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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On going-down for simple overrings. III


Authors: David E. Dobbs and Ira J. Papick
Journal: Proc. Amer. Math. Soc. 54 (1976), 35-38
MSC: Primary 13B20
MathSciNet review: 0417153
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Abstract: Theorem 1. Let $ R$ be an integral domain with quotient field $ K$. The following three conditions are equivalent: (a) $ R \subset R[u]$ satisfies going-down $ (GD)$ for each $ u$ in $ K$; (b) $ R \subset V$ satisfies $ GD$ for each valuation overring $ V$ of $ R$; (c) $ R \subset S$ satisfies $ GD$ for each domain $ S$ containing $ R$. If (d) is the condition obtained by restricting the domains $ S$ in $ ({\text{c)}}$ to be overrings of $ R$, then $ ({\text{a)}} \Leftrightarrow {\text{(d)}}$ has been proved in case $ R$ is Krull or integrally closed finite-conductor (e.g., pseudo-Bézout) or Noetherian. Theorem 2. Let $ R \subset T$ be domains such that either $ \operatorname{Spec} (R)$ or $ \operatorname{Spec} (T)$, as a poset under inclusion, is a tree. If $ R \subset R[u,v]$ satisfies $ GD$ for each $ u$ and $ v$ in $ T$, then $ R \subset T$ satisfies $ GD$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0417153-8
PII: S 0002-9939(1976)0417153-8
Keywords: Going-down, treed domain, flat overring, valuation ring
Article copyright: © Copyright 1976 American Mathematical Society