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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.
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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

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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