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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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 | References | Similar Articles | Additional Information

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 $k$.


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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