Geodesic
On Chain Conditions in Integral Domains
Canadian mathematical bulletin, Tome 27 (1984) no. 3, pp. 351-359

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-1984-053-1
Mots-clés : 13E05, 13F05, 13B20, 13G05 
Barucci, Valentina; Dobbs, David E. On Chain Conditions in Integral Domains. Canadian mathematical bulletin, Tome 27 (1984) no. 3, pp. 351-359. doi: 10.4153/CMB-1984-053-1
@article{10_4153_CMB_1984_053_1,
     author = {Barucci, Valentina and Dobbs, David E.},
     title = {On {Chain} {Conditions} in {Integral} {Domains}},
     journal = {Canadian mathematical bulletin},
     pages = {351--359},
     year = {1984},
     volume = {27},
     number = {3},
     doi = {10.4153/CMB-1984-053-1},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1984-053-1/}
}
TY  - JOUR
AU  - Barucci, Valentina
AU  - Dobbs, David E.
TI  - On Chain Conditions in Integral Domains
JO  - Canadian mathematical bulletin
PY  - 1984
SP  - 351
EP  - 359
VL  - 27
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1984-053-1/
DO  - 10.4153/CMB-1984-053-1
ID  - 10_4153_CMB_1984_053_1
ER  - 
%0 Journal Article
%A Barucci, Valentina
%A Dobbs, David E.
%T On Chain Conditions in Integral Domains
%J Canadian mathematical bulletin
%D 1984
%P 351-359
%V 27
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1984-053-1/
%R 10.4153/CMB-1984-053-1
%F 10_4153_CMB_1984_053_1

[1] 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] 2. Barucci, V., On a class of Mori domains, Comm. in Algebra, 11 (1983), 1989–2001. Google Scholar

[3] 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] 4. Dobbs, D. E. and Fedder, R., Conducive integral domains, J. Algebra, 86 (1984), 494–510. Google Scholar

[5] 5. Fontana, M., Carrés cartésiens et anneaux de pseudo-valuation, Publ. Dept. Math. Lyon 17 (1980), 57–95. Google Scholar

[6] 6. Gilmer, R., Multiplicative ideal theory (Dekker, New York, 1972). Google Scholar

[7] 7. Gilmer, R., Domains in which valuation ideals are prime powers, Arch. Math. 17 (1966), 210–215. Google Scholar

[8] 8. Hedstrom, J. R. and Houston, E. G., Pseudovaluation domains, Pac. J. Math. 75 (1978), 137–147. Google Scholar

[9] 9. Jaffard, P., Théorie de la dimension dans les anneaux de polynômes (Gauthier-Villars, Paris, 1960). Google Scholar

[10] 10. Kaplansky, I., Commutative rings, rev. ed. (Univ. of Chicago Press, Chicago, 1974). Google Scholar

[11] 11. Mott, J. L. and Gilmer, R., On proper overrings of integral domains, Monatsh. Math. 72 (1968), 61–71. Google Scholar

[12] 12. Ohm, J., Some counterexamples related to integral closure in D[[x]], Trans. Amer. Math. Soc. 122 (1966), 321–333. Google Scholar

[13] 13. Querré, J., Sur une propriété des anneaux de Krull, Bull. Se. Math. 95 (1971), 341–354. Google Scholar

[14] 14. Sheldon, P., How changing D[[x]] changes its quotient field, Trans. Amer. Math. Soc. 159 (1971), 223–244. Google Scholar

Cité par Sources :