Open Access
Subscription Access
Open Access
Subscription Access
On Nagata's Result about Height One Maximal Ideals and Depth One Minimal Prime Ideals (I)
Subscribe/Renew Journal
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.
Subscription
Login to verify subscription
User
Font Size
Information
- 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.
Abstract Views: 208
PDF Views: 2