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Quotient overrings of integral domains

Published online by Cambridge University Press:  26 February 2010

William Heinzer
William Heinzer
Affiliation:
Purdue University, Lafayette, Indiana.
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If D is an integral domain with quotient field K, then by an overring of D we shall mean a ring D′ such that D ⊂ D′ ⊂ K. Gilmer and Ohm in [GO], Davis in [D] and Pendleton in [P] have studied the class of integral domains D having the property that each overring D′ of D is of the form Ds for some multiplicative system S of D. In [D] and [P] a domain with this property is called a Q-domain and in [GO] such a domain is said to have the QR-property. Slightly altering the terminology of [D] and [P], we shall say “QR-domain” instead of “Q-domain”. Noetherian QR-domains are precisely the Dedekind domains having torsion class group [GO; p. 97] or [D; p. 200]. Pendleton in [P; p. 500] classifies QR-domains as Prüfer domains satisfying the additional condition that the radical of each finitely generated ideal is the radical of a principal ideal. It is pointed out in [P; p. 500] that this characterization of QR-domains still leaves unresolved the question of whether the class group of a QR-domain is necessarily a torsion group. We show in §1 that a QR-domain need not have torsion class group. Our construction is direct; however, the problem can be viewed in terms of the realization of certain ordered abelian groups as divisibility groups of Prüfer domains, and we conclude §1 with a brief discussion of this approach to the problem.

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Type
Research Article
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Mathematika , Volume 17 , Issue 1 , June 1970 , pp. 139 - 148
Copyright
Copyright © University College London 1970

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References

Akiba, T., “Remarks on generalized rings of quotients”, Proc. Japan Acad., 40 (1964), 801–806.Google Scholar
Bourbaki, N., Algibre Commutative, Chapters 5 and 6 (1964, Hermann, Paris).Google Scholar
Brewer, J. and Gilmer, R., “Integral domains whose overrings are ideal transforms”, to appear.Google Scholar
Davis, E., “Overrings of commutative rings II”, Trans. Amer. Math. Soc., 110 (1964), 196–212.CrossRefGoogle Scholar
Gilmer, R., Multiplicative ideal theory. Queen's papers on pure and applied mathematics (Kingston, Ontario, 1968).Google Scholar
Gilmer, R. and Heinzer, W., “Intersections of quotient rings of an integral domain”, J. Math. Kyoto Univ., 7 (1967), 133–150.Google Scholar
Gilmer, R. and Ohm, J., “Integral domains with quotient overrings”, Math. Ann., 153 (1964), 813–818.CrossRefGoogle Scholar
Goldman, O., “On a special class of Dedekind domains”, Topology, 3 (1964), 113–118.CrossRefGoogle Scholar
Heinzer, W., Ohm, J. and Pendleton, R., “On integral domains of the form ∩ D p, P minimal”, to appear in J. reine angew. Math.Google Scholar
Jaffard, P., Les systemes d'ideaux (Dunod, Paris, 1960).Google Scholar
Nagata, M., Local rings (Interscience, New York, 1962).Google Scholar
Nagata, M., “A treatise on the 14-th problem of Hilbert”, Mem. Coll. Sci. Univ. Kyoto, Ser. A Math., 30 (1956–57), 57–70.Google Scholar
Nagata, M., “A theorem on finite generation of a ring”, Nagoya Math. J., 27 (1966), 193–205.CrossRefGoogle Scholar
Ohm, J., “Some counterexamples related to integral closure in D[[x]]”, Trans. Amer. Math. Soc, 122 (1966), 321–333.Google Scholar
Pendleton, R., “A characterization of Q-domains”, Bull. Amer. Math. Soc., 72 (1966), 499–500.CrossRefGoogle Scholar
Richman, F., “Generalized quotient rings”, Proc. Amer. Math. Soc., 16 (1965), 794–799.CrossRefGoogle Scholar
Zariski, O. and Samuel, P., Commutative Algebra (Van Nostrand, New York, I (1958), II (I960)).Google Scholar