Journal article
On v-Domains and Star Operations
Communications in Algebra, Vol.37(9), pp.3018-3043
08/18/2009
DOI: 10.1080/00927870802502688
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.
Details
- Title: Subtitle
- On v-Domains and Star Operations
- Creators
- D. D Anderson - Department of Mathematics , University of IowaDavid F Anderson - Department of Mathematics , University of Tennessee at KnoxvilleMarco Fontana - Dipartimento di Matematica , Università degli Studi "Roma Tre"Muhammad Zafrullah - Department of Mathematics , Idaho State University
- Resource Type
- Journal article
- Publication Details
- Communications in Algebra, Vol.37(9), pp.3018-3043
- DOI
- 10.1080/00927870802502688
- ISSN
- 0092-7872
- eISSN
- 1532-4125
- Publisher
- Taylor & Francis Group
- Language
- English
- Date published
- 08/18/2009
- Academic Unit
- Mathematics
- Record Identifier
- 9983985852802771
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