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Transactions of the American Mathematical Society

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

 

 

Overrings of commutative rings. III. Normal pairs


Author: Edward D. Davis
Journal: Trans. Amer. Math. Soc. 182 (1973), 175-185
MSC: Primary 13B20
DOI: https://doi.org/10.1090/S0002-9947-1973-0325599-3
MathSciNet review: 0325599
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Abstract: A pair of integral domains (A, B) is a normal (resp., QR-) pair provided that A is a subring of B and all intermediate rings are normal in B (resp., rings of quotients of A). The special case of B the field of fractions of A (e.g., Prüfer domains and Dedekind domains with torsion class group) has been studied in detail. It is shown that any domain A possesses a unique overring B maximal with respect to forming a normal (resp., QR-) pair with A. An explicit description of this overring and all the intermediate rings in terms of localizations A is obtained, and further details are provided in the presence of a noetherian-like condition on A. In addition, the ``overring'' characterizations of Prüfer domains are extended to ``intermediate ring'' characterizations of normal pairs.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0325599-3
Keywords: Prüfer domains, Dedekind domains, valuation rings, rings of quotients, flat overrings, normal (or integrally closed)
Article copyright: © Copyright 1973 American Mathematical Society