Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-16T17:30:29.771Z Has data issue: false hasContentIssue false
  • English
  • Français

Semi-Valuations and Groups of Divisibility

Published online by Cambridge University Press:  20 November 2018

Jack Ohm
Jack Ohm*
Affiliation:
Louisiana State University, Baton Rouge, Louisiana
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Associated with any integral domain R there is a partially ordered group A, called the group of divisibility of R. When R is a valuation ring, A is merely the value group; and in this case, ideal-theoretic properties of R are easily derived from corresponding properties of A, and conversely. Even in the general case, though, it has proved useful on occasion to phrase a ring-theoretic problem in terms of the ordered group A, first solve the problem there, and then pull back the solution if possible to R. Lorenzen (15) originally applied this technique to solve a problem of Krull, and Nakayama (16) used it to produce a counterexample to another question of Krull. More recently, Heinzer (7;8) has used the method to construct other interesting examples of rings.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 576 - 591
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Bourbaki, N., Algèbre, chapitre 6, Groupes et corps ordonnés (Hermann, Paris, 1964).Google Scholar
2. Bourbaki, N., Algèbre commutative, (a) chapitres 5 et 6, (b) chapitre 7 (Hermann, Paris, 1964, 1965).Google Scholar
3. Clifford, A., Note on Hahn's theorem on ordered abelian groups, Proc. Amer. Math. Soc. 5 (1954), 860–863.Google Scholar
4. Estes, D. and Ohm, J., Stable range in commutative rings, J. Algebra 7 (1967), 343–362.Google Scholar
5. Gilmer, R. W., Multiplicative ideal theory, Queen's Papers, Lecture Notes No. 12, Queen's University, Kingston, Ontario, 1968.Google Scholar
6. Gilmer, R. W. and Ohm, J., Primary ideals and valuation ideals, Trans. Amer. Math. Soc. 117 (1965), 237–250.Google Scholar
7. Heinzer, W., J-Noetherian integral domains with 1 in the stable range, Proc. Amer. Math. Soc. 19 (1968), 1369–1372.Google Scholar
8. Heinzer, W., Some remarks on complete integral closure (to appear).Google Scholar
9. Jaffard, P., Contribution à la théorie des groupes ordonnés, J. Math. Pures Appl. 32 (1953), 203–280.Google Scholar
10. Jaffard, P., Extension des groupes réticules et applications, Publ. Sci. Univ. Alger. 1 (1954), 197–222.Google Scholar
11. Jaffard, P., Un contre-exemple concernant les groupes de divisibilité, C. R. Acad. Sci. Paris 243 (1956), 1264–1268.Google Scholar
12. Jaffard, P., Les systèmes d'idéaux (Dunod, Paris, 1960).Google Scholar
13. Krull, W., Allgemeine Bewertungstheorie, J. Reine Angew. Math. 167 (1931), 160–196.Google Scholar
14. Krull, W., Beitràge zur Arithmetik kommutativer Integritdtsbereiche. I, Math. Z. 41 (1936), 544–577.Google Scholar
15. Lorenzen, P., Abstrakte Begrundung der Multiplicativen Idealtheorie, Math. Z. Ifî (1939), 533–553.Google Scholar
16. Nakayama, T., On KrulVs conjecture concerning completely integrally closed integrity domains. I, II, Proc. Imp. Acad. Tokyo 18 (1942), 185-187; 233–236; III, Proc. Japan Acad. 22 (1946), 249-250.Google Scholar
17. Ohm, J., Some counterexamples related to integral closure in D[[x\], Trans. Amer. Math. Soc. 122 (1966), 321–333.Google Scholar
18. Ribenboim, P., Sur les groupes totalement ordonnés et Varithmétique des anneaux de valuation, Summa Brasil. Math. 4 (1958), 1–64.Google Scholar
19. Ribenboim, P., Théorie des groupes ordonnés (Universidad Nacional del Sur, Bahia Blanca, 1959).Google Scholar
20. Zariski, O. and Samuel, P., Commutative algebra. II (Van Nostrand, New York, 1961).Google Scholar
21. Zelinsky, D., Topological characterization of fields with valuations, Bull. Amer. Math. Soc. 54 (1948), 1145–1150.Google Scholar