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Proceedings of the American Mathematical Society

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

 
 

 

On Noetherianness of Nash rings


Authors: Fulvio Mora and Mario Raimondo
Journal: Proc. Amer. Math. Soc. 90 (1984), 30-34
MSC: Primary 13E05; Secondary 14G30, 32B05, 58A07
DOI: https://doi.org/10.1090/S0002-9939-1984-0722409-X
MathSciNet review: 722409
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Abstract: 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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Keywords: Nash rings, noetherian property, algebraic closure of rings
Article copyright: © Copyright 1984 American Mathematical Society