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

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}.

Department(s)

Mathematics

Document Type

Article

DOI

https://doi.org/10.18311/jims/2018/20123

Keywords

integral closure, completion of a local ring, depth one minimal prime ideal, height one maximal ideal

Publication Date

2018

Journal Title

The Journal of the Indian Mathematical Society

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