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

Authors

Paula Kemp, Missouri State UniversityFollow
Louis J. Ratliff Jr.
Kishor Shah, Missouri State UniversityFollow

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

Recommended Citation

Kemp, Paula, Louis J. Ratliff, and Kishor Shah. "On Nagata's Result about Height One Maximal Ideals and Depth One Minimal Prime Ideals (I)." The Journal of the Indian Mathematical Society 85, no. 3-4 (2018): 356-376.

Journal Title

The Journal of the Indian Mathematical Society