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Prime Ideals in GCD-Domains

  • Philip B. Sheldon (a1)

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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.

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References

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1. Arnold, J. and Brewer, J., Kronecker function rings and flat D[X\-modules, Proc. Amer. Math. Soc. 27 (1971), 483485.
2. Bastida, E. and Gilmer, R., Overrings and divisorial ideals of rings of the form D + M, Michigan J. Math. 20(1973), 7995.
3. Bourbaki, N., Algèbre commutative, Chapter 7 (Diviseurs) (Hermann, Paris, 1965).
4. Cohn, P. M., Bezout rings and their subrings, Proc. Cambridge Philos. Soc. 64 (1968), 251264.
5. Dawson, J. and Dobbs, D., On going down in polynomial rings (to appear in Can. J. Math.).
6. Gilmer, R., Multiplicative ideal theory, Queen's Papers in Pure and Applied Math., no. 12 (Queen's University, Kingston, Ontario, 1968).
7. Griffin, M., Some results on v-multiplication rings, Can. J. Math. 19 (1967), 710722.
8. Jaffard, P., Les systèmes d'idéaux (Dunod, Paris, 1960).
9. Kaplansky, I., Commutative rings (Allyn and Bacon, Boston, 1970).
10. McAdam, S., Two conductor theorems, J. Algebra 23 (1972), 239240.
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).
12. Sheldon, P., Two counterexamples involving complete integral closure in finite-dimensional Prilfer domains (to appear in J. Algebra).
13. Vasconcelos, W., The local rings of global dimension two, Proc. Amer. Math. Soc. 35 (1972), 381386.
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Prime Ideals in GCD-Domains

  • Philip B. Sheldon (a1)

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