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Proceedings of the American Mathematical Society

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

 

 

A note on intersections of valuation ideals


Author: Charles H. Brase
Journal: Proc. Amer. Math. Soc. 38 (1973), 37-39
MathSciNet review: 0309915
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Abstract: In this note it is proved that if $ R$ is an integral domain the set of valuation ideals of $ R$ is closed under intersection if and only if the integral closure of $ R$ is a valuation ring. Let $ S$ be a domain which is integrally dependent on $ R$ and contains the integral closure of $ R$. Then the set of valuation ideals of $ R$ is the same as the set of ideals of $ R$ which contract from ideals of $ S$ if and only if the set of valuation ideals of $ R$ is closed under intersection.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0309915-X
Keywords: Valuation ideal, completion, integral closure
Article copyright: © Copyright 1973 American Mathematical Society