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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Lattice-ordered groups and a conjecture for adequate domains

Authors: J. W. Brewer, P. F. Conrad and P. R. Montgomery
Journal: Proc. Amer. Math. Soc. 43 (1974), 31-35
MSC: Primary 06A60; Secondary 13F15
MathSciNet review: 0332616
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Abstract: In this paper, we present a counterexample to show that adequate domains are not characterized by the property that nonzero prime ideals are contained in a unique maximal ideal. The counterexample is obtained by constructing a lattice-ordered group with certain properties and exploiting the relation between Bezout domains and their (lattice-ordered) group of divisibility. The domain constructed is an elementary divisor ring with zero Jacobson radical. The lattice-ordered group constructed also shows that various conjectures about $l$-groups are false.

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Keywords: Lattice ordered abelian group, adequate domain, group of divisibility, elementary divisor ring, lateral completion
Article copyright: © Copyright 1974 American Mathematical Society