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
DOI:
https://doi.org/10.1090/S0002-9939-1974-0332616-X
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 -groups are false.
- [1] Alain Bigard, Groupes archimédiens et hyper-archimédiens, Séminaire P. Dubreil, M.-L. Dubreil-Jacotin, L. Lesieur et G. Pisot: 1967/68, Algèbre et Théorie des Nombres, Secrétariat mathématique, Paris, 1969, pp. Fasc. 1, Exp. 2, 13 (French). MR 0250950
- [2] Roger Bleier and Paul Conrad, The lattice of closed ideals and 𝑎*-extensions of an abelian 𝑙-group, Pacific J. Math. 47 (1973), 329–340. MR 325486
- [3] Paul Conrad and Donald McAlister, The completion of a lattice ordered group, J. Austral. Math. Soc. 9 (1969), 182–208. MR 0249340
- [4] Robert Gilmer, Multiplicative ideal theory, Marcel Dekker, Inc., New York, 1972. Pure and Applied Mathematics, No. 12. MR 0427289
- [5] Olaf Helmer, The elementary divisor theorem for certain rings without chain condition, Bull. Amer. Math. Soc. 49 (1943), 225–236. MR 7744, https://doi.org/10.1090/S0002-9904-1943-07886-X
- [6] Melvin Henriksen, Some remarks on elementary divisor rings. II, Michigan Math. J. 3 (1955/56), 159–163. MR 92772
- [7] Irving Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc. 66 (1949), 464–491. MR 31470, https://doi.org/10.1090/S0002-9947-1949-0031470-3
- [8] Irving Kaplansky, Infinite abelian groups, Revised edition, The University of Michigan Press, Ann Arbor, Mich., 1969. MR 0233887
- [9] Max D. Larsen, William J. Lewis, and Thomas S. Shores, Elementary divisor rings and finitely presented modules, Trans. Amer. Math. Soc. 187 (1974), 231–248. MR 335499, https://doi.org/10.1090/S0002-9947-1974-0335499-1
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1974-0332616-X
Keywords:
Lattice ordered abelian group,
adequate domain,
group of divisibility,
elementary divisor ring,
lateral completion
Article copyright:
© Copyright 1974
American Mathematical Society