The Journal of the Indian Mathematical Society

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


Affiliations
1 Department of Mathematics, Missouri State University, Springeld, Missouri - 65897, United States
2 Department of Mathematics, University of California, Riverside, California - 92521, United States
3 Department of Mathematics, Missouri State University, Springfield, Missouri - 65897, United States
     

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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 = {^V ∈ H}.


Keywords

Integral Closure, Completion of a Local Ring, Depth One Minimal Prime Ideal, Height One Maximal Ideal.
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  • On Nagata's Result about Height One Maximal Ideals and Depth One Minimal Prime Ideals (I)

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Authors

Paula Kemp
Department of Mathematics, Missouri State University, Springeld, Missouri - 65897, United States
Louis J. Ratliff, Jr.
Department of Mathematics, University of California, Riverside, California - 92521, United States
Kishor Shah
Department of Mathematics, Missouri State University, Springfield, Missouri - 65897, 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 = {^V ∈ H}.


Keywords


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

References