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Proceedings of the American Mathematical Society

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Overring properties of $ G$-domains

Authors: Revati Ramaswamy and T. M. Viswanathan
Journal: Proc. Amer. Math. Soc. 58 (1976), 59-66
MSC: Primary 13G05
MathSciNet review: 0407005
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Abstract: A commutative domain $ R$ is called a strong $ G$-domain if every overring between $ R$ and the quotient field $ K$ of $ R$ is of the form $ R[1/t]$ for some nonzero element $ t$ of $ R$. After characterizing valuation rings which are strong $ G$-domains, the authors show that $ R$ is a strong $ G$-domain if and only if it is a finite intersection of valuation rings each of which is a strong $ G$-domain. Using some results of R. W. Gilmer, Jr., the authors identify the strong $ G$-domains in the class of all Prüfer domains. They reprove via Krull domains the theorem characterizing Noetherian $ G$-domains, a result first proved by Artin and Tate. The authors also raise some relevant questions on related overring properties of $ G$-domains.

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Keywords: $ G$-domains, Noetherian $ G$-domains, Prüfer domains, Krull domains, valuation rings, overrings, localization, spectrum of a commutative ring
Article copyright: © Copyright 1976 American Mathematical Society