A Generalization of the Jaffard-Ohm-Kaplansky Theorem

Wolf Iberkleid, Warren William McGovern

    Research output: Contribution to journalArticlepeer-review

    Abstract

    The well-known Jaffard–Ohm–Kaplansky Theorem states that every abelian ℓ-group can be realized as the group of divisibility of a commutative Bézout domain. To date there is no realization (except in certain circumstances) of an arbitrary, not necessarily abelian, ℓ-group as the group of divisibility of an integral domain. We show that using filters on lattices we can construct a nice quantal frame whose “group of divisibility” is the given ℓ-group. We then show that our construction when applied to an abelian ℓ-group gives rise to the lattice of ideals of any Prüfer domain assured by the Jaffard–Ohm–Kaplansky Theorem. Thus, we are assured of the appropriate generalization of the Jaffard–Ohm–Kaplansky Theorem

    Original languageAmerican English
    Pages (from-to)201-212
    Number of pages12
    JournalAlgebra Universalis
    Volume61
    Issue number2
    DOIs
    StatePublished - Aug 29 2009

    ASJC Scopus Subject Areas

    • Algebra and Number Theory

    Keywords

    • Algebraic frame
    • Lattice-ordered group
    • Prüfer domain
    • Quantale

    Disciplines

    • Mathematics

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