Semi-Valuations and Groups of Divisibility

Ohm, Jack

Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 576-591

Résumé

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.

Détail

DOI : 10.4153/CJM-1969-065-9

Bibliographie

[1] 1. Bourbaki, N., Algèbre, chapitre 6, Groupes et corps ordonnés (Hermann, Paris, 1964). Google Scholar

[2] 2. Bourbaki, N., Algèbre commutative, (a) chapitres 5 et 6, (b) chapitre 7 (Hermann, Paris, 1964, 1965). Google Scholar

[3] 3. Clifford, A., Note on Hahn's theorem on ordered abelian groups, Proc. Amer. Math. Soc. 5 (1954), 860–863. Google Scholar

[4] 4. Estes, D. and Ohm, J., Stable range in commutative rings, J. Algebra 7 (1967), 343–362. Google Scholar

[5] 5. Gilmer, R. W., Multiplicative ideal theory, Queen's Papers, Lecture Notes No. 12, Queen's University, Kingston, Ontario, 1968. Google Scholar

[6] 6. Gilmer, R. W. and Ohm, J., Primary ideals and valuation ideals, Trans. Amer. Math. Soc. 117 (1965), 237–250. Google Scholar

[7] 7. Heinzer, W., J-Noetherian integral domains with 1 in the stable range, Proc. Amer. Math. Soc. 19 (1968), 1369–1372. Google Scholar

[8] 8. Heinzer, W., Some remarks on complete integral closure (to appear). Google Scholar

[9] 9. Jaffard, P., Contribution à la théorie des groupes ordonnés, J. Math. Pures Appl. 32 (1953), 203–280. Google Scholar

[10] 10. Jaffard, P., Extension des groupes réticules et applications, Publ. Sci. Univ. Alger. 1 (1954), 197–222. Google Scholar

[11] 11. Jaffard, P., Un contre-exemple concernant les groupes de divisibilité, C. R. Acad. Sci. Paris 243 (1956), 1264–1268. Google Scholar

[12] 12. Jaffard, P., Les systèmes d'idéaux (Dunod, Paris, 1960). Google Scholar

[13] 13. Krull, W., Allgemeine Bewertungstheorie, J. Reine Angew. Math. 167 (1931), 160–196. Google Scholar

[14] 14. Krull, W., Beitràge zur Arithmetik kommutativer Integritdtsbereiche. I, Math. Z. 41 (1936), 544–577. Google Scholar

[15] 15. Lorenzen, P., Abstrakte Begrundung der Multiplicativen Idealtheorie, Math. Z. Ifî (1939), 533–553. Google Scholar

[16] 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] 17. Ohm, J., Some counterexamples related to integral closure in D[[x\], Trans. Amer. Math. Soc. 122 (1966), 321–333. Google Scholar

[18] 18. Ribenboim, P., Sur les groupes totalement ordonnés et Varithmétique des anneaux de valuation, Summa Brasil. Math. 4 (1958), 1–64. Google Scholar

[19] 19. Ribenboim, P., Théorie des groupes ordonnés (Universidad Nacional del Sur, Bahia Blanca, 1959). Google Scholar

[20] 20. Zariski, O. and Samuel, P., Commutative algebra. II (Van Nostrand, New York, 1961). Google Scholar

[21] 21. Zelinsky, D., Topological characterization of fields with valuations, Bull. Amer. Math. Soc. 54 (1948), 1145–1150. Google Scholar