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Convex Directed Subgroups of a Group of Divisibility

Published online by Cambridge University Press:  20 November 2018

Joe L. Mott
Joe L. Mott*
Affiliation:
Florida State University, Tallahassee, Florida
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If D is an integral domain with quotient field K, the group of divisibility G(D) of D is the partially ordered group of non-zero principal fractional ideals with aD ≦ bD if and only if aD contains bD. If K* denotes the multiplicative group of K and U(D) the group of units of D, then G(D) is order isomorphic to K*/U(D), where aU(D) ≦ bU(D) if and only if b/a ∊ D.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 3 , June 1974 , pp. 532 - 542
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Ailing, N. L., Valuation theory of meromorphic function fields over open Riemann surfaces, Acta Math. 110 (1965), 79–95.Google Scholar
2. Banaschewski, B., On lattice-ordered groups, Fund. Math. 55 (1964), 113–123.Google Scholar
3. Banaschewski, B., Zür Idealtheorie der ganzen Functionen, Math. Nachr. 19 (1958), 136–160.Google Scholar
4. Bernau, S. J., Free abelian lattice groups, Math. Ann. 180 (1969), 48–59.Google Scholar
5. Conrad, P. F. and McAlister, D., The completion of a lattice-ordered group, J. Austral. Math. Soc. 9 (1969), 182–208.Google Scholar
6. Enochs, E., A note on the dimension of the ring of entire functions, Collect. Math. 20 (1969), P. 3.Google Scholar
7. Fuster, R. A., Estudio de los idéales del anillo de las functiones enteras, Collect. Math. 17 (1965), 105–134.Google Scholar
8. Gilmer, R. W., Multiplicative ideal theory, Queen's Papers, Lecture Notes No. 12 (Queen's University, Kingston, Ontario, 1968).Google Scholar
9. Heinzer, W., Some remarks on complete integral closure, J. Austral. Math. Soc. 9 (1969), 310–314.Google Scholar
10. Helmer, O., Divisibility properties of integral functions, Duke Math. J. 6 (1940), 345–356.Google Scholar
11. Henriksen, M., On the ideal structure of the ring of entire functions, Pacific J. Math. 2 (1952), 179–184.Google Scholar
12. Henriksen, M., On the prime ideals of the ring of entire functions, Pacific J. Math. 3 (1953), 711–720.Google Scholar
13. Jaffard, P., Contribution a la théorie des groupes ordonnés, J. Math. Pures Appl. 32 (1953), 203–280.Google Scholar
14. Krull, W., Allgemeine Bewertungstheorie, J. Reine Angew. Math. 167 (1931), 160–196.Google Scholar
15. Laplaza, M., Some properties of the ring of entire functions (to appear).Google Scholar
16. Ohm, J., Semi-valuation and groups of divisibility, Can. J. Math. 21 (1969), 576–591.Google Scholar
17. Schilling, O. F. G., Ideal theory on open Riemann surfaces, Bull. Amer. Math. Soc. 52 (1946), 945–963.Google Scholar
18. Sheldon, P., Two counterexamples involving complete integral closure in finite dimensional Prilfer domains (to appear in J. Algebra).Google Scholar
19. Silverman, R. A., Introductory complex analysis (Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967).Google Scholar
20. Weinberg, E. C., Free lattice-ordered abelian groups, Math. Ann. 151 (1963), 187–199; II, Math. Ann. 159 (1965), 217–222.Google Scholar
21. Zariski, O. and Samuel, P., Commutative algebra, Vol. II (van Nostrand, New York, 1960).Google Scholar