Integral Closure of Noetherian Domains and Intersections of Rees Valuation Rings, (I)
DOI:
https://doi.org/10.18311/jims/2017/6108Keywords:
Integral Closure, Noetherian Domain, Local Domain, Rees Valuation RingAbstract
It is shown that the integral closure R' of a local (Noetherian) domain R is equal to the intersection of the Rees valuation rings of all proper ideals in R of the form (b, Ik)R, where b is an arbitrary nonzero nonunit in R and the Ik are an arbitrary descending sequence of ideals (varying with b and with Ik ⊆ (Ik-1 ∩ I1k) for all k > 1, one sequence for each b). Also, this continues to hold when b is restricted to being irreducible and no two distinct b are associates. We prove similar results for a Noetherian domain.Downloads
Metrics
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2017 Paula Kemp, Louis J. Ratliff, Jr., Kishor Shah
This work is licensed under a Creative Commons Attribution 4.0 International License.
Accepted 2016-06-14
Published 2017-01-02
References
M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, MA 1969.
W. J. Heinzer, L. J. Ratli , Jr., and D. E. Rush, Bases of ideals and Rees valuation rings, J. Algebra 323 (2010), 839-853.
I. N. Herstein, Topics in Algebra, Cisdell Publishing Co., New York, 1964.
I. Kaplansky, Commutative Rings, Allyn and Bacon, Boston, 1970.
H. Matsumura, Commutative Algebra, W. A. Benjamin, NY, 1970.
M. Nagata, Local Rings, Interscience, John Wiley, New York, 1962.
L. J. Ratli , Jr., Note on analytically unrami ed semi-local rings, Proc. Amer. Math. Soc. 17 (1966), 274-279.
L. J. Ratli , Jr., On prime divisors of the integral closure of a principal ideal, J. Reine Angew. Math. 255 (1972), 210-220.
D. Rees, Valuations associated with ideals (II), J. London Math. Soc. 36 (1956), 221-228.
I. Swanson and C. Huneke, Integral Closure of Ideals, Rings and Modules, Cambridge Univ. Press, Cambridge, 2006.