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Integral ∨-ideals

Published online by Cambridge University Press:  18 May 2009

David F. Anderson
David F. Anderson
Affiliation:
Department of Mathematics, The University of Tennessee, Knoxvelle, Tennessee 37916
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Let R be an integral domain with quotient field K. A fractional ideal I of R is a ∨-ideal if I is the intersection of all the principal fractional ideals of R which contain I. If I is an integral ∨-ideal, at first one is tempted to think that I is actually just the intersection of the principal integral ideals which contain I.However, this is not true. For example, if R is a Dedekind domain, then all integral ideals are ∨-ideals. Thus a maximal ideal of R is an intersection of principal integral ideals if and only if it is actually principal. Hence, if R is a Dedekind domain, each integral ∨-ideal is an intersection of principal integral ideals precisely when R is a PID.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 22 , Issue 2 , July 1981 , pp. 167 - 172
Copyright
Copyright © Glasgow Mathematical Journal Trust 1981

References

REFERENCES

1.Arnold, J. T. and Sheldon, P. B., Integral domains that satisfy Gauss's lemma, Michigan Math. J., 22 (1975), 39–51.CrossRefGoogle Scholar
2.Bastida, E. and Gilmer, R., Overrings and divisorial ideals of rings of the form D + M, Michigan Math. J., 20 (1973), 79–95.CrossRefGoogle Scholar
3.Cohn, P. M., Bezout rings and their subrings, Proc. Cambridge Phil. Soc., 64 (1968), 251–264.CrossRefGoogle Scholar
4.Cohn, P. M., Unique factorization domains, Amer. Math. Monthly, 80 (1973), 1–18.CrossRefGoogle Scholar
5.Fossum, R. M., The Divisor Class Group of a Krull Domain, (Springer-Verlag, 1973).CrossRefGoogle Scholar
6.Gilmer, R., Multiplicative Ideal Theory, (Dekker, 1972).Google Scholar
7.Heinzer, W. and Ohm, J., An essential ring which is not a ∨ -multiplication ring, Canad. J. Math., 25 (1973), 856–861.CrossRefGoogle Scholar
8.McAdam, S. and Rush, D. E., Schreier rings, Bull. London Math. Soc., 10 (1978), 77–80.CrossRefGoogle Scholar
9.Mott, J. L. and Zafrullah, M., On Prüfer ∨-multiplication domains, preprint.Google Scholar
10.Tang, H. T., Gauss' lemma, Proc. Amer. Math. Soc., 35 (1972), 372–376.Google Scholar
11.Vasconcelos, W. V., Divisor Theory in Module Categories, (North-Holland, 1974).Google Scholar
12.Zafrullah, M., On finite conductor domains, Manuscripta Math., 24 (1978), 191–204.CrossRefGoogle Scholar