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Integral Closure of Noetherian Domains and Intersections of Rees Valuation Rings, (II)
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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 .
Keywords
Integral Closure, Local Domain, Rees Valuation Ring, Unique Factorization Domain.
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- 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.
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