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Prime Ideals in GCD-Domains

Published online by Cambridge University Press:  20 November 2018

Philip B. Sheldon
Philip B. Sheldon*
Affiliation:
Virginia Polytechnic Institute and State University, Blacksburg, Virginia
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A GCD-domain is a commutative integral domain in which each pair of elements has a greatest common divisor (g.c.d.). (This is the terminology of Kaplansky [9]. Bourbaki uses the term ''anneau pseudobezoutien" [3, p. 86], while Cohn refers to such rings as "HCF-rings" [4].) The concept of a GCD-domain provides a useful generalization of that of a unique factorization domain (UFD), since several of the standard results for a UFD can be proved in this more general setting (for example, integral closure, some properties of D[X], etc.). Since the class of GCD-domains contains all of the Bezout domains, and in particular, the valuation rings, it is clear that some of the properties of a UFD do not hold in general in a GCD-domain. Among these are complete integral closure, ascending chain condition on principal ideals, and some of the important properties of minimal prime ideals.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 1 , February 1974 , pp. 98 - 107
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Arnold, J. and Brewer, J., Kronecker function rings and flat D[X\-modules, Proc. Amer. Math. Soc. 27 (1971), 483–485.Google Scholar
2. Bastida, E. and Gilmer, R., Overrings and divisorial ideals of rings of the form D + M, Michigan J. Math. 20(1973), 79–95.Google Scholar
3. Bourbaki, N., Algèbre commutative, Chapter 7 (Diviseurs) (Hermann, Paris, 1965).Google Scholar
4. Cohn, P. M., Bezout rings and their subrings, Proc. Cambridge Philos. Soc. 64 (1968), 251–264.Google Scholar
5. Dawson, J. and Dobbs, D., On going down in polynomial rings (to appear in Can. J. Math.).Google Scholar
6. Gilmer, R., Multiplicative ideal theory, Queen's Papers in Pure and Applied Math., no. 12 (Queen's University, Kingston, Ontario, 1968).Google Scholar
7. Griffin, M., Some results on v-multiplication rings, Can. J. Math. 19 (1967), 710–722.Google Scholar
8. Jaffard, P., Les systèmes d'idéaux (Dunod, Paris, 1960).Google Scholar
9. Kaplansky, I., Commutative rings (Allyn and Bacon, Boston, 1970).Google Scholar
10. McAdam, S., Two conductor theorems, J. Algebra 23 (1972), 239–240.Google Scholar
11. Mott, J., The group of divisibility and its applications, Conference on Commutative Algebra, Lawrence, Kansas, 1972; Lecture Notes in Mathematics, No. 311 (Springer-Verlag, New York, 1973).Google Scholar
12. Sheldon, P., Two counterexamples involving complete integral closure in finite-dimensional Prilfer domains (to appear in J. Algebra).Google Scholar
13. Vasconcelos, W., The local rings of global dimension two, Proc. Amer. Math. Soc. 35 (1972), 381–386.Google Scholar