The Journal of the Indian Mathematical Society

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


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

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It is shown that the integral closure R' of a local (Noetherian) domain R is equal to the intersection of the Rees valuation rings of all proper ideals in R of the form (b, Ik)R, where b is an arbitrary nonzero nonunit in R and the Ik are an arbitrary descending sequence of ideals (varying with b and with Ik ⊆ (Ik-1 ∩ I1k) for all k > 1, one sequence for each b). Also, this continues to hold when b is restricted to being irreducible and no two distinct b are associates. We prove similar results for a Noetherian domain.

Keywords

Integral Closure, Noetherian Domain, Local Domain, Rees Valuation Ring.
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  • Integral Closure of Noetherian Domains and Intersections of Rees Valuation Rings, (I)

Abstract Views: 321  |  PDF Views: 1

Authors

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

Abstract


It is shown that the integral closure R' of a local (Noetherian) domain R is equal to the intersection of the Rees valuation rings of all proper ideals in R of the form (b, Ik)R, where b is an arbitrary nonzero nonunit in R and the Ik are an arbitrary descending sequence of ideals (varying with b and with Ik ⊆ (Ik-1 ∩ I1k) for all k > 1, one sequence for each b). Also, this continues to hold when b is restricted to being irreducible and no two distinct b are associates. We prove similar results for a Noetherian domain.

Keywords


Integral Closure, Noetherian Domain, Local Domain, Rees Valuation Ring.

References