Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-28T23:53:21.875Z Has data issue: false hasContentIssue false
  • English
  • Français

Ascent and descent of going-down rings for integral extensions

Published online by Cambridge University Press:  17 April 2009

David E. Dobbs
David E. Dobbs
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, Tennessee, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The author and Ira J. Papick have termed an integral domain R a going-down ring if R ⊂ T satisfies going-down for each domain T containing R. The present paper investigates conditions which, for an integral extension A ⊂ B of domains, imply that A (respectively B ) is going-down whenever B (respectively A ) is going-down. This explains the “descent” (respectively “ascent”) in the title. Two typical results (the first about descent, the second about ascent) are given next.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 15 , Issue 2 , October 1976 , pp. 253 - 264
Copyright
Copyright © Australian Mathematical Society 1976

References

[1]Brase, Charles H., “A note on intersections of valuation ideals”, Proc. Amer. Math. Soc. 38 (1973), 37–39.CrossRefGoogle Scholar
[2]Brewer, J.W. and Heinzer, W.J., “Ideals I of R[X] for which R[X]/I is R-projective”, Proc. Amer. Math. Soc. 43 (1974), 21–25.Google Scholar
[3]Burnside, W., “On groups of order p αq β”, Proc. London Math. Soc. (2) 1 (1904), 388–392.CrossRefGoogle Scholar
[4]Dawson, Jeffrey and Dobbs, David E., “On going down in polynomial rings”, Canad. J. Math. 26 (1974), 177–184.CrossRefGoogle Scholar
[5]Dobbs, David E., “On going down for simple overrings”, Proc. Amer. Math. Soc. 39 (1973), 515–519.CrossRefGoogle Scholar
[6]Dobbs, David E., “On going down for simple overrings II”, Comm. Algebra 1 (1974), 439–458.CrossRefGoogle Scholar
[7]Dobbs, David E. and Papick, Ira J., “On going-down for simple overrings. III”, Proc. Amer. Math. Soc. 54 (1976), 35–38.CrossRefGoogle Scholar
[8]Gilmer, Robert W., Multiplicative ideal theory, Part I (Queen's Papers on Pure and Applied Mathematics, 12. Queen's University, Kingston, Ontario, 1968).Google Scholar
[9]Heinzer, William J., “Some properties of integral closure”, Proc. Amer. Math. Soc. 18 (1967), 749–753.CrossRefGoogle Scholar
[10]Heinzer, William and Ohm, Jack, “The finiteness of I when R[X]/I is R-flat. II”, Proc. Amer. Math. Soc. 35 (1972), 1–8.Google Scholar
[11]Kaplansky, Irving, Commutative rings (Allyn and Bacon, Boston, 1970).Google Scholar
[12]Krull, Wolfgang, “Beiträge zur Arithmetik kommutativer Integritätsbereiche. III: Zum Dimensionsbegriff der Idealtheorie”, Math. Z. 42 (1936–1937), 715–766.Google Scholar
[13]Lewis, William J., “The spectrum of a ring as a partially ordered set”, J. Algebra 25 (1973), 419–434.CrossRefGoogle Scholar
[14]McAdam, Stephen, “Going down”, Duke Math. J. 39 (1972), 633–636.CrossRefGoogle Scholar
[15]McAdam, Stephen, Private communication, 05 1974.Google Scholar
[16]Papick, Ira J., “Topologically defined classes of going-down rings”, Trans. Amer. Math. Soc. 219 (1976), 1–37.CrossRefGoogle Scholar
[17]Prüfer, Heinz, “Untersuchungen über Teilbarkeitseigenschaften in Kürpern”, J. reine angew. Math. 168 (1932), 1–36.CrossRefGoogle Scholar
[18]Richman, Fred, “Generalized quotient rings”, Proc. Amer. Math. Soc. 16 (1965), 794–799.CrossRefGoogle Scholar
[19]Seidenberg, A., “A note on the dimension theory of rings”, Pacific J. Math. 3 (1953), 505–512.CrossRefGoogle Scholar
[20]Zariski, Oscar and Samuel, Pierre, Commutative algebra, Volume I (Van Hostrand, Princeton, New Jersey; Toronto; New York; London; 1958). See also: Oscar Zariski, Pierre Samuel, Commutative algebra, Volume 1 (Graduate Texts in Mathematics, 28. Springer-Verlag, New York, Heidelberg, Berlin, 1975).Google Scholar