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On Chain Conditions in Integral Domains

Published online by Cambridge University Press:  20 November 2018

Valentina Barucci  and
David E. Dobbs
Valentina Barucci
Affiliation:
Istituto Matematico, Università Di Roma I00185 Roma, Italy
David E. Dobbs
Affiliation:
Department of Mathematics, University of TennesseeKnoxville, Tennessee 37996, U.S.A.
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Abstract

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The following two theorems are proved. If R is an Archimedean conducive integral domain, then R is quasilocal and dim(R) ≤1. If each overring of an integral domain R has ascending chain condition on divisorial ideals, then the integral closure of R is a Dedekind domain. Both theorems sharpen results already known in the Noetherian case. The second theorem leads to a strengthened converse of the Krull-Akizuki Theorem. We also investigate the effect of restricting the hypothesis in the second theorem to the proper overrings of R.

Keywords

13E0513F0513B2013G05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 27 , Issue 3 , 01 September 1984 , pp. 351 - 359
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Anderson, D. D., Anderson, D. F., Dobbs, D. E. and Houston, E. G., Some finiteness and divisibility conditions on the proper overrings of an integral domain, Comm. in Alg. (to appear).Google Scholar
2. Barucci, V., On a class of Mori domains, Comm. in Algebra, 11 (1983), 1989–2001.Google Scholar
3. Beauregard, R. A. and Dobbs, D. E., On a class of Archimedean integral domains, Can. J. Math. 28 (1976), 365–375.Google Scholar
4. Dobbs, D. E. and Fedder, R., Conducive integral domains, J. Algebra, 86 (1984), 494–510.Google Scholar
5. Fontana, M., Carrés cartésiens et anneaux de pseudo-valuation, Publ. Dept. Math. Lyon 17 (1980), 57–95.Google Scholar
6. Gilmer, R., Multiplicative ideal theory (Dekker, New York, 1972).Google Scholar
7. Gilmer, R., Domains in which valuation ideals are prime powers, Arch. Math. 17 (1966), 210–215.Google Scholar
8. Hedstrom, J. R. and Houston, E. G., Pseudovaluation domains, Pac. J. Math. 75 (1978), 137–147.Google Scholar
9. Jaffard, P., Théorie de la dimension dans les anneaux de polynômes (Gauthier-Villars, Paris, 1960).Google Scholar
10. Kaplansky, I., Commutative rings, rev. ed. (Univ. of Chicago Press, Chicago, 1974).Google Scholar
11. Mott, J. L. and Gilmer, R., On proper overrings of integral domains, Monatsh. Math. 72 (1968), 61–71.Google Scholar
12. Ohm, J., Some counterexamples related to integral closure in D[[x]], Trans. Amer. Math. Soc. 122 (1966), 321–333.Google Scholar
13. Querré, J., Sur une propriété des anneaux de Krull, Bull. Se. Math. 95 (1971), 341–354.Google Scholar
14. Sheldon, P., How changing D[[x]] changes its quotient field, Trans. Amer. Math. Soc. 159 (1971), 223–244.Google Scholar