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Proceedings of the American Mathematical Society

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



A note on finite dimensional subrings of polynomial rings

Author: Paul Eakin
Journal: Proc. Amer. Math. Soc. 31 (1972), 75-80
MathSciNet review: 0289498
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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$.

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Keywords: Dedekind domain, Krull ring, polynomial ring, affine ring, noetherian ring, Krull dimension
Article copyright: © Copyright 1972 American Mathematical Society