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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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  • [1] John Conway Adams, Rings with a finitely generated total quotient ring, Canad. Math. Bull. 17 (1974), 1–4. MR 0354639
  • [2] Emil Artin and John T. Tate, A note on finite ring extensions, J. Math. Soc. Japan 3 (1951), 74–77. MR 0044509
  • [3] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
  • [4] Otto Endler, Valuation theory, Springer-Verlag, New York-Heidelberg, 1972. To the memory of Wolfgang Krull (26 August 1899–12 April 1971); Universitext. MR 0357379
  • [5] Robert M. Fossum, The divisor class group of a Krull domain, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 74. MR 0382254
  • [6] Robert W. Gilmer Jr., Overrings of Prüfer domains, J. Algebra 4 (1966), 331–340. MR 0202749
  • [7] Robert W. Gilmer, Multiplicative ideal theory, Queen’s Papers in Pure and Applied Mathematics, No. 12, Queen’s University, Kingston, Ont., 1968. MR 0229624
  • [8] Jean Guérindon, Anneaux de Goldman, Séminaire P. Dubreil, M.-L. Dubreil-Jacotin, L. Lesieur et C. Pisot: 1969/70, Algébre et Théorie des Nombres, Secrétariat mathématique, Paris, 1970, pp. Fasc. 1, Exp. 9, 8 (French). MR 0284423
  • [9] Irving Kaplansky, Commutative rings, Allyn and Bacon, Inc., Boston, Mass., 1970. MR 0254021
  • [10] Paulo Ribenboim, Théorie des valuations, Deuxième édition multigraphiée. Séminaire de Mathématiques Supérieures, No. 9 (Été, vol. 1964, Les Presses de l’Université de Montréal, Montreal, Que., 1968 (French). MR 0249425
  • [11] O. Zariski and P. Samuel, Commutative algebra. Vols. 1, 2, Univ. Ser. in Higher Math., Van Nostrand, Princeton, N. J., 1958, 1960. MR 19, 833; 22 #11006.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0407005-1
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