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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Noetherian intersections of integral domains


Authors: William Heinzer and Jack Ohm
Journal: Trans. Amer. Math. Soc. 167 (1972), 291-308
MSC: Primary 16A02; Secondary 13E05
DOI: https://doi.org/10.1090/S0002-9947-1972-0296095-6
MathSciNet review: 0296095
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Abstract: Let $ D < R$ be integral domains having the same quotient field K and suppose that there exists a family $ {\{ {V_i}\} _{i \in I}}$ of 1-dim quasi-local domains having quotient field K such that $ D = R \cap \{ {V_i}\vert i \in I\} $. The goal of this paper is to find conditions on R and the $ {V_i}$ in order for D to be noetherian and, conversely, conditions on D in order for R and the $ {V_i}$ to be noetherian. An important motivating case is when the set $ \{ {V_i}\} $ consists of a single element V and V is a valuation ring. It is shown, for example, in this case that (i) if V is centered on a finitely generated ideal of D, then V is noetherian and (ii) if V is centered on a maximal ideal of D, then D is noetherian if and only if R and V are noetherian.


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DOI: https://doi.org/10.1090/S0002-9947-1972-0296095-6
Keywords: 1-dim quasi-local integral domain, flat ring extension, Krull domain, essential valuation ring
Article copyright: © Copyright 1972 American Mathematical Society