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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Pairs of domains where all intermediate domains are Noetherian


Author: Adrian R. Wadsworth
Journal: Trans. Amer. Math. Soc. 195 (1974), 201-211
MSC: Primary 13G05
MathSciNet review: 0349665
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Abstract: For Noetherian integral domains R and T with $ R \subseteq T,(R,T)$ is called a Noetherian pair (NP) if every domain A, $ R \subseteq A \subseteq T$, is Noetherian. When $ \dim R = 1$ (Krull dimension) it is shown that the only NP's are those given by the Krull-Akizuki Theorem. For $ \dim R \geq 2$, there is another type of NP besides the finite integral extension, namely $ (R,\tilde R)$ where $ \tilde R = \bigcap {\{ {R_P}\vert{\text{rk}}\;P \geq 2\} } $. Further, for every NP (R, T) with $ \dim R \geq 2$ there is an integral NP extension B of R with $ T \subseteq \tilde B$. In all known examples B can be chosen to be a finite integral extension of R. For such NP's it is shown that the NP relation is transitive. T may itself be an infinite integral extension R, though, and an example of this is given. It is unknown exactly which infinite integral extensions are NP's.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0349665-2
PII: S 0002-9947(1974)0349665-2
Keywords: Finitely generated module, integral extension, Krull dimension, Krull-Akizuki Theorem, Noetherian integral domain
Article copyright: © Copyright 1974 American Mathematical Society