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Integrally closed factor domains

Published online by Cambridge University Press:  17 April 2009

Valentina Barucci ,
David E. Dobbs  and
S.B. Mulay
Valentina Barucci
Affiliation:
Dipartimento di Matematica, Universitá di Roma, “La Sapienza”, 00185 Roma, Italy
David E. Dobbs
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300, USA
S.B. Mulay
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300, USA
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Abstract

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This paper characterises the integral domains R with the property that R/P is integrally closed for each prime ideal P of R. It is shown that Dedekind domains are the only Noetherian domains with this property. On the other hand, each integrally closed going-down domain has this property. Related properties and examples are also studied.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 37 , Issue 3 , June 1988 , pp. 353 - 366
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Boiseis, M.B. Jr. and Sheldon, P.B., ‘CPI extensions: overrings of integral domains with special prime spectrums’, Canad. J. Math. 29 (1977), 722–737.Google Scholar
[2]Bourbaki, N., Algèbre Commutative, Chapitres 8 et 9 (Masson, Paris, 1983).Google Scholar
[3]Dobbs, D.E., ‘On going-down for simple overrings’, Proc. Amer. Math. Soc. 39 (1973), 515–519.CrossRefGoogle Scholar
[4]Dobbs, D.E., ‘On going-down for simple overrings II’, Comm. Algebra 1 (1974), 439–458.Google Scholar
[5]Dobbs, D.E., ‘Divided rings and going-down’, Pacific J. Math. 67 (1976), 353–363.Google Scholar
[6]Dobbs, D.E., ‘Coherence, ascent of going-down, and psuedo-valuation domains’, Houston J. Math. 4 (1978), 551–567.Google Scholar
[7]Dobbs, D.E., ‘On locally divided integral domains and CPI-overrings’, Internat. J. Math. Math. Sci. 4 (1981), 119–135.CrossRefGoogle Scholar
[8]Fontana, M., ‘Topologically defined classes of commutative rings’, Ann. Mat. Pura Appl. 123 (1980), 331–355.CrossRefGoogle Scholar
[9]Gilmer, R. and Heitmann, R.C., ‘On Pic(R[X]) for R seminormal’, J. Pure Appl. Algebra 16 (1980), 251–257.CrossRefGoogle Scholar
[10]Greenberg, B., ‘Coherence in Cartesian squares’, J. Algebra 50 (1978), 12–25.CrossRefGoogle Scholar
[11]Kaplansky, I., Commutative Rings, rev. ed. (University of Chicago Press, Chicago, 1974).Google Scholar
[12]McAdam, S., ‘Simple going-down’, J. London Math. Soc. 13 (1976), 167–173.CrossRefGoogle Scholar
[13]Zariski, O. and Samuel, P., Commutative Algebra, Vol. I (Van Nostrand, Princeton, 1958).Google Scholar
[14]Zariski, O. and Samuel, P., Commutative Algebra, Vol. II (Van Nostrand, Princeton, 1960).CrossRefGoogle Scholar