On Nagata's Result about Height One Maximal Ideals and Depth One Minimal Prime Ideals (I)


Affiliations

  • Missouri State University, Department of Mathematics, Springfield, Missouri, 65897, United States
  • University of California, Riverside, Department of Mathematics, California, 92521, United States

Abstract

It is shown that, for all local rings (R,M), there is a canonical bijection between the set DO(R) of depth one minimal prime ideals ω in the completion ^R of R and the set HO(R/Z) of height one maximal ideals ̅M' in the integral closure (R/Z)' of R/Z, where Z := Rad(R). Moreover, for the finite sets D := {V*/V* := (^R/ω)', ω ∈ DO(R)} and H := {V/V := (R/Z)'̅M', ̅M' ∈ HO(R/Z)}:

(a) The elements in D and H are discrete Noetherian valuation rings.

(b) D = {^VH}.


Keywords

Integral Closure, Completion of a Local Ring, Depth One Minimal Prime Ideal, Height One Maximal Ideal

Subject Discipline

Mathematical Sciences

Full Text:

References

Paula Kemp, Louis J. Ratli, Jr., and Kishor Shah, On Nagata's Result About Height One Maximal Ideals and Depth One Minimal Prime Ideals (II), in preparation.

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

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


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