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Integral domains in which each non-zero ideal is divisorial

Published online by Cambridge University Press:  26 February 2010

William Heinzer
William Heinzer
Affiliation:
Louisiana State University, Baton Rouge, Louisiana.
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Let D be an integral domain with identity having quotient field K. A non-zero fractional ideal F of D is said to be divisorial if F is an intersection of principal fractional ideals of D[4; 2]. Equivalently, F is divisorial if there is a non-zero fractional ideal E of D such that

Divisorial ideals arose in the investigations of Van der Waerden, Artin, and Krull in the 1930's and were called v-ideals by Krull [9; 118]. The concept has played an important role in the development of multiplicative ideal theory.

Type
Research Article
Information
Mathematika , Volume 15 , Issue 2 , December 1968 , pp. 164 - 170
Copyright
Copyright © University College London 1968

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References

1.Bass, H., “Injective dimension in noetherian rings”, Trans. Amer. Soc., 102 (1962), 18–29.CrossRefGoogle Scholar
2.Bass, H., “On the ubiquity of Gorenstein rings”, Math. Z., 82 (1963), 8–28.CrossRefGoogle Scholar
3.Berger, R., “Uber eine Klasse unvergabelter lokaler Ringe”, Math. Ann., 146 (1962), 98–102.CrossRefGoogle Scholar
4.Bourbaki, N., Éléments de mathémathique, Algèbre commutative, XXXI (Hermann, Paris, 1965).Google Scholar
5.Gilmer, R., “Integral domains which are almost Dedekind”, Proc. Amer. Math. Soc., 15 (1964), 813–818.CrossRefGoogle Scholar
6.Gilmer, R. and Heinzer, W., “Overrings of Prüfer domains II”, J. of Algebra, 7 (1967), 281–302.CrossRefGoogle Scholar
7.Gilmer, R. and Heinzer, W., “Intersections of quotient rings of an integral domain”, J. Math. Kyoto, 7 (1967), 133–150.CrossRefGoogle Scholar
8.Jans, J., “Duality in noetherian rings”, Proc. Amer. Math. Soc., 12 (1961), 829–835.CrossRefGoogle Scholar
9.Krull, W., Idealtheorie (Springer, Berlin, 1935).CrossRefGoogle Scholar
10.Nagata, M., Local Rings (Interscience, New York, 1962).Google Scholar
11.Nakano, N., “Idealtheorie in einem speziellen unendlichen algebraischen Zahlkörper”, J. Sci. Hiroshima Univ., 11 (1953), 425–439.Google Scholar
12.Zariski, O. and Samuel, P., Commutative Algebra, Vol. I (Van Nostrand, Princeton, 1958).Google Scholar
13.Zariski, O. and Samuel, P., Commutative Algebra, Vol. II (Van Nostrand, Princeton, 1960).CrossRefGoogle Scholar