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On the Product of Ideals

Published online by Cambridge University Press:  20 November 2018

David F. Anderson  and
David E. Dobbs
David F. Anderson
Affiliation:
Department of Mathematics, University of TennesseeKnoxville, Tennessee 37996, U.S.A
David E. Dobbs
Affiliation:
Department of Mathematics, University of TennesseeKnoxville, Tennessee 37996, U.S.A
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Abstract

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This article introduces the concept of a condensed domain, that is, an integral domain R for which IJ = {ij: i ∊ I, j ∊ J} for all ideals I and J of R. This concept is used to characterize Bézout domains (resp., principal ideal domains; resp., valuation domains) in suitably larger classes of integral domains. The main technical results state that a condensed domain has trivial Picard group and, if quasilocal, has depth at most 1. Special attention is paid to the Noetherian case and related examples.

Keywords

13F0513A1513CT513E0513G05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 26 , Issue 1 , 01 March 1983 , pp. 106 - 114
Copyright
Copyright © Canadian Mathematical Society 1983

References

1. Bourbaki, N., Commutative algebra, Addison-Wesley, Reading, Mass., 1972.Google Scholar
2. Gilmer, R., Multiplicative ideal theory, Dekker, New York, 1972.Google Scholar
3. Kaplansky, I., Commutative rings, rev. edition, Univ. of Chicago Press, Chicago, 1974.Google Scholar
4. Nagata, M., Local rings, Interscience, New York, 1962.Google Scholar
5. Papick, I.J., Topologically defined classes of going down domains, Trans. Amer. Math. Soc. 219 (1976), 1-37.Google Scholar
6. Vasconcelos, W.V., Divisor theory in module categories, North Holland, Amsterdam, 1974.Google Scholar
7. Zafrullah, M., On pre-Schreier domains, preprint.Google Scholar