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


Affiliations
  • Missouri State University, Department of Mathematics, Springeld, United States
  • University of California, Department of Mathematics, Riverside, United States
     

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

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Authors

Paula Kemp
, United States
Louis J. Ratliff
, United States
Kishor Shah
, United States

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 .


Keywords


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

References