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On generalized Dedekind domains. (English) Zbl 0613.13001

A commutative integral domain D is called a generalized Dedekind (G- Dedekind) domain if for every pair of fractional ideals A and B of D, one has \((AB)^{-1}=A^{-1}B^{-1}\); equivalently, for each A, the ideal \(A_ v=(A^{-1})^{-1}\) is invertible. The author discusses the properties of G-Dedekind domains, particularly their relationship with the various types of rings (Prüfer, Mori, Krull,...) that are characterized by similar multiplicative conditions on ideals. Particular results are that a G-Dedekind domain D is completely integrally closed, the polynomial ring D[X] is also G-Dedekind, but a ring of fractions \(D_ s\) need not be; the ring of entire functions provides an example. A set of twelve characterizations of G-Dedekind Krull domains is given, and some open questions are mentioned.
Reviewer: M.E.Keating

MSC:

13A05 Divisibility and factorizations in commutative rings
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13F30 Valuation rings
13G05 Integral domains
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