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A note on finite dimensional subrings of polynomial rings


Author: Paul Eakin
Journal: Proc. Amer. Math. Soc. 31 (1972), 75-80
DOI: https://doi.org/10.1090/S0002-9939-1972-0289498-2
MathSciNet review: 0289498
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Abstract | References | Additional Information

Abstract: Let $ k$ be a field and $ {\{ {X_\alpha }\} _{\alpha \in \Delta }}$ a family of indeterminates over $ k$. We show that if $ A$ is a ring of Krull dimension $ d$ such that $ k \subseteq A \subseteq k[{\{ {A_\alpha }\} _{\alpha \in \Delta }}]$ then there are elements $ {Y_1}, \cdots ,{Y_d}$ which are algebraically independent over $ k$ and a $ k$-isomorphism $ \phi $ such that $ k \subseteq \phi (A) \subseteq k[{Y_1}, \cdots ,{Y_d}]$. This is used to show that a onedimensional ring $ A$ which satisfies the above conditions is necessarily an affine ring over $ k$ and is necessarily a polynomial ring if it is normal. In addition we show that such a ring $ A$ is a normal affine ring of transcendence degree two over $ k$ if and only if it is a two-dimensional Krull ring such that each essential valuation of $ A$ has residue field transcendental over $ k$.


References [Enhancements On Off] (What's this?)

  • [EZ] A. Evyatar and A. Zaks, Rings of polynomials, Proc. Amer. Math. Soc. 25 (1970), 559-562. MR 0258820 (41:3466)
  • [E] P. M. Eakin, Jr., The converse to a well known theorem on Noetherian rings, Math. Ann. 177 (1968), 278-282. MR 37 #1360. MR 0225767 (37:1360)
  • [H] W. Heinzer, On Krull overrings of a Noetherian domain, Proc. Amer. Math. Soc. 22 (1969), 217-222. MR 40 #7235. MR 0254022 (40:7235)
  • [J] N. Jacobson, Lectures in abstract algebra. Vol. 3: Theory of fields and Galois theory, Van Nostrand, Princeton, N.J., 1964. MR 30 #3087. MR 0172871 (30:3087)
  • [K] I. Kaplansky, Commutative rings, Allyn and Bacon, Boston, Mass., 1970. MR 40 #7234. MR 0254021 (40:7234)
  • [N] M. Nagata, Local rings, Interscience Tracts in Pure and Appl. Math., no. 13, Interscience, New York, 1962. MR 27 5790. MR 0155856 (27:5790)
  • [N] ; 14H. -, Lectures on the fourteenth problem of Hilbert, Tata Institute of Fundamental Research Lectures on Math., no. 31, Tata Institute of Fundamental Research, Bombay, 1965. MR 35 #6663. MR 0215828 (35:6663)
  • [ZS] O. Zariski and P. Samuel, Commutative algebra. Vol. II, University Series in Higher Math., Van Nostrand, Princeton, N.J., 1960. MR 22 #11006. MR 0120249 (22:11006)


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0289498-2
Keywords: Dedekind domain, Krull ring, polynomial ring, affine ring, noetherian ring, Krull dimension
Article copyright: © Copyright 1972 American Mathematical Society

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