On Nagata's Result about Height One Maximal Ideals and Depth One Minimal Prime Ideals (I)
Authors
-
Department of Mathematics, Missouri State University, Springeld, Missouri - 65897 ,US
Paula Kemp -
Department of Mathematics, University of California, Riverside, California - 92521 ,USLouis J. Ratliff, Jr.
-
Department of Mathematics, Missouri State University, Springfield, Missouri - 65897 ,USKishor Shah
DOI:
https://doi.org/10.18311/jims/2018/20123Keywords:
Integral Closure, Completion of a Local Ring, Depth One Minimal Prime Ideal, Height One Maximal IdealAbstract
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}.
Downloads
Metrics
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2018 Paula Kemp, Louis J. Ratliff, Jr., Kishor Shah
This work is licensed under a Creative Commons Attribution 4.0 International License.
Accepted 2023-01-30
Published 2018-06-01
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.
7