## 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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