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Mathematics Faculty Articles
A Generalization of the Jaffard-Ohm-Kaplansky Theorem
Document Type
Article
Publication Date
1-1-2009
Publication Title
Algebra Universalis
Keywords
Algebraic frame, Quantale, Prüfer domain, Lattice-ordered group
ISSN
0002-5240
Volume
61
First Page
201
Last Page
212
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
NSUWorks Citation
Iberkleid, Wolf and McGovern, Warren William, "A Generalization of the Jaffard-Ohm-Kaplansky Theorem" (2009). Mathematics Faculty Articles. 21.
https://nsuworks.nova.edu/math_facarticles/21
DOI
10.1007/s00012-009-0012-4.
Comments
© Birkhäuser Verlag Basel/Switzerland 2009