Abstract
This paper considers ten conditions on the set of overrings of an integral domain D with identity. Each of these conditions is satisfied if D is a Prüfer domain. Relations among the conditions are discussed, and several related questions are mentioned.
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References
Akiba, T. Remarks on generalized quotient rings, Proc. Japan Acad. 40, 801–806 (1964).
_____ Remarks on generalized rings of quotients II, J. Math. Kyoto Univ. 5, 39–44 (1965).
_____ Remarks on generalized rings of quotients III, J. Math. Kyoto Univ. 9, 205–212 (1969).
Arnold, J. T. On the dimension theory of overrings of an integral domain, Trans. Amer. Math. Soc. 138, 313–326 (1969).
Arnold, J. T., and Brewer, J. W. On flat overrings, ideal transforms, and generalized transforms of a commutative ring, J. Algebra 18, 254–263 (1971).
Arnold, J. T., and Gilmer, R. The dimension sequence of a commutative ring, in preparation.
Bastida, E., and Gilmer, R. Overrings and divisorial ideals of rings of the form D + M, preprint.
Borho, W. Die torsionsfreien Ringe mit lauter Noetherschen Unterringen, preprint.
Budach, L. Quotientenfunktoren und Erweiterungstheorie, Math. Froschungsberichte 22, VEB Deutscher Verlag der Wiss. Berlin, 1967.
Butts, H. S., and Spaht, C. G. Generalized quotient rings, to appear in Math. Nach.
Cartan, H., and Eilenberg, S. Homological Algebra, Princeton Univ. Press, Princeton, N. J. (1956).
Davis, E. D. Overrings of commutative rings. II. Integrally closed overrings, Trans. Amer. Math. Soc. 110, 196–212 (1964).
Endo, S. On semi-hereditary rings, J. Math. Soc. Japan 13, 109–119 (1961).
Fuchs, L. Über die Ideale arithmetischer Ringe, Comment. Math. Helv. 23, 334–341 (1949).
Gilmer, R. Domains in which valuation ideals are prime powers, Arch. Math. 17, 210–215 (1966).
_____ Multiplicative Ideal Theory, Queen's University, Kinston, Ontario (1968).
_____ On a condition of J. Ohm for integral domains, Canad. J. Math. 20, 970–983 (1968).
_____ Two constructions of Prüfer domains, J. Reine Angew. Math. 239/240, 153–162 (1969).
_____ Integral domains with Noetherian subrings, Comment. Math. Helv. 45, 129–134 (1970).
_____ Domains with integrally closed subrings, Math. Jap. 16 9–11 (1971).
Gilmer, R., and Heinzer, W. Intersections of quotient rings of an integral domain, J. Math. Kyoto Univ. 7, 133–150 (1967).
_____ On the number of generators of an invertible ideal, J. Algebra 14, 139–151 (1970).
Gilmer, R., and Huckaba, J. A. The transform formula for ideals, J. Algebra 21, 191–215 (1972).
_____ Δ-rings, preprint.
Gilmer, R., Lea, R., and O'Malley, M. Rings whose proper subrings have property P, to appear in Acta Sci. Math. (Szeged).
Gilmer, R., and Mott, J. L. Multiplication rings as rings in which ideals with prime radical are primary, Trans. Amer. Math. Soc. 114, 40–52 (1965).
_____ Integrally closed subrings of an integral domain, Trans. Amer. Math. Soc. 154, 239–250 (1971).
Gilmer, R., and Ohm, J. Integral domains with quotient overrings, Math. Ann. 53, 97–103 (1964).
_____ Primary ideals and valuation ideals, Trans. Amer. Math. Soc. 117, 237–250 (1965).
Gilmer, R., and O'Malley, M. Non-Noetherian rings for which each proper subring is Noetherian, to appear in Math. Scand.
Goldman, O. On a special class of Dedekind domains, Topology 3, 113–118 (1964).
Griffin, M. Prüfer rings with zero divisors, J. Reine Angew. Math. 239/240, 55–67 (1969).
Hattori, A. On Prüfer rings, J. Math. Soc. Japan 9, 381–385 (1957).
Heinzer, W. Quotient overrings of an integral domain, Mathematika 17, 139–148 (1970).
Heinzer, W., Ohm, J., and Pendleton, R. On integral domains of the form ∩ DP, P minimal, J. Reine Angew. Math. 241, 147–159 (1970).
Jaffard, P. Dimension des anneaux de polynomes. La notion de dimension valuative, C. R. Acad. Sci. Paris Ser. A-B 246, 3305–3307 (1958).
_____ Theorie de la Dimension dans les Anneaux de Polynomes, Gauthier-Villars, Paris, 1960.
Jensen, C. U. On characterizations of Prüfer rings, Math. Scand. 13, 90–98 (1963).
_____ Arithmetical rings, Acta Math. Acad. Sci. Hungar. 17, 115–123 (1966).
Kaplansky, I. A characterization of Prüfer rings, J. Indian Math. Soc. (N.S.) 24, 279–281 (1960).
Kirby, D. Components of ideals in a commutative ring, Ann. Mat. Pura Appl. (4) 71, 109–125 (1966).
Krull, W. Beiträge zur Arithmetik kommutativer Integritätsbereiche, Math. Z. 41, 545–577 (1936).
_____ Beiträge zur Arithmetik kommutativer Integritätsbereiche. II. v-Ideale und vollständig ganz abegeschlossene Integritätsbereiche, Math. Z. 41, 665–679 (1936).
_____ Beiträge zur Arithmetik kommutativer Integritätsbereiche. VIII. Multiplikativ abgeschlossene Systeme von endlichen Idealen, Math. Z. 48, 533–552 (1943).
Larsen, M. D. Equivalent conditions for a ring to be a P-ring and a note on flat overrings, Duke Math. J. 34, 273–280 (1967).
Larsen, M. D., and McCarthy, P. J. Multiplicative Theory of Ideals, Academic Press, New York, 1971.
Ohm, J. Integral closure and (x, y)n=(xn, yn), Monatsh. Math. 71, 32–39 (1967).
Pendleton, R. L. A characterization of Q-domains, Bull. Amer. Math. Soc. 72, 499–500 (1966).
Prüfer, H. Untersuchungen über die Teilbarkeitseigenschaften in Körpern, J. Reine Angew. Math. 168, 1–36 (1932).
Richman, F. Generalized quotient rings, Proc. Amer. Math. Soc. 16, 794–799 (1965).
Seidenberg, A. A note on the dimension theory of rings, Pacific J. Math. 3, 505–512 (1953).
_____ On the dimension theory of rings II. Pacific J. Math. 4, 603–614 (1954).
Storrer, H. H. A characterization of Prüfer rings, Canad. Math. Bull. 12, 809–812 (1969).
Wadsworth, A. Noetherian pairs and the function field of a quadratic form., Thesis, Chicago, 1972.
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Gilmer, R. (1973). Prüfer-like conditions on the set of overrings of an integral domain. In: Brewer, J.W., Rutter, E.A. (eds) Conference on Commutative Algebra. Lecture Notes in Mathematics, vol 311. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0068922
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DOI: https://doi.org/10.1007/BFb0068922
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