Overrings of commutative rings. III. Normal pairs

Davis, Edward D.

doi:10.1090/S0002-9947-1973-0325599-3

Overrings of commutative rings. III. Normal pairs
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Trans. Amer. Math. Soc. 182 (1973), 175-185 Request permission

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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Grundlagen zur Strukturtheorie der Integritätsbereiche mit eingeschränkter Minimalbedingung

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