Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T05:19:28.570Z Has data issue: false hasContentIssue false
  • English
  • Français

Two notes on ideal-transforms

Published online by Cambridge University Press:  24 October 2008

Daniel Katz  and
Louis J. Ratliff Jr
Daniel Katz
Affiliation:
Department of Mathematics, University of Kansas, Lawrence, KS 66045, U.S.A.
Louis J. Ratliff Jr
Affiliation:
Department of Mathematics, University of California, Riverside, CA 92521, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

The first note gives two new characterization of the ideal-transform T(I) of a finitely generated regular ideal I in a large class of rings. Specifically, if b is a regular element in I, then there exists a regular element c ∈ I and a multiplicatively closed set S of regular elements in R such that T(I) = T((b, c)R) = Rb ∩ Rc = Rb ∩ Rs, so T(I) is the ideal-transform of an ideal generated by two elements, and every ring of the form Rb ∩ Rs is an ideal-transform. The second theorem shows that if T(I) is integrally closed, then it is a Krull ring. As an application of these results we strengthen some known results concerning when certain ideal-transforms of the Rees ring R(R, I) are finite or integral extension rings of R(R, I).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 102 , Issue 3 , November 1987 , pp. 389 - 397
Copyright
Copyright © Cambridge Philosophical Society 1987

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Brewer, J.. The ideal transform and overrings of an integral domain. Math. Z. 107 (1968), 301–306.CrossRefGoogle Scholar
[2]Brewer, J. and Gilmer, R.. Integral domains whose overrings are ideal transforms. Math. Nachr. 51 (1971), 255–267.CrossRefGoogle Scholar
[3]Brodmann, M.. Finiteness of ideal transforms. Preprint.Google Scholar
[4]Katz, D. and Ratliff, L. J. Jr. U-essential prime divisors and sequences over an ideal. Nagoya Math. J. 103 (1986), 39–66.CrossRefGoogle Scholar
[5]Kiyek, K. H.. Anwendung von Ideal-Transformationen. Manuscripta Math. 34 (1981), 327–353.CrossRefGoogle Scholar
[6]Matijevic, J. R.. Maximal ideal transforms of Noetherian rings. Proc. Amer. Math. Soc. 54 (1976), 49–52.CrossRefGoogle Scholar
[7]McAdam, S.. Asymptotic Prime Divisors. Lecture Notes in Math., No. 1023 (Springer-Verlag, 1983).CrossRefGoogle Scholar
[8]McAdam, S.. Filtrations, Rees rings, and ideal transforms. J. Pure Appl. Algebra (to appear).Google Scholar
[9]McAdam, S. and Ratliff, L. J. JrEssential sequences. J. Algebra 95 (1985), 217–235.CrossRefGoogle Scholar
[10]McAdam, S. and Ratliff, L. J. JrFinite transforms of a Noetherian ring. J. Algebra 101 (1986), 479–489.CrossRefGoogle Scholar
[11]Nagata, M.. A treatise on the 14th problem of Hilbert. Mem. Coll. Sci. Kyoto Univ. 30 (1956), 57–82.Google Scholar
[12]Nagata, M.. Note on a chain condition for prime ideals. Mem. Coll. Sci. Kyoto Univ. 32 (1959), 85–90.Google Scholar
[13]Nagata, M.. Some sufficient conditions for the fourteenth problem of Hilbert. Actas Del Coloquio Interrac. Sobre Geometria Algebraica, Madrid 1965 (Inst. Jorge Juan del C.S.I.C., 1966), 107–121.Google Scholar
[14]Nishimura, J.. On ideal transforms of Noetherian rings, I. J. Math. Kyoto Univ. 19 (1979).Google Scholar
[15]Nishimura, J.. On ideal transforms of Noetherian rings, II. J. Math. Kyoto Univ. 20 (1980), 149–154.Google Scholar
[16]Ratliff, L. J. JrOn prime divisors of the integral closure of a principal ideal. J. Reine Angew. Math. 255 (1972), 210–220.Google Scholar
[17]Ratliff, L. J. JrOn asymptotic prime divisors. Pacific J. Math. 111 (1984), 395–413.CrossRefGoogle Scholar
[18]Ratliff, L. J. JrFive notes on asymptotic prime divisors. Math. Z. 190 (1985), 567–581.CrossRefGoogle Scholar
[19]Ratliff, L. J. JrOn linearly equivalent ideal topologies. J. Pure Appl. Algebra 41 (1986), 67–77.CrossRefGoogle Scholar
[20]Rees, D.. On a problem of Zariski. Illinois J. Math. 2 (1968), 145–149.Google Scholar
[21]Ribenboim, P.. Anneaux de Rees integralement clos. J. Reine Angew. Math. 204 (1962), 99–107.Google Scholar
[22]Schenzel, P.. Finiteness of relative Rees rings and asymptotic prime divisors. Math. Nachr. 129 (1986), 123–148.CrossRefGoogle Scholar