Mathematics Faculty Articles

Title

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

Comments

© Birkhäuser Verlag Basel/Switzerland 2009

DOI

10.1007/s00012-009-0012-4.

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Peer Reviewed

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