Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-23T13:56:03.546Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Remarks on Complete Integral Closure

Published online by Cambridge University Press:  09 April 2009

William Heinzer
William Heinzer
Affiliation:
Louisiana State UniversityBâton 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.

This paper continues an investigation of the complete integral closure of an integral domain which was begun in [2]. We recall that if D is an integral domain with quotient field K then an element x of K is said to be almost integral over D if there exists a nonzero element y of D such that yxn is an element of D for each positive integer n. The set D* of elements of K almost integral over D is called the complete integral closure of D and D is said to be completely integrally closed if D* = D.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 9 , Issue 3-4 , May 1969 , pp. 310 - 314
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Gillman, L. and Jerison, M., ‘Rings of Continuous Functions’, (Van Nostrand, Princeton 1960).CrossRefGoogle Scholar
[2]Gilmer, R. and Heinzer, W., ‘On the complete integral closure of an integral domain’, J. Austral. Math. Soc., 6 (1966), 351–361.Google Scholar
[3]Gilmer, R. and Heinzer, W., ‘Overrings of Prüfer domains II’, J. of Algebra, 7 (1967), 281–302.Google Scholar
[4]Gilmer, R. and Ohm, J., ‘Integral domains with quotient overrings’, Math. Ann., 153 (1964), 97–103.CrossRefGoogle Scholar
[5]Heinzer, W., ‘J-noetherian integral domains with 1 in the stable range’, Proc. Amer. Math. Soc. (to appear).Google Scholar
[6]Jaffard, P., Les Systèmes d'Idéaux, (Dunod, Paris, 1960).Google Scholar
[7]Jaffard, P., Théorie de la Dimension dans les Anneaux de Polynomes (Gauthier-Villars, Paris 1960).Google Scholar
[8]Krull, W., ‘Allgemeine Bewertungstheorie’, J. reine angew. Math., 167 (1931), 160–196.Google Scholar
[9]Lorenzen, P., ‘Abstrakte Begründung der Multiplikativen Idealtheorie’, Math. Zeit., 45 (1939), 533–553.CrossRefGoogle Scholar
[10]Nakayama, T., ‘On Krull's conjecture concerning completely integrally closed integrity domains I’, Proc. Imp. Acad. Tokyo 18 (1942), 185–187;Google Scholar
II, Proc. Imp. Acad. Tokyo 18 (1942), 233–236;Google Scholar
III, Proc. Japan Acad. 22 (1946), 249–250.Google Scholar
[11]Ohm, J., ‘Some counterexamples related to integral closure in D[[X]]’, Trans. Amer. Math. Soc., 122 (1966), 321–333.Google Scholar