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Proceedings of the American Mathematical Society

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

 

 

On going down for simple overrings


Author: David E. Dobbs
Journal: Proc. Amer. Math. Soc. 39 (1973), 515-519
MSC: Primary 13B20
DOI: https://doi.org/10.1090/S0002-9939-1973-0417152-3
MathSciNet review: 0417152
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Abstract: Let $ R$ be an integral domain with quotient field $ K$. If $ R$ is Noetherian: then the Krull dimension of $ R$ is at most $ 1 \Leftrightarrow $for all overrings $ S$ of $ R$, $ R \subset S$ satisfies going down. $ R$ is Dedekind $ {\text{(resp}}{\text{., PID)}} \Leftrightarrow R$ is Krull (resp., UFD) and, for all $ u \in K,R \subset R[u]$ satisfies going down. $ R$ is Prüfer $ \Leftrightarrow R$ is integrally closed, every intersection of two principal ideals of $ R$ is finitely generated, and $ R \subset R[u]$ satisfies going down for all $ u \in K$.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0417152-3
Keywords: Going down, Krull domain, FC domain
Article copyright: © Copyright 1973 American Mathematical Society