Article
A Generalization of the Jaffard-Ohm-Kaplansky Theorem
Algebra Universalis
Document Type
Article
Publication Date
1-1-2009
Keywords
- Algebraic frame,
- Quantale,
- Prüfer domain,
- Lattice-ordered group
Disciplines
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
DOI
10.1007/s00012-009-0012-4.
Citation Information
Wolf Iberkleid and Warren William McGovern. "A Generalization of the Jaffard-Ohm-Kaplansky Theorem" Algebra Universalis Vol. 61 (2009) p. 201 - 212 ISSN: 0002-5240 Available at: http://works.bepress.com/wolf-iberkleid/2/
© Birkhäuser Verlag Basel/Switzerland 2009