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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles 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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On Noetherianness of Nash rings
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by Fulvio Mora and Mario Raimondo PDF
Proc. Amer. Math. Soc. 90 (1984), 30-34 Request permission


We introduce a class of rings, called Nash Rings, which generalize the notation of rings of Nash functions. Let $k$ be any field, $X$ be a normal algebraic variety in ${k^n}$, and $U \subset X$. A Nash ring $D$ is the algebraic closure of $\Gamma (X,{\mathcal {O}_X})$ in a suitable domain $B$ such that $U$ is contained in the maximal spectrum of $B$ and $\Gamma (X,{\mathcal {O}_X})$ is analytically isomorphic to $B$ at each $x \in U$. We show that $D$ is a ring of fractions of the integral closure of $\Gamma (X,{\mathcal {O}_X})$ in $B$. Moreover, if $k$ is algebraically nonclosed and if every algebraic subvariety $V \subset X$ intersects $U$ in a finite number of connected components (in the topology induced by $B$), then $D$ is noetherian.
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Additional Information
  • © Copyright 1984 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 90 (1984), 30-34
  • MSC: Primary 13E05; Secondary 14G30, 32B05, 58A07
  • DOI:
  • MathSciNet review: 722409