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Factoring Ideals into Semiprime Ideals

Published online by Cambridge University Press:  20 November 2018

N. H. Vaughan  and
R. W. Yeagy
N. H. Vaughan
Affiliation:
Stephen F. Austin State University, Nacogdoches, Texas
R. W. Yeagy
Affiliation:
Stephen F. Austin State University, Nacogdoches, Texas
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Let D be an integral domain with 1 ≠ 0 . We consider “property SP” in D, which is that every ideal is a product of semiprime ideals. (A semiprime ideal is equal to its radical.) It is natural to consider property SP after studying Dedekind domains, which involve factoring ideals into prime ideals. We prove that a domain D with property SP is almost Dedekind, and we give an example of a nonnoetherian almost Dedekind domain with property SP.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 30 , Issue 6 , 01 December 1978 , pp. 1313 - 1318
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Bourbaki, N., Éléments de mathématique; Algèbre commutative, Chapter 7 (Herman, Paris, 1965).Google Scholar
2. Butts, H. S. and Cranford, R. H., Some containment relations between classes of ideals in an integral domain, J. Sci. Hiroshima Univ. Ser A-1 20 (1965), 1–10.Google Scholar
3. Butts, H. S. and Gilmer, R. W., Primary ideals and prime power ideals, Can. J. Math. 18 (1966), 1183–1195.Google Scholar
4. Butts, H. S. and Phillips, R. C., Almost multiplication rings, Can. J. Math. 17 (1965), 267–277.Google Scholar
5. Gilmer, R. W., Integral domains which are almost Dedekind, Proc. Amer. Math. Soc. 15 (1964), 813–818.Google Scholar
6. Gilmer, R. W., Multiplicative ideal theory I (Queen's University Press, Kingston, Ontario, 1968).Google Scholar
7. Gilmer, R. W. and Ohm, J., Primary ideals and valuation ideals, Trans. Amer. Math. Soc. 117 (1965), 237–250.Google Scholar
8. Heinzer, W. and Ohm, J., Locally noetherian commutative rings, Trans. Amer. Math. Soc. 158 (1971), 273–284.Google Scholar
9. Larsen, M. and McCarthy, P., Multiplicative theory of ideals (Academic Press, New York, 1971).Google Scholar
10. Jaffard, P., Les systèmes d'idéaux (Dunod, Paris, 1960).Google Scholar
11. Zariski, O. and Samuel, P., Commutative algebra, vol. I (Van Nostrand, Princeton, New Jersey, 1958).Google Scholar
12. Zariski, O. and Samuel, P., Commutative algebra, vol. II (Van Nostrand, Princeton, New Jersey, 1960).Google Scholar
13. Yeagy, R. W., Semiprime factorizations in unions of Dedekind domains, submitted for publication.Google Scholar