Integral Closure of Noetherian Domains and Intersections of Rees Valuation Rings, (II)

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Authors

  • Paula Kemp
     ,US
  • Louis J. Ratliff, Jr.
     ,US
  • Kishor Shah
     ,US

DOI:

https://doi.org/10.18311/jims/2017/6133

Keywords:

Integral Closure, Local Domain, Rees Valuation Ring, Unique Factorization Domain
Geometry

Abstract

Let 1 < s1 < . . . < sk be integers, and assume that κ ≥ 2 (so sk ≤ 3). Then there exists a local UFD (Unique Factorization Domain) (R,M) such that:

(1) Height(M) = sk.

(2) R = R' = ∩{VI (V,N) € Vj}, where Vj (j = 1, . . . , κ) is the set of all of the Rees valuation rings (V,N) of the M-primary ideals such that trd((V I N) I (R I M)) = sj - 1.

(3) With V1, . . . , Vκ as in (2), V1 ∪ . . . Vκis a disjoint union of all of the Rees valuation rings of allof the M-primary ideals, and each M-primary ideal has at least one Rees valuation ring in each Vj .

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Published

2017-01-02

How to Cite

Kemp, P., Ratliff, Jr., L. J., & Shah, K. (2017). Integral Closure of Noetherian Domains and Intersections of Rees Valuation Rings, (II). The Journal of the Indian Mathematical Society, 84(1-2), 55–72. https://doi.org/10.18311/jims/2017/6133

Issue

Volume 84, Issue 1-2, January-June 2017

Section

Articles

License

Copyright (c) 2017 Paula Kemp, Louis J. Ratliff, Jr., Kishor Shah

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

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Europe PMC
Received 2016-06-14
Accepted 2016-07-20
Published 2017-01-02

 

References

M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, MA 1969.

R. C. Heitmann, Characterization of completions of unique factorization domains, Trans. Amer. Math. Soc. 337 (1993), 379-387.

I. Kaplansky, Commutative Rings, Allyn and Bacon, Boston, 1970.

D. Katz and J. Validashti, Multiplicities and Rees valuations, Collect. Math. 61 (2010), 1-24.

P. Kemp, L. J. Ratli , Jr., and K. Shah, Integral Closure of Noetherian Domains and Inter- sections of Rees Valuation Rings, (I), J. Indian Math. Soc. (to appear).

H. Matsumura, Commutative Algebra, W. A. Benjamin, NY, 1970.

M. Nagata, Local Rings, Interscience, John Wiley, New York, 1962.

D. G. Northcott, Ideal Theory, Cambridge Tracts in Math. No. 42, Cambridge, 1965.

L. J. Ratli , Jr., On quasi-unmixed local domains, the altitude formula, and the chain con- dition for prime ideals (II), Amer. J. Math. 92 (1970), 99-144.

L. J. Ratli , Jr., On prime divisors of the integral closure of a principal ideal, J. Reine Angew. Math. 255 (1972), 210-220.

I. Swanson and C. Huneke, Integral Closure of Ideals, Rings and Modules, Cambridge Univ. Press, Cambridge, 2006.

O. Zariski and P. Samuel, Commutative Algebra, Vol. 2, D. Van Nostrand, New York, 1960.