. Van Nostrand Company, Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR 0090581
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Keywords:
Rings of polynomials,
rings of power series,
unique factorization domain,
principal ideal domain,
Euclidean domain,
transcendence degree,
Krull dimension
Article copyright:
© Copyright 1970
American Mathematical Society