On v-Domains and Star Operations

Abstract

Let * be a star operation on an integral domain D. Let f (D) be the set of all nonzero finitely generated fractional ideals of D. Call D a *-Prüfer (respectively, (*, v)-Prüfer) domain if (FF ⁻¹)* = D (respectively, (F ^v F ⁻¹)* = D) for all F ∈  f (D). We establish that *-Prüfer domains (and (*, v)-Prüfer domains) for various star operations * span a major portion of the known generalizations of Prüfer domains inside the class of v-domains. We also use Theorem 6.6 of the Larsen and McCarthy book [30], which gives several equivalent conditions for an integral domain to be a Prüfer domain, as a model, and we show which statements of that theorem on Prüfer domains can be generalized in a natural way and proved for *-Prüfer domains, and which cannot be. We also show that in a *-Prüfer domain, each pair of *-invertible *-ideals admits a GCD in the set of *-invertible *-ideals, obtaining a remarkable generalization of a property holding for the “classical” class of Prüfer v-multiplication domains. We also link D being *-Prüfer (or (*, v)-Prüfer) with the group Inv*(D) of *-invertible *-ideals (under *-multiplication) being lattice-ordered.

Key Words:

• Completely integrally closed domain
• v-Domain
• GCD domain
• *-Invertible ideal
• Prüfer domain
• *-Prüfer domain
• Star operation

2000 Mathematics Subject Classification:

• 13A15
• 13F05
• 13G05

ACKNOWLEDGMENTS

We would like to thank Franz Halter-Koch for several helpful comments on a previous version of the present paper and for suggesting lines for further investigation and extension to the general setting of ideal systems on monoids. During the preparation of this paper, the third named author was partially supported by a research grant PRIN-MiUR.

Notes

Communicated by R. Wiegand.