Hostname: page-component-848d4c4894-xfwgj Total loading time: 0 Render date: 2024-06-22T13:12:47.928Z Has data issue: false hasContentIssue false
  • English
  • Français

Integrally Closed Condensed Domains are Bézout

Published online by Cambridge University Press:  20 November 2018

David F. Anderson ,
Jimmy T. Arnold  and
David E. Dobbs
David F. Anderson
Affiliation:
Department of Mathematics, University of Tennessee Knoxville, Tennessee 37996-1300 U.S.A.
Jimmy T. Arnold
Affiliation:
Department of Mathematics, Virginia Polytechnic Instituteand, State University Blacksburg, Virginia 24061 U.S.A.
David E. Dobbs
Affiliation:
Department of Mathematics, University of Tennessee Knoxville, Tennessee 37996-1300 U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

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.

It is proved that an integral domain R is a Bézout domain if (and only if) R is integrally closed and I J = {ij|i ∊ I, j ∊ J} for all ideals I and J of R; that is, if (and only if) R is an integrally closed condensed domain. The article then introduces a weakening of the "condensed" concept which, in the context of the k + M construction, is equivalent to a certain field-theoretic condition. Finally, the field extensions satisfying this condition are classified.

Keywords

13F0513A1513G0513E0512F10
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 28 , Issue 1 , 01 March 1985 , pp. 98 - 102
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Anderson, D.F. and Dobbs, D.E., On the product of ideals, Canad. Math. Bull. 26 (1983), pp. 106–114.Google Scholar
2. Gilmer, R., Multiplicative Ideal Theory, Dekker, New York, 1972.Google Scholar
3. Gilmer, R. and Grams, A., The equality (A ∩ B)n = An ∩ Bn for ideals, Can. J. Math. 24 (1972), pp. 792–798.Google Scholar
4. Hungerford, T.W., Algebra, Springer-Verlag, New York, 1974.Google Scholar
5. Kaplansky, I., Commutative Rings, rev. éd., University of Chicago Press, Chicago, 1974.Google Scholar
6. Lang, S., Algebra, Addison-Wesley, Reading, Mass., 1965.Google Scholar